Finite Element Method - The Basis (Volume 1)

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The Finite Element MethodFifth editionVolume 1: The Basis

The Finite ElementMethodFifth editionVolume 1: The BasisO.C. Zienkiewicz, CBE, FRS, FREngUNESCO Professor of Numerical Methods in EngineeringInternational Centre for Numerical Methods in Engineering, BarcelonaEmeritus Professor of Civil Engineering and Director of the Institute forNumerical Methods in Engineering, University of Wales, SwanseaR.L. TaylorProfessor in the Graduate SchoolDepartment of Civil and Environmental EngineeringUniversity of California at BerkeleyBerkeley, CaliforniaOXFORD AUCKLAND BOSTON JOHANNESBURG MELBOURNE NEW DELHI

DedicationThis book is dedicated to our wives Helen and MaryLou and our families for their support and patienceduring the preparation of this book, and also to all ofour students and colleagues who over the years havecontributed to our knowledge of the ®nite elementmethod. In particular we would like to mentionProfessor Eugenio OnÄ ate and his group at CIMNE fortheir help, encouragement and support during thepreparation process.

ContentsPrefacexv1. Some preliminaries: the standard discrete system 11.1 Introduction 11.2 The structural element and the structural system 41.3 Assembly and analysis of a structure 81.4 The boundary conditions 91.5 Electrical and ¯uid networks 101.6 The general pattern 121.7 The standard discrete system 141.8 Transformation of coordinates 15References 162. A direct approach to problems in elasticity 182.1 Introduction 182.2 Direct formulation of ®nite element characteristics 192.3 Generalization to the whole region 262.4 Displacement approach as a minimization of total potential energy 292.5 Convergence criteria 312.6 Discretization error and convergence rate 322.7 Displacement functions with discontinuity between elements 332.8 Bound on strain energy in a displacement formulation 342.9 Direct minimization 352.10 An example 352.11 Concluding remarks 37References 373. Generalization of the ®nite element concepts. Galerkin-weighted residualand variational approaches 393.1 Introduction 393.2 Integral or `weak' statements equivalent to the di€erential equations 423.3 Approximation to integral formulations 463.4 Virtual work as the `weak form' of equilibrium equations foranalysis of solids or ¯uids 53

viiiContents3.5 Partial discretization 553.6 Convergence 583.7 What are `variational principles'? 603.8 `Natural' variational principles and their relation to governingdi€erential equations 623.9 Establishment of natural variational principles for linear,self-adjoint di€erential equations 663.10 Maximum, minimum, or a saddle point? 693.11 Constrained variational principles. Lagrange multipliers andadjoint functions 703.12 Constrained variational principles. Penalty functions and the leastsquare method 763.13 Concluding remarks 82References 844. Plane stress and plane strain 874.1 Introduction 874.2 Element characteristics 874.3 Examples ± an assessment of performance 974.4 Some practical applications 1004.5 Special treatment of plane strain with an incompressible material 1104.6 Concluding remark 111References 1115. Axisymmetric stress analysis 1125.1 Introduction 1125.2 Element characteristics 1125.3 Some illustrative examples 1215.4 Early practical applications 1235.5 Non-symmetrical loading 1245.6 Axisymmetry ± plane strain and plane stress 124References 1266. Three-dimensional stress analysis 1276.1 Introduction 1276.2 Tetrahedral element characteristics 1286.3 Composite elements with eight nodes 1346.4 Examples and concluding remarks 135References 1397. Steady-state ®eld problems ± heat conduction, electric and magneticpotential, ¯uid ¯ow, etc. 1407.1 Introduction 1407.2 The general quasi-harmonic equation 1417.3 Finite element discretization 1437.4 Some economic specializations 1447.5 Examples ± an assessment of accuracy 1467.6 Some practical applications 149

Contentsix7.7 Concluding remarks 161References 1618. `Standard' and `hierarchical' element shape functions: some generalfamilies of C 0 continuity 1648.1 Introduction 1648.2 Standard and hierarchical concepts 1658.3 Rectangular elements ± some preliminary considerations 1688.4 Completeness of polynomials 1718.5 Rectangular elements ± Lagrange family 1728.6 Rectangular elements ± `serendipity' family 1748.7 Elimination of internal variables before assembly ± substructures 1778.8 Triangular element family 1798.9 Line elements 1838.10 Rectangular prisms ± Lagrange family 1848.11 Rectangular prisms ± `serendipity' family 1858.12 Tetrahedral elements 1868.13 Other simple three-dimensional elements 1908.14 Hierarchic polynomials in one dimension 1908.15 Two- and three-dimensional, hierarchic, elements of the `rectangle'or `brick' type 1938.16 Triangle and tetrahedron family 1938.17 Global and local ®nite element approximation 1968.18 Improvement of conditioning with hierarchic forms 1978.19 Concluding remarks 198References 1989. Mapped elements and numerical integration ± `in®nite' and `singularity'elements 2009.1 Introduction 2009.2 Use of `shape functions' in the establishment of coordinatetransformations 2039.3 Geometrical conformability of elements 2069.4 Variation of the unknown function within distorted, curvilinearelements. Continuity requirements 2069.5 Evaluation of element matrices (transformation in , , coordinates) 2089.6 Element matrices. Area and volume coordinates 2119.7 Convergence of elements in curvilinear coordinates 2139.8 Numerical integration ± one-dimensional 2179.9 Numerical integration ± rectangular (2D) or right prism (3D)regions 2199.10 Numerical integration ± triangular or tetrahedral regions 2219.11 Required order of numerical integration 2239.12 Generation of ®nite element meshes by mapping. Blending functions 2269.13 In®nite domains and in®nite elements 2299.14 Singular elements by mapping for fracture mechanics, etc. 234

xContents9.15 A computational advantage of numerically integrated ®niteelements 2369.16 Some practical examples of two-dimensional stress analysis 2379.17 Three-dimensional stress analysis 2389.18 Symmetry and repeatability 244References 24610. The patch test, reduced integration, and non-conforming elements 25010.1 Introduction 25010.2 Convergence requirements 25110.3 The simple patch test (tests A and B) ± a necessary condition forconvergence 25310.4 Generalized patch test (test C) and the single-element test 25510.5 The generality of a numerical patch test 25710.6 Higher order patch tests 25710.7 Application of the patch test to plane elasticity elements with`standard' and `reduced' quadrature 25810.8 Application of the patch test to an incompatible element 26410.9 Generation of incompatible shape functions which satisfy thepatch test 26810.10 The weak patch test ± example 27010.11 Higher order patch test ± assessment of robustness 27110.12 Conclusion 273References 27411. Mixed formulation and constraints± complete ®eld methods 27611.1 Introduction 27611.2 Discretization of mixed forms ± some general remarks 27811.3 Stability of mixed approximation. The patch test 28011.4 Two-®eld mixed formulation in elasticity 28411.5 Three-®eld mixed formulations in elasticity 29111.6 An iterative method solution of mixed approximations 29811.7 Complementary forms with direct constraint 30111.8 Concluding remarks ± mixed formulation or a test of element`robustness' 304References 30412. Incompressible materials, mixed methods and other procedures ofsolution 30712.1 Introduction 30712.2 Deviatoric stress and strain, pressure and volume change 30712.3 Two-®eld incompressible elasticity (u±p form) 30812.4 Three-®eld nearly incompressible elasticity (u±p±" v form) 31412.5 Reduced and selective integration and its equivalence to penalizedmixed problems 31812.6 A simple iterative solution process for mixed problems: Uzawamethod 323

Contentsxi12.7 Stabilized methods for some mixed elements failing theincompressibility patch test 32612.8 Concluding remarks 342References 34313. Mixed formulation and constraints ± incomplete (hybrid) ®eld methods,boundary/Tre€tz methods 34613.1 General 34613.2 Interface traction link of two (or more) irreducible formsubdomains 34613.3 Interface traction link of two or more mixed form subdomains 34913.4 Interface displacement `frame' 35013.5 Linking of boundary (or Tre€tz)-type solution by the `frame' ofspeci®ed displacements 35513.6 Subdomains with `standard' elements and global functions 36013.7 Lagrange variables or discontinuous Galerkin methods? 36113.8 Concluding remarks 361References 36214. Errors, recovery processes and error estimates 36514.1 De®nition of errors 36514.2 Superconvergence and optimal sampling points 37014.3 Recovery of gradients and stresses 37514.4 Superconvergent patch recovery ± SPR 37714.5 Recovery by equilibration of patches ± REP 38314.6 Error estimates by recovery 38514.7 Other error estimators ± residual based methods 38714.8 Asymptotic behaviour and robustness of error estimators ± theBabusÏ ka patch test 39214.9 Which errors should concern us? 398References 39815. Adaptive ®nite element re®nement 40115.1 Introduction 40115.2 Some examples of adaptive h-re®nement 40415.3 p-re®nement and hp-re®nement 41515.4 Concluding remarks 426References 42616. Point-based approximations; element-free Galerkin ± and othermeshless methods 42916.1 Introduction 42916.2 Function approximation 43116.3 Moving least square approximations ± restoration of continuityof approximation 43816.4 Hierarchical enhancement of moving least square expansions 44316.5 Point collocation ± ®nite point methods 446

xiiContents16.6 Galerkin weighting and ®nite volume methods 45116.7 Use of hierarchic and special functions based on standard ®niteelements satisfying the partition of unity requirement 45716.8 Closure 464References 46417. The time dimension ± semi-discretization of ®eld and dynamic problemsand analytical solution procedures 46817.1 Introduction 46817.2 Direct formulation of time-dependent problems with spatial ®niteelement subdivision 46817.3 General classi®cation 47617.4 Free response ± eigenvalues for second-order problems anddynamic vibration 47717.5 Free response ± eigenvalues for ®rst-order problems and heatconduction, etc. 48417.6 Free response ± damped dynamic eigenvalues 48417.7 Forced periodic response 48517.8 Transient response by analytical procedures 48617.9 Symmetry and repeatability 490References 49118. The time dimension ± discrete approximation in time 49318.1 Introduction 49318.2 Simple time-step algorithms for the ®rst-order equation 49518.3 General single-step algorithms for ®rst- and second-order equations 50818.4 Multistep recurrence algorithms 52218.5 Some remarks on general performance of numerical algorithms 53018.6 Time discontinuous Galerkin approximation 53618.7 Concluding remarks 538References 53819. Coupled systems 54219.1 Coupled problems ± de®nition and classi®cation 54219.2 Fluid±structure interaction (Class I problem) 54519.3 Soil±pore ¯uid interaction (Class II problems) 55819.4 Partitioned single-phase systems ± implicit±explicit partitions(Class I problems) 56519.5 Staggered solution processes 567References 57220. Computer procedures for ®nite element analysis 57620.1 Introduction 57620.2 Data input module 57820.3 Memory management for array storage 58820.4 Solution module ± the command programming language 59020.5 Computation of ®nite element solution modules 597

Volume 2: Solid and structural mechanics1. General problems in solid mechanics and non-linearity2. Solution of non-linear algebraic equations3. Inelastic materials4. Plate bending approximation: thin (Kirchho€) plates and C 1 continuity requirements5. `Thick' Reissner±Mindlin plates ± irreducible and mixed formulations6. Shells as an assembly of ¯at elements7. Axisymmetric shells8. Shells as a special case of three-dimensional analysis ± Reissner±Mindlinassumptions9. Semi-analytical ®nite element processes ± use of orthogonal functions and `®nitestrip' methods10. Geometrically non-linear problems ± ®nite deformation11. Non-linear structural problems ± large displacement and instability12. Pseudo-rigid and rigid±¯exible bodies13. Computer procedures for ®nite element analysisAppendix A: Invariants of second-order tensorsVolume 3: Fluid dynamics1. Introduction and the equations of ¯uid dynamics2. Convection dominated problems ± ®nite element approximations3. A general algorithm for compressible and incompressible ¯ows ± the characteristicbased split (CBS) algorithm4. Incompressible laminar ¯ow ± newtonian and non-newtonian ¯uids5. Free surfaces, buoyancy and turbulent incompressible ¯ows6. Compressible high speed gas ¯ow7. Shallow-water problems8. Waves9. Computer implementation of the CBS algorithmAppendix A. Non-conservative form of Navier±Stokes equationsAppendix B. Discontinuous Galerkin methods in the solution of the convection±di€usion equationAppendix C. Edge-based ®nite element formulationAppendix D. Multi grid methodsAppendix E. Boundary layer ± inviscid ¯ow coupling

xviPrefaceIn Volumes 1, 2 and 3 the chapters on computational methods are much reduced bytransferring the computer source programs to a web site. 1 This has the very substantialadvantage of not only eliminating errors in copying the programs but also inensuring that the reader has the bene®t of the most recent set of programs availableto him or her at all times as it is our intention from time to time to update and expandthe available programs.The authors are particularly indebted to the International Center of NumericalMethods in Engineering (CIMNE) in Barcelona who have allowed their pre- andpost-processing code (GiD) to be accessed from the publisher's web site. Thisallows such di cult tasks as mesh generation and graphic output to be dealt withe ciently. The authors are also grateful to Dr J.Z. Zhu for his careful scrutiny andhelp in drafting Chapters 14 and 15. These deal with error estimation and adaptivity,a subject to which Dr Zhu has extensively contributed. Finally, we thank Peter andJackie Bettess for writing the general subject index.OCZ and RLT1 Complete source code for all programs in the three volumes may be obtained at no cost from thepublisher's web page: http://www.bh.com/companions/fem

1Some preliminaries: the standarddiscrete system1.1 IntroductionThe limitations of the human mind are such that it cannot grasp the behaviour of itscomplex surroundings and creations in one operation. Thus the process of subdividingall systems into their individual components or `elements', whose behaviouris readily understood, and then rebuilding the original system from such componentsto study its behaviour is a natural way in which the engineer, the scientist, or even theeconomist proceeds.In many situations an adequate model is obtained using a ®nite number of wellde®nedcomponents. We shall term such problems discrete. In others the subdivisionis continued inde®nitely and the problem can only be de®ned using the mathematical®ction of an in®nitesimal. This leads to di€erential equations or equivalent statementswhich imply an in®nite number of elements. We shall term such systems continuous.With the advent of digital computers, discrete problems can generally be solvedreadily even if the number of elements is very large. As the capacity of all computersis ®nite, continuous problems can only be solved exactly by mathematical manipulation.Here, the available mathematical techniques usually limit the possibilities tooversimpli®ed situations.To overcome the intractability of realistic types of continuum problems, variousmethods of discretization have from time to time been proposed both by engineersand mathematicians. All involve an approximation which, hopefully, approachesin the limit the true continuum solution as the number of discrete variablesincreases.The discretization of continuous problems has been approached di€erently bymathematicians and engineers. Mathematicians have developed general techniquesapplicable directly to di€erential equations governing the problem, such as ®nite differenceapproximations, 1;2 various weighted residual procedures, 3;4 or approximatetechniques for determining the stationarity of properly de®ned `functionals'. Theengineer, on the other hand, often approaches the problem more intuitively by creatingan analogy between real discrete elements and ®nite portions of a continuumdomain. For instance, in the ®eld of solid mechanics McHenry, 5 Hreniko€, 6Newmark 7 , and indeed Southwell 9 in the 1940s, showed that reasonably good solutionsto an elastic continuum problem can be obtained by replacing small portions

2 Some preliminaries: the standard discrete systemof the continuum by an arrangement of simple elastic bars. Later, in the same context,Argyris 8 and Turner et al. 9 showed that a more direct, but no less intuitive, substitutionof properties can be made much more e€ectively by considering that smallportions or `elements' in a continuum behave in a simpli®ed manner.It is from the engineering `direct analogy' view that the term `®nite element' wasborn. Clough 10 appears to be the ®rst to use this term, which implies in it a directuse of a standard methodology applicable to discrete systems. Both conceptually andfrom the computational viewpoint, this is of the utmost importance. The ®rstallows an improved understanding to be obtained; the second o€ers a uni®edapproach to the variety of problems and the development of standard computationalprocedures.Since the early 1960s much progress has been made, and today the purely mathematicaland `analogy' approaches are fully reconciled. It is the object of this text topresent a view of the ®nite element method as a general discretization procedure of continuumproblems posed by mathematically de®ned statements.In the analysis of problems of a discrete nature, a standard methodology has beendeveloped over the years. The civil engineer, dealing with structures, ®rst calculatesforce±displacement relationships for each element of the structure and then proceedsto assemble the whole by following a well-de®ned procedure of establishing localequilibrium at each `node' or connecting point of the structure. The resulting equationscan be solved for the unknown displacements. Similarly, the electrical orhydraulic engineer, dealing with a network of electrical components (resistors, capacitances,etc.) or hydraulic conduits, ®rst establishes a relationship between currents(¯ows) and potentials for individual elements and then proceeds to assemble thesystem by ensuring continuity of ¯ows.All such analyses follow a standard pattern which is universally adaptable to discretesystems. It is thus possible to de®ne a standard discrete system, and this chapterwill be primarily concerned with establishing the processes applicable to such systems.Much of what is presented here will be known to engineers, but some reiteration atthis stage is advisable. As the treatment of elastic solid structures has been themost developed area of activity this will be introduced ®rst, followed by examplesfrom other ®elds, before attempting a complete generalization.The existence of a uni®ed treatment of `standard discrete problems' leads us to the®rst de®nition of the ®nite element process as a method of approximation to continuumproblems such that(a) the continuum is divided into a ®nite number of parts (elements), the behaviour ofwhich is speci®ed by a ®nite number of parameters, and(b) the solution of the complete system as an assembly of its elements follows preciselythe same rules as those applicable to standard discrete problems.It will be found that most classical mathematical approximation procedures as wellas the various direct approximations used in engineering fall into this category. It isthus di cult to determine the origins of the ®nite element method and the precisemoment of its invention.Table 1.1 shows the process of evolution which led to the present-day concepts of®nite element analysis. Chapter 3 will give, in more detail, the mathematical basiswhich emerged from these classical ideas. 11ÿ20

Table 1.1ENGINEERINGMATHEMATICSTrialfunctionsFinitedifferencesVariationalmethodsWeightedresidualsRichardson 1910 15Liebman 1918 16Southwell 1946 1Rayleigh 1870 11Ritz 1909 12 ÿÿGauss 1795 18Galerkin 1915 19Biezeno±Koch 1923 20Structuralanaloguesubstitution"Piecewisecontinuoustrial functionsÿÿ "ÿÿ "Hreniko€ 1941 6McHenry 1943 5Newmark 1949 7ÿÿ "Courant 1943 13Prager±Synge 1947 14Zienkiewicz 1964 21DirectcontinuumelementsArgyris 1955 8Turner et al. 1956 9ÿÿ "ÿÿ "ÿÿ "VariationalfinitedifferencesVarga 1962 17PRESENT-DAYFINITE ELEMENT METHOD

4 Some preliminaries: the standard discrete system1.2 The structural element and the structural systemY 4X 4p3 4(2)y(1)(4)1 2x(3)5 6yp(1)2V 3U 33Nodes1xA typical element (1)Fig. 1.1 A typical structure built up from interconnected elements.To introduce the reader to the general concept of discrete systems we shall ®rstconsider a structural engineering example of linear elasticity.Figure 1.1 represents a two-dimensional structure assembled from individualcomponents and interconnected at the nodes numbered 1 to 6. The joints at thenodes, in this case, are pinned so that moments cannot be transmitted.As a starting point it will be assumed that by separate calculation, or for that matterfrom the results of an experiment, the characteristics of each element are preciselyknown. Thus, if a typical element labelled (1) and associated with nodes 1, 2, 3 isexamined, the forces acting at the nodes are uniquely de®ned by the displacementsof these nodes, the distributed loading acting on the element (p), and its initialstrain. The last may be due to temperature, shrinkage, or simply an initial `lack of®t'. The forces and the corresponding displacements are de®ned by appropriate components(U, V and u, v) in a common coordinate system.Listing the forces acting on all the nodes (three in the case illustrated) of the element(1) as a matrixy we have8>:q 1 1q 1 2q 1 39>=>;q 1 1 ˆU1V 1; etc: …1:1†y A limited knowledge of matrix algebra will be assumed throughout this book. This is necessary forreasonable conciseness and forms a convenient book-keeping form. For readers not familiar with the subjecta brief appendix (Appendix A) is included in which su cient principles of matrix algebra are given to followthe development intelligently. Matrices (and vectors) will be distinguished by bold print throughout.

and for the corresponding nodal displacements8 9a >< 1 >= a 1 ˆ a 2 a 1 ˆ>: >;a 3u1v 1; etc: …1:2†Assuming linear elastic behaviour of the element, the characteristic relationship willalways be of the formq 1 ˆ K 1 a 1 ‡ f 1 p ‡ f 1 " 0…1:3†in which f 1 p represents the nodal forces required to balance any distributed loads actingon the element and f 1 " 0the nodal forces required to balance any initial strains such asmay be caused by temperature change if the nodes are not subject to any displacement.The ®rst of the terms represents the forces induced by displacement of the nodes.Similarly, a preliminary analysis or experiment will permit a unique de®nition ofstresses or internal reactions at any speci®ed point or points of the element interms of the nodal displacements. De®ning such stresses by a matrix r 1 a relationshipof the formr 1 ˆ Q 1 a 1 ‡ r 1 " 0…1:4†is obtained in which the two term gives the stresses due to the initial strains when nonodal displacement occurs.The matrix K e is known as the element sti€ness matrix and the matrix Q e as theelement stress matrix for an element (e).Relationships in Eqs (1.3) and (1.4) have been illustrated by an example of an elementwith three nodes and with the interconnection points capable of transmittingonly two components of force. Clearly, the same arguments and de®nitions willapply generally. An element (2) of the hypothetical structure will possess only twopoints of interconnection; others may have quite a large number of such points. Similarly,if the joints were considered as rigid, three components of generalized force andof generalized displacement would have to be considered, the last of these correspondingto a moment and a rotation respectively. For a rigidly jointed, three-dimensionalstructure the number of individual nodal components would be six. Quite generally,therefore,8 98 9q e 1>=2 ˆ .>:. >;q e mandThe structural element and the structural system 5a 1>=2 ˆ .>:>;a m…1:5†with each q e i and a i possessing the same number of components or degrees of freedom.These quantities are conjugate to each other.The sti€ness matrices of the element will clearly always be square and of the form2K e ii K e ij K e 3imK e . .ˆ 4.. . .5 …1:6†K e mi K e mm

6 Some preliminaries: the standard discrete systemnyV i (v i )pCE 1 A 1x iiβU i (u i )Ly ixFig. 1.2 A pin-ended bar.in which K e ii, etc., are submatrices which are again square and of the size l l, where lis the number of force components to be considered at each node.As an example, the reader can consider a pin-ended bar of uniform section A andmodulus E in a two-dimensional problem shown in Fig. 1.2. The bar is subject to auniform lateral load p and a uniform thermal expansion strain" 0 ˆ Twhere is the coe cient of linear expansion and T is the temperature change.If the ends of the bar are de®ned by the coordinates x i , y i and x n , y n its length can becalculated asL ˆand its inclination from the horizontal asq‰…x n ÿ x i † 2 ‡…y n ÿ y i † 2 Š ˆ tan ÿ1 y n ÿ y ix n ÿ x iOnly two components of force and displacement have to be considered at thenodes.The nodal forces due to the lateral load are clearly8 9 8 9ÿsin >:U iV iU nV n>= >< cos >=ˆÿÿsin >; >: >;p cos and represent the appropriate components of simple reactions, pL=2. Similarly, torestrain the thermal expansion " 0 an axial force …ETA† is needed, which gives thepL2

componentsf e " 08>:U iV iU nV nFinally, the element displacements9 8 9ÿcos >= >< ÿsin >=…ETA†cos >; >: >;" 0ˆÿsin 8>:will cause an elongation …u n ÿ u i † cos ‡…v n ÿ v i † sin . This, when multiplied byEA=L, gives the axial force whose components can again be found. Rearrangingthese in the standard form gives8 9>=The structural element and the structural system 7>: >;V n 2cos 2 sin cos ÿcos 2 38 ÿsin cos ˆ EA sin cos sin 2 ÿsin cos ÿsin 2 >:ÿsin cos ÿsin 2 sin cos sin 2 The components of the general equation (1.3) have thus been established for theelementary case discussed. It is again quite simple to ®nd the stresses at any sectionof the element in the form of relation (1.4). For instance, if attention is focused onthe mid-section C of the bar the average stress determined from the axial tensionto the element can be shown to ber e ˆ EL ‰ÿcos ; ÿsin ; cos ; sin Šae ÿ ETwhere all the bending e€ects of the lateral load p have been ignored.For more complex elements more sophisticated procedures of analysis are requiredbut the results are of the same form. The engineer will readily recognize that the socalled`slope±de¯ection' relations used in analysis of rigid frames are only a specialcase of the general relations.It may perhaps be remarked, in passing, that the complete sti€ness matrix obtainedfor the simple element in tension turns out to be symmetric (as indeed was the casewith some submatrices). This is by no means fortuitous but follows from the principleof energy conservation and from its corollary, the well-known Maxwell±Bettireciprocal theorem.u iv iu nv n9>=>;u iv iu nv n9>=>;

8 Some preliminaries: the standard discrete systemThe element properties were assumed to follow a simple linear relationship. Inprinciple, similar relationships could be established for non-linear materials, butdiscussion of such problems will be held over at this stage.The calculation of the sti€ness coe cients of the bar which we have given here willbe found in many textbooks. Perhaps it is worthwhile mentioning here that the ®rstuse of bar assemblies for large structures was made as early as 1935 when Southwellproposed his classical relaxation method. 221.3 Assembly and analysis of a structureConsider again the hypothetical structure of Fig. 1.1. To obtain a complete solutionthe two conditions of(a) displacement compatibility and(b) equilibriumhave to be satis®ed throughout.Any system of nodal displacements a:8 9>< a 1 >=a ˆ.…1:7†>: >;listed now for the whole structure in which all the elements participate, automaticallysatis®es the ®rst condition.As the conditions of overall equilibrium have already been satis®ed within an element,all that is necessary is to establish equilibrium conditions at the nodes of thestructure. The resulting equations will contain the displacements as unknowns, andonce these have been solved the structural problem is determined. The internalforces in elements, or the stresses, can easily be found by using the characteristicsestablished a priori for each element by Eq. (1.4).Consider the structure to be loaded by external forces r:8 9>< r 1 >=r ˆ . …1:8†>: >;applied at the nodes in addition to the distributed loads applied to the individualelements. Again, any one of the forces r i must have the same number of componentsas that of the element reactions considered. In the example in question Xir i ˆ…1:9†Y ias the joints were assumed pinned, but at this stage the general case of an arbitrarynumber of components will be assumed.If now the equilibrium conditions of a typical node, i, are to be established, eachcomponent of r i has, in turn, to be equated to the sum of the component forcescontributed by the elements meeting at the node. Thus, considering all the forcea nr n

The boundary conditions 9components we haver i ˆ Xme ˆ 1q e i ˆ q 1 i ‡ q 2 i ‡…1:10†in which q 1 i is the force contributed to node i by element 1, q 2 i by element 2, etc.Clearly, only the elements which include point i will contribute non-zero forces,but for tidiness all the elements are included in the summation.Substituting the forces contributing to node i from the de®nition (1.3) and notingthat nodal variables a i are common (thus omitting the superscript e), we have X m X mr i ˆ a 1 ‡a 2 ‡‡ Xm…1:11†whereK e i 1e ˆ 1K e i 2e ˆ 1f e ˆ f e p ‡ f e " 0f e ie ˆ 1The summation again only concerns the elements which contribute to node i. If allsuch equations are assembled we have simplyKa ˆ r ÿ f…1:12†in which the submatrices areK ij ˆ Xmf i ˆ XmK e ije ˆ 1f e ie ˆ 1…1:13†with summations including all elements. This simple rule for assembly is veryconvenient because as soon as a coe cient for a particular element is found it canbe put immediately into the appropriate `location' speci®ed in the computer. Thisgeneral assembly process can be found to be the common and fundamental feature ofall ®nite element calculations and should be well understood by the reader.If di€erent types of structural elements are used and are to be coupled it must beremembered that the rules of matrix summation permit this to be done only ifthese are of identical size. The individual submatrices to be added have therefore tobe built up of the same number of individual components of force or displacement.Thus, for example, if a member capable of transmitting moments to a node is to becoupled at that node to one which in fact is hinged, it is necessary to complete thesti€ness matrix of the latter by insertion of appropriate (zero) coe cients in therotation or moment positions.1.4 The boundary conditionsThe system of equations resulting from Eq. (1.12) can be solved once theprescribed support displacements have been substituted. In the example of Fig. 1.1,where both components of displacement of nodes 1 and 6 are zero, this will mean

10 Some preliminaries: the standard discrete systemthe substitution of 0a 1 ˆ a 6 ˆ0which is equivalent to reducing the number of equilibrium equations (in this instance12) by deleting the ®rst and last pairs and thus reducing the total number of unknowndisplacement components to eight. It is, nevertheless, always convenient to assemblethe equation according to relation (1.12) so as to include all the nodes.Clearly, without substitution of a minimum number of prescribed displacements toprevent rigid body movements of the structure, it is impossible to solve this system,because the displacements cannot be uniquely determined by the forces in such asituation. This physically obvious fact will be interpreted mathematically as thematrix K being singular, i.e., not possessing an inverse. The prescription of appropriatedisplacements after the assembly stage will permit a unique solution to beobtained by deleting appropriate rows and columns of the various matrices.If all the equations of a system are assembled, their form isK 11 a 1 ‡ K 12 a 2 ‡ˆr 1 ÿ f 1K 21 a 1 ‡ K 22 a 2 ‡ˆr 2 ÿ f 2etc:…1:14†and it will be noted that if any displacement, such as a 1 ˆ a 1 , is prescribed then theexternal `force' r 1 cannot be simultaneously speci®ed and remains unknown. The®rst equation could then be deleted and substitution of known values of a 1 made inthe remaining equations. This process is computationally cumbersome and thesame objective is served by adding a large number, I, to the coe cient K 11 andreplacing the right-hand side, r 1 ÿ f 1 ,bya 1 .If is very much larger than othersti€ness coe cients this alteration e€ectively replaces the ®rst equation by the equationa 1 ˆ a 1…1:15†that is, the required prescribed condition, but the whole system remains symmetricand minimal changes are necessary in the computation sequence. A similar procedurewill apply to any other prescribed displacement. The above arti®ce was introduced byPayne and Irons. 23 An alternative procedure avoiding the assembly of equationscorresponding to nodes with prescribed boundary values will be presented inChapter 20.When all the boundary conditions are inserted the equations of the system can besolved for the unknown displacements and stresses, and the internal forces in each elementobtained.1.5 Electrical and ¯uid networksIdentical principles of deriving element characteristics and of assembly will be foundin many non-structural ®elds. Consider, for instance, the assembly of electricalresistances shown in Fig. 1.3.

Electrical and ¯uid networks 11jP iiJ j ,V jjr eJ i ,V iiFig. 1.3 A network of electrical resistances.If a typical resistance element, ij, is isolated from the system we can write, by Ohm's law,the relation between the currents entering the element at the ends and the end voltages asJ e i ˆ 1r e …V i ÿ V j †J e j ˆ 1r e …V j ÿ V i †or in matrix form Je iJ e jˆ 1 1 ÿ1 Vir e ÿ1 1 V jwhich in our standard form is simplyJ e ˆ K e V e…1:16†This form clearly corresponds to the sti€ness relationship (1.3); indeed if an externalcurrent were supplied along the length of the element the element `force' termscould also be found.To assemble the whole network the continuity of the potential …V† at the nodes isassumed and a current balance imposed there. If P i now stands for the external inputof current at node i we must have, with complete analogy to Eq. (1.11),P i ˆ Xn X m…1:17†j ˆ 1K e ijV je ˆ 1where the second summation is over all `elements', and once again for all the nodesP ˆ KV…1:18†in whichK ij ˆ XmK e ije ˆ 1

12 Some preliminaries: the standard discrete systemMatrix notation in the above has been dropped since the quantities such as voltageand current, and hence also the coe cients of the `sti€ness' matrix, are scalars.If the resistances were replaced by ¯uid-carrying pipes in which a laminar regimepertained, an identical formulation would once again result, with V standing forthe hydraulic head and J for the ¯ow.For pipe networks that are usually encountered, however, the linear laws are ingeneral not valid. Typically the ¯ow±head relationship is of a formJ i ˆ c…V i ÿ V j † …1:19†where the index lies between 0.5 and 0.7. Even now it would still be possible to writerelationships in the form (1.16) noting, however, that the matrices K e are no longerarrays of constants but are known functions of V. The ®nal equations can onceagain be assembled but their form will be non-linear and in general iterative techniquesof solution will be needed.Finally it is perhaps of interest to mention the more general form of an electricalnetwork subject to an alternating current. It is customary to write the relationshipsbetween the current and voltage in complex form with the resistance being replacedby complex impedance. Once again the standard forms of (1.16)±(1.18) will beobtained but with each quantity divided into real and imaginary parts.Identical solution procedures can be used if the equality of the real and imaginaryquantities is considered at each stage. Indeed with modern digital computers it ispossible to use standard programming practice, making use of facilities availablefor dealing with complex numbers. Reference to some problems of this class will bemade in the chapter dealing with vibration problems in Chapter 17.1.6 The general patternAn example will be considered to consolidate the concepts discussed in this chapter.This is shown in Fig. 1.4(a) where ®ve discrete elements are interconnected. Thesemay be of structural, electrical, or any other linear type. In the solution:The ®rst step is the determination of element properties from the geometric materialand loading data. For each element the `sti€ness matrix' as well as the corresponding`nodal loads' are found in the form of Eq. (1.3). Each element has its own identifyingnumber and speci®ed nodal connection. For example:element 1 connection 1 3 42 1 4 23 2 54 3 6 7 45 4 7 8 5Assuming that properties are found in global coordinates we can enter each `sti€ness'or `force' component in its position of the global matrix as shown in Fig.1.4(b), Each shaded square represents a single coe cient or a submatrix of typeK ij if more than one quantity is being considered at the nodes. Here the separatecontribution of each element is shown and the reader can verify the position of

The general pattern 131 2(a)2 313 4 54 56871 2 3 4 5++++a(b)= + + + +[K]{f}a=(c)BANDFig. 1.4 The general pattern.the coe cients. Note that the various types of `elements' considered here present nodi culty in speci®cation. (All `forces', including nodal ones, are here associatedwith elements for simplicity.)The second step is the assembly of the ®nal equations of the type given by Eq. (1.12).This is accomplished according to the rule of Eq. (1.13) by simple addition of allnumbers in the appropriate space of the global matrix. The result is shown inFig. 1.4(c) where the non-zero coe cients are indicated by shading.As the matrices are symmetric only the half above the diagonal shown needs, infact, to be found.All the non-zero coe cients are con®ned within a band or pro®le which can becalculated a priori for the nodal connections. Thus in computer programs onlythe storage of the elements within the upper half of the pro®le is necessary, asshown in Fig. 1.4(c).The third step is the insertion of prescribed boundary conditions into the ®nalassembled matrix, as discussed in Sec. 1.3. This is followed by the ®nal step.The ®nal step solves the resulting equation system. Here many di€erent methodscan be employed, some of which will be discussed in Chapter 20. The general

14 Some preliminaries: the standard discrete systemsubject of equation solving, though extremely important, is in general beyond thescope of this book.The ®nal step discussed above can be followed by substitution to obtain stresses,currents, or other desired output quantities.All operations involved in structural or other network analysis are thus of anextremely simple and repetitive kind.We can now de®ne the standard discrete system as one in which such conditionsprevail.1.7 The standard discrete systemIn the standard discrete system, whether it is structural or of any other kind, we ®ndthat:1. A set of discrete parameters, say a i , can be identi®ed which describes simultaneouslythe behaviour of each element, e, and of the whole system. We shall callthese the system parameters.2. For each element a set of quantities q e i can be computed in terms of the systemparameters a i . The general function relationship can be non-linearq e i ˆ q e i …a†…1:20†but in many cases a linear form exists givingq e i ˆ K e i 1a 1 ‡ K e i 2a 2 ‡‡f e i…1:21†3. The system equations are obtained by a simple additionr i ˆ Xm…1:22†q e ie ˆ 1where r i are system quantities (often prescribed as zero).In the linear case this results in a system of equationsKa ‡ f ˆ rsuch thatK ij ˆ Xmf i ˆ XmK e ije ˆ 1f e ie ˆ 1…1:23†…1:24†from which the solution for the system variables a can be found after imposingnecessary boundary conditions.The reader will observe that this de®nition includes the structural, hydraulic, andelectrical examples already discussed. However, it is broader. In general neitherlinearity nor symmetry of matrices need exist ± although in many problems thiswill arise naturally. Further, the narrowness of interconnections existing in usualelements is not essential.While much further detail could be discussed (we refer the reader to speci®c booksfor more exhaustive studies in the structural context 24ÿ26 ), we feel that the generalexpose given here should su ce for further study of this book.

Transformation of coordinates 15Only one further matter relating to the change of discrete parameters need bementioned here. The process of so-called transformation of coordinates is vital inmany contexts and must be fully understood.1.8 Transformation of coordinatesIt is often convenient to establish the characteristics of an individual element in acoordinate system which is di€erent from that in which the external forces anddisplacements of the assembled structure or system will be measured. A di€erentcoordinate system may, in fact, be used for every element, to ease the computation.It is a simple matter to transform the coordinates of the displacement and forcecomponents of Eq. (1.3) to any other coordinate system. Clearly, it is necessary todo so before an assembly of the structure can be attempted.Let the local coordinate system in which the element properties have been evaluatedbe denoted by a prime su x and the common coordinate system necessary forassembly have no embellishment. The displacement components can be transformedby a suitable matrix of direction cosines L asa 0 ˆ La…1:25†As the corresponding force components must perform the same amount of work ineither systemyq T a ˆ q 0 T a 0…1:26†On inserting (1.25) we haveorq T a ˆ q 0 T Laq ˆ L T q 0…1:27†The set of transformations given by (1.25) and (1.27) is called contravariant.To transform `sti€nesses' which may be available in local coordinates to globalones note that if we writeq 0 ˆ K 0 a 0…1:28†then by (1.27), (1.28), and (1.25)or in global coordinatesq ˆ L T K 0 LaK ˆ L T K 0 L…1:29†The reader can verify the usefulness of the above transformations by reworkingthe sample example of the pin-ended bar, ®rst establishing its sti€ness in its lengthcoordinates.y With () T standing for the transpose of the matrix.

16 Some preliminaries: the standard discrete systemIn many complex problems an external constraint of some kind may be imagined,enforcing the requirement (1.25) with the number of degrees of freedom of a and a 0being quite di€erent. Even in such instances the relations (1.26) and (1.27) continueto be valid.An alternative and more general argument can be applied to many other situationsof discrete analysis. We wish to replace a set of parameters a in which the systemequations have been written by another one related to it by a transformationmatrix T asa ˆ Tb…1:30†In the linear case the system equations are of the formKa ˆ r ÿ f…1:31†and on the substitution we haveKTb ˆ r ÿ f…1:32†The new system can be premultiplied simply by T T , yielding…T T KT†b ˆ T T r ÿ T T f…1:33†which will preserve the symmetry of equations if the matrix K is symmetric. However,occasionally the matrix T is not square and expression (1.30) represents in fact anapproximation in which a larger number of parameters a is constrained. Clearly thesystem of equations (1.32) gives more equations than are necessary for a solutionof the reduced set of parameters b, and the ®nal expression (1.33) presents a reducedsystem which in some sense approximates the original one.We have thus introduced the basic idea of approximation, which will be the subjectof subsequent chapters where in®nite sets of quantities are reduced to ®nite sets.A historical development of the subject of ®nite element methods has been presentedby the author. 27;28References1. R.V. Southwell. Relaxation Methods in Theoretical Physics. Clarendon Press, 1946.2. D.N. de G. Allen. Relaxation Methods. McGraw-Hill, 1955.3. S.H. Crandall. Engineering Analysis. McGraw-Hill, 1956.4. B.A. Finlayson. The Method of Weighted Residuals and Variational Principles. AcademicPress, 1972.5. D. McHenry. A lattice analogy for the solution of plane stress problems. J. Inst. Civ. Eng.,21, 59±82, 1943.6. A. Hreniko€. Solution of problems in elasticity by the framework method. J. Appl. Mech.,A8, 169±75, 1941.7. N.M. Newmark. Numerical methods of analysis in bars, plates and elastic bodies, inNumerical Methods in Analysis in Engineering (ed. L.E. Grinter), Macmillan, 1949.8. J.H. Argyris. Energy Theorems and Structural Analysis. Butterworth, 1960 (reprinted fromAircraft Eng., 1954±5).9. M.J. Turner, R.W. Clough, H.C. Martin, and L.J. Topp. Sti€ness and de¯ection analysisof complex structures. J. Aero. Sci., 23, 805±23, 1956.

10. R.W. Clough. The ®nite element in plane stress analysis. Proc. 2nd ASCE Conf. on ElectronicComputation. Pittsburgh, Pa., Sept. 1960.11. Lord Rayleigh (J.W. Strutt). On the theory of resonance. Trans. Roy. Soc. (London), A161,77±118, 1870.12. W. Ritz. UÈ ber eine neue Methode zur LoÈ sung gewissen Variations ± Probleme der mathematischenPhysik. J. Reine Angew. Math., 135, 1±61, 1909.13. R. Courant. Variational methods for the solution of problems of equilibrium and vibration.Bull. Am. Math. Soc., 49, 1±23, 1943.14. W. Prager and J.L. Synge. Approximation in elasticity based on the concept of functionspace. Q. J. Appl. Math., 5, 241±69, 1947.15. L.F. Richardson. The approximate arithmetical solution by ®nite di€erences of physicalproblems. Trans. Roy. Soc. (London), A210, 307±57, 1910.16. H. Liebman. Die angenaÈ herte Ermittlung: harmonischen, functionen und konformerAbbildung. Sitzber. Math. Physik Kl. Bayer Akad. Wiss. MuÈnchen. 3, 65±75, 1918.17. R.S. Varga. Matrix Iterative Analysis. Prentice-Hall, 1962.18. C.F. Gauss, See Carl Friedrich Gauss Werks. Vol. VII, GoÈ ttingen, 1871.19. B.G. Galerkin. Series solution of some problems of elastic equilibrium of rods and plates'(Russian). Vestn. Inzh. Tech., 19, 897±908, 1915.20. C.B. Biezeno and J.J. Koch. Over een Nieuwe Methode ter Berekening van Vlokke Platen.Ing. Grav., 38, 25±36, 1923.21. O.C. Zienkiewicz and Y.K. Cheung. The ®nite element method for analysis of elasticisotropic and orthotropic slabs. Proc. Inst. Civ. Eng., 28, 471±488, 1964.22. R.V. Southwell. Stress calculation in frame works by the method of systematic relaxationof constraints, Part I & II. Proc. Roy. Soc. London (A), 151, 56±95, 1935.23. N.A. Payne and B.M. Irons, Private communication, 1963.24. R.K. Livesley. Matrix Methods in Structural Analysis. 2nd ed., Pergamon Press, 1975.25. J.S. Przemieniecki. Theory of Matrix Structural Analysis. McGraw-Hill, 1968.26. H.C. Martin. Introduction to Matrix Methods of Structural Analysis. McGraw-Hill, 1966.27. O.C. Zienkiewicz. Origins, milestones and directions of the ®nite element method. Arch.Comp. Methods Eng., 2, 1±48, 1995.28. O.C. Zienkiewicz. Origins, milestones and directions of the ®nite element method ± Apersonal view. Handbook of Numerical Analysis, IV, 3±65. Editors P.C. Ciarlet and J.L.Lions, North-Holland, 1996.17

2A direct approach to problemsin elasticity2.1 IntroductionThe process of approximating the behaviour of a continuum by `®nite elements'which behave in a manner similar to the real, `discrete', elements described in theprevious chapter can be introduced through the medium of particular physical applicationsor as a general mathematical concept. We have chosen here to follow the ®rstpath, narrowing our view to a set of problems associated with structural mechanicswhich historically were the ®rst to which the ®nite element method was applied. InChapter 3 we shall generalize the concepts and show that the basic ideas are widelyapplicable.In many phases of engineering the solution of stress and strain distributions inelastic continua is required. Special cases of such problems may range from twodimensionalplane stress or strain distributions, axisymmetric solids, plate bending,and shells, to fully three-dimensional solids. In all cases the number of interconnectionsbetween any `®nite element' isolated by some imaginary boundariesand the neighbouring elements is in®nite. It is therefore di cult to see at ®rstglance how such problems may be discretized in the same manner as was describedin the preceding chapter for simpler structures. The di culty can be overcome (andthe approximation made) in the following manner:1. The continuum is separated by imaginary lines or surfaces into a number of `®niteelements'.2. The elements are assumed to be interconnected at a discrete number of nodalpoints situated on their boundaries and occasionally in their interior. InChapter 6 we shall show that this limitation is not necessary. The displacementsof these nodal points will be the basic unknown parameters of the problem, justas in simple, discrete, structural analysis.3. A set of functions is chosen to de®ne uniquely the state of displacement within each`®nite element' and on its boundaries in terms of its nodal displacements.4. The displacement functions now de®ne uniquely the state of strain within anelement in terms of the nodal displacements. These strains, together with anyinitial strains and the constitutive properties of the material, will de®ne the stateof stress throughout the element and, hence, also on its boundaries.

Direct formulation of ®nite element characteristics 195. A system of `forces' concentrated at the nodes and equilibrating the boundarystresses and any distributed loads is determined, resulting in a sti€ness relationshipof the form of Eq. (1.3).Once this stage has been reached the solution procedure can follow the standard discretesystem pattern described earlier.Clearly a series of approximations has been introduced. Firstly, it is not always easyto ensure that the chosen displacement functions will satisfy the requirement of displacementcontinuity between adjacent elements. Thus, the compatibility conditionon such lines may be violated (though within each element it is obviously satis®eddue to the uniqueness of displacements implied in their continuous representation).Secondly, by concentrating the equivalent forces at the nodes, equilibrium conditionsare satis®ed in the overall sense only. Local violation of equilibrium conditions withineach element and on its boundaries will usually arise.The choice of element shape and of the form of the displacement function forspeci®c cases leaves many opportunities for the ingenuity and skill of the engineerto be employed, and obviously the degree of approximation which can be achievedwill strongly depend on these factors.The approach outlined here is known as the displacement formulation. 1;2So far, the process described is justi®ed only intuitively, but what in fact has beensuggested is equivalent to the minimization of the total potential energy of the systemin terms of a prescribed displacement ®eld. If this displacement ®eld is de®ned in asuitable way, then convergence to the correct result must occur. The process is thenequivalent to the well-known Rayleigh±Ritz procedure. This equivalence will beproved in a later section of this chapter where also a discussion of the necessary convergencecriteria will be presented.The recognition of the equivalence of the ®nite element method to a minimizationprocess was late. 2;3 However, Courant in 1943 4 y and Prager and Synge 5 in 1947 proposedmethods that are in essence identical.This broader basis of the ®nite element method allows it to be extended to other continuumproblems where a variational formulation is possible. Indeed, general proceduresare now available for a ®nite element discretization of any problem de®ned by a properlyconstituted set of di€erential equations. Such generalizations will be discussed in Chapter3, and throughout the book application to non-structural problems will be made. It willbe found that the processes described in this chapter are essentially an application of trialfunctionand Galerkin-type approximations to a particular case of solid mechanics.2.2 Direct formulation of ®nite element characteristicsThe `prescriptions' for deriving the characteristics of a `®nite element' of a continuum,which were outlined in general terms, will now be presented in more detailedmathematical form.y It appears that Courant had anticipated the essence of the ®nite element method in general, and of a triangularelement in particular, as early as 1923 in a paper entitled `On a convergence principle in the calculus of variations.'KoÈ n. Gesellschaft der Wissenschaften zu GoÈ ttingen, Nachrichten, Berlin, 1923. He states: `We imagine amesh of triangles covering the domain . . . the convergence principles remain valid for each triangular domain.'

20 A direct approach to problems in elasticityyu i (V i )it = t xt yu i (U i )emjxFig. 2.1 A plane stress region divided into ®nite elements.It is desirable to obtain results in a general form applicable to any situation, butto avoid introducing conceptual di culties the general relations will be illustratedwith a very simple example of plane stress analysis of a thin slice. In this a divisionof the region into triangular-shaped elements is used as shown in Fig. 2.1. Relationshipsof general validity will be placed in a box. Again, matrix notation will beimplied.2.2.1 Displacement functionAtypical®niteelement,e, is de®ned by nodes, i, j, m, etc., and straight line boundaries.Let the displacements u at any point within the element be approximated as a columnvector, uà :8u ^u ˆ X>:a ia j.9e>=ˆ Na e>;…2:1†in which the components of N are prescribed functions of position and a e represents alisting of nodal displacements for a particular element.

Direct formulation of ®nite element characteristics 21y1iN ijmxFig. 2.2 Shape function N i for one element.In the case of plane stress, for instance, u…x; y†u ˆv…x; y†represents horizontal and vertical movements of a typical point within the elementand uia i ˆv ithe corresponding displacements of a node i.The functions N i , N j , N m have to be chosen so as to give appropriate nodaldisplacements when the coordinates of the corresponding nodes are inserted inEq. (2.1). Clearly, in general,N i …x i ; y i †ˆI (identity matrix)whileN i …x j ; y j †ˆN i …x m ; y m †ˆ0;which is simply satis®ed by suitable linear functions of x and y.If both the components of displacement are speci®ed in an identical manner thenwe can writeN i ˆ N i Iand obtain N i from Eq. (2.1) by noting that N i ˆ 1atx i , y i but zero at othervertices.The most obvious linear function in the case of a triangle will yield the shape of N iof the form shown in Fig. 2.2. Detailed expressions for such a linear interpolation aregiven in Chapter 4, but at this stage can be readily derived by the reader.The functions N will be called shape functions and will be seen later to play a paramountrole in ®nite element analysis.etc:2.2.2 StrainsWith displacements known at all points within the element the `strains' at any pointcan be determined. These will always result in a relationship that can be written in

22 A direct approach to problems in elasticitymatrix notation asye ^e ˆ Su…2:2†where S is a suitable linear operator. Using Eq. (2.1), the above equation can beapproximated aswithe ^e ˆ Ba…2:3†B ˆ SN…2:4†For the plane stress case the relevant strains of interest are those occurring in theplane and are de®ned in terms of the displacements by well-known relations 6 whichde®ne the operator S:8 9@u2@38 9 @x@x ; 0>: >; ˆ @v>=@ ˆ 0;u @y@yv xy>:@u@y ‡ @v6 74>;@@x @y ; @ 5@xWith the shape functions N i , N j ,andN m already determined, the matrix B caneasily be obtained. If the linear form of these functions is adopted then, in fact, thestrains will be constant throughout the element.2.2.3 StressesIn general, the material within the element boundaries may be subjected to initialstrains such as may be due to temperature changes, shrinkage, crystal growth,and so on. If such strains are denoted by e 0 then the stresses will be caused by thedi€erence between the actual and initial strains.In addition it is convenient to assume that at the outset of the analysis the body isstressed by some known system of initial residual stresses r 0 which, for instance, couldbe measured, but the prediction of which is impossible without the full knowledge ofthe material's history. These stresses can simply be added on to the general de®nition.Thus, assuming general linear elastic behaviour, the relationship between stresses andstrains will be linear and of the formr ˆ D…e ÿ e 0 †‡r 0…2:5†where D is an elasticity matrix containing the appropriate material properties.y It is known that strain is a second-rank tensor by its transformation properties; however, in this bookwe will normally represent quantities using matrix (Voigt) notation. The interested reader is encouragedto consult Appendix B for the relations between tensor forms and matrix quantities.

Direct formulation of ®nite element characteristics 23Again, for the particular case of plane stress three components of stress correspondingto the strains already de®ned have to be considered. These are, in familiar notation8 9>< x >=r ˆ y>: >;and the D matrix may be simply obtained from the usual isotropic stress±strainrelationship 6 " x ÿ…" x † 0 ˆ 1 xyE x ÿ E y" y ÿ…" y † 0 ˆÿ E x ‡ 1 E yi.e., on solving, xy ÿ… xy † 0 ˆ2…1 ‡ †E xy23D ˆ E1 0671 ÿ 2 4 1 0 50 0 …1 ÿ †=22.2.4 Equivalent nodal forcesLet8q e 9i >< >=q e ˆ q e j>: . >;.de®ne the nodal forces which are statically equivalent to the boundary stresses anddistributed body forces on the element. Each of the forces q e i must contain thesame number of components as the corresponding nodal displacement a i and beordered in the appropriate, corresponding directions.The distributed body forces b are de®ned as those acting on a unit volume ofmaterial within the element with directions corresponding to those of the displacementsu at that point.In the particular case of plane stress the nodal forces are, for instance, eq e Uii ˆV iwith components U and V corresponding to the directions of u and v displacements,and the distributed body forces are bxb ˆb yin which b x and b y are the `body force' components.

24 A direct approach to problems in elasticityTo make the nodal forces statically equivalent to the actual boundary stresses anddistributed body forces, the simplest procedure is to impose an arbitrary (virtual)nodal displacement and to equate the external and internal work done by the variousforces and stresses during that displacement.Let such a virtual displacement be a e at the nodes. This results, by Eqs (2.1) and(2.2), in displacements and strains within the element equal tou ˆ N a e and e ˆ B a e …2:6†respectively.The work done by the nodal forces is equal to the sum of the products of the individualforce components and corresponding displacements, i.e., in matrix languagea eT q e…2:7†Similarly, the internal work per unit volume done by the stresses and distributedbody forces isorye T r ÿ u T b…2:8†a T …B T r ÿ N T b†…2:9†Equating the external work with the total internal work obtained by integratingover the volume of the element, V e , we have …a eT q e ˆ a…V eT B T r d…vol†ÿ N T b d…vol†…2:10†e V eAs this relation is valid for any value of the virtual displacement, the multipliersmust be equal. Thus……q e ˆ B T r d…vol†ÿ N T b d…vol†V e V e …2:11†This statement is valid quite generally for any stress±strain relation. With the linearlaw of Eq. (2.5) we can write Eq. (2.11) aswhereandq e ˆ K e a e ‡ f e…K e ˆ B T DB d…vol†V e………f e ˆÿ N T b d…vol†ÿV e B T De 0 d…vol†‡V e B T r 0 d…vol†V e…2:12†…2:13a†…2:13b†y Note that by the rules of matrix algebra for the transpose of products…AB† T ˆ B T A T

Direct formulation of ®nite element characteristics 25In the last equation the three terms represent forces due to body forces, initialstrain, and initial stress respectively. The relations have the characteristics of thediscrete structural elements described in Chapter 1.If the initial stress system is self-equilibrating, as must be the case with normalresidual stresses, then the forces given by the initial stress term of Eq. (2.13b) areidentically zero after assembly. Thus frequent evaluation of this force component isomitted. However, if for instance a machine part is manufactured out of a block inwhich residual stresses are present or if an excavation is made in rock whereknown tectonic stresses exist a removal of material will cause a force imbalancewhich results from the above term.For the particular example of the plane stress triangular element these characteristicswill be obtained by appropriate substitution. It has already been noted that the Bmatrix in that example was not dependent on the coordinates; hence the integrationwill become particularly simple.The interconnection and solution of the whole assembly of elements follows thesimple structural procedures outlined in Chapter 1. In general, external concentratedforces may exist at the nodes and the matrix8 9r 1>< r>=2r ˆ>:. .>;r n…2:14†will be added to the consideration of equilibrium at the nodes.A note should be added here concerning elements near the boundary. If, at theboundary, displacements are speci®ed, no special problem arises as these can be satis-®ed by specifying some of the nodal parameters a. Consider, however, the boundaryas subject to a distributed external loading, say t per unit area. A loading term on thenodes of the element which has a boundary face A e will now have to be added. By thevirtual work consideration, this will simply result in…f e ˆÿ N T t d…area†A e …2:15†with the integration taken over the boundary area of the element. It will be notedthat t must have the same number of components as u for the above expression tobe valid.Such a boundary element is shown again for the special case of plane stressin Fig. 2.1. An integration of this type is sometimes not carried out explicitly.Often by `physical intuition' the analyst will consider the boundary loading to berepresented simply by concentrated loads acting on the boundary nodes and calculatethese by direct static procedures. In the particular case discussed the results will beidentical.Once the nodal displacements have been determined by solution of the overall`structural' type equations, the stresses at any point of the element can be foundfrom the relations in Eqs (2.3) and (2.5), givingr ˆ DBa e ÿ De 0 ‡ r 0…2:16†

26 A direct approach to problems in elasticityin which the typical terms of the relationship of Eq. (1.4) will be immediatelyrecognized, the element stress matrix beingQ e ˆ DB…2:17†To this the stressesr "0 ˆÿDe 0 and r 0 …2:18†have to be added.2.2.5 Generalized nature of displacements, strains, and stressesThe meaning of displacements, strains, and stresses in the illustrative case of planestress was obvious. In many other applications, shown later in this book, this terminologymay be applied to other, less obvious, quantities. For example, in consideringplate elements the `displacement' may be characterized by the lateral de¯ection andthe slopes of the plate at a particular point. The `strains' will then be de®ned as thecurvatures of the middle surface and the `stresses' as the corresponding internalbending moments (see Volume 2).All the expressions derived here are generally valid provided the sum product ofdisplacement and corresponding load components truly represents the externalwork done, while that of the `strain' and corresponding `stress' components resultsin the total internal work.2.3 Generalization to the whole region ±internal nodalforce concept abandonedIn the preceding section the virtual work principle was applied to a single element andthe concept of equivalent nodal force was retained. The assembly principle thusfollowed the conventional, direct equilibrium, approach.The idea of nodal forces contributed by elements replacing the continuousinteraction of stresses between elements presents a conceptual di culty. However,it has a considerable appeal to `practical' engineers and does at times allow an interpretationwhich otherwise would not be obvious to the more rigorous mathematician.There is, however, no need to consider each element individually and the reasoning ofthe previous section may be applied directly to the whole continuum.Equation (2.1) can be interpreted as applying to the whole structure, that is,u ˆ ~Na…2:19†in which a lists all the nodal points and~N i ˆ N e i …2:20†when the point concerned is within a particular element e and i is a point associatedwith that element. If point i does not occur within the element (see Fig. 2.3)~N i ˆ 0 …2:21†

Generalization to the whole region ±internal nodal force concept abandoned 27~N iyixFig. 2.3. A `global' shape function ± N iMatrix B can be similarly de®ned and we shall drop the bar superscript, consideringsimply that the shape functions, etc., are always de®ned over the whole region V.For any virtual displacement a we can now write the sum of internal and externalwork for the whole region as………ÿa T r ˆ u T b dV ‡ u T t dA ÿ e T r dV…2:22†VAVIn the above equation a, u, and e can be completely arbitrary, providing theystem from a continuous displacement assumption. If for convenience we assumethey are simply variations linked by the relations (2.19) and (2.3) we obtain, on substitutionof the constitutive relation (2.5), a system of algebraic equationswhereand…K ˆKa ‡ f ˆ rVB T DB dV…………f ˆÿ N T b dV ÿ N T t dA ÿ B T De 0 dV ‡ B T r 0 dVVAVV…2:23†…2:24a†…2:24b†The integrals are taken over the whole volume V and over the whole surface area Aon which the tractions are given.It is immediately obvious from the above thatK ij ˆ X K e ijf i ˆ X f e i…2:25†by virtue of the property of de®nite integrals requiring that the total be the sum of theparts:…… †dV ˆ X … … †dV …2:26†V eV

28 A direct approach to problems in elasticityThe same is obviously true for the surface integrals in Eq. (2.25). We thus see that the`secret' of the approximation possessing the required behaviour of a `standard discretesystem of Chapter 1' lies simply in the requirement of writing the relationshipsin integral form.The assembly rule as well as the whole derivation has been achieved withoutinvolving the concept of `interelement forces' (i.e., q e ). In the remainder of thisbook the element superscript will be dropped unless speci®cally needed. Also nodi€erentiation between element and system shape functions will be made.However, an important point arises immediately. In considering the virtual workfor the whole system [Eq. (2.22)] and equating this to the sum of the elementcontributions it is implicitly assumed that no discontinuity in displacement betweenadjacent elements develops. If such a discontinuity developed, a contribution equalto the work done by the stresses in the separations would have to be added.∆‘Smoothing’ zoneududxd 2 udx 2 –∞Fig. 2.4 Differentiation of a function with slope discontinuity (C 0 continuous).

Displacement approach as a minimization of total potential energy 29Put in other words, we require that the terms integrated in Eq. (2.26) be ®nite.These terms arise from the shape functions N i used in de®ning the displacement u[by Eq. (2.19)] and its derivatives associated with the de®nition of strain [viz. Eq.(2.3)]. If, for instance, the `strains' are de®ned by ®rst derivatives of the functionsN, the displacements must be continuous. In Fig. 2.4 we see how ®rst derivatives ofcontinuous functions may involve a `jump' but are still ®nite, while second derivativesmay become in®nite. Such functions we call C 0 continuous.In some problems the `strain' in a generalized sense may be de®ned by secondderivatives. In such cases we shall obviously require that both the function N andits slope (®rst derivative) be continuous. Such functions are more di cult to derivebut we shall make use of them in plate and shell problems (see Volume 2). Thecontinuity involved now is called C 1 continuity.2.4 Displacement approach as a minimization of totalpotential energyThe principle of virtual displacements used in the previous sections ensured satisfactionof equilibrium conditions within the limits prescribed by the assumeddisplacement pattern. Only if the virtual work equality for all, arbitrary, variationsof displacement was ensured would the equilibrium be complete.As the number of parameters of a which prescribes the displacement increases withoutlimit then ever closer approximation of all equilibrium conditions can be ensured.The virtual work principle as written in Eq. (2.22) can be restated in a di€erent formif the virtual quantities a, u,ande are considered as variations of the real quantities.Thus, for instance, we can write …… a T r ‡ u T b dV ‡ u T t dA ˆÿW…2:27†Vfor the ®rst three terms of Eq. (2.22), where W is the potential energy of the externalloads. The above is certainly true if r, b, andt are conservative (or independent ofdisplacement).The last term of Eq. (2.22) can, for elastic materials, be written as…U ˆ e T r dV…2:28†Vwhere U is the `strain energy' of the system. For the elastic, linear material describedby Eq. (2.5) the reader can verify thatU ˆ 1 ………e T De dV ÿ e T De20 dV ‡ e T r 0 dV…2:29†VVVwill, after di€erentiation, yield the correct expression providing D is a symmetricmatrix. (This is indeed a necessary condition for a single-valued U to exist.)Thus instead of Eq. (2.22) we can write simply…U ‡ W† ˆ…† ˆ0…2:30†in which the quantity is called the total potential energy.A

30 A direct approach to problems in elasticityThe above statement means that for equilibrium to be ensured the total potentialenergy must be stationary for variations of admissible displacements. The ®nite elementequations derived in the previous section [Eqs (2.23)±(2.25)] are simply thestatements of this variation with respect to displacements constrained to a ®nitenumber of parameters a and could be written as8 9@@@a ˆ>=@ ˆ 0@a 2>: . >;.…2:31†It can be shown that in stable elastic situations the total potential energy is not onlystationary but is a minimum. 7 Thus the ®nite element process seeks such a minimumwithin the constraint of an assumed displacement pattern.The greater the degrees of freedom, the more closely will the solution approximatethe true one, ensuring complete equilibrium, providing the true displacement can, inthe limit, be represented. The necessary convergence conditions for the ®nite elementprocess could thus be derived. Discussion of these will, however, be deferred tosubsequent sections.It is of interest to note that if true equilibrium requires an absolute minimum of thetotal potential energy, , a ®nite element solution by the displacement approach willalways provide an approximate greater than the correct one. Thus a bound on thevalue of the total potential energy is always achieved.If the functional could be speci®ed, a priori, then the ®nite element equationscould be derived directly by the di€erentiation speci®ed by Eq. (2.31).The well-known Rayleigh 8 ±Ritz 9 process of approximation frequently used inelastic analysis uses precisely this approach. The total potential energy expressionis formulated and the displacement pattern is assumed to vary with a ®nite set ofundetermined parameters. A set of simultaneous equations minimizing the totalpotential energy with respect to these parameters is set up. Thus the ®nite elementprocess as described so far can be considered to be the Rayleigh±Ritz procedure.The di€erence is only in the manner in which the assumed displacements areprescribed. In the Ritz process traditionally used these are usually given byexpressions valid throughout the whole region, thus leading to simultaneousequations in which no banding occurs and the coe cient matrix is full. In the ®niteelement process this speci®cation is usually piecewise, each nodal parameterin¯uencing only adjacent elements, and thus a sparse and usually banded matrix ofcoe cients is found.By its nature the conventional Ritz process is limited to relatively simple geometricalshapes of the total region while this limitation only occurs in ®nite elementanalysis in the element itself. Thus complex, realistic, con®gurations can be assembledfrom relatively simple element shapes.A further di€erence in kind is in the usual association of the undetermined parameterwith a particular nodal displacement. This allows a simple physical interpretationinvaluable to an engineer. Doubtless much of the popularity of the ®nite elementprocess is due to this fact.

Convergence criteria 312.5 Convergence criteriaThe assumed shape functions limit the in®nite degrees of freedom of the system, andthe true minimum of the energy may never be reached, irrespective of the ®neness ofsubdivision. To ensure convergence to the correct result certain simple requirementsmust be satis®ed. Obviously, for instance, the displacement function should be able torepresent the true displacement distribution as closely as desired. It will be found thatthis is not so if the chosen functions are such that straining is possible when theelement is subjected to rigid body displacements. Thus, the ®rst criterion that thedisplacement function must obey is as follows:Criterion 1. The displacement function chosen should be such that it does notpermit straining of an element to occur when the nodal displacements are causedby a rigid body motion.This self-evident condition can be violated easily if certain types of function are used;care must therefore be taken in the choice of displacement functions.A second criterion stems from similar requirements. Clearly, as elements getsmaller nearly constant strain conditions will prevail in them. If, in fact, constantstrain conditions exist, it is most desirable for good accuracy that a ®nite size elementis able to reproduce these exactly. It is possible to formulate functions that satisfy the®rst criterion but at the same time require a strain variation throughout the elementwhen the nodal displacements are compatible with a constant strain solution. Suchfunctions will, in general, not show good convergence to an accurate solution andcannot, even in the limit, represent the true strain distribution. The second criterioncan therefore be formulated as follows:Criterion 2. The displacement function has to be of such a form that if nodaldisplacements are compatible with a constant strain condition such constantstrain will in fact be obtained. (In this context again a generalized `strain' de®nitionis implied.)It will be observed that Criterion 2 in fact incorporates the requirement of Criterion 1,as rigid body displacements are a particular case of constant strain ± with a value ofzero. This criterion was ®rst stated by Bazeley et al. 10 in 1965. Strictly, both criterianeed only be satis®ed in the limit as the size of the element tends to zero. However,the imposition of these criteria on elements of ®nite size leads to improved accuracy,although in certain situations (such as illustrated by the axisymmetric analysis ofChapter 5) the imposition of the second one is not possible or essential.Lastly, as already mentioned in Sec. 2.3, it is implicitly assumed in this derivationthat no contribution to the virtual work arises at element interfaces. It thereforeappears necessary that the following criterion be included:Criterion 3. The displacement functions should be chosen such that the strains atthe interface between elements are ®nite (even though they may be discontinuous).This criterion implies a certain continuity of displacements between elements. Inthe case of strains being de®ned by ®rst derivatives, as in the plane stress examplequoted here, the displacements only have to be continuous. If, however, as in the

32 A direct approach to problems in elasticityplate and shell problems, the `strains' are de®ned by second derivatives of de¯ections,®rst derivatives of these have also to be continuous. 2The above criteria are included mathematically in a statement of `functional completeness'and the reader is referred elsewhere for full mathematical discussion. 11ÿ16The `heuristic' proof of the convergence requirements given here is su cient forpractical purposes in all but the most pathological cases and we shall generalize allof the above criteria in Section 3.6 and more fully in Chapter 10, where we shallpresent a universal test which justi®es convergence even if some of the above criteriaare violated.2.6 Discretization error and convergence rateIn the foregoing sections we have assumed that the approximation to the displacementas represented by Eq. (2.1) will yield the exact solution in the limit as the sizeh of elements decreases. The arguments for this are simple: if the expansion is capable,in the limit, of exactly reproducing any displacement form conceivable in thecontinuum, then as the solution of each approximation is unique it must approach,in the limit of h ! 0, the unique exact solution. In some cases the exact solution isindeed obtained with a ®nite number of subdivisions (or even with one elementonly) if the polynomial expansion is used in that element and if this can ®t exactlythe correct solution. Thus, for instance, if the exact solution is of the form of aquadratic polynomial and the shape functions include all the polynomials of thatorder, the approximation will yield the exact answer.The last argument helps in determining the order of convergence of the ®niteelement procedure as the exact solution can always be expanded in the vicinity ofany point (or node) i as a polynomial:u ˆ u i ‡ @u@x…x ÿ x i †‡i @u…y ÿ y@yi †‡i…2:32†If within an element of `size' h a polynomial expansion of degree p is employed, thiscan ®t locally the Taylor expansion up to that degree and, as x ÿ x i and y ÿ y i are ofthe order of magnitude h, the error in u will be of the order O…h p ‡ 1 †. Thus, forinstance, in the case of the plane elasticity problem discussed, we used a linear expansionand p ˆ 1. We should therefore expect a convergence rate of order O…h 2 †, i.e., theerror in displacement being reduced to 1 4for a halving of the mesh spacing.By a similar argument the strains (or stresses) which are given by the mth derivativesof displacement should converge with an error of O…h p ‡ 1 ÿ m †, i.e., as O…h† inthe example quoted, where m ˆ 1. The strain energy, being given by the square ofthe stresses, will show an error of O…h 2…p ‡ 1 ÿ m† † or O…h 2 † in the plane stress example.The arguments given here are perhaps a tri¯e `heuristic' from a mathematical viewpoint± they are, however, true 15;16 and correctly give the orders of convergence,which can be expected to be achieved asymptotically as the element size tends tozero and if the exact solution does not contain singularities. Such singularities mayresult in in®nite values of the coe cients in terms omitted in the Taylor expansionof Eq. (2.32) and invalidate the arguments. However, in many well-behaved problemsthe mere determination of the order of convergence often su ces to extrapolate the

Displacement functions with discontinuity between elements 33solution to the correct result. Thus, for instance, if the displacement converges atO…h 2 † and we have two approximate solutions u 1 and u 2 obtained with meshes ofsize h and h=2, we can write, with u being the exact solution,u 1 ÿ uu 2 ÿ u ˆ O…h2 †O…h=2† 2 ˆ 4…2:33†From the above an (almost) exact solution u can be predicted. This type of extrapolationwas ®rst introduced by Richardson 17 and is of use if convergence is monotonicand nearly asymptotic.We shall return to the important question of estimating errors due to the discretizationprocess in Chapter 14 and will show that much more precise methodsthan those arising from convergence rate considerations are possible today. Indeedautomatic mesh re®nement processes are being introduced so that the speci®edaccuracy can be achieved (viz. Chapter 15).Discretization error is not the only error possible in a ®nite element computation.In addition to obvious mistakes which can occur when using computers, errors due toround-o€ are always possible. With the computer operating on numbers rounded o€to a ®nite number of digits, a reduction of accuracy occurs every time di€erencesbetween `like' numbers are being formed. In the process of equation solving manysubtractions are necessary and accuracy decreases. Problems of matrix conditioning,etc., enter here and the user of the ®nite element method must at all times be aware ofaccuracy limitations which simply do not allow the exact solution ever to be obtained.Fortunately in many computations, by using modern machines which carry a largenumber of signi®cant digits, these errors are often small.The question of errors arising from the algebraic processes will be stressed inChapter 20 dealing with computation procedures.2.7 Displacement functions with discontinuity betweenelements ±non-conforming elements and the patchtestIn some cases considerable di culty is experienced in ®nding displacement functionsfor an element which will automatically be continuous along the whole interfacebetween adjacent elements.As already pointed out, the discontinuity of displacement will cause in®nite strainsat the interfaces, a factor ignored in this formulation because the energy contributionis limited to the elements themselves.However, if, in the limit, as the size of the subdivision decreases continuity isrestored, then the formulation already obtained will still tend to the correct answer.This condition is always reached if(a) a constant strain condition automatically ensures displacement continuity and(b) the constant strain criterion of the previous section is satis®ed.To test that such continuity is achieved for any mesh con®guration when usingsuch non-conforming elements it is necessary to impose, on an arbitrary patch ofelements, nodal displacements corresponding to any state of constant strain. If

34 A direct approach to problems in elasticitynodal equilibrium is simultaneously achieved without the imposition of external, nodal,forces and if a state of constant stress is obtained, then clearly no external work has beenlost through interelement discontinuity.Elements which pass such a patch test will converge, and indeed at times nonconformingelements will show a superior performance to conforming elements.The patch test was ®rst introduced by Irons 10 and has since been demonstrated togive a su cient condition for convergence. 16;18ÿ22 The concept of the patch test can begeneralized to give information on the rate of convergence which can be expectedfrom a given element.We shall return to this problem in detail in Chapter 10 where the test will be fullydiscussed.2.8 Bound on strain energy in a displacementformulationWhile the approximation obtained by the ®nite element displacement approachalways overestimates the true value of , the total potential energy (the absolute minimumcorresponding to the exact solution), this is not directly useful in practice. It is,however, possible to obtain a more useful limit in special cases.Consider in particular the problem in which no `initial' strains or initial stressesexist. Now by the principle of energy conservation the strain energy will be equalto the work done by the external loads which increase uniformly from zero. 23 Thiswork done is equal to ÿ 1 2W where W is the potential energy of the loads.ThusU ‡ 1 2 W ˆ 0…2:34†or ˆ U ‡ W ˆÿU…2:35†whether an exact or approximate displacement ®eld is assumed.Thus in the above case the approximate solution always underestimates the value ofU and a displacement solution is frequently referred to as the lower bound solution.If only one external concentrated load R is present the strain energy bound immediatelyinforms us that the de¯ection under this load has been underestimated (asU ˆÿ 1 2 W ˆ 12 rT a). In more complex loading cases the usefulness of this bound islimited as neither local de¯ections nor stresses, i.e., the quantities of real engineeringinterest, can be bounded.It is important to remember that this bound on strain energy is only valid in theabsence of any initial stresses or strains.The expression for U in this case can be obtained from Eq. (2.29) as…U ˆ 12e T De d…vol†…2:36†which becomes by Eq. (2.2) simply …U ˆ 12 aTVVB T DB d…vol† a ˆ 12 aT Ka …2:37†a `quadratic' matrix form in which K is the `sti€ness' matrix previously discussed.

The above energy expression is always positive from physical considerations. It followstherefore that the matrix K occurring in all the ®nite element assemblies is notonly symmetric but is `positive de®nite' (a property de®ned in fact by the requirementsthat the quadratic form should always be greater than or equal to zero).This feature is of importance when the numerical solution of the simultaneousequations involved is considered, as simpli®cations arise in the case of `symmetricpositive de®nite' equations.An example 352.9 Direct minimizationThe fact that the ®nite element approximation reduces to the problem of minimizingthe total potential energy de®ned in terms of a ®nite number of nodal parametersled us to the formulation of the simultaneous set of equations given symbolically byEq. (2.31). This is the most usual and convenient approach, especially in linear solutions,but other search procedures, now well developed in the ®eld of optimization,could be used to estimate the lowest value of . In this text we shall continue withthe simultaneous equation process but the interested reader could well bear the alternativepossibilities in mind. 24;252.10 An exampleThe concepts discussed and the general formulation cited are a little abstract andreaders may at this stage seek to test their grasp of the nature of the approximationsderived. While detailed computations of a two-dimensional element system are performedusing the computer, we can perform a simple hand calculation on a onedimensional®nite element of a beam. Indeed, this example will allow us to introducethe concept of generalized stresses and strains in a simple manner.Consider the beam shown in Fig. 2.5. The generalized `strain' here is the curvature.Thus we havee ˆÿ d2 wdx 2where w is the de¯ection, which is the basic unknown. The generalized stress (in theabsence of shear deformation) will be the bending moment M, which is related to the`strain' asr M ˆÿEI d2 wdx 2Thus immediately we have, using the general notation of previous sections,D EIIf the displacement w is discretized we can writew Nafor the whole system or, for an individual element, ij.

36 A direct approach to problems in elasticityPk i jxwL L Lijd i =w iw xiw i≡ θ id i = w jw xj≡w jθ jShape functionsFor w iFor θ iFig. 2.5 A beam element and its shape functions.In this example the strains are expressed as the second derivatives of displacementand it is necessary to ensure that both w and its slopew x dwdx ˆ be continuous between elements. This is easily accomplished if the nodal parametersare taken as the values of w and the slope, . Thus, wia i ˆ iThe shape functions will now be derived. If we accept that in an element two nodes(i.e., four variables) de®ne the de¯ected shape we can assume this to be given by a cubicw ˆ 1 ‡ 2 s ‡ 3 s 2 ‡ 4 s 3 where s ˆ xL :This will de®ne the shape functions corresponding to w i and i by taking for each acubic giving unity for the appropriate points …x ˆ 0, L or s ˆ 0; 1† and zero forother quantities, as shown in Fig. 2.5.The expressions for the shape functions can be written for the element shown asImmediately we can writeN i ˆ‰1 ÿ 3s 2 ‡ 2s 3 ; L…s ÿ 2s 2 ‡ s 3 †ŠN j ˆ‰3s 2 ÿ 2s 3 ; L…ÿs 2 ‡ s 3 †ŠB i ˆÿ d2 N idx 2 ˆ 1 ‰6 ÿ 12s; L…4 ÿ 6s†Š2LB i ˆÿ d2 N jdx 2 ˆ 1 ‰ÿ6 ‡ 12s; L…2 ÿ 6s†Š2L

and the sti€ness matrices for the element can be written as2312 6L ÿ12 6L… LK e ij ˆ B T i EI B j dx ˆ EI6L 4L 2 ÿ6L 2L 20L 3 674 ÿ12 ÿ6L 12 ÿ6L 56L 2L 2 ÿ6L 4L 2We shall leave the detailed calculation of this and the `forces' corresponding to auniformly distributed load p (assumed constant on ij and zero elsewhere) to thereader. It will be observed that the ®nal assembled equations for a node i are of theform linking three nodal displacements i; j; k. Explicitly these equations are forelements of equal length L: ÿ12=L 3 ; ÿ6=L 2EI6=L 2 ; 2=L ÿ12=L 3 ; ‡6=L 2‡ EIÿ6=L 2 ; 2=L wk 24=L 3 ; 0‡ EI k 0; 8=Lwj j‡ pL=2ÿpL 2 =12wi iIt is of interest to compare these with the exact form represented by the so-called`slope±de¯ection' equations which can be found in standard texts on structuralanalysis.Here it will be found that the ®nite element approximation has achieved the exactsolution at nodes for a uniform load. We show in Chapter 3 and in Appendix Hreasons for this unexpected result.ˆ 0References 372.11 Concluding remarksThe `displacement' approach to the analysis of elastic solids is still undoubtedly themost popular and easily understood procedure. In many of the following chapterswe shall use the general formulae developed here in the context of linear elasticanalysis (Chapters 4, 5, and 6). These are also applicable in the context of nonlinearanalysis, the main variants being the de®nitions of the stresses, generalizedstrains, and other associated quantities. It is thus convenient to summarize theessential formulae, and this is done in Appendix C.In Chapter 3 we shall show that the procedures developed here are but a particularcase of ®nite element discretization applied to the governing equilibrium equationswritten in terms of displacements. 26 Clearly, alternative starting points are possible.Some of these will be mentioned in Chapters 11 and 12.References1. R.W. Clough. The ®nite element in plane stress analysis. Proc. 2nd ASCE Conf. onElectronic Computation. Pittsburgh, Pa., Sept. 1960.2. R.W. Clough. The ®nite element method in structural mechanics. Chapter 7 of StressAnalysis (eds O.C. Zienkiewicz and G.S. Holister), Wiley, 1965.

38 A direct approach to problems in elasticity3. J. Szmelter. The energy method of networks of arbitrary shape in problems of the theory ofelasticity. Proc. IUTAM Symposium on Non-Homogeneity in Elasticity and Plasticity (ed.W. Olszak), Pergamon Press, 1959.4. R. Courant. Variational methods for the solution of problems of equilibrium and vibration.Bull. Am. Math. Soc., 49, 1±23, 1943.5. W. Prager and J.L. Synge. Approximation in elasticity based on the concept of functionspace. Quart. Appl. Math., 5, 241±69, 1947.6. S. Timoshenko and J.N. Goodier. Theory of Elasticity. 2nd ed., McGraw-Hill, 1951.7. K. Washizu. Variational Methods in Elasticity and Plasticity. 2nd ed., Pergamon Press,1975.8. J.W. Strutt (Lord Rayleigh). On the theory of resonance. Trans. Roy. Soc. (London), A161,77±118, 1870.9. W. Ritz. UÈ ber eine neue Methode zur LoÈ sung gewissen Variations ± Probleme dermathematischen Physik. J. Reine angew. Math., 135, 1±61, 1909.10. G.P. Bazeley, Y.K. Cheung, B.M. Irons, and O.C. Zienkiewicz. Triangular elements inbending ± conforming and non-conforming solutions. Proc. Conf. Matrix Methods inStructural Mechanics. Air Force Inst. Tech., Wright-Patterson AF Base, Ohio, 1965.11. S.C. Mikhlin. The Problem of the Minimum of a Quadratic Functional. Holden-Day, 1966.12. M.W. Johnson and R.W. McLay. Convergence of the ®nite element method in the theoryof elasticity. J. Appl. Mech.. Trans. Am. Soc. Mech. Eng., 274±8, 1968.13. P.G. Ciarlet. The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam,1978.14. T.H.H. Pian and Ping Tong. The convergence of ®nite element method in solving linearelastic problems. Int. J. Solids Struct., 3, 865±80, 1967.15. E.R. de Arrantes Oliveira. Theoretical foundations of the ®nite element method. Int.J. Solids Struct., 4, 929±52, 1968.16. G. Strang and G.J. Fix. An Analysis of the Finite Element Method. p. 106, Prentice-Hall,1973.17. L.F. Richardson. The approximate arithmetical solution by ®nite di€erences of physicalproblems. Trans. Roy. Soc. (London), A210, 307±57, 1910.18. B. N. Irons and A. Razzaque. Experience with the patch test, in Mathematical Foundationsof the Finite Element Method (ed. A.R. Aziz), pp. 557±87, Academic Press, 1972.19. B. Fraeijs de Veubeke. Variational principles and the patch test. Int. J. Num. Meth. Eng., 8,783±801, 1974.20. R.L. Taylor, O.C. Zienkiewicz, J.C. Simo, and A.H.C. Chan. The patch test ± a conditionfor assessing FEM convergence. Int. J. Numer. Methods Engrg., 22, 39±62, 1986.21. O.C. Zienkiewicz, S. Qu, R.L. Taylor, and S. Nakazawa. The patch test for mixed formulations.Int. J. Numer. Methods Engrg., 23, 1873±83, 1986.22. O.C. Zienkiewicz and R.L. Taylor. The ®nite element patch test revisited. A computer testfor convergence, validation and error estimates. Comp. Meth. Appl. Mech. and Engrg., 149,223±54, 1997.23. B. Fraeijs de Veubeke. Displacement and equilibrium models in the ®nite element method.Chapter 9 of Stress Analysis (eds O.C. Zienkiewicz and G.S. Holister), Wiley, 1965.24. R.L. Fox and E.L. Stanton. Developments in structural analysis by direct energy minimization.JAIAA. 6, 1036±44, 1968.25. F.K. Bogner, R.H. Mallett, M.D. Minich, and L.A. Schmit. Development and evaluationof energy search methods in non-linear structural analysis. Proc. Conf. Matrix Methods inStructural Mechanics. Air Force Inst. Tech., Wright-Patterson AF Base, Ohio, 1965.26. O.C. Zienkiewicz and K. Morgan. Finite Elements and Approximation. Wiley, 1983.

3Generalization of the ®nite elementconcepts. Galerkin-weightedresidual and variational approaches3.1 IntroductionWe have so far dealt with one possible approach to the approximate solution of theparticular problem of linear elasticity. Many other continuum problems arise inengineering and physics and usually these problems are posed by appropriate di€erentialequations and boundary conditions to be imposed on the unknown function orfunctions. It is the object of this chapter to show that all such problems can be dealtwith by the ®nite element method.Posing the problem to be solved in its most general terms we ®nd that we seek anunknown function u such that it satis®es a certain di€erential equation set8 9A 1 …u† >< >=A…u† ˆ A 2 …u†>: . >; ˆ 0 …3:1†in a `domain' (volume, area, etc.) (Fig. 3.1), together with certain boundaryconditions8 9B 1 …u† >< >=B…u† ˆ B 2 …u†>:. >; ˆ 0 …3:2†.on the boundaries ÿ of the domain (Fig. 3.1).The function sought may be a scalar quantity or may represent a vector of severalvariables. Similarly, the di€erential equation may be a single one or a set of simultaneousequations and does not need to be linear. It is for this reason that we haveresorted to matrix notation in the above.The ®nite element process, being one of approximation, will seek the solution in theapproximate formu ^u ˆ XnN i a i ˆ Na…3:3†i ˆ 1

40 Generalization of the ®nite element conceptsΓ eyB (u) = 0ΩA (u) = 0Subdomain Ω e(element)ΓxFig. 3.1 Problem domain and boundary ÿ.where N i are shape functions prescribed in terms of independent variables (such as thecoordinates x, y, etc.) and all or most of the parameters a i are unknown.We have seen that precisely the same form of approximation was used in thedisplacement approach to elasticity problems in the previous chapter. We alsonoted there that (a) the shape functions were usually de®ned locally for elements orsubdomains and (b) the properties of discrete systems were recovered if theapproximating eqations were cast in an integral form [viz. Eqs (2.22)±(2.26)].With this object in mind we shall seek to cast the equation from which the unknownparameters a i are to be obtained in the integral form……G j …^u† d ‡ g j …^u† dÿ ˆ 0 j ˆ 1ton …3:4†ÿin which G j and g j prescribe known functions or operators.These integral forms will permit the approximation to be obtained element byelement and an assembly to be achieved by the use of the procedures developed forstandard discrete systems in Chapter 1, since, providing the functions G j and g j areintegrable, we have… … … … G j d ‡ g j dÿ ˆ XmG j d ‡ g j dÿ ˆ 0 …3:5†ÿ e ÿ ee ˆ 1where e is the domain of each element and ÿ e its part of the boundary.Two distinct procedures are available for obtaining the approximation in suchintegral forms. The ®rst is the method of weighted residuals (known alternatively asthe Galerkin procedure); the second is the determination of variational functionalsfor which stationarity is sought. We shall deal with both approaches in turn.If the di€erential equations are linear, i.e., if we can write (3.1) and (3.2) asA…u† Lu ‡ p ˆ 0 in …3:6†B…u† Mu ‡ t ˆ 0 on ÿ …3:7†

Introduction 41then the approximating equation system (3.4) will yield a set of linear equations of theformKa ‡ f ˆ 0…3:8†withK ij ˆ XmK e ije ˆ 1f i ˆ Xmf e ie ˆ 1…3:9†The reader not used to abstraction may well now be confused about the meaning ofthe various terms. We shall introduce here some typical sets of di€erential equationsfor which we will seek solutions (and which will make the problems a little morede®nite).Example 1. Steady-state heat conduction equations in a two-dimensional domain:A…† ˆ @ k @ ‡ @ k @ ‡ Q ˆ 0@x @x @y @yB…† ˆ ÿ ˆ 0 on ÿ …3:10†orB…† ˆk @@n ‡ q ˆ 0on ÿ qwhere u indicates temperature, k is the conductivity, Q is a heat source, and qare the prescribed values of temperature and heat ¯ow on the boundaries and n is thedirection normal to ÿ.In the above problem k and Q can be functions of position and, if the problem isnon-linear, of or its derivatives.Example 2. Steady-state heat conduction±convection equation in two dimensions:A…† ˆ @ k @ ‡ @ k @ @‡ u@x @x @y @y x@x ‡ u @y@y ‡ Q ˆ 0 …3:11†with boundary conditions as in the ®rst example. Here u x and u y are known functionsof position and represent velocities of an incompressible ¯uid in which heat transferoccurs.Example 3. A system of three ®rst order equations equivalent to Example 1:8@q x@x ‡ @q 9y@y ‡ Q>=ˆ 0@x>: q y ‡ k @ >;@y…3:12†

42 Generalization of the ®nite element conceptsin andB…u† ˆ ÿ ˆ 0ˆ q n ÿ q ˆ 0on ÿ on ÿ qwhere q n is the ¯ux normal to the boundary.Here the unknown function vector u corresponds to the set8 9>< >=u ˆ q x>: >;This last example is typical of a so-called mixed formulation. In such problems thenumber of dependent unknowns can always be reduced in the governing equations bysuitable algebraic operations, still leaving a solvable problem [e.g., obtaining Eq.(3.10) from (3.12) by eliminating q x and q y ].If this cannot be done [viz. Eq. (3.10)] we have an irreducible formulation.Problems of mixed form present certain complexities in their solution which weshall discuss in Chapters 11±13.In Chapter 7 we shall return to detailed examples of the above ®eld problems, andother examples will be introduced throughout the book. The three sets of problemswill, however, be useful in their full form or reduced to one dimension (by suppressingthe y variable) to illustrate the various approaches used in this chapter.q yWeighted residual methods3.2 Integral or `weak' statements equivalent to thedifferential equationsAs the set of di€erential equations (3.1) has to be zero at each point of the domain ,it follows that……v T A…u† d ‰v 1 A 1 …u†‡v 2 A 2 …u†‡Šd 0 …3:13†where8>:v 1v 2..9>=>;…3:14†is a set of arbitrary functions equal in number to the number of equations (or componentsof u) involved.The statement is, however, more powerful. We can assert that if (3.13) is satis®ed forall v then the di€erential equations (3.1) must be satis®ed at all points of the domain. Theproof of the validity of this statement is obvious if we consider the possibility thatA…u† 6ˆ0 at any point or part of the domain. Immediately, a function v can befound which makes the integral of (3.13) non-zero, and hence the point is proved.

If the boundary conditions (3.12) are to be simultaneously satis®ed, then we requirethat……v T B…u† dÿ ‰v 1 B 1 …u†‡v 2 B 2 …u†‡Šdÿ ˆ 0 …3:15†ÿfor any set of functions v.Indeed, the integral statement that…v T A…u† d ‡Integral or `weak' statements equivalent to the differential equations 43ÿ…ÿv T B…u† dÿ ˆ 0…3:16†is satis®ed for all v and v is equivalent to the satisfaction of the di€erential equations(3.1) and their boundary conditions (3.2).In the above discussion it was implicitly assumed that integrals such as those in Eq.(3.16) are capable of being evaluated. This places certain restrictions on the possiblefamilies to which the functions v or u must belong. In general we shall seekto avoidfunctions which result in any term in the integrals becoming in®nite.Thus, in Eq. (3.16) we generally limit the choice of v and v to bounded functionswithout restricting the validity of previous statements.What restrictions need to be placed on the functions? The answer obviouslydepends on the order of di€erentiation implied in the equations A…u† [or B…u†].Consider, for instance, a function u which is continuous but has a discontinuousslope in the x-direction, as shown in Fig. 3.2 which is identical to Fig. 2.4 but is reproducedhere for clarity. We imagine this discontinuity to be replaced by a continuousvariation in a very small distance (a process known as `moli®cation') and study thebehaviour of the derivatives. It is easy to see that although the ®rst derivative is notde®ned here, it has ®nite value and can be integrated easily but the second derivativetends to in®nity. This therefore presents di culties if integrals are to be evaluatednumerically by simple means, even though the integral is ®nite. If such derivativesare multiplied by each other the integral does not exist and the function is knownas non-square integrable. Such a function is said to be C 0 continuous.In a similar way it is easy to see that if nth-order derivatives occur in any term of Aor B then the function has to be such that its n ÿ 1 derivatives are continuous (C n ÿ 1continuity).On many occasions it is possible to perform an integration by parts on Eq. (3.16)and replace it by an alternative statement of the form……C…v† T D…u† d ‡ E…v† T F…u† dÿ ˆ 0…3:17†In this the operators C to F usually contain lower order derivatives than those occurringin operators A or B. Now a lower order of continuity is required in the choice ofthe u function at a price of higher continuity for v and v.The statement (3.17) is now more `permissive' than the original problem posed byEqs (3.1), (3.2), or (3.16) and is called a weakform of these equations. It is a somewhatsurprising fact that often this weak form is more realistic physically than the originaldi€erential equation which implied an excessive `smoothness' of the true solution.Integral statements of the form of (3.16) and (3.17) will form the basis of ®niteelement approximations, and we shall discuss them later in fuller detail. Beforedoing so we shall apply the new formulation to an example.ÿ

44 Generalization of the ®nite element concepts∆‘Smoothing’ zoneududxd 2 udx 2 –∞Fig. 3.2 Differentiation of function with slope discontinuity (C 0 continuous).Example. Weak form of the heat conduction equation ± forced and natural boundaryconditions. Consider now the integral form of Eq. (3.10). We can write the statement(3.16) as… @v k @ ‡ @ k @ … ‡ Q dx dy ‡ v k @ @x @x @y @yÿ q@n ‡ q dÿ ˆ 0 …3:18†noting that v and v are scalar functions and presuming that one of the boundaryconditions, i.e., ÿ ˆ 0is automatically satis®ed by the choice of the functions on ÿ .Equation (3.18) can now be integrated by parts to obtain a weak form similar toEq. (3.17). We shall make use here of general formulae for such integration (Green'sformulae) which we derive in Appendix G and which on many occasions will be

Integral or `weak' statements equivalent to the differential equations 45useful, i.e.…v @ @x…v @ @yk @@xk @@y …dx dy ÿdx dy ÿ…@v@x@v@yk @@xk @@y ‡dx dy ‡dx dy ‡‡ÿÿv k @@xv k @@yn x dÿn y dÿWe have thus in place of Eq. (3.18)… @vÿ @x k @@x ‡ @v@y k @ ‡ @@y ÿ vQ dx dy ‡ vkÿ @x n x ‡ @ @y n y dÿ… ‡ v k @ ÿ q@n ‡ q dÿ ˆ 0…3:19†…3:20†Noting that the derivative along the normal is given as@@n @@x n x ‡ @@y n y…3:21†and, further, makingv ˆÿv on ÿ …3:22†without loss of generality (as both functions are arbitrary), we can write Eq. (3.20) as…… … …r T vkr d ÿ vQd ÿ v q dÿ ÿ vk @ÿ q ÿ @n dÿ ˆ 0 …3:23†where the operator r is simply8 9@>< >=@xr ˆ@ >: >;@yWe note that(a) the variable has disappeared from the integrals taken along the boundary ÿ qand that the boundary conditionB…† ˆk @@n ‡ q ˆ 0on that boundary is automatically satis®ed ± such a condition is known as anatural boundary condition ± and(b) if the choice of is restricted so as to satisfy the forced boundary conditions ÿ ˆ 0, we can omit the last term of Eq. (3.23) by restricting the choice of vto functions which give v ˆ 0onÿ .The form of Eq. (3.23) is the weakform of the heat conduction statement equivalentto Eq. (3.17). It admits discontinuous conductivity coe cients k and temperature which show discontinuous ®rst derivatives, a real possibility not admitted in thedi€erential form.

46 Generalization of the ®nite element concepts3.3 Approximation to integral formulations: theweighted residual Galerkin methodIf the unknown function u is approximated by the expansion (3.3), i.e.,u ^u ˆ XnN i a i ˆ Nai ˆ 1then it is clearly impossible to satisfy both the di€erential equation and the boundaryconditions in the general case. The integral statements (3.16) or (3.17) allow anapproximation to be made if, in place of any function v, we put a ®nite set of approximatefunctionsv ˆ Xnw j a j v ˆ Xnw j a j…3:24†j ˆ 1in which a j are arbitrary parameters and n is the number of unknowns entering theproblem.Inserting the above approximations into Eq. (3.16) we have ……a T j w T j A…Na† d ‡ w T j B…Na† dÿ ˆ 0ÿand since a j is arbitrary we have a set of equations which is su cient to determine theparameters a j as… …w T j A…Na† d ‡ w T j B…Na† dÿ ˆ 0 j ˆ 1ton …3:25†ÿor, from Eq. (3.17),……C…w j † T D…Na† d ‡ E…w j † T F…Na† dÿ ˆ 0 j ˆ 1ton …3:26†ÿIf we note that A…Na† represents the residual or error obtained by substitution ofthe approximation into the di€erential equation [and B…Na†, the residual of theboundary conditions], then Eq. (3.25) is a weighted integral of such residuals. Theapproximation may thus be called the method of weighted residuals.In its classical sense it was ®rst described by Crandall, 1 who points out the variousforms used since the end of the last century. More recently a very full expose of themethod has been given by Finlayson. 2 Clearly, almost any set of independent functionsw j could be used for the purpose of weighting and, according to the choice offunction, a di€erent name can be attached to each process. Thus the variouscommon choices are:1. Point „ collocation. 3 w j ˆ d j , where d j is such that for x 6ˆ x j ; y 6ˆ y j , w j ˆ 0 but w j d ˆ I (unit matrix). This procedure is equivalent to simply making theresidual zero at n points within the domain and integration is `nominal' (incidentallyalthough w j de®ned here does not satisfy all the criteria of Sec. 3.2, it is neverthelessadmissible in view of its properties).2. Subdomain collocation. 4 w j ˆ I in j and zero elsewhere. This essentially makes theintegral of the error zero over the speci®ed subdomains.j ˆ 1

Approximation to integral formulations: the weighted residual Galerkin method 473. The Galerkin method (Bubnov±Galerkin). 5;6 w j ˆ N j . Here simply the originalshape (or basis) functions are used as weighting. This method, as we shall see,frequently (but by no means always) leads to symmetric matrices and for thisand other reasons will be adopted in our ®nite element work almost exclusively.The name of `weighted residuals' is clearly much older than that of the `®nite elementmethod'. The latter uses mainly locally based (element) functions in the expansion ofEq. (3.3) but the general procedures are identical. As the process always leads to equationswhich, being of integral form, can be obtained by summation of contributionsfrom various subdomains, we choose to embrace all weighted residual approximationsunder the name of generalized ®nite element method. Frequently, simultaneous use ofboth local and `global' trial functions will be found to be useful.In the literature the names of Petrov and Galerkin 5 are often associated with theuse of weighting functions such that w j 6ˆ N j . It is important to remark that thewell-known ®nite di€erence method of approximation is a particular case of collocationwith locally de®ned basis functions and is thus a case of a Petrov±Galerkinscheme. We shall return to such unorthodox de®nitions in more detail in Chapter 16.To illustrate the procedure of weighted residual approximation and its relation tothe ®nite element process let us consider some speci®c examples.Example 1. One-dimensional equation of heat conduction (Fig. 3.3). The problem herewill be a one-dimensional representation of the heat conduction equation [Eq. (3.10)]with unit conductivity. (This problem could equally well represent many otherphysical situations, e.g., deformation of a loaded string.) Here we haveA…† ˆd2 ‡ Q ˆ 0 …0 4 x 4 L† …3:27†dx2 with Q ˆ Q…x† given by Q ˆ 1 …0 4 x < L=2† and Q ˆ 0 …L=2 4 x 4 L†. The boundaryconditions assumed will be simply ˆ 0atx ˆ 0 and x ˆ L.In the ®rst case we shall consider a one- or two-term approximation of the Fourierseries form, i.e., ^ ˆ X a i sin x iLN i ˆ sin x iL…3:28†with i ˆ 1andi ˆ 1 and 2. These satisfy the boundary conditions exactly and arecontinuous throughout the domain. We can thus use either Eq. (3.16) or Eq. (3.17)for the approximation with equal validity. We shall use the former, which allowsvarious weighting functions to be adopted. In Fig. 3.3 we present the problem andits solution using point collocation, subdomain collocation, and the Galerkin method.yAs the chosen expansion satis®es a priori the boundary conditions there is no needto introduce them into the formulation, which is given simply by… L d2 X w j Ni0 dx 2 a i ‡ Q dx ˆ 0…3:29†The full working out of this problem is left as an exercise to the reader.y In the case of point collocation using i ˆ 1 …x i ˆ L=2† a di culty arises about the value of Q (as this iseither zero or one). The value of 1 2was therefore used for the example.

48 Generalization of the ®nite element conceptsQ = 1Q = 0φ = 0 L/2L/2φ = 0N 1 = W 1 (Galerkin)1N 1 = sin πxLxW 1 Point collocation∞xW 1 Subdomain collocation1x2.5φ 10πL 22.01.5ExactSubdomain collocationGalerkin1.0Pointcollocation0.500 x/L1.0Fig. 3.3 One-dimensional heat conduction. (a) One-term solution using different weighting procedures.

Approximation to integral formulations: the weighted residual Galerkin method 49φ 10πGalerkinL 2 1 N 1 = W 1 (Galerkin)1N 2 = W 2 (Galerkin)1W 1 Point collocation W 21 W 1SubdomaincollocationW 212.5Subdomain collocationPoint collocation2.01.5N 1 = sin πxLN 2 = sin 2πxL1.0Exact0.500 x/L1.0Fig. 3.3 (cont.) (b) Two-term solutions using different weighting procedures.

50 Generalization of the ®nite element conceptsOf more interest to the standard ®nite element ®eld is the use of piecewise de®ned(locally based) functions in place of the global functions of Eq. (3.28). Here, to avoidimposing slope continuity, we shall use the equivalent of Eq. (3.17) obtained byintegrating Eq. (3.29) by parts. This yields… L0 dwjdx XidN idx a i ÿ w j Q dx ˆ 0The boundary terms disappear identically if w j ˆ 0 at the two ends.The above equations can be written asKa ‡ f ˆ 0where for each `element' of length L e ,K e ji ˆ… Lef e j ˆÿdw jdx0… Le0dN idx dxw j Q dxwith the usual rules of addition pertaining, i.e.,K ji ˆ Xf j ˆ XeK e jief e j…3:30†…3:31†…3:32†…3:33†In the computation we shall use the Galerkin procedure, i.e. w j ˆ N j , and the readerwill observe that the matrix K is then symmetric, i.e., K ij ˆ K ji .As the shape functions need only be of C 0 continuity, a piecewise linear approximationis conveniently used, as shown in Fig. 3.4. Considering a typical element ij shown,we can write (moving the origin of x to point i)N j ˆ xL egiving, for a typical element,N i ˆ Le ÿ xL eK ii ˆ K jj ˆ 1L e ˆÿK ji ˆÿK ijf e j ˆÿQ e L e =2 ˆ f e i…3:34†…3:35†where Q e is the value for element e.Assembly of a typical equation at a node i is left to the reader, who is well advised tocarry out the calculations leading to the results shown in Fig. 3.4 for a two- and fourelementsubdivision.Some points of interest immediately arise if the results of Figs 3.3 and 3.4 arecompared. With smooth global shape functions the Galerkin method gives betteroverall results than those achieved for the same number of unknown parameters awith locally based functions. This we shall ®nd to be the general case with higherorder approximations, yielding better accuracy. Further, it will be observed thatthe linear approximation has given the exact answers at the interelement nodalpoints. This is a property of the particular equation being solved and unfortunatelydoes not carry over to general problems. 7 (See also Appendix H.) Lastly, the

Approximation to integral formulations: the weighted residual Galerkin method 51N iiN j1jL ex2.5φ 10πL 22.01.5FourelementsExact1.00.5Twoelements00 x/L1.0Fig. 3.4 Galerkin ®nite element solution of problem of Fig. 3.3 using linear locally based shaped functions.reader will observe how easy it is to create equations with any degree of subdivisiononce the element properties [Eq. (3.35)] have been derived. This is not the case withglobal approximation where new integrations have to be carried out for each newparameter introduced. It is this repeatability feature that is one of the advantagesof the ®nite element method.Example 2. Steady-state heat conduction±convection in two dimensions. The Galerkinformulation. We have already introduced the problem in Sec. 3.1 and de®ned it byEq. (3.11) with appropriate boundary conditions. The equation di€ers only in theconvective terms from that of simple heat conduction for which the weak form hasalready been obtained in Eq. (3.23). We can write the weighted residual equationimmediately from this, substituting v ˆ w j a j and adding the convective terms.Thus we have……r T w j kr ^ d ÿ@w j u ^x@x ‡ u @ ^ ……y d ÿ w@yj Q d ÿ w j q dÿ ˆ 0 …3:36†ÿ q

52 Generalization of the ®nite element conceptswith ^ ˆ P N i a i being such that the prescribed values of are given on the boundaryÿ and that a j ˆ 0 on that boundary (ignoring that term in (3.36)).Specializing to the Galerkin approximation, i.e., putting w j ˆ N j , we haveimmediately a set of equations of the formwith…K ji ˆ…ˆ…ÿ…f j ˆÿKa ‡ f ˆ 0… r T @NN j krN i d ÿ N j u ix @Nj@x k @N i@x ‡ @N j@y k @N id@y@NN j u ix@x ‡ N @N iju y@y…N j Q d ÿÿ qN j q dÿ@x ‡ N @N iju y@ydd…3:37†…3:38a†…3:38b†Once again the components K ji and f j can be evaluated for a typical element or subdomainand systems of equations built up by standard methods.At this point it is important to mention that to satisfy the boundary conditionssome of the parameters a i have to be prescribed and the number of approximationequations must be equal to the number of unknown parameters. It is neverthelessoften convenient to form all equations for all parameters and prescribe the ®xedvalues at the end using precisely the same techniques as we have described inChapter 1 for the insertion of prescribed boundary conditions in standard discreteproblems.A further point concerning the coe cients of the matrix K should be noted here.The ®rst part, corresponding to the pure heat conduction equation, is symmetric…K ij ˆ K ji † but the second is not and thus a system of non-symmetric equationsneeds to be solved. There is a basic reason for such non-symmetries which will bediscussed in Sec. 3.9.To make the problem concrete consider the domain to be divided into regularsquare elements of side h (Fig. 3.5). To preserve C 0 continuity with nodes placed atcorners, shape functions given as the product of the linear expansions can be written.For instance, for node i, as shown in Fig. 3.5,N i ˆ x yh hand for node j,N j ˆ…h ÿ x†hWith these shape functions the reader is invited to evaluate typical element contributionsand to assemble the equations for point 1 of the mesh numbered asyh ;etc:

Virtual work as the `weak form' of equilibrium equations for analysis of solids or ¯uids 53N jyN iji1hhx(a)54361 27 8 9(b)Fig. 3.5 A linear square element of C 0 continuity. (a) Shape functions for a square element. (b) `Connected'equation for node 1.shown in Fig. 3.5. The result will be (if no boundary of type ÿ q is present and Q isassumed to be constant)8 13 a 1 ÿ3 ÿ u xh3k ÿ u yh 1a6k 2 ÿ3 ÿ u xh12k ÿ u yh 1a12k 3 ÿ3 ÿ u xh6k ÿ u yha3k 4 1ÿ3 ‡ u xh12k ÿ u yh 1a12k 5 ÿ3 ‡ u xh3k ÿ u yh 1a6k 6 ÿ3 ‡ u xh12k ‡ u yha12k 7 1ÿ3 ÿ u xh6k ‡ u yh 1a3k 8 ÿ3 ‡ u xh12k ‡ u yha12k 9 ˆ 4h 2 Q…3:39†This equation is similar to those that would be obtained by using ®nite di€erenceapproximations to the same equations in a fairly standard manner. 8;9 In the examplediscussed some di culties arise when the convective terms are large. In such cases theGalerkin weighting is not acceptable and other forms have to be used. This isdiscussed in detail in Chapter 2 of the third volume.3.4 Virtual work as the `weak form' of equilibriumequations for analysis of solids or ¯uidsIn Chapter 2 we introduced a ®nite element by way of an application to thesolid mechanics problem of linear elasticity. The integral statement necessary for

54 Generalization of the ®nite element conceptsformulation in terms of the ®nite element approximation was supplied via theprinciple of virtual work, which was assumed to be so basic as not to merit proof.Indeed, to many this is so, and the virtual work principle is considered as a statementof mechanics more fundamental than the traditional equilibrium conditions ofNewton's laws of motion. Others will argue with this view and will point out thatall work statements are derived from the classical laws pertaining to the equilibriumof the particle. We shall therefore show in this section that the virtual work statementis simply a `weak form' of equilibrium equations.In a general three-dimensional continuum the equilibrium equations of an elementaryvolume can be written in terms of the components of the symmetric cartesianstress tensor as 10 8>< A 1>:A 2A 39>=>; ˆ8@ x@x ‡ @ xy@y ‡ @ 9xz@z 8>= >:>:@ z@z ‡ @ xz@x ‡ @ yz@y>;b xb yb z9>=>; ˆ 0 …3:40†where b T ˆ‰b x ; b y ; b z Š stands for the body forces acting per unit volume (which maywell include acceleration e€ects).In solid mechanics the six stress components will be some general functions of thecomponents of the displacementu ˆ‰u; v; wŠ T…3:41†and in ¯uid mechanics of the velocity vector u, which has identical components. ThusEq. (3.40) can be considered as a general equation of the form Eq. (3.1), i.e., A…u† ˆ0.To obtain a weak form we shall proceed as before, introducing an arbitrary weightingfunction vector, de®ned asv u ˆ‰u;v;wŠ TWe can now write the integral statement of Eq. (3.13) as…… u T @xA…u† dV ˆ u@x ‡ @ xy@y ‡ @ xz@z ‡ b x ‡ v…A 2 †‡w…A 3 † dwhere V, the volume, is the problem domain.Integrating each term by parts and rearranging we can write this as… @@ÿ x @x …u†‡ xy@y …u†‡ @ @x …v† ‡ÿub x ÿ vb y ÿ wb z d…‡ ‰u… x n x ‡ xy n y ‡ xz n z †‡v… † ‡ w… †Š dÿ ˆ 0ÿ…3:42†…3:43†…3:44†where ÿ is the surface area of the solid (here again Green's formulae of Appendix Gare used).

In the ®rst set of bracketed terms we can recognize immediately the small strainoperators acting on u, which can be termed a virtual displacement (or virtualvelocity). We can therefore introduce a virtual strain (or strain rate) de®ned as8>:@@x …u†@@y …v†@@z …w†.9>=ˆ S uwhere the strain operators are de®ned as in Chapter 2 [Eqs (2.2)±(2.4)].Similarly, the terms in the second integral will be recognized as forces t:t ˆ‰t x ; t y ; t z Š T>;…3:45†…3:46†acting per unit area of the surface A. Arranging the six stress components in a vector rand similarly the six virtual strain (or rate of virtual strain) components in a vector e,we can write Eq. (3.44) simply as………e T r d ÿ u T b d ÿ u T t dÿ ˆ 0…3:47†ÿwhich is the three dimensional equivalent virtual work statement used in Eqs (2.10)and (2.22) of Chapter 2.We see from the above that the virtual workstatement is precisely the weakform ofthe equilibrium equations and is valid for non-linear as well as linear stress±strain (orstress±rate of strain) relations.The ®nite element approximation which we have derived in Chapter 2 is in fact aGalerkin formulation of the weighted residual process applied to the equilibrium equation.Thus, if we take u as the shape function times arbitrary parametersu ˆ N a…3:48†where the displacement ®eld is discretized, i.e.,Partial discretization 55u ˆ X N i a i…3:49†together with the constitutive relation of Eq. (2.5), we shall determine once again allthe basic expressions of Chapter 2 which are so essential to the solution of elasticityproblems.Similar expressions are vital to the formulation of equivalent ¯uid mechanicsproblems as discussed further in the third volume.3.5 Partial discretizationIn the approximation to the problem of solving the di€erential equation (3.1) byan expression of the standard form of Eq. (3.3), we have assumed that theshape functions N included in them are all independent coordinates of the problem

56 Generalization of the ®nite element conceptsand that a was simply a set of constants. The ®nal approximation equations werethus always of an algebraic form, from which a unique set of parameters could bedetermined.In some problems it is convenient to proceed di€erently. Thus, for instance, if theindependent variables are x, y and z we could allow the parameters a to be functionsof z and do the approximate expansion only in the domain of x, y, say . Thus, inplace of Eq. (3.3) we would haveu ˆ NaN ˆ N…x; y†…3:50†a ˆ a…z†Clearly the derivatives of a with respect to z will remain in the ®nal discretization andthe result will be a set of ordinary di€erential equations with z as the independentvariable. In linear problems such a set will have the appearanceKa ‡ C_a ‡‡f ˆ 0…3:51†where _a da=dz, etc.Such a partial discretization can obviously be used in di€erent ways, but is particularlyuseful when the domain is not dependent on z, i.e., when the problem isprismatic. In such a case the coe cient matrices of the ordinary di€erential equations,(3.51), are independent of z and the solution of the system can frequently be carriedout e ciently by standard analytical methods.This type of partial discretization has been applied extensively by Kantorovitch 11and is frequently known by his name. In the second volume we shall discuss suchsemi-analytical treatments in the context of prismatic solids where the ®nal solutionis obtained in terms of Fourier series. However, the most frequently encountered`prismatic' problem is one involving the time variable, where the space domain isnot subject to change. We shall address such problems in Chapter 17 of thisvolume. It is convenient by way of illustration to consider here heat conduction ina two-dimensional equation in its transient state. This is obtained from Eq. (3.10)by addition of the heat storage term c…@=@t†, where c is the speci®c heat per unitvolume. We now have a problem posed in a domain …x; y; t† in which the followingequation holds:A…† @ k @ ‡ @ k @ ‡ Q ÿ c @@x @x @y @y @t ˆ 0 …3:52†with boundary conditions identical to those of Eq. (3.10) and the temperature is zeroat time zero. Taking ^ ˆ X N i a i…3:53†with a i ˆ a i …t† and N i ˆ N i …x; y† and using the Galerkin weighting procedure wefollow precisely the steps outlined in Eqs (3.36)±(3.38) and arrive at a system ofordinary di€erential equationsKa ‡ C dadt ‡ f ˆ 0…3:54†

Here the expression for K ij is identical with that of Eq. (3.38a) (convective termsneglected), f i identical to Eq. (3.38b), and the reader can verify that the matrix C isde®ned by…C ij ˆ N i cN j dx dy…3:55†Once again the matrix C can be assembled from its element contributions. Variousanalytical and numerical procedures can be applied simply to the solution of suchtransient, ordinary, di€erential equations which, again, we shall discuss in detail inChapters 17 and 18. However, to illustrate the detail and the possible advantage ofthe process of partial discretization, we shall consider a very simple problem.Example. Consider a square prism of size L in which the transient heat conductionequation (3.52) applies and assume that the rate of heat generation varies with time asQ ˆ Q 0 e ÿt…3:56†(this approximates a problem of heat development due to hydration of concrete).We assume that at t ˆ 0, ˆ 0 throughout. Further, we shall take ˆ 0 on allboundaries throughout all times.As a ®rst approximation a shape function for a one-parameter solution is taken: ˆ N 1 a 1N 1 ˆ cos xLcosyL…3:57†with x and y measured from the centre (Fig. 3.6). Evaluating the coe cients, we have… L=2… L=2 2 2 @N1 @N1K 11 ˆk ‡ k dx dy ˆ 2 k@x @y2C 11 ˆf 1 ˆÿL=2 ÿL=2… L=2… L=2ÿL=2 ÿL=2… L=2… L=2ÿL=2ÿL=2cN 2 1 dx dy ˆ L2 c4N 1 Q 0 e ÿt dx dy ˆ 4Q 0L 2 2e ÿtThus leads to an ordinary di€erential equation with one parameter a 1 :…3:58†daC 111dt ‡ K 11a 1 ‡ f 1 ˆ 0…3:59†with a 1 ˆ 0 when t ˆ 0. The exact solution of this is easy to obtain, as is shown inFig. 3.6 for speci®c values of the parameters and k=L 2 c.On the same ®gure we show a two-parameter solution withN 2 ˆ cos 3xL3ycosLPartial discretization 57…3:60†which readers can pursue to test their grasp of the problem. The second component ofthe Fourier series is here omitted due to the required symmetry of solution.The remarkable accuracy of the one-term approximation in this example should benoted.

58 Generalization of the ®nite element concepts0.3Two-term approximationπ 2 cφ (x = 0, y = 0, t)16 Q 00.2One-termapproximationy2π 2 kL 2 c = 1.5α = 1Q = Q 0 e –αtφ = 0, t = 00.1xφ = 000 1 2 3 4tFig. 3.6 Two-dimensional transient heat development in a square prism: plot of temperature at centre.3.6 ConvergenceIn the previous sections we have discussed how approximate solutions can beobtained by use of an expansion of the unknown function in terms of trial or shapefunctions. Further, we have stated the necessary conditions that such functionshave to ful®l in order that the various integrals can be evaluated over the domain.Thus if various integrals contain only the values of N or its ®rst derivatives then Nhas to be C 0 continuous. If second derivatives are involved, C 1 continuity isneeded, etc. The problem to which we have not yet addressed ourselves consists ofthe questions of just how good the approximation is and how it can be systematicallyimproved to approach the exact answer. The ®rst question is more di cult toanswer and presumes knowledge of the exact solution (see Chapter 14). The secondis more rational and can be answered if we consider some systematic way in whichthe number of parameters a in the standard expansion of Eq. (3.3),^u ˆ Xn 1N i a iis presumed to increase.In some of the examples we have assumed, in e€ect, a trigonometric Fourier-typeseries limited to a ®nite number of terms with a single form of trial function assumedover the whole domain. Here addition of new terms would be simply an extension ofthe number of terms in the series included in the analysis, and as the Fourier series isknown to be able to represent any function within any accuracy desired as the numberof terms increases, we can talk about convergence of the approximation to the truesolution as the number of terms increases.

Convergence 59In other examples of this chapter we have used locally based functions which arefundamental in the ®nite element analysis. Here we have tacitly assumed that convergenceoccurs as the size of elements decreases and, hence, the number of a parametersspeci®ed at nodes increases. It is with such convergence that we need to be concernedand we have already discussed this in the context of the analysis of elastic solids inChapter 2 (Sec. 2.6).We have now to determine(a) that, as the number of elements increases, the unknown functions can be approximatedas closely as required, and(b) how the error decreases with the size, h, of the element subdivisions (h is heresome typical dimension of an element).The ®rst problem is that of completeness of the expansion and we shall here assumethat all trial functions are polynomials (or at least include certain terms of a polynomialexpansion).Clearly, as the approximation discussed here is to the weak, integral form typi®edby Eqs (3.13) or (3.17) it is necessary that every term occurring under the integral be inthe limit capable of being approximated as nearly as possible and, in particular, givinga single constant value over an in®nitesimal part of the domain .If a derivative of order m exists in any such term, then it is obviously necessary forthe local polynomial to be at least of the order m so that, in the limit, such a constantvalue can be obtained.We will thus state that a necessary condition for the expansion to be covergent isthe criterion of completeness: that a constant value of the mth derivative be attainablein the element domain (if mth derivatives occur in the integral form) when the size ofany element tends to zero.This criterion is automatically ensured if the polynomials used in the shapefunction N are complete to mth order. This criterion is also equivalent to the oneof constant strain postulated in Chapter 2 (Sec. 2.5). This, however, has to be satis®edonly in the limit h ! 0.If the actual order of a complete polynomial used in the ®nite element expansion isp 5 m, then the order of convergence can be ascertained by seeing how closely such apolynomial can follow the local Taylor expansion of the unknown u. Clearly the orderof error will be simply O…h p ‡ 1 † since only terms of order p can be rendered correctly.Knowledge of the order of convergence helps in ascertaining how good the approximationis if studies on several decreasing mesh sizes are conducted. Though, inChapter 15, we shall see this asymptotic convergence rate is seldom reached if singularitiesoccur in the problem. Once again we have reestablished some of the conditionsdiscussed in Chapter 2.We shall not discuss, at this stage, approximations which do not satisfy thepostulated continuity requirements except to remark that once again, in manycases, convergence and indeed improved results can be obtained (see Chapter 10).In the above we have referred to the convergence of a given element type as its sizeis reduced. This is sometimes referred to as h convergence.On the other hand, it is possible to consider a subdivision into elements of a givensize and to obtain convergence to the exact solution by increasing the polynomialorder p of each element. This is referred to as p convergence, which is obviously

60 Generalization of the ®nite element conceptsassured. In general p convergence is more rapid per degree of freedom introduced. Weshall discuss both types further in Chapter 15.Variational principles3.7 What are `variational principles'?What are variational principles and how can they be useful in the approximation tocontinuum problems? It is to these questions that the following sections are addressed.First a de®nition: a `variational principle' speci®es a scalar quantity (functional) ,which is de®ned by an integral form… ˆFu; @u@x ; ... d ‡…ÿE u; @u @x ; ... dÿ…3:61†in which u is the unknown function and F and E are speci®ed di€erential operators.The solution to the continuum problem is a function u which makes stationary withrespect to arbitrary changes u. Thus, for a solution to the continuum problem, the`variation' is ˆ 0…3:62†for any u, which de®nes the condition of stationarity. 12If a `variational principle' can be found, then means are immediately established forobtaining approximate solutions in the standard, integral form suitable for ®niteelement analysis.Assuming a trial function expansion in the usual form [Eq. (3.3)]u ^u ˆ Xn 1N i a iwe can insert this into Eq. (3.61) and write ˆ @ a@a 1 ‡ @ a1 @a 2 ‡‡ @ a2 @a n ˆ 0nThis being true for any variations a yields a set of equations8 9@@@a ˆ>=.ˆ 0>:@ >;@a n…3:63†…3:64†from which parameters a i are found. The equations are of an integral form necessaryfor the ®nite element approximation as the original speci®cation of was given interms of domain and boundary integrals.The process of ®nding stationarity with respect to trial function parameters a is anold one and is associated with the names of Rayleigh 13 and Ritz. 14 It has become

What are `variational principles'? 61extremely important in ®nite element analysis which, to many investigators, is typi®edas a `variational process'.If the functional is `quadratic', i.e., if the function u and its derivatives occur inpowers not exceeding 2, then Eq. (3.64) reduces to a standard linear form similar toEq. (3.8), i.e.,@@a Ka ‡ f ˆ 0…3:65†It is easy to show that the matrix K will now always be symmetric. To do this let usconsider a linearization of the vector @=@a. This we can write as2 3 @ @ @ @@ 6a ˆ @a 1 @a 1 ;a1 @a 2 @a 2 ; ...1745 K@aT a …3:66†.in which K T is generally known as the tangent matrix, of signi®cance in non-linearanalysis, and a j are small incremental changes to a. Now it is easy to see thatK Tij ˆ@2 @a i @a jˆ K T Tji…3:67†Hence K T is symmetric.For a quadratic functional we have, from Eq. (3.65), @ ˆ K a or K ˆ K T …3:68†@aand hence symmetry must exist.The fact that symmetric matrices will arise whenever a variational principle exists isone of the most important merits of variational approaches for discretization. However,symmetric forms will frequently arise directly from the Galerkin process. In suchcases we simply conclude that the variational principle exists but we shall not needto use it directly.How then do `variational principles' arise and is it always possible to constructthese for continuous problems?To answer the ®rst part of the question we note that frequently the physical aspectsof the problem can be stated directly in a variational principle form. Theorems such asminimization of total potential energy to achieve equilibrium in mechanical systems,least energy dissipation principles in viscous ¯ow, etc., may be known to the readerand are considered by many as the basis of the formulation. We have already referredto the ®rst of these in Sec. 2.4 of Chapter 2.Variational principles of this kind are `natural' ones but unfortunately they do notexist for all continuum problems for which well-de®ned di€erential equations may beformulated.However, there is another category of variational principles which we may call`contrived'. Such contrived principles can always be constructed for any di€erentiallyspeci®ed problems either by extending the number of unknown functions u byadditional variables known as Lagrange multipliers, or by procedures imposing ahigher degree of continuity requirements such as least square problems. In subsequent

62 Generalization of the ®nite element conceptssections we shall discuss, respectively, such `natural' and `contrived' variationalprinciples.Before proceeding further it is worth noting that, in addition to symmetry occurringin equations derived by variational means, sometimes further motivation arises. When`natural' variational principles exist the quantity may be of speci®c interest itself. Ifthis arises a variational approach possesses the merit of easy evaluation of thisfunctional.The reader will observe that if the functional is `quadratic' and yields Eq. (3.65),then we can write the approximate `functional' simply asBy simple di€erentiation ˆ 12 aT Ka ‡ a T f …3:69†As K is symmetric,Hencewhich is true for all a and hence ˆ 12 …aT †Ka ‡ 1 2 aT K a ‡ a T fa T Ka a T K a ˆ a T …Ka ‡ f† ˆ0Ka ‡ f ˆ 03.8 `Natural' variational principles and their relation togoverning differential equations3.8.1 Euler equationsIf we consider the de®nitions of Eqs (3.61) and (3.62) we observe that for stationaritywe can write, after performing some di€erentiations,…… ˆ u T A…u† d ‡ u T B…u† dÿ ˆ 0…3:70†As the above has to be true for any variations u, we must haveA…u† ˆ0 in and (3.71)B…u† ˆ0 on ÿIf A corresponds precisely to the di€erential equations governing the problem and Bto its boundary conditions, then the variational principle is a natural one. Equations(3.71) are known as the Euler di€erential equations corresponding to the variationalprinciple requiring the stationarity of . It is easy to show that for any variationalprinciple a corresponding set of Euler equations can be established. The reverse isunfortunately not true, i.e., only certain forms of di€erential equations are Eulerÿ

`Natural' variational principles and their relation to governing differential equations 63equations of a variational functional. In the next section we shall consider the conditionsnecessary for the existence of variational principles and give a prescriptionfor the establishment of from a set of suitable linear di€erential equations. Inthis section we shall continue to assume that the form of the variational principle isknown.To illustrate the process let us now consider a speci®c example. Suppose we specifythe problem by requiring the stationarity of a functional… ˆ 12 k @@x 2‡ 1 2 k @@y 2ÿ Qd ÿ…ÿ qq dÿ…3:72†in which k and Q depend only on position and is de®ned such that ˆ 0onÿ ,where ÿ and ÿ q bound the domain .We now perform the variation. 12 This can be written following the rules ofdi€erentiation as… ˆ k @ @ @x ‡ k @ …@@x @y ÿ Q d ÿ …q † dÿ …3:73†@yÿ qAs @@xˆ @@x …†…3:74†we can integrate by parts (as in Sec. 3.3) and, noting that ˆ 0onÿ , obtain… @ ˆÿ k @ ‡ @ k @ ‡ Q d @x @x @y @y… ‡ k @ ÿ q@n ÿ q dÿ ˆ 0…3:75a†This is of the form of Eq. (3.70) and we immediately observe that the Eulerequations areA…† ˆ @ k @ ‡ @ k @ ‡ Q in @x @y @y @yB…† ˆk @…3:75b†@n ÿ q ˆ 0on ÿ qIf is prescribed so that ˆ on ÿ and ˆ 0 on that boundary, then theproblem is precisely the one we have already discussed in Sec. 3.2 and the functional(3.72) speci®es the two-dimensional heat conduction problem in an alternative way.In this case we have `guessed' the functional but the reader will observe that thevariation operation could have been carried out for any functional speci®ed andcorresponding Euler equations could have been established.Let us continue the process to obtain an approximate solution of the linear heatconduction problem. Taking, as usual, ^ ˆ X N i a i ˆ Na…3:76†

64 Generalization of the ®nite element conceptswe substitute this approximation into the expression for the functional [Eq. (3.72)]and obtain… 1 X 2 … ˆ 2 k @Ni@x a 1 X 2i d ‡ 2 k @Ni@y a i d…ÿ Q X …N i a i d ÿ q X N i a i dÿ…3:77†ÿ qOn di€erentiation with respect to a typical parameter a j we have… @ X … @Niˆ k@a j @x a @NjXi@x d ‡ @Nik@y a @Nji@y d……ÿ QN j d ÿ qN j dÿÿ q…3:78†and a system of equations for the solution of the problem iswith…K ij ˆ K ji ˆk @N i@x…f j ˆÿ N j Q d ÿKa ‡ f ˆ 0@N j@x…d ‡ …N j q dÿÿ qk @N i@y@N j@y d…3:79†…3:80†The reader will observe that the approximation equations are here identical withthose obtained in Sec. 3.5 for the same problem using the Galerkin process. No specialadvantage accrues to the variational formulation here, and indeed we can predict nowthat Galerkin and variational procedures must give the same answer for cases wherenatural variational principles exist.3.8.2 Relation of the Galerkin method to approximation viavariational principlesIn the preceding example we have observed that the approximation obtained by theuse of a natural variational principle and by the use of the Galerkin weighting processproved identical. That this is the case follows directly from Eq. (3.70), in which thevariation was derived in terms of the original di€erential equations and the associatedboundary conditions.If we consider the usual trial function expansion [Eq. (3.3)]u ^u ˆ Nawe can write the variation of this approximation as^u ˆ N a…3:81†

`Natural' variational principles and their relation to governing differential equations 65and inserting the above into (3.70) yields…… ˆ a T N T A…Na† d ‡ a TÿN T B…Na† dÿ ˆ 0…3:82†The above form, being true for all a, requires that the expression under theintegrals should be zero. The reader will immediately recognize this as simply theGalerkin form of the weighted residual statement discussed earlier [Eq. (3.25)], andidentity is hereby proved.We need to underline, however, that this is only true if the Euler equations of thevariational principle coincide with the governing equations of the original problem.The Galerkin process thus retains its greater range of applicability.At this stage another point must be made, however. If we consider a system ofgoverning equations [Eq. (3.1)]8 9A 1 …u† >< >=A…u† ˆ A 2 …u†>: . >; ˆ 0.with ^u ˆ Na, the Galerkin weighted residual equation becomes (disregarding theboundary conditions)…N T A…^u† d ˆ 0…3:83†This form is not unique as the system of equations A can be ordered in a number ofways. Only one such ordering will correspond precisely with the Euler equations of avariational principle (if this exists) and the reader can verify that for an equationsystem weighted in the Galerkin manner at best only one arrangement of thevector A results in a symmetric set of equations.As an example, consider, for instance, the one-dimensional heat conductionproblem (Example 1, Sec. 3.3) rede®ned as an equation system with two unknowns, being the temperature and q the heat ¯ow. Disregarding at this stage the boundaryconditions we can write these equations as8>: dqdx ‡ Q >; ˆ 0 …3:84†or as a linear equation system,A…u† Lu ‡ b ˆ 0in which21; ÿ d 3 L 6 dx7 0 q4 5 b ˆ u ˆ…3:85†Qddx ; 0Writing the trial function in which a di€erent interpolation is used for each functionu ˆ X N1 N i a i N i ˆ i 009>=N 2 i

66 Generalization of the ®nite element conceptsand applying the Galerkin process, we arrive at the usual linear equation system with2…… NK ij ˆ N T i 1 Nj 1 ; ÿN 1 dN 2 3jii LN j dx ˆdx674Ni2 dNj1 5 dx …3:86†dx ; 0After integration by parts, this form yields a symmetric equationy system andK ij ˆ K ji…3:87†If the order of equations were simply reversed, i.e., using2 3dqA…u† ˆdx ‡ Q64q ÿ d75 ˆ 0 …3:88†dxapplication of the Galerkin process would now lead to non-symmetric equations quitedi€erent from those arising using the variational principle. The second type ofGalerkin approximation would clearly be less desirable due to loss of symmetry inthe ®nal equations. It is easy to show that the ®rst system corresponds precisely tothe Euler equations of the variational functional deduced in the next section.3.9 Establishment of natural variational principles forlinear, self-adjoint differential equations3.9.1 General theoremsGeneral rules for deriving natural variational principles from non-linear di€erentialequations are complicated and even the tests necessary to establish the existence ofsuch variational principles are not simple. Much mathematical work has beendone, however, in this context by Vainberg, 15 Tonti, 16 Oden, 17 and others.For linear di€erential equations the situation is much simpler and a thorough studyis available in the works of Mikhlin, 18;19 and in this section a brief presentation ofsuch rules is given.We shall consider here only the establishment of variational principles for a linearsystem of equations with forced boundary conditions, implying only variation offunctions which yield u ˆ 0 on their boundaries. The extension to include naturalboundary conditions is simple and will be omitted.Writing a linear system of di€erential equations asA…u† Lu ‡ b ˆ 0…3:89†y As…Ni1 dNj2 … dN1dx dx ÿ idx N2 j dx ‡ boundary terms

Establishment of natural variational principles for linear, self-adjoint differential equations 67in which L is a linear di€erential operator it can be shown that natural variationalprinciples require that the operator L be such that……w T …Lc† d ˆ c T …Lw† d ‡ b:t:…3:90†for any two function sets w and c. In the above, `b.t.' stands for boundary terms whichwe disregard in the present context. The property required in the above operator iscalled that of self-adjointness or symmetry.If the operator L is self-adjoint, the variational principle can be writtenimmediately as… ˆ ‰ 1 2 uT Lu ‡ u T bŠ d ‡ b:t: …3:91†To prove the veracity of the last statement a variation needs to be considered. Wethus write… ˆ ‰ 1 2 uT Lu ‡ 1 2 uT …Lu†‡u T bŠ d ‡ b:t: …3:92†Noting that for any linear operator…Lu† L u…3:93†and that u and u can be treated as any two independent functions, by identity (3.90)we can write Eq. (3.92) as… ˆ u T ‰Lu ‡ bŠ d ‡ b:t:…3:94†We observe immediately that the term in the brackets, i.e. the Euler equation of thefunctional, is identical with the original equation postulated, and therefore thevariational principle is veri®ed.The above gives a very simple test and a prescription for the establishment ofnatural variational principles for di€erential equations of the problem.Consider, for instance, two examples.Example 1. This is a problem governed by the di€erential equation similar to the heatconduction equation, e.g.,r 2 ‡ c ‡ Q ˆ 0…3:95†with c and Q being dependent on position only.The above can be written in the general form of Eq. (3.89), with @2L @x 2 ‡ @2@y 2 ‡ c b Q…3:96†Verifying that self-adjointness applies (which we leave to the reader as an exercise),we immediately have a variational principle… 1 @ 2 ˆ2 @x 2 ‡ @2 @y 2 ‡ c ‡ Q dx dy…3:97†

68 Generalization of the ®nite element conceptswith satisfying the forced boundary condition, i.e., ˆ on ÿ . Integration byparts of the ®rst two terms results in… 1 @ 2 ˆÿ‡ 1 @ 2ÿ 1 2 @x 2 @y 2 c2 ÿ Qdx dy …3:98†on noting that boundary terms with prescribed do not alter the principle.Example 2. This problem concerns the equation system discussed in the previoussection [Eqs (3.84) and (3.85)]. Again self-adjointness of the operator can be tested,and found to be satis®ed. We now write the functional as2… 1 q T 1; ÿ d 3 dx ˆ6 7 q q T 0 2 4d5‡dxdx ; 0 Q… 1ˆ 2 q2 ÿ 1 2 q d@x ‡ 1 2 dq dx ‡ Q dx…3:99†The veri®cation of the correctness of the above, by executing a variation, is left to thereader.These two examples illustrate the simplicity of application of the general expressions.The reader will observe that self-adjointness of the operator will generallyexist if even orders of di€erentiation are present. For odd orders self-adjointness isonly possible if the operator is a `skew'-symmetric matrix such as occurs in thesecond example.3.9.2 Adjustment for self-adjointnessOn occasion a linear operator which is not self-adjoint can be adjusted so that selfadjointnessis achieved without altering the basic equation. Consider, for instance,the problem governed by the following di€erential equation of a standard linear form:d 2 dx 2 ‡ ddx ‡ ‡ Q ˆ 0…3:100†In this equation and are functions of x. It is easy to see that the operator L is nowa scalar:L d2dx 2 ‡ ddx ‡ …3:101†and is not self-adjoint.Let p be some, as yet undetermined, function of x. We shall show that it is possibleto convert Eq. (3.100) to a self-adjoint form by multiplying it by this function. Thenew operator becomesL ˆ pL…3:102†

To test for symmetry with any two functions and we write…… …pL† dx ˆ p d2 d‡ p dx2 dx ‡ p dx …3:103†On integration of the ®rst term, by parts, we have (b.t. denoting boundary terms)… d… p† d dÿ ‡ p dx dx dx ‡ p dx dx ‡ b:t:… ˆ ÿ d dx p ddx ‡ d p ÿ dp ‡ p dx ‡ b:t: …3:104†dx dxSymmetry (and therefore self-adjointness) is now achieved in the ®rst and lastterms. The middle term will only be symmetric if it disappears, i.e., ifp ÿ dpdx ˆ 0…3:105†or„dpp ˆ dx; p ˆ e dx…3:106†By using this value of p the operator is made self-adjoint and a variational principlefor the problem of Eq. (3.100) is easily found.A procedure of this kind has been used by Guymon et al. 20 to derive variationalprinciples for a convective di€usion equation which is not self-adjoint. (We havenoted such lack of symmetry in the equation in Example 2, Sec. 3.3.)A similar method for creating variational functionals can be extended to the specialcase of non-linearity of Eq. (3.89) whenb ˆ b…u; x; ...†…3:107†If Eq. (3.92) is inspected we note that we could write…u T b†ˆ…g†…3:108†if…g ˆ b T duThis integration is often quite easy to accomplish.Maximum, minimum, or a saddle point? 693.10 Maximum, minimum, or a saddle point?In discussing variational principles so far we have assumed simply that at the solutionpoint ˆ 0, that is the functional is stationary. It is often desirable to know whether is at a maximum, minimum, or simply at a `saddle point'. If a maximum or aminimum is involved, then the approximation will always be `bounded', i.e., willprovide approximate values of which are either smaller or larger than the correctones.y This in itself may be of practical signi®cance.y Provided all integrals are exactly evaluated.

70 Generalization of the ®nite element conceptsΠd 2 ΠAda 2 > 0 Ad 2 Πda 2 = 0Ad 2 Πda 2 < 0AFig. 3.7 Maximum, minimum, and a `saddle' point for a functional of one variable.When, in elementary calculus, we consider a stationary point of a function of onevariable a, we investigate the rate of change of d with da and write @d…d† ˆd@a da ˆ @2 @a 2 …da†2…3:109†The sign of the second derivative determines whether is a minimum, maximum, orsimply stationary (saddle point), as shown in Fig. 3.7. By analogy in the calculus ofvariations we shall consider changes of . Noting the general form of this quantitygiven by Eq. (3.63) and the notion of the second derivative of Eq. (3.66) we can write,in terms of discrete parameters, @ T @…† a ˆ a T ˆ a T K@a@aT a …3:110†If, in the above, …† is always negative then is obviously reaching a maximum,if it is always positive then is a minimum, but if the sign is indeterminate this showsonly the existence of a saddle point.As a is an arbitrary vector this statement is equivalent to requiring the matrix K Tto be negative de®nite for a maximum or positive de®nite for a minimum. The form ofthe matrix K T (or in linear problems of K which is identical to it) is thus of greatimportance in variational problems.a3.11 Constrained variational principles. Lagrangemultipliers and adjoint functions3.11.1 Lagrange multipliersConsider the problem of making a functional stationary, subject to the unknown uobeying some set of additional di€erential relationshipsC…u† ˆ0 in …3:111†

Constrained variational principles. Lagrange multipliers and adjoint functions 71We can introduce this constraint by forming another functional……u; k† ˆ…u†‡ k T C…u† d…3:112†in which k is some set of functions of the independent coordinates in the domain known as Lagrange multipliers. The variation of the new functional is now…… ˆ ‡ k T C…u† d ‡ k T C…u† d…3:113†and this is zero providing C…u† ˆ0 and, simultaneously, ˆ 0…3:114†In a similar way, constraints can be introduced at some points or over boundaries ofthe domain. For instance, if we require that u obeyE…u† ˆ0 on ÿ …3:115†we would add to the original functional the term…k T E…u† dÿ…3:116†ÿwith k now being an unknown function de®ned only on ÿ. Alternatively, if theconstraint C is applicable only at one or more points of the system, then the simpleaddition of k T C…u† at these points to the general functional will introduce a discretenumber of constraints.It appears, therefore, possible to always introduce additional functions k andmodify a functional to include any prescribed constraints. In the `discretization'process we shall now have to use trial functions to describe both u and k.Writing, for instance,we shall obtain a set of equations8@>:@@a@@b9>=ˆ 0>; ac ˆb…3:118†from which both the sets of parameters a and b can be obtained. It is somewhatparadoxical that the `constrained' problem has resulted in a larger number ofunknown parameters than the original one and, indeed, has complicated the solution.We shall, nevertheless, ®nd practical use for Lagrange multipliers in formulatingsome physical variational principles, and will make use of these in a more generalcontext in Chapters 11 and 12.Example. The point about increasing the number of parameters to introduce aconstraint may perhaps be best illustrated in a simple algebraic situation in whichwe require a stationary value of a quadratic function of two variables a 1 and a 2 : ˆ 2a 2 1 ÿ 2a 1 a 2 ‡ a 2 2 ‡ 18a 1 ‡ 6a 2…3:119†

72 Generalization of the ®nite element conceptssubject to a constrainta 1 ÿ a 2 ˆ 0…3:120†The obvious way to proceed would be to insert directly the equality `constraint' andobtain ˆ a 2 1 ‡ 24a 1…3:121†and write, for stationarity,@ˆ 0 ˆ 2a@a 1 ‡ 241a 1 ˆ a 2 ˆÿ12 …3:122†Introducing a Lagrange multiplier we can alternatively ®nd the stationarity of ˆ 2a 2 1 ÿ 2a 1 a 2 ‡ a 2 2 ‡ 18a 1 ‡ 6a 2 ‡ …a 1 ÿ a 2 † …3:123†and write three simultaneous equations@ @ @ ˆ 0 ˆ 0@a 1 @a 2 @ ˆ 0…3:124†The solution of the above system again yields the correct answera 1 ˆ a 2 ˆÿ12 ˆ 6but at considerably more e€ort. Unfortunately, in most continuum problems directelimination of constraints cannot be so simply accomplished.yBefore proceeding further it is of interest to investigate the form of equations resultingfrom the modi®ed functional of Eq. (3.112). If the original functional gave asits Euler equations a systemA…u† ˆ0…3:125†then we have… ˆ……u T A…u† d ‡ k T C…u† d ‡ k T C d…3:126†Substituting the trial functions (3.117) we can write for a linear set of constraintsC…u† ˆL 1 u ‡ C 1…… ˆ a T N T A…^u† d ‡ b T N T …L 1^u ‡ C 1 † d…3:127†…‡ a T …L 1 N† T^k d ˆ 0As this has to be true for all variations a and b, we have a system of equations……N T A…^u† d ‡ …L 1 N† T^k d ˆ 0……3:128†N T …L 1^u ‡ C 1 † d ˆ 0y In the ®nite element context, Szabo and Kassos 21 use such direct elimination; however, this involvesconsiderable algebraic manipulation.

Constrained variational principles. Lagrange multipliers and adjoint functions 73For linear equations A, the ®rst term of the ®rst equation is precisely the ordinary,unconstrained, variational approximationK aa a ‡ f a…3:129†and inserting again the trial functions (3.117) we can write the approximated Eq.(3.128) as a linear system:K c c ˆ Kaa; K ab aK T ‡ f aˆ 0…3:130†ab; 0 b f bwith……K T ab ˆ N T L 1 N d; f b ˆÿ N T C 1 d…3:131†Clearly the system of equations is symmetric but now possesses zeros on the diagonal,and therefore the variational principle is merely stationary. Further, computationaldi culties may be encountered unless the solution process allows for zero diagonalterms.3.11.2 Identi®cation of Lagrange multipliers. Forced boundaryconditions and modi®ed variational principlesAlthough the Lagrange multipliers were introduced as a mathematical ®ctionnecessary for the enforcement of certain external constraints required to satisfy theoriginal variational principle, we shall ®nd that in most physical situations theycan be identi®ed with certain physical quantities of importance to the originalmathematical model. Such an identi®cation will follow immediately from thede®nition of the variational principle established in Eq. (3.112) and through thesecond of the Euler equations corresponding to it. The variation , written in Eq.(3.113), supplies through its third term the constraint equation. The ®rst two termscan always be rewritten as……k T C…u† d ‡ u T A…u† d ‡ b:t: ˆ 0…3:132†This supplies the identi®cation of k.In the literature of variational calculation such identi®cation arises frequently andthe reader is referred to the excellent text by Washizu 22 for numerous examples.Example. Here we shall introduce this identi®cation by means of the exampleconsidered in Sec. 3.8.1. As we have noted, the variational principle of Eq. (3.72)established the governing equation and the natural boundary conditions of theheat conduction problem providing the forced boundary conditionC…† ˆ ÿ ˆ 0…3:133†was satis®ed on ÿ in the choice of the trial function for .

74 Generalization of the ®nite element conceptsThe above forced boundary condition can, however, be considered as a constrainton the original problem. We can write the constrained variational principle as… ˆ ‡ … ÿ † dÿÿ …3:134†where is given by Eq. (3.72).Performing the variation we have…… ˆ ‡ … ÿ † dÿ ‡ÿ dÿÿ …3:135† is now given by the expression (3.75a) augmented by an integral…k @ÿ @n dÿ…3:136†which was previously disregarded (as we had assumed that ˆ 0onÿ ). In additionto the conditions of Eq. (3.75b), we now require that…… … ÿ † dÿ ‡ ‡ k @ dÿ ˆ 0…3:137†ÿ ÿ @nwhich must be true for all variations and . The ®rst simply reiterates theconstraint ÿ ˆ 0 on ÿ …3:138†The second de®nes as ˆÿk @@n…3:139†Noting that k…@=@n† is equal to the ¯ux q n on the boundary ÿ , the physical identi-®cation of the multiplier has been achieved.The identi®cation of the Lagrange variable leads to the possible establishment of amodi®ed variational principal in which is replaced by the identi®cation.We could thus write a new principle for the above example:… ˆ ÿÿ k @@n … ÿ † dÿ…3:140†in which once again is given by the expression (3.72) but is not constrained tosatisfy any boundary conditions. Use of such modi®ed variational principles can bemade to restore interelement continuity and appears to have been ®rst introducedfor that purpose by Kikuchi and Ando. 23 In general these present interesting newprocedures for establishing useful variational principles.A further extension of such principles has been made use of by Chen and Mei 24 andZienkiewicz et al. 25 Washizu 22 discusses many such applications in the context ofstructural mechanics. The reader can verify that the variational principle expressedin Eq. (3.140) leads to automatic satisfaction of all the necessary boundary conditionsin the example considered.The use of modi®ed variational principles restores the problem to the original numberof unknown functions or parameters and is often computationally advantageous.

Constrained variational principles. Lagrange multipliers and adjoint functions 753.11.3 A general variational principle: adjoint functions andoperatorsThe Lagrange multiplier method leads to an obvious procedure of `creating' avariational principle for any set of equations even if the operators are not self-adjoint:A…u† ˆ0…3:141†Treating all the above equations as a set of constraints we can obtain such a generalvariational functional simply by putting ˆ 0 in Eq. (3.112) and writing… ˆ k T A…u† d…3:142†now requiring stationarity for all variations of k and u. The new variationalprinciple has, however, been introduced at the expense of doubling the number ofvariables in the discretized situation. Treating the case of linear equations only, i.e.,A…u† ˆLu ‡ g ˆ 0…3:143†and discretizing we note, going through the steps involved in Eqs (3.126) to (3.130),that the ®nal system of equations now takes the form 0 K ab a 0K T ‡ ˆ 0…3:144†ab0 b fwith…K T ab ˆ N T LN d……3:145†f ˆ N T g dThe equations are completely decoupled and the second set can be solved independentlyfor all the parameters a describing the unknowns in which we were originallyinterested without consideration of the parameters b. It will be observed that thissecond set of equations is identical with an, apparently arbitrary, weighted residualprocess. We have thus completed the full circle and obtained the weighted residualforms of Sec. 3.3 from a general variational principle.The function k which appears in the variational principle of Eq. (3.142) is known asthe adjoint function to u.By performing a variation on Eq. (3.142) it is easy to show that the Euler equationsof the principle are such thatA…u† ˆ0…3:146†andA …u† ˆ0…3:147†where the operator A is such that……k T …Au† d ˆ u T A …k† d…3:148†

76 Generalization of the ®nite element conceptsThe operator A is known as the adjoint operator and will exist only in linearproblems (see Appendix H).For the full signi®cance of the adjoint operator the reader is advised to consultmathematical texts. 263.12 Constrained variational principles. Penalty functionsand the least square method3.12.1 Penalty functionsIn the previous section we have seen how the process of introducing Lagrange multipliersallows constrained variational principles to be obtained at the expense ofincreasing the total number of unknowns. Further, we have shown that even inlinear problems the algebraic equations which have to be solved are now complicatedby having zero diagonal terms. In this section we shall consider an alternative procedureof introducing constraints which does not possess these drawbacks.Considering once again the problem of obtaining stationarity of with a set ofconstraint equations C…u† ˆ0 in domain , we note that the productwhereC T C ˆ C 2 1 ‡ C 2 2 ‡…3:149†C T ˆ‰C 1 ; C 2 ; ...Šmust always be a quantity which is positive or zero. Clearly, the latter value is foundwhen the constraints are satis®ed and clearly the variation…C T C†ˆ0as the product reaches that minimum.We can now immediately write a new functional… ˆ ‡ C T …u†C…u† d…3:150†…3:151†in which is a `penalty number' and then require the stationarity for the constrainedsolution. If is itself a minimum of the solution then should be a positive number.The solution obtained by the stationarity of the functional will satisfy theconstraints only approximately. The larger the value of the better will be theconstraints achieved. Further, it seems obvious that the process is best suited tocases where is a minimum (or maximum) principle, but success can be obtainedeven with purely saddle point problems. The process is equally applicable toconstrants applied on boundaries or simple discrete constraints. In this latter caseintegration is dropped.Example. To clarify ideas let us once again consider the algebraic problem ofSec. 3.11, in which the stationarity of a functional given by Eq. (3.119) was soughtsubject to a constraint. With the penalty function approach we now seek the

Constrained variational principles. Penalty functions and the least square method 77Table 3.1 ˆ 1 2 6 10 100a 1 ˆÿ12:00 ÿ12:00 ÿ12:00 ÿ12:00 ÿ12:00a 2 ˆÿ13:50 ÿ13:00 ÿ12:43 ÿ12:78 ÿ12:03minimum of a functional ˆ 2a 2 1 ÿ 2a 1 a 2 ‡ a 2 2 ‡ 18a 1 ‡ 6a 2 ‡ …a 1 ÿ a 2 † 2…3:152†with respect to the variation of both parameters a 1 and a 2 . Writing the two simultaneousequations@ @ ˆ 0 ˆ 0…3:153†@a 1 @a 2we ®nd that as is increased we approach the correct solution. In Table 3.1 the resultsare set out demonstrating the convergence.The reader will observe that in a problem formulated in the above manner theconstraint introduces no additional unknown parameters ± but neither does itdecrease their original number. The process will always result in strongly positivede®nite matrices if the original variational principle is one of a minimum.In practical applications the method of penalty functions has proved to be quitee€ective, 27 and indeed is often introduced intuitively. One such `intuitive' applicationwas already made when we enforced the value of boundary parameters in the mannerindicated in Chapter 1, Sec. 1.4.In the example presented here (and frequently practised in the real assembly ofdiscretized ®nite element equations), the forced boundary conditions are notintroduced a priori and the problem gives, on assembly, a singular system ofequationsKa ‡ f ˆ 0…3:154†which can be obtained from a functional (providing K is symmetric) ˆ 12 aT Ka ‡ a T f …3:155†Introducing a prescribed value of a 1 , i.e., writinga 1 ÿ a 1 ˆ 0…3:156†the functional can be modi®ed toyielding ˆ ‡ …a 1 ÿ a 1 † 2…3:157†K 11 ˆ K 11 ‡ 2 f 1 ˆ f 1 ÿ 2a 1…3:158†and giving no change in any of the other matrix coe cients. This is precisely theprocedure adopted in Chapter 1 (page 10) for modifying the equations, to introduceprescribed values of a 1 (2 here replacing , the `large number' of Sec. 1.4). Manyapplications of such a `discrete' kind are discussed by Campbell. 28

78 Generalization of the ®nite element conceptsIt is easy to show in another context 27;29 that the use of a high Poisson's ratio… ! 0:5† for the study of incompressible solids or ¯uids is in fact equivalent to theintroduction of a penalty term to suppress any compressibility allowed by anarbitrary displacement variation.The use of the penalty function in the ®nite element context presents certaindi culties.Firstly, the constrained functional of Eq. (3.151) leads to equations of the form…K 1 ‡ K 2 †a ‡ f ˆ 0…3:159†where K 1 derives from the original functions and K 2 from the constraints. As increases the above equation degenerates:K 2 a ˆÿf= ! 0and a ˆ 0 unless the matrix K 2 is singular. The phenomenon where a ) 0 is known aslocking and has often been encountered by researchers who failed to recognize itssource. This singularity in the equations does not always arise and we shall discussmeans of its introduction in Chapters 11 and 12.Secondly, with large but ®nite values of numerical di culties will be encountered.Noting that discretization errors can be of comparable magnitude to those due to notsatisfying the constraint, we can make ˆ constant …1=h† nensuring a limiting convergence to the correct answer. Fried 30;31 discusses thisproblem in detail.A more general discussion of the whole topic is given in reference 32 and in Chapter12 where the relationship between Lagrange constraints and penalty forms is madeclear.3.12.2 Least square approximationsIn Sec. 3.11.3 we have shown how a constrained variational principle procedure couldbe used to construct a general variational principle if the constraints become simplythe governing equations of the problemC…u† ˆA…u†…3:160†Obviously the same procedure can be used in the context of the penalty functionapproach by setting ˆ 0 in Eq. (3.151). We can thus write a `variationalprinciple'…… ˆ …A 2 1 ‡ A 2 2 ‡†d ˆ A T …u†A…u† d…3:161†for any set of di€erential equations. In the above equation the boundary conditionsare assumed to be satis®ed by u (forced boundary condition) and the parameter isdropped as it becomes a multiplier.Clearly, the above statement is a requirement that the sum of the squares of theresiduals of the di€erential equations should be a minimum at the correct solution.

This minimum is obviously zero at that point, and the process is simply the wellknownleast square method of approximation.It is equally obvious that we could obtain the correct solution by minimizing anyfunctional of the form…… ˆ …p 1 A 2 1 ‡ p 2 A 2 2 ‡†d ˆ A T …u†pA…u† d …3:162†in which p 1 , p 2 , ..., etc., are positive valued weighting functions or constants and p isa diagonal matrix:23p 1 0p 2 p ˆ6 p437…3:163†5.0. .The above alternative form is sometimes convenient as it puts di€erent importanceon the satisfaction of individual components of the equation and allows additionalfreedom in the choice of the approximate solution. Once again this weighting functioncould be chosen so as to ensure a constant ratio of terms contributed by variouselements, although this has not yet been put into practice.Least square methods of the kind shown above are a very powerful alternativeprocedure for obtaining integral forms from which an approximate solution can bestarted, and have been used with considerable success. 33;34 As the least square variationalprinciples can be written for any set of di€erential equations without introducingadditional variables, we may well enquire what is the di€erence between theseand the natural variational principles discussed previously. On performing a variationin a speci®c case the reader will ®nd that the Euler equations which are obtained nolonger give the original di€erential equations but give higher order derivatives ofthese. This introduces the possibility of spurious solutions if incorrect boundary conditionsare used. Further, higher order continuity of trial function is now generallyneeded. This may be a serious drawback but frequently can be by-passed by statingthe original problem as a set of lower order equations.We shall now consider the general form of discretized equations resulting from theleast square approximation for linear equation sets (again neglecting boundary conditionswhich are enforced). Thus, if we takeA…u† ˆLu ‡ b…3:164†and take the usual trial function approximation^u ˆ Na…3:165†we can write, substituting into (3.162),… ˆ ‰…LN†a ‡ bŠ T p‰…LN†a ‡ bŠ d…3:166†and obtain… ˆConstrained variational principles. Penalty functions and the least square method 79…a T …LN† T p‰…LN†a ‡ bŠ d ‡ ‰…LN†a ‡ bŠ T p…LN† a d ˆ 0…3:167†

80 Generalization of the ®nite element conceptsor, as p is symmetric, … ˆ 2a T…LN† T p…LN† d a ‡……LN† T pb dˆ 0…3:168†This immediately yields the approximation equation in the usual form:Ka ‡ f ˆ 0…3:169†and the reader can observe that the matrix K is symmetric and positive de®nite.To illustrate an actual example, consider the problem governed by Eq. (3.95) forwhich we have already obtained a natural variational principle [Eq. (3.98)] in whichonly ®rst derivatives were involved requiring C 0 continuity for u. Now, if we usethe operator L and term b de®ned by Eq. (3.96), we have a set of approximatingequations with…K ij ˆ …r 2 N i ‡ cN i †…r 2 N j ‡ cN j † dx dy……3:170†f i ˆ …r 2 N i ‡ cN i †Q dx dyThe reader will observe that now C 1 continuity is needed for the trial functions N.An alternative avoiding this di culty is to write Eq. (3.95) as a ®rst-order system.This can be written as8@q x@x ‡ @q 9y@y ‡ c ‡ Q>=@A…u† ˆ@x ÿ q xˆ 0…3:171†@>:@y ÿ q y>;or, introducing the vector u,u ˆ‰; q x ; q y Š T ˆ…Na†…3:172†as the unknown we can write the standard linear form (3.164) aswhereLu ‡ b ˆ 02@c;@x ; @3@y@L ˆ@x ; ÿ1; 0674 @5; 0; ÿ1@y8 9>< Q >=b ˆ 0>: >;0…3:173†The reader can now perform the substitution into Eq. (3.168) to obtain theapproximation equations in a form requiring only C 0 continuity ± introduced,

Constrained variational principles. Penalty functions and the least square method 81however, at the expense of additional variables. Use of such forms has been madeextensively in the ®nite element context. 33;343.12.3 Galerkin least squares, stabilizationIt is interesting to note that the concept of penalty formulation introduced earlier inthis section was anticipated as early as 1943 by Courant 35 in a somewhat di€erentmanner. He used the original variational principle augmented by the di€erentialequations of the problem employed as least square constraints. In this manner heclaimed, though never proved, that the convergence rate could be accelerated.The suggestion put forward by Courant has been used e€ectively by othersthough in a somewhat di€erent manner. Noting that the Galerkin process is, forself-adjoint equations, equivalent to that of minimizing a functional, the leastsquare formulation using the original equation is simply added to the Galerkinform. Here it allows non-self adjoint operators to be used, for instance, and thisfeature has been exploited with success. Consider, for instance, the problem whichwe have discussed in Section 3.9.2 [viz. Eq. (3.100)] with ˆ 0. This equation, aswe have already pointed out, is non-self adjoint but Galerkin methods have beensuccessfully used in its solution providing the convection term ( d=dx) remainsrelatively small compared to the second derivative term (the di€usion term). However,it is found that as the convection term increases the solution becomes highlyoscillatory. We shall discuss the stabilization of such problems in a generalmanner exhaustively in Volume 3 as such problems are frequently encountered in¯uid mechanics. But here it is easy to consider the problem in a preliminarymanner. Suppose in a Galerkin form given by… dv ddx dx ÿ v d dx ‡ Q dx ˆ 0…3:174†we add a multiple of the minimization of the least square of the total equation. Theresult is… dv ddx dx ÿ v d dx ‡ Q dx…‡ d 2 vdx 2 ‡ dvdx d 2 dx 2 ‡ d dx ‡ Q dx ˆ 0…3:175†and we see immediately that an additional di€usive term has been added whichdepends on the parameter , though at the expense of having higher derivativesappearing in the integrals. If only linear elements are used and the discontinuitiesignored at element interfaces, the process of adding the di€usive terms can stabilizethe oscillations which would otherwise occur. The idea appears to have ®rst beenused by Hughes 36 . This process in the view of the authors is somewhat unorthodoxas discontinuity of derivatives is ignored, and alternatives to this will be discussedat length in Chapter two of Volume 3.

82 Generalization of the ®nite element conceptsIt interesting to note also that another application of the same Galerkin leastsquare process can be made to the mixed formulation with two variables u and pfor incompressible problems. We shall discuss such problems in Chapter 12 of thisvolume and show how this process can be made applicable there.Finally, it is of interest to note that the simple procedure introduced by Courantcan also be e€ective in the prevention of locking of other problems. The treatmentfor beams has been studied by Freund and Salonen 40 and it appears that quite ane€ective process can be reached.3.13 Concluding remarks ± ®nite difference andboundary methodsThis very extensive chapter presents the general possibilities of using the ®nite elementprocess in almost any mathematical or mathematically modelled physical problem.The essential approximation processes have been given in as simple a form aspossible, at the same time presenting a fully comprehensive picture which shouldallow the reader to understand much of the literature and indeed to experimentwith new permutations. In the chapters that follow we shall apply to various physicalproblems a limited selection of the methods to which allusion has been made. In somewe shall show, however, that certain extensions of the process are possible (Chapters12 and 16) and in another (Chapter 10) how a violation of some of the rules hereexpounded can be accepted.The numerous approximation procedures discussed fall into several categories. Toremind the reader of these, we present in Table 3.2 a comprehensive catalogue of themethods used here and in Chapter 2. The only aspect of the ®nite element processmentioned in that table that has not been discussed here is that of a direct physicalmethod. In such models an `atomic' rather than continuum concept is the startingpoint. While much interest exists in the possibilities o€ered by such models, theirdiscussion is outside the scope of this book.In all the continuum processes discussed the ®rst step is always the choice ofsuitable shape or trial functions. A few simple forms of such functions have beenintroduced as the need demanded and many new forms will be introduced in subsequentchapters. Indeed, the reader who has mastered the essence of the presentchapter will have little di culty in applying the ®nite element method to any suitablyde®ned physical problem. For further reading references 41±45 could be consulted.The methods listed do not include speci®cally two well-known techniques, i.e., ®nitedi€erence methods and boundary solution methods (sometimes known as boundaryelements). In the general sense these belong under the category of the generalized®nite element method discussed here. 411. Boundary solution methods choose the trial functions such that the governingequation is automatically satis®ed in the domain . Thus starting from the generalapproximation equation (3.25), we note that only boundary terms are retained. Weshall return to such approximations in Chapter 13.2. Finite di€erence procedures can be interpreted as an approximation based onlocal, discontinuous, shape functions with collocation weighting applied (although

Table 3.2 Finite element approximationConcluding remarks ± ®nite difference and boundary methods 83ÿÿ "Integral forms of continuum problemstrial functionsÿÿ "Direct physicalmodelu ˆ X N i a iÿÿ "ÿÿÿVariational principlesÿÿ "Meaningfulphysicalprinciplesÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ ÿÿÿ"Constrainedlangragianformsÿÿ "Weighted integrals of partialdi€erential equation governing(weak formulations)ÿÿÿÿÿÿÿÿÿ ÿÿÿ ÿÿÿ ÿÿÿ ÿÿ "ÿÿÿ ÿÿÿ ÿÿÿ ÿÿÿ ÿÿÿ ÿÿÿ ÿÿ "ÿÿÿ ÿÿÿÿÿÿÿÿÿÿÿÿ ÿÿÿ ÿÿÿ ÿÿÿ ÿÿÿ ÿÿÿ ÿÿ "ÿÿ "MiscellaneousweightfunctionsPenaltyCollocationAdjoint function (point orfunctions formssubdomain)ÿÿ ÿÿ"3Leastÿÿ "Global physicalstatements(e.g. virtual work)ÿÿÿ ÿÿÿ ÿÿÿ ÿÿÿ ÿÿÿ ÿÿÿ ÿÿÿ ÿÿÿÿÿÿ ÿÿÿ ÿÿÿ ÿÿ "ÿÿ " 3 ÿÿGalerkinsquare …W j ˆ N j †formsÿÿ ""usually the derivation of the approximation algorithm is based on a Taylorexpansion).As Galerkin or variational approaches give, in the energy sense, the bestapproximation, this method has only the merit of computational simplicity andoccasionally a loss of accuracy.To illustrate this process we discuss an approximation carried out for the onedimensionalequation (3.27) (viz. p. 47). Here we represent a localized approximationthrough equally spaced nodal point values by 1 …x ÿ xi † 2…x† ˆ28>:h 2 i ÿ 1 i i ‡ 19>=>;ÿ x ÿ x i; 1 ÿ …x ÿ x i† 2hh 2 ; 1 …x ÿ xi † 22 h 2 ‡ x ÿ x ih 2…3:176†where h ˆ x i ‡ 1 ÿ x i (shown in Fig. 3.8). It is clear that adjacent parabolic approximationsin this case are discontinuous between the nodes. Values of the function and its

84 Generalization of the ®nite element conceptsφ ii – 2 i – 1 i i+1 i +2xFig. 3.8. A local, discontinuous shape function by parabolic segments used to obtain a ®nite differenceapproximation.®rst two derivatives at a typical node i are given by…x i †ˆ i@@x ˆ 1x ˆ xi 2h … i ‡ 1 ÿ i ÿ 1 †@ 2 x@x 2 ˆ 1ˆ xi h 2 … i ‡ 1 ÿ 2 i ‡ i ÿ 1 †…3:177†If we insert these into the governing equation at node i, we note immediately that theapproximating equation at the node becomes1h 2 … i ÿ 1 ÿ 2 i ‡ i ‡ 1 †‡Q i ˆ 0…3:178†This is identical (within a multiple of h) to the assembled ®nite element equations(which we did not do explicitly) for the approximation with linear elements discussedin Eq. (3.35). This is indeed one of the cases in which the approximation is identicalrather than di€erent. In Chapter 16 we shall be discussing such ®nite di€erence andpoint approximations in more detail. However, the reader will note the present exerciseis simply given to underline the similarity of ®nite element and ®nite di€erenceprocesses.Many textbooks deal exclusively with these types of approximations. References46±50 discuss ®nite di€erence approximation and references 51±54 relate to boundarymethods.References1. S.H. Crandall. Engineering Analysis. McGraw-Hill, 1956.2. B.A. Finlayson. The Method of Weighted Residuals and Variational Principles. AcademicPress, 1972.3. R.A. Frazer, W.P. Jones, and S.W. Sken. Approximations to functions and to the solutions ofdi€erential equations. Aero. Research Committee Report 1799, 1937.4. C.B. Biezeno and R. Grammel. Technische Dynamik, p. 142, Springer-Verlag, 1933.5. B.G. Galerkin. Series solution of some problems of elastic equilibrium of rods and plates(Russian). Vestn. Inzh. Tech., 19, 897±908, 1915.

6. Also attributed to Bubnov, 1913: see S.C. Mikhlin. Variational Methods in MathematicalPhysics. Macmillan, 1964.7. P. Tong. Exact solution of certain problems by the ®nite element method. J. AIAA, 7, 179±80, 1969.8. R.V. Southwell. Relaxation Methods in Theoretical Physics. Clarendon Press, 1946.9. R.S. Varga. Matrix Iterative Analysis. Prentice-Hall, 1962.10. S. Timoshenko and J.N. Goodier. Theory of Elasticity. 2nd ed., McGraw-Hill, 1951.11. L.V. Kantorovitch and V.I. Krylov. Approximate Methods of Higher Analysis. Wiley(International), 1958.12. F.B. Hildebrand. Methods of Applied Mathematics, 2nd edn. Dover Publications, 1992.13. J.W. Strutt (Lord Rayleigh). On the theory of resonance. Trans. Roy. Soc. (London), A161,77±118, 1870.14. W. Ritz. UÈ ber eine neue Methode zur LoÈ sung gewissen Variations ± Probleme derMathematischen Physik. J. Reine angew. Math., 135, 1±61, 1909.15. M.M. Vainberg. Variational Methods for the Study of Nonlinear Operators. Holden-Day,1964.16. E. Tonti. Variational formulation of non-linear di€erential equations. Bull. Acad. Roy.Belg. (Classe Sci.), 55, 137±65 and 262±78, 1969.17. J.T. Oden. A general theory of ®nite elements ± I: Topological considerations, pp. 205±21,and II: Applications, pp. 247±60. Int. J. Num. Meth. Eng., 1, 1969.18. S.C. Mikhlin. Variational Methods in Mathematical Physics. Macmillan, 1964.19. S.C. Mikhlin. The Problems of the Minimum of a Quadratic Functional. Holden-Day,1965.20. G.L. Guymon, V.H. Scott, and L.R. Herrmann. A general numerical solution of the twodimensionaldi€erential±convection equation by the ®nite element method. Water Res., 6,1611±15, 1970.21. B.A. Szabo and T. Kassos. Linear equation constraints in ®nite element approximations.Int. J. Num. Meth. Eng., 9, 563±80, 1975.22. K. Washizu. Variational Methods in Elasticity and Plasticity. 2nd ed., Pergamon Press,1975.23. F. Kikuchi and Y. Ando. A new variational functional for the ®nite element method and itsapplication to plate and shell problems. Nucl. Eng. Des., 21, 95±113, 1972.24. H.S. Chen and C.C. Mei. Oscillations and water forces in an o€shore harbour. Ralph M.Parsons Laboratory for Water Resources and Hydrodynamics, Report 190, Cambridge,Mass., 1974.25. O.C. Zienkiewicz, D.W. Kelly, and P. Bettess. The coupling of the ®nite element methodand boundary solution procedures. Int. J. Num. Meth. Eng., 11, 355±75, 1977.26. I. Stakgold. Boundary Value Problems of Mathematical Physics. Macmillan, 1967.27. O.C. Zienkiewicz. Constrained variational principles and penalty function methods in the®nite element analysis. Lecture Notes in Mathematics. No. 363, pp. 207±14, Springer-Verlag, 1974.28. J. Campbell. A ®nite element system for analysis and design. Ph.D. thesis, Swansea, 1974.29. D.J. Naylor. Stresses in nearly incompressible materials for ®nite elements with applicationto the calculation of excess pore pressures. Int. J. Num. Meth. Eng., 8, 443±60, 1974.30. I. Fried. Finite element analysis of incompressible materials by residual energy balancing.Int. J. Solids Struct., 10, 993±1002, 1974.31. I. Fried. Shear in C 0 and C 1 bending ®nite elements. Int. J. Solids Struct., 9, 449±60, 1973.32. O.C. Zienkiewicz and E. Hinton. Reduced integration, function smoothing and nonconformityin ®nite element analysis. J. Franklin Inst., 302, 443±61, 1976.33. P.P. Lynn and S.K. Arya. Finite elements formulation by the weighted discrete leastsquares method. Int. J. Num. Meth. Eng., 8, 71±90, 1974.References 85

86 Generalization of the ®nite element concepts34. O.C. Zienkiewicz, D.R.J. Owen, and K.N. Lee. Least square ®nite element for elasto-staticproblems ± use of reduced integration. Int. J. Num. Meth. Eng., 8, 341±58, 1974.35. R. Courant. Variational methods for the solution of problems of equilibrium and vibration.Bull. Amer Math. Soc., 49, 1±61, 1943.36. T.J.R. Hughes, L.P. Franca, and M. Balestra. A new ®nite element formulation for computational¯uid dynamics: V. Circumventing the BabusÏ ka±Brezzi condition: A stablePetrov±Galerkin formulation of the Stokes problem accommodating equal-order interpolations.Comp. Meth. Appl. Mech. Eng., 59, 85±99, 1986.37. T.J.R. Hughes and L.P. Franca. A new ®nite element formulation for computational ¯uiddynamics: VII. The Stokes problem with various well-posed boundary conditions: Symmetricformulations that converge for all velocity/pressure spaces. Comp. Meth. Appl.Mech. Eng., 65, 85±96, 1987.38. T.J.R. Hughes, L.P. Franca, and G.M. Hulbert. A new ®nite element formulation forcomputational ¯uid dynamics: VIII. The Galerkin/least-squares method for advective±di€usive equations. Comp. Meth. Appl. Mech. Eng., 73, 173±189, 1989.39. R. Codina. A comparison of some ®nite element methods for solving the di€usion±convection±reaction equation. Comp. Meth. Appl. Mech. Eng., 156, 185±210, 1998.40. Jouni Freund and Eero-Matti Salonen. Sensitizing according to Courant the Timoshenkobeam ®nite element solution. Int. J. Num. Meth. Eng., x, 129±60, 1999.41. O.C. Zienkiewicz and K. Morgan. Finite Elements and Approximation. Wiley, 1983.42. E.B. Becker, G.F. Carey, and J.T. Oden. Finite Elements: An Introduction. Vol. 1, Prentice-Hall, 1981.43. I. Fried. Numerical Solution of Di€erential Equations. Academic Press, New York, 1979.44. A.J. Davies. The Finite Element Method. Clarendon Press, Oxford, 1980.45. C.A.T. Fletcher. Computational Galerkin Methods. Springer-Verlag, 1984.46. R.V. Southwell. Relaxation Methods in Theoretical Physics. 1st edn., Clarendon Press,Oxford, 1946.47. R.V. Southwell. Relaxation Methods in Theoretical Physics. 2nd edn., Clarendon Press,Oxford, 1956.48. D.N. de G. Allen. Relaxation Methods. McGraw-Hill, London, 1955.49. F.B. Hildebrand. Introduction to Numerical Analysis. 2nd edn., Dover Publications, 1987.50. A.R. Mitchell and D. Gri ths. The Finite Di€erence Method in Partial Di€erential Equations.John Wiley & Sons, London, 1980.51. J. MacKerle and C.A. Brebbia, editors. The Boundary Element Reference Book. ComputationalMechanics, Southampton, 1988.52. G. Beer and J.O. Watson. Introduction to Finite and Boundary Element Methods forEngineers. John Wiley & Sons, London, 1993.53. P.K. Banerjee. The Boundary Element Methods in Engineering. McGraw-Hill, London,1994.54. Prem K. Kythe. An Introduction to Boundary Element Methods. CRC Press, 1994.

4Plane stress and plane strain4.1 IntroductionTwo-dimensional elastic problems were the ®rst successful examples of the applicationof the ®nite element method. 1;2 Indeed, we have already used this situation toillustrate the basis of the ®nite element formulation in Chapter 2 where the generalrelationships were derived. These basic relationships are given in Eqs (2.1)±(2.5)and (2.23) and (2.24), which for quick reference are summarized in Appendix C.In this chapter the particular relationships for the plane stress and plane strainproblem will be derived in more detail, and illustrated by suitable practical examples,a procedure that will be followed throughout the remainder of the book.Only the simplest, triangular, element will be discussed in detail but the basicapproach is general. More elaborate elements to be discussed in Chapters 8 and 9could be introduced to the same problem in an identical manner.The reader not familiar with the applicable basic de®nitions of elasticity is referredto elementary texts on the subject, in particular to the text by Timoshenko andGoodier, 3 whose notation will be widely used here.In both problems of plane stress and plane strain the displacement ®eld is uniquely givenby the u and v displacement in the directions of the cartesian, orthogonal x and y axes.Again, in both, the only strains and stresses that have to be considered are the threecomponents in the xy plane. In the case of plane stress, by de®nition, all other componentsof stress are zero and therefore give no contribution to internal work. In planestrain the stress in a direction perpendicular to the xy plane is not zero. However, byde®nition, the strain in that direction is zero, and therefore no contribution to internalwork is made by this stress, which can in fact be explicitly evaluated from the threemain stress components, if desired, at the end of all computations.4.2 Element characteristics4.2.1 Displacement functionsFigure 4.1 shows the typical triangular element considered, with nodes i, j, m

88 Plane stress and plane strainymv i (V i )x iiu i (U i )jy ixFig. 4.1 An element of a continuum in plane stress or plane strain.numbered in an anticlockwise order. The displacements of a node have twocomponents uia i ˆ…4:1†v iand the six components of element displacements are listed as a vector8 9>< a i >=a e ˆ a j…4:2†>: >;The displacements within an element have to be uniquely de®ned by these sixvalues. The simplest representation is clearly given by two linear polynomialsu ˆ 1 ‡ 2 x ‡ 3 y…4:3†v ˆ 4 ‡ 5 x ‡ 6 yThe six constants can be evaluated easily by solving the two sets of three simultaneousequations which will arise if the nodal coordinates are inserted and the displacementsequated to the appropriate nodal displacements. Writing, for example,a mu i ˆ 1 ‡ 2 x i ‡ 3 y iu j ˆ 1 ‡ 2 x j ‡ 3 y j…4:4†u m ˆ 1 ‡ 2 x m ‡ 3 y mwe can easily solve for 1 , 2 , and 3 in terms of the nodal displacements u i , u j , u m andobtain ®nallyu ˆ 12 ‰…a i ‡ b i x ‡ c i y†u i ‡…a j ‡ b j x ‡ c j y†u j ‡…a m ‡ b m x ‡ c m y†u m Š …4:5a†

Element characteristics 89in whicha i ˆ x j y m ÿ x m y jb i ˆ y j ÿ y m…4:5b†c i ˆ x m ÿ x jwith the other coe cients obtained by a cycle permutation of subscripts in the order,i, j, m, and wherey1 x i y i2 ˆ det1 x j y jˆ 2 …area of triangle ijm† …4:5c† 1 x m y mAs the equations for the vertical displacement v are similar we also havev ˆ 12 ‰…a i ‡ b i x ‡ c i y†v i ‡…a j ‡ b j x ‡ c j y†v j ‡…a m ‡ b m x ‡ c m y†v m Š …4:6†Though not strictly necessary at this stage we can represent the above relations, Eqs(4.5a) and (4.6), in the standard form of Eq. (2.1): uu ˆ ˆ Na e ˆ‰INv i ; IN j ; IN m Ša e…4:7†with I a two by two identity matrix, andN i ˆ ai ‡ b i x ‡ c i y; etc: …4:8†2The chosen displacement function automatically guarantees continuity of displacementwith adjacent elements because the displacements vary linearly along any sideof the triangle and, with identical displacement imposed at the nodes, the samedisplacement will clearly exist all along an interface.4.2.2 Strain (total)The total strain at any point within the element can be de®ned by its three componentswhich contribute to internal work. Thus2@38 9" @x ; 0>< x >= e ˆ " y>: >; ˆ@0;u 6 @yˆ Su…4:9†7 v xy 4 @@y ; @ 5@xy Note: If coordinates are taken from the centroid of the element thenx i ‡ x j ‡ x m ˆ y i ‡ y j ‡ y m ˆ 0 and a i ˆ 2=3 ˆ a j ˆ a mSee also Appendix D for a summary of integrals for a triangle.

90 Plane stress and plane strainSubstituting Eq. (4.7) we have8>:with a typical matrix B i given by2B i ˆ SN i ˆ64@N i@x ; 00;@N i@y ;3a ia ja m9>=>;2 3@N i@yˆ 1b i ; 064 0; c7 2 i@N 5 c i ; b ii@x…4:10a†75 …4:10b†This de®nes matrix B of Eq. (2.4) explicitly.It will be noted that in this case the B matrix is independent of the position withinthe element, and hence the strains are constant throughout it. Obviously, the criterionof constant strain mentioned in Chapter 2 is satis®ed by the shape functions.4.2.3 Elasticity matrixThe matrix D of Eq. (2.5)8>: x y xy9 08>= >< " x>; ˆ D B@ " y>: xy9 1>=>; ÿ e C0A…4:11†can be explicitly stated for any material (excluding here r 0 which is simply additive).To consider the special cases in two dimensions it is convenient to start from the forme ˆ D ÿ1 r ‡ e 0and impose the conditions of plane stress or plane strain.Plane stress ± isotropic materialFor plane stress in an isotropic material we have by de®nition," x ˆ xE ÿ yE ‡ " x0" y ˆÿ xE ‡ yE ‡ " y0 xy ˆ 2…1 ‡ † xyE‡ xy0…4:12†Solving the above for the stresses, we obtain the matrix D as23D ˆ E1 0671 ÿ 2 4 1 0 5 …4:13†0 0 …1 ÿ †=2

and the initial strains as8>:" x0" x0 xy09>=>;Element characteristics 91…4:14†in which E is the elastic modulus and is Poisson's ratio.Plane strain ± isotropic materialIn this case a normal stress z exists in addition to the other three stress components.Thus we now have" x ˆ xE ÿ yE ÿ zE ‡ " x0and in additionwhich yields" y ˆÿ xE ‡ yE ÿ zE ‡ " y0 xy ˆ 2…1 ‡ † xyE‡ xy0" z ˆÿ xE ÿ yE ‡ zE ‡ " z0 ˆ 0ÿ z ˆ x ‡ y ÿ E"z0…4:15†On eliminating z and solving for the three remaining stresses we obtain the matrix Das231 ÿ 0E 67D ˆ4 1 ÿ 0 5 …4:16†…1 ‡ †…1 ÿ 2†0 0 …1 ÿ 2†=2and the initial strains8 9>=e 0 ˆ " y0 ‡ " z0>: >; xy0…4:17†Anisotropic materialsFor a completely anisotropic material, 21 independent elastic constants are necessaryto de®ne completely the three-dimensional stress±strain relationship. 4;5If two-dimensional analysis is to be applicable a symmetry of properties must exist,implying at most six independent constants in the D matrix. Thus, it is always possibleto write23d 11 d 12 d 1367D ˆ 4 d 22 d 23 5 …4:18†sym: d 33

92 Plane stress and plane strainyzxPlane of strata parallel to xzFig. 4.2 A strati®ed (transversely isotropic) material.to describe the most general two-dimensional behaviour. (The necessary symmetry ofthe D matrix follows from the general equivalence of the Maxwell±Betti reciprocaltheorem and is a consequence of invariant energy irrespective of the path taken toreach a given strain state.)A case of particular interest in practice is that of a `strati®ed' or transversely isotropicmaterial in which a rotational symmetry of properties exists within the planeof the strata. Such a material possesses only ®ve independent elastic constants.The general stress±strain relations give in this case, following the notation ofLekhnitskii 4 and taking now the y-axis as perpendicular to the strata (neglectinginitial strain) (Fig. 4.2)," x ˆ xÿ 2 yÿ 1 zE 1 E 2 E 1" y ˆÿ 2 xE 2" z ˆÿ 1 xE 1 xz ˆ 2…1 ‡ 1†E xz1 xy ˆ 1G 2 xy yz ˆ 1G 2 yz‡ yE 2ÿ 2 zE 2ÿ 2 yE 2‡ zE 1…4:19†in which the constants E 1 , 1 (G 1 is dependent) are associated with the behaviour inthe plane of the strata and E 2 , G 2 , 2 with a direction normal to the plane.The D matrix in two dimensions now becomes, taking E 1 =E 2 ˆ n and G 2 =E 2 ˆ m,23n n 2 0D ˆE2 671 ÿ n22 4 n 2 1 0 5 …4:20†0 0 m…1 ÿ n2†2

Element characteristics 93for plane stress orE 2D ˆ…1 ‡ 1 †…1 ÿ 1 ÿ 2n2 2†23n…1 ÿ n2† 2 n 2 …1 ‡ 1 † 0 6 n 2 …1 ‡ 1 † …1 ÿ 1† 2 0745 …4:21†0 0 m…1 ‡ 1 †…1 ÿ 1 ÿ 2n2†2for plane strain.When, as in Fig. 4.3, the direction of the strata is inclined to the x-axis then toobtain the D matrices in universal coordinates a transformation is necessary.Taking D 0 as relating the stresses and strains in the inclined coordinate system…x 0 ; y 0 † it is easy to show thatD ˆ TD 0 T T…4:22†where23T ˆ64cos 2 sin 2 ÿ2 sin cos sin 2 cos 2 2 sin cos sin cos ÿ sin cos cos 2 ÿ sin 2 75 …4:23†with as de®ned in Fig. 4.3.If the stress systems r 0 and r correspond to e 0 and e respectively then by equalityof workr 0T e 0 ˆ r T eore 0T D 0 e 0 ˆ e T Deyy'mx'βijxFig. 4.3 An element of a strati®ed (transversely isotropic) material.

94 Plane stress and plane strainfrom which Eq. (4.22) follows on noting (see also Chapter 1)e 0 ˆ T T e…4:24†4.2.4 Initial strain (thermal strain)`Initial' strains, i.e., strains which are independent of stress, may be due to manycauses. Shrinkage, crystal growth, or, most frequently, temperature change will, ingeneral, result in an initial strain vector:e 0 ˆ‰" x0 " y0 " z0 xy0 yz0 zx0 Š T …4:25†Although this initial strain may, in general, depend on the position within theelement, it will here be de®ned by average, constant values to be consistent withthe constant strain conditions imposed by the prescribed displacement function.For an isotropic material in an element subject to a temperature rise e with acoe cient of thermal expansion we will havee 0 ˆ e ‰1 1 1 0 0 0Š T …4:26†as no shear strains are caused by a thermal dilatation. Thus, for plane stress, Eq.(4.14) yields the initial strains given by8 9>< 1 >=e 0 ˆ e 1>: >; ˆ e m …4:27†0In plane strain the z stress perpendicular to the xy plane will develop due to thethermal expansion as shown above. Using Eq. (4.17) the initial thermal strains forthis case are given bye 0 ˆ…1 ‡ † e m…4:28†Anisotropic materials present special problems, since the coe cients of thermalexpansion may vary with direction. In the general case the thermal strains aregiven bye 0 ˆ a e…4:29†where a has properties similar to strain. Accordingly, it is always possible to ®ndorthogonal directions for which a is diagonal. If we let x 0 and y 0 denote the principalthermal directions of the material, the initial strain due to thermal expansion for aplane stress state becomes (assuming z 0 is a principal direction)8>: x 0 y 0 09 8>= >< 1>; ˆ e 29>=>: >;0…4:30†where 1 and 2 are the expansion coe cients referred to the x 0 and y 0 axes,respectively.

Element characteristics 95To obtain strain components in the x; y system it is necessary to use the straintransformatione 0 0 ˆ T T e 0…4:31†where T is again given by Eq. (4.23). Thus, e 0 can be simply evaluated. It will be notedthat the shear component of strain is no longer equal to zero in the x; y coordinates.4.2.5 The stiffness matrixThe sti€ness matrix of the element ijm is de®ned from the general relationship (2.13)with the coe cients…K e ij ˆ B T i DB j t dx dy…4:32†where t is the thickness of the element and the integration is taken over the area of thetriangle. If the thickness of the element is assumed to be constant, an assumption convergentto the truth as the size of elements decreases, then, as neither of the matricescontains x or y we have simplyK e ij ˆ B T i DB j t…4:33†where is the area of the triangle [already de®ned by Eq. (4.5)]. This form is nowsu ciently explicit for computation with the actual matrix operations being left tothe computer.4.2.6 Nodal forces due to initial strainThese are given directly by the expression Eq. (2.13b) which, on performing the integration,becomes…f i † e e 0 ˆÿB T i De 0 t; etc: …4:34†These `initial strain' forces contribute to the nodes of an element in an unequalmanner and require precise evaluation. Similar expressions are derived for initialstress forces.4.2.7 Distributed body forcesIn the general case of plane stress or strain each element of unit area in the xy plane issubject to forces bxb ˆb yin the direction of the appropriate axes.

96 Plane stress and plane strainAgain, by Eq. (2.13b), the contribution of such forces to those at each node is given by… f e bxi ˆÿ N i dx dyb yor by Eq. (4.7), …f e bxi ˆÿ Nb i dx dy; etc: …4:35†yif the body forces b x and b y are constant. As N i is not constant the integration has tobe carried out explicitly. Some general integration formulae for a triangle are given inAppendix D.In this special case the calculation will be simpli®ed if the origin of coordinates istaken at the centroid of the element. Now… …x dx dy ˆ y dx dy ˆ 0and on using Eq. (4.8)bx …f e ai dx dyi ˆÿb y 2by relations noted on page 89.Explicitly, for the whole element8>:f e if e jf e mˆÿ bxb yai2 ˆÿ bxb y 38b x9 b y>= >; ˆÿ b x>:b yb xb y9>=>;3…4:36†…4:37†which means simply that the total forces acting in the x and y directions due to thebody forces are distributed to the nodes in three equal parts. This fact correspondswith physical intuition, and was often assumed implicitly.4.2.8 Body force potentialIn many cases the body forces are de®ned in terms of a body force potential asb x ˆÿ @@xb y ˆÿ @@y…4:38†and this potential, rather than the values of b x and b y , is known throughout the regionand is speci®ed at nodal points. If f e lists the three values of the potential associatedwith the nodes of the element, i.e.,8>: i j m9>=>;…4:39†

Examples ± an assessment of performance 97and has to correspond with constant values of b x and b y , must vary linearly withinthe element. The `shape function' of its variation will obviously be given by a procedureidentical to that used in deriving Eqs (4.4)±(4.6), and yields ˆ‰N i ; N j ; N m Šf e…4:40†Thus,andb x ˆÿ @@x ˆÿ‰b i; b j ; b m Š fe2b y ˆÿ @@y ˆÿ‰c i; c j ; c m Š fe…4:41†2The vector of nodal forces due to the body force potential will now replace Eq. (4.37)by23b i ; b j ; b mc i ; c j ; c mf e ˆ 1b i ; b j ; b m6c i ; c j ; c f e…4:42†m674 b i ; b j ; b m 5c i ; c j ; c m4.2.9 Evaluation of stressesThe derived formulae enable the full sti€ness matrix of the structure to be assembled,and a solution for displacements to be obtained.The stress matrix given in general terms in Eq. (2.16) is obtained by the appropriatesubstitutions for each element.The stresses are, by the basic assumption, constant within the element. It is usual toassign these to the centroid of the element, and in most of the examples in this chapterthis procedure is followed. An alternative consists of obtaining stress values at thenodes by averaging the values in the adjacent elements. Some `weighting' procedureshave been used in this context on an empirical basis but their advantage appearssmall.It is also usual to calculate the principal stresses and their directions for everyelement. In Chapter 14 we shall return to the problem of stress recovery and showthat better procedures of stress recovery exist. 6;74.3 Examples ± an assessment of performanceThere is no doubt that the solution to plane elasticity problems as formulated inSec. 4.2 is, in the limit of subdivision, an exact solution. Indeed at any stage of a

98 Plane stress and plane strain®nite subdivision it is an approximate solution as is, say, a Fourier series solution witha limited number of terms.As explained in Chapter 2, the total strain energy obtained during any stage ofapproximation will be below the true strain energy of the exact solution. In practiceit will mean that the displacements, and hence also the stresses, will be underestimatedby the approximation in its general picture. However, it must be emphasized that thisis not necessarily true at every point of the continuum individually; hence the value ofsuch a bound in practice is not great.What is important for the engineer to know is the order of accuracy achievable intypical problems with a certain ®neness of element subdivision. In any particular casethe error can be assessed by comparison with known, exact, solutions or by a study ofthe convergence, using two or more stages of subdivision.With the development of experience the engineer can assess a priori the order ofapproximation that will be involved in a speci®c problem tackled with a given elementsubdivision. Some of this experience will perhaps be conveyed by the examplesconsidered in this book.In the ®rst place attention will be focused on some simple problems for which exactsolutions are available.4.3.1 Uniform stress ®eldIf the exact solution is in fact that of a uniform stress ®eld then, whatever the elementsubdivision, the ®nite element solution will coincide exactly with the exact one. This isan obvious corollary of the formulation; nevertheless it is useful as a ®rst check ofwritten computer programs.4.3.2 Linearly varying stress ®eldHere, obviously, the basic assumption of constant stress within each element meansthat the solution will be approximate only. In Fig. 4.4 a simple example of a beamsubject to constant bending moment is shown with a fairly coarse subdivision. It isreadily seen that the axial … y † stress given by the element `straddles' the exactvalues and, in fact, if the constant stress values are associated with centroids of theelements and plotted, the best `®t' line represents the exact stresses.The horizontal and shear stress components di€er again from the exact values(which are simply zero). Again, however, it will be noted that they oscillate byequal, small amounts around the exact values.At internal nodes, if the average of the stresses of surrounding elements is taken itwill be found that the exact stresses are very closely represented. The average atexternal faces is not, however, so good. The overall improvement in representingthe stresses by nodal averages, as shown in Fig. 4.4, is often used in practice forcontour plots. However, we shall show in Chapter 14 a method of recovery whichgives much improved values at both interior and boundary points.

Examples ± an assessment of performance 99y–0.440.070.07–0.44–0.07–0.07x0.00–0.070.070.00–0.07–0.06–0.86–0.07–0.07–0.86–0.07–0.08–0.450.07–0.08–0.48–0.08–0.06–0.45–0.06–0.06–0.48–0.060.080.00–0.09–0.09–0.830.04–0.04lNodal averagesValue atcentres oftrianglesσ y = –1Exact(σ x = τ xy = 0)ν = 0.15Fig. 4.4 Pure bending of a beam solved by a coarse subdivision into elements of triangular shape. (Values of y , x , and xy listed in that order.)4.3.3 Stress concentrationA more realistic test problem is shown in Figs 4.5 and 4.6. Here the ¯ow of stressaround a circular hole in an isotropic and in an anisotropic strati®ed material is consideredwhen the stress conditions are uniform. 8 A graded division into elements isused to allow a more detailed study in the region where high stress gradients areexpected. The accuracy achievable can be assessed from Fig. 4.6 where some of theresults are compared against exact solutions. 3;9In later chapters we shall see that even more accurate answers can be obtained withthe use of more elaborate elements; however, the principles of the analysis remainidentical.

100 Plane stress and plane strainyσ x = –1xFig. 4.5 A circular hole in a uniform stress ®eld: (a) isotropic material; (b) strati®ed (orthotropic) material;E x ˆ E 1 ˆ 1, E y ˆ E 2 ˆ 3, 1 ˆ 0:1, 2 ˆ 0, G xy ˆ 0:42.Exact solution for infinite plateFinite element solution–1.0 –1.0σ xσ x–3.0 –2.831.0σ y1.75σ y(a) Isotropic(b) OrthotropicFig. 4.6 Comparison of theoretical and ®nite element results for cases (a) and (b) of Fig. 4.5.4.4 Some practical applicationsObviously, the practical applications of the method are limitless, and the ®niteelement method has superseded experimental technique for plane problems becauseof its high accuracy, low cost, and versatility. The ease of treatment of materialanisotropy, thermal stresses, or body force problems add to its advantages.A few examples of actual early applications of the ®nite element method to complexproblems of engineering practice will now be given.

y521004910047CBA48100xRestrained in y direction from movementFig. 4.7 A reinforced opening in a plate. Uniform stress ®eld at a distance from opening x ˆ 100, y ˆ 50. Thickness of plate regions A, B, and C is in the ratio of1 : 3 : 23.

102 Plane stress and plane strain4.4.1 Stress ¯ow around a reinforced opening (Fig. 4.7)In steel pressure vessels or aircraft structures, openings have to be introduced in thestressed skin. The penetrating duct itself provides some reinforcement round the edgeand, in addition, the skin itself is increased in thickness to reduce the stresses due toconcentration e€ects.Analysis of such problems treated as cases of plane stress present no di culties.The elements are chosen so as to follow the thickness variation, and appropriatevalues of this are assigned.The narrow band of thick material near the edge can be represented either byspecial bar-type elements, or by very thin triangular elements of the usual type, towhich appropriate thickness is assigned. The latter procedure was used in the problemshown in Fig. 4.7 which gives some of the resulting stresses near the opening itself.The fairly large extent of the region introduced in the analysis and the grading ofthe mesh should be noted.4.4.2 An anisotropic valley subject to techtonic stress 8 (Fig. 4.8)A symmetrical valley subject to a uniform horizontal stress is considered. Thematerial is strati®ed, and hence is `transversely isotropic', and the direction ofstrata varies from point to point.The stress plot shows the tensile region that develops. This phenomenon is ofconsiderable interest to geologists and engineers concerned with rock mechanics.(See reference 10 for additional applications on this topic.)4.4.3 A dam subject to external and internal water pressures 11;12A buttress dam on a somewhat complex rock foundation is shown in Fig. 4.9 andanalysed. This dam (completed in 1964) is of particular interest as it is the ®rst towhich the ®nite element method was applied during the design stage. The heterogeneousfoundation region is subject to plane strain conditions while the dam itselfis considered in a state of plane stress of variable thickness.With external and gravity loading no special problems of analysis arise.When pore pressures are considered, the situation, however, requires perhaps someexplanation.It is well known that in a porous material the water pressure is transmitted to thestructure as a body force of magnitudeb x ˆÿ @p@xb y ˆÿ @p@y…4:43†and that now the external pressure need not be considered.The pore pressure p is, in fact, now a body force potential, as de®ned in Eq. (4.38).Figure 4.9 shows the element subdivision of the region and the outline of the dam.Figure 4.10(a) and(b) shows the stresses resulting from gravity (applied to the damonly) and due to water pressure assumed to be acting as an external load or, alternatively,

Some practical applications 103012–0.66–7.493–4.50–0.27–0.55–2.03–1.89–0.88–0.56–0.89–0.49–0.99–2.61–0.38 –1.86–0.51–0.27–0.37–0.71–1.58–0.02 –0.03–2.08 –1.44–0.13–2.20–0.36–0.32–1.15–0.40–1.06–0.260.02–0.12–0.01–1.95–0.270.16–0.040.090.08–1.78–0.14–0.27–0.73–0.970.020.04 –1.100.05–1.24–0.161 2 3–0.670.02 –0.83–0.23–1.350.20–0.020.21–0.020.07–0.01 –0.130.00 0.03–0.02–0.97 –0.20–0.01 0.04–0.010.14–0.300.01–0.460.02–0.18–0.01–1.090.01–1.25–0.04–1.15–0.12–0.11–0.75–0.530.08–0.340.04–0.470.040.04 –0.810.08 –0.74–1.03–0.01–0.960.07–1.05–0.05–0.540.05–0.710.04–0.920.08–0.960.12–0.970.04–0.98–0.05–0.20–0.01–0.280.04–0.360.11–0.660.110.14–0.540.11–0.860.13–0.290.06–0.410.10–0.780.17–0.870.06–0.620.11–0.670.20–0.510.08–0.910.04–0.32–0.03–0.410.06–0.860.03–0.760.04–0.970.00–0.510.06Tensile zone–0.630.05–0.440.0310 tonne/m 2–0.860.06–0.790.08–0.720.08E 2 = 1 tonne/m 20 1 2 3 4 5 6 7 80x = 2(1 – y 2 )1y2345E 1 = 10E 2 = 1σ y = –1678xu = 0 Region analysedFig. 4.8 A valley with curved strata subject to a horizontal tectonic stress (plane strain 170 nodes, 298 elements).as an internal pore pressure. Both solutions indicate large tensile regions, but the increaseof stresses due to the second assumption is important.The stresses calculated here are the so-called `e€ective' stresses. These represent theforces transmitted between the solid particles and are de®ned in terms of the totalstresses r and the pore pressures p byr 0 ˆ r ‡ mp m T ˆ‰1; 1; 0Š …4:44†i.e., simply by removing the hydrostatic pressure component from the total stress. 10;13The e€ective stress is of particular importance in the mechanics of porous mediasuch as those that occur in the study of soils, rocks, or concrete. The basic assumption

104 Plane stress and plane strainButtress webthickness 9 ft207 ft 0 in60°125 ft 0 inConstant web taper1 ft in 82 ft 6 in60°A10 ft121 ft 11 in 121 ft 11 inA10 ft 6 in47 ft 0 in 12 ft 0 in(a)Sectional plan AADam E CFault(no restraintassumed)1Altered grit E =10E C1Mudstone E =2E CGrit E G = 1 E 4 CZero displacementsassumed(b)Zero displacementsassumedFig. 4.9 Stress analysis of a buttress dam. A plane stress condition is assumed in the dam and plane strain inthe foundation. (a) The buttress section analysed. (b) Extent of foundation considered and division into ®niteelements.in deriving the body forces of Eq. (4.43) is that only the e€ective stress is of anyimportance in deforming the solid phase. This leads immediately to anotherpossibility of formulation. 14 If we examine the equilibrium conditions of Eq. (2.10)we note that this is written in terms of total stresses. Writing the constitutive relation,Eq. (2.5), in terms of e€ective stresses, i.e.,r 0 ˆ D 0 …e ÿ e 0 †‡r 0 0…4:45†and substituting into the equilibrium equation (2.10) we ®nd that Eq. (2.12) is againobtained, with the sti€ness matrix using the matrix D 0 and the force terms of

–300+200Arrow indicatestension–92200 lb/in 2Tensile zoneCompression (–)+37–84300 lb/in 2–99–178+17–70 +136+17 –85+8–4–65–47–6–16–11–18 –8 –22–37–15–84–29 –39 –29–202–198–31 –41 –18–5–60 –66–10–199–273 –256 –283 –260–39–108–71–328 –309–142–29–8 –9–375–95–355–7–2–5–337–8–12–14–249–16–327–7–84–9–8–289–16–317–11–2–289 –289–11 –1–288–18–293–19–392–13–5–84–13 –54–8–77Tensile zone–4 –22–9 –6–33 –15–66–3–42–8–43–17 –282–11–4–205–14 –34 –25 –18 –15 –8–48 –52–2–188 –253 –240 –238 –246–170 –304–46 –14–288 –285 –290 –202–18200 lb/in 2–2–14–27 –78–8 –2–9–345–310 –347–329 -320 –315 –310+16+23+12–30+20 +4 –6 –13 –21 –17 –7–10–211–473–13–333 –332 –296+144+59 +26 +5 +4+10–118–7 –14 –23 –25 –54 –39 –7–265–358 –313 –332–303–9–15 –29–59–100–80 –97–17 +3 –17 +3–1 –8 –14 –11–78 –79–70 –70–28+38–38+31+20–59–51+8–46+22–54+17–67+9–71 –75–8 –16–61+8–62–2–69+3–87–4–300+200–710Compression (–)–47 –1–42+3300 lb/in 2Arrow indicatestensionBelow the foundation initial rock stresses should be superimposed(a)(b)Fig. 4.10 Stress analysis of the buttress dam of Fig. 4.9. Principal stresses for gravity loads are combined with water pressures, which are assumed to act (a) as externalloads, (b) as body forces due to pore pressure.

106 Plane stress and plane strainEq. (2.13b) being augmented by an additional force…ÿ B T mp d…vol†…4:46†V eor, if p is interpolated by shape functions Ni, 0 the force becomes…ÿ B T mN 0 d…vol†p e…4:47†V eThis alternative form of introducing pore pressure e€ects allows a discontinuousinterpolation of p to be used [as in Eq. (4.46) no derivatives occur] and this is nowfrequently used in practice.4.4.4 CrackingThe tensile stresses in the previous example will doubtless cause the rock to crack. If astable situation can develop when such a crack spreads then the dam can beconsidered safe.Cracks can be introduced very simply into the analysis by assigning zero elasticityvalues to chosen elements. An analysis with a wide cracked wedge is shown inFig. 4.11, where it can be seen that with the extent of the crack assumed no tensionwithin the dam body develops.Compression ( – )UpstreamTensile zoneTension ( + )200 lb/in 2400 lb/in 2Cracked zone(materialconsidered)as E = 0Arrow indicatestensionStresses exclude theoriginal rock stressesin foundationTensile zone buttension less thanprobable initialcompression in rockFig. 4.11 Stresses in a buttress dam. The introduction of a `crack' modi®es the stress distribution [same loadingas Fig. 4.10(b)].

Some practical applications 107Temperature in this region –15°FTemperature dropslinearly hereTension ( + )+200–100Compression ( – )200 lb/in 2Arrow indicatestension100 lb/in 2No temperature charge in foundationFig. 4.12 Stress analysis of a buttress dam. Thermal stresses due to cooling of the shaded area by 15 8F…E ˆ 3 10 6 lb/in 2 , ˆ 6 10 ÿ6 =8F†.Water level–8–2512–5 –311.3 x 10 6 lb–94–9–147–8–15 –323–29105 –520–374–407 5531–144054–70Load effectof prestressing–57 –62 2–8 –42–49133–51–80 4 x 10 6 lb–59 –3217–21–13 795 5 –37–359–2 103 7–98155'–53–1 –6–29 1–62187 5 –42–49 –5–52–5–2 6–19 –6 –6 –59 –8–13–6–56 –71–124–85–13–85–12–3–98–10737–48 –40–11–75–10–2 79 10 –2741–135–61–108–80 10 20 30 40 ft –183–1051416 012–7527–8728 121–54 –79112–156 0–113–14934 14 –26–5361 –1 674852–92–1–2–10352 –6 49 8–49–42–8954 –33–119–18–117–432–7–116–22–101–111–130–88–26–89–21–2535 15–1–10–16–20–21Prestressing cablesand gate reactions2–12–19860'25' 25'10'Tailwater level0 20 40 60 80 100 ftFig. 4.13 A large barrage with piers and prestressing cables.

Fig. 4.14 An underground power station. Mesh used in analysis.

yx250000 lb/ft 2Indicates tensionFig. 4.15 An underground power station. Plot of principal stresses.

110 Plane stress and plane strainA more elaborate procedure for allowing crack propagation and resulting stressredistribution can be developed (see Volume 2).4.4.5 Thermal stressesAs an example of thermal stress computation the same dam is shown under simpletemperature distribution assumptions. Results of this analysis are given in Fig. 4.12.4.4.6 Gravity damsA buttress dam is a natural example for the application of ®nite element methods.Other types, such as gravity dams with or without piers and so on, can also besimply treated. Figure 4.13 shows an analysis of a large dam with piers and crest gates.In this case the approximation of assuming a two-dimensional treatment in thevicinity of the abrupt change of section, i.e., where the piers join the main body ofthe dam, is clearly involved, but this leads to localized errors only.It is important to note here how, in a single solution, the grading of element size isused to study concentration of stress at the cable anchorages, the general stress ¯ow inthe dam, and the foundation behaviour. The linear ratio of size of largest to smallestelements is of the order of 30 to 1 (the largest elements occurring in the foundation arenot shown in the ®gure).4.4.7 Underground power stationThis last example, illustrated in Figs 4.14 and 4.15, shows an interesting application.Here principal stresses are plotted automatically. In this analysis many di€erentcomponents of r 0 , the initial stress, were used due to uncertainty of knowledgeabout geological conditions. The rapid solution and plot of many results enabledthe limits within which stresses vary to be found and an engineering decision arrivedat. In this example, the exterior boundaries were taken far enough and `®xed'…u ˆ v ˆ 0†. However, a better treatment could be made using in®nite elements asdescribed in Sec. 9.13.4.5 Special treatment of plane strain with anincompressible materialIt will have been noted that the relationship (4.16) de®ning the elasticity matrix D foran isotropic material breaks down when Poisson's ratio reaches a value of 0.5 as thefactor in parentheses becomes in®nite. A simple way of side-stepping this di culty isto use values of Poisson's ratio approximating to 0.5 but not equal to it. Experienceshows, however, that if this is done the solution deteriorates unless special formulationssuch as those discussed in Chapter 12 are used.

References 1114.6 Concluding remarkIn subsequent chapters, we shall introduce elements which give much greateraccuracy for the same number of degrees of freedom in a particular problem. Thishas led to the belief that the simple triangle used here is completely superseded.In recent years, however, its very simplicity has led to its revival in practical usein combination with the error estimation and adaptive procedures discussed inChapters 14 and 15.References1. M.J. Turner, R.W. Clough, H.C. Martin, and L.J. Topp. Sti€ness and de¯ection analysisof complex structures. J. Aero. Sci., 23, 805±23, 1956.2. R.W. Clough. The ®nite element in plane stress analysis. Proc. 2nd ASCE Conf. on ElectronicComputation. Pittsburgh, Pa., Sept. 1960.3. S. Timoshenko and J.N. Goodier. Theory of Elasticity. 2nd ed., McGraw-Hill, 1951.4. S.G. Lekhnitskii. Theory of Elasticity of an Anisotropic Elastic Body (Translation fromRussian by P. Fern). Holden Day, San Francisco, 1963.5. R.F.S. Hearmon. An Introduction to Applied Anisotropic Elasticity. Oxford UniversityPress, 1961.6. O.C. Zienkiewicz and J.Z. Zhu. The superconvergent patch recovery (SPR) and adaptive®nite element re®nement. Comp. Methods Appl. Mech. Eng., 101, 207±24, 1992.7. B. Boroomand and O.C. Zienkiewicz. Recovery by equilibrium patches (REP). Internat.J. Num. Meth. Eng., 40, 137±54, 1997.8. O.C. Zienkiewicz, Y.K. Cheung, and K.G. Stagg. Stresses in anisotropic media withparticular reference to problems of rock mechanics. J. Strain Analysis, 1, 172±82, 1966.9. G.N. Savin. Stress Concentration Around Holes (Translation from Russian). PergamonPress, 1961.10. O.C. Zienkiewicz, A.H.C. Chan, M. Pastor, B. Schre¯er, and T. Shiomi. ComputationalGeomechanics. John Wiley and Sons, Chichester, 1999.11. O.C. Zienkiewicz and Y.K. Cheung. Buttress dams on complex rock foundations. WaterPower, 16, 193, 1964.12. O.C. Zienkiewicz and Y.K. Cheung. Stresses in buttress dams. Water Power, 17, 69, 1965.13. K. Terzhagi. Theoretical Soil Mechanics. Wiley, 1943.14. O.C. Zienkiewicz, C. Humpheson, and R.W. Lewis. A uni®ed approach to soil mechanicsproblems, including plasticity and visco-plasticity. Int. Symp. on Numerical Methods in Soiland Rock Mechanics. Karlsruhe, 1975. See also Chapter 4 of Finite Elements in Geomechanics(ed. G. Gudehus), pp. 151±78, Wiley, 1977.

5Axisymmetric stress analysis5.1 IntroductionThe problem of stress distribution in bodies of revolution (axisymmetric solids) underaxisymmetric loading is of considerable practical interest. The mathematical problemspresented are very similar to those of plane stress and plane strain as, onceagain, the situation is two dimensional. 1;2 By symmetry, the two components ofdisplacements in any plane section of the body along its axis of symmetry de®necompletely the state of strain and, therefore, the state of stress. Such a cross-sectionis shown in Fig. 5.1. If r and z denote respectively the radial and axial coordinates of apoint, with u and v being the corresponding displacements, it can readily be seen thatprecisely the same displacement functions as those used in Chapter 4 can be used tode®ne the displacements within the triangular element i; j; m shown.The volume of material associated with an `element' is now that of a body ofrevolution indicated in Fig. 5.1, and all integrations have to be referred to this.The triangular element is again used mainly for illustrative purposes, the principlesdeveloped being completely general.In plane stress or strain problems it was shown that internal work was associatedwith three strain components in the coordinate plane, the stress component normal tothis plane not being involved due to zero values of either the stress or the strain.In the axisymmetrical situation any radial displacement automatically induces a strainin the circumferential direction, and as the stresses in this direction are certainly non-zero,this fourth component of strain and of the associated stress has to be considered. Here liesthe essential di€erence in the treatment of the axisymmetric situation.The reader will ®nd the algebra involved in this chapter somewhat more tediousthan that in the previous one but, essentially, identical operations are once againinvolved, following the general formulation of Chapter 2.5.2 Element characteristics5.2.1 Displacement functionUsing the triangular shape of element (Fig. 5.1) with the nodes i; j; m numbered in the

Element characteristics 113z (v)mijr (u)Fig. 5.1 Element of an axisymmetric solid.anticlockwise sense, we de®ne the nodal displacement by its two components as uia i ˆ…5:1†v iand the element displacements by the vector8 9>< a i >=a e ˆ a j…5:2†>: >;Obviously, as in Sec. 4.2.1, a linear polynomial can be used to de®ne uniquely thedisplacements within the element. As the algebra involved is identical to that ofChapter 4 it will not be repeated here. The displacement ®eld is now given again byEq. (4.7): uu ˆ ˆ‰INv i ; IN j ; IN m Ša e…5:3†withN i ˆ ai ‡ b i r ‡ c i z; etc:2and I a two-by-two identity matrix. In the abovea ma i ˆ r j z m ÿ r m z jb i ˆ z j ÿ z m…5:4†c i ˆ r m ÿ r jetc., in cyclic order. Once again is the area of the element triangle.

114 Axisymmetric stress analysis5.2.2 Strain (total)As already mentioned, four components of strain have now to be considered. Theseare, in fact, all the non-zero strain components possible in an axisymmetric deformation.Figure 5.2 illustrates and de®nes these strains and the associated stresses.The strain vector de®ned below lists the strain components involved and de®nesthem in terms of the displacements of a point. The expressions involved are almostself-evident and will not be derived here. The interested reader can consult a standardelasticity textbook 3 for the full derivation. We thus have8>:" r" z" rz8 9@u9 @r@v>= >< >=ˆ @z ˆ Suu>; r@u >:@z ‡ @v >;@rUsing the displacement functions de®ned by Eqs (5.3) and (5.4) we havee ˆ Ba e ˆ‰B i ; B j ; B m Ša ein which2B i ˆ64@N i@r ; 00;@N i@zN ir ; 0@N i@z ;@N i@r32ˆ6475…5:5†b i ;300; c i;75etc: …5:6†c i ;a ir ‡ b i ‡ c izr ; 0b izγ rz (τ rz )ε r (σ r )θrε θ (σ θ )ε z (σ z )Fig. 5.2 Strains and stresses involved in the analysis of axisymmetric solids.

Element characteristics 115With the B matrix now involving the coordinates r and z, the strains are no longerconstant within an element as in the plane stress or strain case. This strain variation isdue to the " term. If the imposed nodal displacements are such that u is proportionalto r then indeed the strains will all be constant. In addition, constant " z and rz strainsmay be deduced from a linear v displacement. This is the only state of displacementcoincident with a constant strain condition and it is clear that the displacementfunction satis®es the basic criterion of Chapter 2.5.2.3 Initial strain (thermal strain)In general, four independent components of the initial strain vector can beenvisaged:8 9>:" r0" z0" 0 rz0>=>;…5:7†Although this can, in general, be variable within the element, it will be convenient totake the initial strain as constant there.The most frequently encountered case of initial strain will be that due to thermalexpansion. For an isotropic material we shall have then8 91>=ˆ e m…5:8†1>: >;0where e is the average temperature rise in an element and is the coe cient ofthermal expansion.The general case of anisotropy need not be considered since axial symmetry wouldbe impossible to achieve under such circumstances. A case of some interest in practiceis that of a `strati®ed' material, similar to the one discussed in Chapter 4, in which theplane of isotropy is normal to the axis of symmetry (Fig. 5.3). Here, two di€erentexpansion coe cients are possible: one in the axial direction z and another in theplane normal to it, r .Now the initial thermal strain becomes8 9 r>:0>=>;…5:9†Practical cases of such `strati®ed' anisotropy often arise in laminated or ®breglassconstruction of machine components.

116 Axisymmetric stress analysiszrFig. 5.3 Axisymmetrically strati®ed material.5.2.4 Elasticity matrixThe elasticity matrix D linking the strains e and the stresses r in the standard form[Eq. (2.5)],8 9>: r z rz>=ˆ D…e ÿ e 0 †‡r 0>;needs now to be derived.The anisotropic `strati®ed' material will be considered ®rst, as the isotropic case canbe simply presented as a special form.Anisotropic, strati®ed, material (Fig. 5.3)With the z-axis representing the normal to the planes of strati®cation we can rewriteEqs (4.19) (again ignoring the initial strains and stresses for convenience) as" r ˆ rÿ 2 zÿ 1 E 1 E 2 E 1" ˆÿ 1 rE 1ÿ 2 zE 2‡ zE 1" z ˆÿ 2 rE 2 rz ˆ rzG 2‡ zE 2ÿ 2 E 2…5:10†

Element characteristics 117Writing againE 1E 2ˆ n;G 2D 2ˆ m and d ˆ…1 ‡ 1 †…1 ÿ 1 ÿ 2n 2 2†we have on solving for the stresses, that2n…1 ÿ n 2 2†; n 2 …1 ‡ 1 †; n… 1 ‡ n 2 32†; 0D ˆ E21 ÿ 2 1; n 2 …1 ‡ †; 06dn…1 ÿ n 2 742†; 0 5sym: ; mdIsotropic materialFor an isotropic material we can obtain the D matrix by takingE 1 ˆ E 2 ˆ E or n ˆ 1and 1 ˆ 2 ˆ and using the well-known relationship between isotropic elastic constantsG 2ˆ GE 2 E ˆ m ˆ 12…1 ‡ †Substituting in Eq. (5.11) we now have231 ÿ ; ; ; 0E; 1 ÿ ; ; 0D ˆ67…1 ‡ †…1 ÿ 2† 4 ; ; 1 ÿ ; 0 50; 0; 0; …1 ÿ 2†=2…5:11†…5:12†5.2.5 The stiffness matrixThe sti€ness matrix of the element ijm can now be computed according to the generalrelationship (2.13). Remembering that the volume integral has to be taken over thewhole ring of material we have…K e ij ˆ 2 B T i DB j r dr dz…5:13†with B given by Eq. (5.6) and D by either Eq. (5.11) or Eq. (5.12), depending on thematerial.The integration cannot now be performed as simply as was the case in the planestress problem because the B matrix depends on the coordinates. Two possibilitiesexist: the ®rst is that of numerical integration and the second of an explicit multiplicationand term-by-term integration.The simplest numerical integration procedure is to evaluate all quantities for a centroidalpointr ˆ ri ‡ r j ‡ r m3andz ˆ zi ‡ z j ‡ z m3

118 Axisymmetric stress analysisIn this case we have simply as a ®rst approximationK e ij ˆ 2B T i DB j r…5:14†with being the triangle area and B the value of the strain-displacement matrix at thecentroidal point.More elaborate numerical integration schemes could be used by evaluating the integrandat several points of the triangle. Such methods will be discussed in detail inChapter 9. However, it can be shown that if the numerical integration is of such anorder that the volume of the element is exactly determined by it, then in the limitof subdivision, the solution will converge to the exact answer. 4 The `one point' integrationsuggested here is of this type, as it is well known that the volume of a body ofrevolution is given exactly by the product of the area and the path swept around by itscentroid. With the simple triangular element used here a fairly ®ne subdivision is inany case needed for accuracy and most practical programs use the simple approximationwhich, surprisingly perhaps, is in fact usually superior to exact integration (seeChapter 10). One reason for this is the occurrence of logarithmic terms in the exactformulation. These involve ratios of the type r i =r m and, when the element is at alarge distance from the axis, such terms tend to unity and evaluation of the logarithmis inaccurate.5.2.6 External nodal forcesIn the case of the two-dimensional problems of the previous chapter the question ofassigning of the external loads was so obvious as not to need further comment. In thepresent case, however, it is important to realize that the nodal forces represent a combinede€ect of the force acting along the whole circumference of the circle forming theelement `node'. This point was already brought out in the integration of the expressionsfor the sti€ness of an element, such integrations being conducted over thewhole ring.Thus, if R represents the radial component of force per unit length of the circumferenceof a node at a radius r, the external `force' which will have to be introduced inthe computation is2r RIn the axial direction we shall, similarly, have2r Zto represent the combined e€ect of axial forces.5.2.7 Nodal forces due to initial strainAgain, by Eq. (2.13),…f e ˆÿ2 B T De 0 r dr dz…5:15†

Element characteristics 119or noting that e 0 is constant, …f e i ˆÿ2B T i r dr dz De 0…5:16†The integration should be performed in a similar manner to that used in the determinationof the sti€ness.It will readily be seen that, again, an approximate expression using a centroidalvalue isf e i ˆÿ2B T i De 0 rInitial stress forces are treated in an identical manner.…5:17†5.2.8 Distributedbody forcesDistributed body forces, such as those due to gravity (if acting along the z-axis), centrifugalforce in rotating machine parts, or pore pressure, often occur in axisymmetricproblems.Let such forces be denoted by brb ˆ…5:18†b zper unit volume of material in the directions of r and z respectively. By the generalequation (2.13) we have… f e bri ˆÿ2 IN i r dr dz…5:19†b zUsing a coordinate shift similar to that of Sec. 4.2.7 it is easy to show that the ®rstapproximation, if the body forces are constant, results inbr rf e i ˆÿ2…5:20†b z 3Although this is not exact the error term will be found to decrease with reduction ofelement size and, as it is also self-balancing, it will not introduce inaccuracies. Indeed,as will be shown in Chapter 10, the convergence rate is maintained.If the body forces are given by a potential similar to that de®ned in Sec. 4.2.8, i.e.,b r ˆÿ @@rb z ˆÿ @@z…5:21†and if this potential is de®ned linearly by its nodal values, an expression equivalent toEq. (4.42) can again be determined.In many problems the body forces vary proportionately to r. For example in rotatingmachinery we have centrifugal forcesb r ˆ ! 2 rwhere ! is the angular velocity and the density of the material.…5:22†

120 Axisymmetric stress analysis2.01.0Exact(a)(b)(c)σ θ0σ rAxis of revolution–1.0Stresses on section AA–0.13–0.37–0.802.042.202.33UnitpressureA2.362.012.17–0.77–0.40A–0.11–0.162.332.242.08 1.932.21–0.70–0.852.26–0.46–0.34–0.76 –0.56–0.32 –0.01–0.011.932.06 2.092.49–0.772.35 2.16 2.01–0.40 –0.12(a) (b) (c)Fig. 5.4 Stresses in a sphere subject to an internal pressure (Poisson's ratio ˆ 0:3:(a) triangular mesh ± centroidalvalues; (b) triangular mesh ± nodal averages; (c) quadrilateral mesh obtained by averaging adjacenttriangles.

Some illustrative examples 1215.2.9 Evaluation of stressesThe stresses now vary throughout the element, as will be appreciated from Eqs (4.5)and (4.6). It is convenient now to evaluate the average stress at the centroid of theelement. The stress matrix resulting from Eqs (5.6) and (2.3) gives there, as usual,r e ˆ DBa e ÿ De 0 ‡ r 0…5:23†It will be found that a certain amount of oscillation of stress values betweenelements occurs and better approximation can be achieved by averaging nodalstresses or recovery procedures of Chapter 14.5.3 Some illustrative examplesTest problems such as those of a cylinder under constant axial or radial stress give, asindeed would be expected, solutions which correspond to exact ones. This is again anobvious corollary of the ability of the displacement function to reproduce constantstrain conditions.A problem for which an exact solution is available and in which almost linear stressgradients occur is that of a sphere subject to internal pressure. Figure 5.4(a) shows thecentroidal stresses obtained using rather a coarse mesh, and the stress oscillationaround the exact values should be noted. (This oscillation becomes even more pronouncedat larger values of Poisson's ratio although the exact solution is independentof it.) In Fig. 5.4(b) the very much better approximation obtained by averaging thestresses at nodal points is shown, and in Fig. 5.4(c) a further improvement is givenby element averaging. The close agreement with the exact solution even for thevery coarse subdivision used here shows the accuracy achievable. The displacementsat nodes compared with the exact solution are given in Fig. 5.5.Computed valueExact value5.275.19Axis of revolutionComputed valueExact value6.306.34Fig. 5.5 Displacements of internal and external surfaces of sphere under loading of Fig. 5.4.

122 Axisymmetric stress analysis100 °ΧT (°C)400200σ θ00σ r–200σ r–50σ t100 °C 0 °C–32 –51–23–400–1006–600340σ t(a)σ r–422ExactTriangular averageQuadrilateral average(b)Fig. 5.6 Sphere subject to steady-state heat ¯ow (100 8C internal temperature, 0 8C external temperature):(a) temperature and stress variation on radial section; (b) `quadrilateral' averages.Axis of revolutionInternalpressure p2pStress scale(arrow fortension)(b)(a)qFig. 5.7 A reactor pressure vessel. (a) `Quadrilateral' mesh used in analysis; this was generated automaticallyby a computer. (b) Stresses due to a uniform internal pressure (automatic computer plot). Solution based onquadrilateral averages. (Poisson's ratio ˆ 0:15).

Early practical applications 123In Fig. 5.6 thermal stresses in the same sphere are computed for the steady-statetemperature variation shown. Again, excellent accuracy is demonstrated by comparisonwith the exact solution.5.4 Early practical applicationsTwo examples of practical applications of the programs available for axisymmetricalstress distribution are given here.5.4.1 A prestressedconcrete reactor pressure vesselFigure 5.7 shows the stress distribution in a relatively simple prototype pressurevessel. Due to symmetry only one-half of the vessel is analysed, the results givenhere referring to the components of stress due to internal pressure.In Fig. 5.8 contours of equal major principal stresses caused by temperature areshown. The thermal state is due to steady-state heat conduction and was itselffound by the ®nite element method in a way described in Chapter 7.Axis ofsymmetry(a)20001000500500(Zero contourcoincides withboundary)00–50010002000300010001000200030001000200030002000C L3000Fig. 5.8 A reactor pressure vessel. Thermal stresses due to steady-state heat conduction. Contours of majorprincipal stress in pounds per square inch. (Interior temperature 400 8C, exterior temperature 0 8C, ˆ 5 10 ÿ6 =8C. E ˆ 2:58 10 6 lb=in 2 ;ˆ 0:15†.

124 Axisymmetric stress analysis5.4.2 Foundation pileFigure 5.9 shows the stress distribution around a foundation pile penetrating twodi€erent strata. This non-homogeneous problem presents no di culties and is treatedby the standard approach given in this chapter in which the `quadrilateral' elementsshown are assemblies of two triangles and the results are averaged.5.5 Non-symmetrical loadingThe method described in the present chapter can be extended to deal with nonsymmetricalloading. If the circumferential loading variation is expressed in circularharmonics then it is still possible to focus attention on one axial section although thenodal degrees of freedom are now increased to three.Details of this process are described in references 5 and 6 and in Chapter 9 ofVolume 2.5.6 Axisymmetry ± plane strain and plane stressIn the previous chapter we noted that plane stress and strain analysis was done interms of three stress and strain components and, indeed, both cases could be generallyincorporated in a single program with an indicator changing appropriate constants inthe matrix D. Doing this loses track of the z component in the plane strain casewhich has to be separately evaluated. Further, special expressions [viz. Eq. 4.28]had to be used to introduce initial strains. This is inconvenient (especially whennon-linear constitutive laws are used), and an alternative of writing the plane straincase in terms of four stress±strain components as a special case of axisymmetric analysisis highly recommended.If the axisymmetric strain de®nition of Eq. (5.5) is examined, we note that r ˆ1gives " 0 and plane strain conditions are obtained. Thus, if we ignore the termsin B associated with " , replace the coordinatesr and z by x and yand further change the volume of integration2r to 1the plane strain formulation becomes available from the axisymmetric plane straindirectly.Plane stress conditions can similarly be incorporated, requiring in additionsubstitution of the axisymmetric D matrix by Eqs (4.13) or (4.19) augmented byan appropriate zero row and column. Thus, at the cost of additional storage ofthe fourth stress and strain component, all the cases discussed can be incorporatedin a single format.

Axisymmetry ± plane strain and plane stress 125p = 162000π lbE 1 = 1ν 1 = 0.3590 ftE pile = 600ν pile = 0.25100 ftRadius of pile = 1.5 ftE 2 = 40ν 2 = 0.30Bottom of pile0 10 20 30 40 50 60Distance scale (ft)Finite element subdivisionC LpExact solution for Boussinesq problemFinite element solution forBossinesq problemFinite element solution for pile problem0 5 10 15 20 25Distance scale (ft)E 138000 42940 52200200 300 4000 100Stress scale (lb/ft 2 )E 2Vertical stresses onhorizontal sectionsFig. 5.9 (a) A pile in strati®ed soil. Irregular mesh and data for the problem. (b) A pile in strati®ed soil. Plot ofvertical stresses on horizontal sections. Solution also plotted for Boussinesq problem obtained by makingE 1 ˆ E 2 ˆ E pile , and this is compared with exact values.

126 Axisymmetric stress analysisReferences1. R.W. Clough. Chapter 7 of Stress Analysis (eds O.C. Zienkiewicz and G.S. Holister), Wiley,1965.2. R.W. Clough and Y.R. Rashid. Finite element analysis of axi-symmetric solids. Proc.ASCE, 91, EM.1, 71, 1965.3. S. Timoshenko and J.N. Goodier. Theory of Elasticity. 2nd ed., McGraw-Hill, 1951.4. B.M. Irons. Comment on `Sti€ness matrices for section element' by I.R. Raju and A.K.Rao. JAIAA, 7, 156±7, 1969.5. E.L. Wilson. Structural analysis of axisymmetric solids. JAIAA, 3, 2269±74, 1965.6. O.C. Zienkiewicz. The Finite Element Method. 3rd ed., McGraw-Hill, 1977.

6Three-dimensional stress analysis6.1 IntroductionIt will have become obvious to the reader by this stage of the book that there is but onefurther step to apply the general ®nite element procedure to fully three-dimensionalproblems of stress analysis. Such problems embrace clearly all the practical cases,though for some, the various two-dimensional approximations give an adequateand more economical `model'.The simplest two-dimensional continuum element is a triangle. In three dimensionsits equivalent is a tetrahedron, an element with four nodal corners{, and this chapterwill deal with the basic formulation of such an element. Immediately, a di culty notencountered previously is presented. It is one of ordering of the nodal numbers and, infact, of a suitable representation of a body divided into such elements.The ®rst suggestions for the use of the simple tetrahedral element appear to be thoseof Gallagher et al. 1 and Melosh. 2 Argyris 3;4 elaborated further on the theme andRashid and Rockenhauser 5 were the ®rst to apply three-dimensional analysis torealistic problems.It is immediately obvious, however, that the number of simple tetrahedral elementswhich has to be used to achieve a given degree of accuracy has to be very large. Thiswill result in very large numbers of simultaneous equations in practical problems,which may place a severe limitation on the use of the method in practice. Further,the bandwidth of the resulting equation system becomes large, leading to increaseduse of iterative solution methods.To realize the order of magnitude of the problems presented let us assume that theaccuracy of a triangle in two-dimensional analysis is comparable to that of a tetrahedronin three dimensions. If an adequate stress analysis of a square, twodimensionalregion requires a mesh of some 20 20 ˆ 400 nodes, the total numberof simultaneous equations is around 800 given two displacement variables at anode. (This is a fairly realistic ®gure.) The bandwidth of the matrix involves 20nodes (Chapter 20), i.e., some 40 variables.{ The simplest polygonal shape which permits the approximation of the domain is known as the simplex.Thus a triangular and tetrahedral element constitute the simplex in two and three dimensions, respectively.

128 Three-dimensional stress analysisAn equivalent three-dimensional region is that of a cube with 20 20 20 ˆ8000 nodes. The total number of simultaneous equations is now some 24 000 asthree displacement variables have to be speci®ed. Further, the bandwidth nowinvolves an interconnection of some 20 20 ˆ 400 nodes or 1200 variables.Given that with direct solution techniques the computation e€ort is roughlyproportional to the number of equations and to the square of the bandwidth, themagnitude of the problems can be appreciated. It is not surprising therefore thate€orts to improve accuracy by use of complex elements with many degrees of freedomhave been strongest in the area of three-dimensional analysis. 6ÿ10 The developmentand practical application of such elements will be described in the following chapters.However, the presentation of this chapter gives all the necessary ingredients of theformulation for three-dimensional elastic problems and so follows directly from theprevious ones. Extension to more elaborate elements will be self-evident.6.2 Tetrahedral element characteristics6.2.1 Displacement functionsFigure 6.1 illustrates a tetrahedral element i, j, m, p in space de®ned by x, y, and zcoordinates.zimjpyxFig. 6.1 A tetrahedral volume. (Always use a consistent order of numbering, e.g., for p count the other nodesin an anticlockwise order as viewed from p, giving the element as ijmp, etc.).

Tetrahedral element characteristics 129The state of displacement of a point is de®ned by three displacement components,u, v, and w, in the directions of the three coordinates x, y, and z. Thus8 9>< u >=u ˆ v…6:1†>: >;wJust as in a plane triangle where a linear variation of a quantity was de®ned by itsthree nodal values, here a linear variation will be de®ned by the four nodal values.In analogy to Eq. (4.3) we can write, for instance,u ˆ 1 ‡ 2 x ‡ 3 y ‡ 4 z…6:2†Equating the values of the displacement at the nodes we have four equations of thetypeu i ˆ 1 ‡ 2 x i ‡ 3 y i ‡ 4 z i ; etc: …6:3†from which 1 to 4 can be evaluated.Again, it is possible to write this solution in a form similar to that of Eq. (4.5) byusing a determinant form, i.e.,u ˆ 16V ‰…a i ‡ b i x ‡ c i y ‡ d i z†u i ‡…a j ‡ b j x ‡ c j y ‡ d j z†u j‡…a m ‡ b m x ‡ c m y ‡ d m z†u m ‡…a p ‡ b p x ‡ c p y ‡ d p z†u p Š…6:4†with1 x i y i z i1 x j y j z j6V ˆ det1 x m y m z m 1 x p y p z p…6:5a†in which, incidentally, the value V represents the volume of the tetrahedron. Byexpanding the other relevant determinants into their cofactors we havex j y j z j1 y j z ja i ˆ detx m y m z mb i ˆÿdet1 y m z m x p y p z p 1 y p z px j 1 z jx j y j 1c i ˆÿdetx m 1 z md i ˆÿdetx m y m 1…6:5b† x p 1 z p x p y p 1 with the other constants de®ned by cyclic interchange of the subscripts in the order i,j, m, p.The ordering of nodal numbers i, j, m, p must follow a `right-hand'rule obviousfrom Fig. 6.1. In this the ®rst three nodes are numbered in an anticlockwisemanner when viewed from the last one.

130 Three-dimensional stress analysisThe element displacement is de®ned by the 12 displacement components of thenodes as8 9with>:8>:u iv iw i9>=>;a ia ja ma p>=>;etc:We can write the displacements of an arbitrary point asu ˆ‰IN i ; IN j ; IN m ; IN p Ša e ˆ Na ewith shape functions de®ned as…6:6†…6:7†N i ˆ ai ‡ b i x ‡ c i y ‡ d i z;6Vetc: …6:8†and I being a three by three identity matrix.Once again the displacement functions used will obviously satisfy continuityrequirements on interfaces between various elements. This fact is a direct corollaryof the linear nature of the variation of displacement.6.2.2 Strain matrixSix strain components are relevant in full three-dimensional analysis. The strainmatrix can now be de®ned as8 9@u@x8 9 @v" x@y" y @w>< ">= >< >=z @ze ˆ ˆ xy@u @y ‡ @v ˆ Su…6:9†@xyz >: >; @v zx@z ‡ @w@y@w >:@x ‡ @u >;@zfollowing the standard notation of Timoshenko's elasticity text. 11 Using Eqs (6.4)±(6.8) it is an easy matter to verify thate ˆ SNa e ˆ Ba e ˆ‰B i ; B j ; B m ; B p Ša e…6:10†

in which23@N i@x ; 0; 0@N0; i@y ; 023b i ; 0; 0@N 0; c 0; 0; ii ; 0@zB i ˆˆ 10; 0; d i@N i@y ; @N i@x ; 06Vc i ; b i ; 0674 0; d i ; c i5@N0; i@z ; @N i6@yd i 0; b i745@N i@z ; 0; @N i@x…6:11†with other submatrices obtained in a similar manner simply by interchange ofsubscripts.Initial strains, such as those due to thermal expansion, can be written in the usualway as a six-component vector which, for example, in an isotropic thermal expansionis simply8 911>=ˆ e m…6:12†00>: >;0with being the expansion coeTetrahedral element characteristics 131cient and e the average element temperature rise.6.2.3 Elasticity matrixWith complete anisotropy the D matrix relating the six stress components to the straincomponents can contain 21 independent constants (see Sec. 4.2.3).In general, thus,8 9 x>=ˆ D…e ÿ e 0 †‡r 0…6:13†>: yz zx>;Although no di culty presents itself in computation when dealing with suchmaterials, it is convenient to recapitulate here the D matrix for an isotropic material.This, in terms of the usual elastic constants E (modulus) and (Poisson's ratio),

132 Three-dimensional stress analysiscan be written as231 ÿ ; ; ; 0; 0; 01 ÿ ; ; 0; 0; 0E1 ÿ ; 0; 0; 0D ˆ…1 ‡ †…1 ÿ 2†…1 ÿ 2†=2; 0; 0674 Sym: …1 ÿ 2†=2; 0 5…1 ÿ 2†=2…6:14†6.2.4 Stiffness, stress, and load matricesThe sti€ness matrix de®ned by the general relationship (2.10) can now be explicitlyintegrated since the strain and stress components are constant within the element.The general ij submatrix of the sti€ness matrix will be a three by three matrixde®ned asK e ij ˆ B T i DB j V e…6:15†where V e represents the volume of the elementary tetrahedron.The nodal forces due to the initial strain become, similarly to Eq. (4.34),f e i ˆÿB T i De 0 V e…6:16†with a similar expression for forces due to initial stresses.Distributed body forces can once again be expressed in terms of their b x , b y , and b zcomponents or in terms of the body force potential. Not surprisingly, it will oncemore be found that if the body forces are constant the nodal components of thetotal resultant are distributed in four equal parts [see Eq. (4.36)].In fact, the similarity with the expressions and results of Chapter 4 is such thatfurther explicit formulation is unnecessary. The reader will ®nd no di culty in repeatingthe various steps needed for the formulation of a computer program.Fig. 6.2 A systematic way of dividing a three-dimensional object into `brick'-type elements.

Tetrahedral element characteristics 133zyx(a)zyx(b)Fig. 6.3 Composite element with eight nodes and its subdivision into ®ve tetrahedra by alternatives (a) or (b).

134 Three-dimensional stress analysis6.3 Composite elements with eight nodesThe division of a space volume into individual tetrahedra sometimes presents di cultiesof visualization and could easily lead to errors in nodal numbering, etc., unless afully automatic code is available. A more convenient subdivision of space is intoeight-cornered brick elements (bricks being the natural way to build a universe!).By sectioning a three-dimensional body parallel sections can be drawn and, eachone being subdivided into quadrilaterals, a systematic way of element de®nitioncould be devised as in Fig. 6.2.Such elements could be assembled automatically from several tetrahedra and theprocess of creating these tetrahedra left to a simple logical program. For instance,Fig. 6.3 shows how a typical brick can be divided into ®ve tetrahedra in two (andonly two) distinct ways. Stresses could well be presented as averages for a wholebrick-like element or as ®nal nodal averages. We shall discuss again a rationalprocedure for stress recovery in Chapter 14.In Fig. 6.4 a more convenient subdivision of a brick into six tetrahedra is shown.Here obviously the number of alternatives is very great; however (contrary to theTwo partseach dividedthusFig. 6.4 A systematic way of splitting an eight-cornered brick into six tetrahedra.

Examples and concluding remarks 1355-element subdivision) diagonals on adjacent faces of elements for a mesh type shownin Fig. 6.2 can always be made to match. Thus the 6-element subdivision creates aconforming approximation.In later chapters it will be seen how the basic bricks can be obtained directly withmore complex types of shape function.6.4 Examples and concluding remarksA simple, illustrative example of the application of simple, tetrahedral, elements isshown in Figs 6.5 and 6.6. Here the well-known Boussinesq problem of an elastichalf-space with a point load is approximated by analysing a cubic volume of space.Use of symmetry is made to reduce the size of the problem and the boundarydisplacements are prescribed in a manner shown in Fig. 6.5. 12 As zero displacementswere prescribed at a ®nite distance below the load a correction obtained from theexact expression was applied before executing the plots shown in Fig. 6.6. Comparisonof both stresses and displacement appears reasonable although it will beappreciated that the division is very coarse. However, even this trivial probleminvolved the solution of some 375 equations. More ambitious problems treatedwith simple tetrahedra are given in references 5 and 12. Figure 6.7, taken from theformer, illustrates an analysis of a complex pressure vessel. Some 10 000 degrees offreedom are involved in this analysis. In Chapter 8 it will be seen how the use ofcomplex elements permits a su ciently accurate analysis to be performed with amuch smaller total number of degrees of freedom for a very similar problem.zEHFGADyBoundary conditionsu = v = w = 0 on ABCDu = 0 on AEHDv = 0 on AEFBsymmetryAll other boundaries freexB10 ftCFig. 6.5 The Boussinesq problem as one of three-dimensional stress analysis.

Computed valueExact value45 tonsSectionz = –0.5 ft45 tonsx5 1010–5σ z (T/ft 2 )–wE (ft 2 )20–1030z = –1.75 ft10–520z = –3.75 ft(a)10(b)Fig. 6.6 The Boussineq problem: (a) vertical stresses ( z ); (b) vertical displacements (w).

Prestressing Concretesystem E = 5 x 106 lb/in 23.52 x 10 6 kg/cm 28 ft (2.43 m)ν = 0.15–700–500–10000200200–400–500–700–100–300400–2000–300–20000–100–400–600–700–800300400–500–1200 –600–1400–80040 ft(12.2 m)1.5 ft(0.46 m)σ rr–300–1000σ θθ–1000–700–400–20022 ft (6.71 m)Z15 ft(4.57 m)9 ft (2.74 m)θ3 ft (0.91 m)R–700–500–300–100σ zz–800–900–1000–600–1400–1400–1000–1000–700–600–400–2000 –200–900–1000–200–600–9002001200–500800–100 2000000σ rz–200–10000–100–300Fig. 6.7 A nuclear pressure vessel analysis using simple tetrahedral elements. 5 Geometry, subdivision, and some stress results.

138 Three-dimensional stress analysisAlthough we have in this chapter emphasized the easy visualization of a tetrahedralmesh through the use of brick-like subdivision, it is possible to generate automaticallyarbitrary tetrahedral meshes of great complexity with any prescribed mesh densitydistribution. The procedures follow the general pattern of automatic trianglegeneration 13 to which we shall refer in Chapter 15 when discussing e cient, adaptivelyconstructed meshes, but, of course, the degree of complexity introduced ismuch greater in three dimensions. Some details of such a generator are describedby Peraire et al., 14 and Fig. 6.8 illustrates an intersection of such an automatically(a)(b)Fig. 6.8 An automatically generated mesh of tetrahedra for a speci®ed mesh density in the exterior region onaircraft (a) and (b) an intersection of the mesh with the centreline plane.

References 139generated mesh with an outline of an aircraft. It is impractical to show the full plot ofthe mesh which contains over 30 000 nodes. The important point to note is that suchmeshes can be generated for any con®guration which can be suitably describedgeometrically. 15ÿ17 Although this example concerns aerodynamics rather thanelasticity, similar meshes can be generated in the latter context.References1. R.H. Gallagher, J. Padlog, and P.P. Bijlaard. Stress analysis of heated complex shapes.ARS Journal, 700±7, 1962.2. R.J. Melosh. Structural analysis of solids. Proc. Am. Soc. Civ. Eng., ST 4, 205±23, Aug.1963.3. J.H. Argyris. Matrix analysis of three-dimensional elastic media ± small and largedisplacements. JAIAA, 3, 45±51, Jan. 1965.4. J.H. Argyris. Three-dimensional anisotropic and inhomogeneous media ± matrix analysisfor small and large displacements. Ingenieur Archiv., 34, 33±55, 1965.5. Y.R. Rashid and W. Rockenhauser. Pressure vessel analysis by ®nite element techniques.Proc. Conf. on Prestressed Concrete Pressure Vessels. Inst. Civ. Eng., 1968.6. J.H. Argyris. Continua and discontinua. Proc. Conf. Matrix Methods in StructuralMechanics. Wright Patterson Air Force Base, Ohio, Oct. 1965.7. B.M. Irons. Engineering applications of numerical integration in sti€ness methods.JAIAA, 4, 2035±7, 1966.8. J.G. Ergatoudis, B.M. Irons, and O.C. Zienkiewicz. Three dimensional analysis of archdams and their foundations. Proc. Symp. Arch Dams. Inst. Civ. Eng., 1968.9. J.H. Argyris and J.C. Redshaw. Three dimensional analysis of two arch dams by a ®niteelement method. Proc. Symp. Arch Dams. Inst. Civ. Eng., 1968.10. S. Fjeld. Three dimensional theory of elastics. Finite Element Methods in Stress Analysis(eds I. Holand and K. Bell), Tech. Univ. of Norway, Tapir Press, Trondheim, 1969.11. S. Timoshenko and J.N. Goodier. Theory of Elasticity. 2nd ed., McGraw-Hill, 1951.12. Oliveira Pedro. Thesis, Laboratorio Nacional de Engenharia Civil, Lisbon, 1967.13. J. Peraire, M. Vahdati, K. Morgan and O.C. Zienkiewicz. Adaptive remeshing forcompressible ¯ow computations. J. Comp. Physics, 72, 449±66, 1987.14. J. Peraire, J. Peiro, L. Formaggia, K. Morgan, and O.C. Zienkiewicz. Finite element Eulercomputations in three dimensions. Int. J. Num. Meth. Eng. 1988 (to be published).15. N.P. Weatherill, P.R. Eiseman, J. Hause, and J.F. Thompson. Numerical Grid Generationin Computational Fluid Dynamics and Related Fields. Pineridge Press, Swansea, 1994.16. J.F. Thompson, B.K. Soni, and N.P. Weatherill, editors. Handbook of Grid Generation.CRC Press, January 1999.17. GiD ± The Personal Pre/Postprocessor (CIMNE). Barcelona, Spain, 1999.

7Steady-state ®eld problems ± heatconduction, electric and magneticpotential, ¯uid ¯ow, etc.7.1 IntroductionWhile, in detail, most of the previous chapters dealt with problems of an elastic continuumthe general procedures can be applied to a variety of physical problems.Indeed, some such possibilities have been indicated in Chapter 3 and here moredetailed attention will be given to a particular but wide class of such situations.Primarily we shall deal with situations governed by the general `quasi-harmonic'equation, the particular cases of which are the well-known Laplace and Poissonequations. 1ÿ6 The range of physical problems falling into this category is large. Tolist but a few frequently encountered in engineering practice we have:Heat conductionSeepage through porous mediaIrrotational ¯ow of ideal ¯uidsDistribution of electrical (or magnetic) potentialTorsion of prismatic shaftsBending of prismatic beams,Lubrication of pad bearings, etc.The formulation developed in this chapter is equally applicable to all, and hencelittle reference will be made to the actual physical quantities. Isotropic or anisotropicregions can be treated with equal ease.Two-dimensional problems are discussed in the ®rst part of the chapter. Ageneralization to three dimensions follows. It will be observed that the same, C 0 ,`shape functions'as those used previously in two- or three-dimensional formulationsof elasticity problems will again be encountered. The main di€erence will be that nowonly one unknown scalar quantity (the unknown function) is associated with eachpoint in space. Previously, several unknown quantities, represented by the displacementvector, were sought.In Chapter 3 we indicated both the `weak form'and a variational principle applicableto the Poisson and Laplace equations (see Secs 3.2 and 3.8.1). In the followingsections we shall apply these approaches to a general, quasi-harmonic equation andindicate the ranges of applicability of a single, uni®ed, approach by which one computerprogram can solve a large variety of physical problems.

7.2 The general quasi-harmonic equation7.2.1 The general statementThe general quasi-harmonic equation 141In many physical situations we are concerned with the di€usion or ¯ow of somequantity such as heat, mass, or a chemical, etc. In such problems the rate of transferper unit area, q, can be written in terms of its cartesian components asq T ˆ‰q x ; q y ; q z Š…7:1†If the rate at which the relevant quantity is generated (or removed) per unit volumeis Q, then for steady-state ¯ow the balance or continuity requirement gives@q x@x ‡ @q y@y ‡ @q z@z ‡ Q ˆ 0…7:2†Introducing the gradient operatorwe can write the above as8 9@@x >=r ˆ@y>: @ >;@zr T q ‡ Q ˆ 0…7:3†…7:4†Generally the rates of ¯ow will be related to gradients of some potential quantity .This may be temperature in the case of heat ¯ow, etc. A very general linear relationshipwill be of the form8 98>:q xq yq z@9@x>= >; ˆÿk @>=ˆÿk r@y>: @>;@z…7:5†where k is a three by three matrix. This is generally of a symmetric form due to energyarguments and is variously referred to as Fourier's, Fick's, or Darcy's law dependingon the physical problem.The ®nal governing equation for the `potential' is obtained by substitution ofEq. (7.5) into (7.4), leading toÿr T k r ‡ Q ˆ 0…7:6†

142 Steady-state ®eld problemswhich has to be solved in the domain . On the boundaries of such a domain we shallusually encounter one or other of the following conditions:1. On ÿ , ˆ i.e., the potential is speci®ed.2. On ÿ q the normal component of ¯ow, q n , is given asq n ˆ q ‡ where is a transfer or radiation coeAsq n ˆ q T ncient.n T ˆ‰n x ; n y ; n z Š…7:7a†…7:7b†where n is a vector of direction cosines of the normal to the surface, this conditioncan immediately be rewritten asin which q and are given.…k r† T n ‡ q ‡ ˆ 0…7:7c†7.2.2 Particular formsIf we consider the general statement of Eq. (7.5) as being determined for an arbitraryset of coordinate axes x; y; z we shall ®nd that it is always possible to determine locallyanother set of axes x 0 ; y 0 ; z 0 with respect to which the matrix k 0 becomes diagonal.With respect to such axes we have23k 0 ˆ64k x 0 0 00 k y 0 00 0 k z 075 …7:8†and the governing equation (7.6) can be written (now dropping the prime) @ @ÿ k@x x ‡ @ @k@x @y y ‡ @ @k@y @z z ‡ Q ˆ 0 …7:9†@zwith a suitable change of boundary conditions.Lastly, for an isotropic material we can writek ˆ kI…7:10†where I is an identity matrix. This leads to the simple form of Eq. (3.10) which wasdiscussed in Chapter 3.7.2.3 Weak form of general quasi-harmonic equation [Eq. (7.6)]Following the principles of Chapter 3, Sec. 3.2, we can obtain the weak form of

Finite element discretization 143Eq. (7.6) by writing……v…ÿr T k r ‡ Q† d ‡ v‰…k r† T n ‡ q ‡ Š dÿ ˆ 0ÿ q…7:11†for all functions v which are zero on ÿ .Integration by parts (see Appendix G) will result in the following weak statementwhich is equivalent to satisfying the governing equations and the natural boundaryconditions (7.7b):…… ……rv† T k r d ‡ vQ d ‡ v… ‡ q† dÿ ˆ 0ÿ qThe forced boundary condition (7.7a) still needs to be imposed.…7:12†7.2.4 The variational principleWe shall leave as an exercise to the reader the veri®cation that the functional ˆ 1 ………r† T k r d ‡ Q d ‡ 1 …… 2 dÿ ‡ q dÿ …7:13†2 2 ÿ q ÿ qgives on minimization [subject to the constraint of Eq. (7.7a)] the satisfaction of theoriginal problem set in Eqs (7.6) and (7.7).The algebraic manipulations required to verify the above principle follow preciselythe lines of Sec. 3.8 of Chapter 3 and can be carried out as an exercise.7.3 Finite element discretizationThis can now proceed on the assumption of a trial function expansion ˆ X N i a i ˆ Na…7:14†using either the weak formulation of Eq. (7.12) or the variational statement ofEq. (7.13). If, in the ®rst, we takev ˆ X W i a i with W i ˆ N i …7:15†according to the Galerkin principle, an identical form will arise with that obtainedfrom the minimization of the variational principle.Substituting Eq. (7.15) into (7.12) we have a typical statement giving …… ………rN i † T k rN d ‡ N i N dÿ a ‡ N i Q d ‡ N i q dÿ ˆ 0ÿ q ÿ qi ˆ 1; ...; n …7:16†or a set of standard discrete equations of the formHa ‡ f ˆ 0…7:17†

144 Steady-state ®eld problemswith…H ij ˆ……rN i † T k rN j d ‡ N i N j dÿÿ q…f i ˆ…N i Q d ‡ N i q dÿÿ qon which prescribed values of have to be imposed on boundaries ÿ .We note now that an additional `sti€ness'is contributed on boundaries for which aradiation constant is speci®ed but that otherwise a complete analogy with the elasticstructural problem exists.Indeed in a computer program the same standard operations will be followed evenincluding an evaluation of quantities analogous to the stresses. These, obviously, arethe ¯uxesq ÿk r ˆÿ…k rN†a…7:18†and, as with stresses, the best recovery procedure is discussed in Chapter 14.7.4 Some economic specializations7.4.1 Anisotropic and non-homogeneous mediaClearly material properties de®ned by the k matrix can vary from element to element ina discontinuous manner. This is implied in both the weak and variational statements ofthe problem.The material properties are usually known only with respect to the principal (or symmetry)axes, and if these directions are constant within the element it is convenient to usethem in the formulation of local axes speci®ed within each element, as shown in Fig. 7.1.yy'x'StratificationFig. 7.1 Anisotropic material. Local coordinates coincide with the principal directions of strati®cation.x

Some economic specializations 145With respect to such axes only three coe cients k x , k y , and k z need be speci®ed, andnow only a multiplication by a diagonal matrix is needed in formulating the coecients of the matrix H [Eq. (7.17)].It is important to note that as the parameters a correspond to scalar values, no transformationof matrices computed in local coordinates is necessary before assembly of theglobal matrices.Thus, in many computer programs only a diagonal speci®cation of the k matrix isused.7.4.2 Two-dimensional problemThe two-dimensional plane case is obtained by taking the gradient in the formr ˆ @@x ; @ @y T…7:19†and taking the ¯ux as8 9@ qx kx 0>< >=@xq ˆ ˆÿq y 0 k y@ >: >;@y…7:20†On discretization by Eq. (7.16) a slightly simpli®ed form of the matrices will now befound. Dropping the terms with and q we can writeHij e @Nˆ k i @N jx…V e @x @x ‡ k @N i @N jy dx dy…7:21†@y @yNo further discussion at this point appears necessary. However, it may be worthwhileto particularize here to the simplest yet still useful triangular element (Fig. 7.2).WithN i ˆ ai ‡ b i x ‡ c i y2as in Eq. (4.8) of Chapter 4, we can write down the element `sti€ness'matrix as23 23b i b i b i b j b i b mH e ˆ kx 674 b4j b j b j b m 5 ‡ k c i c i c i c j c i c my 674 c4j c j c j c m 5 …7:22†symmetric b m b m symmetric c m c mThe load matrices follow a similar simple pattern and thus, for instance, the readercan show that due to Q we have8 9f e ˆÿ Q >< 1 >=1…7:23†3 >: >;1a very simple (almost `obvious') result.

146 Steady-state ®eld problemsymsLq nikjrxFig. 7.2 Division of a two-dimensional region into triangular elements.Alternatively the formulation may be specialized to cylindrical coordinates andused for the solution of axisymmetric situations by introducing the gradient @r ˆ@r ; @ T…7:24†@zwhere r; z replace x; y. With the ¯ux now given by8 9 @qr kr 0>< >=q ˆ ˆÿ@rq z 0 k z@ >: >;@z…7:25†the discretization of Eq. (7.16) is now performed with the volume element expressedbyd ˆ 2r dr dzand integration carried out as described in Chapter 5, Section 5.2.5.7.5 Examples ± an assessment of accuracyIt is very easy to show that by assembling explicitly worked out `sti€nesses'of triangularelements for `regular'meshes shown in Fig. 7.3a, the discretized plane equationsare identical with those that can be derived by well-known ®nite di€erence methods. 7

Examples ± an assessment of accuracy 147(a)(b)Fig. 7.3 `Regular' and `irregular' subdivision patterns.Obviously the solutions obtained by the two methods will be identical, and so alsowill be the orders of approximation.yIf an `irregular'mesh based on a square arrangement of nodes is used a di€erencebetween the two aproaches will be evident [Fig. 7.3(b)]. This is con®ned to the `load'vector f e . The assembled equations will show `loads'which di€er by small amountsfrom node to node, but the sum of which is still the same as that due to the ®nitedi€erence expressions. The solutions therefore di€er only locally and will representthe same averages.In Fig. 7.4 a test comparing the results obtained on an `irregular'mesh with arelaxation solution of the lowest order ®nite di€erence approximation is shown.Both give results of similar accuracy, as indeed would be anticipated. However, itcan be shown that in one-dimensional problems the ®nite element algorithm givesexact answers of nodes, while the ®nite di€erence method generally does not. Ingeneral, therefore, superior accuracy is available with the ®nite element discretization.Further advantages of the ®nite element process are:1. It can deal simply with non-homogeneous and anisotropic situations (particularlywhen the direction of anisotropy is variable).2. The elements can be graded in shape and size to follow arbitrary boundaries and toallow for regions of rapid variation of the function sought, thus controlling theerrors in a most e cient way (viz. Chapters 14 and 15).3. Speci®ed gradient or `radiation'boundary conditions are introduced naturally andwith a better accuracy than in standard ®nite di€erence procedures.y This is only true in the case where the boundary values are prescribed.

148 Steady-state ®eld problems2042(2085)3041(3095)3352(3400)1921 3132 32513123(3190)4793(4850)5317(5380)3214 4712 53973656(3695)5684(5745)6332(6405)3556 5764 62473818(3855)5957(6027)6644(6715)3914 5876 6728(a)(b)Fig. 7.4 Torsion of a rectangular shaft. Numbers in parentheses show a more accurate solution due to Southwellusing a 12 16 mesh (values of =GL 2 ).4. Higher order elements can be readily used to improve accuracy without complicatingboundary conditions ± a di culty always arising with ®nite di€erence approximationsof a higher order.5. Finally, but of considerable importance in the computer age, standard programsmay be used for assembly and solution.Two more realistic examples are given at this stage to illustrate the accuracy attainablein practice. The ®rst is the problem of pure torsion of a non-homogeneous shaftillustrated in Fig. 7.5. The basic di€erential equation here is @ 1 @‡ @ 1 @‡ 2 ˆ 0 …7:26†@x G @x @y G @yφ = 0 on external boundary229 444 547 608563631 623495 475 438445351654671676707 888 1063 1145 1160 1113 965 895 8328901073117012161235180012001400397 350 290 2186001285 1485 1550 1540 1463 1378 1347 1260 1143 9891505161017121765177515801743161117431743174316001000735800200400609 4807651150174114329831743 17431740143017401132148917401231174036648062172580685599217289347391421L/2G 2 G 1LG 2 /G 1 = 3L67016171292436666 1240 1605 1743174317401300871435C LFig. 7.5 Torsion of a hollow bimetallic shaft. =GL 2 10 4 .

in which is the stress function, G is the shear modulus, and the angle of twist perunit length of the shaft.In the ®nite element solution presented, the hollow section was represented by amaterial for which G has a value of the order of 10 ÿ3 compared with the othermaterials.y The results compare well with the contours derived from an accurate®nite di€erence solution. 8An example concerning ¯ow through an anisotropic porous foundation is shown inFig. 7.6.Here the governing equation is@@xk x@H@x‡ @ @Hk@y y ˆ 0@ySome practical applications 149…7:27†in which H is the hydraulic head and k x and k y represent the permeability coe cientsin the direction of the (inclined) principal axes. The answers are here comparedagainst contours derived by an exact solution. The possibilities of the use of agraded size of subdivision are evident in this example.7.6 Some practical applications7.6.1 Anisotropic seepageThe ®rst of the problems is concerned with the ¯ow through highly non-homogeneous,anisotropic, and contorted strata. The basic governing equation is stillEq. (7.27). However, a special feature has to be incorporated to allow for changesof x 0 and y 0 principal directions from element to element.No di culties are encountered in computation, and the problem together with itssolution is given in Fig. 7.7. 37.6.2 Axisymmetric heat ¯owThe axisymmetric heat ¯ow equation results by using (7.24) and (7.25) with replacedby T. Now T is the temperature and k the conductivity.In Fig. 7.8 the temperature distribution in a nuclear reactor pressure vessel 1 isshown for steady-state heat conduction when a uniform temperature increase isapplied on the inside.7.6.3 Hydrodynamic pressures on moving surfacesIf a submerged surface moves in a ¯uid with prescribed accelerations and a smallamplitude of movement, then it can be shown 9 that if compressibility is ignored they This was done to avoid distandard program.culties due to the `multiple connection'of the region and to permit the use of a

H = 100H = 094.093.694.2 93.7 94.489.389.2 85.289.280.5 76.0 70.760.37.88.28.918.634.018.432.6 79.68.67.650.350.491.694.189.389.689.485.185.380.585.080.876.080.676.370.470.269.733.261.549.961.950.362.520.234.835.020.219.88.27.78.4k u80.950.885.276.569.762.135.020.216.7Stratificationk x = 4k y81.071.762.950.719.8Computed values76.351.235.325.075.071.7Equipotentials of exact solution66.763.035.233.358.352.841.7Fig. 7.6 Flow under an inclined pile wall in a strati®ed foundation. A ®ne mesh near the tip of the pile is not shown. Comparison with exact solution given by contours.50.0

Some practical applications 151H = 100 H = 0Dam90.0k = 180.089.1k = 477.583.290.791.284.376.092.279.483.570.187.371.891.181.079.395.690.511.283.577.767.556.5 36.7 34.726.119.270.648.971.1 59.5 42.3 29.249.4 35.941.755.554.345.347.640.835.140.035.736.322.529.46.724.316.929.032.731.218.69.7 8.927.826.713.121.524.331.417.27.27.710.015.820.022.830.030.073.171.170.068.7k = 1k = 266.063.758.869.160.761.753.956.550.450.747.745.441.541.633.736.233.933.139.335.634.334.137.135.0 34.232.7 33.334.240.860.0k = 150.0k = 443.9Equipotentials40.0k = 939.2Imperviousk = 3Fig. 7.7 Flow under a dam through a highly non-homogeneous and contorted foundation.excess pressures that are developed obey the Laplace equationr 2 p ˆ 0On moving (or stationary) boundaries the boundary condition is of type 2 [seeEq. (7.7b)] and is given by@p@n ˆÿa n…7:28†in which is the density of the ¯uid and a n is the normal component of acceleration ofthe boundary.On free surfaces the boundary condition is (if surface waves are ignored) simplyp ˆ 0…7:29†The problem clearly therefore comes into the category of those discussed in thischapter.As an example, let us consider the case of a vertical wall in a reservoir, shown inFig. 7.9, and determine the pressure distribution at points along the surface of thewall and at the bottom of the reservoir for any prescribed motion of the boundarypoints 1 to 7.The division of the region into elements (42 in number) is shown. Here elements ofrectangular shape are used (see Sect 3.3) and combined with quadrilaterals composed

C L0 0 0 0 0 0152 Steady-state ®eld problems31.0 31.264.564.429.713.863.841.423.6204030.61013.60Axis of rotational symmetrical10010010010063.56080 9062.710010010022.40 054.930.714.2062.437.313.30 010064.734.610.8010065.68.336.50 010064.334.308.410064.706.710029.265.2010063.430.20Fig. 7.8 Temperature distribution in steady-state conduction for an axisymmetrical pressure vessel.Moving wallH1211111x (a n ) LρL = H/612123456714 21 28 35 42 49 56Element subdivision505152535455Fig. 7.9 Problem of a wall moving horizontally in a reservoir.

Some practical applications 15312345Exact solution(Westergaard)00.20.467 14 2128 35 42 49 56a 00.60.8a 0ConstantaccelerationLinearly varyingaccelerationFig. 7.10 Pressure distribution on a moving wall and reservoir bottom.of two triangles near the sloping boundary. The pressure distribution on the wall andthe bottom of the reservoir for a constant acceleration of the wall is shown in Fig. 7.10.The results for the pressures on the wall agree to within 1 per cent with the wellknown,exact solution derived by Westergaard. 10For the wall hinged at the base and oscillating around this point with the top(point 1) accelerating by a 0 , the pressure distribution obtained is also plotted inFig. 7.10.In the study of vibration problems the interaction of the ¯uid pressure withstructural accelerations may be determined using Eq. (7.28) and the formulationgiven above. This and related problems will be discussed in more detail in Chapter 19.In Fig. 7.11 the solution of a similar problem in three dimensions is shown. 4 Heresimple tetrahedral elements combined as bricks as described in Chapter 6 were usedand very good accuracy obtained.In many practical problems the computation of such simpli®ed `added'massesis su cient, and the process described here has become widely used in thiscontext. 11ÿ137.6.4 Electrostatic and magnetostatic problemsIn this area of activity frequent need arises to determine appropriate ®eld strengthsand the governing equations are usually of the standard quasi-harmonic typediscussed here. Thus the formulations are directly transferable. One of the ®rstapplications made as early as 1967 4 was to fully three-dimensional electrostatic®eld distributions governed by simple Laplace equations (Fig. 7.12).In Fig. 7.13 a similar use of triangular elements was made in the context ofmagnetic two-dimensional ®elds by Winslow 6 in 1966. These early works stimulatedconsiderable activity in this area and much work has now been published. 14ÿ17

154 Steady-state ®eld problemsφ = constant1.25 H‘Non-conducting’H‘Non-conducting’4 HR = 1.27 HTypical volume elementθ = 58°–6'(a)67°–48'q Specified on this faceθ = 67°–48'θ = 58°–6'θ = 38°–44'θ = 0°θ = 19°–22'C LZHPresent solutionElectrolytic tank solution(b)p 1a 1 ρgHp 1a 1ρExcess pressureRelative accelerationDensityFig. 7.11 Pressures on an accelerating surface of a dam in an incompressible ¯uid.The magnetic problem is of particular interest as its formulation usually involvesthe introduction of a vector potential with three components which leads to aformulation di€erent from those discussed in this chapter. It is, therefore, worthwhileintroducing a variant which allows the standard programs of this section to be utilizedfor this problem. 18ÿ20

Some practical applications 15590eac907050703010503010fdbFig. 7.12 A three-dimensional distribution of electrostatic potential around a porcelain insulator in anearthed trough 4 .In electromagnetic theory for steady-state ®elds the problem is governed byMaxwell's equations which arer H ˆÿJB ˆ Hr T B ˆ 0…7:30†with the boundary condition speci®ed at an in®nite distance from the disturbance,requiring H and B to tend to zero there. In the above J is a prescribed electric currentdensity con®ned to conductors, H and B are vector quantities with three componentsdenoting the magnetic ®eld strength and ¯ux density respectively, is the magneticpermeability which varies (in an absolute set of units) from unity in vacuo to severalthousand in magnetizing materials and denotes the vector product, de®ned inAppendix F.

156 Steady-state ®eld problemsFig. 7.13 Field near a magnet (after Winslow 6 ).The formulation presented here depends on the fact that it is a relatively simplematter to determine the ®eld H s which exactly solves Eq. (7.30) when 1 everywhere.This is given at any point de®ned by a vector coordinate r by an integral:…H s ˆ 14 J …r ÿ r 0 † …r ÿ r 0 † T …r ÿ r 0 † d…7:31†In the above, r 0 refers to the coordinates of d and obviously the integration domainonly involves the electric conductors where J 6ˆ 0.With H s known we can writeH ˆ H s ‡ H mand, on substitution into Eq. (7.30), we have a systemr H m ˆ 0B ˆ …H s ‡ H m †r T B ˆ 0If we now introduce a scalar potential , de®ning H m asH m r…7:32†…7:33†

Some practical applications 157we ®nd the ®rst of Eqs (7.36) to be automatically satis®ed and, on eliminating B in theother two, the governing equation becomesr T r ‡ r T H s ˆ 0…7:34†with ! 0 at in®nity. This is precisely of the standard form discussed in this chapter[Eq. (7.6)] with the second term, which is now speci®ed, replacing Q.An apparent di culty exists, however, if varies in a discontinuous manner, asindeed we would expect it to do on the interfaces of two materials.Here the term Q is now unde®ned and, in the standard discretization of Eq. (7.16)or (7.17), the term……N i Q d ÿ N i r T H s d…7:35†apparently has no meaning.Integration by parts comes once again to the rescue and we note that………N i r T H s d ÿ r T N i H s ‡ N i H s n dÿ …7:36†As in regions of constant , r T H s 0, the only contribution to the forcing termscomes as a line integral of the second term at discontinuity interfaces.Introduction of the scalar potential makes both two- and three-dimensional magnetostaticproblems solvable by a standard program used for all the problems in thissection. Figure 7.14 shows a typical three-dimensional solution for a transformer.Here isoparametric quadratic brick elements of the type which will be described inChapter 8 were used. 18In typical magnetostatic problems a high non-linearity exists withq ˆ …jHj† where jHj ˆ Hx 2 ‡ Hy 2 ‡ Hz2 …7:37†The treatment of such non-linearities will be discussed in Volume 2.Considerable economy in this and other problems of in®nite extent can be achievedby the use of in®nite elements to be discussed in Chapter 9.ÿ7.6.5 Lubrication problemsOnce again a standard Poisson type of equation is encountered in the twodimensionaldomain of a bearing pad. In the simplest case of constant lubricantdensity and viscosity the equation to be solved is the Reynolds equation@@xh 3 @p@x‡ @ @yh 3 @p@yˆ 6V @h@x…7:38†where h is the ®lm thickness, p the pressure developed, the viscosity and V thevelocity of the pad in the x-direction.Figure 7.15 shows the pressure distribution in the typical case of a stepped pad. 21The boundary condition is simply that of zero pressure and it is of interest to note that

158 Steady-state ®eld problemsElement subdivision5000.0CoilB'Field due tocurrent H sTotal field H(with magnetization)C LB A'4000.0AC5.260.85 L0.213000.012.02000.010.08.06.0 y (cm)1000.02.0 4.00.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0x (cm)(a)z300.0y200.0100.00.04.0 cmx(b)Fig. 7.14 Three-dimensional transformer. (a) Field strength H. (b) Scalar potential on plane z ˆ 4:0 cm.the step causes an equivalent of a `line load'on integration by parts of the right-handside of Eq. (7.38), just as in the case of magnetic discontinuity mentioned above.More general cases of lubrication problems, including vertical pad movements(squeeze ®lms) and compressibility, can obviously be dealt with, and much workhas been done here. 22ÿ29Total field strength HA-t/mScalar potential φ Amp-t

Some practical applications 1592.625 in 2.375 inLContours of ph 2 1 /6µULOInletC L1.182 inUC Lh = 1 16in116inFig. 7.15 A stepped pad bearing. Pressure distribution.7.6.6 Irrotational and free surface ¯owsThe basic Laplace equation which governs the ¯ow of viscous ¯uid in seepageproblems is also applicable in the problem of irrotational ¯uid ¯ow outside theboundary layer created by viscous e€ects. The examples already given are adequateto illustrate the general applicability in this context. Further examples are quotedby Martin 30 and others. 31ÿ36If no viscous e€ects exist, then it can be shown that for a ¯uid starting at rest themotion must be irrotational, i.e.,! z @u@y ÿ @v ˆ 0 etc: …7:39†@xwhere u and v are appropriate velocity components.This implies the existence of a velocity potential, givingu ˆÿ @@xv ˆÿ @@y…7:40†…or u ˆÿr†If, further, the ¯ow is incompressible the continuity equation [see Eq. (7.2)] has tobe satis®ed, i.e.,r T u ˆ 0…7:41†

160 Steady-state ®eld problemsand thereforer T r ˆ 0…7:42†Alternatively, for two-dimensional ¯ow a stream function may be introduced de®ningthe velocities asu ˆÿ @@yv ˆ @@x…7:43†and this identically satis®es the continuity equation. The irrotationality conditionmust now ensure thatr T r ˆ 0…7:44†and thus problems of ideal ¯uid ¯ow can be posed in one form or the other. As thestandard formulation is again applicable, there is little more that needs to beadded, and for examples the reader can well consult the literature cited. We shallalso discuss further such examples in Volume 3.The similarity with problems of seepage ¯ow, which has already been discussed, isobvious. 37;38A particular class of ¯uid ¯ow deserves mention. This is the case when a free surfacelimits the extent of the ¯ow and this surface is not known a priori.The class of problem is typi®ed by two examples ± that of a freely over¯owing jet[Fig. 7.16(a)] and that of ¯ow through an earth dam [Fig. 7.16(b)]. In both, the freesurface represents a streamline and in both the position of the free surface is unknowna priori but has to be determined so that an additional condition on this surface is satis-®ed. For instance, in the second problem, if formulated in terms of the potential H,Eq. (7.27) governs the problem.p = 0(a)p = 0yp = 0(b)xFig. 7.16 Typical free surface problems with a streamline also satisfying an additional condition ofpressure ˆ 0. (a) Jet over¯ow. (b) Seepage through an earth dam.

References 161The free surface, being a streamline, imposes the condition@H@n ˆ 0…7:45†to be satis®ed there. In addition, however, the pressure must be zero on the surface asthis is exposed to atmosphere. AsH ˆ p ‡ y…7:46†where is the ¯uid speci®c weight, p is the ¯uid pressure, and y the elevation abovesome (horizontal) datum, we must have on the surfaceH ˆ y…7:47†The solution may be approached iteratively. Starting with a prescribed free surfacestreamline the standard problem is solved. A check is carried out to see if Eq. (7.47) issatis®ed and, if not, an adjustment of the surface is carried out to make the new yequal to the H just found. A few iterations of this kind show that convergence isreasonably rapid. Taylor and Brown 39 show such a process. Alternative methodsincluding special variational principles for dealing with this problem have beendevised over the years and interested readers can consult references 40±48.7.7 Concluding remarksWe have shown how a general formulation for the solution of a steady-state quasiharmonicproblem can be written, and how a single program of such a form can beapplied to a wide variety of physical situations. Indeed, the selection of problemsdealt with is by no means exhaustive and many other examples of application areof practical interest. Readers will doubtless ®nd appropriate analogies for theirown problems.References1. O.C. Zienkiewicz and Y.K. Cheung. Finite elements in the solution of ®eld problems. TheEngineer. 507±10, Sept. 1965.2. W. Visser. A ®nite element method for the determination of non-stationary temperaturedistribution and thermal deformations. Proc. Conf. on Matrix Methods in StructuralMechanics. Air Force Inst. Tech., Wright-Patterson AF Base, Ohio, 1965.3. O.C. Zienkiewicz, P. Mayer, and Y.K. Cheung. Solution of anisotropic seepage problemsby ®nite elements. Proc. Am. Soc. Civ. Eng. 92, EM1, 111±20, 1966.4. O.C. Zienkiewicz, P.L. Arlett, and A.L. Bahrani. Solution of three-dimensional ®eldproblems by the ®nite element method. The Engineer. 27 October 1967.5. L.R. Herrmann. Elastic torsion analysis of irregular shapes. Proc. Am. Soc. Civ. Eng. 91,EM6, 11±19, 1965.6. A.M. Winslow. Numerical solution of the quasi-linear Poisson equation in a non-uniformtriangle `mesh'. J. Comp. Phys. 1, 149±72, 1966.7. D.N. de G. Allen. Relaxation Methods. p. 199, McGraw-Hill, 1955.

162 Steady-state ®eld problems8. J.F. Ely and O.C. Zienkiewicz. Torsion of compound bars ± a relaxation solution. Int. J.Mech. Sci. 1, 356±65, 1960.9. O.C. Zienkiewicz and B. Nath. Earthquake hydrodynamic pressures on arch dams ± anelectric analogue solution. Proc. Inst. Civ. Eng. 25, 165±76, 1963.10. H.M. Westergaard. Water pressure on dams during earthquakes. Trans. Am. Soc. Civ. Eng.98, 418±33, 1933.11. O.C. Zienkiewicz and R.E. Newton. Coupled vibrations of a structure submerged in a compressible¯uid. Proc. Symp. on Finite Element Techniques. pp. 359±71, Stuttgart, 1969.12. R.E. Newton. Finite element analysis of two-dimensional added mass and damping, inFinite Elements in Fluids (eds R.H. Gallagher, J.T. Oden, C. Taylor, and O.C. Zienkiewicz),Vol. I, pp. 219±32, Wiley, 1975.13. P.A.A. Back, A.C. Cassell, R. Dungar, and R.T. Severn. The seismic study of a doublecurvature dam. Prov. Inst. Civ. Eng. 43, 217±48, 1969.14. P. Silvester and M.V.K. Chari. Non-linear magnetic ®eld analysis of D.C. machines. Trans.IEEE. No. 7, 5±89, 1970.15. P. Silvester and M.S. Hsieh. Finite element solution of two dimensional exterior ®eld problems.Proc. IEEE. 118, 1971.16. B.H. McDonald and A. Wexler. Finite element solution of unbounded ®eld problems.Proc. IEEE. MTT-20, No. 12, 1972.17. E. Munro. Computer design of electron lenses by the ®nite element method, in Image Processingand Computer Aided Design in Electron Optics. p. 284, Academic Press, 1973.18. O.C. Zienkiewicz, J.F. Lyness, and D.R.J. Owen. Three dimensional magnetic ®eld determinationusing a scalar potential. A ®nite element solution. IEEE, Trans. Magnetics MAG.13, 1649±56, 1977.19. J. Simkin and C.W. Trowbridge. On the use of the total scalar potential in the numericalsolution of ®eld problems in electromagnets. Int. J. Num. Meth. Eng. 14, 423±40, 1979.20. J. Simkin and C.W. Trowbridge. Three-dimensional non-linear electromagnetic ®eldcomputations using scalar potentials. Proc. Inst. Elec. Eng. 127, B(6), 1980.21. D.V. Tanesa and I.C. Rao. Student project report on lubrication. Royal Naval College,Dartmouth, 1966.22. M.M. Reddi. Finite element solution of the incompressible lubrication problem. Trans.Am. Soc. Mech. Eng. 91 (Ser. F), 524, 1969.23. M.M. Reddi and T.Y. Chu. Finite element solution of the steady state compressible lubricationproblem. Trans. Am. Soc. Mech. Eng. 92 (Ser. F), 495, 1970.24. J.H. Argyris and D.W. Scharpf. The incompressible lubrication problem. J. Roy. Aero.Soc. 73, 1044±6, 1969.25. J.F. Booker and K.H. Huebner. Application of ®nite element methods to lubrication: anengineering approach. J. Lubr. Techn., Trans. Am. Soc. Mech. Eng. 14 (Ser. F), 313, 1972.26. K.H. Huebner. Application of ®nite element methods to thermohydrodynamic lubrication.Int. J. Num. Meth. Eng. 8, 139±68, 1974.27. S.M. Rohde and K.P. Oh. Higher order ®nite element methods for the solution of compressibleporous bearing problems. Int. J. Num. Meth. Eng. 9, 903±12, 1975.28. A.K. Tieu. Oil ®lm temperature distributions in an in®nitely wide glider bearing: an applicationof the ®nite element method. J. Mech. Eng. Sci. 15, 311, 1973.29. K.H. Huebner. Finite element analysis of ¯uid ®lm lubrication ± a survey, in FiniteElements in Fluids (eds R.H. Gallagher, J.T. Oden, C. Taylor, and O.C. Zienkiewicz).Vol. II, pp. 225±54, Wiley, 1975.30. H.C. Martin. Finite element analysis of ¯uid ¯ows. Proc. 2nd Conf. on Matrix Methods inStructural Mechanics. Air Force Inst. Tech., Wright-Patterson AF Base, Ohio, 1968.31. G. de Vries and D.H. Norrie. Application of the ®nite element technique to potential ¯owproblems. Reports 7 and 8, Dept. Mech. Eng., Univ. of Calgary, Alberta, Canada, 1969.

32. J.H. Argyris, G. Mareczek, and D.W. Scharpf. Two and three dimensional ¯ow using ®niteelements. J. Roy. Aero. Soc. 73, 961±4, 1969.33. L.J. Doctors. An application of ®nite element technique to boundary value problems ofpotential ¯ow. Int. J. Num. Meth. Eng. 2, 243±52, 1970.34. G. de Vries and D.H. Norrie. The application of the ®nite element technique to potential¯ow problems. J. Appl. Mech., Am. Soc. Mech. Eng. 38, 978±802, 1971.35. S.T.K. Chan, B.E. Larock, and L.R. Herrmann. Free surface ideal ¯uid ¯ows by ®niteelements. Proc. Am. J. Civ. Eng. 99, HY6, 1973.36. B.E. Larock. Jets from two dimensional symmetric nozzles of arbitrary shape. J. FluidMech. 37, 479±83, 1969.37. C.S. Desai. Finite element methods for ¯ow in porous media, in Finite Elements in Fluids(ed. R.H. Gallagher). Vol. 1, pp. 157±82, Wiley, 1975.38. I. Javandel and P.A. Witherspoon. Applications of the ®nite element method to transient¯ow in porous media. Trans. Soc. Petrol. Eng. 243, 241±51, 1968.39. R.L. Taylor and C.B. Brown. Darcy ¯ow solutions with a free surface. Proc. Am. Soc. Civ.Eng. 93, HY2, 25±33, 1967.40. J.C. Luke. A variational principle for a ¯uid with a free surface. J. Fluid Mech. 27, 395±7,1957.41. K. Washizu, Variational Methods in Elasticity and Plasticity. 2nd ed., Pergamon Press,1975.42. J.C. Bruch. A survey of free-boundary value problems in the theory of ¯uid ¯ow throughporous media. Advances in Water Resources. 3, 65±80, 1980.43. C. Baiocchi, V. Comincioli, and V. Maione. Uncon®ned ¯ow through porous media.Meccanice. Ital. Ass. Theor. Appl. Mech. 10, 51±60, 1975.44. J.M. Sloss and J.C. Bruch. Free surface seepage problem. Proc. ASCE. 108, EM5, 1099±1111, 1978.45. N. Kikuchi. Seepage ¯ow problems by variational inequalities. Int. J. Num. Anal. Meth.geomech. 1, 283±90, 1977.46. C.S. Desai. Finite element residual schemes for uncon®ned ¯ow. Int. J. Num. Meth. Eng.10, 1415±18, 1976.47. C.S. Desai and G.C. Li. A residual ¯ow procedure and application for free surface, andporous media. Advances in Water Resources. 6, 27±40, 1983.48. K.J. Bathe and M. Koshgoftar. Finite elements from surface seepage analysis without meshiteration. Int. J. Num. Anal. Meth. Geomech. 3, 13±22, 1979.References 163

8`Standard' and `hierarchical'element shape functions: somegeneral families of C 0 continuity8.1 IntroductionIn Chapters 4, 5, and 6 the reader was shown in some detail how linear elasticityproblems could be formulated and solved using very simple ®nite element forms. InChapter 7 this process was repeated for the quasi-harmonic equation. Although thedetailed algebra was concerned with shape functions which arose from triangularand tetrahedral shapes only it should by now be obvious that other element formscould equally well be used. Indeed, once the element and the corresponding shapefunctions are determined, subsequent operations follow a standard, well-de®nedpath which could be entrusted to an algebraist not familiar with the physical aspectsof the problem. It will be seen later that in fact it is possible to program a computerto deal with wide classes of problems by specifying the shape functions only. Thechoice of these is, however, a matter to which intelligence has to be applied and inwhich the human factor remains paramount. In this chapter some rules for thegeneration of several families of one-, two-, and three-dimensional elements will bepresented.In the problems of elasticity illustrated in Chapters 4, 5, and 6 the displacementvariable was a vector with two or three components and the shape functions werewritten in matrix form. They were, however, derived for each component separatelyand in fact the matrix expressions in these were derived by multiplying a scalarfunction by an identity matrix [e.g., Eqs (4.7), (5.3), and (6.7)]. This scalar formwas used directly in Chapter 7 for the quasi-harmonic equation. We shall thereforeconcentrate in this chapter on the scalar shape function forms, calling these simply N i .The shape functions used in the displacement formulation of elasticity problemswere such that they satisfy the convergence criteria of Chapter 2:(a) the continuity of the unknown only had to occur between elements (i.e., slopecontinuity is not required), or, in mathematical language, C 0 continuity wasneeded;(b) the function has to allow any arbitrary linear form to be taken so that theconstant strain (constant ®rst derivative) criterion could be observed.The shape functions described in this chapter will require the satisfaction of thesetwo criteria. They will thus be applicable to all the problems of the preceding chapters

Standard and hierarchical concepts 165and also to other problems which require these conditions to be obeyed. Indeed theyare applicable to any situation where the functional or (see Chapter 3) is de®nedby derivatives of ®rst order only.The element families discussed will progressively have an increasing number ofdegrees of freedom. The question may well be asked as to whether any economic orother advantage is gained by thus increasing the complexity of an element. Theanswer here is not an easy one although it can be stated as a general rule that as theorder of an element increases so the total number of unknowns in a problem can bereduced for a given accuracy of representation. Economic advantage requires, however,a reduction of total computation and data preparation e€ort, and this does not followautomatically for a reduced number of total variables because, though equation-solvingtimes may be reduced, the time required for element formulation increases.However, an overwhelming economic advantage in the case of three-dimensionalanalysis has already been hinted at in Chapters 6 and 7 for three-dimensional analyses.The same kind of advantage arises on occasion in other problems but in general theoptimum element may have to be determined from case to case.In Sec. 2.6 of Chapter 2 we have shown that the order of error in the approximationto the unknown function is O…h p ‡ 1 †, where h is the element `size' and p is the degree ofthe complete polynomial present in the expansion. Clearly, as the element shape functionsincrease in degree so will the order of error increase, and convergence to theexact solution becomes more rapid. While this says nothing about the magnitudeof error at a particular subdivision, it is clear that we should seek element shape functionswith the highest complete polynomial for a given number of degrees of freedom.8.2 Standard and hierarchical conceptsThe essence of the ®nite element method already stated in Chapters 2 and 3 is inapproximating the unknown (displacement) by an expansion given in Eqs (2.1) and(3.3). For a scalar variable u this can be written asu ^u ˆ XnN i a i ˆ Na…8:1†i ˆ 1where n is the total number of functions used and a i are the unknown parameters to bedetermined.We have explicitly chosen to identify such variables with the values of the unknownfunction at element nodes, thus makingu i ˆ a i…8:2†The shape functions so de®ned will be referred to as `standard' ones and are the basisof most ®nite element programs. If polynomial expansions are used and the elementsatis®es Criterion 1 of Chapter 2 (which speci®es that rigid body displacements causeno strain), it is clear that a constant value of a i speci®ed at all nodes must result in aconstant value of ^u: X n^u ˆ N iu 0 ˆ u 0 …8:3†i ˆ 1

166 `Standard' and `hierarchical' element shape functionswhen a i ˆ u 0 . It follows thatX ni ˆ 1N i ˆ 1…8:4†at all points of the domain. This important property is known as a partition of unity 1which we will make extensive use of in Chapter 16. The ®rst part of this chapter willdeal with such standard shape functions.A serious drawback exists, however, with `standard' functions, since when elementre®nement is made totally new shape functions have to be generated and hence allcalculations repeated. It would be of advantage to avoid this di culty by consideringthe expression (8.1) as a series in which the shape function N i does not depend on thenumber of nodes in the mesh n. This indeed is achieved with hierarchic shape functionsto which the second part of this chapter is devoted.The hierarchic concept is well illustrated by the one-dimensional (elastic bar)problem of Fig. 8.1. Here for simplicity elastic properties are taken as constant(D ˆ E) and the body force b is assumed to vary in such a manner as to producethe exact solution shown on the ®gure (with zero displacements at both ends).Two meshes are shown and a linear interpolation between nodal points assumed.For both standard and hierarchic forms the coarse mesh givesK11a c c 1 ˆ f 1…8:5†For a ®ne mesh two additional nodes are added and with the standard shapefunction the equations requiring solution are2389 8 9K11 F K12 F 0 >< a 1 >= >< f 1 >=64 K21 F K22 F K23F 75 a 2>: >; ˆ f 2…8:6†>: >;0 K32 F K33F a 3 f 3In this form the zero matrices have been automatically inserted due to element interconnectionwhich is here obvious, and we note that as no coe cients are the same, thenew equations have to be resolved. [Equation (2.13) shows how these coe cients arecalculated and the reader is encouraged to work these out in detail.]With the `hierarchic' form using the shape functions shown, a similar form ofequation arises and an identical approximation is achieved (being simply given bya series of straight segments). The ®nal solution is identical but the meaning of theparameters a i is now di€erent, as shown in Fig. 8.1.Quite generally,K11 F ˆ K11c …8:7†as an identical shape function is used for the ®rst variable. Further, in this particularcase the o€-diagonal coe cients are zero and the ®nal equations become, for the®ne mesh,2K11 c 380 0 a 9 8 9>< 1 >= >< f 1 >=64 0 K22 F 70 5 a 2>: >; ˆ f 2…8:8†>: >;0 0 K33F f 3a 3

Standard and hierarchical concepts 167Coarsea c 1FineExactApproximateaa 12a 31 2 1 3N 2 N 1 N 3N 11 1(a)a* 2a* 1a* 31 2 1 3N 2 N 1 N 31(b)Fig. 8.1 A one-dimensional problem of stretching of a uniform elastic bar by prescribed body forces. (a) `Standardapproximation. (b) Hierarchic approximation.The `diagonality' feature is only true in the one-dimensional problem, but ingeneral it will be found that the matrices obtained using hierarchic shape functionsare more nearly diagonal and hence imply better conditioning than those withstandard shape functions.Although the variables are now not subject to the obvious interpretation (as localdisplacement values), they can be easily transformed to those if desired. Though it isnot usual to use hierarchic forms in linearly interpolated elements their derivation inpolynomial form is simple and very advantageous.The reader should note that with hierarchic forms it is convenient to consider the®ner mesh as still using the same, coarse, elements but now adding additional re®ningfunctions.Hierarchic forms provide a link with other approximate (orthogonal) series solutions.Many problems solved in classical literature by trigonometric, Fourier series,expansion are indeed particular examples of this approach.

168 `Standard' and `hierarchical' element shape functionsIn the following sections of this chapter we shall consider the development of shapefunctions for high order elements with many boundary and internal degree offreedoms. This development will generally be made on simple geometric forms andthe reader may well question the wisdom of using increased accuracy for suchsimple shaped domains, having already observed the advantage of generalized®nite element methods in ®tting arbitrary domain shapes. This concern is wellfounded, but in the next chapter we shall show a general method to map highorder elements into quite complex shapes.Part 1 `Standard' shape functionsTwo-dimensional elements8.3 Rectangular elements ± some preliminaryconsiderationsConceptually (especially if the reader is conditioned by education to thinking in thecartesian coordinate system) the simplest element form of a two-dimensional kindis that of a rectangle with sides parallel to the x and y axes. Consider, for instance,the rectangle shown in Fig. 8.2 with nodal points numbered 1 to 8, located asshown, and at which the values of an unknown function u (here representing, forinstance, one of the components of displacement) form the element parameters.How can suitable C 0 continuous shape functions for this element be determined?Let us ®rst assume that u is expressed in polynomial form in x and y. To ensureinterelement continuity of u along the top and bottom sides the variation must belinear. Two points at which the function is common between elements lying aboveor below exist, and as two values uniquely determine a linear function, its identityall along these sides is ensured with that given by adjacent elements. Use of thisfact was already made in specifying linear expansions for a triangle.Similarly, if a cubic variation along the vertical sides is assumed, continuity will bepreserved there as four values determine a unique cubic polynomial. Conditions forsatisfying the ®rst criterion are now obtained.To ensure the existence of constant values of the ®rst derivative it is necessary thatall the linear polynomial terms of the expansion be retained.Finally, as eight points are to determine uniquely the variation of the function onlyeight coe cients of the expansion can be retained and thus we could writeu ˆ 1 ‡ 2 x ‡ 3 y ‡ 4 xy ‡ 5 y 2 ‡ 6 xy 2 ‡ 7 y 3 ‡ 8 xy 3 …8:9†The choice can in general be made unique by retaining the lowest possible expansionterms, though in this case apparently no such choice arises.y The reader will easilyverify that all the requirements have now been satis®ed.y Retention of a higher order term of expansion, ignoring one of lower order, will usually lead to a poorerapproximation though still retaining convergence, 2 providing the linear terms are always included.

Rectangular elements ± some preliminary considerations 169y18273645xFig. 8.2 A rectangular element.Substituting coordinates of the various nodes a set of simultaneous equations willbe obtained. This can be written in exactly the same manner as was done for a trianglein Eq. (4.4) as8 9 2389>< u 1 >= 1; x 1 ; y 1 ; x 1 y 1 ; y 2 1; x 1 y 2 1; y 3 1; x 1 y 3 1>< 1 >=.. >: >; ˆ 6: : : : : : : :7 .45 . …8:10†u 8 1; x 8 ; y 8 ; : : : : x 8 y 3 >: >;8 8or simply asu e ˆ Ca…8:11†Formally,a ˆ C ÿ1 u e…8:12†and we could write Eq. (8.9) asu ˆ Pa ˆ PC ÿ1 u e…8:13†in whichP ˆ‰1; x; y; xy; y 2 ; xy 2 ; y 3 ; xy 3 Š…8:14†Thus the shape functions for the element de®ned byu ˆ Nu e ˆ‰N 1 ; N 2 ; ...; N 8 Šu e…8:15†can be found asN ˆ PC ÿ1…8:16†This process has, however, some considerable disadvantages. Occasionally aninverse of C may not exist 2;3 and always considerable algebraic di culty is experiencedin obtaining an expression for the inverse in general terms suitable for allelement geometries. It is therefore worthwhile to consider whether shape functionsN i …x; y† can be written down directly. Before doing this some general properties ofthese functions have to be mentioned.

170 `Standard' and `hierarchical' element shape functionsN 111yxN 212Fig. 8.3 Shape functions for elements of Fig. 8.2.Inspection of the de®ning relation, Eq. (8.15), reveals immediately some importantcharacteristics. Firstly, as this expression is valid for all components of u e , 1; i ˆ jN i …x j ; y j †ˆ ij ˆ0; i 6ˆ jwhere ij is known as the Kronecker delta. Further, the basic type of variation alongboundaries de®ned for continuity purposes (e.g., linear in x and cubic in y in theabove example) must be retained. The typical form of the shape functions for theelements considered is illustrated isometrically for two typical nodes in Fig. 8.3. Itis clear that these could have been written down directly as a product of a suitablelinear function in x with a cubic function in y. The easy solution of this example isnot always as obvious but given su cient ingenuity, a direct derivation of shapefunctions is always preferable.It will be convenient to use normalized coordinates in our further investigation.Such normalized coordinates are shown in Fig. 8.4 and are chosen so that theirvalues are 1 on the faces of the rectangle: ˆ x ÿ x ca ˆ y ÿ y cbd ˆ dxad ˆ dyb…8:17†Once the shape functions are known in the normalized coordinates, translation intoactual coordinates or transformation of the various expressions occurring, forinstance, in the sti€ness derivation is trivial.

Completeness of polynomials 171yaaη = 1ηcξ = –1y c η = –1x cξξ = 1bbxFig. 8.4 Normalized coordinates for a rectangle.8.4 Completeness of polynomialsThe shape function derived in the previous section was of a rather special form [seeEq. (8.9)]. Only a linear variation with the coordinate x was permitted, while in y afull cubic was available. The complete polynomial contained in it was thus of order1. In general use, a convergence order corresponding to a linear variation wouldoccur despite an increase of the total number of variables. Only in situations wherethe linear variation in x corresponded closely to the exact solution would a higherorder of convergence occur, and for this reason elements with such `preferential'directions should be restricted to special use, e.g., in narrow beams or strips. Ingeneral, we shall seek element expansions which possess the highest order of acomplete polynomial for a minimum of degrees of freedom. In this context it isuseful to recall the Pascal triangle (Fig. 8.5) from which the number of terms1xyorder 1x 3 x 2x 2 yxyxy 2y 2y 323x 4 x 3 y x 2 y 2 xy 3 y 44x 5 y 55Fig. 8.5 The Pascal triangle. (Cubic expansion shaded ± 10 terms).

172 `Standard' and `hierarchical' element shape functionsoccurring in a polynomial in two variables x, y can be readily ascertained. Forinstance, ®rst-order polynomials require three terms, second-order require sixterms, third-order require ten terms, etc.8.5 Rectangular elements ± Lagrange family 4ÿ6An easy and systematic method of generating shape functions of any order can beachieved by simple products of appropriate polynomials in the two coordinates.Consider the element shown in Fig. 8.6 in which a series of nodes, external andinternal, is placed on a regular grid. It is required to determine a shape function forthe point indicated by the heavy circle. Clearly the product of a ®fth-orderpolynomial in which has a value of unity at points of the second column of nodesand zero elsewhere and that of a fourth-order polynomial in having unity on thecoordinate corresponding to the top row of nodes and zero elsewhere satis®es allthe interelement continuity conditions and gives unity at the nodal point concerned.Polynomials in one coordinate having this property are known as Lagrange polynomialsand can be written down directly aslk…† n ˆ … ÿ 0†… ÿ 1 †… ÿ k ÿ 1 †… ÿ k ‡ 1 †… ÿ n †…8:18†… k ÿ 0 †… k ÿ 1 †… k ÿ k ÿ 1 †… k ÿ k ‡ 1 †… k ÿ n †giving unity at k and passing through n points.(0, m) (I, J)(n, m)1(0, 0) (n, 0)11N iFig. 8.6 A typical shape function for a Lagrangian element (n ˆ 5, m ˆ 4, I ˆ 1, J ˆ 4).

Rectangular elements ± Lagrange family 173(a)(b)Fig. 8.7 Three elements of the Lagrange family: (a) linear, (b) quadratic, and (c) cubic.(c)Thus in two dimensions, if we label the node by its column and row number, I, J,we haveN i N IJ ˆ l n I …†l m J …†…8:19†where n and m stand for the number of subdivisions in each direction.Figure 8.7 shows a few members of this unlimited family where m ˆ n.Indeed, if we examine the polynomial terms present in a situation where n ˆ m weobserve in Fig. 8.8, based on the Pascal triangle, that a large number of polynomialterms is present above those needed for a complete expansion. 7 However, whenmapping of shape functions is considered (Chapter 9) some advantages occur forthis family.11xy1x 2 xy y 2x 3 x 2 y xy 2 y 3x ny nx 3 y x 2 y 2 xy 3 y 4x 3 y 2 x 2 y 3x n y nx 3 y 3Fig. 8.8 Terms generated by a lagrangian expansion of order 3 3 (or n n). Complete polynomials of order3 (or n).

174 `Standard' and `hierarchical' element shape functions8.6 Rectangular elements ± `serendipity' family 4;5It is usually more e cient to make the functions dependent on nodal values placed onthe element boundary. Consider, for instance, the ®rst three elements of Fig. 8.9. Ineach a progressively increasing and equal number of nodes is placed on the elementboundary. The variation of the function on the edges to ensure continuity is linear,parabolic, and cubic in increasing element order.To achieve the shape function for the ®rst element it is obvious that a product oflinear lagrangian polynomials of the form14… ‡ 1†… ‡ 1† …8:20†gives unity at the top right corners where ˆ ˆ 1 and zero at all the other corners.Further, a linear variation of the shape function of all sides exists and hencecontinuity is satis®ed. Indeed this element is identical to the lagrangian one with n ˆ 1.Introducing new variables 0 ˆ i 0 ˆ i …8:21†in which i , i are the normalized coordinates at node i, the formN i ˆ 14 …1 ‡ 0†…1 ‡ 0 † …8:22†allows all shape functions to be written down in one expression.As a linear combination of these shape functions yields any arbitrary linear variationof u, the second convergence criterion is satis®ed.The reader can verify that the following functions satisfy all the necessary criteriafor quadratic and cubic members of the family.`Quadratic' elementCorner nodes:N i ˆ 14 …1 ‡ 0†…1 ‡ 0 †… 0 ‡ 0 ÿ 1† …8:23†η = 1ξ = –1ηξξ = 1(a)η = –1(b)(c)(d)Fig. 8.9 Rectangles of boundary node (serendipity) family: (a) linear, (b) quadratic, (c) cubic, (d) quartic.

Rectangular elements ± `serendipity' family 175Mid-side nodes:`Cubic' elementCorner nodes:Mid-side nodes: i ˆ 0 N i ˆ 12 …1 ÿ 2 †…1 ‡ 0 † i ˆ 0 N i ˆ 12 …1 ‡ 0†…1 ÿ 2 †N i ˆ 132 …1 ‡ 0†…1 ‡ 0 †‰ÿ10 ‡ 9… 2 ‡ 2 †Š …8:24† i ˆ1 and i ˆ 1 3N i ˆ 932 …1 ‡ 0†…1 ÿ 2 †…1 ‡ 9 0 †with the remaining mid-side node expression obtained by changing variables.In the next, quartic, member 8 of this family a central node is added so that all termsof a complete fourth-order expansion will be available. This central node adds a shapefunction …1 ÿ 2 †…1 ÿ 2 † which is zero on all outer boundaries.The above functions were originally derived by inspection, and progression to yethigher members is di cult and requires some ingenuity. It was therefore appropriate1 5 24 7 38 61.01(a) N 5 = (1 – ξ 2 12 ) (1 – η)(b) N 8 = (1 – ξ ) (1 – η 2 2)1.0Step 11.00.5ηξNˆ1 = (1 – ξ) (1 – η)/40.5Step 2Nˆ 11 – 2 N 5(c)Step 31.00.5N 1 = Nˆ 11 – 2 N 5 – 1 2 N 8Fig. 8.10 Systematic generation of `serendipity' shape functions.

176 `Standard' and `hierarchical' element shape functionsto name this family `serendipity' after the famous princes of Serendip noted for theirchance discoveries (Horace Walpole, 1754).However, a quite systematic way of generating the `serendipity' shape functions canbe devised, which becomes apparent from Fig. 8.10 where the generation of aquadratic shape function is presented. 7;9As a starting point we observe that for mid-side nodes a lagrangian interpolation ofa quadratic linear type su ces to determine N i at nodes 5 to 8. N 5 and N 8 are shownat Fig. 8.10(a) and (b). For a corner node, such as Fig. 8.10(c), we start with a bilinearlagrangian family ^N 1 and note immediately that while ^N 1 ˆ 1 at node 1, it is not zeroat nodes 5 or 8 (step 1). Successive subtraction of 1 2 N 5 (step 2) and 1 2 N 8 (step 3) ensuresthat a zero value is obtained at these nodes. The reader can verify that the expressionsobtained coincide with those of Eq. (8.23).Indeed, it should now be obvious that for all higher order elements the mid-side andcorner shape functions can be generated by an identical process. For the former asimple multiplication of mth-order and ®rst-order lagrangian interpolations su ces.For the latter a combination of bilinear corner functions, together with appropriatefractions of mid-side shape functions to ensure zero at appropriate nodes, isnecessary.Similarly, it is quite easy to generate shape functions for elements with di€erentnumbers of nodes along each side by a systematic algorithm. This may be very431 5 6 24 31.01.01 5 6 26N 5 N 61.01N 1 = Nˆ 1 – 2 3 N 5 – 1 3 N 6Fig. 8.11 Shape functions for a transition `serendipity' element, cubic/linear.

Elimination of internal variables before assembly ± substructures 177x1y1x yx 2 y 2x 3 y 3 x m y mx m yxy mFig. 8.12 Terms generated by edge shape functions in serendipity-type elements (3 3 and m m†:desirable if a transition between elements of di€erent order is to be achieved, enablinga di€erent order of accuracy in separate sections of a large problem to be studied.Figure 8.11 illustrates the necessary shape functions for a cubic/linear transition.Use of such special elements was ®rst introduced in reference 9, but the simplerformulation used here is that of reference 7.With the mode of generating shape functions for this class of elements available it isimmediately obvious that fewer degrees of freedom are now necessary for a givencomplete polynomial expansion. Figure 8.12 shows this for a cubic element whereonly two surplus terms arise (as compared with six surplus terms in a lagrangian ofthe same degree).It is immediately evident, however, that the functions generated by nodes placedonly along the edges will not generate complete polynomials beyond cubic order.For higher order ones it is necessary to supplement the expansion by internalnodes (as was done in the quartic element of Fig. 8.9) or by the use of `nodeless'variables which contain appropriate polynomial terms.8.7 Elimination of internal variables before assembly ±substructuresInternal nodes or nodeless internal parameters yield in the usual way the elementproperties (Chapter 2)@ e@a e ˆ Ke a e ‡ f e…8:25†As a e can be subdivided into parts which are common with other elements, a e ,andothers which occur in the particular element only, a e , we can immediately write@@a e ˆ @e@a e ˆ 0

178 `Standard' and `hierarchical' element shape functionsand eliminate a e from further consideration. Writing Eq. (8.25) in a partitioned formwe have8@ e 98@ e >< >= @a e ˆ @a e K e ^Ke ae f e @ e 9< =@ >:e ˆ^K >;eT K e a e ‡ feˆ @a e …8:26†: ;0@a eFrom the second set of equations given above we can writewhich on substitution yieldsa e ˆÿ…K e † ÿ1 …^K eT a e ‡ f e †@ e@a e ˆ Ke a e ‡ f e…8:27†…8:28†in whichK e ˆ K e ÿ ^K e …K e † ÿ1 ^K eTf e ˆ f e ÿ ^K e … K …8:29†e † ÿ1 feAssembly of the total region then follows, by considering only the element boundaryvariables, thus giving a considerable saving in the equation-solving e€ort at theexpense of a few additional manipulations carried out at the element stage.Perhaps a structural interpretation of this elimination is desirable. What in fact isinvolved is the separation of a part of the structure from its surroundings anddetermination of its solution separately for any prescribed displacements at the interconnectingboundaries. K e is now simply the overall sti€ness of the separatedstructure and f e the equivalent set of nodal forces.If the triangulation of Fig. 8.13 is interpreted as an assembly of pin-jointed bars thereader will recognize immediately the well-known device of `substructures' usedfrequently in structural engineering.Such a substructure is in fact simply a complex element from which the internaldegrees of freedom have been eliminated.Immediately a new possibility for devising more elaborate, and presumably moreaccurate, elements is presented.(a)(b)Fig. 8.13 Substructure of a complex element.

Triangular element family 179Fig. 8.14 A quadrilateral made up of four simple triangles.Figure 8.13(a) can be interpreted as a continuum ®eld subdivided into triangularelements. The substructure results in fact in one complex element shown in Fig.8.13(b) with a number of boundary nodes.The only di€erence from elements derived in previous sections is the fact that theunknown u is now not approximated internally by one set of smooth shape functionsbut by a series of piecewise approximations. This presumably results in a slightlypoorer approximation but an economic advantage may arise if the total computationtime for such an assembly is saved.Substructuring is an important device in complex problems, particularly where arepetition of complicated components arises.In simple, small-scale ®nite element analysis, much improved use of simpletriangular elements was found by the use of simple subassemblies of the triangles(or indeed tetrahedra). For instance, a quadrilateral based on four triangles fromwhich the central node is eliminated was found to give an economic advantageover direct use of simple triangles (Fig. 8.14). This and other subassemblies basedon triangles are discussed in detail by Doherty et al. 108.8 Triangular element familyThe advantage of an arbitrary triangular shape in approximating to any boundaryshape has been amply demonstrated in earlier chapters. Its apparent superiorityhere over rectangular shapes needs no further discussion. The question of generatingmore elaborate higher order elements needs to be further developed.Consider a series of triangles generated on a pattern indicated in Fig. 8.15. Thenumber of nodes in each member of the family is now such that a complete polynomialexpansion, of the order needed for interelement compatibility, is ensured.This follows by comparison with the Pascal triangle of Fig. 8.5 in which we see thenumber of nodes coincides exactly with the number of polynomial terms required.This particular feature puts the triangle family in a special, privileged position, inwhich the inverse of the C matrices of Eq. (8.11) will always exist. 3 However, onceagain a direct generation of shape functions will be preferred ± and indeed will beshown to be particularly easy.Before proceeding further it is useful to de®ne a special set of normalized coordinatesfor a triangle.

180 `Standard' and `hierarchical' element shape functions33(a)1 26 53(b)1 4 28 79 610(c)1 4 5 2Fig. 8.15 Triangular element family: (a) linear, (b) quadratic, and (c) cubic.8.8.1 Area coordinatesWhile cartesian directions parallel to the sides of a rectangle were a natural choice forthat shape, in the triangle these are not convenient.A new set of coordinates, L 1 , L 2 , and L 3 for a triangle 1, 2, 3 (Fig. 8.16), is de®nedby the following linear relation between these and the cartesian system:x ˆ L 1 x 1 ‡ L 2 x 2 ‡ L 3 x 3y ˆ L 1 y 1 ‡ L 2 y 2 ‡ L 3 y 3…8:30†1 ˆ L 1 ‡ L 2 ‡ L 3To every set, L 1 , L 2 , L 3 (which are not independent, but are related by the third equation),there corresponds a unique set of cartesian coordinates. At point 1, L 1 ˆ 1andL 2 ˆ L 3 ˆ 0, etc. A linear relation between the new and cartesian coordinates impliesL 1 = 0.25L 1 = 03(x 3 y 3 )L 1 = 0.75L 1 = 0.5L 1 = 1P (L 1 L 2 L 3 )1 2(x 1 y 1 ) (x 2 y 2 )Fig. 8.16 Area coordinates.

that contours of L 1 are equally placed straight lines parallel to side 2±3 on whichL 1 ˆ 0, etc.Indeed it is easy to see that an alternative de®nition of the coordinate L 1 of a pointP is by a ratio of the area of the shaded triangle to that of the total triangle:area P23L 1 ˆ …8:31†area 123Hence the name area coordinates.Solving Eq. (8.30) givesL 1 ˆ a1 ‡ b 1 x ‡ c 1 y2L 2 ˆ a2 ‡ b 2 x ‡ c 2 y…8:32†2L 3 ˆ a3 ‡ b 3 x ‡ c 3 y2in whichand1 x 1 y 1 ˆ 12 det 1 x 2 y 2ˆ area 123 1 x 3 y 3a 1 ˆ x 2 y 3 ÿ x 3 y 2 b 1 ˆ y 2 ÿ y 3 c 1 ˆ x 3 ÿ x 2Triangular element family 181…8:33†etc., with cyclic rotation of indices 1, 2, and 3.The identity of expressions with those derived in Chapter 4 [Eqs (4.5b) and (4.5c)] isworth noting.8.8.2 Shape functionsFor the ®rst element of the series [Fig. 8.15(a)], the shape functions are simply the areacoordinates. ThusN 1 ˆ L 1 N 2 ˆ L 2 N 3 ˆ L 3 …8:34†This is obvious as each individually gives unity at one node, zero at others, and varieslinearly everywhere.To derive shape functions for other elements a simple recurrence relation can bederived. 3 However, it is very simple to write an arbitrary triangle of order M in amanner similar to that used for the lagrangian element of Sec. 8.5.Denoting a typical node i by three numbers I, J, andK corresponding to theposition of coordinates L 1i , L 2i ,andL 3i we can write the shape function in termsof three lagrangian interpolations as [see Eq. (8.18)]N i ˆ l I I…L 1 †l J J…L 2 †l K K…L 3 †…8:35†In the above l I I, etc., are given by expression (8.18), with L 1 taking the place of ,etc.

182 `Standard' and `hierarchical' element shape functions(0, 0, M )i = (I, J, K )(M, 0, 0)(0, M, 0)Fig. 8.17 A general triangular element.It is easy to verify that the above expression givesN i ˆ 1 at L 1 ˆ L 1I ; L 2 ˆ L 2J ; L 3 ˆ L 3Kand zero at all other nodes.The highest term occurring in the expansion isL I 1L J 2L K 3and asI ‡ J ‡ K Mfor all points the polynomial is also of order M.Expression (8.35) is valid for quite arbitrary distributions of nodes of the patterngiven in Fig. 8.17 and simpli®es if the spacing of the nodal lines is equal (i.e., 1=m).The formula was ®rst obtained by Argyris et al. 11 and formalized in a di€erentmanner by others. 7;12The reader can verify the shape functions for the second-and third-order elementsas given below and indeed derive ones of any higher order easily.Quadratic triangle [Fig. 8.15(b)]Corner nodes:N 1 ˆ…2L 1 ÿ 1†L 1 ;Mid-side nodes:N 4 ˆ 4L 1 L 2 ;etc:etc:Cubic triangle [Fig. 8.15(c)]Corner nodes:N 1 ˆ 12 …3L 1 ÿ 1†…3L 1 ÿ 2†L 1 ; etc: …8:36†Mid-side nodes:N 4 ˆ 92 L 1L 2 …3L 1 ÿ 1†; etc: …8:37†and for the internal node:N 10 ˆ 27L 1 L 2 L 3

The last shape again is a `bubble' function giving zero contribution along boundaries± and this will be found to be useful in many other contexts (see the mixedforms in Chapter 12).The quadratic triangle was ®rst derived by Veubeke 13 and used later in the contextof plane stress analysis by Argyris. 14When element matrices have to be evaluated it will follow that we are faced withintegration of quantities de®ned in terms of area coordinates over the triangularregion. It is useful to note in this context the following exact integration expression:……L a 1L b 2L c a! b! c!3 dx dy ˆ…a ‡ b ‡ c ‡ 2†! 2…8:38†Line elements 183One-dimensional elements8.9 Line elementsSo far in this book the continuum was considered generally in two or three dimensions.`One-dimensional' members, being of a kind for which exact solutions aregenerally available, were treated only as trivial examples in Chapter 2 and inSec. 8.2. In many practical two-or three-dimensional problems such elements do infact appear in conjunction with the more usual continuum elements ± and a uni®edtreatment is desirable. In the context of elastic analysis these elements may representlines of reinforcement (plane and three-dimensional problems) or sheets of thin liningmaterial in axisymmetric bodies. In the context of ®eld problems of the type discussedin Chapter 7 lines of drains in a porous medium of lesser conductivity can beenvisaged.Once the shape of such a function as displacement is chosen for an element of thiskind, its properties can be determined, noting, however, that derived quantities suchas strain, etc., have to be considered only in one dimension.Figure 8.18 shows such an element sandwiched between two adjacent quadratictypeelements. Clearly for continuity of the function a quadratic variation of theFig. 8.18 A line element sandwiched between two-dimensional elements.

184 `Standard' and `hierarchical' element shape functionsunknown with the one variable is all that is required. Thus the shape functions aregiven directly by the Lagrange polynomial as de®ned in Eq. (8.18).Three-dimensional elements8.10 Rectangular prisms ± Lagrange familyIn a precisely analogous way to that given in previous sections equivalent elements ofthree-dimensional type can be described.Now, for interelement continuity the simple rules given previously have to bemodi®ed. What is necessary to achieve is that along a whole face of an element thenodal values de®ne a unique variation of the unknown function. With incompletepolynomials, this can be ensured only by inspection.Shape function for such elements, illustrated in Fig. 8.19, will be generated by adirect product of three Lagrange polynomials. Extending the notation of Eq. (8.19)we now haveN i N IJK ˆ l n I l m J l p K…8:39†for n, m, andp subdivisions along each side.This element again is suggested by Zienkiewicz et al. 5 and elaborated upon byArgyris et al. 6 All the remarks about internal nodes and the properties of the formulationwith mappings (to be described in the next chapter) are applicable here.Fig. 8.19 Right prism of Lagrange family.

Rectangular prisms ± `serendipity' family 1858.11 Rectangular prisms ± `serendipity' family 4;9;15The family of elements shown in Fig. 8.20 is precisely equivalent to that of Fig. 8.9.Using now three normalized coordinates and otherwise following the terminology ofSec. 8.6 we have the following shape functions:`Linear' element (8 nodes)N i ˆ 18 …1 ‡ 0†…1 ‡ 0 †…1 ‡ 0 † …8:40†which is identical with the linear lagrangian element.`Quadratic' element (20 nodes)Corner nodes:N i ˆ 18 …1 ‡ 0†…1 ‡ 0 †…1 ‡ 0 †… 0 ‡ 0 ‡ 0 ÿ 2† …8:41†ξ = –18 nodesζηξζ = 1η = 1η = –1ζ = –1ξ = 120 nodes32 nodesFig. 8.20 Right prisms of boundary node (serendipity) family with corresponding sheet and line elements.

186 `Standard' and `hierarchical' element shape functionsTypical mid-side node: i ˆ 0 i ˆ1 i ˆ1N i ˆ 14 …1 ÿ 2 †…1 ‡ 0 †…1 ‡ 0 †`Cubic' elements (32 nodes)Corner node:N i ˆ 164 …1 ‡ 0†…1 ‡ 0 †…1 ‡ 0 †‰9… 2 ‡ 2 ‡ 2 †ÿ19Š …8:42†Typical mid-side node: i ˆ 1 3 i ˆ1 i ˆ1N i ˆ 964 …1 ÿ 2 †…1 ‡ 9 0 †…1 ‡ 0 †…1 ‡ 0 †When ˆ 1 ˆ 0 the above expressions reduce to those of Eqs (8.22)±(8.24).Indeed such elements of three-dimensional type can be joined in a compatiblemanner to sheet or line elements of the appropriate type as shown in Fig. 8.20.Once again the procedure for generating the shape functions follows that describedin Figs 8.10 and 8.11 and once again elements with varying degrees of freedom alongthe edges can be derived following the same steps.The equivalent of a Pascal triangle is now a tetrahedron and again we can observethe small number of surplus degrees of freedom ± a situation of even greater magnitudethan in two-dimensional analysis.8.12 Tetrahedral elementsThe tetrahedral family shown in Fig. 8.21 not surprisingly exhibits properties similarto those of the triangle family.Firstly, once again complete polynomials in three coordinates are achieved at eachstage. Secondly, as faces are divided in a manner identical with that of the previoustriangles, the same order of polynomial in two coordinates in the plane of the face isachieved and element compatibility ensured. No surplus terms in the polynomial occur.8.12.1 Volume coordinatesOnce again special coordinates are introduced de®ned by (Fig. 8.22):x ˆ L 1 x 1 ‡ L 2 x 2 ‡ L 3 x 3 ‡ L 4 x 4Solving Eq. (8.43) givesy ˆ L 1 y 1 ‡ L 2 y 2 ‡ L 3 y 3 ‡ L 4 y 4…8:43†z ˆ L 1 z 1 ‡ L 2 z 2 ‡ L 3 z 3 ‡ L 4 z 41 ˆ L 1 ‡ L 2 ‡ L 3 ‡ L 4L 1 ˆ a1 ‡ b 1 x ‡ c 1 y ‡ d 1 z6Vetc:

Tetrahedral elements 18711(a) 4 nodes341x 4x 2 yxz2y6x 2 xzz 2xyyz3x 3yx 2 z2 yz 2 z 3x 2 yxyzyz 2xy 2y 2 zx 3 z x 2 z 2 y 3 xz 3z 4x 3 yx 2 yzxyz 2yz 32 xy 3 zy 2 z 2xy 3y 3 zy 4859(b) 10 nodes714(c) 20 nodes1079258101917 1863 413 1412 20 1611152Fig. 8.21 The tetrahedron family: (a) linear, (b) quadratic, and (c) cubic.where the constants can be identi®ed from Chapter 6, Eq. (6.5). Again the physicalnature of the coordinates can be identi®ed as the ratio of volumes of tetrahedrabased on an internal point P in the total volume, e.g., as shown in Fig. 8.22:volume P234L 1 ˆ ; etc: …8:44†volume 12348.12.2 Shape functionAs the volume coordinates vary linearly with the cartesian ones from unity at onenode to zero at the opposite face then shape functions for the linear element[Fig. 8.21(a)] are simplyN 1 ˆ L 1 N 2 ˆ L 2 ; etc: …8:45†Formulae for shape functions of higher order tetrahedra are derived in precisely thesame manner as for the triangles by establishing appropriate Lagrange-type formulaesimilar to Eq. (8.35). Leaving this to the reader as a suitable exercise we quote thefollowing:

188 `Standard' and `hierarchical' element shape functions4P312Fig. 8.22 Volume coordinates.`Quadratic' tetrahedron [Fig. 8.21(b)]For corner nodes:N 1 ˆ…2L 1 ÿ 1†L 1 ; etc: …8:46†For mid-edge nodes:N 5 ˆ 4L 1 L 2 ;etc:`Cubic' tetrahedronCorner nodes:Mid-edge nodes:N 1 ˆ 12 …3L 1 ÿ 1†…3L 1 ÿ 2†L 1 ; etc: …8:47†Mid-face nodes:N 5 ˆ 92 L 1L 2 …3L 1 ÿ 1†; etc:N 17 ˆ 27L 1 L 2 L 3 ;etc:A useful integration formula may again be here quoted:………L a 1L b 2L c 3L d a! b! c! d!4 dx dy dz ˆ…a ‡ b ‡ c ‡ d ‡ 3†! 6Vvol…8:48†

ζ = 1123(a) 6 nodesζ46511211371029(b) 15 nodesIζ = –14II138515614 1817125 161315142781039(c) 26 nodes12411 2324 2619 21202265Fig. 8.23 Triangular prism elements (serendipity) family: (a) linear, (b) quadratic, and (c) cubic.

190 `Standard' and `hierarchical' element shape functions8.13 Other simple three-dimensional elementsThe possibilities of simple shapes in three dimensions are greater, for obvious reasons,than in two dimensions. A quite useful series of elements can, for instance, be basedon triangular prisms (Fig. 8.23). Here again variants of the product, Lagrange,approach or of the `serendipity' type can be distinguished. The ®rst element ofboth families is identical and indeed the shape functions for it are so obvious asnot to need quoting.For the `quadratic' element illustrated in Fig. 8.23(b) the shape functions areCorner nodes L 1 ˆ 1 ˆ 1:N 1 ˆ 12 L 1…2L 1 ÿ 1†…1 ‡ †ÿ 1 2 L 1…1 ÿ 2 †…8:49†Mid-edge of triangles:N 10 ˆ 2L 1 L 2 …1 ‡ †; etc: …8:50†Mid-edge of rectangle:N 7 ˆ L 1 …1 ÿ 2 †;etc:Such elements are not purely esoteric but have a practical application as `®llers' inconjunction with 20-noded serendipity elements.Part 2 Hierarchical shape functions8.14 Hierarchic polynomials in one dimensionThe general ideas of hiearchic approximation were introduced in Sect. 8.2 in thecontext of simple, linear, elements. The idea of generating higher order hierarchicforms is again simple. We shall start from a one-dimensional expansion as this hasbeen shown to provide a basis for the generation of two-and three-dimensionalforms in previous sections.To generate a polynomial of order p along an element side we do not need tointroduce nodes but can instead use parameters without an obvious physical meaning.As shown in Fig. 8.24, we could use here a linear expansion speci®ed by `standard'functions N 0 and N 1 and add to this a series of polynomials always designed so asto have zero values at the ends of the range (i.e. points 0 and 1).Thus for a quadratic approximation, we would write over the typical onedimensionalelement, for instance,^u ˆ u 0 N 0 ‡ u 1 N 1 ‡ a 2 N 2…8:51†whereN 0 ˆÿ ÿ 12N 1 ˆ ‡ 12N 2 ˆÿ… ÿ 1†… ‡ 1†using in the above the normalized x-coordinate [viz. Eq. (8.17)].…8:52†

Hierarchic polynomials in one dimension 191N e 0 N e 1Elementnodes 0 11011dN e 1dξdN e 0dξ0 10 12dN e 2dξN e 2N e 30 120 1dN e 3dξN e 40 10 16dN e 4dξFig. 8.24 Hierarchical element shape functions of nearly orthogonal form and their derivatives.We note that the parameter a 2 does in fact have a meaning in this case as it is themagnitude of the departure from linearity of the approximation ^u at the elementcentre, since N 2 has been chosen here to have the value of unity at that point.In a similar manner, for a cubic element we simply have to add a 3 N 3 to the quadraticexpansion of Eq. (8.51), where N 3 is any cubic of the formN e 3 ˆ 0 ‡ 1 ‡ 2 2 ‡ 3 3…8:53†and which has zero values at ˆ1 (i.e., at nodes 0 and 1). Again an in®nity of choicesexists, and we could select a cubic of a simple form which has a zero value at the centreof the element and for which dN 3 =d ˆ 1 at the same point. Immediately we can writeN e 3 ˆ …1 ÿ 2 †…8:54†as the cubic function with the desired properties. Now the parameter a 3 denotesthe departure of the slope at the centre of the element from that of the ®rstapproximation.

192 `Standard' and `hierarchical' element shape functionsWe note that we could proceed in a similar manner and de®ne the fourth-orderhierarchical element shape function asN e 4 ˆ 2 …1 ÿ 2 †…8:55†but a physical identi®cation of the parameter associated with this now becomes moredi cult (even though it is not strictly necessary).As we have already noted, the above set is not unique and many other possibilitiesexist. An alternative convenient form for the hierarchical functions is de®ned by81>:p! …p ÿ † p oddwhere p ( 52) is the degree of the introduced polynomial. 16 This yields the set of shapefunctions:N2 e ˆ 12 …2 ÿ 1† N3 e ˆ 16 …3 ÿ †…8:57†N4 e ˆ 124 …4 ÿ 1† N5 e ˆ 1120 …5 ÿ † etc:We observe that all derivatives of Np e of second or higher order have the value zeroat ˆ 0, apart from d p Np=d e p , which equals unity at that point, and hence, whenshape functions of the form given by Eq. (8.57) are used, we can identify theparameters in the approximation asa e p ˆ dp^u d p p 5 2 …8:58†ˆ 0This identi®cation gives a general physical signi®cance but is by no means necessary.In two-and three-dimensional elements a simple identi®cation of the hierarchicparameters on interfaces will automatically ensure C 0 continuity of the approximation.As mentioned previously, an optimal form of hierarchical function is one thatresults in a diagonal equation system. This can on occasion be achieved, or at leastapproximated, quite closely.In the elasticity problems which we have discussed in the preceding chapters theelement matrix K e possesses terms of the form [using Eq. (8.17)]…Klm e ˆ e k dNe ldxdNmedx dx ˆ 1 … 1k dNe la ÿ1 ddN e mdd …8:59†If shape function sets containing the appropriate polynomials can be found for whichsuch integrals are zero for l 6ˆ m, then orthogonality is achieved and the couplingbetween successive solutions disappears.One set of polynomial functions which is known to possess this orthogonalityproperty over the range ÿ1 4 4 1 is the set of Legendre polynomials P p …†, andthe shape functions could be de®ned in terms of integrals of these polynomials. 9Here we de®ne the Legendre polynomial of degree p byP p …† ˆ1 1 d p…p ÿ 1†! 2 p ÿ 1 d p ‰…2 ÿ 1† p Š…8:60†

Triangle and tetrahedron family 16;17 193and integrate these polynomials to de®ne…Np e 1 d p ÿ 1‡ 1 ˆ P p …† d ˆ…p ÿ 1†! 2 p ÿ 1 d p ÿ 1 ‰…2 ÿ 1† p Š …8:61†Evaluation for each p in turn givesN2 e ˆ 2 ÿ 1 N3 e ˆ 2… 3 ÿ † etc:These di€er from the element shape functions given by Eq. (8.57) only by a multiplyingconstant up to N3, e but for p 5 3 the di€erences become signi®cant. The reader caneasily verify the orthogonality of the derivatives of these functions, which is useful incomputation. A plot of these functions and their derivatives is given in Fig. 8.24.8.15 Two- and three-dimensional, hierarchic, elements ofthe `rectangle' or `brick' typeIn deriving `standard' ®nite element approximations we have shown that all shapefunctions for the Lagrange family could be obtained by a simple multiplication ofone-dimensional ones and those for serendipity elements by a combination of suchmultiplications. The situation is even simpler for hierarchic elements. Here all theshape functions can be obtained by a simple multiplication process.Thus, for instance, in Fig. 8.25 we show the shape functions for a lagrangian ninenodedelement and the corresponding hierarchical functions. The latter not only havesimpler shapes but are more easily calculated, being simple products of linear andquadratic terms of Eq. (8.56), (8.57), or (8.61). Using the last of these the threefunctions illustrated are simplyN 1 ˆ…1 ÿ †…1 ‡ †=4N 2 ˆ…1 ÿ †…1 ÿ 2 †=2N 3 ˆ…1 ÿ 2 †…1 ÿ 2 †…8:62†The distinction between lagrangian and serendipity forms now disappears as forthe latter in the present case the last shape function …N 3 † is simply omitted.Indeed, it is now easy to introduce interpolation for elements of the type illustratedin Fig. 8.11 in which a di€erent expansion is used along di€erent sides. This essentialcharacteristic of hierarchical elements is exploited in adaptive re®nement (viz.Chapter 15) where new degrees of freedom (or polynomial order increase) is madeonly when required by the magnitude of the error.8.16 Triangle and tetrahedron family 16;17Once again the concepts of multiplication can be introduced in terms of area (volume)coordinates.Returning to the triangle of Fig. 8.16 we note that along the side 1±2, L 3 is identicallyzero, and therefore we have…L 1 ‡ L 2 † 1ÿ2 ˆ 1…8:63†

194 `Standard' and `hierarchical' element shape functions12 31231N s 11N h 1N s 2N h 211N s 3N h 311(a) Standard(b) HierarchicalFig. 8.25 Standard and hierarchic shape functions corresponding to a lagrangian, quadratic element.If , measured along side 1±2, is the usual non-dimensional local element coordinateof the type we have used in deriving hierarchical functions for one-dimensionalelements, we can writeL 1 j 1ÿ2 ˆ 12 …1 ÿ † L 2j 1ÿ2 ˆ 12…1 ‡ † …8:64†from which it follows that we have ˆ…L 2 ÿ L 1 † 1ÿ2…8:65†This suggests that we could generate hierarchical shape functions over the triangleby generalizing the one-dimensional shape function forms produced earlier. For

Triangle and tetrahedron family 16;17 195example, using the expressions of Eq. (8.56), we associate with the side 1±2 thepolynomial of degree p ( 52) de®ned by81>:p! ‰…L 2 ÿ L 1 † p ÿ…L 2 ÿ L 1 †…L 1 ‡ L 2 † p ÿ 1 Š p oddIt follows from Eq. (8.64) that these shape functions are zero at nodes 1 and 2.In addition, it can easily be shown that Np…1ÿ2† e will be zero all along the sides 3±1and 3±2 of the triangle, and so C 0 continuity of the approximation ^u is assured.It should be noted that in this case for p 5 3 the number of hierarchical functionsarising from the element sides in this manner is insu cient to de®ne a completepolynomial of degree p, and internal hierarchical functions, which are identicallyzero on the boundaries, need to be introduced; for example, for p ˆ 3 the functionL 1 L L 3 could be used, while for p ˆ 4 the three additional functions L 2 1L 2 L 3 ,L 1 L 2 2L 3 , L 1 L 2 L 2 3 could be adopted.In Fig. 8.26 typical hierarchical linear, quadratic, and cubic trial functions for atriangular element are shown. Similar hierarchical shape functions could be generated11N e 2e(a)2 31–2N e 2(I – 2)(b)e2 316N e 3(I – 2)(c)e2 3Fig. 8.26 Triangular elements and associated hierarchical shape functions of (a) linear, (b) quadratic, and(c) cubic form.

196 `Standard' and `hierarchical' element shape functionsfrom the alternative set of one-dimensional shape functions de®ned in Eq. (8.61).Identical procedures are obvious in the context of tetrahedra.8.17 Global and local ®nite element approximationThe very concept of hierarchic approximations (in which the shape functions are nota€ected by the re®nement) means that it is possible to include in the expansionu ˆ XnN i a i…8:67†i ˆ 1functions N which are not local in nature. Such functions may, for instance, be theexact solutions of an analytical problem which in some way resembles the problemdealt with, but do not satisfy some boundary or inhomogeneity conditions. The`®nite element', local, expansions would here be a device for correcting this solutionto satisfy the real conditions. This use of the global±local approximation was ®rstsuggested by Mote 18 in a problem where the coe cients of this function were ®xed.The example involved here is that of a rotating disc with cutouts (Fig. 8.27). Theglobal, known, solution is the analytical one corresponding to a disc withoutcutout, and ®nite elements are added locally to modify the solution. Other examplesof such `®xed' solutions may well be those associated with point loads, where the useof the global approximation serves to eliminate the singularity modelled badly by thediscretization.(a)‘Local’ elements(b)Fig. 8.27 Some possible uses of the local±global approximation: (a) rotating slotted disc, (b) perforatedbeam.

Improvement of conditioning with hierarchic forms 197In some problems the singularity itself is unknown and the appropriate functioncan be added with an unknown coe cient.8.18 Improvement of conditioning with hierarchic formsWe have already mentioned that hierarchic element forms give a much improvedequation conditioning for steady-state (static) problems due to their form which ismore nearly diagonal. In Fig. 8.28 we show the `condition number' (which is ameasure of such diagonality and is de®ned in standard texts on linear algebra; seeAppendix A) for a single cubic element and for an assembly of four cubic elements,using standard and hierarchic forms in their formulation. The improvement of theconditioning is a distinct advantage of such forms and allows the use of iterative solutiontechniques to be more easily adopted. 19 Unfortunately much of this advantagedisappears for transient analysis as the approximation must contain speci®c modes(see Chapter 17).Single element (Reduction of condition number = 10.7)ABλ max /λ min = 390 λ max /λ min = 36Four element assembly (Reduction of condition number = 13.2)ABλ max /λ min = 1643 λ max /λ min = 124Cubic order elementsA Standard shape functionBHierarchic shape functionFig. 8.28 Improvement of condition number (ratio of maximum to minimum eigenvalue of the stiffnessmatrix) by use of a hierarchic form (elasticity isotropic ˆ 0:15).

198 `Standard' and `hierarchical' element shape functions8.19 Concluding remarksAn unlimited selection of element types has been presented here to the reader ± andindeed equally unlimited alternative possibilities exist. 4;9 What of the use of suchcomplex elements in practice? The triangular and tetrahedral elements are limitedto situations where the real region is of a suitable shape which can be representedas an assembly of ¯at facets and all other elements are limited to situations representedby an assembly of right prisms. Such a limitation would be so severe thatlittle practical purpose would have been served by the derivation of such shape functionsunless some way could be found of distorting these elements to ®t realisticcurved boundaries. In fact, methods for doing this are available and will be describedin the next chapter.References1. W. Rudin. Principles of Mathematical Analysis. 3rd ed, McGraw-Hill, 1976.2. P.C. Dunne. Complete polynomial displacement ®elds for ®nite element methods. Trans.Roy. Aero. Soc. 72, 245, 1968.3. B.M. Irons, J.G. Ergatoudis, and O.C. Zienkiewicz. Comment on ref. 1. Trans. Roy. Aero.Soc. 72, 709±11, 1968.4. J.G. Ergatoudis, B.M. Irons, and O.C. Zienkiewicz. Curved, isoparametric, quadrilateralelements for ®nite element analysis. Int. J. Solids Struct. 4, 31±42, 1968.5. O.C. Zienkiewicz et al. Iso-parametric and associated elements families for two andthree dimensional analysis. Chapter 13 of Finite Element Methods in Stress Analysis(eds I. Holand and K. Bell), Tech. Univ. of Norway, Tapir Press, Norway, Trondheim,1969.6. J.H. Argyris, K.E. Buck, H.M. Hilber, G. Mareczek, and D.W. Scharpf. Some newelements for matrix displacement methods. 2nd Conf. on Matrix Methods in Struct.Mech. Air Force Inst. of Techn., Wright Patterson Base, Ohio, Oct. 1968.7. R.L. Taylor. On completeness of shape functions for ®nite element analysis. Int. J. Num.Meth. Eng. 4, 17±22, 1972.8. F.C. Scott. A quartic, two dimensional isoparametric element. Undergraduate Project,Univ. of Wales, Swansea, 1968.9. O.C. Zienkiewicz, B.M. Irons, J. Campbell, and F.C. Scott. Three dimensional stressanalysis. Int. Un. Th. Appl. Mech. Symposium on High Speed Computing in Elasticity.LieÂge, 1970.10. W.P. Doherty, E.L. Wilson, and R.L. Taylor. Stress Analysis of Axisymmetric SolidsUtilizing Higher-Order Quadrilateral Finite Elements. Report 69±3, Structural EngineeringLaboratory, Univ. of California, Berkeley, Jan. 1969.11. J.H. Argyris, I. Fried, and D.W. Scharpf. The TET 20 and the TEA 8 elements for thematrix displacement method. Aero. J. 72, 618±25, 1968.12. P. Silvester. Higher order polynomial triangular ®nite elements for potential problems. Int.J. Eng. Sci. 7, 849±61, 1969.13. B. Fraeijs de Veubeke. Displacement and equilibrium models in the ®nite element method.Chapter 9 of Stress Analysis (eds O.C. Zienkiewicz and G.S. Holister), Wiley, 1965.14. J.H. Argyris. Triangular elements with linearly varying strain for the matrix displacementmethod. J. Roy. Aero. Soc. Tech. Note. 69, 711±13, Oct. 1965.

15. J.G. Ergatoudis, B.M. Irons, and O.C. Zienkiewicz. Three dimensional analysis of archdams and their foundations. Symposium on Arch Dams. Inst. Civ. Eng., London, 1968.16. A.G. Peano. Hierarchics of conforming ®nite elements for elasticity and plate bending.Comp. Math. and Applications. 2, 3±4, 1976.17. J.P. de S.R. Gago. A posteri error analysis and adaptivity for the ®nite element method. Ph.Dthesis, University of Wales, Swansea, 1982.18. C.D. Mote, Global±local ®nite element. Int. J. Num. Meth. Eng. 3, 565±74, 1971.19. O.C. Zienkiewicz, J.P. de S.R. Gago, and D.W. Kelly. The hierarchical concept in ®niteelement analysis. Computers and Structures. 16, 53±65, 1983.References 199

9Mapped elements and numericalintegration ± `in®nite' and`singularity' elements9.1 IntroductionIn the previous chapter we have shown how some general families of ®nite elementscan be obtained for C 0 interpolations. A progressively increasing number of nodesand hence improved accuracy characterizes each new member of the family andpresumably the number of such elements required to obtain an adequate solutiondecreases rapidly. To ensure that a small number of elements can represent a relativelycomplex form of the type that is liable to occur in real,rather than academic,problems,simple rectangles and triangles no longer su ce. This chapter is thereforeconcerned with the subject of distorting such simple forms into others of morearbitrary shape.Elements of the basic one-,two-,or three-dimensional types will be `mapped' intodistorted forms in the manner indicated in Figs 9.1 and 9.2.In these ®gures it is shown that the , , ,or L 1 L 2 L 3 L 4 coordinates can be distortedto a new,curvilinear set when plotted in cartesian x, y, z space.Not only can two-dimensional elements be distorted into others in two dimensionsbut the mapping of these can be taken into three dimensions as indicated by the ¯atsheet elements of Fig. 9.2 distorting into a three-dimensional space. This principleapplies generally,providing a one-to-one correspondence between cartesian andcurvilinear coordinates can be established,i.e.,once the mapping relations of the type8 9 8 9 89>< x >= >< f x …; ; † >= >< f x …L 1 ; L 2 ; L 3 ; L 4 † >=y>: >; ˆ f y …; ; † or f y …L 1 ; L 2 ; L 3 ; L 4 † …9:1†>: >; >:>;z f z …; ; †f z …L 1 ; L 2 ; L 3 ; L 4 †can be established.Once such coordinate relationships are known,shape functions can be speci®ed inlocal coordinates and by suitable transformations the element properties establishedin the global coordinate system.In what follows we shall ®rst discuss the so-called isoparametric form of relationship(9.1) which has found a great deal of practical application. Full details ofthis formulation will be given,including the establishment of element matrices bynumerical integration.

yη(x, y)(–1, 1) (1, 1)ηξξ(–1, –1)(1, –1)L 2 = 03L 2 = 031L 1 = 0L 1 = 0L 3 = 02Local coordinates1L 3 = 02Cartesian mapxFig. 9.1 Two-dimensional `mapping' of some elements.

yη = 1 ξ = 1ηη = 1ζηζξξη = –144L 1 = 0L 1 = 113132zLocal coordinates2Cartesian mapxFig. 9.2 Three-dimensional `mapping' of some elements.

Use of `shape functions' in the establishment of coordinate transformations 203In the ®nal section we shall show that many other coordinate transformations canbe used e€ectively.Parametric curvilinear coordinates9.2 Use of `shape functions' in the establishment ofcoordinate transformationsA most convenient method of establishing the coordinate transformations is to usethe `standard' type of C 0 shape functions we have already derived to represent thevariation of the unknown function.If we write,for instance,for each element8 9x >< 1 >=x ˆ N1x 0 1 ‡ N2x 0 2 ‡ˆN 0 x 2>: . >; ˆ N0 x8 9y >< 1 >=y ˆ N1 0 y 1 ‡ N2 0 y 2 ‡ˆN 0 y 2>: . >; ˆ N0 y…9:2†8 9z >< 1 >=z ˆ N1z 0 1 ‡ N2z 0 2 ˆˆN 0 z 2>:. >; ˆ N0 z.in which N 0 are standard shape functions given in terms of the local coordinates,thena relationship of the required form is immediately available. Further,the points withcoordinates x 1 , y 1 , z 1 ,etc.,will lie at appropriate points of the element boundary (asfrom the general de®nitions of the standard shape functions we know that these havea value of unity at the point in question and zero elsewhere). These points canestablish nodes a priori.To each set of local coordinates there will correspond a set of global cartesian coordinatesand in general only one such set. We shall see,however,that a non-uniquenessmay arise sometimes with violent distortion.The concept of using such element shape functions for establishing curvilinearcoordinates in the context of ®nite element analysis appears to have been ®rst introducedby Taig. 1 In his ®rst application basic linear quadrilateral relations were used.Irons 2;3 generalized the idea for other elements.Quite independently the exercises of devising various practical methods of generatingcurved surfaces for purposes of engineering design led to the establishment ofsimilar de®nitions by Coons 4 and Forrest, 5 and indeed today the subjects of surfacede®nitions and analysis are drawing closer together due to this activity.In Fig. 9.3 an actual distortion of elements based on the cubic and quadraticmembers of the two-dimensional `serendipity' family is shown. It is seen here that aone-to-one relationship exists between the local …; † and global …x; y† coordinates.

204 Mapped elements and numerical integrationηη = 1ξξ = 1ξ = –1ηη = –1ξFig. 9.3 Computer plots of curvilinear coordinates for cubic and parabolic elements (reasonable distortion).If the ®xed points are such that a violent distortion occurs then a non-uniqueness canoccur in the manner indicated for two situations in Fig. 9.4. Here at internal points ofthe distorted element two or more local coordinates correspond to the same cartesiancoordinate and in addition to some internal points being mapped outside the element.Care must be taken in practice to avoid such gross distortion.Figure 9.5 shows two examples of a two-dimensional …; † element mapped into athree-dimensional …x; y; z† space.We shall often refer to the basic element in undistorted,local,coordinates as a`parent' element.ηξηξFig. 9.4 Unreasonable element distortion leading to a non-unique mapping and `overspill'. Cubic andparabolic elements.

Use of `shape functions' in the establishment of coordinate transformations 205ξηiξηizZ iyxyzxFig. 9.5 Flat elements (of parabolic type) mapped into three-dimensions.In Sec. 9.5 we shall de®ne a quantity known as the jacobian determinant. The wellknowncondition for a one-to-one mapping (such as exists in Fig. 9.3 and does not inFig. 9.4) is that the sign of this quantity should remain unchanged at all the points ofthe mapped element.It can be shown that with a parametric transformation based on bilinear shapefunctions,the necessary condition is that no internal angle [such as in Fig. 9.6(a)]ηξαα < 180°(a) Linear elementα < 180°α13 L(b) Quadratic elementFig. 9.6 Rules for uniqueness of mapping (a) and (b).13 LLSafe zonefor midpoint

206 Mapped elements and numerical integrationbe greater than 1808. 6 In transformations based on parabolic-type `serendipity' functions,itis necessary in addition to this requirement to ensure that the mid-side nodesare in the `middle half' of the distance between adjacent corners but a `middle third'shown in Fig. 9.6 is safer. For cubic functions such general rules are impractical andnumerical checks on the sign of the jacobian determinant are necessary. In practice aparabolic distortion is usually su cient.9.3 Geometrical conformability of elementsWhile it was shown that by the use of the shape function transformation each parentelement maps uniquely a part of the real object,it is important that the subdivision ofthis into the new,curved,elements should leave no gaps. The possibility of such gapsis indicated in Fig. 9.7.(a)(b)Fig. 9.7 Compatibilityrequirements in a real subdivision of space.Theorem 1. If two adjacent elements are generated from `parents' in which the shapefunctions satisfy C 0 continuity requirements then the distorted elements will be contiguous(compatible).This theorem is obvious,as in such cases uniqueness of any function u required bycontinuity is simply replaced by that of uniqueness of the x, y,or z coordinate. Asadjacent elements are given the same sets of coordinates at nodes,continuity isimplied.9.4 Variation of the unknown function within distorted,curvilinear elements. Continuity requirementsWith the shape of the element now de®ned by the shape functions N 0 the variation ofthe unknown, u,has to be speci®ed before we can establish element properties. This is

Variation of the unknown function within distorted, curvilinear elements 207(a)(b)(c)Fig. 9.8 Various element speci®cations: k point at which coordinate is speci®ed; h points at which thefunction parameter is speci®ed. (a) Isoparametric, (b) superparametric, (c) subparametric.most conveniently given in terms of local,curvilinear coordinates by the usualexpressionu ˆ Na e…9:3†where a e lists the nodal values.Theorem 2. If the shape functions N used in (9.3) are such that C 0 continuity of u ispreserved in the parent coordinates then C 0 continuity requirements will be satis®ed indistorted elements.The proof of this theorem follows the same lines as that in the previous section.The nodal values may or may not be associated with the same nodes as used tospecify the element geometry. For example,in Fig. 9.8 the points marked with acircle are used to de®ne the element geometry. We could use the values of the functionde®ned at nodes marked with a square to de®ne the variation of the unknown.In Fig. 9.8(a) the same points de®ne the geometry and the ®nite element analysispoints. If thenN ˆ N 0…9:4†i.e.,the shape functions de®ning the geometry and the function are the same,theelements will be called isoparametric.We could,however,use only the four corner points to de®ne the variation of u[Fig. 9.8(b)]. We shall refer to such an element as superparametric,noting that thevariation of geometry is more general than that of the actual unknown.Similarly,if for instance we introduce more nodes to de®ne u than are used to de®nethe geometry, subparametric elements will result [Fig. 9.8(c)].

208 Mapped elements and numerical integrationWhile for mapping it is convenient to use `standard' forms of shape functions theinterpolation of the unknown can,of course,use hierarchic forms de®ned in theprevious chapter. Once again the de®nitions of sub- and superparametric variationsare applicable.Transformations9.5 Evaluation of element matrices (transformation in n,g, f coordinates)To perform ®nite element analysis the matrices de®ning element properties,e.g.,sti€ness,etc.,haveto be found. These will be of the form…G dV…9:5†Vin which the matrix G depends on N or its derivatives with respect to global coordinates.As an example of this we have the sti€ness matrix…B T DB dV…9:6†and associated body force vectorsV…VN T b dV…9:7†For each particular class of elastic problems the matrices of B are given explicitly bytheir components [see the general form of Eqs (4.10),(5.6),and (6.11)]. Quoting the®rst of these,Eq. (4.10),valid for plane problems we have2B i ˆ64@N i@x ; 00;@N i@y ;3@N i@y7@N 5i@x…9:8†In elasticity problems the matrix G is thus a function of the ®rst derivatives of Nand this situation will arise in many other classes of problem. In all, C 0 continuityis needed and,as we have already noted,this is readily satis®ed by the functions ofChapter 8,written now in terms of curvilinear coordinates.To evaluate such matrices we note that two transformations are necessary. In the®rst place,as N i is de®ned in terms of local (curvilinear) coordinates,it is necessaryto devise some means of expressing the global derivatives of the type occurring inEq. (9.8) in terms of local derivatives.In the second place the element of volume (or surface) over which the integrationhas to be carried out needs to be expressed in terms of the local coordinates with anappropriate change of limits of integration.

Consider,for instance,the set of local coordinates , , and a corresponding set ofglobal coordinates x, y, z. By the usual rules of partial di€erentiation we can write,forinstance,the derivative as@N i@ ˆ @N i @x@x @ ‡ @N i @y@y @ ‡ @N i @z…9:9†@z @Performing the same di€erentiation with respect to the other two coordinates andwriting in matrix form we have8 9 2@N i @x@@ ;>=i@xˆ@@ ;6@N >: i4 @x >;@ @ ;@y@ ;@y@ ;@y@ ;3@z8@@z>:@@N i@x@N i@y@N i@z9>= >;8>:@N i@x@N i@y@N i@z9>=>;…9:10†In the above,the left-hand side can be evaluated as the functions N i are speci®ed inlocal coordinates. Further,as x, y, z are explicitly given by the relation de®ning thecurvilinear coordinates [Eq. (9.2)],the matrix J can be found explicitly in terms ofthe local coordinates. This matrix is known as the jacobian matrix.To ®nd now the global derivatives we invert J and write8 9 8@N i@x >= >: @N i >; >:@z@N i@@N i@@N i@9>=>;…9:11†In terms of the shape function de®ning the coordinate transformation N 0 (which aswe have seen are only identical with the shape functions N when the isoparametricformulation is used) we have2P @Ni0 @ x i;P @Ni0 J ˆ@ x i;64 P @Ni0@ x i;P @Ni0@ y i;P @Ni0@ y i;P @Ni0@ y i;9.5.1 Volume integralsEvaluation of element matrices (transformation in n, g, f coordinates) 209P @Ni0 3 2@ z iP @Ni0 @ z iˆ7 6P @Ni0 5 4@ z i@N 0 1@ ; @N 0 2@N10@ ; @N20@N10@ ; @N20@ ...3@ ...23x 1 ; y 1 ; z 1@ ...6x 2 ; y 2 ; z 7427 .5. . 5…9:12†To transform the variables and the region with respect to which the integration ismade,a standard process will be used which involves the determinant of J. Thus,for instance,a volume element becomesdx dy dz ˆ det J d d d…9:13†

210 Mapped elements and numerical integrationThis type of transformation is valid irrespective of the number of coordinates used.For its justi®cation the reader is referred to standard mathematical texts.y (See alsoAppendix F.)Assuming that the inverse of J can be found we now have reduced the evaluation ofthe element properties to that of ®nding integrals of the form of Eq. (9.5).More explicitly we can write this as… 1ÿ1… 1ÿ1… 1ÿ1G…; ; † d d d…9:14†if the curvilinear coordinates are of the normalized type based on the right prism.Indeed the integration is carried out within such a prism and not in the complicateddistorted shape,thus accounting for the simple integration limits. One- and twodimensionalproblems will similarly result in integrals with respect to one or twocoordinates within simple limits.While the limits of integration are simple in the above case,unfortunately theexplicit form of G is not. Apart from the simplest elements,algebraic integrationusually de®es our mathematical skill,and numerical integration has to be used.This,as will be seen from later sections,is not a severe penalty and has the advantagethat algebraic errors are more easily avoided and that general programs,not tied to aparticular element,can be written for various classes of problems. Indeed in suchnumerical calculations the analytical inverses of J are never explicitly found.9.5.2 Surface integralsIn elasticity and other applications,surface integrals frequently occur. Typical hereare the expressions for evaluating the contributions of surface tractions [see Chapter2,Eq. (2.24b)]:…f ˆÿ N T t dAAThe element dA will generally lie on a surface where one of the coordinates (say ) isconstant.The most convenient process of dealing with the above is to consider dA as avector oriented in the direction normal to the surface (see Appendix F). For threedimensionalproblems we form the vector product8 9 8 9@x @x@ @>: >;@>= >: >;@>=d dand on substitution integrate within the domain ÿ1 4 ; 4 1.y The determinant of the jacobian matrix is known in the literature simply as `the jacobian' and is oftenwritten as@…x; y; z†det J @…; ; †

Element matrices. Area and volume coordinates 211For two dimensions a line length dS arises and here the magnitude is simply8 98 9@x 8 9 @y>< @ >= >< 0 >= >< @ >=n dS ˆ dS ˆ @y 0>:@>: >; d ˆ ÿ @x d>; 1 >:@ >;00on constant surfaces. This may now be reduced to two components for the twodimensionalproblem.9.6 Element matrices. Area and volume coordinatesThe general relationship (9.2) for coordinate mapping and indeed all the followingtheorems are equally valid for any set of local coordinates and could relate thelocal L 1 , L 2 , ...coordinates used for triangles and tetrahedra in the previous chapter,to the global cartesian ones.Indeed most of the discussion of the previous chapter is valid if we simply renamethe local coordinates suitably. However,two important di€erences arise.The ®rst concerns the fact that the local coordinates are not independent and in factnumber one more than the cartesian system. The matrix J would apparently thereforebecome rectangular and would not possess an inverse. The second is simply thedi€erence of integration limits which have to correspond with a triangular ortetrahedral `parent'.The simplest,though perhaps not the most elegant,way out of the ®rst di culty isto consider the last variable as a dependent one. Thus,for example,we can introduceformally,in the case of the tetrahedra, ˆ L 1 ˆ L 2…9:15† ˆ L 31 ÿ ÿ ÿ ˆ L 4(by de®nition in the previous chapter) and thus preserve without change Eq. (9.9) andall the equations up to Eq. (9.14).As the functions N i are given in fact in terms of L 1 , L 2 ,etc.,we must observethat@N i@ ˆ @N i @L 1@L 1 @ ‡ @N i @L 2@L 2 @ ‡ @N i @L 3@L 3 @ ‡ @N i @L 4@L 4 @…9:16†On using Eq. (9.15) this becomes simply@N i@ ˆ @N iÿ @N i@L 1 @L 4with the other derivatives obtainable by similar expressions.

212 Mapped elements and numerical integrationThe integration limits of Eq. (9.14) now change,however,to correspond with thetetrahedron limits,typically…1 … 1 ÿ … 1 ÿ ÿ G…; ; † d d d…9:17†0 0 0The same procedure will clearly apply in the case of triangular coordinates.It must be noted that once again the expression G will necessitate numericalintegration which,however,is carried out over the simple,undistorted,parentregion whether this be triangular or tetrahedral.An alternative to the above is to express the coordinates and constraint asr x ˆ x ÿ x 1 N1 0 ÿ x 2 N2 0 ÿ x 3 N3 0 ÿˆ0r y ˆ y ÿ y 1 N1 0 ÿ y 2 N2 0 ÿ y 3 N3 0 ÿˆ0…9:18†r z ˆ z ÿ z 1 N1 0 ÿ z 2 N2 0 ÿ z 3 N3 0 ÿˆ0r 1 ˆ 1 ÿ L 1 ÿ L 2 ÿ L 3 ÿ L 4 ˆ 0where Ni 0 ˆ N 0 …L 1 ; L 2 ; L 3 ; L 4 †,etc. Now derivatives of the above with respect to x andy may be written directly as23@r x @r x @r x2 P @x @y @z@N k P @N k P @N k P @N3k2 3 xk xk xk xk @r y @r y @r y1 0 0@L 1 @L 2 @L 3 @L 4@x @y @z0 1 0P @N k P @N k P @N k P @N kyk yk yk yk ˆÿ@L 1 @L 2 @L 3 @L 4@r z @r z @r z64 0 0 175@x @y @zP @N k P @N k P @N k P @N k667 0 0 0 zk zk zk zk 74 @L 1 @L 2 @L 3 @L 454 @r 1 @r 1 @r 151 1 1 1@x @y @z264@L 1@x@L 2@x@L 3@x@L 4@x@L 1@y@L 2@y@L 3@y@L 4@y@L 1@z@L 2@z@L 3@z@L 4@z3ˆ 075…9:19†The above may be solved for the partial derivatives of L i with respect to the x, y, zcoordinates and used directly with the chain rule written as@N i@x ˆ @N i @L 1@L 1 @x ‡ @N i @L 2@L 2 @x ‡ @N i @L 3@L 3 @x ‡ @N i @L 4…19:20†@L 4 @xThe above has advantages when the coordinates are written using mapping functionsas the computation can still be more easily carried out. Also,the calculation ofintegrals will normally be performed numerically (as described in Sec. 9.10) wherethe points for integration are de®ned directly in terms of the volume coordinates.

Convergence of elements in curvilinear coordinates 213Fig. 9.9 A distorted triangular prism.Finally it should be remarked that any of the elements given in the previous chapterare capable of being mapped. In some,such as the triangular prism,both area andrectangular coordinates are used (Fig. 9.9). The remarks regarding the dependenceof coordinates apply once again with regard to the former but the processes of thepresent section should make procedures clear.9.7 Convergence of elements in curvilinear coordinatesTo consider the convergence aspects of the problem posed in curvilinear coordinatesit is convenient to return to the starting point of the approximation where an energyfunctional ,or an equivalent integral form (Galerkin problem statement),wasde®ned by volume integrals essentially similar to those of Eq. (9.5),in which theintegrand was a function of u and its ®rst derivatives.Thus,for instance,the variational principles of the energy kind discussed inChapter 2 (or others of Chapter 3) could be stated for a scalar function u as… ˆ F u; @u @x ; @u …@y ; x; y d ‡ E…u; ...† dÿ…9:21†ÿThe coordinate transformation changes the derivatives of any function by thejacobian relation (9.11). Thus8 9 8 9@u@u>< >= >=…; †…9:22†@u@u>: >; >: >;@y@and the functional can be stated simply by a relationship of the form (9.21) with x, y,etc.,replaced by , ,etc.,with the maximum order of di€erentiation unchanged.It follows immediately that if the shape functions are chosen in curvilinear coordinatespace so as to observe the usual rules of convergence (continuity and presence ofcomplete ®rst-order polynomials in these coordinates),then convergence will occur.Further,all the arguments concerning the order of convergence with the elementsize h still hold,providing the solution is related to the curvilinear coordinate system.Indeed,all that has been said above is applicable to problems involving higherderivatives and to most unique coordinate transformations. It should be noted that

214 Mapped elements and numerical integrationthe patch test as conceived in the x, y, ...coordinate system (see Chapters 2 and 10) isno longer simply applicable and in principle should be applied with polynomial ®eldsimposed in the curvilinear coordinates. In the case of isoparametric (or subparametric)elements the situation is more advantageous. Here a linear (constant derivativex, y) ®eld is always reproduced by the curvilinear coordinate expansion,and thusthe lowest order patch test will be passed in the standard manner on such elements.The proof of this is simple. Consider a standard isoparametric expansionu ˆ XnN i a i Na N ˆ N…; ; † …9:23†i ˆ 1with coordinates of nodes de®ning the transformation asx ˆ X N i x i y ˆ X N i y i z ˆ X N i z i …9:24†The question is under what circumstances is it possible for expression (9.23) to de®nea linear expansion in cartesian coordinates:u ˆ 1 ‡ 2 x ‡ 3 y ‡ 4 zIf we take 1 ‡ 2XNi x i ‡ 3XNi y i ‡ 4XNi z i…9:25†a i ˆ 1 ‡ 2 x i ‡ 3 y i ‡ 4 z iand compare expression (9.23) with (9.25) we note that identity is obtained betweenthese providingXNi ˆ 1As this is the usual requirement of standard element shape functions [see Eq. (8.4)] wecan conclude that the following theorem is valid.Theorem 3. The constant derivative condition will be satis®ed for all isoparametricelements.As subparametric elements can always be expressed as speci®c cases of an isoparametrictransformation this theorem is obviously valid here also.It is of interest to pursue the argument and to see under what circumstances higherpolynomial expansions in cartesian coordinates can be achieved under various transformations.The simple linear case in which we `guessed' the solution has now to bereplaced by considering in detail the polynomial terms occurring in expressions suchas (9.23) and (9.25) and establishing conditions for equating appropriate coe cients.Consider a speci®c problem: the circumstances under which the bilinearly mappedquadrilateral of Fig. 9.10 can fully represent any quadratic cartesian expansion. Wenow havex ˆ X4 1N 0 ix i y ˆ X4 1N 0 i y i…9:26†and we wish to be able to reproduceu ˆ 1 ‡ 2 x ‡ 3 y ‡ 4 x 2 ‡ 5 xy ‡ 6 y 2…9:27†

Convergence of elements in curvilinear coordinates 215Mapping nodes(a)(b)(c)Fig. 9.10 Bilinear mapping of subparametric quadratic eight- and nine-noded element.Noting that the bilinear form of N 0 i contains terms such as 1, , and ,the above canbe written asu ˆ 1 ‡ 2 ‡ 3 ‡ 4 2 ‡ 5 ‡ 6 2 ‡ 7 2 ‡ 8 2 ‡ 9 2 2…9:28†where 1 to 9 depend on the values of 1 to 6 .We shall now try to match the terms arising from the quadratic expansions of theserendipity and lagrangian kinds shown in Fig. 9.10(b) and (c):u ˆ X8 1N i a i…9:29a†u ˆ X9 1N i a i…9:29b†where the appropriate terms are of the kind de®ned in the previous chapter.For the eight-noded element (serendipity) [Fig. 9.10(b)] we can write (9.29(a))directly using polynomial coe cients b i , i ˆ 1, ...,8,in place of the nodal variablesa i (noting the terms occurring in the Pascal triangle) asu ˆ b 1 ‡ b 2 ‡ b 3 ‡ b 4 2 ‡ b 5 ‡ b 6 2 ‡ b 7 2 ‡ b 8 2 …9:30†

216 Mapped elements and numerical integrationRegular (R) mesh0.5M5.0AIrregular (I) meshAC LM(a)8- and 9-noded elementsExactR = 8/9I = 9I = 8= ExactC L deflection(b)Exact1000900800700600σ x stresson AA(Gauss point)(c)500400Fig. 9.11 Quadratic serendipityand Lagrange eight- and nine-noded elements in regular and distorted form.Elastic de¯ection of a beam under constant moment. Note the poor results of the eight-noded element.It is immediately evident that for arbitrary values of 1 to 9 it is impossible tomatch the coe cients b 1 to b 8 due to the absence of the term 2 2 in Eq. (9.30).[However if higher order (quartic,etc.) expansions of the serendipity kind wereused such matching would evidently be possible and we could conclude that for

Numerical integration ± one-dimensional 217linearly distorted elements the serendipity family of order four or greater will alwaysrepresent quadratics.]For the nine-noded,lagrangian,element [Fig. 9.10(c)] the expansion similar to(9.30) givesu ˆ b 1 ‡ b 2 ‡ b 3 ‡ b 4 2 ‡‡b 8 2 ‡ b 9 2 2…9:31†and the matching of the coe cients of Eqs (9.31) and (9.28) can be made directly.We can conclude therefore that nine-noded elements represent better cartesianpolynomials (when distorted linearly) and therefore are generally preferable inmodelling smooth solutions. This matter was ®rst presented by Wachspress but thesimple proof presented above is due to Crochet. 8 An example of this is given inFig. 9.11 where we consider the results of a ®nite element calculation with eightandnine-noded elements respectively used to reproduce a simple beam solution inwhich we know that the exact answers are quadratic. With no distortion both elementsgive exact results but when distorted only the nine-noded element does so,with the eight-noded element giving quite wild stress ¯uctuation.Similar arguments will lead to the conclusion that in three dimensions again onlythe lagrangian 27-noded element is capable of reproducing fully the quadratic incartesian coordinates when trilinearly distorted.Lee and Bathe 9 investigate the problem for cubic and quartic serendipity andlagrangian quadrilateral elements and show that under bilinear distortions the fullorder cartesian polynomial terms remain in Lagrange elements but not in serendipityones. They also consider edge distortion and show that this polynomial order isalways lost. Additional discussion of such problems is also given by Wachspress. 79.8 Numerical integration ± one-dimensionalIn Chapter 5,dealing with a relatively simple problem of axisymmetric stress distributionand simple triangular elements,it was noted that exact integration of expressionsfor element matrices could be troublesome. Now for the more complex distortedelements numerical integration is essential.Some principles of numerical integration will be summarized here together withtables of convenient numerical coe cients.To ®nd numerically the integral of a function of one variable we can proceed in oneof several ways. 109.8.1 Newton±Cotes quadratureyIn the most obvious procedure,points at which the function is to be found aredetermined a priori ± usually at equal intervals ± and a polynomial passed throughthe values of the function at these points and exactly integrated [Fig. 9.12(a)].y `Quadrature' is an alternative term to `numerical integration'.

218 Mapped elements and numerical integrationff (ξ 8 )f (ξ 1 )(a)–1 0 1ξ∆ = 2 7ff (ξ 1 )f (ξ 2 ) f (ξ 3 )f (ξ 4 )–1 0 1ξ(b)0.339980.86114Fig. 9.12 (a) Newton±Cotes and (b) Gauss integrations. Each integrates exactlya seventh-order polynomial[i.e., error O…h 8 †].As n values of the function de®ne a polynomial of degree n ÿ 1,the errors will beof the order O…h n † where h is the element size. This leads to the well-knownNewton±Cotes `quadrature' formulae. The intregrals can be written asI ˆ… 1f …† d ˆ Xnÿ11H i f … i †…9:32†for the range of integration between ÿ1 and‡1 [Fig. 9.12(a)]. For example,if n ˆ 2,we have the well-known trapezoidal rule:I ˆ f …ÿ1†‡f …1†…9:33†for n ˆ 3,the Simpson `one-third' rule:I ˆ 13‰ f …ÿ1†‡4f …0†‡f …1†Š …9:34†

Numerical integration ± rectangular (2D) or right prism (3D) regions 219and for n ˆ 4:I ˆ 14 ‰ f …ÿ1†‡3f …ÿ 1 3 †‡3f …1 3†‡f …1†Š …9:35†Formulae for higher values of n are given in reference 10.9.8.2 Gauss quadratureIf in place of specifying the position of sampling points aprioriwe allow these to be locatedat points to be determined so as to aim for best accuracy,then for a given number ofsampling points increased accuracy can be obtained. Indeed,if we again considerI ˆ… 1f …† d ˆ Xnÿ11H i f … i †…9:36†and again assume a polynomial expression,it is easy to see that for n samplingpoints we have 2n unknowns (H i and i ) and hence a polynomial of degree 2n ÿ 1could be constructed and exactly integrated [Fig. 9.12(b)]. The error is thus oforder O…h 2n †.The simultaneous equations involved are di cult to solve,but some mathematicalmanipulation will show that the solution can be obtained explicitly in terms ofLegendre polynomials. Thus this particular process is frequently known as Gauss±Legendre quadrature. 10Table 9.1 shows the positions and weighting coe cients for gaussian integration.For purposes of ®nite element analysis complex calculations are involved in determiningthe values of f ,the function to be integrated. Thus the Gauss-type processes,requiring the least number of such evaluations,are ideally suited and from now onwill be used exclusively.Other expressions for integration of functions of the typeI ˆ… 1w…† f …† d ˆ Xnÿ11H i f … i †…9:37†can be derived for prescribed forms of w…†,again integrating up to a certain orderof accuracy a polynomial expansion of f …†. 109.9 Numerical integration ± rectangular (2D) or rightprism (3D) regionsThe most obvious way of obtaining the integralI ˆ… 1ÿ1… 1f …; † d dÿ1…9:38†is to ®rst evaluate the inner integral keeping constant,i.e.,… 1f …; † d ˆ XnH j f … j ;†ˆ …†ÿ1j ˆ 1…9:39†

220 Mapped elements and numerical integrationTable 9.1 Abscissae and weight coe cients of the gaussianquadrature formula „ 1ÿ1 f …x† dx ˆ Pnj ˆ 1 H i f …a j †aHn ˆ 10 2.000 000 000 000 000pn ˆ 21= 31.000 000 000 000 000pn ˆ 30:65/90.000 000 000 000 000 8/9n ˆ 40.861 136 311 594 953 0.347 854 845 137 4540.339 981 043 584 856 0.652 145 154 862 546n ˆ 50.906 179 845 938 664 0.236 926 885 056 1890.538 469 310 105 683 0.478 628 670 499 3660.000 000 000 000 000 0.568 888 888 888 889n ˆ 60.932 469 514 203 152 0.171 324 492 379 1700.661 209 386 466 265 0.360 761 573 048 1390.238 619 186 083 197 0.467 913 934 572 691n ˆ 70.949 107 912 342 759 0.129 484 966 168 8700.741 531 185 599 394 0.279 705 391 489 2770.405 845 151 377 397 0.381 830 050 505 1190.000 000 000 000 000 0.417 959 183 673 469n ˆ 80.960 289 856 497 536 0.101 228 536 290 3760.796 666 477 413 627 0.222 381 034 453 3740.525 532 409 916 329 0.313 706 645 877 8870.183 434 642 495 650 0.362 683 783 378 362n ˆ 90.968 160 239 507 626 0.081 274 388 361 5740.836 031 107 326 636 0.180 648 160 694 8570.613 371 432 700 590 0.260 610 696 402 9350.324 253 423 403 809 0.312 347 077 040 0030.000 000 000 000 000 0.330 239 355 001 260n ˆ 100.973 906 528 517 172 0.066 671 344 308 6880.865 063 366 688 985 0.149 451 349 150 5810.679 409 568 299 024 0.219 086 362 515 9820.433 395 394 129 247 0.269 266 719 309 9960.148 874 338 981 631 0.295 524 224 714 753Evaluating the outer integral in a similar manner,we haveI ˆ… 1ÿ1…† d ˆ XnH i … i †i ˆ 1ˆ XnX nH i H j f … j ; i †i ˆ 1 j ˆ 1ˆ Xn X ni ˆ 1 j ˆ 1H i H j f … j ; i †…9:40†

Numerical integration ± triangular or tetrahedral regions 2211–17 8 9η4 5 ξ 61 2 31–1Fig. 9.13 Integrating points for n ˆ 3 in a square region. (Exact for polynomial of ®fth order in eachdirection).For a right prism we have similarlyI ˆ… 1ÿ1… 1ÿ1… 1f …; ; † d d dÿ1ˆ Xn X n X nm ˆ 1 j ˆ 1 i ˆ 1H i H j H m f … i ; j ; m †…9:41†In the above,the number of integrating points in each direction was assumed to be thesame. Clearly this is not necessary and on occasion it may be an advantage to usedi€erent numbers in each direction of integration.It is of interest to note that in fact the double summation can be readily interpretedas a single one over …n n† points for a rectangle (or n 3 points for a cube). Thus inFig. 9.13 we show the nine sampling points that result in exact integrals of order 5in each direction.However,we could approach the problem directly and require an exact integrationof a ®fth-order polynomial in two dimensions. At any sampling point two coordinatesand a value of f have to be determined in a weighting formula of typeI ˆ… 1ÿ1… 1f …; † d d ˆ Xmÿ11w i f … i ; i †…9:42†There it would appear that only seven points would su ce to obtain the same order ofaccuracy. Some formulae for three-dimensional bricks have been derived by Irons 11and used successfully. 129.10 Numerical integration ± triangular or tetrahedralregionsFor a triangle,in terms of the area coordinates the integrals are of the formI ˆ… 10… 1 ÿ L10f …L 1 L 2 L 3 † dL 2 dL 1 L 3 ˆ 1 ÿ L 1 ÿ L 2 …9:43†

222 Mapped elements and numerical integrationTable 9.2 Numerical integration formulae for trianglesTriangularOrder Figure Error Points coordinates WeightsLinear • a R ˆ O…h 2 † a13 ; 1 3 ; 1 31Quadratic• ab•c•1a2 ; 1 2 ; 0 13R ˆ O…h 3 † b 0; 1 2 ; 1 1231c2 ; 0; 1 123Cubicb•c• •ad•1a3 ; 1 3 ; 1 3ÿ 2748R ˆ O…h 4 † b 0:6; 0:2; 0:2)25c 0:2; 0:6; 0:248d 0:2; 0:2; 0:6Quintic1a3 ; 1 3 ; 1 30.225 000 000 0b 1 ; 1 ;)•1b fc 1 ; 1 ; 1 0.132 394 152 7•g d•R ˆ O…h• •6 † d 1 ; 1 ; 1ae 2 ; 2 ;)2• c e•f 2 ; 2 ; 2 0.125 939 180 5g 2 ; 2 ; 2with 1 ˆ 0:059 715 871 7 1 ˆ 0:470 142 064 1 2 ˆ 0:797 426 985 3 2 ˆ 0:101 286 507 3Once again we could use n Gauss points and arrive at a summation expression ofthe type used in the previous section. However,the limits of integration now involvethe variable itself and it is convenient to use alternative sampling points for the secondintegration by use of a special Gauss expression for integrals of the type given byEq. (9.37) in which w is a linear function. These have been devised by Radau 13 andused successfully in the ®nite element context. 14 It is,however,much more desirable(and aesthetically pleasing) to use special formulae in which no bias is given to any ofthe natural coordinates L i . Such formulae were ®rst derived by Hammer et al. 15 andFelippa 16 and a series of necessary sampling points and weights is given in Table 9.2. 17(A more comprehensive list of higher formulae derived by Cowper is given on p. 184of reference 17.)A similar extension for tetrahedra can obviously be made. Table 9.3 presents someformulae based on reference 15.

Required order of numerical integration 223Table 9.3 Numerical integration formulae for tetrahedraTetrahedralNo. Order Figure Error Points coordinates Weights1 Linear a • R ˆ O…h 2 † a14 ; 1 4 ; 1 4 ; 1 412 Quadraticb •a •d •• cR ˆ O…h 3 †abcd9; ; ; ; ; ; >=; ; ; >;; ; ; ˆ 0:585 410 20 ˆ 0:138 196 60143 Cubice•b a • c R ˆ O…h 4 † c• •d•de1ab 114 ; 1 4 ; 1 4 ; 1 4ÿ 4 52 ; 1 6 ; 1 6 ; 9166 ; 1 2 ; 1 6 ; 1 >=616 ; 1 6 ; 1 2 ; 1 616 ; 1 6 ; 1 6 ; 1 2>;9209.11 Required order of numerical integrationWith numerical integration used in place of exact integration,an additional error isintroduced into the calculation and the ®rst impression is that this should be reducedas much as possible. Clearly the cost of numerical integration can be quite signi®cant,and indeed in some early programs numerical formulation of element characteristicsused a comparable amount of computer time as in the subsequent solution of theequations. It is of interest,therefore,to determine (a) the minimum integrationrequirement permitting convergence and (b) the integration requirements necessaryto preserve the rate of convergence which would result if exact integation were used.It will be found later (Chapters 10 and 12) that it is in fact often a positivedisadvantage to use higher orders of integration than those actually needed under(b) as,for very good reasons,a `cancellation of errors' due to discretization anddue to inexact integration can occur.9.11.1 Minimum order of integration for convergenceIn problems where the energy functional (or equivalent Galerkin integral statements)de®nes the approximation we have already stated that convergence can occurproviding any arbitrary constant value of the mth derivatives can be reproduced.

224 Mapped elements and numerical integrationIn the present case m ˆ 1 and we thus require that in integrals of the form (9.5) aconstant value of G be correctly integrated. Thus the volume of the element „ V dVneeds to be evaluated correctly for convergence to occur. In curvilinear coordinateswe can thus argue that „ V det jJj d d d has to be evaluated exactly. 3;69.11.2 Order of integration for no loss of convergenceIn a general problem we have already found that the ®nite element approximateevaluation of energy (and indeed all the other integrals in a Galerkin-type approximation,seeChapter 3) was exact to the order 2… p ÿ m†,where p was the degree of thecomplete polynomial present and m the order of di€erentials occurring in theappropriate expressions.Providing the integration is exact to order 2… p ÿ m†,or shows an error ofO…h 2… p ÿ m†‡1 †,or less,then no loss of convergence order will occur.y If in curvilinearcoordinates we take a curvilinear dimension h of an element,the same rule applies.For C 0 problems (i.e., m ˆ 1) the integration formulae should be as follows:p ˆ 1; linear elements O…h†p ˆ 2; quadratic elements O…h 3 †p ˆ 3; cubic elements O…h 5 †We shall make use of these results in practice,as will be seen later,but it should benoted that for a linear quadrilateral or triangle a single-point integration is adequate.For parabolic quadrilaterals (or bricks) 2 2 (or 2 2 2),Gauss point integrationis adequate and for parabolic triangles (or tetrahedra) three-point (and four-point)formulae of Tables 9.2 and 9.3 are needed.The basic theorems of this section have been introduced and proved numerically inpublished work. 18ÿ209.11.3 Matrix singularity due to numerical integrationThe ®nal outcome of a ®nite element approximation in linear problems is an equationsystemKa ‡ f ˆ 0…9:44†in which the boundary conditions have been inserted and which should,on solutionfor the parameter a,give an approximate solution for the physical situation. If a solutionis unique,as is the case with well-posed physical problems,the equation matrix Kshould be non-singular. We have a priori assumed that this was the case with exactintegration and in general have not been disappointed. With numerical integrationsingularities may arise for low integration orders,and this may make such ordersimpractical. It is easy to show how,in some circumstances,a singularity of K musty For an energy principle use of quadrature may result in loss of a bound for …a†.

Required order of numerical integration 225arise,but it is more di cult to prove that it will not. We shall,therefore,concentrateon the former case.With numerical integration we replace the integrals by a weighted sum of independentlinear relations between the nodal parameters a. These linear relations supply theonly information from which the matrix K is constructed. If the number of unknowns aexceeds the number of independent relations supplied at all the integrating points, thenthe matrix K must be singular.Integrating point (3 independent relations)Nodal point with 2 degrees of freedom(a)Both d.o.f.suppressedOne d.o.f.suppressed(b)(c)(a)(b)Degree offreedomLinearIndependentrelation4 x 2 – 3 = 5 > 1 x 3 = 3singular6 x 2 – 3 = 9 > 2 x 3 = 6singularDegree offreedomQuadraticIndependentrelation2 x 8 – 3 = 13 > 4 x 3 = 12singular13 x 2 – 3 = 23 < 8 x 3 = 24(c)25 x 2 – 18 = 32 < 16 x 3 = 48 48 x 2 = 96< 64 x 3 = 192Fig. 9.14 Check on matrix singularityin two-dimensional elasticityproblems (a), (b), and (c).

226 Mapped elements and numerical integrationTo illustrate this point we shall consider two-dimensional elasticity problems usinglinear and parabolic serendipity quadrilateral elements with one- and four-pointquadratures respectively.Here at each integrating point three independent `strain relations' are used and thetotal number of independent relations equals 3 (number of integration points). Thenumber of unknowns a is simply 2 (number of nodes) less restrained degrees offreedom.In Fig. 9.14(a) and(b) we show a single element and an assembly of two elementssupported by a minimum number of speci®ed displacements eliminating rigid bodymotion. The simple calculation shows that only in the assembly of the quadraticelements is elimination of singularities possible,all the other cases remaining strictlysingular.In Fig. 9.14(c) a well-supported block of both kinds of elements is considered andhere for both element types non-singular matrices may arise although local,nearsingularity may still lead to unsatisfactory results (see Chapter 10).The reader may well consider the same assembly but supported again by the minimumrestraint of three degrees of freedom. The assembly of linear elements with asingle integrating point will be singular while the quadratic ones will,in fact,usuallybe well behaved.For the reason just indicated,linear single-point integrated elements are usedinfrequently in static solutions,though they do ®nd wide use in `explicit' dynamicscodes ± but needing certain remedial additions (e.g.,hourglass control 21;22 ) ± whilefour-point quadrature is often used for quadratic serendipity elements.yIn Chapter 10 we shall return to the problem of convergence and will indicatedangers arising from local element singularities.However,it is of interest to mention that in Chapter 12 we shall in fact seek matrixsingularities for special purposes (e.g.,incompressibility) using similar arguments.9.12 Generation of ®nite element meshes by mapping.Blending functionsIt would have been observed that it is an easy matter to obtain a coarse subdivision ofthe analysis domain with a small number of isoparametric elements. If second- orthird-degree elements are used,the ®t of these to quite complex boundaries is reasonable,asshown in Fig. 9.15(a) where four parabolic elements specify a sectorial region.This number of elements would be too small for analysis purposes but a simple subdivisioninto ®ner elements can be done automatically by,say,assigning new positionsof nodes of the central points of the curvilinear coordinates and thus deriving a largernumber of similar elements,as shown in Fig. 9.15(b). Indeed,automatic subdivisioncould be carried out further to generate a ®eld of triangular elements. The processthus allows us,with a small amount of original input data,to derive a ®nite elementmesh of any re®nement desirable. In reference 23 this type of mesh generation isdeveloped for two- and three-dimensional solids and surfaces and is reasonablyy Repeating the test for quadratic lagrangian elements indicates a singularity for 2 2 quadrature (seeChapter 10 for dangers).

Generation of ®nite element meshes by mapping 227(a)(b)(c)Fig. 9.15 Automatic mesh generation byparabolic isoparametric elements. (a) Speci®ed mesh points.(b) Automatic subdivision into a small number of isoparametric elements. (c) Automatic subdivision intolinear triangles.e cient. However,elements of predetermined size and/or gradation cannot be easilygenerated.The main drawback of the mapping and generation suggested is the fact that theoriginally circular boundaries in Fig. 9.15(a) are approximated by simple parabolaeand a geometric error can be developed there. To overcome this di culty anotherform of mapping,originally developed for the representation of complex motor-carbody shapes,can be adopted for this purpose. 24 In this mapping blending functionsinterpolate the unknown u in such a way as to satisfy exactly its variations along theedges of a square , domain. If the coordinates x and y are used in a parametricexpression of the type given in Eq. (9.1),then any complex shape can be mappedby a single element. In reference 24 the region of Fig. 9.15 is in fact so mappedand a mesh subdivision obtained directly without any geometric error on theboundary.The blending processes are of considerable importance and have been used toconstruct some interesting element families 25 (which in fact include the standardserendipity elements as a subclass). To explain the process we shall show how afunction with prescribed variations along the boundaries can be interpolated.Consider a region ÿ1 4 ; 4 1,shown in Fig. 9.16,on the edges of which anarbitrary function is speci®ed [i.e., …ÿ1;†;…1;†;…; ÿ1†;…; 1† are given].The problem presented is that of interpolating a function …; † so that a smoothsurface reproducing precisely the boundary values is obtained. WritingN 1 …† ˆ 1 ÿ 2N 1 …† ˆ 1 ÿ 2N 2 …† ˆ1 ‡ 2N 2 …† ˆ1 ‡ 2for our usual one-dimensional linear interpolating functions,we note thatP N 1 …†…; ÿ1†‡N 2 …†…; 1†…9:45†…9:46†

228 Mapped elements and numerical integrationN 1 (ξ) N 2 (ξ)1 1(a)(–1, 1)ηξ(1, 1)(–1, –1)=(+1, 1)(b)+(c)–(d)Fig. 9.16 Stages of construction of a blending interpolation (a), (b), (c), and (d).interpolates linearly between the speci®ed functions in the direction,as shown inFig. 9.16(b). Similarly,P N 1 …†…; ÿ1†‡N 2 …†…; 1†…9:47†interpolates linearly in the direction [Fig. 9.16(c)]. Constructing a third functionwhich is a standard linear,bilinear interpolation of the kind we have already encountered[Fig. 9.16(d)],i.e.,P P ˆ N 2 …†N 2 …†…1; 1†‡N 2 …†N 1 …†…1; ÿ1†‡ N 1 …†N 2 …†…ÿ1; 1†‡N 1 …†N 1 …†…ÿ1; ÿ1† …9:48†

In®nite domains and in®nite elements 229we note by inspection that ˆ P ‡ P ÿ P P …9:49†is a smooth surface interpolating exactly the boundary functions.Extension to functions with higher order blending is almost evident,and immediatelythe method of mapping the quadrilateral region ÿ1 4 , 4 1 to any arbitraryshape is obvious.Though the above mesh generation method derives from mapping and indeed hasbeen widely applied in two and three dimensions,we shall see in the chapter devotedto adaptivity (Chapter 15) that the optimal solution or speci®cation of mesh density orsize should guide the mesh generation. We shall discuss this problem in that chapter tosome extent,but the interested reader is directed to references 26,27 or books thathave appeared on the subject. 28ÿ31 The subject has now grown to such an extentthat discussion in any detail is beyond the scope of this book. In the programsmentioned at the end of each volume of this book we shall refer to the GiD systemwhich is available to readers. 329.13 In®nite domains and in®nite elements9.13.1 IntroductionIn many problems of engineering and physics in®nite or semi-in®nite domains exist. Atypical example from structural mechanics may,for instance,be that of threedimensional(or axisymmetric) excavation,illustrated in Fig. 9.17. Here the problemis one of determining the deformations in a semi-in®nite half-space due to the removalof loads with the speci®cation of zero displacements at in®nity. Similar problemsabound in electromagnetics and ¯uid mechanics but the situation illustrated is typical.The question arises as to how such problems can be dealt with by a method of approximationin which elements of decreasing size are used in the modelling process. The®rst intuitive answer is the one illustrated in Fig. 9.17(a) where the in®nite boundarycondition is speci®ed at a ®nite boundary placed at a large distance from the object.This,however,begs the question of what is a `large distance' and obviously substantialerrors may arise if this boundary is not placed far enough away. On the otherhand,pushing this out excessively far necessitates the introduction of a largenumber of elements to model regions of relatively little interest to the analyst.To overcome such `in®nite' di culties many methods have been proposed. In somea sequence of nesting grids is used and a recurrence relation derived. 33;34 In others aboundary-type exact solution is used and coupled to the ®nite element domain. 35;36However,without doubt,the most e€ective and e cient treatment is the use of`in®nite elements' 37ÿ40 pioneered originally by Bettess. 41 In this process the conventional,®niteelements are coupled to elements of the type shown in Fig. 9.17(b)which model in a reasonable manner the material stretching to in®nity.The shape of such two-dimensional elements and their treatment is best accomplishedby mapping 39ÿ41 these onto a bi-unit square (or a ®nite line in one dimensionor cube in three dimensions). However,it is essential that the sequence of trial

230 Mapped elements and numerical integration(a)Conventionaltreatment(b)Solution usinginfinite elementsu = 0Imposed at anarbitrary boundary‘Infinite’ elementwith two nodes at ∞ru = 0 at r = ∞Fig. 9.17 A semi-in®nite domain. Deformations of a foundation due to removal of load following anexcavation. (a) Conventional treatment and (b) use of in®nite elements.functions introduced in the mapped domain be such that it is complete and capable ofmodelling the true behaviour as the radial distance r increases. Here it would beadvantageous if the mapped shape functions could approximate a sequence of thedecaying formC 1r ‡ C 2r 2 ‡ C 3r 3 ‡…9:50†where C i are arbitrary constants and r is the radial distance from the `focus' of theproblem.In the next subsection we introduce a mapping function capable of doing just this.9.13.2 The mapping functionFigure 9.18 illustrates the principles of generation of the derived mapping function.We shall start with a one-dimensional mapping along a line CPQ coinciding withthe x-direction. Consider the following function:x ˆÿ1 ÿ x C ‡1 ‡ 1 ÿ x Q ˆ N C x C ‡ N Q x Q…9:51a†

In®nite domains and in®nite elements 231yxCPξQR–1 1MapPQR at ∞PC 1Q 1Q1P 1RMapQ 1R 1 at ∞η–1 ξ1–1P 1 R 1Fig. 9.18 In®nite line and element map. Linear interpolation.and we immediately observe that ˆÿ1 ˆ 0 ˆ 1corresponds to x ˆ xQ ‡ x C2corresponds to x ˆ x Qcorresponds to x ˆ1 x Pwhere x P is a point midway between Q and C.Alternatively the above mapping could be written directly in terms of the Q and Pcoordinates by simple elimination of x C . This gives,using our previous notation:x ˆ N Q x Q ‡ N P x Pˆ 1 ‡ 2 x1 ÿ Q ÿ21 ÿ x P …9:51b†Both forms give a mapping that is independent of the origin of the x-coordinate asN Q ‡ N P ˆ 1 ˆ N C ‡ N Q…9:52†The signi®cance of the point C is,however,of great importance. It represents thecentre from which the `disturbance' originates and,as we shall now show,allowsthe expansion of the form of Eq. (9.50) to be achieved on the assumption that r ismeasured from C. Thusr ˆ x ÿ x C…9:53†

232 Mapped elements and numerical integrationIf,for instance,the unknown function u is approximated by a polynomial functionusing,say,hierarchical shape functions and givingu ˆ 0 ‡ 1 ‡ 2 2 ‡ 3 3 ‡…9:54†we can easily solve Eqs (9.51a) for ,obtaining ˆ 1 ÿ x Q ÿ x Cx ÿ x Cˆ 1 ÿ x Q ÿ x Cr…9:55†Substitution into Eq. (9.54) shows that a series of the form given by Eq. (9.50) isobtained with the linear shape function in corresponding to 1=r terms,quadratic to1=r 2 ,etc.In one dimenson the objectives speci®ed have thus been achieved and the element willyield convergence as the degree of the polynomial expansion, p,increases. Now a generalizationto two or three dimensions is necessary. It is easy to see that this can be achievedby simple products of the one-dimensional in®nite mapping with a `standard' type ofshape function in (and ) directions in the manner indicated in Fig. 9.18.Firstly we generalize the interpolation of Eqs (9.51) for any straight line in x, y, zspace and write (for such a line as C 1 P 1 Q 1 in Fig. 9.18)x ˆÿ1 ÿ x C 1‡y ˆÿ1 ÿ y C 1‡z ˆÿ1 ÿ z C 1‡1 ‡ 1 ÿ 1 ‡ 1 ÿ 1 ‡ 1 ÿ x Q1y Q1z Q1…in three dimensions†…9:56†Secondly we complete the interpolation and map the whole …† domain by adding a`standard' interpolation in the …† directions. Thus for the linear interpolation shownwe can write for elements PP 1 QQ 1 RR 1 of Fig. 9.18,asx ˆ N 1 …† ÿ 1 ÿ x C 1 ‡ x1 ÿ Q‡ N 0 …† ÿ1 ÿ x C 1‡ 1 ÿ x Q 1; etc: …9:57†withN 1 …† ˆ1 ‡ N20 …† ˆ1 ÿ 2and map the points as shown.In a similar manner we could use quadratic interpolations and map an element asshown in Fig. 9.19 by using quadratic functions in .Thus it is an easy matter to create in®nite elements and join these to a standardelement mesh as shown in Fig. 9.17(b). The reader will observe that in the generationof such element properties only the transformation jacobian matrix di€ers fromstandard forms,hence only this has to be altered in conventional programs.The `origin' or `pole' of the coordinates C can be ®xed arbitrarily for each radialline,as shown in Fig. 9.18. This will be done by taking account of the knowledgeof the physical solution expected.

In®nite domains and in®nite elements 233Rat ∞QPCR 1at ∞Q 1C 2C 1P 1P 2Q 2R 2at ∞PQRηQ 1P 1 R 1ξP 2 Q 2R 2Fig. 9.19 In®nite element map. Quadratic interpolation.In Fig. 9.20 we show a solution of the Boussinesq problem (a point load on anelastic half-space). Here results of using a ®xed displacement or in®nite elementsare compared and the big changes in the solution noted. In this example the poleof each element was taken at the load point for obvious reasons. 40Figure 9.21 shows how similar in®nite elements (of the linear kind) can give excellentresults,even when combined with very few standard elements. In this example where asolution of the Laplace equation is used (see Chapter 7) for an irrotational ¯uid ¯ow,thepoles of the in®nite elements are chosen at arbitrary points of the aerofoil centre-line.In concluding this section it should be remarked that the use of in®nite elements (asindeed of any other ®nite elements) must be tempered by background analyticalknowledge and `miracles' should not be expected. Thus the user should not expect,for instance,such excellent results as those shown in Fig. 9.20 for a plane elasticityproblem for the displacements. It is `well known' that in this case the displacementsunder any load which is not self-equilibrated will be in®nite everywhere and thenumbers obtained from the computation will not be,whereas for the three-dimensionalcase it is in®nite only at a point load.Extensive use of in®nite elements is made in Volume 3 in the context of the solutionof wave problems.

234 Mapped elements and numerical integrationPStandardelement2 4PStandard andinfinite elements2 4224u = 04u = 0 at ∞0.6zzVertical Displacement on z axis0.40.2Exact00 1 2 3 4zFig. 9.20 A point load on an elastic half-space (Boussinesq problem). Standard linear elements and in®niteline elements (E ˆ 1, ˆ 0:1, P ˆ 1).9.14 Singular elements by mapping for fracturemechanics, etc.In the study of fracture mechanics interest is often focused on the singularity pointwhere quantities such as stress become (mathematically,but not physically) in®nite.Near such singularities normal,polynomial-based,®nite element approximationsperform badly and attempts have frequently been made here to include special functionswithin an element which can model the analytically known singular function.References 42±69 give an extensive literature survey of the problem and ®nite elementsolution techniques. An alternative to the introduction of special functions within anelement ± which frequently poses problems of enforcing continuity requirements withadjacent,standard,elements ± lies in the use of special mapping techniques.An element of this kind,shown in Fig. 9.22(a),was introduced almost simultaneouslyby Henshell and Shaw 65 and Barsoum 66;67 for quadrilaterals by a simple shift of themid-side node in quadratic,isoparametric elements to the quarter point.It can now be shown (and we leave this exercise to the curiouspreader) that along theelement edges the derivatives @u=@x (or strains) vary as 1= r where r is the distance

Singular elements by mapping for fracture mechanics, etc. 235Infinite elementsC Lv ∞P PCC1.5v ∞u1.00.5Analyticsolution0Fig. 9.21 Irrotational ¯ow around a NACA 0018 wing section. 36 (a) Mesh of bilinear isoparametric and in®niteelements. (b) Computed k and analytical Ð results for velocity parallel to surface.from the corner node at which the singularity develops. Although good results areachievable with such elements the singularity is,in fact,not well modelled on linesother than element edges. A development suggested by Hibbitt 68 achieves a betterresult by using triangular second-order elements for this purpose [Fig. 9.22(b)].14 a14 aaa(a)(b)(c)Fig. 9.22 Singular elements from degenerate isoparameters (a), (b), and (c).

236 Mapped elements and numerical integrationIndeed,the use of distorted or degenerate isoparametrics is not con®ned to elasticsingularities. Rice 56 shows that in the case of plasticity a shear strain singularity of1=r type develops and Levy et al. 49 use an isoparametric,linear quadrilateral togenerate such a singularity by the simple device of coalescing two nodes but treatingthese displacements independently. A variant of this is developed by Rice and Tracey. 45The elements just described are evidently simple to implement without any changesin a standard ®nite element program. However,in Chapter 16 we introduce a methodwhereby any singularity (or other function) can be modelled directly. We believe themethods to be described there supercede the above described techniques.9.15 A computational advantage of numericallyintegrated ®nite elements 70One considerable gain that is possible in numerically integrated ®nite elements is theversatility that can be achieved in a single computer program.It will be observed that for a given class of problems the general matrices are alwaysof the same form [see the example of Eq. (9.8)] in terms of the shape function and itsderivatives.To proceed to evaluation of the element properties it is necessary ®rst to specify theshape function and its derivatives and,second,to specify the order of integration.The computation of element properties is thus composed of three distinct parts asshown in Fig. 9.23. For a given class of problems it is only necessary to change theprescription of the shape functions to achieve a variety of possible elements.Conversely,the same shape function routines can be used in many di€erent classesof problem,as is shown in Chapter 20.Use of di€erent elements,testing the e ciency of a new element in a given context,or extension of programs to deal with new situations can thus be readily achieved,andconsiderable algebra (with its inherent possibilities of mistakes) avoided.The computer is thus placed in the position it deserves,i.e.,of being the obedientslave capable of saving routine work.The greatest practical advantage of the use of universal shape function routines isthat they can be checked decisively for errors by a simple program with the patch test(viz. Chapter 10) playing the crucial role.Shape functionspecificationOrder ofintegrationGeneralformulationfor aparticulartype ofelementmatrixFig. 9.23 Computation scheme for numericallyintegrated elements.

Some practical examples of two-dimensional stress analysis 237The incorporation of simple,exactly integrable,elements in such a system is,incidentally,not penalized as the time of exact and numerical integration in manycases is almost identical.9.16 Some practical examples of two-dimensional stressanalysis 71ÿ77Some possibilities of two-dimensional analysis o€ered by curvilinear elements areillustrated in the following axisymmetric examples.9.16.1 Rotating disc (Fig. 9.24)Here only 18 elements are needed to obtain an adequate solution. It is of interest toobserve that all mid-side nodes of the cubic elements are generated within a programand need not be speci®ed.9.16.2 Conical water tank (Fig. 9.25)In this problem cubic elements are again used. It is worth noting that single-elementthickness throughout is adequate to represent the bending e€ects in both the thickand thin parts of the container. With simple triangular elements,several layers ofelements would have been needed to give an adequate solution.10.5910.188.88Density 0.283 lb/in 322500 r/minE = 10.300 T/in 2v = 0.310.3510.2010.25 9.989.69 9.791.1209.168.727.6310.2610.269.909.499.309.710.549.319.3610.9010.3510.099.558.859.547.82 8.27 7.4 5.908.27 7.587.589.38 8.367.566.457.517.316.97 5.636.486.945.494.594.379.556.90 4.30 3.93 3.32 3.13 2.852.91010.09.08.06.67Fig. 9.24 A rotating disc analysed with cubic elements.8.637.645.927.06.06.364.164.975.05.284.04.233.03.112.822.532.295.395.715.285.035.395.064.564.2118 Elements119 Nodes238 Degrees of freedom

238 Mapped elements and numerical integration330 270 210 150 90 30Hoop stress – lb/in 20 5 ftFig. 9.25 Conical water tank.9.16.3 A hemispherical dome (Fig. 9.26)The possibilities of dealing with shells approached in the previous example are herefurther exploited to show how a limited number of elements can adequately solve athin shell problem,with precisely the same program. This type of solution can befurther improved upon from the economy viewpoint by making use of the wellknownshell assumptions involving a linear variation of displacements across thethickness. Thus the number of degrees of freedom can be reduced. Methods of thiskind will be dealt with in detail in the second volume of this text.9.17 Three-dimensional stress analysisIn three-dimensional analysis,as was already hinted at in Chapter 6,the complexelement presents a considerable economic advantage. Some typical examples areshown here in which the quadratic,serendipity-type formulation is used almostexclusively. In all problems integration using three Gauss points in each directionwas used.9.17.1 Rotating sphere (Fig. 9.27)This example,in which the stresses due to centrifugal action are compared with exact

Three-dimensional stress analysis 239600E = 10 7 lb/in 2v = 0.2t = 0.5 in400M θ lb/inθp = 100R = 100 in15200Exact05 15 20θdegrees–200t1L L/t varies for 2 to 20Typical elementFig. 9.26 Encastre, thin hemispherical shell. Solution with 15 and 24 cubic elements.values,is perhaps a test on the e ciency of highly distorted elements. Seven elementsare used here and results show good agreement with exact stresses.9.17.2 Arch dam in rigid valleyThis problem,perhaps a little unrealistic from the engineer's viewpoint,was thesubject of a study carried out by a committee of the Institution of Civil Engineersand provided an excellent test for a convergence evaluation of three-dimensionalanalysis. 75 In Fig. 9.28 two subdivisions into quadratic and two into cubic elementsare shown. In Fig. 9.29 the convergence of displacements in the centre-line sectionis shown,indicating that quite remarkable accuracy can be achieved with even oneelement.The comparison of stresses in Fig. 9.30 is again quite remarkable,though showing agreater `oscillation' with coarse subdivision. The ®nest subdivision results can betaken as `exact' from checks by models and alternative methods of analysis.The above test problems illustrate the general applicability and accuracy. Twofurther illustrations typical of real situations are included.

1010zyσ z(r = 0)0–10 0lb/in 24030lbin 2 208642z inches8642σ r = σ θ(r = 0)00 10 20 30 40lb/in 2TheoreticalNumericalpw 2= 1 rad lb/in4gν = 0.31010 inxσ r (z = 0) σ ρ (z = 0) 5 σ z (z = 0)00 2 4 6 8 10r inches–0 2 4 6 8 10r inches–50 2 4 6 8 10r inches+Fig. 9.27 A rotating sphere as a three-dimensional problem. Seven parabolic elements. Stresses along z ˆ 0andr ˆ 0.

Three-dimensional stress analysis 241(a)(b)(c)(d)Fig. 9.28 Arch dam in a rigid valley± various element subdivisions.

242 Mapped elements and numerical integration120Level 6100(a)(b)(c)(d)32 EI9A EI9B EI1(96D)EI80Level 46040Level 22000 10 20 30 40 50 60 mmFig. 9.29 Arch dam in a rigid valley± centre-line displacements.Air face32 EI9A EI9B EI1(96D)EI(a)(b)(c)(d)Water face–80 –60 –40 –20 0 20 40 60 80 100 kg/cm 2CompressionTensionFig. 9.30 Arch dam in a rigid valley± vertical stresses on centre-line.

Three-dimensional stress analysis 2439.17.3 Pressure vessel (Fig. 9.31): an analysis of a biomechanicproblem (Fig. 9.32)Both show subdivisions su cient to obtain reasonable engineering accuracy. Thepressure vessel,somewhat similar to the one indicated in Chapter 6,Fig. 6.7,showsthe very considerable reduction of degrees of freedom possible with the use ofmore complex elements to obtain similar accuracy.123456789101112131415Total No. of elementsTotal No. of NodesTotal No. of Freedoms= 96= 707= 2121SupportFig. 9.31 Three-dimensional analysis of a pressure vessel.

244 Mapped elements and numerical integrationFig. 9.32 A problem of biomechanics. Plot of linear element form only; curvature of elements omitted. Notedegenerate element shapes.The example of Fig. 9.32 shows a perspective view of the elements used. Such plotsare not only helpful in visualization of the problem but also form an essential part ofdata correctness checks as any gross geometric error can be easily discovered.The importance of avoiding data errors in complex three-dimensional problems shouldbe obvious in view of their large usage of computer time. These,and indeed other, 76checking methods must form an essential part of any computation system.9.18 Symmetry and repeatabilityIn most of the problems shown,the advantage of symmetry in loading and geometrywas taken when imposing the boundary conditions,thus reducing the whole problemto manageable proportions. The use of symmetry conditions is so well known to the

Symmetry and repeatability 245Analysis domainA BIIIA Bu I ≡ u IIFig. 9.33 Repeatabilitysegments and analysis domain (shaded).engineer and physicist that no statement needs to be made about it explicitly. Lessknown,however,appears to be the use of repeatability 78 when an identical structure(and) loading is continuously repeated,as shown in Fig. 9.33 for an in®nite bladecascade. Here it is evident that a typical segment shown shaded behaves identicallyto the next one,and thus functions such as velocities and displacements atcorresponding points of AA and BB are simply identi®ed,i.e.,u I ˆ u IIThis identi®cation is made directly in a computer program.Similar repeatability,in radial coordinates,occurs very frequently in problemsinvolving turbine or pump impellers. Figure 9.34 shows a typical three-dimensionalanalysis of such a repeatable segment.Fig. 9.34 Repeatable sector in analysis of an impeller.

246 Mapped elements and numerical integrationReferences1. I.C. Taig. Structural analysis by the matrixdisplacement method. Engl. Electric AviationReport No. S017,1961.2. B.M. Irons. Numerical integration applied to ®nite element methods. Conf. Use of DigitalComputers in Struct. Eng. Univ. of Newcastle,1966.3. B.M. Irons. Engineering application of numerical integration in sti€ness method. JAIAA.14,2035±7,1966.4. S.A. Coons. Surfaces for computer aided design of space form. MIT Project MAC,MAC-TR-41,1967.5. A.R. Forrest. Curves and Surfaces for Computer Aided Design. Computer Aided DesignGroup,Cambridge,England,1968.6. G. Strang and G.J. Fix. An Analysis of the Finite Element Method. pp. 156±63,Prentice-Hall,1973.7. E.L. Wachspress. High order curved ®nite elements. Int. J. Num. Meth. Eng. 17,735±45,1981.8. M. Crochet. Personal communication. 1988.9. Nam-Sua Lee and K.-J. Bathe. E€ects of element distortion on the performance of isoparametricelements. Internat. J. Num. Meth. Eng. 36,3553±76,1993.10. N. Abramowitz and I.A. Stegun. Handbook of Mathematical Functions. Dover,New York,1965.11. B.M. Irons. Quadrature rules for brick based ®nite elements. Int. J. Num. Meth. Eng. 3,293±4,1971.12. T.K. Hellen. E€ective quadrature rules for quadratic solid isoparametric ®nite elements.Int. J. Num. Meth. Eng. 4,597±600,1972.13. Radau. Journ. de Math. 3,283,1880.14. R.G. Anderson,B.M. Irons,and O.C. Zienkiewicz. Vibration and stability of plates using®nite elements. Int. J. Solids Struct. 4,1031±55,1968.15. P.C. Hammer,O.P. Marlowe,and A.H. Stroud. Numerical integration over simplexes andcones. Math. Tables Aids Comp. 10,130±7,1956.16. C.A. Felippa. Re®ned ®nite element analysis of linear and non-linear two-dimensional structures.Structures Materials Research Report No. 66±22,Oct. 1966,Univ. of California,Berkeley.17. G.R. Cowper. Gaussian quadrature formulas for triangles. Int. J. Num. Mech. Eng. 7,405±8,1973.18. I. Fried. Accuracy and condition of curved (isoparametric) ®nite elements. J. Sound Vibration.31,345±55,1973.19. I. Fried. Numerical integration in the ®nite element method. Comp. Struc. 4,921±32,1974.20. M. Zlamal. Curved elements in the ®nite element method. SIAM J. Num. Anal. 11,347±62,1974.21. D. Koslo€ and G.A. Frasier. Treatment of hour glass patterns in low order ®nite elementcodes. Int. J. Num. Anal. Meth. Geomechanics. 2,57±72,1978.22. T. Belytschko and W.E. Bachrach. The e cient implementation of quadrilaterals with highcoarse mesh accuracy. Comp. Methods Appl. Mech and Engineering. 54,276±301,1986.23. O.C. Zienkiewicz and D.V. Phillips. An automatic mesh generation scheme for plane andcurved element domains. Int. J. Num. Meth. Eng. 3,519±28,1971.24. W.J. Gordon and C.A. Hall. Construction of curvilinear co-ordinate systems and applicationto mesh generation. Int. J. Num. Meth. Eng. 7,461±77,1973.25. W.J. Gordon and C.A. Hall. Trans®nite element methods ± blending-function interpolationover arbitrary curved element domains. Numer. Math. 21,109±29,1973.

References 24726. J. Peraire,M. Vahdati,K. Morgan,and O.C. Zienkiewicz. Adaptive remeshing for compressible¯ow computations. J. Comp. Phys. 72,449±66,1987.27. J. Peraire,K. Morgan,M. Vahdati,and O.C. Zienkiewicz. Finite element Euler computationsin 3-d. Internat. J. Num. Meth. Eng. 26,2135±59,1988.28. P. Kaupp and S. Steinberg. Fundamentals of Grid Generation. CRC Press,1993.29. N.P. Weatherill,P.R. Eiseman,J. Hause,and J.P. Thompson. Numerical Grid Generationin Fluid Dynamics and Related Fields. Pineridge Press,Swansea,1994.30. P.L. George and H. Borouchaki. Delaunay Triangulation and Meshing. Hermes,Paris,1998.31. J.F. Thompson,B.K. Soni,and N.P. Weatherill,eds. Handbook of Grid Generation. CRCPress,January 1999.32. GiD±The Personal Pre/Postprocessor,CIMNE,Barcelona,Spain,1999.33. R.W. Thatcher. On the ®nite element method for unbounded regions. SIAM J. NumericalAnalysis. 15,3,pp. 466±76,June 1978.34. P. Silvester,D.A. Lowther,C.J. Carpenter,and E.A. Wyatt. Exterior ®nite elements for 2-dimensional ®eld problems with open boundaries. Proc. IEE. 123,No. 12,December 1977.35. S.F. Shen. An aerodynamicist looks at the ®nite element method,in Finite Elements inFluids (eds R.H. Gallagher et al.). Vol. 2,pp. 179±204,Wiley,1975.36. O.C. Zienkiewicz,D.W. Kelly,and P. Bettess. The coupling of the ®nite element andboundary solution procedures. Int. J. Num. Meth. Eng. 11,355±75,1977.37. P. Betess. In®nite elements. Int. J. Num. Meth. Eng. 11,53±64,1977.38. P. Bettess and O.C. Zienkiewicz. Di€raction and refraction of surface waves using ®niteand in®nite elements. Int. J. Num. Meth. Eng. 11,1271±90,1977.39. G. Beer and J.L. Meek. In®nite domain elements. Int. J. Num. Meth. Eng. 17,43±52,1981.40. O.C. Zienkiewicz,C. Emson,and P. Bettess. A novel boundary in®nite element. Int. J.Num. Meth. Eng. 19,393±404,1983.41. P. Bettess. In®nite Elements. Penshaw Press,1992.42. R.H. Gallagher. Survey and evaluation of the ®nite element method in fracture mechanicsanalysis,in Proc. 1st Int. Conf. on Structural Mechanics in Reactor Technology. Vol. 6,PartL,pp. 637±53,Berlin,1971.43. N. Levy,P.V. MarcË al,and J.R. Rice. Progress in three-dimensional elastic±plastic stressanalysis for fracture mechanics. Nucl. Eng. Des. 17,64±75,1971.44. J.J. Oglesby and O. Lomacky. An evaluation of ®nite element methods for the computationof elastic stress intensity factors. J. Eng. Ind. 95,177±83,1973.45. J.R. Rice and D.M. Tracey. Computational fracture mechanics,in Numerical and ComputerMethods in Structural Mechanics (eds S.J. Fenves et al.). pp. 555±624,Academic Press,1973.46. A.A. Gri ths. The phenomena of ¯ow and rupture in solids. Phil. Trans. Roy. Soc.(London). A221,163±98,Oct. 1920.47. J.L. Swedlow. Elasto-plastic cracked plates in plane strain. Int. J. Fract. Mech. 4,33±44,March 1969.48. T. Yokobori and A. Kamei. The size of the plastic zone at the tip of a crack in plane strainstate by the ®nite element method. Int. J. Fract. Mech. 9,98±100,1973.49. N. Levy,P.V. MarcË al,W.J. Ostergren,and J.R. Rice. Small scale yielding near a crack inplane strain: a ®nite element analysis. Int. J. Fract. Mech. 7,143±57,1967.50. J.R. Dixon and L.P. Pook. Stress intensity factors calculated generally by the ®nite elementtechnique. Nature. 224,166,1969.51. J.R. Dixon and J.S. Strannigan. Determination of energy release rates and stress-intensityfactors by the ®nite element method. J. Strain Analysis. 7,125±31,1972.52. V.B. Watwood. Finite element method for prediction of crack behavior. Nucl. Eng. Des. II(No. 2),323±32,March 1970.

248 Mapped elements and numerical integration53. D.F. Mowbray. A note on the ®nite element method in linear fracture mechanics. Eng.Fract. Mech. 2,173±6,1970.54. D.M. Parks. A sti€ness derivative ®nite element technique for determination of elasticcrack tip stress intensity factors. Int. J. Fract. 10,487±502,1974.55. T.K. Hellen. On the method of virtual crack extensions. Int. J. Num. Meth. Eng. 9 (No. 1),187±208,1975.56. J.R. Rice. A path-independent integral and the approximate analysis of strain concentrationby notches and cracks. J. Appl. Mech.. Trans. Am. Soc. Mech. Eng. 35,379±86,1968.57. P. Tong and T.H.H. Pian. On the convergence of the ®nite element method for problemswith singularity. Int. J. Solids Struct. 9,313±21,1972.58. T.A. Cruse and W. Vanburen. Three dimensional elastic stress analysis of a fracture specimenwith edge crack. Int. J. Fract. Mech. 7,1±15,1971.59. E. Byskov. The calculation of stress intensity factors using the ®nite element method withcracked elements. Int. J. Fract. Mech. 6,159±67,1970.60. P.F. Walsh. Numerical analysis in orthotropic linear fracture mechanics. Inst. Eng.Australia. Civ. Eng.. Trans. 15,115±19,1973.61. P.F. Walsh. The computation of stress intensity factors by a special ®nite element technique.Int. J. Solids Struct. 7,1333±42,Oct. 1971.62. A.K. Rao,I.S. Raju,and A. Murthy Krishna. A powerful hybrid method in ®nite elementanalysis. Int. J. Num. Meth. Eng. 3,389±403,1971.63. W.S. Blackburn. Calculation of stress intensity factors at crack tips using special ®niteelements,in The Mathematics of Finite Elements (ed. J.R. Whiteman),pp. 327±36,Academic Press,1973.64. D.M. Tracey. Finite elements for determination of crack tip elastic stress intensity factors.Eng. Fract. Mech. 3,255±65,1971.65. R.D. Henshell and K.G. Shaw. Crack tip elements are unnecessary. Int. J. Num. Meth.Eng. 9,495±509,1975.66. R.S. Barsoum. On the use of isoparametric ®nite elements in linear fracture mechanics. Int.J. Num. Meth. Eng. 10,25±38,1976.67. R.S. Barsoum. Triangular quarter point elements as elastic and perfectly elastic crack tipelements. Int. J. Num. Meth. Eng. 11,85±98,1977.68. H.D. Hibbitt. Some properties of singular isoparametric elements. Int. J. Num. Meth. Eng.11,180±4,1977.69. S.E. Benzley. Representation of singularities with isoparametric ®nite elements. Int. J.Num. Meth. Eng. 8 (No. 3),537±45,1974.70. B.M. Irons. Economical computer techniques for numerically integrated ®nite elements.Int. J. Num. Meth. Eng. 1,201±3,1969.71. O.C. Zienkiewicz,B.M. Irons,J.G. Ergatoudis,S. Ahmad,and F.C. Scott. Isoparametricand associated element families for two and three dimensional analysis,in Proc. Course onFinite Element Methods in Stress Analysis (eds I. Holland and K. Bell). Trondheim Tech.University,1969.72. B.M. Irons and O.C. Zienkiewicz. The isoparametric ®nite element system ± a new concept in®nite element analysis. Proc. Conf. Recent Advances in Stress Analysis. Royal Aero Soc.,1968.73. J.G. Ergatoudis,B.M. Irons,and O.C. Zienkiewicz. Curved,isoparametric. `quadrilateral'elements for ®nite element anlysis. Int. J. Solids Struct. 4,31±42,1968.74. J.G. Ergatoudis. Isoparametric elements in two and three dimensional analysis. Ph.D. thesis,University of Wales,Swansea,1968.75. J.G. Ergatoudis,B.M. Irons,and O.C. Zienkiewicz. Three dimensional analysis of archdams and their foundations. Symposium on Arch Dams. Inst. Civ. Eng.,London,1968.76. O.C. Zienkiewicz,B.M. Irons,J. Campbell,and F.C. Scott. Three dimensional stressanalysis. Int. Un. Th. Appl. Mech. Symp. on High Speed Computing in Elasticity.LieÁge,1970.

77. O.C. Zienkiewicz. Isoparametric and other numerically integrated elements,in Numericaland Computer Methods in Structural Mechanics (eds S.J. Fenves,N. Perrone,A.R. Robinson,andW.C. Schnobrich). pp. 13±41,Academic Press,1973.78. O.C. Zienkiewicz and F.C. Scott. On the principle of repeatability and its application inanalysis of turbine and pump impellers. Int. J. Num. Meth. Eng. 9,445±52,1972.References 249

10The patch test, reduced integration,and non-conforming elements10.1 IntroductionWe have brie¯y referred in Chapter 2 to the patch test as a means of assessing convergenceof displacement-type elements for elasticity problems in which the shapefunctions violate continuity requirements. In this chapter we shall deal in moredetail with this test which is applicable to all ®nite element forms and will show that(a) it is a necessary condition for assessing the convergence of any ®nite elementapproximation and further that, if properly extended and interpreted, it canprovide(b) a su cient requirement for convergence,(c) an assessment of the (asymptotic) convergence rate of the element tested,(d) a check on the robustness of the algorithm, and(e) a means of developing new and accurate ®nite element forms which violatecompatibility (continuity) requirements.While for elements which a priori satisfy all the continuity requirements, havecorrect polynomial expansions, and are exactly integrated such a test is super¯uousin principle, it is nevertheless useful as it gives(f ) a check that correct programming was achieved.For all the reasons cited above the patch test has been, since its inception, and continuesto be the most important check for practical ®nite element codes.The original test was introduced by Irons et al. 1ÿ3 in a physical way and could beinterpreted as a check which ascertained whether a patch of elements (Fig. 10.1)subject to a constant strain reproduced exactly the constitutive behaviour of thematerial and resulted in correct stresses when it became in®nitesimally small. If itdid, it could then be argued that the ®nite element model represented the real materialbehaviour and, in the limit, as the size of the elements decreased would thereforereproduce exactly the behaviour of the real structure.Clearly, although this test would only have to be passed when the size of theelement patch became in®nitesimal, for most elements in which polynomials areused the patch size did not in fact enter the consideration and the requirement thatthe patch test be passed for any element size became standard.

Convergence requirements 251σ constant~ε x1ε x dxdxFig. 10.1 A patch of element and a volume of continuum subject to constant strain " x . A physical interpretationof the constant strain or linear displacement ®eld patch test.Quite obviously a rigid body displacement of the patch would cause no strain, andif the proper constitutive laws were reproduced no stress changes would result. Thepatch test thus guarantees that no rigid body motion straining will occur.When curvilinear coordinates are used the patch test is still required to be passed inthe limit but generally will not do so for a ®nite size of the patch. (An exception here isthe isoparametric coordinate system in problems discussed in Chapter 9 since it isguaranteed to contain linear polynomials in the global coordinates.) Thus for manyproblems such as shells, where local curvilinear coordinates are used, this test hasto be restricted to in®nitesimal patch sizes and, on physical grounds alone, appearsto be a necessary and su cient condition for convergence.Numerous publications on the theory and practice of the test have followed theoriginal publications cited 4ÿ6 and mathematical respectability was added to thoseby Strang. 7;8 Although some authors have cast doubts on its validity 9;10 these havebeen fully refuted 11ÿ13 and if the test is used as described here it ful®ls the requirements(a)±(f) stated above.In the present chapter we consider the patch test applied to irreducible forms (seeChapter 3) but an extension to mixed forms is more important. This has been studiedin references 13, 14 and 15 and made use of in many subsequent publications. Thematter of mixed form patch tests will be fully discussed in the next chapter; however,the consistency and stability tests developed in the present chapter are always required.One additional use of the patch test was suggested by BabusÏ ka et al. 16 with ashorter description given by Boroomand and Zienkiewicz. 17 This test can establishthe e ciency of gradient (stress) recovery processes which are so important in errorestimation as will be discussed in Chapter 14.10.2 Convergence requirementsWe shall consider in the following the patch test as applied to a ®nite element solutionof a set of di€erential equationsA…u† L…u†‡g ˆ 0…10:1†in the domain together with the conditionsB…u† ˆ0…10:2†on the boundary of the domain, ÿ.

252 The patch test, reduced integration, and non-conforming elementsThe ®nite element approximation is given in the formu ^u ˆ Na…10:3†where N are shape functions de®ned in each element, e , and a are unknownparameters.By applying standard procedures of ®nite element approximation (see Chapters 2and 3) the problem reduces in a linear case to a set of algebraic equationsKa ˆ f…10:4†which when solved give an approximation to the di€erential equation and its boundaryconditions.What is meant by `convergence' in the approximation sense is that the approximatesolution, ^u, should tend to the exact solution u when the size of the elements happroaches zero (with some speci®ed subdivision pattern). Stated mathematicallywe must ®nd that the error at any point becomes (when h is su ciently small)ju ÿ ^uj ˆO…h q † 4 Ch q…10:5†where q > 0 and C is a positive constant, depending on the position.This must also be true for all the derivatives of u de®ned in the approximation.By the order of convergence in the variable u we mean the value of the index q in theabove de®nition.To ensure convergence it is necessary that the approximation ful®l both consistencyand stability conditions. 18The consistency requirement ensures that as the size of the elements h tends to zero,the approximation equation (10.4) will represent the exact di€erential equation (10.1)and the boundary conditions (10.2) (at least in the weak sense).The stability condition is simply translated as a requirement that the solution of thediscrete equation system (10.4) be unique and avoid spurious mechanisms which maypollute the solution for all sizes of elements. For linear problems in which we solve thesystem of algebraic equations (10.4) asa ˆ K ÿ1 f…10:6†this means simply that the matrix K must be non-singular for all possible elementassemblies (subject to imposing minimum stable boundary conditions).The patch test traditionally has been used as a procedure for verifying theconsistency requirement; the stability was checked independently by ensuring nonsingularityof matrices. 19 Further, it generally tested only the consistency in satisfactionof the di€erential equation (10.1) but not of its natural boundary conditions. Inwhat follows we shall show how all the necessary requirements of convergence can betested by a properly conceived patch test.A `weak' singularity of a single element may on occasion be permissible and someelements exhibiting it have been, and still are, successfully used in practice. One suchcase is given by the eight-node isoparametric element with a 2 2 Gauss quadrature,to which we shall refer later here. This element is on occasion observed to showpeculiar behaviour (though its use has advantages as discussed in Chapter 11). Anelement that occasionally fails is termed non-robust and the patch test provides ameans of assessing the degree of robustness.

The simple patch test (tests Aand B) ± a necessary condition for convergence 25310.3 The simple patch test (tests Aand B) ± a necessarycondition for convergenceWe shall ®rst consider the consistency condition which requires that in the limit (as htends to zero) the ®nite element approximation of Eq. (10.4) should model exactly thedi€erential equation (10.1) and the boundary conditions (10.2). If we consider a`small' region of the domain (of size 2h) we can expand the unknown function uand the essential derivatives entering the weak approximation in a Taylor series.From this we conclude that for convergence of the function and its ®rst derivativein typical problems of a second-order equation and two dimensions, we requirethat around a point i assumed to be at the coordinate origin, @u @uu ˆ u i ‡ x ‡ y ‡‡O…h p †@x i @y i@u@x ˆ@u@y ˆ @u‡‡O…h p ÿ 1 †@x i @u‡‡O…h p ÿ 1 †@yi…10:7†with p 5 2. The ®nite element approximation should therefore reproduce exactly theproblem posed for any linear forms of u as h tends to zero. Similar conditions canobviously be written for higher order problems. This requirement is tested by thecurrent interpretation of the patch test illustrated in Fig. 10.2. We refer to this asthe base solution.In this we compute ®rst an arbitrary linear solution of the di€erential equation andthe corresponding set of parameters a [see Eq. (10.3)] at all `nodes' of a patch whichassembles completely the nodal variable a i (i.e., provides all the equation termscorresponding to it).In test A we simply insert the exact value of the parameters a into the ith equationand verify thatK ij a j ÿ f i 0…10:8†iiTest Aa prescribed on all nodesK ij a j = f i verified at node iTest Ba prescribed at edges of patcha i = K ii–1 (f i – K ij a j ) (j = i) solvedFig. 10.2 Patch test of forms A and B.

254 The patch test, reduced integration, and non-conforming elementswhere f i is a force which results from any `body force' required to satisfy the base solutiondi€erential equation (10.1). Generally in problems given in cartesian coordinatesthe required body force is zero; however, in curvilinear coordinates (e.g., axisymmetricelasticity problems) it can be non-zero.In test B only the values of a corresponding to the boundaries of the `patch' areinserted and a i is found asa i ˆ K ÿ1ii …f i ÿ K ij a j † j 6ˆ i …10:9†and compared against the exact value.Both patch tests verify only the satisfaction of the basic di€erential equation andnot of the boundary approximations, as these have been explicitly excluded here.We mentioned earlier that the test is, in principle, required only for anin®nitesimally small patch of elements; however, for di€erential equations withconstant coe cients and with a mapping involving constant jacobian the size ofthe patch is immaterial and the test can be carried out on a patch of arbitrarydimensions.Indeed, if the coe cients are not constant the same size independence exists providingthat a constant set of such coe cients is used in the formulation of the test. (Thisapplies, for instance, in axisymmetric problems where coe cients of the type 1/radiusenter the equations and when the patch test is here applied, it is simply necessary toenter the computation with such quantities assumed constant.)If mapped curvilinear elements are used it is not obvious that the patch test posed inglobal coordinates needs to be satis®ed. Here, in general, convergence in the mappingcoordinates may exist but a ®nite patch test may not be satis®ed. However, once againif we specify the nature of the subdivision without changing the mapping function, inthe limit the jacobian becomes locally constant and the previous remarks apply. Toillustrate this point consider, for instance, a set of elements in which local coordinatesare simply the polar coordinates as shown in Fig. 10.3. With shape functions usingpolynomial expansions in the r, terms the patch test of the kind we have describedabove will not be satis®ed with elements of ®nite size ± nevertheless in the limit as theelement size tends to zero it will become true. Thus it is evident that patch testsatisfaction is a necessary condition which has always to be achieved providing thesize of the patch is in®nitesimal.rrθθxFig. 10.3 Polar coordinate mapping.

Generalized patch test (test C) and the single-element test 255This proviso which we shall call weak patch test satisfaction is not always simple toverify, particularly if the element coding does not easily permit the insertion ofconstant coe cients or a jacobian. In Sec. 10.10 we shall discuss in some detail itsimplementation, which, however, is only necessary in very special element forms. Itis indeed fortunate that the standard isoparametric element form reproduces exactlythe linear polynomial global coordinates (see Chapter 9) and for this reason does notrequire special treatment unless some other crime (such as selective or reducedintegration) is introduced.10.4 Generalized patch test (test C) and the singleelementtestThe patch test described in the preceding section was shown to be a necessary conditionfor convergence of the formulation but did not establish su cient conditions for it. Inparticular, it omitted the testing of the boundary `load' approximation for the casewhen the `natural' (e.g. `traction of elasticity') conditions are speci®ed. Further it didnot verify the stability of the approximation. A test including a check on the above conditionsis easily constructed. We show this in Fig. 10.4 for a two-dimensional planeproblem as test C. In this the patch of elements is assembled as before but subject toprescribed natural boundary conditions (or tractions around its perimeter) correspondingto the base function. The assembled matrix of the whole patch is written asKa ˆ fFixing only the minimum number of parameters a necessary to obtain a physicallyvalid solution (e.g., eliminating the rigid body motion in an elasticity example or asingle value of temperature in a heat conduction problem) a solution is sought forremaining a values and compared with the exact base solution assumed.Now any singularity of the K matrix will be immediately observed and, as thevector f includes all necessary source and boundary traction terms, the formulationwill be completely tested (providing of course a su cient number of test states isused). The test described is now not only necessary but su cient for convergence.Natural boundaryconditions specifiedy(a)xMinimum esentialboundary conditions(b)Fig. 10.4 (a) Patch test of form C. (b) The single-element test.

256 The patch test, reduced integration, and non-conforming elementsGauss integrationpointsNine-nodedelements onlyEight and nine nodes(a)(b)Fig. 10.5 (a) Zero energy (singular) modes for eight- and nine-noded quadratic elements and (b) for a patchof bilinear elements with single integration points.With boundary traction included it is of course possible to reduce the size of thepatch to a single element and an alternative form of test C is illustrated in Fig.10.4(b), which is termed the single-element test. 11 This test is indeed one requirementof a good ®nite element formulation as, on occasion, a larger patch may not reveal theinherent instabilities of a single element. This happens in the well-documented case ofthe plane strain±stress eight-noded isoparametric element with (reduced) four-pointGauss quadrature i.e., where the singular deformation mode of a single element(see Fig. 10.5) disappears when several elements are assembled.y It should be noted,however, that satisfaction of a single element test is not a su cient condition for convergence.Forsu ciency we require at least one internal element boundary to test thatconsistency of a patch solution is maintained between elements.y This ®gure also shows a similar singularity for a patch of four bilinear elements with single-pointquadrature, and we note the similar shape of zero energy modes (see Chapter 9, Sec. 9.11.3).

10.5 The generality of a numerical patch testHigher order patch tests 257In the previous section we have de®ned in some detail the procedures for conducting apatch test. We have also asserted the fact that such tests if passed guarantee thatconvergence will occur. However all the tests are numerical and it is impractical totest all possible combinations.In particular let us consider the base solutions used. These will invariably be a set ofpolynomials given in two dimensions asu ˆ X a i P i …x; y†…10:10†where P i are a suitable set of low order polynomials (e.g., 1, x, y for Galerkin formspossessing only ®rst-order derivatives) and a i are parameters. It is fairly obvious thatif patch tests are conducted on each of these polynomials individually any base functionof the form given in Eq. (10.10) can be reproduced and the generality preservedfor the particular combination of elements tested. This must always be done and isalmost a standard procedure in engineering tests, necessitating only a limitednumber of combinations.However, as various possible patterns of elements can occur and it is possible toincrease the size without limit the reader may well ask whether the test is completefrom the geometrical point of view. We believe it is necessary in a numerical test toconsider the possibility of several pathological arrangements of elements but that ifthe test is purely limited to a single element and a complete patch around a nodewe can be con®dent about the performance on more general geometric patterns.Indeed even mathematical assessments of convergence are subject to limits oftenimposed a posteriori. Such limits may arise if for instance a singular mapping is used.The procedures referred to in this section satisfy most readers as to the validity andgenerality of the test.On some limited occasions it is possible to perform the test purely algebraically andthen its validity cannot be doubted. Some such algebraic tests will be referred to laterin connection with incompatible elements.In this chapter we have only considered linear di€erential equations and linearmaterial behaviour. In Volume 2 non-linear problems will be fully discussed andon some occasions the patch test can well be used and extended to cover such areas.10.6 Higher order patch tests 6;8While the patch tests discussed in the last three sections ensure (when satis®ed) thatconvergence will occur, they did not test the order of this convergence, beyondassuring us that in the case of Eq. (10.7) the errors were, at least, of order O…h 2 † inu. It is an easy matter to determine the actual highest asymptotic rate of convergenceof a given element by simply imposing, instead of a linear solution, exact higherorder polynomial solutions. The highest value of such polynomials for which completesatisfaction of the patch test is achieved automatically evaluates the corresponding convergencerate. It goes without saying that for such exact solutions generally non-zerosource (e.g., body force) terms in the original equation (10.1) will need to be involved.

258 The patch test, reduced integration, and non-conforming elementsIn addition, test C in conjunction with a higher order patch test may be used toillustrate any tendency for `locking' to occur (see Chapter 11). Accordingly, elementrobustness with regard to various parameters (e.g., Poisson's ratios near one-half forelasticity problems in plane strain) may be established.In such higher order patch tests it will of course ®rst be assumed that the patch issubject to the base expansion solution as described. Thus, for higher order terms itwill be necessary to start and investigate solutions of the typea 3 x 2 ‡ a 4 xy ‡ a 5 y 2 ‡each of which should be applied individually or as linearly independent combinationsand for each the solution should be appropriately tested.In particular, we shall expect higher order elements to exactly satisfy certain ordersolutions. However in Chapter 14 we shall use this idea to ®nd the error between theexact solution and the recovery using precisely the same type of formulation.10.7 Application of the patch test to plane elasticityelements with `standard' and `reduced' quadratureIn the next few sections we consider several applications of the patch test in theevaluation of ®nite element models. In each case we consider only one of the necessarytests which need to be implemented. For a complete evaluation of a formulation it isnecessary to consider all possible independent base polynomial solutions as well as avariety of patch con®gurations which test the e€ects of element distortion or alternativemeshing interconnections which will be commonly used in analysis. As we shallemphasize, it is important that both consistency and stability be evaluated in aproperly conducted test.In Chapter 9 (Sec. 9.11) we have discussed the minimum required order ofnumerical integration for various ®nite element problems which results in no lossof convergence rate. However, it was also shown that for some elements such aminimum integration order results in singular matrices. If we de®ne the standardintegration as one which evaluates the sti€ness of an element exactly (at least in theundistorted form) then any lower order of integration is called reduced.Such reduced integration has some merits in certain problems for reasons which weshall discuss in Chapter 12 (Sec. 12.5), but it can cause singularities which should bediscovered by a patch test (which supplements and veri®es the arguments of Sec.9.11.3). Application of the patch test to some typical problems will now be shown.10.7.1 Example 1: Patch test for base solutionWe consider ®rst a plane stress problem on the patch shown in Fig. 10.6(a). Thematerial is linear, isotropic elastic with properties E ˆ 1000 and ˆ 0:3. The ®niteelement procedure used is based on the displacement form using four-noded isoparametricshape functions and numerical integration. Analyses are conducted using theplane element and program described in Chapter 20. Since the sti€ness computation

Application of the patch test to plane elasticity elements 2593 34 48756(a)1 2 12Fig. 10.6 Patch for evaluation of numerically integrated plane stress problems. (a) Five-element patch.(b) One-element patch.(b)includes only ®rst derivatives of displacements, the formulation converges providedthat the patch test is satis®ed for all linear polynomial solutions of displacementsin the base solution. Here we consider only one of the six independent linear polynomialsolutions necessary to verify satisfaction of the patch test. The solutionconsidered isu ˆ 0:002xv ˆÿ0:0006y…10:11†which produces zero body forces and zero stresses except for x ˆ 2…10:12†The solution given in Table 10.1 is obtained for the nodal displacements and satis®esEq. (10.10) exactly.The patch test is performed ®rst using 2 2 gaussian `standard' quadrature tocompute each element sti€ness and resulting reaction forces at nodes. For patchtest A all nodes are restrained and nodal displacement values are speci®ed accordingto Table 10.1. Stresses are computed at speci®ed Gauss points (1 1, 2 2, and 3 3Gauss points were sampled) and all are exact to within round-o€ error (double precisionwas used which produced round-o€ errors less than 10 ÿ15 in the quantitiescomputed). Reactions were also computed at all nodes and again produced theTable 10.1 Patch solution for Fig. 10.6Coordinates Computed displacements ForcesNode i x i y i u i v i F xi F yi1 0.0 0.0 0.0 0:0 ÿ2 02 2.0 0.0 0.0040 0:0 3 03 2.0 3.0 0.0040 ÿ0:00186 2 04 0.0 2.0 0.0 ÿ0:00120 ÿ3 05 0.4 0.4 0.0008 ÿ0:00024 0 06 1.4 0.6 0.0028 ÿ0:00036 0 07 1.5 2.0 0.0030 ÿ0:00120 0 08 0.3 1.6 0.0006 ÿ0:00096 0 0

260 The patch test, reduced integration, and non-conforming elementsforce values shown in Table 10.1 to within round-o€ limits. This approximation satis-®es all conditions required for a ®nite element procedure (i.e., conforming shape functionsand normal-order quadrature). Accordingly, the patch test merely veri®es thatthe programming steps used contain no errors. Patch test A does not require explicituse of the sti€ness matrix to compute results; consequently the above patch test wasrepeated using patch test B where only nodes 1 to 4 are restrained with theirdisplacements speci®ed according to Table 10.1. This tests the accuracy of the sti€nessmatrix and, as expected, exact results are once again recovered to within round-o€errors. Finally, patch test C was performed with node 1 fully restrained and node 4restrained only in the x-direction. Nodal forces were applied to nodes 2 and 3 inaccordance with the values generated through the boundary tractions by x (i.e.,nodal forces shown in Table 10.1). This test also produced exact solutions for allother nodal quantities in Table 10.1 and recovered x of 2 at all Gauss points ineach element.The above test was repeated for patch tests A, B, and C but using a 1 1 `reduced'Gauss quadrature to compute the element sti€ness and nodal force quantities. Patchtest C indicated that the global sti€ness matrix contained two global `zero energymodes' (i.e., the global sti€ness matrix was rank de®cient by 2), thus producingincorrect nodal displacements whose results depend solely on the round-o€ errorsin the calculations. These in turn produced incorrect stresses except at the 1 1Gauss point used in each element to compute the sti€ness and forces. Thus, basedupon stability considerations, the use of 1 1 quadrature on four-noded elementsproduces a failure in the patch test. The element does satisfy consistency requirements,however, and provided a proper stabilization scheme is employed (e.g., sti€nessor viscous methods are used in practice) this element may be used for practicalcalculations. 20;21It should be noted that a one-element patch test may be performed using the meshshown in Fig. 10.6(b). The results are given by nodes 1 to 4 in Table 10.1. For the oneelementpatch, patch tests A and B coincide and neither evaluates the accuracy orstability of the sti€ness matrix. On the other hand, patch test C leads to theconclusions reached using the ®ve-element patch: namely, 2 2 gaussian quadraturepasses a patch test whereas 1 1 quadrature fails the stability part of the test (asindeed we would expect by the arguments of Chapter 9, Sec. 9.11).A simple test on cancellation of a diagonal during the triangular decompositionstep is su cient to warn of rank de®ciencies in the sti€ness matrix. In the pro®lemethod, described in Chapter 20, this is easily monitored as compact eliminationconverts the initial value of a diagonal element to the ®nal value in one step. Thusonly one extra scalar variable is needed to test the initial and ®nal values.10.7.2 Example 2: Patch test for quadratic elements: quadratureeffectsIn Fig. 10.7 we show a two-element patch of quadratic isoparametric quadrilaterals.Both eight-noded serendipity and nine-noded lagrangian types are considered and abasic patch test type C is performed for load case 1. For the eight-noded element both2 2 (`reduced') and 3 3 (`standard') gaussian quadrature satisfy the patch test,

Application of the patch test to plane elasticity elements 261151521510 Load 1 Load 2555A20dB5Load 1 Load 25Fig. 10.7 Patch test for eight- and nine-noded isoparametric quadrilaterals.whereas for the nine-noded element only 3 3 quadrature is satisfactory, with 2 2reduced quadrature leading to failure in rank of the sti€ness matrix. However, if weperform a one-element test for the eight-noded and 2 2 quadrature element, wediscover the spurious zero-energy mode shown in Fig. 10.5 and thus the one-elementtest has failed. We consider such elements suspect and to be used only with thegreatest of care. To illustrate what can happen in practice we consider the simpleproblem shown in Fig. 10.8(a). In this example the `structure' modelled by a singleelement is considered rigid and interest is centred on the `foundation' response.Accordingly only one element is used to model the structure. Use of 2 2 quadraturethroughout leads to answers shown in Fig. 10.8(b) while results for 3 3 quadratureare shown in Fig. 10.8(c). It should be noted that no zero-energy mode exists sincemore than one element is used. There is, however, here a spurious response due tothe large modulus variation between structure and foundation. This suggests thatproblems in which non-linear response may lead to a large variation in materialparameters could also induce such performance, and thus use of the eight-noded2 2 integrated element should always be closely monitored to detect such anomalousbehaviour.Indeed, support or loading conditions may themselves induce very suspectresponses for elements in which near singularity occurs. Figure 10.9 shows someamusing peculiarities which can occur for reduced integration elements and whichdisappear entirely if full integration is used. 22 In all cases the assembly of elementsis non-singular even though individual elements are rank de®cient.10.7.3 Example 3: Higher order patch test ± assessment of orderIn order to demonstrate a higher order patch test we consider the two-element planestress problem shown in Fig. 10.7 and subjected to bending loading shown as Load 2.As above, two di€erent types of element are considered: (a) an eight-noded serendipityquadrilateral elenent and (b) a nine-noded lagrangian quadrilateral element. In our

262 The patch test, reduced integration, and non-conforming elementsPE = 10 6v = 0.3StructureE = 10 2v = 0.3ZerodisplacementFoundation(a)(b)(c)Fig. 10.8 A propagating spurious mode from a single unsatisfactory element. (a) Problem and mesh. (b) 2 2integration. (c) 3 3 integration.test we wish to demonstrate a feature for nine-noded element mapping discussed inChapter 9 (see Sec. 9.7) and ®rst shown by Wachspress. 23 In particular we restrictthe mapping into the xy plane to be that produced by the four-noded isoparametricbilinear element, but permit the dependent variable to assume the full range of variationsconsistent with the eight- or nine-noded shape functions. In Chapter 9 weshowed that the nine-noded element can approximate a complete quadratic displacementfunction in x, y whereas the eight-noded element cannot. Thus we expect thatthe nine-noded element when restricted to the isoparametric mappings of the fournodedelement will pass a higher order patch test for all arbitrary quadratic displacement®elds. The pure bending solution in elasticity is composed of polynomial terms

(b)(a)(c)Fig. 10.9 Peculiar response of near singular assemblies of elements. 22 (a) A column of nine-noded elements with point load response of full 3 3and2 2 integration. Thewhole assembly is non-singular but singular element modes are apparent. (b) A fully constrained assembly of nine-noded elements with no singularity ± ®rst six eigenmodeswith full (3 3) integration. (c) As (b) but with 2 2 integration. Note the appearance of `wild' modes called `Escher' modes named so in reference 22 after this graphic artist.

264 The patch test, reduced integration, and non-conforming elementsTable 10.2 Bending load case …E ˆ 100; ˆ 0:3†Element Quadrature d v A u B v BEight-node)3 3 0.750 0.150 0.75225Eight-node 2 20 0.750 0.150 0.75225Nine-node 3 3 0.750 0.150 0.75225)Eight-node 3 3 0.7448 0.1490 0.74572Eight-node 2 21 0.750 0.150 0.75100Nine-node 3 3 0.750 0.150 0.75225)Eight-node 3 3 0.6684 0.1333 0.66364Eight-node 2 22 0.750 0.150 0.75225Nine-node 3 3 0.750 0.150 0.75225Exact Ð Ð 0.750 0.150 0.75225up to quadratic order. Furthermore, no body force loadings are necessary to satisfythe equilibrium equations. For the mesh considered the nodal loadings are equal andopposite on the top and bottom nodes as shown in Fig. 10.7. The results for the twoelements are shown in Table 10.2 for the indicated quadratures with E ˆ 100 and ˆ 0:3.From this test we observe that the nine-noded element does pass the higher ordertest performed. Indeed, provided the mapping is restricted to the four-noded shape itwill always pass a patch test for displacements with terms no higher than quadratic.On the other hand, the eight-noded element passes the higher order patch test performedonly for rectangular element (or constant jacobian) mappings. Moreover,the accuracy of the eight-noded element deteriorates very rapidly with increaseddistortions de®ned by the parameter d in Fig. 10.7.The use of 2 2 reduced quadrature improves results for the higher order patchtest performed. Indeed, two of the points sampled give exact results and the third isonly slightly in error. As noted previously, however, a single element test for the2 2 integrated eight-noded element will fail the stability part of the patch test andit should thus be used with great care.10.8 Application of the patch test to an incompatibleelementIn order to demonstrate the use of the patch test for a ®nite element formulationwhich violates the usually stated requirements for shape function continuity, weconsider the plane strain incompatible modes ®rst introduced by Wilson et al. 24and discussed by Taylor et al. 25 The speci®c incompatible formulation considereduses the element displacement approximations:^u ˆ N i a i ‡ N n 1a 1 ‡ N n 2a 2…10:13†where N i (i ˆ 1; ...; 4) are the usual conforming bilinear shape functions and the lasttwo terms are incompatible modes of deformation de®ned by the hierarchical functionsde®ned independently for each element.N n 1 ˆ 1 ÿ 2 and N n 2 ˆ 1 ÿ 2 …10:14†

Application of the patch test to an incompatible element 265(a)(b)(c)Fig. 10.10 (a) Linear quadrilateral with auxiliary incompatible shape functions; (b) pure bending and lineardisplacements causing shear; (c) auxiliary `bending' shape functions with internal variables.The shape functions used are illustrated in Fig. 10.10. The ®rst, a set of standardbilinear type, gives a displacement pattern which, as shown in Fig. 10.10(b),introduces spurious shear strains in pure bending. The second, in which the parametersa 1 and a 2 are strictly associated with a speci®c element, therefore introducesincompatibility but assures correct bending behaviour in an individual element.The excellent performance of this element in the bending situation is illustrated inFig. 10.11.In reference 25 the ®nite element approximation is computed by summing thepotential energies of each element and computing the nodal loads due to boundarytractions from the conforming part of the displacement ®eld only. Thus for thepurposes of conducting patch tests we compute the strains using all parts of thedisplacement ®eld leading to a generalization of (10.4) which may be written asK11K 21K 12K 22 aaˆf1f 2…10:15†Here K 11 and f 1 are the sti€ness and loads of the four-noded (conforming) bilinearelement, K 12 and K 21 (ˆ K T 12) are coupling sti€nesses between the conforming and

266 The patch test, reduced integration, and non-conforming elements1501000150Mesh 11000150256.2510100056.25187.50jLoad BMesh 21000iLoad A187.5056.25Load B2Displacement at iDisplacement at jLoad A Load B Load A Load BBeam theoryMesh 1(a)Mesh 2Mesh 1(b)Mesh 210.006.817.0610.0010.00103.070.172.3101.5101.3300.0218.2218.8300.0300.040502945295440504050Fig. 10.11 Performance of the non-conforming quadrilateral in beam bending treated as plane stress:(a) conforming linear quadrilateral, (b) non-conforming quadrilateral.non-conforming displacements, and K 22 and f 2 are the sti€ness and loads of the nonconformingdisplacements. We note that, according to the algorithm of reference 24,f 2 must vanish from the patch test solutions.For a patch test in plane strain or plane stress, only linear polynomials need beconsidered for which all non-conforming displacements must vanish. Thus for asuccessful patch test we must haveK 11 a ˆ f 1…10:16a†andK 21 a ˆ f 2…10:16b†If we carry out a patch test for the mesh shown in Fig. 10.12(a) we ®nd that all threeforms (i.e., patch tests A, B, and C) satisfy these conditions and thus pass the patchtest. If we consider the patch shown in Fig. 10.12(b), however, the patch test is notsatis®ed. The lack of satisfaction shows up in di€erent ways for each form of thepatch test. Patch test A produces non-zero f 2 values when a is set to zero and a accordingto the displacements considered. In form B the values of the nodal displacements

Application of the patch test to an incompatible element 2677 8 94 5 6(a)1 2 3789456(b)123789546(c)123Fig. 10.12 Patch test for an incompatible element form. (a) Regular discretization. (b) Irregular discretizationabout node 5. (c) Constant jacobian discretization about node 5.a 5 are in error and a are non-zero, also leading to erroneous stresses in each element.In form C all unspeci®ed displacements are in error as well as the stresses.It is interesting to note that when a patch is constructed according to Fig. 10.12(c)in which all elements are parallelograms all three forms of the patch test are onceagain satis®ed. Accordingly we can note that if any mesh is systematically re®nedby subdivision of each element into four elements whose sides are all along , lines in the original element with values of ÿ1, 0, or 1 (i.e., by bisections) the meshconverges to constant jacobian approximations of the type shown in Fig. 10.12(c).Thus, in this special case the incompatible mode element satis®es a weak patch testand will thus converge. In general, however, it may be necessary to use a very ®nediscretization to achieve su cient accuracy, and hence the element probably has nopractical (nor e cient) engineering use.

268 The patch test, reduced integration, and non-conforming elementsA simple arti®ce to ensure that an element passes the patch test is to replace thederivatives of the incompatible modes by8@Njn 98@N n 9j>< >=>:n ˆj0 @>=j…; † >;Jÿ1 0@N >:jn …10:17†>;@y@where j…; † is the determinant of the jacobian J…; † and J 0 and j 0 are the values ofthe inverse jacobian and jacobian determinant evaluated at the element centre… ˆ ˆ 0†. This ensures satisfaction of the patch test for all element shapes, andwith this alteration of the algorithm the incompatible element proves convergentand quite accurate. 2510.9 Generation of incompatible shape functions whichsatisfy the patch testIn the previous section we have shown how an incompatible element can, on occasion, producesuperior results despite its violation of the rules generally postulated. In solving platesand shells we deal with problems requiring C 1 continuity; the use of such incompatiblefunctions is widespread not only because these produce superior results but also due tothe di culty of developing functions which satisfy not only the continuity of the functionsbut also their slope (viz. Volume 2, Chapter 4). In this section we address the problem ofhow to generate incompatible shape functions in a manner that will automatically ensurethe satisfaction of the patch test and hence convergence. The rules for doing this have beendeveloped 26;27 and applied to the derivation of plate bending elements. We derive theserules here in a simple example of a second-order partial di€erential equation problembut the results are easy to generalize to other situations.Consider the ®nite element solution to the following equation:A…u† ÿTr 2 u ‡ ku ÿ q ˆ 0 in the domain …10:18†with boundary conditionsu ˆ u on ÿ uand (10.19)T @u@n ˆ t on ÿ tThis may represent the displacement u of an elastic membrane with an initialtension T on an elastic foundation with spring constant k. Let the unknown u beapproximated by two sets of (hierarchic) expansionsu ˆ u c ‡ u n…10:20a†u c ˆ N c a c and u n ˆ N n a …10:20b†in which N c and N n are, respectively, compatible and non-conforming shape functions.It must be stressed that these are linearly independent as otherwise stabilityconditions (i.e., the non-singularity of matrices) would be violated as was the casein the counterexample of Stummel. 9

When a patch of elements is subject to a linear variation of u such that Eq. (10.18) issatis®ed, the approximation u c is capable of yielding this solution and satisfying allthe patch test requirements. (Now, of course, q ˆÿku has to be assumed.)It follows therefore that in the patch test u n will be zero. However, it is important toconsider here a single element test in which the constant traction t (deduced fromu ˆ u c ) is applied. The Galerkin equation corresponding to the incompatible modenow yields …Ni n T @u … …ÿ e@n dÿ Ni n @uT n xÿ e@x ‡ n @uy dÿ ˆ Ni n t@y dÿ …10:21†ÿ teand this equation has to be satis®ed identically with t, T, and @u=@n being constants.In the above ÿ e represents the total boundary of the element and n x and n y are componentsof the boundary normal vector (see Appendix G).The above condition can be easily achieved by ensuring that……N n @ui n xÿ e@x dÿ ˆ N n @ui n yÿ e@y dÿ ˆ 0…10:22†for each element, thus imposing the constraint…Ni n t dÿ ˆ 0…10:23†ÿ tewhich implies (as originally suggested by Wilson et al. 24 ) that the e€ects of boundaryloads (and loads q) from the incompatible displacements must vanish or be ignored.In order to illustrate the use of the above procedure in developing incompatiblemode shape functions, we consider the case of a non-conforming four-noded quadrilateralelement which in the special case of a rectangle reproduces the non-conformingelement of reference 24. The convergence of this non-conforming element for therectangular or constant jacobian case has been illustrated in the previous section.We take the conforming part of the shape function for each displacement componentas the four-noded isoparametric functionsu c ˆ N I a c I…10:24†whereN I ˆ 14 …1 ‡ I†…1 ‡ I † …10:25†and , are natural coordinates on the interval …ÿ1; 1† with values at each cornernode I given by I , I . The non-conforming functions will be constructed from theremaining four shape functions for the eight-noded isoparametric serendipity element(Chapter 8). Accordingly we take for the non-conforming ®eldu n ˆ 12 …1 ÿ 2 †…1 ÿ † 1 ‡ 1 2 …1 ‡ †…1 ÿ 2 † 2 ‡ 1 2 …1 ÿ 2 †…1 ‡ † 3‡ 1 2 …1 ÿ †…1 ‡ 2 † 4 …10:26†Substitution into the constraint conditions (10.23) yields the two scalar conditionsandGeneration of incompatible shape functions which satisfy the patch test 269X 4i ˆ 1X 4i ˆ 1b i i ˆ 0c i i ˆ 0…10:27†…10:28†

270 The patch test, reduced integration, and non-conforming elementswhere b i , c i depend on the geometry of the element throughb i ˆ x i ÿ x jandc i ˆ y j ÿ y i…10:29†withj ˆ mod …i; 4†‡1The two constraint conditions may be used to express two of the i in terms of theother two. The result gives two incompatible displacement modes which may beadded to the conforming ®eld with the satisfaction of a strong patch test still ensured.For elements which are rectangular the two resulting modes are identical to thoseproposed and used in Eq. (10.14).Other possibilities exist for constructing non-conforming or incompatible functions. 510.10 The weak patch test ± exampleThe problems described above yield exact solutions for the patch tests performed andaccordingly satisfy strong conditions. In order to illustrate the performance of anelement which only satis®es a weak patch test we consider an axisymmetric linearelastic problem modelled by four-noded isoparametric elements. The material is assumedisotropic and the ®nite element sti€ness and reaction force matrices are computed using aselective integration method where terms associated with the bulk modulus are evaluatedby a single-point Gauss quadrature, whereas all other terms are computed using a 2 2(normal) gaussian quadrature (such as will be discussed in Chapter 12). It may be readilyveri®ed that the sti€ness matrix is of proper rank and thus stability of solutions is not anissue. On the other hand, consistency must still be evaluated.In order to assess the performance of a selective reduced quadrature formulationwe consider the patch of elements shown in Fig. 10.13. The patch is not as generallyshaped as desirable and is only used to illustrate performance of an element thatsatis®es a weak patch test. The polynomial solution considered isu ˆ 2r…10:30†w ˆ 0hhz6 5 41 2 3hr = 1Fig. 10.13 Patch for selective, reduced quadrature on axisymmetric four-noded elements.

Table 10.3 Exact solution for patchHigher order patch test ± assessment of robustness 271DisplacementForceNode RadiusI r I U I W I F rI F zI1, 4 1 ÿ h 2…1 ÿ h† 0 ÿ…1 ÿ h†h 02, 5 1 2 0 0 03, 6 1 ‡ h 2…1 ‡ h† 0 …1 ‡ h†h 0and material constants E ˆ 1 and ˆ 0 are used in the analysis. The resulting stress®eld is given by r ˆ ˆ 2…10:31†with other components identically zero. The exact solution for the nodal quantities ofthe mesh shown in Fig. 10.13 are summarized in Table 10.3. Patch tests have beenperformed for this problem using the selective reduced integration scheme describedabove and values of h of 0.8, 0.4, 0.2, 0.1, and 0.05. The result for the radialdisplacement at nodes 2 and 5 (reported to six digits) is given in Table 10.4. Allother quantities (displacements, strains, and stresses) have a similar performancewith convergence rates of at least O…h† or more. Based on this assessment we concludethe element passes a weak patch test.Table 10.4 Radial displacement atnodes 2 and 5hu0.8 2.011140.4 2.000490.2 2.000030.1 2.000000.05 2.0000010.11 Higher order patch test ± assessment ofrobustnessA higher order patch test may also be used to assess element `robustness'. An elementis termed robust if its performance is not sensitive to physical parameters of thedi€erential equation. For example, the performance of many elements for solutionof plane strain linear elasticity problems is sensitive to Poisson's ratio values near0.5 (called `near incompressibility'). Indeed, for Poisson ratios near 0.5 the energystored by a unit volumetric strain is many orders larger than the energy stored by aunit deviatoric strain. Accordingly ®nite elements which exhibit a strong couplingbetween volumetric and deviatoric strains often produce poor results in the nearlyincompressible range, a problem discussed further in Chapter 12.This may be observed using a four-noded element to solve a problem with aquadratic displacement ®eld (i.e., a higher order patch test). If we again consider a

272 The patch test, reduced integration, and non-conforming elements1.0Incompatible–regular (2 x 2)(3 x 3)Incompatible–distorted (3x3)Incompatible–distorted (2 x 2)0.8Limit ≈ 0.83V A /V exact0.60.4Bilinear–regular (2 x 2)Bilinear–distorted (3 x 3)(n x n) quadratureorder0.200 0.1 0.2 0.3 0.4 0.5vy (v)5ARegular mesh5x (u)5ADistorted mesh5Fig. 10.14 Plane strain four-noded quadrilaterals with and without incompatible modes (higher order patchtest for performance evaluation).pure bending example and an eight-element mesh shown in Fig. 10.14 we can clearlyobserve the deterioration of results as Poisson's ratio approaches a value of one-half.Also shown in Fig. 10.14 are results for the incompatible modes derived in Sec. 10.9. Itis evident that the response is considerably improved by adding these modes,especially if 2 2 quadrature is used.If we consider the regular mesh and four-noded elements and further keep thedomain constant and successively re®ne the problem using meshes of 8, 32, 128,and 512 elements, we observe that the answers do converge as guaranteed by thepatch test. However, as shown in Fig. 10.15, the rate of convergence in energy for

Conclusion 2730–1Element size ln (h)–2 –1 0v = 0.4999v = 0.2500v = 0.2500v = 0.4999Energy error ln (∆E/E)–2–3–4–512Four node (2 x 2) on all termsFour node (2 x 2) on shear termsFour node (1 x 1) on bulk termsFig. 10.15 Higher order patch test on element robustness (see Fig. 10.14) (convergence test under subdivisionof elements).Poisson ratio values of 0.25 and 0.4999 is quite di€erent. For 0.25 the rate of convergenceis nearly a straight line for all meshes, whereas for 0.4999 the rate starts outquite low and approaches an asymptotic value of 2 as h tends towards zero. For near 0.25 the element is called robust, whereas for near 0.5 it is not. If we useselective reduced integration (which for the plane strain case passes strong patchtests) and repeat the experiment, both values of produce a similar response andthus the element becomes robust for all values of Poisson's ratio less than 0.5.The use of higher order patch tests can thus be very important to separate robustelements from non-robust elements. For methods which seek to automatically re®ne amesh adaptively in regions with high errors, as discussed in Chapter 15, it is extremelyimportant to use robust elements.10.12 ConclusionIn the preceding sections we have described the patch test and its use in practice byconsidering several example problems. The patch test described has two essentialparts: (a) a consistency evaluation and (b) a stability check. In the consistency test aset of linearly independent essential polynomials (i.e., all independent terms up tothe order needed to describe the ®nite element model) is used as a solution to the differentialequations and boundary conditions, and in the limit as the size of a patchtends to zero the ®nite element model must exactly satisfy each solution. We presentedthree forms to perform this portion of the test which we call forms A, B, and C.

274 The patch test, reduced integration, and non-conforming elementsThe use of form C, where all boundary conditions are the natural ones (e.g., tractionsfor elasticity) except for the minimum number of essential conditions needed toensure a unique solution to the problem (e.g., rigid body modes for elasticity), isrecommended to test consistency and stability simultaneously. Both one-elementand more-than-one-element tests are necessary to ensure that the patch test issatis®ed. With these conditions and assuming that the solution procedure used candetect any possible rank de®ciencies the stability of solution is also tested. If nosuch condition is included in the program a stability test must be conducted independently.This can be performed by computing the number of zero eigenvalues in thecoe cient matrix for methods that use a solution of linear equations to computethe ®nite element parameters, a. Alternatively, the loading used for the patch solutionmay be perturbed at one point by a small value (say square root of the round-o€ limit,e.g., by 10 ÿ8 for round-o€s of order 10 ÿ15 ) and the solution tested to ensure that itdoes not change by a large amount.Once an element has been shown to pass all of the essential patch tests for bothconsistency and stability, convergence is assured as the size of elements tends tozero. However, in some situations (e.g., the nearly incompressible elastic problem)convergence may be very slow until a very large number of elements is used. Accordingly,we recommend that higher order patch tests be used to establish elementrobustness. Higher order patch tests involve the use of polynomial solutions of thedi€erential equation and boundary conditions with the order of terms larger thanthe basic polynomials used in a patch test. Indeed, the order of polynomials usedshould be increased until the patch test is satis®ed only in a weak sense (i.e., as htends to zero). The advantage of using a higher order patch test, as opposed toother boundary value problems, is that the exact solution may be easily computedeverywhere in the model.In some of the examples we have tested the use of incompatible function and inexactnumerical integration procedures (reduced and selective integration). Some of theseviolations of the rules previously stipulated have proved justi®ed not only by yieldingimproved performance but by providing methods for which convergence is guaranteed.We shall discuss in Chapter 12 some of the reasons for such improved performance.References1. B.M. Irons. Numerical integration applied to ®nite element methods. Conf.on Use ofDigital Computers in Structural Engineering. Univ. of Newcastle, 1966.2. G.P. Bazeley, Y.K. Cheung, B.M. Irons, and O.C. Zienkiewicz. Triangular elements inplate bending. Conforming and nonconforming solutions. Proc.1st Conf.on MatrixMethods in Structural Mechanics. pp. 547±76, AFFDLTR-CC-80, Wright-Patterson AFBase, Ohio, 1966.3. B.M. Irons and A. Razzaque. Experience with the patch test for convergence of ®niteelement method, in Mathematical Foundations of the Finite Element Method (ed. A.K.Aziz). pp. 557±87, Academic Press, 1972.4. B. Fraeijs de Veubeke. Variational principles and the patch test. Int.J.Num.Meth.Eng.8,783±801, 1974.5. G. Sander and P. Beckers. The in¯uence of the choice of connectors in the ®nite elementmethod. Int.J.Num.Meth.Eng.11, 1491±505, 1977.

6. E.R. de Arantes Oliveira. The patch test and the general convergence criteria of the ®niteelement method. Int.J.Solids Struct.13, 159±78, 1977.7. G. Strang. Variational crimes and the ®nite element method, in Proc.Foundations of theFinite Element Method (ed. A.K. Aziz). pp. 689±710, Academic Press, 1972.8. G. Strang and G.J. Fix. An Analysis of the Finite Element Method. Prentice-Hall, 1973.9. F. Stummel. The limitations of the patch test. Int.J.Num.Meth.Eng.15, 177±88, 1980.10. J. Robinson et al. Correspondence on patch test. Finite Element News. 1, 30±4, 1982.11. R.L. Taylor, O.C. Zienkiewicz, J.C. Simo, and A.H.C. Chan. The patch test ± a conditionfor assessing f.e.m. convergence. Int.J.Num.Meth.Eng.22, 39±62, 1986.12. R.E. Gri ths and A.R. Mitchell. Non-conforming elements, in Mathematical Basis ofFinite Element Methods. Inst. Math. and Appl. Conference series, pp. 41±69, ClarendonPress, Oxford, 1984.13. O.C. Zienkiewicz and R.L. Taylor. The ®nite element patch test revisited. A computer testfor convergence, validation and error estimates. Comp.Meth.Appl.Mech.Eng.149, 223±54, 1997.14. O.C. Zienkiewicz, S. Qu, R.L. Taylor, and S. Nakazawa. The patch test for mixed formulations.Internat.J.Num.Meth.Eng.23, 1873±83, 1986.15. W.X. Zhong. Convergence of fem and the conditions of the patch test. Technical Report97-3002, Research Institute Engineering Mechanics, Dalian University of Technology,1997 (in Chinese).16. I. BabusÏ ka, T. Strouboulis, and C.S. Upadhyay. A model study of the quality of aposteriori error estimators for linear elliptic problems. Error estimation in the interior ofpatchwise uniform grids of triangles. Comp.Meth.Appl.Mech.Eng. 114, 307±78, 1994.17. B. Boroomand and O.C. Zienkiewicz. An improved REP recovery and the e€ectivityrobustness test. Internat.J.Num.Meth.Eng.40, 3247±77, 1997.18. A. Ralston, A First Course in Numerical Analysis. McGraw-Hill, New York, 1965.19. B.M. Irons and S. Ahmad. Techniques of Finite Elements. Horwood, Chichester, 1980.20. D. Koslo€ and G.A. Fraser. Treatment of hour glass patterns in low order ®nite elementcodes. Int.J.Num.Anal.Meth.Geomechanics. 2, 57±72, 1978.21. T. Belytchko and W.E. Bachrach. The e cient implementation of quadrilaterals with highcoarse mesh accuracy. Comp.Meth.Appl.Mech.Eng.54, 276±301, 1986.22. N. Bi^cani^c and E. Hinton. Spurious modes in two dimensional isoparametric elements. Int.J.Num.Meth.Eng.14, 1545±57, 1979.23. E.L. Wachspress. High-order curved ®nite elements. Int.J.Num.Meth.Eng.17, 735±45,1981.24. E.L. Wilson, R.L. Taylor, W.P. Doherty, and J. Ghaboussi. Incompatible displacementmodels, in Num.and Comp.Meth.in Struct.Mech.(eds S.T. Fenves et al,). pp. 43±57,Academic Press, 1973.25. R.L. Taylor, P.J. Beresford, and E.L. Wilson. A non-conforming element for stressanalysis. Int.J.Num.Meth.Eng.10, 1211±20, 1976.26. A. Samuelsson. The global constant strain condition and the patch test. Chapter 3 ofEnergy Methods in Finite Element Methods (eds R. Glowinski, E.Y. Rodin, and O.C.Zienkiewicz). pp. 47±58, Wiley, 1979.27. B. Specht. Modi®ed shape functions for the three-node plate bending element passing thepatch test. Int.J.Num.Mech.Eng.26, 705±15, 1988.References 275

11Mixed formulation and constraints± complete ®eld methods11.1 IntroductionThe set of di€erential equations from which we start the discretization process willdetermine whether we refer to the formulation as mixed or irreducible. Thus if we consideran equation system with several dependent variables u written as [see Eqs (3.1)and (3.2)]A…u† ˆ0 in domain and (11.1)B…u† ˆ0 on boundary ÿin which none of the components of u can be eliminated still leaving a well-de®nedproblem, then the formulation will be termed irreducible. If this is not the case theformulation will be called mixed. These de®nitions were given in Chapter 3 (p. 421).This de®nition is not the only one possible 1 but appears to the authors to be widelyapplicable 2;3 if in the elimination process referred to we are allowed to introducepenalty functions. Further, for any given physical situation we shall ®nd that morethan one irreducible form is usually possible.As an example we shall consider the simple problem of heat conduction (or thequasi-harmonic equation)to which we have referred in Chapters 3 and 7. In thiswe start with a physical constitutive relation de®ning the ¯uxes [see Eq. (7.5)] interms of the potential (temperature)gradients, i.e., qxq ˆÿk r q ˆq yThe continuity equation can be written as [see Eq. (7.7)]r T q @q x@x ‡ @q y@y ˆÿQIf the above equations are satis®ed in and the boundary conditionsare obeyed then the problem is solved.…11:2†…11:3† ˆ on ÿ or q n ˆ q n on ÿ q …11:4†

Introduction 277Clearly elimination of the vector q is possible and simple substitution of Eq. (11.2)into Eq. (11.3)leads toÿr T …k r†‡Q ˆ 0 in …11:5†with appropriate boundary conditions expressed in terms of or its gradient.In Chapter 7 we showed discretized solutions starting from this point and clearly,as no further elimination of variables is possible, the formulation was irreducible.On the other hand, if we start the discretization from Eqs (11.2)±(11.4) the formulationwould be mixed.An alternative irreducible form is also possible in terms of the variables q. Here wehave to introduce a penalty form and write in place of Eq. (11.3)r T q ‡ Q ˆ …11:6†where is a penalty number which tends to in®nity. Clearly in the limit both equationsare the same and in general if is very large but ®nite the solutions should beapproximately the same.Now substitution into Eq. (11.2)gives the single governing equationrr T q ‡ 1 kÿ1 q ‡ rQ ˆ 0…11:7†which again could be used for the start of a discretization process as a possibleirreducible form. 4The reader should observe that, by the de®nition given, the formulations so farused in this book were irreducible. In subsequent sections we will show how elasticityproblems can be dealt with in mixed form and indeed will show how such formulationsare essential in certain problems typi®ed by the incompressible elasticityexample to which we have referred in Chapter 4. In Chapter 3 (Sec. 3.8.2)we haveshown how discretization of a mixed problem can be accomplished.Before proceeding to a discussion of such discretization (which will reveal theadvantages and disadvantages of mixed methods)it is important to observe that ifthe operator specifying the mixed form is symmetric or self-adjoint (see Sec. 3.9.1)the formulation can proceed from the basis of a variational principle which can bedirectly obtained for linear problems. We invite the reader to prove by using themethods of Chapter 3 that stationarity of the variational principle given below isequivalent to the di€erential equations (11.2)and (11.3)together with the boundaryconditions (11.4):for ˆ 12……… …q T k ÿ1 q d ‡ q T r d ‡ Q d ÿ q n dÿÿ q ˆ on ÿ …11:8†The establishment of such variational principles is a worthy academic pursuit andhad led to many famous forms given in the classical work of Washizu. 5 However, wealso know (see Sec. 3.7)that if symmetry of weighted residual matrices is obtained in alinear problem then a variational principle exists and can be determined. As such symmetrycan be established by inspection we shall, in what follows, proceed with suchweighting directly and thus avoid some unwarranted complexity.

278 Mixed formulation and constraints ± complete ®eld methods11.2 Discretization of mixed forms ± some generalremarksWe shall demonstrate the discretization process on the basis of the mixed form of theheat conduction equations (11.2)and (11.3). Here we start by assuming that each ofthe unknowns is approximated in the usual manner by appropriate shape functionsand corresponding unknown parameters. Thusyq ^q ˆ N q ~q and ^ ˆ N ~ f …11:9†where ~q and ~ f stand for nodal or element parameters that have to be determined.Similarly the weighting functions are given byv q ^v q ˆ W q ~q and v ^v ˆ W ~ f …11:10†where ~q and ~ f are arbitrary parameters.Assuming that the boundary conditions for ˆ are satis®ed by the choice of theexpansion, the weighted statement of the problem is, for Eq. (11.2)after eliminationof the arbitrary parameters,…W T q …k ÿ1^q ‡ r ^† d ˆ 0…11:11†and, for Eq. (11.3)and the `natural' boundary conditions,……ÿ W T …r T^q ‡ Q† d ‡W T …^q n ÿ q n † dÿ ˆ 0ÿ q…11:12†The reason we have premultiplied Eq. (11.2)by k ÿ1 is now evident as the choiceW q ˆ N q W ˆ N …11:13†will yield symmetric equations [using Green's theorem to perform integration by partson the gradient term in Eq. (11.12)] of the form A C ~q f1C T 0 f ~ˆf 2…11:14†with…A ˆ N T q k ÿ1 N q d…C ˆ N T q rN d…11:15†f 1 ˆ 0…f 2 ˆÿ N T Q d ÿ…N T q dÿÿ qy The reader will note that we have now changed the notation slightly, having previously used a di€erentsymbol such as a i for nodal quantities. We do this because now more than one variable occurs and it isconvenient to denote this variable with a similarly denoted nodal parameter.

Discretization of mixed forms ± some general remarks 279This problem, which we shall consider as typifying a large number of mixedapproximations, illustrates the main features of the mixed formulation, includingits advantages and disadvantages. We note that1. The continuity requirements on the shape functions chosen are di€erent. It is easilyseen that those given for N can be C 0 continuous while those for N q can bediscontinuous in or between elements (C ÿ1 continuity)as no derivatives of thisare present. Alternatively, this discontinuity can be transferred to N (usingGreen's theorem on the integral in C)while maintaining C 0 continuity for N q .This relaxation of continuity is of particular importance in plate and shell bendingproblems (see Volume 2)and indeed many important early uses of mixed formshave been made in that context. 6ÿ92. If interest is focused on the variable q rather than , use of an improved approximationfor this may result in higher accuracy than possible with the irreducibleform previously discussed. However, we must note that if the approximationfunction for q is capable of reproducing precisely the same type of variation asthat determinable from the irreducible form then no additional accuracy will resultand, indeed, the two approximations will yield identical answers.Thus, for instance, if we consider the mixed approximation to the ®eld problemsdiscussed using a linear triangle to determine N and piecewise constant N q ,asshown in Fig. 11.1, we will obtain precisely the same results as those obtainedby the irreducible formulation with the same N applied directly to Eq. (11.5),providing k is constant within each element. This is evident as the second of Eqs(11.14)is precisely the weighted continuity statement used in deriving the irreducibleformulation in which the ®rst of the equations is identically satis®ed.Indeed, should we choose to use a linear but discontinuous approximation formof N q in the interior of such a triangle, we would still obtain precisely the sameanswers, with the additional coe cients becoming zero. This discovery was madeOR + =Constant q Linear q Linear φ=Linear φFig. 11.1 A mixed approximation to the heat conduction problem yielding identical results as the correspondingirreducible form (the constant k is assumed in each element).

280 Mixed formulation and constraints ± complete ®eld methodsby Fraeijs de Veubeke 10 and is called the principle of limitation, showing that undersome circumstances no additional accuracy is to be expected from a mixed formulation.In a more general case where k is, for instance, discontinuous and variablewithin an element, the results of the mixed approximation will be di€erent and onoccasion superior. 2 Note that a C 0 -continuous approximation for q does not fallinto this category as it is not capable of reproducing the discontinuous ones.3. The equations resulting from mixed formulations frequently have zero diagonalterms as indeed in the case of Eq. (11.14).We noted in Chapter 3 that this is a characteristic of problems constrained by aLagrange multiplier variable. Indeed, this is the origin of the problem, which addssome di culty to a standard gaussian elimination process used in equation solving(see Chapter 20). As the form of Eq. (11.14) is typical of many two-®eld problemswe shall refer to the ®rst variable (here ~q)as the primary variable and the second(here f)as ~ the constraint variable.4. The added number of variables means that generally larger size algebraic problemshave to be dealt with. However, in Sec. 11.6 we shall show how such di culties canoften be avoided by a suitable iterative solution.The characteristics so far discussed did not mention one vital point which weelaborate in the next section.11.3 Stability of mixed approximation. The patch test11.3.1 Solvability requirementDespite the relaxation of shape function continuity requirements in the mixedapproximation, for certain choices of the individual shape functions the mixedapproximation will not yield meaningful results. This limitation is indeed muchmore severe than in an irreducible formulation where a very simple `constant gradient'(or constant strain)condition su ced to ensure a convergent form once continuityrequirements were satis®ed.The mathematical reasons for this di culty are discussed by BabusÏ ka 11 andBrezzi, 12 who formulated a mathematical criterion associated with their names. However,some sources of the di culties (and hence ways of avoiding them)follow fromquite simple reasoning.If we consider the equation system (11.14)to be typical of many mixed systemsin which ~q is the primary variable and f ~ is the constraint variable (equivalent to alagrangian multiplier), we note that the solution can proceed by eliminating ~q fromthe ®rst equation and by substituting into the second to obtain…C T A ÿ1 C† f ~ ˆÿf 2 ‡ C T A ÿ1 f 1…11:16†which requires the matrix A to be non-singular (or A~q 6ˆ 0 for all ~q 6ˆ 0). To calculate~f it is necessary to ensure that the bracketed matrix, i.e.is non-singular.H ˆ C T A ÿ1 C…11:17†

Stability of mixed approximation. The patch test 281Singularity of the H matrix will always occur if the number of unknowns in thevector ~q, which we call n q , is less than the number of unknowns n in the vector f. ~Thus for avoidance of singularityn q 5 n …11:18†is necessary though not su cient as we shall ®nd later.The reason for this is evident as the rank of the matrix (11.17), which needs to be n ,cannot be greater than n q , i.e., the rank of A ÿ1 .In some problems the matrix A may well be singular. It can normally be made nonsingularby addition of a multiple of the second equation, thus changing the ®rstequation toA ˆ A ‡ CC T f1 ˆ f 1 ‡ Cf 2where is an arbitrary number.Although both the matrices A and CC T are singular their combination A shouldnot be, providing we ensure that for all vectors ~q 6ˆ 0 eitherA~q 6ˆ 0 or C T ~q 6ˆ 0In mathematical terminology this means that A is non-singular in the null space of CC T .The requirement of Eq. (11.18)is a necessary but not su cient condition for nonsingularityof the matrix H. An additional requirement evident from Eq. (11.16)isC f ~ 6ˆ 0 for all f ~ 6ˆ 0If this is not the case the solution would not be unique.The above requirements are inherent in the BabusÏ ka±Brezzi condition previouslymentioned, but can always be veri®ed algebraically.11.3.2 LockingThe condition (11.18)ensures that non-zero answers for the variables ~q are possible. Ifit is violated locking or non-convergent results will occur in the formulation, givingnear-zero answers for ~q [see Chapter 3, Eq. (3.159)€.].To show this, we shall replace Eq. (11.14)by its penalized form:" #A C ~qC T ÿ 1 f1 with !1 I f ~ˆ…11:19†f 2 and I ˆ identity matrixElimination of f ~ leads to…A ‡ CC T †~q ˆ f 1 ‡ Cf 2As !1the above becomes simply…11:20†…CC T †~q ˆ Cf 2…11:21†Non-zero answers for ~q should exist even when f 2 is zero and hence the matrix CC Tmust be singular. This singularity will always exist if n q > n .

282 Mixed formulation and constraints ± complete ®eld methodsThe stability conditions derived on the particular example of Eq. (11.14)aregenerally valid for any problem exhibiting the standard Lagrange multiplier form.In particular the necessary count condition will in many cases su ce to determineelement acceptability; however, ®nal conclusions for successful elements which passall count conditions must be evaluated by rank tests on the full matrix.In the example just quoted ~q denote ¯uxes and f ~ temperatures and perhaps theconcept of locking was not clearly demonstrated. It is much more de®nite wherethe ®rst primary variable is a displacement and the second constraining one is astress or a pressure. There locking is more evident physically and simply means anoccurrence of zero displacements throughout as the solution approaches a limit.This unfortunately will happen on occasion.11.3.3 The patch testThe patch test for mixed elements can be carried out in exactly the way we havedescribed in the previous chapter for irreducible elements. As consistency is easilyassured by taking a polynomial approximation for each of the variables, only stabilityneeds generally to be investigated. Most answers to this can be obtained by simplyensuring that count condition (11.18)is satis®ed for any isolated patch on the boundariesof which we constrain the maximum number of primary variables and theminimum number of constraint variables. 13In Fig. 11.2 we illustrate a single element test for two possible formulations with C 0continuous N (quadratic)and discontinuous N q , assumed to be either constant orlinear within an element of triangular form. As no values of ~q can here be speci®edon the boundaries, we shall ®x a single value of ~ f only, as is necessary to ensure+n q < n φTest failed(a)n q = 2n φ = 6 – 1 = 5Restrainedn q > n φTest passed(but results equivalentto irreducible form)(b)n q = 6n φ = 6 – 1=5Fig. 11.2 Single element patch test for mixed approximations to the heat conduction problem with discontinuous¯ux q assumed: (a) quadratic C 0 , ; constant q; (b) quadratic C 0 , ; linear q.

Stability of mixed approximation. The patch test 283n q > n φn q = 12 n φ = 6 – 1=5RestrainedFig. 11.3 As Fig. 11.2 but with C 0 continuous q.uniqueness, on the patch boundary, which is here simply that of a single element. Acount shows that only one of the formulations, i.e., that with linear ¯ux variation,satis®es condition (11.18)and therefore may be acceptable.In Fig. 11.3 we illustrate a similar patch test on the same element but with identicalC 0 continuous shape functions speci®ed for both ~q and f ~ variables. This exampleshows satisfaction of the basic condition of Eq. (11.18)and therefore is apparentlya permissible formulation. The permissible formulation must always be subjectedto a numerical rank test. Clearly condition (11.18)will need to be satis®ed andmany useful conclusions can be drawn from such counts. These eliminate elementswhich will not function and on many occasions will give guidance to elementswhich will.Even if the patch test is satis®ed occasional di culties can arise, and these areindicated mathematically by the BabusÏ ka±Brezzi condition already referred to: 14These di culties can be due to excessive continuity imposed on the problem byrequiring, for instance, the ¯ux condition to be of C 0 continuity class. In Fig. 11.4we illustrate some cases in which the imposition of such continuity is physicallyincorrect and therefore can be expected to produce erroneous (and usually highlyoscillating)results. In all such problems we recommend that the continuity be relaxedat least locally.We shall discuss this problem further in Sec. 11.4.3.q I nk(a)q Iq I n = q IInk IIq IInq IIOnly q ncontinuousq changes abruptly(discontinuity)q(b)qFig. 11.4 Some situations for which C 0 continuity of ¯ux q is inappropriate: (a) discontinuous change ofmaterial properties; (b) singularity.

284 Mixed formulation and constraints ± complete ®eld methods11.4 Two-®eld mixed formulation in elasticity11.4.1 GeneralIn all the previous formulations of elasticity problems in this book we have used anirreducible formulation, using the displacement u as the primary variable. The virtualwork principle was used to establish the equilibrium conditions which were written as(see Chapter 2)………e T r d ÿ u T b d ÿu T t dÿ ˆ 0ÿ t…11:22†where t are the tractions prescribed on ÿ t and withr ˆ De…11:23†as the constitutive relation (omitting here initial strains and stresses for clarity).We recall that statements such as Eq. (11.22)are equivalent to weighted residualforms (see Chapter 3)and in what follows we shall use these frequently. In theabove the strains are related to displacement by the matrix operator S introducedin Chapter 2, givinge ˆ Su…11:24†e ˆ S u…11:25†with the displacement expansions constrained to satisfy the prescribed displacementson ÿ u . This is, of course, equivalent to Galerkin-type weighting.With the displacement u approximated asu ^u ˆ N u ~u…11:26†the required sti€ness equations were obtained in terms of the unknown displacementvector ~u and the solution obtained.It is possible to use mixed forms in which either r or e, or, indeed, both these variables,are approximated independently. We shall discuss such formulations below.11.4.2 The u±r mixedformIn this we shall assume that Eq. (11.22)is valid but that we approximate r independentlyasr ^r ˆ N ~r…11:27†and approximately satisfy the constitutive relationr ˆ DSu…11:28†which replaces (11.23)and (11.24). The approximate integral form is written as…r T …Su ÿ D ÿ1 r† d ˆ 0…11:29†

Two-®eld mixed formulation in elasticity 285where the expression in the brackets is simply Eq. (11.28)premultiplied by D ÿ1 toestablish symmetry and r is introduced as a weighting variable.Indeed, Eqs (11.22)and (11.29)which now de®ne the problem are equivalent to thestationarity of the functional… HR ˆ r T Su d ÿ 1 …… …r T D ÿ1 r d ÿ u T b d ÿ u T t dÿ …11:30† 2 ÿ twhere the boundary displacementu ˆ uis enforced on ÿ u , as the reader can readily verify. This is the well-known Hellinger±Reissner 15;16 variational principle, but, as we have remarked earlier, it is unnecessaryin deriving approximate equations. UsingN u ~u in place of uB~u SN u ~u in place of eN ~r in place of rwe write the approximate equations (11.29)and (11.22)in the standard form [see Eq.(11.14)] A C ~r f1C Tˆ…11:31†0 ~u f 2with…A ˆÿ N T D ÿ1 N d…C ˆ‡ N T B d…11:32†f 1 ˆ 0……f 2 ˆ‡ N T u b d ‡N T u t dÿÿ tIn the form given above the N u shape functions have still to be of C 0 continuity,though N can be discontinuous. However, integration by parts of the expressionfor C allows a reduction of such continuity and indeed this form has been used byHerrmann 6;17;18 for problems of plates and shells.11.4.3 Stability of two-®eldapproximation in elasticity (u±r)Before attempting to formulate practical mixed approach approximations in detail,identical stability problems to those discussed in Sec. 11.3 have to be considered.For the u±r forms it is clear that r is the primary variable and u the constraintvariable (see Sec. 11.2), and for the total problem as well as for element patches wemust have as a necessary, though not su cient conditionn 5 n u…11:33†where n and n u stand for numbers of degrees of freedom in appropriate variables.

286 Mixed formulation and constraints ± complete ®eld methodsT 1/3T 3/3T 3/6n σ = 3n u = 3 x 2 – 3=3(pass)= 3 x 3 = 9= 3 x 2 – 3=3(pass)= 3 x 3 = 9= 6 x 2 – 3=9(pass)Q 1/4Q 3/8Q 4/8Q 4/9n σ = 3n u = 4 x 2 – 3=5(fail)= 3 x 3 = 9= 8 x 2 – 3=13(fail)= 4 x 3 = 12= 8 x 2 – 3=13(fail)= 4 x 3 = 12= 9 x 2 – 3=15(fail)Two-element Q 4/8 assembly patch testn σ = 8 x 3 = 24n u = 13 x 2 – 3=23(pass)Fig. 11.5 Elasticity by the mixed r±u formulation. Discontinuous stress approximation. Single element patchtest. No restraint on ~r variables but three ~u degrees of freedom restrained on patch. Test conditionn 5 n u (X denotes ~r (3 DOF) and the ~u (2 DOF) variables).In Fig. 11.5 we consider a two-dimensional plane problem and show a series ofelements in which N is discontinuous while N u has C 0 continuity. We note again,by invoking the Veubeke `principle of limitation', that all the elements that passthe single-element test here will in fact yield identical results to those obtained byusing the equivalent irreducible form, providing the D matrix is constant withineach element. They are therefore of little interest. However, we note in passing thatthe Q 4/8, which fails in a single-element test, passes that patch test for assembliesof two or more elements, and performs well in many circumstances. We shall seelater that this is equivalent to using four-point Gauss, reduced integration (seeSec. 12.5), and as we have mentioned in Chapter 10 such elements will not alwaysbe robust.It is of interest to note that if a higher order of interpolation is used for r than for uthe patch test is still satis®ed, but in general the results will not be improved because ofthe principle of limitation.We do not show the similar patch test for the C 0 continuous N assumption butstate simply that, similarly to the example of Fig. 11.3, identical interpolation of

Two-®eld mixed formulation in elasticity 287+(a)σ linearu linearσ xxσ nnτ ntσ ttτ xy(b)σ yyσttFig. 11.6 Elasticity by the mixed r±u formulation. Partially continuous r (continuity at nodes only). (a) rlinear, u linear; (b) possible transformation of interface stresses with tt disconnected.N and N u is acceptable from the point of view of stability. However, as in Fig. 11.4,restriction of excessive continuity for stresses has to be avoided at singularities and atabrupt material property change interfaces, where only the normal and tangentialtractions are continuous.The disconnection of stress variables at corner nodes can only be accomplished forall the stress variables. For this reason an alternative set of elements with continuousstress nodes at element interfaces can be introduced (see Fig. 11.6). 19In suchs elements excessive continuity can easily be avoided by disconnecting onlythe direct stress components parallel to an interface at which material changes occur.It should be noted that even in the case when all stress components are connected at amid-side node such elements do not ensure stress continuity along the whole interface.Indeed, the amount of such discontinuity can be useful as an error measure. However,we observe that for the linear element [Fig. 11.6(a)] the interelement stresses arecontinuous in the mean.It is, of course, possible to derive elements that exhibit complete continuity of theappropriate components along interfaces and indeed this was achieved by Raviartand Thomas 20 in the case of the heat conduction problem discussed previously.Extension to the full stress problem is di cult 21 and as yet such elements have notbeen successfully noted.11.4.4 Pian±Sumihara quadrilateralToday very few two-®eld elements based on interpolation of the full stress anddisplacement ®elds are used. One, however, deserves to be mentioned. We begin by®rst considering a rectangular element where interpolations may be given directlyin terms of cartesian coordinates. A four-node plane rectangular element with side

288 Mixed formulation and constraints ± complete ®eld methodsybx(x 0 , y 0 )baaFig. 11.7 Geometry of rectangular r±u element.lengths 2a in the x-direction and 2b in the y-direction, shown in Fig. 11.7, hasdisplacement interpolation given byThe shape functions are given byN 1 …x; y† ˆ 14N 2 …x; y† ˆ 14N 3 …x; y† ˆ 14N 4 …x; y† ˆ 14u ˆ X4i ˆ 1N i …x; y†~u i1 ÿ x ÿ x 0a1 ‡ x ÿ x 0a1 ‡ x ÿ x 0a1 ÿ x ÿ x 0a1 ÿ y ÿ y 0b1 ÿ y ÿ y 0b1 ‡ y ÿ y 0b1 ‡ y ÿ y 0bwhere x 0 and y 0 are the cartesian coordinates of the element centre. The strainsgenerated from this interpolation will be such that" x ˆ 1 ‡ 2 y" y ˆ 3 ‡ 4 x xy ˆ 5 ‡ 6 x ‡ 7 ywhere j are expressed in terms of ~u. For isotropic linear elasticity problems thesestrains will lead to stresses which have a complete linear polynomial variation ineach element (except for the special case when ˆ 0).

Here the stress interpolation is restricted to each element individually and, thus, canbe discontinuous between adjacent elements. The limitation principle restricts thepossible choices which lead to di€erent results from the standard displacementsolution. Namely, the approximation must be less than a complete linear polynomial.To satisfy the stability condition given by Eq. (11.18)we need at least ®ve stressparameters in each element. A viable choice for a ®ve-term approximation is onewhich has the same variation in each element as the normal strains given above butonly a constant shear stress. Accordingly,8 98>: x y xy9>=>; ˆ231 0 0 y ÿ y 0 0 >:Two-®eld mixed formulation in elasticity 289Indeed, this approximation satis®es Eq. (11.18)and leads to excellent results for arectangular element. We now rewrite the formulation to permit a general quadrilateralshape to be used.The element coordinate and displacement ®eld are given by a standard bilinearisoparametric expansion 1 2 3 4 5>=>;where nowx ˆ X4i ˆ 1N i …; †~x i^u ˆ X4i ˆ 1N i …; † ~u iN i …; † ˆ14 …1 ‡ i†…1 ‡ i †in which i and i are the values of the parent coordinates at the nodes.The problem remains to deduce an approximation for stresses for the generalquadrilateral element. Here this is accomplished by ®rst assuming stresses on theparent element (for convenience in performing the coordinate transformation thetensor form is used, see Appendix B)in an analogous manner as the rectangleabove:" #…; † ˆ ˆ 1 ‡ 4 3 3 2 ‡ 5 In the above the normal stresses again produce constant and bending terms whileshear stress is only constant. These stresses are then mapped (transformed)tocartesian space usingr ˆ T T …; †TIt remains now only to select an appropriate transformation. The transformationmust1. produce stresses in cartesian space which satisfy the patch test (i.e., can produceconstant stresses and be stable);

290 Mixed formulation and constraints ± complete ®eld methods2. be independent of the orientation of the initially chosen element coordinate systemand numbering of element nodes (frame invariance requirement).Pian and Sumihara 22 use a constant array (to preserve constant stresses)deducedfrom the jacobian matrix at the centre of the element. Accordingly, with J 0 ˆ J0;11 J 0;12ˆJ 0;21 J 0;222@x6@4 @x@3@y@7@y 5@; ˆ 0the elements of the jacobian matrix at the centre are given by [see Eq. (8.10)]J 0;11 ˆ 14 x i i J 0;12 ˆ 14 x i iJ 0;21 ˆ 14 y i i J 0;22 ˆ 14 y i iUsing T ˆ J 0 gives the stresses (in matrix form)8 9 8 9 2>< x >= >< 1 >= J0;11 2 J 2 30;12 y>: >; ˆ 2>: >; ‡ 64 J0;21 2 J0;222 75 4 5 xy 3 J 0;12 J 0;21 J 0;12 J 0;22 where the parameters i , i ˆ 1; 2; 3; replace the transformed quantities for theconstant part of the stresses. This approximation clearly satis®es the constant stresscondition (Condition 1)and can also be shown to satisfy the frame invariancecondition (Condition 2). The development is now complete and the arrays indicated1.02a5 5E = 75, ν = 00.50.50.8v (a)0.60.40.2P–S: JP–S: J–inverseQ–40 0 1 2 3 4aFig. 11.8 Pian±Sumihara quadrilateral (P-S) compared with displacement quadrilateral (Q-4). Effect ofelement distortion (Exact ˆ 1.0).

Three-®eld mixed formulations in elasticity 291in Eq. (11.32)may be computed. We note that the integrals are computed exactly forall quadrilateral elements (with constant D)using 2 2 gaussian quadrature.An alternative to the above de®nition for T 0 is to use the transpose of the jacobianinverse at the centre of the element (i.e., T 0 ˆ J ÿT0 ). This has also been suggestedrecently by several authors as a frame invariant transformation. However, asshown in Fig. 11.8, the sensitivity to element distortion is much greater for thisform than the original one given by Pian and Sumihara for the above two-®eldapproximation. The other two options (e.g., T ˆ J T 0 and T ˆ J ÿ10 )do not satisfythe frame invariance requirement, thus giving elements which depend on the orientationof the element with respect to the global coordinates.11.5 Three-®eld mixed formulations in elasticity11.5.1 The u±r±e formIt is, of course, possible to use an independent approximation to all the essentialvariables entering the elasticity problem. We can then write the three equations(11.24), (11.23), and (11.22) in their weak form as…e T …De ÿ r† d ˆ 0…r T …Su ÿ e† d ˆ 0…11:34†…………Su† T r d ÿ u T b d ÿ u T t dÿ ˆ 0ÿ twith a corresponding variational principle requiring the stationarity of……… …1 HW ˆ2 eT De d ÿ r T …e ÿ Su† d ÿ u T b d ÿ u T t dÿ …11:35†ÿ twhere u u on ÿ u is enforced.y This principle is known by the name of Hu±Washizu. 5 However, again we can proceed directly, using Eq. (11.34), taking thefollowing approximationsu ^u ˆ N u ~u r ^r ˆ N ~r and e ^e ˆ N " ~ewith corresponding `variations' (i.e., the Galerkin form W u ˆ N u , etc.)and writingthe approximating equations in a similar fashion as we have in the previous section.This yields an equation system of the following form:2389 8 9A C 0 >< ~e >= >< f 1 >=64 C T 70 E5~r>: >; ˆ f 2…11:36†>: >;0 E T 0 ~uf 3y It is possible to include the displacement boundary conditions in Eq. (11.35)as a natural rather thanimposed constraint; however, most ®nite element applications of the principle are in the form shown.

292 Mixed formulation and constraints ± complete ®eld methodswhere…A ˆ N T " DN " d…E ˆ N T B d…C ˆÿ N T " N df 1 ˆ f 2 ˆ 0……f 3 ˆ N T u b d ‡ N T u t dÿÿ t…11:37†The reader will have observed again that in this section we have quoted the variationalprinciple purely as a matter of interest and that all the approximations have beenmade directly.11.5.2 Stability condition of three-®eld approximation (u±r±e)The stability condition derived in Sec. 11.3 [Eq. (11.18)] for two-®eld problems,which we later used in Eq. (11.33)for the simple mixed elasticity form, needs to bemodi®ed when three-®eld approximations of the form given in Eq. (11.36)areconsidered.Many other problems fall into a similar category (for instance, plate bending)and hence the conditions of stability are generally useful. The requirement now isthatn " ‡ n u 5 n …11:38†n 5 n uThis was ®rst stated in reference 23 and follows directly from the two-®eld criterion asshown below.The system of Eq. (11.36)can be `regularized' by adding E times the thirdequation to the second, with being an arbitrary constant. We now have2389 8 9A C 0 >< ~e >= >< f 1 >=64 C T EE T 7E 5 ~r>: >; ˆ f 2 ‡ Ef 3>: >;0 E T 0 ~uf 3On elimination of e using the ®rst of the above we have EE T ÿ C T A ÿ1 C; E ~r f2 ‡ EfE T ˆ3 ÿ C T A ÿ1 f 1; 0 ~uFrom the two-®eld requirement [Eq. (11.18)] it follows that we require for nosingularityn 5 n u…11:39†f 3

Three-®eld mixed formulations in elasticity 293Rearranging Eq. (11.36)we can write2389 8 9A 0 C < ~e = < f 1 =4 0 0 E T 5 ~uC T : ; ˆ f: 3;E 0 ~r f 2This again can be regularized by adding multiples C and E T of the third of theabove equations to the ®rst and second respectively obtaining2A ‡ CC T 389 8 9; CE C >= >=64 E T C T ; E T E E T 75 ~u>: >; ˆ f 3 ‡ E T f 2>: >;C T ; E 0 ~rBy partitioning as above it is evident that we requiren " ‡ n u 5 n …11:40†We shall not discuss in detail any of the possible approximations to the e±r±uformulation or their corresponding patch tests as the arguments are similar tothose of two-®eld problems.In some practical applications of the three-®eld form the approximation of thesecond and third equations in (11.34)is used directly to eliminate all but the displacementterms. This leads to a special form of the displacement method which has beencalled a B (B-bar)form. 24;25 In the B form the shape function derivatives are replacedby approximations resulting from the mixed form. We shall illustrate this conceptwith an example of a nearly incompressible material in Sec. 12.4.f 211.5.3 The u±r±e en form. Enhancedstrain formulationIn the previous two sections the general form and stability conditions of the three-®eldformulation for elasticity problems is given in Eqs (11.34)and (11.38). Here we considera special case of this form from which several useful elements may be deduced.In the special form considered the strain approximation is split into two parts: onethe usual displacement-gradient term; and, second, an added or enhanced strain part.Accordingly, we writee ˆ Su ‡ e en e ˆ …Su†‡e en …11:41†Substitution into Eq. (11.34)yields the weak forms as……Su† … DSu‡ … e en †ÿr†d ˆ 0…e T en… DSu‡ … e en †ÿr†d ˆ 0…r T e en d ˆ 0…………Su† r d ÿ u T b d ÿu T t dÿ ˆ 0ÿ t…11:42†

294 Mixed formulation and constraints ± complete ®eld methodswith, for completeness, a corresponding variational principle requiring the stationarityof……1 en ˆ … 2 Su ‡ e en† T D… Su‡ e en †d ‡ re en d… …ÿ u T b d ÿ u T t dÿ…11:43†ÿ twhere, as before, u ˆ u is enforced on ÿ u .We can directly discretize Eq. (11.42)by taking the following approximationsu ^u ˆ N u ~u r ^r ˆ N ~r e en ^e en ˆ N en ~e en …11:44†with corresponding expressions for variations. Substituting the approximations intoEq. (11.42)yields the discrete equation system2389 8 9A C G >< ~e en >= >< f 1 >=64 C T 70 0 5 ~r>: >; ˆ f 2…11:45†>: >;G T 0 K ~uwhere…A ˆ N T enDN en d…C ˆÿ N T enN d…G ˆ N T enDB d…K ˆ B T DB df 3f 1 ˆ f 2 ˆ 0……f 3 ˆ N T u b d ‡ N T u t dÿÿ t…11:46†In this form there is only one zero diagonal term and the stability condition reduces tothe single conditionn u ‡ n en 5 n …11:47†Further, the use of the strains deduced from the displacement interpolation leads to amatrix which is identical to that from the irreducible form and we have thus includedthis in Eq. (11.46)as K.11.5.4 Simo±Rifai quadrilateralAn enhanced strain formulation for application to problems in plain elasticity wasintroduced by Simo and Rifai. 26 The element has four nodes and employs isoparametricinterpolation for the displacement ®eld. The derivatives of the shape

functions yield a form8>:@N j@x@N j@y9 8>= >; >:a x; j …y i †‡b x; j …y i † ‡ c x; j …y i †j…; †a y; j …x i †‡b y; j …x i † ‡ c y; j …x i †j…; †where a j , b j and c j depend on the nodal coordinates, and the jacobian determinant forthe 4-node quadrilateral is given byydet J ˆ j…; † ˆj 0 ‡ j 1 ‡ j 2 The enhanced strains are ®rst assumed in the parent coordinate frame and transformedto the cartesian frame using a transformation similar to that used in developingthe Pian±Sumihara quadrilateral in Sec. 11.4.2. Due to the presence of thejacobian determinant in the strains computed from the displacements (as well asthe requirement to later pass the patch test for constant stress states)the enhancedstrains are computed frome en ˆ 1j…; † TT E…; †TIn matrix form this may be written as8 9 2389>: >; ˆ 1 T11 2 T21 2 T 11 T 21 >=6T12 2 T22 2 74Tj…; †12 T 22 5 E >: >; xy2T 11 T 12 2T 21 T22 2 T 11 T 22 ‡ T 12 T 212E The parent strains (strains with components in the parent element frame)are assumedas8 98 9 23 1>= 0 0 0 >: >; ˆ 67 >=24 0 0 05 30 0 >: >;2E The above is motivated by the fact that the derivatives of the shape functions withrespect to parent coordinates yields@N j@ ˆ a ‡ b Three-®eld mixed formulations in elasticity 295 4@N j@ ˆ a ‡ b and these may be combined to form strains in the usual manner, but in the parentframe. Thus, by design, the above enhanced strains are speci®ed to generate completepolynomials in the parent coordinates for each strain component. References27 and 28 discuss the relationship between the design of assumed stress elementsusing the two-®eld form and the selection of enhanced strain modes so as to producethe same result.9>=>;y In general, the determinant of the jacobian for the two-dimensional Lagrange family of elements will notcontain the term with the product of the highest order polynomial, e.g., for the 4-node element, 2 2 forthe 9-node element, etc.

296 Mixed formulation and constraints ± complete ®eld methodsRemarks1. The above enhanced strains are de®ned so that the C array is identically zero forconstant assumed stresses in each element.2. Parent normal strains have linearly independent terms added. However, theassumed parent shear strains are linearly dependent. Due to this linear dependencethe ®nal shearing strain will usually be nearly constant in each element. Accordingly,to be more explicit, normal strains are enhanced while shearing strain isde-enhanced.Since the C array vanishes, the equation set to be solved becomes A G ~eenG Tˆf1K ~uand in this form no additional count conditions are apparently needed. The solutionmay be accomplished partly at the element level by eliminating the equation associatedwith the enhanced strain parameters. Accordingly,f 3whereandK ~u ˆ f 3K ˆ K ÿ G T A ÿ1 Gf 3 ˆ f 3 ÿ G T A ÿ1 f 1The sensitivity of the enhanced strain element to geometric distortion is evaluatedusing the problem shown in Fig. 11.8. The transformation from the parent to theglobal frame is assessed using T ˆ J 0 and T ˆ J ÿT0 . These are the only optionswhich maintain frame invariance for the element. As observed in Fig. 11.9 the results1.00.8v (a)0.60.40.2S–R: JS–R: J–inverseQ–40 0 1 2 3 4aFig. 11.9 Simo±Rifai enhanced strain quadrilateral (S-R) compared with displacement quadrilateral (Q-4).Effect of element distortion (Exact ˆ 1.0).

Three-®eld mixed formulations in elasticity 297Fig. 11.10 Mesh with 4 4 elements for shear load.are now better using the inverse transpose. Since the stress and strain are conjugates inan energy sense, this result could be anticipated from the equivalence relationshipE ˆ 1 …r T e d 1 … T E dh2 2 hwhere E is energy and h denotes the domain of the element in the parent coordinatesystem (i.e., the bi-unit square for a quadrilateral element).The performance of the enhanced element is compared to the Pian±Sumiharaelement for a shear loading on the mesh shown in Fig. 11.10. In Fig. 11.11 theconvergence results for various order meshes are shown for linear elastic, planestrain conditions with: (a) E ˆ 70 and ˆ 1=3 and(b)for E ˆ 70 and ˆ 0:499995.The results shown in Fig. 11.11 clearly show the strong dependence of the displacement353530302525v (a)201510S–RP–SQ–4v (a)201510S–RP–SQ–45500 10 20 30 40 50 60n – elements/side (ν = 1/3)00 10 20 30 40 50 60n – elements/side (ν = 0.499995)Fig. 11.11 Convergence behaviour for: (a) ˆ 1=3; (b) ˆ 0:499995.

298 Mixed formulation and constraints ± complete ®eld methodsformulation on Poisson's ratio ± namely the tendency for the element to lock for valueswhich approach the incompressibility limit of ˆ 1=2. On the other hand, theperformance of both the enhanced strain and the Pian±Sumihara element are nearlyinsensitive to the value of Poisson's ratio selected, with somewhat better performanceof the enhanced element on coarse meshing.11.6 An iterative method solution of mixedapproximationsIt is of interest to consider here the procedure ®rst suggested by Cantin et al. 29;30 inwhich the authors aimed at an iterative improvement of the displacement type solution.This iterative process in fact solves two equations. In this the ®rst equation replaces thediscontinuous stresses computed from a displacement type solution by continuousstresses calculated by a least square smoothing. The continuous stress is expressed usingr ˆ N~r…11:48†where N are the same shape functions used in the displacement solution and ~r arenodal values of stresses. The least square problem is then expressed as…… r ÿ ^r † T … r ÿ ^r †d ˆ min …11:49†whose solution for a typical iteration i may be written asA~r …k ‡ 1† ÿ C ~u …k† ˆ 0with…A ˆ N T N d…C ˆ N T DB d…11:50†This type of stress smoothing was suggested by Brauchli and Oden in 1973. 31 Thoughwe shall discuss its achievements later in Chapter 14 on recovery methods it has beenquite successfully used in the iterative improvement discussed here.The second stage of the calculation takes the stresses computed above r andcalculates the out-of-balance residual…r …k ‡ 1† ˆ B T r ;…k ‡ 1† d ‡ f…11:51†The correction to the displacements using this residual is then expressed byK~u …k ‡ 1† ˆ K~u …k† …k ‡ 1†ÿ r …11:52†The iteration may now proceed by incrementing k and computing new smoothedstresses followed by new displacements.The two steps may be written in a matrix setting as ( )A 0 ~r…k ‡ 1†ÿC Tˆ 0 ( )C ~r…k†‡ 0 …11:53†ÿK0 ÿKÿf…k ‡ 1†uu …k†

where…C ˆAt convergence the solutions becomeAn iterative method solution of mixed approximations 299N T B d~u …k† ˆ ~u …k ‡ 1† ˆ ~u ~r …k† ˆ ~r …k ‡ 1† ˆ ~r…11:54†Combining the two sides of the above equation yields A C ~rÿC T 0 u ˆ 0 …11:55†ÿfThe reader will notice that the equations which result at the end of this process are infact a mixed problem in stress and displacement form.The convergence of the process is quite rapid and very often considerable improvementin the answers is obtained. In Fig. 11.12 we show some results by Nakazawaet al. 32ÿ34 using the bilinear displacement element and it is seen how much the resultsare improved. In Fig. 11.13 a similar iteration is carried out using now triangularBab a/b = 10Stress σ x /σ x (exact)at point B1.51.00.5Displacement solutionwith stress recovery00 5 10Equilibrium iterationFull integration (consistent A)Full integration (lumped A)Nodal integration throughoutat point BDeflection v/v (exact)1.51.00.5Displacement solutionwith stress recovery00 5 10Equilibrium iterations (with line search)Fig. 11.12 Iterative solution of the mixed r/u formulation for a beam. Bilinear u and r.

300 Mixed formulation and constraints ± complete ®eld methodsA1248(a)p/2p/2(b)Approximate/Exact tipdisplacementTC 3/31.21.00.8ExactTC 3/3TCR 3/3T3Approximate/Exact tipstress at AT30.60 10 20IterationsMax deflection of beam (32 elements)Byxσp (10)uT=1CSigma xx1.21.00.8T30.60 10 20IterationsNormal stress of point A9080706050403020 Reference*TC 3/310TCR 3/300 20 40 60 80 100x coordinateSigma xx along B–CTCR 3/3uT3Fig. 11.13 Iterative solution of the mixed r/u formulation using two triangular element forms TC 3/3and TCR 3/3. (a) A beam showing convergence with iterations. (b) An L-shaped domain showing theimproved results of stress distribution when no continuity of stress is imposed at singularity (elementTCR 3/3).

Complementary forms with direct constraint 301elements. Here various combinations of displacement and stress variation have beenused and, in particular, the reader should note that at the singularity point somemeans of stress disconnection is used as di culties in C 0 stress continuity exist.The very simplest procedure of disconnecting all components of stress at suchpoints has proven to be optimal. Details of such calculations are given in reference 19.In a subsequent chapter, where we shall deal with problems of incompressibility, weshall deal with an iteration due to Uzawa. 35 The particular iteration used in the aboveiteration process is in fact a form of the Uzawa algorithm to which we will refer inmore detail later.11.7 Complementary forms with direct constraint11.7.1 General formsIn the introduction to this chapter we de®ned the irreducible and mixed forms andindicated that on occasion it is possible to obtain more than one `irreducible' form.To illustrate this in the problem of heat transfer given by Eqs (11.2)and (11.3)weintroduced a penalty function in Eq. (11.6)and derived a corresponding singlegoverning equation (11.7)given in terms of q. This penalty function here has noobvious physical meaning and served simply as a device to obtain a close enoughapproximation to the satisfaction of the continuity of ¯ow equations.On occasion it is possible to solve the problem as an irreducible one assuming apriori that the choice of the variable satis®es one of the equations. We call suchforms directly constrained and obviously the choice of the shape function becomesdi cult.We shall consider two examples.The complementary heat transfer problemIn this we assume a priori that the choice of q is such that it satis®es Eq. (11.3)and thenatural boundary conditionsr T q ˆÿQ in and q n ˆ q n on ÿ q …11:56†Thus we only have to satisfy the constitutive relation (11.2), i.e.,k ÿ1 q ‡ r ˆ 0 in with ˆ on ÿ …11:57†A weak statement of the above is……q T …k ÿ1 q ‡ r† d ÿ q n … ÿ † dÿ ˆ 0ÿ in which q n represents the variation of normal ¯ux on the boundary.Use of Green's theorem transforms the above into………q T k ÿ1 q d ÿ r T q d ‡ q n dÿ ‡ q n dÿ ˆ 0…ÿ ÿ q…11:58†…11:59†

302 Mixed formulation and constraints ± complete ®eld methodsIf we further assume that r T q 0in and q n ˆ 0onÿ q , i.e., that the weightingfunctions are simply the variations of q, the equation reduces to……q T k ÿ1 q d ‡ q n dÿ ˆ 0…11:60†ÿ This is in fact the variation of a complementary ¯ux principle……1 ˆ2 qT k ÿ1 q d ‡ q n dÿ…11:61†ÿ Numerical solutions can obviously be started from either of the above equationsbut the di culty is the choice of the trial function satisfying the constraints. Weshall return to this problem in Sec. 11.7.2.The complementary elastic energy principleIn the elasticity problem speci®ed in Sec. 11.4 we can proceed similarly, assumingstress ®elds which satisfy the equilibrium conditions both on the boundary ÿ t andin the domain .Thus in an analogous manner to that of the previous example we impose on thepermissible stress ®eld the constraints which we assume to be satis®ed by theapproximation identically, i.e.,S T r ‡ b ˆ 0 in and t ˆ t on ÿ t …11:62†Thus only the constitutive relations and displacement boundary conditions remain tobe satis®ed, i.e.,D ÿ1 r ÿ Su ˆ 0 in and u ˆ u on ÿ u …11:63†The weak statement of the above can be written as……r T …D ÿ1 r ÿ Su† d ‡ t T …u ÿ u† dÿ ˆ 0…11:64†ÿ uwhich on integration by Green's theorem gives…………r T D ÿ1 r d ‡ …S T r† T u d ÿ t T u dÿ ÿ t T u dÿ ˆ 0 …11:65†ÿ u ÿ tAgain assuming that the test functions are complete variations satisfying the homogeneousequilibrium equation, i.e.,S T r ˆ 0 in and t ˆ 0 on ÿ t …11:66†we have as the weak statement……r T D ÿ1 r d ÿt T u dÿ ˆ 0ÿ u…11:67†The corresponding complementary energy variational principle is ˆ 1 ……r T D ÿ1 r d ÿ t T u dÿ2 ÿ u…11:68†Once again in practical use the di culties connected with the choice of the approximatingfunction arise but on occasion a direct choice is possible. 30

11.7.2 Solution using auxiliary functionsBoth the complementary forms can be solved using auxiliary functions to ensure thesatisfaction of the constraints.In the heat transfer problem it is easy to verify that the homogeneous equationr T q @q x@x ‡ @q y@y ˆ 0is automatically satis®ed by de®ning a function such thatq x ˆ @@yq y ˆÿ @@x…11:69†…11:70†Thus we de®neq ˆ L ‡ q 0 and q ˆ L …11:71†where q 0 is any ¯ux chosen so thatr T q 0 ˆÿQ…11:72†and @L ˆ@y ; ÿ @ T…11:73†@xthe formulations of Eqs (11.60)and (11.61)can be used without any constraints and,for instance, the stationarity…… 1@ ˆ 2 …L ‡ q 0† T k ÿ1 …L ‡ q 0 † d ÿ dÿÿ @s…11:74†will su ce to so formulate the problem (here s is the tangential direction to theboundary).The above form will require shape functions for satisfying C 0 continuity.In the corresponding elasticity problem a similar two-dimensional form can beobtained by the use of the so-called Airy stress function . 36Now the equilibrium equations8@ xS T @x ‡ @ 9xy>=r ‡ b @ xy@x ‡ @ ˆ 0y>:@y ‡ b >;y…11:75†are identically solved by choosingr ˆ L ‡ r 0where @2L ˆ@y 2 ; @ 2 T@x 2 ; ÿ @2@x @yand r 0 is an arbitrary stress chosen so thatS T r 0 ‡ b ˆ 0Complementary forms with direct constraint 303…11:76†…11:77†…11:78†

304 Mixed formulation and constraints ± complete ®eld methodsAgain the substitution of (11.76)into the weak statement (11.67)or the complementaryvariational problem (11.68)will yield a direct formulation to which noadditional constraints need be applied. However, use of the above forms does leadto further complexity in multiply connected regions where further conditions areneeded. The reader will note that in Chapter 7 we encountered this in a similarproblem in torsion and suggested a very simple procedure of avoidance (see Sec. 7.5).The use of this stress function formulation in the two-dimensional context was ®rstmade by de Veubeke and Zienkiewicz 37 and Elias, 38 but the reader should note thatnow with second-order operators present, C 1 continuity of shape functions is neededin a similar manner to the problems which we have to consider in plate bending (seeVolume 2).Incidentally, analogies with plate bending go further here and indeed it can beshown that some of these can be usefully employed for other problems. 3911.8 Concluding remarks ± mixed formulation or a testof element `robustness'The mixed form of ®nite element formulation outlined in this chapter opens a newrange of possibilities, many with potentially higher accuracy and robustness thanthose o€ered by irreducible forms. However, an additional advantage arises even insituations where, by the principle of limitation, the irreducible and mixed formsyield identical results. Here the study of the behaviour of the mixed form canfrequently reveal weaknesses or lack of `robustness' in the irreducible form whichotherwise would be di cult to determine.The mixed approximation, if properly understood, expands the potential of the®nite element method and presents almost limitless possibilities of detailed improvement.Some of these will be discussed further in the next two chapters, and others inVolumes 2 and 3.References1. S.N. Atluri, R.H. Gallagher, and O.C. Zienkiewicz (eds). Hybrid and Mixed Finite ElementMethods. Wiley, 1983.2. O.C. Zienkiewicz, R.L. Taylor, and J.A.W. Baynham. Mixed and irreducible formulationsin ®nite element analysis. Chapter 21 of Hybrid and Mixed Finite Element Methods (edsS.N. Atluri, R.H. Gallagher, and O.C. Zienkiewicz), pp. 405±31, Wiley, 1983.3. I. BabusÏ ka and J.E. Osborn. Generalized ®nite element methods and their relations tomixed problems. SIAM J. Num. Anal. 20, 510±36, 1983.4. R.L. Taylor and O.C. Zienkiewicz. Complementary energy with penalty function in ®niteelement analysis. Chapter 8 of Energy Methods in Finite Element Analysis (eds R.Glowinski, E.Y. Rodin, and O.C. Zienkiewicz). Wiley, 1979.5. K. Washizu, Variational Methods in Elasticity and Plasticity. 2nd ed., Pergamon Press, 1975.6. L.R. Herrmann. Finite element bending analysis of plates. Proc. 1st Conf. Matrix Methodsin Structural Mechanics. AFFDL-TR-80-66, Wright-Patterson AF Base, Ohio, 1965.7. K. Hellan. Analysis of plates in ¯exure by a simpli®ed ®nite element method. Acta PolytechnicaScandinavia. Civ. Eng. Series 46, Trondheim, 1967.

8. R.S. Dunham and K.S. Pister. A ®nite element application of the Hellinger Reissnervariational theorem. Proc. 1st Conf. Matrix Methods in Structural Mechanics. Wright-Patterson AF Base, Ohio, 1965.9. R.L. Taylor and O.C. Zienkiewicz. Mixed ®nite element solution of ¯uid ¯ow problems.Chapter 1 of Finite Elements in Fluid. Vol. 4 (eds R.H. Gallagher, D.N. Norrie, J.T.Oden, and O.C. Zienkiewicz), pp. 1±20, Wiley, 1982.10. B. Fraeijs de Veubeke. Displacement and equilibrium models in ®nite element method.Chapter 9 of Stress Analysis (eds O.C. Zienkiewciz and C.S. Holister), pp. 145±97,Wiley, 1965.11. I. BabusÏ ka. The ®nite element method with Lagrange multipliers. Num. Math. 20, 179±92,1973; also `Error bounds for ®nite element methods. Num. Math. 16, 322±33, 1971.12. F. Brezzi. On the existence, uniqueness and approximation of saddle point problemsarising from lagrangian multipliers. RAIRO. 8-R2, 129±51, 1974.13. O.C. Zienkiewicz, S. Qu, R.L. Taylor, and S. Nakazawa. The patch test for mixed formulation.Int. J. Num. Meth. Eng. 23, 1873±83, 1986.14. J.T. Oden and N. Kikuchi. Finite element methods for constrained problems of elasticity.Int. J. Num. Mech. Eng. 18, 701±25, 1982.15. E. Hellinger. Die allgemeine Aussetze der Mechanik der Kontinua, in Encyclopedia derMatematischen Wissenschaften. Vol. 4 (eds F. Klein and C. Muller). Tebner, Leipzig, 1914.16. E. Reissner. On a variational theorem in elasticity. J. Math. Phys. 29, 90±5, 1950.17. L.R. Herrmann. Finite element bending analysis of plates. Proc. Am. Soc. Civ. Eng. 94,EM5, 13±25, 1968.18. L.R. Herrmann and D.M. Campbell. A ®nite element analysis for thin shells. JAIAA. 6,1842±7, 1968.19. O.C. Zienkiewicz and D. Lefebvre. Mixed methods for FEM and the patch test. Somerecent developments, in Analyse Mathematique of Application (eds F. Murat and O. Pirenneau).Gauthier Villars, Paris, 1988.20. P.A. Raviart and J.M. Thomas. A mixed ®nite element method for second order ellipticproblems. Lect. Notes in Math. no. 606, pp. 292±315, Springer-Verlag, 1977.21. D. Arnold, F. Brezzi, and J. Douglas. PEERS, a new mixed ®nite element for planeelasticity. Japan J. Appl. Math. 1, 347±67, 1984.22. T.H.H. Pian and K. Sumihara. Rational approach for assumed stress ®nite elements. Int. J.Num. Meth. Eng., 20, 1685±95, 1985.23. O.C. Zienkiewicz and D. Lefebvre. Three ®eld mixed approximation and the plate bendingproblem. Comm. Appl. Num. Math. 3, 301±9, 1987.24. T.J.R. Hughes. Generalization of selective integration procedures to anisotropic and nonlinearmedia. Int. J. Num. Meth. Eng. 15, 1413±18, 1980.25. J.C. Simo, R.L. Taylor, and K.S. Pister. Variational and projection methods for thevolume constraint in ®nite deformation plasticity. Comp. Meth. App. Mech. Eng. 51,177±208, 1985.26. J.C. Simo and M.S. Rifai. A class of mixed assumed strain methods and the method ofincompatible modes. Int. J. Num. Meth. Eng., 29, 1595±638, 1990.27. U. Andel®nger and E. Ramm. EAS-elements for two-dimensional, three-dimensional,plate and shell structures and their equivalence to HR-elements. Int. J. Num. Meth.Eng., 36, 1311±37, 1993.28. M. Bischo€, E. Ramm, and D. Braess. A class of equivalent enhanced assumed strain andhybrid stress ®nite elements. Comp. Mech., 22, 443±49, 1999.29. G. Cantin, C. Loubignac, and C. Touzot. An iterative scheme to build continuous stressand displacement solutions. Int. J. Num. Meth. Eng., 12, 1493±506, 1978.30. C. Loubibnac, G. Cantin, and C. Touzot. Continuous stress ®elds in ®nite element analysis.J. AIAA, 15, 1645±47, 1978.References 305

306 Mixed formulation and constraints ± complete ®eld methods31. H.J. Brauchli and J.T. Oden. On the calculation of consistent stress distributions in ®niteelement applications. Int. J. Num. Meth. Eng., 3, 317±25, 1971.32. S. Nakazawa. Mixed ®nite elements and iterative solution procedures. In W.K. Liu et al.,editor, Innovative Methods in Non-linear Problems. Pineridge Press, Swansea, 1984.33. O.C. Zienkiewicz, J.P. Vilotte, S. Toyoshima, and S. Nakazawa. Iterative method for constrainedand mixed approximation. An inexpensive improvement of FEM performance.Comp. Meth. Appl. Mech. Eng., 51, 3±29, 1985.34. O.C. Zienkiewicz, Xi Kui Li, and S. Nakazawa. Iterative solution of mixed problems andstress recovery procedures. Comm. Appl. Num. Meth., 1, 3±9, 1985.35. K.J. Arrow, L. Hurwicz, and H. Uzawa. Studies in Non-Linear Programming. StanfordUniversity Press, Stanford, CA, 1958.36. S.P. Timoshenko and J.N. Goodier. Theory of Elasticity. 3rd edn, McGraw-Hill, NewYork, 1969.37. B. Fraeijs de Veubeke and O.C. Zienkiewicz. Strain energy bounds in ®nite elementanalysis by slab analogy. J. Strain Anal., 2, 265±71, 1967.38. Z.M. Elias. Duality in ®nite element methods. Proc. Am. Soc. Civ. Eng., 94(EM4), 931±46,1968.39. R.V. Southwell. On the analogues relating ¯exure and displacement of ¯at plates. Quart. J.Mech. Appl. Math., 3, 257±70, 1950.

12Incompressible materials,mixed methods and otherprocedures of solution12.1 IntroductionWe have noted earlier that the standard displacement formulation of elastic problemsfails when Poisson's ratio becomes 0.5 or when the material becomes incompressible.Indeed, problems arise even when the material is nearly incompressible with>0:4 and the simple linear approximation with triangular elements gives highlyoscillatory results in such cases.The application of a mixed formulation for such problems can avoid the di culties andis of great practical interest as nearly incompressible behaviour is encountered in a varietyof real engineering problems ranging from soil mechanics to aerospace engineering. Identicalproblems also arise when the ¯ow of incompressible ¯uids is encountered.In this chapter we shall discuss fully the mixed approaches to incompressibleproblems, generally using a two-®eld manner where displacement (or ¯uid velocity)u and the pressure p are the variables. Such formulation will allow us to deal withfull incompressibility as well as near incompressibility as it occurs. However, whatwe will ®nd is that the interpolations used will be very much limited by the stabilityconditions of the mixed patch test. For this reason much interest has been focusedon the development of so-called stabilized procedures in which the violation of themixed patch test (or BabusÏ ka±Brezzi conditions) is arti®cially compensated. A partof this chapter will be devoted to such stabilized methods.12.2 Deviatoric stress and strain, pressure and volumechangeThe main problem in the application of a `standard' displacement formulation toincompressible or nearly incompressible problems lies in the determination of themean stress or pressure which is related to the volumetric part of the strain (forisotropic materials). For this reason it is convenient to separate this from the totalstress ®eld and treat it as an independent variable. Usingthe `vector' notation ofstress, the mean stress or pressure is given byÿp ˆ 13 x ‡ y ‡ z ˆ 13m T r …12:1†

308 Incompressible materials, mixed methods and other procedures of solutionwhere m for the general three-dimensional state of stress is given bym ˆ ‰ 1; 1; 1; 0; 0; 0 Š TFor isotropic behaviour the `pressure' is related to the volumetric strain, " v , by thebulk modulus of the material, K. Thus," v ˆ " x ‡ " y ‡ " z ˆ m T e …12:2†" v ˆ pK…12:3†For an incompressible material K ˆ1… 0:5† and the volumetric strain is simplyzero.The deviatoric strain e d is de®ned bye d ˆ e ÿ 1 3 m" ÿ v I ÿ 1 3 mmT e ˆ Id e …12:4†where I d is a deviatoric projection matrix which proves useful later and in Volume 2.In isotropic elasticity the deviatoric strain is related to the deviatoric stress by theshear modulus G asr d ˆ I d r ˆ 2GI 0 e d ÿ ˆ 2G I 0 ÿ 1 3 mmT e …12:5†where the diagonal matrix22322I 0 ˆ 1216741 51is introduced because of the vector notation. A deviatoric form for the elastic moduliof an isotropic material is written asÿ D d ˆ 2G I 0 ÿ 1 3 mmT…12:6†for convenience in writingsubsequent equations.The above relationships are but an alternate way of determiningthe stress strainrelations shown in Chapters 2 and 4±6, with the material parameters related throughG ˆK ˆE21‡ … †E31ÿ … 2†…12:7†and indeed Eqs (12.5) and (12.3) can be used to de®ne the standard D matrix in analternative manner.12.3 Two-®eld incompressible elasticity(u±p form)In the mixed form considered next we shall use as variables the displacement u and thepressure p.

Two-®eld incompressible elasticity(u±p form) 309Now the equilibrium equation (11.22) is rewritten using(12.5), treatingp as anindependent variable, as…………e T D d e d ‡ e T m p d ÿ u T b d ÿ u T t dÿ ˆ 0 …12:8†ÿ tand in addition we shall impose a weak form of Eq. (12.3), i.e.,… p m T e ÿ p d ˆ 0…12:9† Kwith e ˆ Su. Independent approximation of u and p asu ^u ˆ N u ~u and p ^p ˆ N p ~p …12:10†immediately gives the mixed approximation in the form A C ~uC Tˆf1…12:11†ÿV ~p f 2where……A ˆ B T D d B d C ˆ B T mN p d…………12:12†V ˆ N T 1p K N p d f 1 ˆ N T u b d ‡ N T u t dÿ f 2 ˆ 0ÿ tWe note that for incompressible situations the equations are of the `standard' form,see Eq. (11.14) with V ˆ 0 (as K ˆ1), but the formulation is useful in practice whenK has a high value (or ! 0:5).A formulation similar to that above and usingthe correspondingvariational theoremwas ®rst proposed by Herrmann 1 and later generalized by Key 2 for anisotropicT 3/1T 6/1T 6/3T 10/3n u = 0n p = 0(pass)= 0= 0(pass)= 0= 2(fail)= 2= 2(pass)Q 4/1Q 8/4Q 8/3Q 9/4Q 9/3(a)n u = 0n p = 0(pass)= 0= 3(fail)= 0= 0(fail)= 2= 3(fail)= 2= 2(pass)Fig. 12.1 Incompressible elasticity u±p formulation. Discontinuous pressure approximation. (a) Singleelementpatch tests.

310 Incompressible materials, mixed methods and other procedures of solutionT 3/1T 6/1T 6/3T 10/3n u = 2n p = 5(fail)= 7 x 2 = 14= 5(pass)= 7 x 2 = 14= 17(fail)= 19 x 2 = 38= 17(pass)Q 4/1Q 8/4Q 9/3n u = 2n p = 3(fail)= 5 x 2 = 10= 15(fail)= 9 x 2 = 18= 11(pass)Q 8/3 Q 9/4(b)n u = 5 x 2 = 10n p = 11(fail)= 9 x 2 = 18= 15(pass)Fig. 12.1 (continued) Incompressible elasticity u±p formulation. Discontinuous pressure approximation.(b) Multiple-element patch tests.elasticity. The arguments concerning stability (or singularity) of the matrices which wepresented in Sec. 11.3 are again of great importance in this problem.Clearly the mixed patch condition about the number of degress of freedom nowyields [see Eq. (11.18)]n u 5 n p…12:13†

and is necessary for prevention of locking(or instability) with the pressure actingnowas the constraint variable of the lagrangian multiplier enforcing incompressibility.In the form of a patch test this condition is most critical and we show in Figs 12.1and 12.2 a series of such patch tests on elements with C 0 continuous interpolation of uand either discontinuous or continuous interpolation of p. For each we have includedall combinations of constant, linear and quadratic functions.In the test we prescribe all the displacements on the boundaries of the patch andone pressure variable (as it is well known that in fully incompressible situationspressure will be indeterminate by a constant for the problem with all boundarydisplacements prescribed).The single-element test is very stringent and eliminates most continuous pressureapproximations whose performance is known to be acceptable in many situations.For this reason we attach more importance to the assembly test and it wouldappear that the followingelements could be permissible accordingto the criteria ofEq. (12.13) (indeed all pass the B-B condition fully):Triangles: T6/1; T10/3; T6/C3Quadrilaterals: Q9/3; Q8/C4; Q9/C4We note, however, that in practical applications quite adequate answers have beenreported with Q4/1, Q8/3 and Q9/4 quadrilaterals, although severe oscillations of pmay occur. If full robustness is sought the choice of the elements is limited. 3It is unfortunate that in the present `acceptable' list, the linear triangle andquadrilateral are missing. This appreciably restricts the use of these simplest elements.A possible and indeed e€ective procedure here is to not apply the pressure constraintat the level of a single element but on an assembly. This was done by Herrmann in hisoriginal presentation 1 where four elements were chosen for such a constraint asshown in Fig. 12.3(a). This composite `element' passes the single-element (andmultiple-element) patch tests but apparently so do several others ®ttinginto thiscategory. In Fig. 12.3(b) we show how a single triangle can be internally subdividedinto three parts by the introduction of a central node. This coupled with constantpressure on the assembly allows the necessary count condition to be satis®ed and astandard element procedure applies to the original triangle treating the centralnode as an internal variable. Indeed, the same e€ect could be achieved by the introductionof any other internal element function which gives zero value on the maintriangle perimeter. Such a bubble function can simply be written in terms of thearea coordinates (see Chapter 8) asL 1 L 2 L 3However, as we have stated before, the degree of freedom count is a necessary but notsu cient condition for stability and a direct rank test is always required. In particularit can be veri®ed by algebra that the conditions stated in Sec. 11.3 are not ful®lled forthis triple subdivision of a linear triangle (or the case with the bubble function) andthusindicatinginstability.Cp ˆ 0 for some non-zero values of pTwo-®eld incompressible elasticity(u±p form) 311

312 Incompressible materials, mixed methods and other procedures of solutionu variable (restrained, free) 2 DOFp variable (restrained, free) 1 DOFT 3/C3T 6/C6T 6/C3n u = 0n p = 2(fail)= 0= 5(fail)= 0= 2(fail)Q 4/C4Q 8/C8Q 8/C4Q 9/C4(a)n u = 0n p = 3(fail)= 0= 7(fail)= 2= 3(fail)= 2= 3(fail)T 3/C3T 6/C6T 6/C3n u = 2n p = 6(fail)= 7 x 2 = 14= 18(fail)= 7 x 2 = 14= 6(pass)Q 4/C4Q 8/C4Q 9/C4(b)n u = 2n p = 8(fail)= 5 x 2 = 10= 8(pass)= 9 x 2 = 18= 8(pass)Fig. 12.2 Incompressible elasticity u±p formulation. Continuous (C 0 ) pressure approximation. (a) Singleelementpatch tests. (b) Multiple-element patch tests.

Two-®eld incompressible elasticity(u±p form) 313(a) +n u = 2 n p = 0+(b)n u = 2 n p = 0OR(Bubble function)+(c)n u = 2 n p = 2OR(Bubble function)(d)+(Bubble function)n u = 2 n p = 2Fig. 12.3 Some simple combinations of linear triangles and quadrilaterals that pass the necessary patch testcounts. Combinations (a), (c), and (d) are successful but (b) is still singular and not usable.

314 Incompressible materials, mixed methods and other procedures of solution12LoadTriangle 1Only verticalmovementpossible forno volumechangeTriangle 1Only horizontalmovementpossible forno volumechangeFig. 12.4 Locking (zero displacements) of a simple assembly of linear triangles for which incompressibility isfully required (n p ˆ n u ˆ 24).In Fig. 12.3(c) we show, however, that the same concept can be used with goode€ect for C 0 continuous p. 4 Similar internal subdivision into quadrilaterals or theintroduction of bubble functions in quadratic triangles can be used, as shown inFig. 12.3(d), with success.The performance of all the elements mentioned above has been extensively discussed5ÿ10 but detailed comparative assessment of merit is di cult. As we haveobserved, it is essential to have n u 5 n p but if near equality is only obtained in alarge problem no meaningful answers will result for u as we observe, for example,in Fig. 12.4 in which linear triangles for u are used with the element constant p.Here the only permissible answer is of course u ˆ 0 as the triangles have to preserveconstant volumes.The ratio n u =n p which occurs as the ®eld of elements is enlarged gives some indicationof the relative performance, and we show this in Fig. 12.5. This approximates tothe behaviour of a very large element assembly, but of course for any practicalproblem such a ratio will depend on the boundary conditions imposed.We see that for the discontinuous pressure approximation this ratio for `good'elements is 2±3 while for C 0 continuous pressure it is 6±8. All the elements shownin Fig. 12.5 perform very well, though two (Q4/1 and Q9/4) can on occasion lockwhen most boundary conditions are on u.12.4 Three-®eld nearlyincompressible elasticity(u±p±e vform)A direct approximation of the three-®eld form leads to an important method in ®niteelement solution procedures for nearly incompressible materials which has sometimesbeen called the B-bar method. The methodology can be illustrated for the nearly

Three-®eld nearlyincompressible elasticity(u±p±e v form) 315(a)6/1 γ = 4 4/1 γ = 210/3 γ = 3 9/3 γ = 2.666B1/3 γ = 2 9/4 γ = 2(b)6/3C γ = 8 9/4C γ = 83B1/3C γ = 6Fig. 12.5 The freedom index or in®nite patch ratio for various u±p elements for incompressible elasticity( ˆ n u =n p ). (a) Discontinuous pressure. (b) Continuous pressure.incompressible isotropic problem. For this problem the method often reduces to thesame two-®eld form previously discussed. However, for more general anisotropicor inelastic materials and in ®nite deformation problems the method has distinctadvantages as will be discussed further in Volume 2. The usual irreducible form(displacement method) has been shown to `lock' for the nearly incompressibleproblem. As shown in Sec. 12.3, the use of a two-®eld mixed method can avoid thislockingphenomenon when properly implemented (e.g., usingthe Q9/3 two-®eldform). Below we present an alternative which leads to an e cient and accurateimplementation in many situations. For the development shown we shall assume

316 Incompressible materials, mixed methods and other procedures of solutionthat the material is isotropic linear elastic but it may be extended easily to includeanisotropic materials.Assumingan independent approximation to " v and p we can formulate theproblem by use of Eq. (12.8) and the weak statement of relations (12.2) and (12.3)written as…p m T Su ÿ " v d ˆ 0 …12:14†…" v ‰ K" v ÿ pŠd ˆ 0 …12:15†If we approximate the u and p ®elds by Eq. (12.10) and" v ^" v ˆ N v ~e v …12:16†we obtain a mixed approximation in the form2389 8 9A C 0 >< ~u >= >< f 1 >=64 C T 70 ÿE 5 ~p>: >; ˆ f 2…12:17†>: >;0 ÿE T H ~e v f 3where A, C, f 1 , f 2 are given by Eq. (12.12) and…E ˆ N T v N p d f 3 ˆ 0 …12:18†with…H ˆN T v KN v d…12:19†For completeness we give the variational theorem whose ®rst variation gives Eqs(12.8), (12.14) and (12.15). First we de®ne the strain deduced from the standarddisplacement approximation ase u ˆ Su B~u…12:20†The variational theorem is then given as ˆ 1 …2…ÿ…ÿe T u D d e u ‡ " v K" v d ‡…u T b d ÿ u T t dÿÿ tÿp m T e u ÿ " v d…12:21†12.4.1 The B-bar method for nearly incompressible problemsThe second of (12.17) has the solution~e v ˆ E ÿ1 C T ~u ˆ W~u …12:22†In the above we assume that E may be inverted, which implies that N v and N p have thesame number of terms. Furthermore, the approximations for the volumetric strainand pressure are constructed for each element individually and are not continuous

Three-®eld nearlyincompressible elasticity(u±p±e v form) 317across element boundaries. Thus, the solution of Eq. (12.22) may be performed foreach individual element. In practice N v is normally assumed identical to N p so thatE is symmetric positive de®nite. The solution of the third of (12.17) yields the pressureparameters in terms of the volumetric strain parameters and is given by~p ˆ E ÿT H T ~e v …12:23†Substitution of (12.22) and (12.23) into the ®rst of (12.17) gives a solution that is interms of displacements only. Accordingly,where for isotropyA~u ˆ f 1…A ˆ B T D d B d ‡ W T HWˆ A ‡ W T HW…12:24†…12:25†The solution of (12.24) yields the nodal parameters for the displacements. Use of(12.22) and (12.23) then gives the approximations for the volumetric strain andpressure.The result given by (12.25) may be further modi®ed to obtain a form that is similarto the standard displacement method. Accordingly, we write…A ˆ B T DB d…12:26†where the strain±displacement matrix is nowB ˆ I d B ‡ 1 3 mN vWFor isotropy the modulus matrix isD ˆ D d ‡ Kmm T…12:27†…12:28†We note that the above form is identical to a standard displacement model except thatB is replaced by B. The method has been discussed more extensively in references 11,12 and 13.The equivalence of (12.25) and (12.26) can be veri®ed by simple matrix multiplication.Extension to treat general small strain formulations can be simply performed byreplacingthe isotropic D matrix by an appropriate form for the general materialmodel. The formulation shown above has been implemented into an element includedas part of the program referred to in Chapter 20. The elegance of the method is morefully utilized when consideringnon-linear problems, such as plasticity and ®nitedeformation elasticity (see Volume 2).We note that elimination startingwith the third equation could be accomplishedleadingto a u±p two-®eld form using K as a penalty number. This is convenient forthe case where p is continuous but " v remains discontinuous ± this is discussedfurther in Sec. 12.7.3. Such an elimination, however, points out that precisely thesame stability criteria operate here as in the two-®eld approximation discussedearlier.

318 Incompressible materials, mixed methods and other procedures of solution12.5 Reduced and selective integration and itsequivalence to penalized mixed problemsIn Chapter 9 we mentioned the lowest order numerical integration rules that stillpreserve the required convergence order for various elements, but at the sametime pointed out the possibility of a singularity in the resulting element matrices.In Chapter 10 we again referred to such low order integration rules, introducingthe name `reduced integration' for those that did not evaluate the sti€ness exactlyfor simple elements and pointed out some dangers of its indiscriminate use due toresulting instability. Nevertheless, such reduced integration and selective integration(where the low order approximation is only applied to certain parts of the matrix)has proved its worth in practice, often yieldingmuch more accurate results thanthe use of more precise integration rules. This was particularly noticeable innearly incompressible elasticity (or Stokes ¯uid ¯ow which is similar) 14ÿ16 and inproblems of plate and shell ¯exure dealt with as a case of a degenerate solid 17;18(see Volume 2).The success of these procedures derived initially by heuristic arguments provedquite spectacular ± though some consider it somewhat verging on immorality toobtain improved results while doingless work! Obviously fuller justi®cation ofsuch processes is required. 19 The main reason for success is associated with the factthat it provides the necessary singularity of the constraint part of the matrix [viz.Eqs (11.19)±(11.21)] which avoids locking. Such singularity can be deduced from acount of integration points, 19;20 but it is simpler to show that there is a completeequivalence between reduced (of selective) integration procedures and the mixedformulation already discussed. This equivalence was ®rst shown by Malkus andHughes 21 and later in a general context by Zienkiewicz and Nakazawa. 22We shall demonstrate this equivalence on the basis of the nearly incompressibleelasticity problem for which the mixed weak integral statement is given by Eqs(12.8) and (12.9). It should be noted, however, that equivalence holds only for thediscontinuous pressure approximation.The correspondingirreducible form can be written by satisfyingthe second of theseequations exactly by implyingp ˆ Km T eand substitutingabove into (12.8) as… e T 2G I 0 ÿ 1 …3 mT m e d ‡On substitutingwe have…ÿe T mKm T e d…u T b d ÿ u T t dÿ ˆ 0ÿ t…12:29†…12:30†u ^u ˆ N u ~u and e ^e ˆ SN u ~u ˆ B~u …12:31†ÿA ‡ A ~u ˆ f1 …12:32†

Reduced and selective integration and its equivalence to penalized mixed problems 319where A and f 1 are exactly as given in Eq. (12.12) and…A ˆ B T mKm T B d…12:33†The solution of Eq. (12.32) for ~u allows the pressures to be determined at all pointsby Eq. (12.29). In particular, if we have used an integration scheme for evaluating(12.33) which samples at points ( k ) we can writep… k †ˆKm T e… k †ˆKm T B… k †~u ˆ XN pj … k †~p j …12:34†Now if we turn our attention to the penalized mixed form of Eqs (12.8)±(12.12) wenote that the second of Eqn. (12.11) is explicitly… m T B~u ÿ 1 K N p~p d ˆ 0…12:35†N T pIf a numerical integration is applied to the above sampling at the pressure nodeslocated at coordinate ( l ), previously de®ned in Eq. (12.34), we can write for eachscalar component of N pXN pj … l † m T B… l †~u ÿ 1 K N p… l †~p W l ˆ 0…12:36†lin which the summation is over all integration points ( l ) and W l are the appropriateweights and jacobian determinants.Now asN pj … k †ˆ jkif l is at the pressure node j and zero at other pressure nodes, Eq. (12.36) reducessimply to the requirement that at all pressure nodesm T B… l †~u ˆ 1K N p… l †~p…12:37†This is precisely the same condition as that given by Eq. (12.34) and the equivalenceof the procedures is proved, providing the integrating scheme used for evaluating A gives an identical integral of the mixed form of Eq. (12.35).This is true in many cases and for these the reduced integration-mixed equivalenceis exact. In all other cases this equivalence exists for a mixed problem in which aninexact rule of integration has been used in evaluating equations such as (12.35).For curved isoparametric elements the equivalence is in fact inexact, and slightlydi€erent results can be obtained usingreduced integration and mixed forms. This isillustrated in examples given in reference 23.We can conclude without detailed proof that this type of equivalence is quitegeneral and that with any problem of a similar type the application of numericalquadrature at n p points in evaluatingthe matrix A within each element is equivalentto a mixed problem in which the variable p is interpolated element-by-element usingas p-nodal values the same integrating points.The equivalence is only complete for the selective integration process, i.e., applicationof reduced numerical quadrature only to the matrix A, and ensures that thisj

320 Incompressible materials, mixed methods and other procedures of solutionmatrix is singular, i.e., no locking occurs if we have satis®ed the previously statedconditions (n u > n p ).The full use of reduced integration on the remainder of the matrix determining ~u,i.e., A, is only permissible if that remains non-singular ± the case which we havediscussed previously for the Q8/4 element.It can therefore be concluded that all the elements with discontinuous interpolationof p which we have veri®ed as applicable to the mixed problem (viz. Fig. 12.1, forinstance) can be implemented for nearly incompressible situations by a penalizedirreducible form usingcorrespondingselective integration.yIn Fig. 12.6 we show an example which clearly indicates the improvement ofdisplacements achieved by such reduced integration as the compressibility modulusK increases (or the Poisson ratio tends to 0.5). We note also in this example thedramatically improved performance of such points for stress sampling.For problems in which the p (constraint) variable is continuously interpolated (C 0 )the arguments given above fail as quantities such as m T e are not interelementcontinuous in the irreducible form.A very interestingcorollary of the equivalence just proved for (nearly) incompressiblebehaviour is observed if we note the rapid increase of order of integratingformulae with the number of quadrature points (viz. Chapter 9). For high orderelements the number of quadrature points equivalent to the p constraint permissiblefor stability rapidly reaches that required for exact integration and hence their performancein nearly incompressible situations is excellent, even if exact integration isused. This was observed on many occasions 24ÿ26 and Sloan and Randolf 27 haveshown good performance with the quintic triangle. Unfortunately such high orderelements pose other di culties and are seldom used in practice.A ®nal remark concerns the use of `reduced' integration in particular and ofpenalized, mixed, methods in general. As we have pointed out in Sec. 11.3.1 it ispossible in such forms to obtain sensible results for the primary variable (u in thepresent example) even though the general stability conditions are violated, providingsome of the constraint equations are linearly dependent. Now of course the constraintvariable (p in the present example) is not determinate in the limit.This situation occurs with some elements that are occasionally used for the solutionof incompressible problems but which do not pass our mixed patch test, such as Q8/4and Q9/4 of Fig. 12.1. If we take the latter number to correspond to the integratingpoints these will yield acceptable u ®elds, though not p.Figure 12.7 illustrates the point on an application involving slow viscous ¯owthrough an ori®ce ± a problem that obeys identical equations to those of incompressibleelasticity. Here elements Q8/4, Q8/3, Q9/4 and Q9/3 are compared althoughonly the last completely satis®es the stability requirements of the mixed patchtest. All elements are found to give a reasonable velocity (u) ®eld but pressuresare acceptable only for the last one, with element Q8/4 failingto give resultswhich can be plotted. 3y The Q9/3 element would involve three-point quadrature which is somewhat unnatural for quadrilaterals.It is therefore better to simply use the mixed form here ± and, indeed, in any problem which has non-linearbehaviour between p and u (see Volume 2).

3 x 3 Gauss point integration2 x 2 Gauss point integrationv =0.451.0v = 0.499992 x 2 only(3 x 3 breaks down)p0.5Aσ θpa = 1.00.5v = 0.3v = 0.4Radial displacements at A pa / EInteg. v = 0.3 v = 0.4 v = 0.45 v = 0.499993x32x20.37790.37760.39040.39100.39500.3977No result0.4041ExactExact0.38090.39450.40130.4081v =0.4500.50 0.75 1.00RadiusPosition of 2 x 2 Gauss pointsFig. 12.6 Sphere under internal pressure. Effect of numerical integration rules on results with different Poisson ratios.

322 Incompressible materials, mixed methods and other procedures of solutionOrifice(a) Mesh of rectangularelementsC L(e) Axial velocities,elements Q 8/3, Q 9/4, Q9/3(b) Pressure variation,element Q 8/3(f) Radial velocities,elements Q 8/3, Q 9/4, Q9/3(c) Pressure variation,element Q 9/4(g) Axial velocities,element Q 8/4(h) Radial velocities,element Q 8/4(d) Pressure variation,element Q 9/3Fig. 12.7 Steady-state, low Reynolds number ¯ow through an ori®ce. Note that pressure variation forelement Q8/4 is so large it cannot be plotted. Solution with u=p elements Q8/3, Q8/4, Q9/3, Q9/4.It is of passinginterest to note that a similar situation develops if four triangles ofthe T3/1 type are assembled to form a quadrilateral in the manner of Fig. 12.8.Although the original element locks, as we have previously demonstrated, a lineardependence of the constraint equation allows the assembly to be used quite e€ectivelyin many incompressible situations, as shown in reference 25.

A simple iterative solution process for mixed problems: Uzawa method 323Fig. 12.8 A quadrilateral with intersecting diagonals forming an assembly of four T3/1 elements. This allowsdisplacements to be determined for nearly incompressible behaviour but does not yield pressure results.12.6 A simple iterative solution process for mixedproblems: Uzawa method12.6.1 GeneralIn the general remarks on the algebraic solution of mixed problems characterized byequations of the type [viz. Eq. (11.14)]A C xC T 0 y ˆf1…12:38†f 2we have remarked on the di culties posed by the zero diagonal and the increasednumber of unknowns (n x ‡ n y ) as compared with the irreducible form (n x or n y ).A general iterative form of solution is possible, however, which substantiallyreduces the cost. 28 In this we solve successivelyy …k ‡ 1† ˆ y …k† ‡ qr …k†…12:39†where r …k† is the residual of the second equation computed asr …k† ˆ C T x …k† ÿ f 2…12:40†and follow with solution of the ®rst equation, i.e.,x …k ‡ 1† ˆ A ÿ1 …f 1 ÿ Cy …k ‡ 1† †…12:41†In the above q is a `convergence accelerator matrix' and is chosen to be e cient andsimple to use.The algorithm is similar to that described initially by Uzawa 29 and has been widelyapplied in an optimization context. 30ÿ35Its relative simplicity can best be grasped when a particular example is considered.12.6.2 Iterative solution for incompressible elasticityIn this case we start from Eq. (12.11) now written with V ˆ 0, i.e., completeincompressibility is assumed. The various matrices are de®ned in (12.12), resulting

324 Incompressible materials, mixed methods and other procedures of solutionin the form A C ~uC T 0 ~p ˆf1…12:42†0Now, however, for three-dimensional problems the matrix A is singular (asvolumetric changes are not restrained) and it is necessary to augment it to make itnon-singular. We can do this in the manner described in Sec. 11.3.1, or equivalentlyby the addition of a ®ctitious compressibility matrix, thus replacing A by…A ˆ A ‡ B T …Gmm T †B d…12:43†If the second matrix uses an integration consistent with the number of discontinuouspressure parameters assumed, then this is precisely equivalent to writingA ˆ A ‡ GCC T…12:44†and is simpler to evaluate. Clearly this addition does not change the equation system.The iteration of the algorithm (12.39)±(12.41) is now conveniently taken with the`convergence accelerator' being simply de®ned asq ˆ GI…12:45†We now have the iterative system given aswherer …k† ˆ C T ~u …k†the residual of the incompressible constraint, and~p …k ‡ 1† ˆ ~p …k† ‡ Gr …k† …12:46†…12:47†~u …k ‡ 1† ˆ A ÿ1 …f 1 ÿ C~p …k ‡ 1† † …12:48†In this A can be interpreted as the sti€ness matrix of a compressible material withbulk modulus K ˆ G and the process may be interpreted as the successive additionof volumetric `initial' strains designed to reduce the volumetric strain to zero. Indeedthis simple approach led to the ®rst realization of this algorithm. 36ÿ38 Alternativelythe process can be visualized as an amendment of the original equation (12.42) bysubtractingthe term p=…G† from each side of the second to give (this is oftencalled an augmented lagrangian form) 28;34" A C# 8 9 ~u < f 1 =C T ÿ 1 G I ~p ˆ: ÿ 1; G ~p…12:49†;and adoptingthe iteration238 9A C 4C T ÿ 1G I5 ~u …k ‡ < f 1 =1†ˆ~p : ÿ 1; G ~p…k† ;…12:50†With this, on elimination, a sequence similar to Eqs (12.46)±(12.48) will beobtained provided A is de®ned by Eq. (12.44).

A simple iterative solution process for mixed problems: Uzawa method 32512.0u = 1 10.05.010 020.0λ = 1010 –5λ = 10 2||div(u)|| x10 –10λ = 10 310 –15 0 5 10λ = 10 4Round-offlimitNumber of iterationsFig. 12.9 Convergence of iterations in an extrusion problem for different values of the penalty ratio ˆ =.Startingthe iteration from~u …0† ˆ 0 and ~p …0† ˆ 0in Fig. 12.9 we show the convergence of the maximum div u computed at any of theintegrating points used. We note that this convergence becomes quite rapid for largevalues of ˆ…10 3 ±10 4 †.For smaller values the process can be accelerated by usingdi€erent q 28 but forpractical purposes the simple algorithm su ces for many problems, includingapplications in large strain. 39 Clearly much better satisfaction of the incompressibilityconstraint can now be obtained than by the simple use of a `large enough' bulkmodulus or penalty parameter. With ˆ 10 4 , for instance, in ®ve iterations the initialdiv u is reduced from the value 10 ÿ4 to 10 ÿ16 , which is at the round-o€ limit of theparticular computer used.The reader will note that the solution improvement strategy discussed in Sec. 11.6 isindeed a similar example of the above iteration process.Finally, we remind the reader that the above iterative process solves the equationsof a mixed problem. Accordingly, it is fully e€ective only when the element usedsatis®es the stability and consistency conditions of the mixed patch test.

326 Incompressible materials, mixed methods and other procedures of solution12.7 Stabilized methods for some mixed elements failingthe incompressibilitypatch test12.7.1 IntroductionIt has been observed earlier in this chapter that many of the two ®eld u±p elements donot pass the stability conditions imposed by the mixed patch test at the incompressiblelimit(or the BabusÏ ka±Brezzi conditions). Here in particular we have such methods inwhich the displacement and pressure are interpolated in an identical manner (forinstance, linear triangles, linear quadrilaterals, quadratic triangles, etc.) and manyattempts for stabilization of such elements have been introduced.The most obvious stabilized element can be directly achieved from the formulationsuggested in Fig. 12.3(b) of the triangle with a displacement bubble introduced. If thisinternal displacement is eliminated, then we have a stable element which has atriangular shape with linear displacement and pressure interpolations from nodalvalues. However, alternatives to this exist and these form several categories. The®rst category is the introduction of non-zero diagonal terms by adding a leastsquareform to the Galerkin formulation. This was ®rst suggested by Courant 40and it appears that Brezzi and Pitkaranta in 1984 41 have produced an element ofthis kind. Numerous further suggestions have been proposed by Hughes et al.between 1986 and 1989. 42ÿ44 More recently, an alternative proposal of achievingsimilar answers has been proposed by OnÄ ate 45 which gains the addition of diagonalterms by the introduction of so-called ®nite increment calculus to the formulation.There is, however, an alternative possibility introduced by time integration of thefull incompressible formulation. Here many of the algorithms will yield, whensteady-state conditions are recovered, a stabilized form. A number of such algorithmshave been discussed by Zienkiewicz and Wu in 1991 46 and more recently a verye cient method has appeared as a by-product of a ¯uid mechanics algorithmnamed the characteristic based split (CBS) procedure 47ÿ50 which will be discussed atlength in Volume 3.In the latter algorithm there exists a free parameter. This parameter depends on thesize of the time increment. In the other methods (with the exception of the bubbleformulation) there is a weighting parameter applied to the additional terms introduced.We shall discuss each of these algorithms in the following subsections andcompare the numerical results obtainable.One may question, perhaps, that resort to stabilization procedures is not worthwhilein view of the relative simplicity of the full mixed form. But this is a matterpractice will decide and is clearly in the hands of the analyst applyingthe necessarysolutions.12.7.2 Simple triangle with bubble eliminatedIn Fig. 12.3(c) we indicated that the simple triangle with C 0 linear interpolationand an added bubble for the displacements u together with continuous C 0 linear

Stabilized methods for some mixed elements failing the incompressibilitypatch test 327interpolation for the pressure p satis®ed the count test part of the mixed patch test andcan be used with success. Here we consider this element further to develop someunderstandingabout its performance at the incompressible limit.The displacement ®eld with the bubble is written asu ^u ˆ XN i ~u i ‡ N b ~u b…12:51†iwhere hereN b ˆ L 1 L 2 L 3…12:52†~u i are nodal parameters of displacement and ~u b are parameters of the hierarchicalbubble function. The pressures are similarly given byp ^p ˆ XN i ~p i…12:53†where ~p i are nodal parameters of the pressure. In the above the shape functions aregiven by (e.g., see Eq. (8.34) and (8.32))N i ˆ L i ˆ 1 …2 a i ‡ b i x ‡ c i y† …12:54†wherea i ˆ x j y k ÿ x k y j ; b i ˆ y j ÿ y k ; c i ˆ x k ÿ y jj; k are cyclic permutations of i and1 x 1 y 12 ˆ det1 x 2 y 2ˆ a 1 ‡ a 2 ‡ a 3 1 x 3 y 3The derivatives of the shape functions are thus given by@N i@x ˆ bi2iand@N i@y ˆ ci2Similarly the derivatives of the bubble are given by@N b@x ˆ 1 …2 b 1L 2 L 3 ‡ b 2 L 3 L 1 ‡ b 3 L 1 L 2 †@N b@y ˆ 1 …2 c 1L 2 L 3 ‡ c 2 L 3 L 1 ‡ c 3 L 1 L 2 †The strains may be expressed in terms of the above and the nodal parameters asy2 32 3e ˆ X b i 01 64 0 c2 iic ib i75~u i ‡ X iwhere again j; k are cyclic permutations of i.L j L k2b i 064 0 c ic ib i75~u b …12:55†y At this point it is also possible to consider the term added to the derivatives to be enhanced modes anddelete the bubble mode from displacement terms.

328 Incompressible materials, mixed methods and other procedures of solutionSubstitutingthe above strains into Eq. (12.12) and evaluatingthe integrals give23A 11 A 12 A 13 0A 21 A 22 A 23 0A ˆ 67…12:56†4 A 31 A 32 A 33 0 50 0 0 A bbwhere"A ij ˆ Gÿ ÿ#4b i b j ‡ 3c i c j 3c i b j ÿ 2b i c jÿ ÿ6 3b i c j ÿ 2c i b j 3b i b j ‡ 4c i c jA bb ˆG2160" ÿ4b T b ‡ 3c T c b T c#b T cÿ3b T b ‡ 4c T candb ˆ ‰ b 1 ; b 2 ; b 3 Š and c ˆ ‰ c 1 ; c 2 ; c 3 ŠNote in the above that all terms except A bb are standard displacement sti€nesses forthe deviatoric part. Similarly,23C 11 C 12 C 13C 21 C 22 C 23C ˆ 67…12:57†4 C 31 C 32 C 33 5C b1 C b2 C b3whereC ij ˆ 1 6b jc jand C bj ˆÿ 1 120In all the above arrays i and j have values from 1 to 3 and b denotes the bubble mode.We note that the bubble mode is decoupled from the other entries in the A array ± itis precisely for this reason that the discontinuous constant pressure case shown in Fig.12.3(b) cannot be improved by the addition of the internal parameters associated with~u b . Also, the parameters ~u b are de®ned separately for each element. Consequently, wemay perform a partial solution at the element level to obtain the set of equations in theform Eq. (12.11) where now23 23 23A 11 A 12 A 13C 11 C 12 C 13V 11 V 12 V 13A ˆ 4withandA 21 A 22 A 23A 31 A 32 A 335; C ˆ 4C 21 C 22 C 23C 31 C 32 C 33b jc j5; V ˆ 4V 21 V 22 V 23V 31 V 32 V 338 9 V ij ˆbi c i 11 >=12 2 2 2 21 ci22 >: >; …12:58†2 " ÿs ˆ 32 3b T b ‡ 4c T #c ÿb T c10Gc ÿb T ÿc 4b T b ‡ 3c T c5…12:59†

Stabilized methods for some mixed elements failing the incompressibilitypatch test 329in whichÿc ˆ 12 b T b 2‡25 ÿb T b …c T ÿc†‡12 c T 2ÿ ÿc b T 2cThe reader may recognize the V array given above as that for the two-dimensional,steady heat equation with conductivity k ˆ s and discretized by linear triangularelements. The direct reduction of the bubble matrix A bb as given above leads to ananisotropic stabilization matrix s. A diagonal form of the stabilization results if theweak form for the bubble terms is given by expressing the equilibrium equation interms of the laplacian of each displacement component and the gradient of thepressure. This is permitted only for bubble terms which vanish identically on theboundary of each element. In Sec. 12.7.4 we indicate how such a reduction couldbe performed and leave as an exercise to the reader the construction of the weakform terms and the resultingdiagonal matrix A bb . Numerical experiments indicatethat very little di€erence is achieved between the two approaches. Since the constructionof the diagonal form requires substitution of the constitutive equations into theequilibrium equation it is very limited in the type of applications which can bepursued (e.g., consideration of non-linear problems will preclude such simplesubstitution).12.7.3 An enhanced strain stabilizationIn the previous section we presented a simple two-®eld formulation usingcontinuous u and p approximations together with added bubble modes to the displacements.For more general applications this form is not the most convenient.For example, if transient problems are considered the accelerations will alsoinvolve the bubble mode and a€ect the inertial terms. We will also ®nd in the comparisonssection that use of the above bubble is not fully e€ective in eliminatingpressure oscillations in solutions. An alternative form is discussed in this section.In the alternative form we use a three-®eld approximation involving u, p and " vdiscussed in Sec. 12.4 together with an enhanced strain formulation as discussedin Sec. 11.5.3.The enhanced strains are added to those computed from displacements ase ˆ e u ‡ e e…12:60†in which e e represents a set of enhanced strain terms. The internal strain energy isrepresented byÿW…e;" v †ˆ12 eT D d e ‡ " v K" v…12:61†Usingthe above notation a Hu±Washizu type variational theorem for thedeviatoric-spherical split may be written as… ÿ me ˆ W…e;" v †‡p m T e ÿ " v ‡ r T … e u ÿ e † d ‡ ext …12:62†where ext represents the terms associated with body and traction forces.

330 Incompressible materials, mixed methods and other procedures of solutionAfter substitution for the mixed enhanced strain the last term in the integralsimpli®es as:……r T … e u ÿ e †d ˆÿ r T e e d…12:63†Takingvariations with respect to u, p, " v , e e and r yields… me ˆ u T B T ‰ D d e ‡ mpŠd ‡ ext……‡ " v ‰ K" v ÿ pŠd ‡ p m T e ÿ " v d …12:64†……‡ e T e ‰ D d e ‡ mp ÿ rŠd ‡ r T e e d ˆ 0Equal order interpolation with shape functions N are used to approximate u, p and " vasu ^u ˆ N~up ^p ˆ N~p…12:65†" v ^" v ˆ N~e vHowever, only approximations for u and p are C 0 continuous between elements. Theapproximation for " v may be discontinuous between elements. The stress r in eachelement is assumed constant. Thus, only the approximation for e e remains to beconstructed in such a way that Eq. (11.49) is satis®ed. For the present we shallassume that this approximation may be represented bye e ^e e ˆ B e ~a e…12:66†and will satisfy Eq. (11.49) so that the terms involving r and its variation in Eq. (12.64)are zero and thus do not appear in the ®nal discrete equations.With the above approximations, Eq. (12.63) may be evaluated as2389 8 9A uu A ue C u 0 ~u f 1 A eu A ee C e 0>= >=64 C T u C T 7 ˆ…12:67†e 0 ÿE 5 ~p f 2>: >; >: >;0 0 ÿE T H ~e v f 3where A uu ˆ A, C u ˆ C, f i , E and H are as de®ned in Eqs (12.12), (12.18) and (12.19)and…A ue ˆ BD d B e d ˆ A T eu…A ee ˆ B e D d B e d…12:68†…C e ˆ B e mN d

Since the approximations for " v and e e are discontinuous between elements we canagain perform a partial solution for ~e v and ~a e usingthe second and fourth row of(12.67). After eliminatingthese variables from the ®rst and third equation weagain, as in the simple triangle with bubble eliminated, obtain a form identical toEq. (12.11).As an example we consider again the three-noded triangular element with linearapproximations for N in terms of area coordinates L i . We will construct enhancedstrain terms from the derivatives of a function. The simplest such approximation isthe bubble mode used in Sec. 12.7.2 where the function is given asN e …n† ˆL 1 L 2 L 3…12:69†and the enhanced strain part is given bye e …L i †ˆB e …L i †~a e…12:70†where ~a e are two enhanced strain parameters and B e is computed usingEq. (12.69) inthe usual strain±displacement matrix2B e ˆ64@N e@x0@N e@y30@N e@y7@N 5 e@x…12:71†The result usingEq. (12.69) is identical to the bubble mode since here we are only consideringstaticproblems in the absence of body loads. If we considered the transientcase or added body loads there would be a di€erence since the displacement in theenhanced form contains only the linear interpolations in N.While this is an admissible form we have noted above that it does not eliminate alloscillations for problems where strong pressure gradients occur. Accordingly, we alsoconsider here an alternative form resultingfrom three enhanced functionsNe i ˆ aL i ‡ L j L k…12:72†in which i; j; k is a cyclic permutation and a is a parameter to be determined. Notethat this form only involves quadratic terms and thus gives linear strains which arefully consistent with the linear interpolations for p and . The derivatives of theenhanced function are given by@Nei@x ˆ 1 2 ab i ‡ L j b k ‡ L k b jwhereStabilized methods for some mixed elements failing the incompressibilitypatch test 331@Nei@y ˆ 1 2 ac i ‡ L j c k ‡ L k c jb i ˆ y j ÿ y k and c i ˆ x k ÿ x j…12:73†and is the area of a triangular element. The requirement imposed by Eq. 11.49 givesa ˆ 1=3.

332 Incompressible materials, mixed methods and other procedures of solutionWhile the use of added enhanced modes leads to increased cost in eliminatingthe ~e vand a e parameters in Eq. (12.67) the results obtained are free of pressure oscillationsin the problems considered in Sec. 12.7.7. Furthermore, this form leads to improvedconsistency between the pressure and strain.12.7.4 A pressure stabilizationIn the ®rst part of this chapter we separated the stress into the deviatoric and pressurecomponents asr ˆ r d ‡ mpUsingthe tensor form described in Appendix B this may be written in index form as ij ˆ d ij ‡ ij pThe deviatoric stresses are related to the deviatoric strains through the relation d ij ˆ 2G" d @uiij ˆ G ‡ @u jÿ 2 @x j @x i 3 @u kij…12:74†@x kThe equilibrium equations (in the absence of inertial forces) are:@ d ij‡ @p ‡ b@x i @x j ˆ 0jSubstitutingthe constitutive equations for the deviatoric part yields the equilibriumform (assuming G is constant) @ 2 u jG ‡ 1 @ 2 u i‡ @p ‡ b@x i @x i 3 @x i @x j @x j ˆ 0…12:75†jIn intrinsic form this is given asG‰r 2 u ‡ 1 3r…div u†Š ‡ rp ‡ b ˆ 0where r 2 is the laplacian operator and r the gradient operator. The constitutiveequation (12.2) is expressed in terms of the displacement as" v ˆ @u iˆ div u ˆ 1@x i K p…12:76†where div…† is the divergence of the quantity. A single equation for pressure may bededuced from the divergence of the equilibrium equation. Accordingly, from Eq.(12.75) we obtain4G3 r2 … div u†‡r 2 p ‡ div b ˆ 0 …12:77†Upon noting(12.76) we obtain1 ‡ 4G3Kr 2 p ‡ div b ˆ 0…12:78†

Stabilized methods for some mixed elements failing the incompressibilitypatch test 333Thus, in general, the pressure must satisfy a Poisson equation, or in the absence ofbody forces, a Laplace equation. We have noted the dangers of arti®cially raisingthe order of the di€erential equation in introducingspurious solutions, however,in the context of constructingapproximate solutions to the incompressible problemthe above is useful in providingadditional terms to the weak form which otherwisewould be zero. Brezzi and Pitkaranta 41 suggested adding a weighted Eq. (12.78) toEq. (12.8) and (on settingthe body force to zero for simplicity) obtain…pm T e ÿ 1 … K p d ‡ pr 2 p d ˆ 0…12:79† eThe last term may be integrated by parts to yield a form which is more amenable tocomputation as…pm T e ÿ 1 … K p @p @pd ‡ d ˆ 0…12:80† e@x i @x iin which the resultingboundary terms are ignored. Upon discretization usingequalorder linear interpolation on triangles for u and p we obtain a form identical tothat for the bubble with the exception that s is now given bys ˆ I…12:81†On dimensional considerations with the ®rst term in Eq. (12.80) the parameter should have a value proportional to L 4 =F, where L is length and F is force.12.7.5 Galerkin least square methodIn Chapter 3, Sec. 3.12.3 we introduced the Galerkin least square (GLS) approach as amodi®cation to constructinga weak form. As a general scheme for solvingthe di€erentialequations (3.1) by a ®nite element method we may write the GLS form as……u T A…u† d ‡ A…u† T sA…u† d ˆ 0…12:82† ewhere the ®rst term represents the normal Galerkin form and the added terms arecomputed for each element individually includinga weight s to provide dimensionalbalance and scaling. Generally, the s will involve parameters which have to be selectedfor good performance. Discontinuous terms on boundaries between elements thatarise from higher order terms in A…u† are commonly omitted.The form given above has been used by Hughes 44 as a means of stabilizingthe ¯uid¯ow equations, which for the case of the incompressible Stokes problem coincide withthose for incompressible linear elasticity. For this problem only the momentumequation is used in the least square terms. After substitutingEq. (12.75) into Eq.(12.76) the momentum equation may be written as (assumingthat G and K areconstant in each element)G @2 u j@x 2 i‡ 1 ‡ G @pˆ 03K @x j…12:83†

334 Incompressible materials, mixed methods and other procedures of solutionA more convenient form results by usinga single parameter de®ned asGG ˆ…12:84†1 ‡ G=3KWith this form the least square term to be appended to each element may be written asG… @2 u ie@x 2 ‡ @p k@x ijG @2 u ji @x 2 ‡ @p d…12:85†m @x jThis leads to terms to be added to the standard Galerkin equations and is expressed asA s C s ~u~pwhere…A s ij ˆ…C s ij ˆ…Vij s ˆC s;T e e eV sG 2 r 2 N i sr 2 N j dGr 2 N i srN j d… rN i † T srN j dand the operators on the shape functions are given in two dimensions byr 2 N i ˆ @2 N i‡ @2 N i@x 2 1rN i ˆ @N i@x 1@x 2 2 T@N i@x 2Note again that all in®nite terms between elements are ignored.For linear triangular elements the second derivatives of the shape functions are identicallyzero within the element and only the V term remains and is now nearly identical to theform obtained by eliminatingthe bubble mode. In the work of Hughes et al, s is given bys ˆÿ h22G I…12:86†where is a parameter which is recommended to be of O…1† for linear triangles andquadrilaterals.12.7.6 Incompressibility by time steppingThe fully incompressible case (i.e., K ˆ1) has been studied by Zienkiewicz and Wu 46usingvarious time steppingprocedures. Their applications concern the solution of¯uid problems in which the rate e€ects for the Stokes problem appear as ®rst derivativesof time. We can consider such a method here as a procedure to obtain the staticsolutions of elasticity problems in the limit as the rate terms become zero. Thus, thisapproach is considered here as a method for either the Stokes problem or the case ofstatic incompressible elasticity.

Stabilized methods for some mixed elements failing the incompressibilitypatch test 335The governing equations for slightly compressible Stokes ¯ow may be written as@u i0@t ÿ @d ijÿ @p ˆ 0…12:87†@x j @x i1 @p 0 c 2 @t ÿ @u iˆ 0…12:88†@x iwhere 0 is density (taken as unity in subsequent developments), c ˆ…K= 0 † 1=2 is thespeed of compressible waves, p is the pressure (here taken as positive in tension), andu i is a velocity (or for elasticity interpretations a displacement) in the i-coordinatedirection. Note that the above form assumes some compressibility in order tointroduce the pressure rate term. At the steady limit this term is not involved,consequently, the solution will correspond to the incompressible case. Deviatoricstresses d ij are related to deviatoric strains (or strain rates for ¯uids) as describedby Eq. (12.74).Zienkiewicz and Wu consider many schemes for integrating the above equationsin time. Here we introduce only one of the forms, which will also be used in thesolution of the ¯uid equations which include transport e€ects (see Volume 3). Forthe full ¯uid equations the algorithm is part of the characteristic based split (CBS)method. 47ÿ50The equations are discretized in time usingthe approximations u…t n †u n and timederivatives@u i@t un ‡ 1 ÿ u n…12:89†twhere t ˆ t n ‡ 1 ÿ t n . The time discretized equations are given byu n ‡ 1iÿ u n itˆ @d;n ij@x j‡ @pn@x i‡ 2@p@x i…12:90†1 p n ‡ 1 ÿ p nc 2 ˆ @un i @u‡ it @x 1 …12:91†i @x iwhere p ˆ p n ‡ 1 ÿ p n ; u i ˆ u n i‡ 1 ÿ u n i ; 1 can vary between 1=2 and 1; and 2 canvary between 0 and 1. In all that follows we shall use 1 ˆ 1.The form to be considered uses a split of the equations by de®ningan intermediateapproximate velocity u i at time t n ‡ 1 when integrating the equilibrium equation(12.90). Accordingly, we consideru n ‡ 1iu i ÿ u n itÿ u itˆ @d;n ij@x jˆ @pn@x i‡ 2@p@x iDi€erentiatingthe second of these with respect to x i to get the divergence of uin ‡ 1combiningwith the discrete pressure equation (12.91) results in1 pc 2 t ÿ 2t @2 p @ 2 p nˆ t ‡ @u i@x i @x i @x i @x i @x i…12:92†…12:93†and…12:94†

336 Incompressible materials, mixed methods and other procedures of solutionThus, the original problem has been replaced by a set of three equations which need tobe solved successively.Equations (12.92), (12.93) and (12.94) may be written in a weak form usingasweighting functions u , u and p, respectively (viz. Chapter 3). They are then discretizedin space usingthe approximationsu n ^u n ˆ N u ~u n and u n ^u ˆ N u ~u nu ^u ˆ N u ~u and u ^u ˆ N u ~u p n ^p n ˆ N p ~p n and p ^p ˆ N p ~pwith similar expressions for u n ‡ 1 and p n ‡ 1 . The ®nal discrete form is given by thethree equation sets1t M u … ~u ÿ ~u n † ˆ ÿA~u n ‡ f 1 …12:95†1t M ÿu ~u n ‡ 1 ÿ ~u ˆÿC T … ~p n ‡ 2 ~p † …12:96† 1t M p ‡ 2 tH~p ˆÿC~u ÿ tH~p n ‡ f 3…12:97†In the above we have integrated by parts all the terms which involve derivatives ondeviator stress ( d ij), pressure (p) and displacements (velocities). In addition we consideronly the case where u n i ‡ 1 ˆ u i ˆ u i on the boundary ÿ u (thus requiringu i ˆ u i ˆ 0onÿ u ). Accordingly, the matrices are de®ned as……M u ˆ N T 1u N u dM p ˆ c 2 NT p N p d……A ˆ B T @N pD d B dC ˆ N @x u di…@N T ……12:98†p @N pH ˆd f @x i @x 1 ˆ N T u …t ÿ knp n † dÿi ÿ t…f 3 ˆ N T p n T u dÿÿ uin which D d are the deviatoric moduli de®ned previously. The parameter k denotes anoption on alternative methods to split the boundary traction term and is taken aseither zero or unity. We note that a choice of zero simpli®es the computation ofboundary contributions, however, some would argue that unity is more consistentwith the integration by parts.The boundary pressure actingon ÿ t is computed from the speci®ed surface tractions(t i ) and the `best' estimate for the deviator stress at step-n ‡ 1 which is givenby d;ij . Accordingly,p n ‡ 1 n it i ÿ n i d;ij n jis imposed at each node on the boundary ÿ t .

Stabilized methods for some mixed elements failing the incompressibilitypatch test 337In general we require that t < t crit where the critical time step is h 2 =2G (in whichh is the element size). Such a quantity is obviously calculated independently for eachelement and the lowest value occurringin any element governs the overall stability. Itis possible and useful to use here the value of t calculated for each element separatelywhen calculatingincompressible stabilizingterms in the pressure calculationand the overall time step elsewhere (we shall label the time increments multiplyingH in Eq. (12.97) as t int ). A ratio of ˆ t int =t greater than unity improvesconsiderably the stabilizingproperties. As Eq. (12.97) has greater stability than Eqs(12.95) and (12.96), and for 2 5 1=2 is unconditionally stable, we recommend thatthe time step used in this equation be t cr for each node. Generally a value of 2 isgood as we shall show in the examples (for details see reference 50).Equation (12.95) de®nes a value of ~u entirely in terms of known quantities at the n-step. If the mass matrix M u is made diagonal by lumping (see Chapter 17 andAppendix I) the solution is thus trivial. Such an equation is called explicit. Theequation for ~p, on the other hand depends on both M p and H and it is not possibleto make the latter diagonal easily.y It is possible to make M p diagonal using a similarmethod as that employed for M u . Thus, if 2 is zero this equation will also be explicit,otherwise it is necessary to solve a set of algebraic equations and the method for thisequation is called implicit. Once the value of ~p is known the solution for ~u n ‡ 1 isagain explicit. In practice the above process is quite simple to implement, however,it is necessary to satisfy stability requirements by limitingthe size of the timeincrement. This is discussed further in Chapter 18 and in reference 47. Here weonly wish to show the limit result as the changes in time go to zero (i.e., for a constantin time load value) and when full incompressibility is imposed.At the steady limit the solutions become~u n ˆ ~u n ‡ 1 ˆ ~u and ~p n ˆ ~p n ‡ 1 ˆ ~p …12:99†Eliminating u the discrete equations reduce to the mixed problem AC~uC T ÿt C T M ÿ1 ‡ f ˆ 0C ÿ 1 H ~p 0u…12:100†At the steady limit we again recover a term on the diagonal which stabilizes thesolution. This term is again of a Laplace equation type ± indeed, it is now the di€erencebetween two discrete forms for the Laplace equation. The term C T M ÿ1u C makesthe bandwidth of the resultingequations larger ± thus this form is di€erent from allthe previous methods discussed above.12.7.7 ComparisonsTo provide some insight into the behaviour of the above methods we consider twoexample problems. The ®rst is a problem often used to assess the performance ofy It is possible to diagonalize the matrix by solving an eigenproblem as shown in Chapter 17 ± for largeproblems this requires more e€ort than is practical.

338 Incompressible materials, mixed methods and other procedures of solutioncodes to solve steady-state Stokes ¯ow problems ± which is identical to the case forincompressible linear elasticity. The second example is a problem in nearly incompressiblelinear elasticity.Example: Driven cavity A two-dimensional plane (strain) case is considered for asquare domain with unit side lengths. The material properties are assumed to befully incompressible ( ˆ 0:5) with unit viscosity (elastic shear modulus, G, of unity).All boundaries of the domain are restrained in the x and y directions with the topboundary havinga unit tangential velocity (displacement) at all nodes except thecorner ones. Since the problem is incompressible it is necessary to prescribe the pressureat one point in the mesh ± this is selected as the centre node alongthe bottom edge. The10 10 element mesh of triangular elements (200 elements total) used for the comparisonis shown in Fig. 12.10(a). The elements used for the analysis use linear velocity(displacement) and pressure on three-noded triangles. Results are presented for thehorizontal velocity alongthe vertical centre line AA and for vertical velocity and pressurealongthe horizontal centre line BB. Three forms of stabilization are considered:1. Galerkin least square (GLS) Brezzi±Pitkaranta (BP) where the e€ect of on isassessed. The results for the horizontal velocity are given in Fig. 12.10(b) and forthe vertical velocity and pressure in Figs 12.10(c) and (d), respectively. From theanalysis it is assessed that the stabilization parameter should be about 0.5 to 1(as also indicated by Hughes et al. 44 ). Use of lower values leads to excessive oscillationin pressure and use of higher values to strong dissipation of pressure results.2. Cubic bubble (MINI) element stabilization. Results for vertical velocity are nearlyindistinguishable from the GLS results as indicated in Fig. 12.11; however, thosefor pressure show oscillation. Such oscillation has also been observed by othersalongwith some suggested boundary modi®cations. 51 No free parameters existfor this element (except possible modi®cation of the bubble mode used), thus,no arti®cial `tuning' is possible. Use of more re®ned meshes leads to a strongdecrease in the oscillation.3. Enhanced strain stabilization with quadratic modes. In Fig. 12.11 we show resultsobtained usingthe enhanced formulation presented in Eq. (12.73). These resultsare free of oscillation in pressures and require no tuningparameters. For use insolvinglinear elasticity and Stokes problems they prove to be the most robust;however, when used with other material models there are limitations in their use.4. The CBS algorithm. Finally in Fig. 12.11 we present results using the CBS solutionwhich may be compared with GLS, ˆ 0:5. Once again the reader will observethat with ˆ 2, the results of CBS reproduce very closely those of GLS, ˆ 0:5. However, in results for ˆ 1 no oscillations are observed and they arequite reasonable. This ratio for is where the algorithm gives excellent resultsin incompressible ¯ow modellingas will be demonstrated further in resultspresented in Volume 3.Example: Tension strip with slot As a second example we consider a plane strainlinear problem on a square domain with a central slot. The domain is two unitssquare and the central slot has a total width of 0.4 units and a height of 0.1 units.The ends of the slot are semicircular. Lateral boundaries have speci®ed normal

Stabilized methods for some mixed elements failing the incompressibilitypatch test 339yu = 1; v = 00.5BAB1.21.00.8GLS 0.1GLS 0.5GLS 1.0GLS 10.0xu velocity0.60.40.50.2p = 0A0.5 0.5(a) Problem definition0–0.2–0.4 –0.2 0 0.2 0.4y coordinate(b) Horizontal velocity on AAv velocity0.200.150.100.050–0.05–0.10–0.15GLS 0.1GLS 0.5GLS 1.0GLS 10.0Pressure1.51.00.50–0.5–1.0GLS 0.1GLS 0.5GLS 1.0GLS 10.0–0.20–0.4 –0.2 0 0.2 0.4x coordinate(c) Vertical velocity on BB–1.5–0.4 –0.2 0 0.2 0.4x coordinate(d) Pressure on BBFig. 12.10 Mesh and GLS/Brezzi±Pitkaranta results.displacement and zero tangential traction. The top and bottom boundaries areuniformly stretched by a uniform axial loadingand lateral boundaries are maintainedat zero displacement. We consider the linear elastic problem with elastic propertiesE ˆ 24 and ˆ 0:499995; thus, giving a nearly incompressible situation. An unstructuredmesh of triangles is constructed as shown in Fig. 12.12(b). Results for thepressure alongthe horizontal and vertical centre lines (i.e., the x and y axes) arepresented in Figs 12.13(a) and 12.13(b) and the distribution of the vertical displacementis shown in Fig. 12.13(c). We note that the results for this problem cause verystronggradients in stress near the ends of the slot. The mesh used for the analysisis not highly re®ned in this region and hence results from di€erent analyses can be

340 Incompressible materials, mixed methods and other procedures of solution0.200.150.10GLS/BP (α = 0.5)BubbleEnhancedCBS (γ = 1)CBS (γ = 2)0.05v velocity0–0.05–0.10–0.15–0.20–0.5 –0.4 –0.3 –0.2 –0.1 0 0.1 0.2 0.3 0.4 0.5(a)x coordinateFig. 12.11 Vertical velocity and pressure for driven cavity problem.

Stabilized methods for some mixed elements failing the incompressibilitypatch test 341yσ y = 11.0x1.00.050.30.051.0 1.0(a) Problem descriptiony(b) Mesh for quadrantxFig. 12.12 Region and mesh used for slotted tension strip.expected to di€er in this region. The results obtained using all formulations aresimilar in distribution. However, the bubble form does show some oscillations inpressure indicatingthat the stabilization achieved is not completely adequate. Resultsfor the CBS algorithm show an oscillation in the pressure along the x-axis at theboundary of the slot. This is caused, we believe, by an inadequate resolution ofthe pressure condition at this point of the curved boundary. In general, however,

342 Incompressible materials, mixed methods and other procedures of solutionPressure3.02.52.01.51.00.500GLS/BP (α = 0.5)BubbleEnhancedCBS (γ = 2)0.2 0.4 0.6 0.8 1.0x coordinatey coordinate1.00.90.80.70.60.50.40.30.20.100GLS/BP (α = 0.5)BubbleEnhancedCBS (γ = 2)0.5 1.0 1.5Pressure(a) Pressure at x-axis(b) Pressure at y-axisv displacement (x10 –3 )4.03.53.02.52.01.51.00.500GLS/BP (α = 0.5)BubbleEnhancedCBS (γ = 2)0.2 0.4 0.6 0.8 1.0x coordinate(c) Displacement on x-axisFig. 12.13 Pressures and displacements for slot problems.the results achieved with all forms are satisfactory and indicate that stabilizedmethods may be considered for use in problems where constraints, such as incompressibility,are encountered.12.8 Concluding remarksIn this chapter we have considered in some detail the application of mixed methods toincompressible problems and also we have indicated some alternative procedures.The extension to non-isotropic problems and non-linear problems will be presentedin Volume 2, but will follow similar lines. In Volume 3 we shall note how importantthe problem is in the context of ¯uid mechanics and it is there that much of theattention to it has been given.In concludingthis chapter we would like to point out two matters:1. The mixed formulation discovers immediately the non-robustness of certainirreducible (displacement) elements and, indeed, helps us to isolate those which

perform well from those that do not. Thus, it has merit which as a test is applicableto many irreducible forms at all times.2. In elasticity, certain mixed forms work quite well at the near incompressible limitwithout resort to splits into deviatoric and mean parts. These include the two-®eldquadrilateral element of Pian±Sumihara and the enhanced strain quadrilateralelement of Simo±Rifai which were presented in the previous chapter. There wenoted how well such elements work for Poisson's ratio approachingone-half ascompared to the standard irreducible element of a similar type.References 343References1. L.R. Herrmann. Finite element bendinganalysis of plates. In Proc. 1st Conf. MatrixMethods in Structural Mechanics. AFFDL-TR-66-80, pages 577±602, Wright-PattersonAir Force Base, Ohio, 1965.2. S.W. Key. Variational principle for incompressible and nearly incompressibly anisotropicelasticity. Intern. J. Solids Struct., 5, 951±964, 1969.3. O.C. Zienkiewicz, R.L. Taylor, and J.A.W. Baynham. Mixed and irreducible formulationsin ®nite element analysis. In S.N. Atluri, R.H. Gallagher, and O.C. Zienkiewicz,editors, Hybrid and Mixed Finite Element Methods, pages 405±431. John Wiley &Sons, 1983.4. D.N. Arnold, F. Brezzi, and M. Fortin. A stable ®nite element for the Stokes equations.Calcolo, 21, 337±344, 1984.5. M. Fortin and N. Fortin. Newer and newer elements for incompressible ¯ow. In R.H.Gallagher, G.F. Carey, J.T. Oden, and O.C. Zienkiewicz, editors, Finite Elements inFluids, volume 6, chapter 7, pages 171±188. John Wiley & Sons, 1985.6. J.T. Oden. R.I.P. methods for Stokesian ¯ow. In R.H. Gallagher, D.N. Norrie, J.T. Oden,and O.C. Zienkiewicz, editors, Finite Elements in Fluids, volume 4, chapter 15, pages 305±318. John Wiley & Sons, 1982.7. M. Crouzcix and P.A. Raviart. Conformingand non-conforming®nite element methodsfor solvingstationary Stokes equations. RAIRO, 7-R3, 33±76, 1973.8. D.S. Malkus. Eigenproblems associated with the discrete LBB condition for incompressible®nite elements. Int. J. Eng. Sci., 19, 1299±1370, 1981.9. M. Fortin. Old and new ®nite elements for incompressible ¯ow. International Journal forNumerical Methods in Fluids, 1, 347±364, 1981.10. C. Taylor and P. Hood. A numerical solution of the Navier-Stokes equations usingthe®nite element technique. Comput. Fluids, 1, 73±100, 1973.11. T.J.R. Hughes. Generalization of selective integration procedures to anisotropic and nonlinearmedia. Intern. J. Num. Meth. Engng, 15, 1413±1418, 1980.12. J.C. Simo, R.L. Taylor, and K.S. Pister. Variational and projection methods for thevolume constraint in ®nite deformation plasticity. Comput. Meth. Appl. Mech. Engng,51, 177±208, 1985.13. J.C. Simo and T.J.R. Hughes. Computational Inelasticity, volume 7 of InterdisciplinaryApplied Mathematics. Springer-Verlag, Berlin, 1998.14. D.J. Naylor. Stresses in nearly incompressible materials for ®nite elements with applicationto the calculation of excess pore pressures. Intern. J. Num. Meth. Engng, 8, 443±460,1974.15. O.C. Zienkiewicz and P.N. Godbole. Viscous incompressible ¯ow with special reference tonon-Newtonian (plastic) ¯ows. In R.H. Gallagher et al., editor, Finite Elements in Fluids,volume 1, chapter 2, pages 25±55. John Wiley & Sons, 1975.

344 Incompressible materials, mixed methods and other procedures of solution16. T.J.R. Hughes, R.L. Taylor, and J.F. Levy. High Reynolds number, steady, incompressible¯ows by a ®nite element method. In R.H. Gallagher et al., editor, Finite Elements in Fluids,volume 3. John Wiley & Sons, 1978.17. O.C. Zienkiewicz, J. Too, and R.L. Taylor. Reduced integration technique in generalanalysis of plates and shells. Intern. J. Num. Meth. Engng, 3, 275±290, 1971.18. S.F. Pawsey and R.W. Clough. Improved numerical integration of thick slab ®niteelements. Intern. J. Num. Meth. Engng, 3, 575±586, 1971.19. O.C. Zienkiewicz and E. Hinton. Reduced integration, function smoothing and nonconformityin ®nite element analysis. J. Franklin Inst., 302, 443±461, 1976.20. O.C. Zienkiewicz. The Finite Element Method. McGraw-Hill, London, 3rd edition, 1977.21. D.S. Malkus and T.J.R. Hughes. Mixed ®nite element methods in reduced and selectiveintegration techniques: A uni®cation of concepts. Comput. Meth. Appl. Mech. Engng, 15,63±81, 1978.22. O.C. Zienkiewicz and S. Nakazawa. On variational formulations and its modi®cation fornumerical solution. Comput. Struct., 19, 303±313, 1984.23. M.S. Engleman, R.L. Sani, P.M. Gresho, and H. Bercovier. Consistent vs. reduced integrationpenalty methods for incompressible media usingseveral old and new elements.Internat. J. Num. Meth. Fluids, 2, 25±42, 1982.24. D.N. Arnold. Discretization by ®nite elements of a model parameter dependent problem.Num. Meth., 37, 405±421, 1981.25. J.C. Nagtegaal, D.M. Parks, and J.R. Rice. On numerical accurate ®nite element solutionsin the fully plastic range. Comput. Meth. Appl. Mech. Engng, 4, 153±177, 1974.26. M. Vogelius. An analysis of the p-version of the ®nite element method for nearlyincompressible materials; uniformly optimal error estimates. Num. Math., 41, 39±53,1983.27. S.W. Sloan and M.F. Randolf. Numerical prediction of collapse loads using®nite elementmethods. Internat. J. Num. Anal. Meth. Geomech., 6, 47±76, 1982.28. O.C. Zienkiewicz, J.P. Vilotte, S. Toyoshima, and S. Nakazawa. Iterative method for constrainedand mixed approximation. An inexpensive improvement of FEM performance.Comput. Meth. Appl. Mech. Engng, 51, 3±29, 1985.29. K.J. Arrow, L. Hurwicz, and H. Uzawa. Studies in Non-Linear Programming. StanfordUniversity Press, Stanford, CA, 1958.30. M.R. Hestenes. Multiplier and gradient methods. J. Opt. Theory Appl., 4, 303±320,1969.31. M.J.D. Powell. A method for nonlinear constraints in minimization problems. In R.Fletcher, editor, Optimization. Academic Press, London, 1969.32. C.A. Felippa. Iterative procedure for improvingpenalty function solutions of algebraicsystems. Intern. J. Num. Meth. Engng, 12, 165±185, 1978.33. M. Fortin and F. Thomasset. Mixed ®nite element methods for incompressible ¯ow problems.J. Comp. Physics, 31, 113±145, 1973.34. M. Fortin and R. Glowinski. Augmented Lagrangian Methods: Applications to NumericalSolution of Boundary- Value Problems. North-Holland, Amsterdam, 1983.35. D.G. Luenberger. Linear and Nonlinear Programming. Addison-Wesley, Reading, Mass.1984.36. J. H. Argyris. Three-dimensional anisotropic and inhomogeneous media ± matrix analysisfor small and large displacements. Ingenieur Archiv, 34, 33±55, 1965.37. O.C. Zienkiewicz and S. Valliappan. Analysis of real structures for creep plasticity andother complex constitutive laws. In M. Te'eni, editor, Structure of Solid Mechanics andEngineering Design, volume Part 1, pages 27±48. 1971.38. O.C. Zienkiewicz. The Finite Element Method in Engineering Science. McGraw-Hill,London, 2nd edition, 1971.

References 34539. J.C. Simo and R.L. Taylor. Quasi-incompressible ®nite elasticity in principal stretches:Continuum basis and numerical algorithms. Comput. Meth. Appl. Mech. Engng, 85,273±310, 1991.40. R. Courant. Variational methods for the solution of problems of equilibrium and vibration.Bull. Amer. Math. Soc., 49, 1±61, 1943.41. F. Brezzi and J. PitkaÈranta. On the stabilization of ®nite element approximations of theStokes problem. In W. Hackbusch, editor, E cient solution of Elliptic Problems, Noteson Numerical Fluid Mechanics, volume 10. Vieweg, Wiesbaden, 1984.42. T.J.R. Hughes, L.P. Franca, and M. Balestra. A new ®nite element formulation forcomputational ¯uid dynamics: V. Circumventingthe BabusÏ ka-Brezzi condition: A stablePetrov-Galerkin formulation of the Stokes problem accommodatingequal-order interpolations.Comput. Meth. Appl. Mech. Engng, 59, 85±99, 1986.43. T.J.R. Hughes and L.P. Franca. A new ®nite element formulation for computational ¯uiddynamics: VII. The Stokes problem with various well-posed boundary conditions:Symmetric formulation that converge for all velocity/pressure spaces. Comput. Meth.Appl. Mech. Engng, 65, 85±96, 1987.44. T.J.R. Hughes, L.P. Franca, and G.M. Hulbert. A new ®nite element formulation for computational¯uid dynamics: VIII. The Galerkin/least-squares method for advective-di€usiveequations. Comput. Meth. Appl. Mech. Engng, 73, 173±189, 1989.45. E. Onate. Derivation of stabilized equations for numerical solution of advective-di€usivetransport and ¯uid ¯ow problems. Comput. Meth. Appl. Mech. Engng, 151, 233±265, 1998.46. O.C. Zienkiewicz and J. Wu. Incompressibility without tears! How to avoid restrictions ofmixed formulations. Intern. J. Num. Meth. Engng, 32, 1184±1203, 1991.47. O.C. Zienkiewicz and R. Codina. A general algorithm for compressible and incompressible¯ow ± Part 1: The split, characteristic-based scheme. Internat. J. Num. Meth. Fluids, 20,869±885, 1995.48. O.C. Zienkiewicz, P. Nithiarasu, R. Codina, M. Vazquez, and P. Ortiz. The characteristicbased-splitprocedure: An e cient and accurate algorithm for ¯uid problems. Internat. J.Num. Meth. Fluids, 31, 359 ± 392, 1999.49. O.C. Zienkiewicz, K. Morgan, B.V.K. Satya Sai, R. Codina, and M. Vasquez. A generalalgorithm for compressible and incompressible ¯ow Part II: Tests on the explicit form.Internat. J. Num. Meth. Fluids, 20, 887±913, 1995.50. P. Nithiarasu and O.C. Zienkiewicz. On stabilization of the CBS algorithm. Internal andexternal time steps. Intern. J. Num. Meth. Engng, 48, 875±880.51. R. Pierre. Simple c 0 approximations for the computation of incompressible ¯ows. Comput.Meth. Appl. Mech. Engng, 68, 205±227, 1988.

13Mixed formulation and constraints± incomplete (hybrid) ®eldmethods, boundary/Trefftz methods13.1 GeneralIn the previous two chapters we have assumed in the mixed approximation that all thevariables were de®ned and approximated in the same manner throughout the domainof the analysis. This process can, however, be conveniently abandoned on occasionwith di€erent formulations adopted in di€erent subdomains and with some variablesbeing only approximated on surfaces joining such subdomains. In this part we shalldiscuss such incomplete or partial ®eld approximations which include various socalledhybrid formulations.In all the examples given here we shall consider elastic solid body approximationsonly, but extension to the heat transfer or other ®eld problems, etc., can be readilymade as a simple exercise following the procedures outlined.13.2 Interface traction link of two (or more) irreducibleform subdomainsOne of the most obvious and frequently encountered examples of an `incomplete ®eld'approximation is the subdivision of a problem into two (or more) subdomains in eachof which an irreducible (displacement) formulation is used. Independently approximatedLagrange multipliers (tractions) are used on the interface to join the subdomains,as in Fig. 13.1(a).In this problem we formulate the approximation in domain 1 in terms of displacementsu 1 and the interface tractions t 1 ˆ k. With the weak form using the standardvirtual work expression [see Eqs (11.22)±(11.24)] we have……………Su 1 † T D 1 Su 1 d ÿ u 1T k dÿ ÿ u 1T b d ÿ u 1T t dÿ ˆ 0 …13:1† 1 ÿ I 1in which as usual we assume that the satisfaction of the prescribed displacement onÿ u 1 is implied by the approximation for u 1 . Similarly in domain 2 we can write,now putting the interface traction as t 2 ˆÿk to ensure equilibrium between theÿ 1 t

Interface traction link of two (or more) irreducible form subdomains 347Γ 1 (t = t) t(b)Variable u 1Γ 1 tΩ 1Γ lt 2t 1Γ 1 u(u = u)Variable u 1and 1 Ωσ1Γ lt 2t 1Γ 1 ut 1 = – t 2 = λon interfaceΩ 2Ω 2Γ 2 uΓ 2 u(a)Fig. 13.1 Linking of two (or more) domains by traction variables de®ned only on the interfaces. (a) Variablesin each domain are displacements u (internal irreducible form). (b) Variables in each domain are displacementsand stresses r±u (mixed form).two domains,……………Su 2 † T D 2 Su 2 d ‡ 2 u 2T k dÿ ÿÿ Iu 2T b d ÿ 2ÿ 2 tu 2T t dÿ ˆ 0…13:2†The two subdomain equations are completed by a weak statement of displacementcontinuity on the interface between the two domains, i.e.,…k T …u 2 ÿ u 1 † dÿ ˆ 0…13:3†ÿ IDiscretization of displacements in each domain and of the tractions k on theinterface yields the ®nal system of equations. Thus putting the independentapproximations aswe haveu 1 ˆ N u 1 u 1u 2 ˆ N u 2 u 2k ˆ N k k2389 8 9K 1 0 Q 1 >< ~u 1 >= >< f 1 >=64 0 K 2 Q 2 75 ~u 2>: >; ˆ f 2 >: >;Q 1T Q 2T 0 ~k 0…13:4†…13:5†…13:6†…13:7a†

348 Mixed formulation and constraintswhere…K 1 ˆ B 1T D 1 B 1 d 1…K 2 ˆ B 2T D 2 B 2 d 2 …Q 1 ˆÿ N T u 1N dÿÿ 1…Q 2 ˆ N T u 2N dÿÿ 1……f 1 ˆ N T 1 u 1b1 dÿ ‡……f 2 ˆ N T 2 u 2b2 dÿ ‡ÿ 1 tÿ 2 tN T u 1 t 1 dÿN T u 2 t 2 dÿ…13:7b†We note that in the derivation of the above matrices the shape function N andhence k itself are only speci®ed along the interface line ± hence complying with ourde®nition of partial ®eld approximation.The formulation just outlined can obviously be extended to many subdomains andin many cases of practical analysis is useful in ensuring a better matrix conditioningand allowing the solution to be obtained with reduced computational e€ort. 1The variables u 1 and u 2 , etc., appear as internal variables within each subdomain(or superelement) and can be eliminated locally providing the matrices K 1 and K 2are non-singular. Such non-singularity presupposes, however, that each of the subdomainshas enough prescribed displacements to eliminate rigid body modes. Ifthis is not the case partial elimination is always possible, retaining the rigid bodymodes until the complete solution is achieved.The process described here is very similar to that introduced by Kron 2 ataveryearlydate and, more recently, used by Farhat et al. 3 in the FETI method which uses the processon many individual element partitions as a means of iteratively solving large problems.The formulation just used can, of course, be applied to a single ®eld displacement formulationin which we are required to specify the displacement on the boundaries in aweak sense (rather than imposing these directly on the displacement shape functions).This problem can be approached directly or can be derived simply via the ®rstequation of (13.7a) in which we put u 2 ˆ u, the speci®ed displacement on ÿ I .Now the equation system is simply" #( )K 1 Q 1 ~u 1 Q 1T ˆf1…13:8†0 ~kwhere…f ˆÿ N T u dÿ…13:9†ÿ IThis formulation is often convenient for imposing a prescribed displacement on adisplacement element ®eld when the boundary values cannot ®t the shape function ®eld.f

Interface traction link of two or more mixed form subdomains 349We have approached the above formulation directly via weak forms or weightedresiduals. Of course, a variational principle could be given here simply as the minimizationof total potential energy (see Chapter 2) subject to a Lagrange multiplier kimposing subdomain continuity. The stationarity of…… … … ˆ 12…Su† T D…Su† d ÿ u T b d ÿ u T t dÿ ‡ k T …u 1 ÿ u 2 † dÿ …13:10†ÿ t ÿ Iwould result in the equation set (13.1)±(13.3). The formulation is, of course, subject tolimitations imposed by the stability and consistency conditions of the mixed patch testfor selection of the appropriate number of k variables.13.3 Interface traction link of two or more mixed formsubdomainsThe problem discussed in the previous section could of course be tackled by assuminga mixed type of two-®eld approximation …r=u† in each subdomain, as illustrated inFig. 13.1(b).Now in each subdomain variables u and r will appear, but the linking will becarried out again with the interface traction k.We now have, using the formulation of Sec. 11.4.2 for domain 1 [see Eqs (11.29)and (11.22)],…r 1T ‰…D 1 † ÿ1 r 1 ÿ Su 1 Š d ˆ 0 …13:11a† 1……………Su 1 † T r 1 d ÿ u 1T k dÿ ÿ u 1T b d ÿ u 1T t dÿ ˆ 0 …13:11b† 1 ÿ I 1ÿ 1 tand for domain 2 similarly…r 2T ‰…D 2 † ÿ1 r 2 ÿ Su 2 Š d ˆ 0 2……………Su 2 † T r 2 d ‡ u 2T k dÿ ÿ u 2T b d ÿ u 2T t dÿ ˆ 0 2 ÿ I 2ÿ 2 t…13:12a†…13:12b†With interface tractions in equilibrium the restoration of continuity demands that…k T …u 2 ÿ u 1 † dÿ ˆ 0…13:13†ÿ IOn discretization we now haveu 1 ˆ N u 1~u 1 u 2 ˆ N u 2~u 2r 1 ˆ N 1 ~r 1 r 2 ˆ N 2 ~r 2k ˆ N k ~

350 Mixed formulation and constraintsand2389 8 9A 1 C 1 0 0 0 ~r 1 f 1 1C 1T 0 0 0 Q 1>< ~u 1 >= >< f 2 1 >=0 0 A 2 C 2 0~r 2 ˆ f 1 264 0 0 C 2T 0 Q 2 75 ~u>:2 f>;2 2>: >;0 Q 1T 0 Q 2T 0 ~k 0…13:14†with A, C, f 1 , and f 2 de®ned similarly to Eq. (11.32) with appropriate subdomainsubscripts and Q 1 and Q 2 given as in (13.7b).All the remarks made in the previous section apply here once again ± though use ofthe above form does not appear frequently.13.4 Interface displacement `frame'13.4.1 GeneralIn the preceding examples we have used traction as the interface variable linking twoor more subdomains. Due to lack of rigid body constraints the elimination of localsubdomain displacements has generally been impossible. For this and other reasonsit is convenient to accomplish the linking of subdomains via a displacement ®eldde®ned only on the interface [Fig. 13.2(a)] and to eliminate all the interior variablesso that this linking can be accomplished via a standard sti€ness matrix procedureusing only the interface variables.The displacement frame can be made to surround the subdomain completely and ifall internal variables are eliminated will yield a sti€ness matrix of a new `element'Γ 1 lΓ 1 lΩ 1Ω 1Γ lInterface frameu = v ≈ N l vNodes defining v(a)(b)Fig. 13.2 Interface displacement ®eld speci®ed on a `frame' linking subdomains: (a) two-domain link; (b) a`superelement' (hybrid) which can be linked to many other similar elements.

which can be used directly in coupling with any other element with similar displacementassumptions on the interface, irrespective of the procedure used for derivingsuch an element [Fig. 13.2(b)].In all the examples of this section we shall approximate the frame displacements asv ˆ N v ~v on ÿ I …13:15†and consider the `nodal forces' contributed by a single subdomain 1 to the `nodes' onthis frame. Using virtual work (or weak) statements we have with discretization…N T v t dÿ ˆ q 1…13:16†ÿ 1 twhere t are the tractions the interior exerts on the imaginary frame and q 1 are thenodal forces developed. The balance of the nodal forces contributed by each subdomainnow provides the weak condition for traction continuity.As ®nally the tractions t can be expressed in terms of the frame parameters ~v only,we shall arrive atq 1 ˆ K 1 ~v ‡ f 1 0…13:17†where K 1 is the sti€ness matrix of the subdomain 1 and f 1 0 its internally contributed`forces'.From this point onwards the standard assembly procedures are valid and the subdomaincan be treated as a standard element which can be assembled with others byensuring thatXq j ˆ 0…13:18†jwhere the sum includes all subdomains (elements!). We thus have only to consider asingle subdomain in what follows.13.4.2 Linking two or more mixed form subdomainsWe shall assume as in Sec. 13.3 that in each subdomain, now labelled e for generality,the stresses r e and displacements u e are independently approximated. The equations(13.11) are rewritten adding to the ®rst the weak statement of displacement continuity.We now have in place of (13.11a) and (13.13) (dropping superscripts)……r T …D ÿ1 r ÿ Su† d ÿ t T …u ÿ v† dÿ ˆ 0 …13:19† e ÿ I eEquation (13.11b) will be rewritten as the weighted statement of the equilibriumrelation, i.e.,……ÿ u T …S T r ‡ b† d ‡ u T …t ÿ t† dÿ ˆ 0 e ÿ t eor, after integration by parts…………Su† T r d ÿ u T b d ÿ e eÿ I e…u T t dÿ ÿ u T t dÿ ˆ 0ÿ t eInterface displacement `frame' 351…13:20†

352 Mixed formulation and constraintsIn the above, t are the tractions corresponding to the stress ®eld r [see Eq. (11.30)]:t ˆ Gr…13:21†In what follows ÿ t e, i.e, the boundary with prescribed tractions, will generally be takenas zero.On approximating Eqs (13.19), (13.20) and (13.16) withu ˆ N u ~u r ˆ N ~r and v ˆ N v ~vwe can write, using Galerkin weighting and limiting the variables to the `element' e,2A e C e Q e 38~r e 9 8 9>< >= >< 0 >=64 C eT 70 0 5 ~u e>: >; ˆ f e…13:22a†>: >;Q eT 0 0 ~v q ewhere…A e ˆ N T D ÿ1 N d e……C e ˆ N T B d ÿ …GN † T N u dÿ e ÿ I e…Q e ˆ …GN † T N v dÿÿ I e…f e ˆ N T u b d e…13:22b†Elimination of ~r e and ~u e from the above yields the sti€ness matrix of the elementand the internally contributed force [see Eq. (13.17)].Once again we can note that the simple stability criteria discussed in Chapter 11 willhelp in choosing the number of r, u, and v parameters. As the ®nal sti€ness matrix ofan element should be singular for three rigid body displacements we must have [byEq. (11.18)]n 5 n u ‡ n v ÿ 3…13:23†in two-dimensional applications.Various alternative variational forms of the above formulation exist. A particularlyuseful one is developed by Pian et al. 4;5 In this the full mixed representation can bewritten completely in terms of a single variational principle (for zero body forces)and no boundary of type ÿ t present:……1 ˆÿ2 rDÿ1 r d ÿ……S T r† T u I d ‡ r T Sv d…13:24†In the above it is assumed that the compatible ®eld of v is speci®ed throughout theelement domain and not only on its interfaces and u I stands for an incompatible®eld de®ned only inside the element domain.yy In this form, of course, the element could well ®t into Chapter 11 and the subdivision of hybrid andmixed forms is not unique here.

We note that in the present de®nitionu ˆ u I ‡ vInterface displacement `frame' 353…13:25†To show the validity of this variational principle, which is convenient as no interfaceintegrals need to be evaluated, we shall derive the weak statement correspondingto Eqs (13.19) and (13.20) using the condition (13.25).We can now write in place of (13.19) (noting that for interelement compatibility wehave to ensure that u I ˆ 0 on the interfaces)………r T …D ÿ1 r ÿ Sv† d ÿ r T Su I d ‡ t T u I dÿ ˆ 0 e e ÿ I e…13:26†After use of Green's theorem the above becomes simply……r T …D ÿ1 r ÿ Sv† d ‡ e …S T r† T u I dÿ ˆ 0 e …13:27†In place of (13.20) we write (in the absence of body forces b and boundary ÿ t )…u T I …S T r† d ‡ e …v T …S T r† d ˆ 0 e …13:28†and again after use of Green's theorem……u T I S T r d ÿ e …Sv† T r d ˆ 0 e …if v ˆ 0 on ÿ I † …13:29†These equations are precisely the variations of the functional (13.24).Of course, the procedure developed in this section can be applied to other mixed orirreducible representations with `frame' links. Tong and Pian 6;7 developed severalalternative element forms by using this procedure.13.4.3 Linking of equilibrating form subdomainsIn this form we shall assume a priori that the stress ®eld expansion is such thatr T ˆ r ‡ r 0…13:30†and that the equilibrium equations are identically satis®ed. ThusS T r 0; S T r 0 b in and Gr ˆ 0; Gr 0 ˆ t on ÿ t eIn the absence of ÿ t e, Eq. (13.20) is identically satis®ed and we write (13.19) as (seeChapter 11, Sec. 11.7)……r T …D ÿ1 r T ÿ Su† d ‡ t T …u ÿ v† dÿ e ÿ I e…… r T D ÿ1 …r ‡ r 0 † d ÿ …Gr† T v dÿ ˆ 0 …13:31† e ÿ I eOn discretization, noting that the ®eld u does not enter the problemr ˆ N ~r v ˆ N v ~v

354 Mixed formulation and constraintswe have, on including Eq. (13.16)Q e f e 1Q eT 0 ˆq e ÿ f e 2 A e where……A e ˆ N D ÿ1 N d f 1 ˆ e …N D ÿ1 r 0 d eQ e ˆ …GN † T N v dÿand…f e 2 ˆÿ I eÿ I eN v Gr 0 dÿ…13:32†Here elimination of ~r is simple and we can write directlyK e ~v ˆ q e ÿ f e 2 ÿ Q eT …A e † ÿ1 f e 1 and K e ˆ Q eT …A e † ÿ1 Q e …13:33†In Sec. 11.7 we have discussed the possible equilibration ®elds and have indicatedthe di culties in choosing such ®elds for a ®nite element, subdivided, ®eld. In thepresent case, on the other hand, the situation is quite simple as the parametersdescribing the equilibrating stresses inside the element can be chosen arbitrarily ina polynomial expression.For instance, if we use a simple polynomial expression in two dimensions: x ˆ 0 ‡ 1 x ‡ 2 y y ˆ 0 ‡ 1 x ‡ 2 y…13:34† xy ˆ 0 ‡ 1 x ‡ 2 ywe note that to satisfy the equilibrium we require23@ @0 S T @x @yr ˆ 674 @ @ 5 r ˆ 1 ‡ 2ˆ 0…13:35†02 ‡ 1@y @xand this simply means 2 ˆÿ 1 1 ˆÿ 2Thus a linear expansion in terms of 9 ÿ 2 ˆ 7 independent parameters is easilyachieved. Similar expansions can of course be used with higher order terms.It is interesting to observe that:1. n 5 n v ÿ 3 is needed to preserve stability.2. By the principle of limitation, the accuracy of this approximation cannot be betterthan that achieved by a simple displacement formulation with compatible expansionof v throughout the element, providing similar polynomial expressions arise instress component variations.

Linking of boundary (or Trefftz)-type solution by the `frame' of speci®ed displacements 355However, in practice two advantages of such elements, known as hybrid-stresselements, are obtained. In the ®rst place it is not necessary to construct compatibledisplacement ®elds throughout the element (a point useful in their application to,say, a plate bending problem). In the second for distorted (isoparametric) elementsit is easy to use stress ®elds varying with the global coordinates and thus achievehigher order accuracy.The ®rst use of such elements was made by Pian 8 and many successful variants arein use today. 9ÿ2213.5 Linking of boundary (or Trefftz)-type solution by the`frame' of speci®ed displacementsWe have already referred to boundary (Tre€tz)-type solutions 23 earlier (Chapter 3).Here the chosen displacement/stress ®elds are such that a priori the homogeneousequations of equilibrium and constitutive relation are satis®ed indentically in thedomain under consideration (and indeed on occasion some prescribed boundarytraction or displacement conditions).Thus in Eqs (13.19) and (13.20) the subdomain (element e) e integral termsdisappear and, as the internal t and u variations are linked, we combine all into asingle statement (in the absence of body force terms) as……ÿ t T …u ÿ v† dÿ ÿ u T …t ÿ t† dÿ ˆ 0…13:36†ÿ I eThis coupled with the boundary statement (13.16) provides the means of devisingsti€ness matrix statements of such subdomains.For instance, if we express the approximate ®elds asu ˆ N~a…13:37†implyingr ˆ D…SN†~a and t ˆ Gr ˆ GD…SN†~awe can write in place of (13.22) ÿH e Q e ~aQ eT 0 ~vwhere…H e ˆ…Q e ˆÿ I eÿ I e…f e 1 ˆÿÿ t eˆ f e 1q…‰GD…SN†Š T N dÿ ‡ N T GD…SN† dÿÿ t e‰GD…SN†Š T N v dÿÿ t eN T t dÿ…13:38†…13:39†In Eqs (13.38) and (13.39) we have omitted the domain integral of the particularsolution r 0 corresponding to the body forces b but have allowed a portion of the

356 Mixed formulation and constraintsboundary ÿ t e to be subject to prescribed tractions. Full expressions including theparticular solution can easily be derived.Equation (13.38) is immediately available for solution of a single boundary problemin which v and t are described on portions of the boundary. More importantly,however, it results in a very simple sti€ness matrix for a full element enclosed bythe frame. We now haveK e ~v ˆ q ÿ f e…13:40†in whichK e ˆ Q eT …H e † ÿ1 Q ef e ˆ Q eT …H e † ÿ1 f e 1…13:41†This form is very similar to that of Eq. (13.33) except that now only integrals on theboundaries of the subdomain element need to be evaluated.Much has been written about so-called `boundary elements' and their merits anddisadvantages. 24ÿ36 Very frequently singular Green's functions are used to satisfythe governing ®eld equations in the domain. 31ÿ35 The singular functiondistributions used do not lend themselves readily to the derivation of symmetriccoupling forms of the type given in Eq. (13.38). Zienkiewicz et al. 36ÿ39 show that itis possible to obtain symmetry at a cost of two successive integrations. Further itshould be noted that the singular distributions always involve di cult integrationover a point of singularity and special procedures need to be used for numericalimplementation. For this reason the use of generally non-singular Tre€tz functionsis preferable and it is possible to derive complete sets of functions satisfying thegoverning equations without introducing singularities, 36ÿ39 and simple integrationthen su ces.While boundary solutions are con®ned to linear homogeneous domains these givevery accurate solutions for a limited range of parameters, and their combination with`standard' ®nite elements has been occasionally described. Several coupling procedureshave been developed in the past, 36ÿ39 but the form given here coincideswith the work of Zielinski and Zienkiewicz, 40 Jirousek 41ÿ44 and Piltner. 45 Jirouseket al. have developed very general two-dimensional elasticity and plate bendingelements which can be enclosed by a many-sided polygonal domain (element) thatcan be directly coupled to standard elements providing that same-displacementinterpolation along the edges is involved, as shown in Fig. 13.3. Here both interiorelements with a frame enclosing an element volume and exterior elements satisfyingtractions at free surface and in®nity are illustrated.Rather than combining in a ®nite element mesh the standard and the Tre€tz-typeelements (`T-elements' 28 ) it is often preferable to use the T-elements alone. Thisresults in the whole domain being discretized by elements of the same nature ando€ering each about the same degree of accuracy. The subprogram of such elementscan include an arsenal of homogeneous `shape functions' N e [see Eq. (13.37)] whichare exact solutions to di€erent types of singularities as well as those which automaticallysatisfy traction boundary conditions on internal boundaries, e.g., circlesor ellipses inscribed within large elements as shown in Fig. 13.4. Moreover, by com-

Linking of boundary (or Trefftz)-type solution by the `frame' of speci®ed displacements 357DDDTD(a)DDDt = 0 t = 0D D DTD D D(b)D D DFig. 13.3 Boundary±Trefftz-type elements (T) with complex-shaped `frames' allowing combination withstandard, displacement elements (D): (a) an interior element; (b) an exterior element.pleting the set of homogeneous shape functions by suitable `load terms' representingthe non-homogeneous di€erential equation solution, u 0 , one may account accuratelyfor various discontinuous or concentrated loads without laborious adjustment of the®nite element mesh.Clearly such elements can perform very well when compared with standard ones, asthe nature of the analytical solution has been essentially included. Figure 13.5 showst = 0t = 0Boundarydisplacementinterpolatedin polynomialformt = 0u = 0Displacement/stressfields defined by‘shape’ functionssatisfying governingequations andparameters aFig. 13.4 Boundary±Trefftz-type elements. Some useful general forms. 43

358 Mixed formulation and constraints(a) 21 Trefftz elements 252 DOFσ x σ y τ xy 1500 kN+––A+–++++y+xThickness t = 1 cm E = 21000 kN/cm 2ν = 00 20 40 60kN/cm 2(b) 920 Q8 standard elements 5960 DOF(a)(b)σ xAkN/cm 277.977.2(74.2)σ yAkN/cm 21.0(2.6)τ xyAkN/cm 20.0(0.1)AFig. 13.5 Application of Trefftz-type elements to a problem of a plane-stress tension bar with a circular hole.(a) Trefftz element solution. (b) Standard displacement element solution. (Numbers in parentheses indicatestandard solution with 230 elements, 1600 DOF).excellent results which can be obtained using such complex elements. The number ofdegrees of freedom is here much smaller than with a standard displacement solutionbut, of course, the bandwidth is much larger. 43Two points come out clearly in the general formulation of Eqs (13.36)±(13.39).

Linking of boundary (or Trefftz)-type solution by the `frame' of speci®ed displacements 359First, the displacement ®eld, u given by parameters ~a, can only be determined byexcluding any rigid body modes. These can only give strains SN identically equalto zero and hence make no contribution to the H matrix.Second, stability conditions require that (in two dimensions)n a 5 n v ÿ 3and thus the minimum n a can be readily found (viz. Chapter 11). Once again there islittle point in increasing the number of internal parameters substantially above theminimum number as additional accuracy may not be gained.1.61.20.80.40–0.42.0(a)1.51.00.50–0.5–1.0–1.5–2.00.60.21.61.20.80.40–0.4–1.0–0.6–0.24.03.53.02.52.01.51.00.502.0(a)1.51.00.50–0.5–1.0–1.5–2.0 1.00.60.24.03.53.02.52.01.51.00.50–1.0–0.6–0.2Fig. 13.6 Boundary±Trefftz-type `elements' linking two domains of different materials in an elliptic barsubject to torsion (Poisson equations). 40 (a) Stress function given by internal variables showing almost completecontinuity. (b) x component of shear stress (gradient of stress function showing abrupt discontinuityof material junction).

360 Mixed formulation and constraintsWe have said earlier that the `translation' of the formulation discussed to problemsgoverned by the quasi-harmonic equations is almost evident. Now identical relationswill hold if we replaceu ! r ! q…13:42†t ! q nS !rFor the Poisson equationr 2 ˆ Q…13:43†a complete series of analytical solutions in two dimensions can be written asRe …z n †ˆ1; x; x 2 ÿ y 2 ; x 3 ÿ 3xy 3 ; ...Im …z n for z ˆ x ‡ iy …13:44††ˆy; 2xy; ...With the above we getN e ˆ 1; x; y; x 2 ÿ y 2 ; 2xy; x 3 ÿ 3xy 2 ; 3x 2 y; ... …13:45†A simple solution involving two subdomains with constant but di€erent values of Qand a linking on the boundary is shown in Fig. 13.6, indicating the accuracy of thelinking procedures.13.6 Subdomains with `standard' elements and globalfunctionsThe procedure just described can be conveniently used with approximations madeinternally with standard (displacement) elements and global functions helping todeal with singularities or other internal problems. Now simply an additional termwill arise inside nodes placed internally in the subdomain but the e€ect of globalfunctions can be contained inside the subdomain. The formulation is somewhatsimpler as complicated Tre€tz-type functions need not be used.We leave details to the reader and in Fig. 13.7 show some possible, useful subdomainassemblies. We shall return to this again in Chapter 16.Fig. 13.7 `Superelements' built from assembly of standard displacement elements with global functionseliminating singularities con®ned to the assembly.

13.7 Lagrange variables or discontinuous Galerkinmethods?In all of the preceding examples we have linked the various element subdomains by aline on which the additional Lagrange multipliers have been speci®ed. These multiplierscould well be displacements or tractions which in fact were the same variablesas those inside the element domain.The lagrangian variables which are so identi®ed can be directly substituted interms of the variables given inside each subdomain. For instance the interfacedisplacement can be reproduced as the average displacement of those given in eachsubdomainu ˆ 12 …u 1 ‡ u 2 †Concluding remarks 361The total number of variables occurring in the problem is thus reduced (though nowelement variables have to be carried in the solution and the solution cost may well beincreased). The idea was ®rst used by Kikuchi and Ando 46 who used it to improve theperformance of non-conforming plate bending elements.Recently a revival of such methods has taken place. The basic idea appear to bepresented by Makridakis and BabusÏ ka et al. 47 and in the context of a `discontinuousGalerkin method' is demonstrated by Oden and co-workers. 48ÿ50 Weshall refer to the discontinuous Galerkin method in Volume 3 when dealing withconvection dominated problems and in a di€erent context in Sec. 18.6 ofChapter 18 for discrete time approximation problems. The process has practicaladvantages such as:1. di€erent local interpolations can be used;2. the stress (¯ux) continuity is preserved on each individual element.We shall discuss these properties further when we address the method inVolume 3.13.8 Concluding remarksThe possibilities of elements of `superelements' constructed by the mixed-incomplete®eld methods of this chapter are very numerous. Many have found practical use inexisting computer codes as `hybrid elements'; others are only now being madewidely available. The use of a frame of speci®ed displacements is only one of thepossible methods for linking Tre€tz-type solutions. As an alternative, a frame ofspeci®ed boundary tractions t has also been successfully investigated. 26;30 In addition,the so-called `frameless formulation' 27;29 has been found to be another e cientsolution (for a review see reference 28) in the Tre€tz-type element approach. All ofthe above mentioned alternative approaches may be implemented into standard®nite element computer codes. Much further research will elucidate the advantagesof some of the forms discovered and we expect the use of such developments tocontinue to increase in the future.

362 Mixed formulation and constraintsReferences1. N.E. Wiberg. Matrix structural analysis with mixed variables. Int. J. Num. Meth. Eng., 8,167±94, 1974.2. G. Kron. Tensor Analysis of Networks. John Wiley & Sons, New York, 1939.3. Ch. Farhat and F.-X. Roux. A method of ®nite element tearing and interconnecting and itsparallel solution algorithm. Intern. J. Num. Meth. Engng, 32, 1205±27, 1991.4. T.H.H. Pian and D.P. Chen. Alternative ways for formulation of hybrid elements. Int. J.Num. Meth. Eng., 18, 1679±84, 1982.5. T.H.H. Pian, D.P. Chen, and D. Kong. A new formulation of hybrid/mixed ®nite elements.Comp. Struct., 16, 81±7, 1983.6. P. Tong. A family of hybrid elements. Int. J. Num. Meth. Eng., 18, 1455±68, 1982.7. T.H.H. Pian and P. Tong. Relations between incompatible displacement model and hybridstrain model. Int. J. Num. Meth. Eng., 22, 173±181, 1986.8. T.H.H. Pian. Derivation of element sti€ness matrices by assumed stress distributions.JAIAA, 2, 1333±5, 1964.9. S.N. Atluri, R.H. Gallagher, and O.C. Zienkiewicz (Eds). Hybrid and Mixed Finite ElementMethods. Wiley, 1983.10. T.H.H. Pian. Element sti€ness matrices for boundary compatibility and for prescribedboundary stresses. Proc. Conf. Matrix Methods in Structural Mechanics. AFFDL-TR-66-80, pp. 457±78, 1966.11. R.D. Cook and J. At-Abdulla. Some plane quadrilateral `hybrid' ®nite elements. JAIAA, 7,1969.12. T.H. Pian and P. Tong. Basis of ®nite element methods for solid continua. Int. J. Num.Meth. Eng., 1, 3±28, 1969.13. S.N. Atluri. A new assumed stress hydrid ®nite element model for solid continua. JAIAA,9, 1647±9, 1971.14. R.D. Henshell. On hybrid ®nite elements, in The Mathematics of Finite Elements andApplications (ed. J.R. Whiteman), pp. 299±312, Academic Press, 1973.15. R. Dungar and R.T. Severn. Triangular ®nite elements of variable thickness. J. StrainAnalysis, 4, 10±21, 1969.16. R.J. Allwood and G.M.M. Cornes. A polygonal ®nite element for plate bending problemsusing the assumed stress approach. Int. J. Num. Meth. Eng., 1, 135±49, 1969.17. T.H.H. Pian. Hybrid models, in Numerical and Computer Methods in Applied Mechanics(eds S.J. Fenves et al.). Academic Press, 1971.18. Y. Yoshida. A hybrid stress element for thin shell analysis, in Finite Element Methods inEngineering (eds V. Pulmano and A. Kabaila), pp. 271±86, University of New SouthWales, Australia, 1974.19. R.D. Cook and S.G. Ladkany. Observations regarding assumed-stress hybrid plateelements. Int. J. Num. Meth. Eng., 8(3), 513±20, 1974.20. J.P. Wolf. Generated hybrid stress ®nite element models. JAIAA, 11, 1973.21. P.L. Gould and S.K. Sen. Re®ned mixed method ®nite elements for shells of revolution.Proc. 3rd Air Force Conf. Matrix Methods in Structural Mechanics. Wright-PattersonAF Base, Ohio, 1971.22. P. Tong. New displacement hybrid ®nite element models for solid continua. Int. J. Num.Meth. Eng., 2, 73±83, 1970.23. E. Tre€tz. Ein Gegenstruck zum Ritz'schem Verfohren. Proc. 2nd Int. Cong. Appl. Mech.Zurich, 1926.24. P.K. Banerjee and R. Butter®eld. Boundary Element Methods in Engineering Science.McGraw-Hill, London and New York, 1981.

25. J.A. Ligget and P.L-F. Liu. The Boundary Integral Equation Method for Porous MediaFlow. Allen and Unwin, London, 1983.26. C.A. Brebbia and S. Walker. Boundary Element Technique in Engineering. Newnes-Butterworth,London, 1980.27. J. Jirousek and A. WroÂblewski. Least-squares T-elements: Equivalent FE and BE forms ofa substructure-oriented boundary solution approach. Comm. Num. Meth. Eng., 10, 21±32,1994.28. J. Jirousek and A. WroÂblewski. T-elements: State of the art and future trends. Arch. Comp.Meth. Eng., 3(4), 1996.29. J. Jirousek and A.P. ZielinÂski. Study of two complementary hybrid-Tre€tz p-elementformulations, in Numerical Methods in Engineering 92, 583±90. Elsevier, 1992.30. J. Jirousek and A.P. ZielinÂski. Dual hybrid-Tre€tz element formulation based on independentboundary traction frame. Internat. J. Num. Meth. Eng., 36, 2955±80, 1993.31. I. Herrera. Boundary methods: a criteria for completeness. Proc. Nat. Acad. Sci. USA,77(8), 4395±8, August 1980.32. I. Herrera. Boundary methods for ¯uids, Chapter 19 of Finite Elements in Fluids. Vol. 4(eds R.H. Gallagher, H.D. Norrie, J.T. Oden, and O.C. Zienkiewicz). Wiley, New York,1982.33. I. Herrera. Tre€tz method, in Progress in Boundary Element Methods. Vol. 3 (ed. C.A.Brebbia). Wiley, New York, 1983.34. I. Herrera and H. Gourgeon. Boundary methods, C-complete system for Stokes problems.Comp. Meth. Appl. Mech. Eng., 30, 225±44, 1982.35. I. Herrera and F.J. Sabina. Connectivity as an alternative to boundary integral equations:construction of bases. Proc. Nat. Acad. Sci. USA, 75(5), 2059±63, May 1978.36. O.C. Zienkiewicz, D.W. Kelly, and P. Bettess. The coupling of the ®nite element methodand boundary solution procedures. Int. J. Num. Meth. Eng., 11, 355±75, 1977.37. O.C. Zienkiewicz, D.W. Kelly, and P. Bettess. Marriage a la mode ± the best of both worlds(®nite elements and boundary integrals). Chapter 5 of Energy Methods in Finite ElementAnalysis (eds R. Glowinski, E.Y. Rodin, and O.C. Zienkiewicz), pp. 81±107, Wiley,London and New York, 1979.38. O.C. Zienkiewicz and K. Morgan. Finite Elements and Approximation. Wiley, London andNew York, 1983.39. O.C. Zienkiewicz. The generalized ®nite element method ± state of the art and futuredirections. J. Appl. Mech. 50th anniversary issue, 1983.40. A.P. Zielinski and O.C. Zienkiewicz. Generalized ®nite element analysis with T completeboundary solution functions. Int. J. Num. Mech. Eng., 21, 509±28, 1985.41. J. Jirousek. A powerful ®nite element for plate bending. Comp. Meth. Appl. Mech. Eng., 12,77±96, 1977.42. J. Jirousek. Basis for development of large ®nite elements locally satisfying all ®eld equations.Comp. Meth. Appl. Mech. Eng., 14, 65±92, 1978.43. J. Jirousek and P. Teodorescu. Large ®nite elements for the solution of problems in thetheory of elasticity. Comp. Struct., 15, 575±87, 1982.44. J. Jirousek and Lan Guex. The hybrid Tre€tz ®nite element model and its application toplate bending. Int. J. Num. Mech. Eng., 23, 651±93, 1986.45. R. Piltner. Special elements with holes and internal cracks. 21, 1471±85, 1985.46. F. Kikichi and Y. Ando. A new variational functional for the ®nite-element method and itsapplication to plate and shell problems. Nucl. Engng and Design, 21(1), 95±113, 1972.47. C.G. Makridakis and I. BabusÏ ka. On the stability of the discontinuous Galerkin methodfor the heat equation. SIAM J. Num. Anal., 34, 389±401, 1997.48. J.T. Oden, I. BabusÏ ka, and C.E. Baumann. A discontinuous hp ®nite element method fordi€usion problems. J. Comp. Physics, 146(2), 491±519, 1998.References 363

364 Mixed formulation and constraints49. J.T. Oden and C.E. Baumann. A discontinuous hp ®nite element method for convectiondi€usionproblems. Comput. Meth. Appl. Mech. Engng, 175(3-4), 311±41, 1999.50. C.E. Baumann and J.T. Oden. A discontinuous hp ®nite element method for convectiondi€usionproblems. Comput. Meth. Appl. Mech. Engng, 175, 311±41, 1999.

14Errors, recovery processes anderror estimates14.1 De®nition of errorsWe have stressed from the beginning of this book the approximate nature of the ®niteelement method and on many occasions to show its capabilities we have compared itwith exact solutions when these were known. Also on many occasions we have spokenabout the `accuracy' of the procedures we suggested and discussed the manner bywhich this accuracy could be improved. Indeed one of the objectives of this chapteris concerned with the question of accuracy and a possible improvement on it by ana posteriori treatment of the ®nite element data. We refer to such processes asrecovery. We shall also consider the discretization error of the ®nite elementapproximation and a posteriori estimates of such error. In particular, we describetwo distinct types of error estimators, recovery based error estimators and residualbased error estimators. The critical role that the recovery processes play in thecomputation of these error estimators will be discussed.Before proceeding further it is necessary to de®ne what we mean by error. This weconsider to be the di€erence between the exact solution and the approximate one.This can apply to the basic function, such as displacement which we have called uand can be given ase ˆ u ÿ ^u…14:1†In a similar way, however, we could focus on the error in the strains (i.e., gradients inthe solution), such as e or stresses r and describe an error in those quantities ase " ˆ e ÿ ^ee ˆ r ÿ ^r…14:2†…14:3†The speci®cation of local error in the manner given in Eqs (14.1)±(14.3) is generallynot convenient and occasionally misleading. For instance, under a point load botherrors in displacements and stresses will be locally in®nite but the overall solutionmay well be acceptable. Similar situations will exist near re-entrant corners where,as is well known, stress singularities exist in elastic analysis and gradient singularitiesdevelop in ®eld problems. For this reason various `norms' representing someintegral scalar quantity are often introduced to measure the error.

366 Errors, recovery processes and error estimatesIf, for instance, we are concerned with a general linear equation of the form of Eq.(3.6) (cf. Chapter 3), i.e.,Lu ‡ p ˆ 0…14:4†we can de®ne an energy norm written for the error as … 1 jjejj ˆ e T 2…Le d … u ÿ ^u † T 2Luÿ … ^u †d1…14:5†This scalar measure corresponds in fact to the square root of the quadraticfunctional such as we have discussed in Sec. 3.8 of Chapter 3 and where we soughtits minimum in the case of a self-adjoint operator L.For elasticity problems the energy norm is identically de®ned and yields, …jjejj ˆ … Se† T 2DSe d1…14:6†(with symbols as used in Chapter 2).Here e is given by Eq. (14.1) and the operator S de®nes the strains ase ˆ Su and ^e ˆ S^u …14:7†and D is the elasticity matrix (see Chapter 2), giving the stress asr ˆ De and ^r ˆ D^e …14:8†in which for simplicity we ignore initial stresses and strains.The energy norm of Eq. (14.6) can thus be written alternatively as …jjejj ˆ …ˆ …ˆ… e ÿ ^e † T D…e ÿ ^e12†d… e ÿ ^e † T … r ÿ ^r †d1…14:9†2… r ÿ ^r † T D ÿ1 … r ÿ ^r12†dand its relation to strain energy is evident.Other scalar norms can easily be devised. For instance, the L 2 norm of displacementand stress error can be written as …jjejj L2 ˆ … u ÿ ^u † T 2… u ÿ ^u †d1…14:10† …jje jj L2 ˆ… r ÿ ^r † T … r ÿ ^r †d1…14:11†2Such norms allow us to focus on the particular quantity of interest and indeed it ispossible to evaluate `root mean square' (RMS) values of its error. For instance, theRMS error in displacement, u, becomes for the domain juj ˆ jjejj2L212…14:12†

De®nition of errors 367Similarly, the RMS error in stress, , becomes for the domain jje jjjj ˆ 2 L 2…14:13†Any of the above norms can be evaluated over the whole domain or over subdomainsor even individual elements.We note thatjjejj 2 ˆ Xmi ˆ 1jjejj 2 i12…14:14†where i refers to individual elements i such that their sum (union) is .We note further that the energy norm given in terms of the stresses, the L 2 stressnorm and the RMS stress error have a very similar structure and that these aresimilarly approximated.At this stage it is of interest to invoke the discussion of Chapter 2 (Sec. 2.6)concerning the rates of convergence. We noted there that with trial functions in thedisplacement formulation of degree p, the errors in the stresses were of the orderO…h p †. This order of error should therefore apply to the energy norm error jjejj.While the arguments are correct for well-behaved problems with no singularity, itis of interest to see how the above rule is violated when singularities exist.To describe the behaviour of stress analysis problems we de®ne the variation of therelative energy norm error (percentage) as ˆ jjejjjjujj 100%…14:15†where …jjujj ˆ e T 2De d1…14:16†is the energy norm of the solution. In Figs 14.1 and 14.2 we consider two similar stressanalysis problems, in the ®rst of which a strong singularity is, however, present. Inboth ®gures we show the relative energy norm error for an h re®nement constructedby uniform subdivision of the initial mesh and of a p re®nement in which polynomialorder is increased throughout the original mesh.We note two interesting facts. First, the h convergence rates for various polynomialorders of the shape functions are nearly the same in the example with singularity (Fig.14.1) and are well below the theoretically predicted optimal order O…h p †, [orO…NDF† ÿp=2 as the NDF (number of degrees of freedom) is approximately inverselyproportional to h 2 for a two-dimensional problem].Secondly, in the case shown in Fig. 14.2, where the singularity is avoided by roundingthe corner, the convergence rates improve for elements of higher order, thoughagain the theoretical (asymptotic) rates are not achieved.The reason for this behaviour is clearly the singularity, and in general it can beshown that the rate of convergence for problems with singularity isÿ‰min…; p†Š=2O…NDF† …14:17†

5000300020001000500300200100NDF400.451(0)Linear(1)18Cubic(1)(2)(1)(0)(0)(0)QuarticQuadratic5.6Error η (%)Subdivision 1(2)(2)Average pconvergence1.8(a)Subdivision 2(b)Fig. 14.1 Analysis of L-shaped domain with singularity.

5000300020001000500300200100NDF(0)40Linear(2)(1)10.518Quadratic(0)(0)5.6Subdivision 1(1)Average pconvergence1.8Error η (%)(a)(2)Cubic1.00.75(1)11.251(b)(2)0.17Fig. 14.2 Analysis of L-shaped domain without singularity.

370 Errors, recovery processes and error estimateswhere is a number associated with the intensity of the singularity. For elasticityproblems ranges from 0:5 for a nearly closed crack to 0:71 for a 908 corner. Therate of convergence illustrated in Fig. 14.2 approaches the value controlled by thesingularity for all values of p used in the elements.14.2 Superconvergence and optimal sampling pointsIn this section we shall consider the matter of points at which the stresses, or displacements,give their most accurate values in typical problems of a self-adjointkind. We shall note that on many occasions the displacements, or the functionitself, are most accurately sampled at the nodes de®ning an element and that thegradients or stresses are best sampled at some interior points. Indeed in onedimension at least we shall ®nd that such points often exhibit the quality knownas superconvergence (i.e., the values sampled at these points show an error whichdecreases more rapidly than elsewhere). Obviously, the user of ®nite elementanalysis should be encouraged to employ such points but at the same time notethat the errors overall may be much larger. To clarify ideas we shall start with atypical problem of second order in one dimension.14.2.1 A one-dimensional exampleHere we consider a problem of a second-order equation such as we have frequentlydiscussed in Chapter 3 and which may be typical of either one-dimensional heatconduction or the displacements of an elastic bar with varying cross-section. Thisequation can readily be written asddxk dudx‡ u ‡ Q ˆ 0 …14:18†with the boundary conditions either de®ning the values of the function u or of itsgradients at the ends of the domain.Let us consider a typical problem shown in Fig. 14.3. Here we show an exactsolution for u and du=dx for a span of several elements and indicate the type ofsolution which will result from a ®nite element calculation using linear elements.We have already noted that on occasions we shall obtain exact solutions for u atnodes (see Fig. 3.4). This will happen when the shape functions contain the exactsolution of the homogeneous di€erential equation (Appendix H) ± a situationwhich happens for Eq. (14.18) when ˆ 0 and polynomial shape functions areused. In all cases, even when is non-zero and linear shape functions are used, thenodal values generally will be much more accurate than those elsewhere, Fig.14.3(a). For the gradients shown in Fig. 14.3(b) we observe large discrepancies ofthe ®nite element solution from the exact solution but we note that somewherewithin each element the results are nearly exact.It would be useful to locate such points and indeed we have already remarked in thecontext of two-dimensional analysis that values obtained within the elements tend tobe more accurate for gradients (strains and stresses) than those values calculated at

Superconvergence and optimal sampling points 371ExactOptimum samplingpoint for uPiecewise linearapproximationLinear approximationExact at nodesh 1 h 2(a)ExactLinear approximationExact at centreFE solutionOptimal samplingpoints for du/dx(b)Fig. 14.3 Optimal sampling points for the function (a) and its gradient (b) in one dimension (linear elements).nodes. Clearly, for the problem illustrated in Fig 14.3(b) we should sample somewherenear the centre of each element.Pursuing this problem further in a heuristic manner we shall note that if higherorder elements (e.g., quadratic elements) are used the solution still remains exact ornearly exact at the end nodes of an element but may depart from exactness at theinterior nodes, as shown in Fig. 14.4(a). The stresses, or gradients, in this case willbe optimal at points which correspond to the two Gauss quadrature points foreach element as indicated in Fig. 14.4(b). This fact was observed experimentally byBarlow 1 , and such points are frequently referred to as Barlow points.We shall now state in an axiomatic manner that:(a) the displacements are best sampled at the nodes of the element, whatever theorder of the element is, and(b) the best accuracy is obtainable for gradients or stresses at the Gauss pointscorresponding, in order, to the polynomial used in the solution.At such points the order of the convergence of the function or its gradients is one orderhigher than that which would be anticipated from the appropriate polynomial andthus such points are known as superconvergent. The reason for such superconvergencewill be shown in the next section where we introduce the reader to a theoremdeveloped by Herrmann. 2

372 Errors, recovery processes and error estimatesExactQuadraticapproximationQuadratic approximationExact at end nodes(a)h 1 h 2 h 3ExactQuadratic approximationExact at some point inthe elementQuadraticapproximation(b)Optimal samplingpoint for dwdxFig. 14.4 Optimal sampling points for the function (a) and its gradient (b) in one dimension (quadratic14.2.2 The Herrmann theorem and optimal sampling pointsThe concept of least square ®tting has additional justi®cation in self-adjoint problemsin which an energy functional is minimized. In such cases, typical of a displacementformulation of elasticity, it can be readily shown that the minimization isequivalent to a least square ®t of approximation stresses to the exact ones. Thusquite generally we can start from a theory which states that minimization of anenergy functional de®ned as ˆ 1 ……… Su† T ASu d ‡ u T p d…14:19†2

which at an absolute minimum gives the exact solution ^u ˆ ~u this is equivalent to minimizationof another functional de®ned as… ˆ 12‰ Suÿ … u † Š T AS… u ÿ u †d …14:20†In the above, S is a self-adjoint operator and A and p are prescribed matrices ofposition. The above quadratic form [Eq. (14.19)] arises in the majority of linearself-adjoint problems.For elasticity problems this theorem is given by Herrmann 2 and shows that theapproximate solution for Su approaches the exact one Su as a weighted least squareapproximation.The proof of the Herrmann theorem is as follows. The variation of de®ned inEq. (14.19) gives, at ^u ˆ ~u (the exact solution),……… ˆ 12 … S^u † T ASu d ‡ 1 2… Su† T AS^u d ‡ ^u T p d ˆ 0 …14:21†or as A is symmetric… ˆ…… S^u † T ASu d ‡^u T p d ˆ 0in which u is any arbitrary variation. Thus we can writeu ˆ uand……… S^u † T ASu d ‡ ^u T p d ˆ 0…14:22†…14:23†…14:24†Subtracting the above from Eq. (14.19) and noting the symmetry of the A matrix, wecan write…… ˆ 12‰ S… ^u ÿ u†Š T AS… ^u ÿ u†d ÿ 1 2‰ Su … † Š T ASu d …14:25†Superconvergence and optimal sampling points 373where the last term is not subject to variation. Thus ˆ ‡ constant…14:26†and its stationarity is equivalent to the stationarity of .It follows directly from the Herrmann theorem that, for one dimension and by awell-known property of the Gauss±Legendre quadrature points, if the approximategradients are de®ned by a polynomial of degree p ÿ 1, where p is the degree of thepolynomial used for the unknown function u, then stresses taken at these quadraturepoints must be superconvergent. The single point at the centre of an elementintegrates precisely all linear functions passing through that point and, hence, if thestresses are exact to the linear form they will be exact at that point of integration.For any higher order polynomial of order p, the Gauss±Legendre points numberingp will provide points of superconvergent sampling. We see this from Fig. 14.5 directly.Here we indicate one, two, and three point Gauss±Legendre quadrature showing whyexact results are recovered there for gradients and stresses.

374 Errors, recovery processes and error estimatesp = 1–101p = 2p = 3Fig. 14.5 The integration property of Gauss points: p ˆ 1, p ˆ 2, and p ˆ 3 which guaranteessuperconvergence.For points based on rectangles and products of polynomial functions it is clearthat the exact integration points will exist at the product points as shown inFig. 14.6 for various rectangular elements assuming that the weighting matrix A isdiagonal. In the same ®gure we show, however, some triangles and what appear tobe `good' but not necessarily superconvergent sampling points. These are suggestedby Moan. 3 Though we ®nd that superconvergent points do not exist in triangles,the points shown in Fig. 14.6 are optimal. In Fig. 14.6 we contrast these pointswith the minimum number of quadrature points necessary for obtaining an accurate(though not always stable) sti€ness representation and ®nd these to be almostcoincident at all times.In Fig. 14.7 representing an analysis of a cantilever by four rectangular quadraticserendipity elements we see how well the stresses sampled at superconvergent pointsbehave compared to the overall stress pattern computed in each element. It is fromresults like this that many suggestions have been made to obtain improved nodalvalues and one method proposed by Hinton and Campbell has proved to be quitewidely used. 4 However, we shall discuss better recovery procedures later.

Recovery of gradients and stresses 375pOptimal error O (h 2(p – m)+ 2 )Minimal quadrature O(h 2(p – m)+ 1 )1O (h 2 ) ≥ O (h 2 )O (h 2 )O (h 2 )O (h 2 )O (h 2 )2O (h 4 )≥ O (h 3 )O (h 4 )O (h 3 )O (h 4 )O (h 4 )O (h 4 )O (h 4 )Fig. 14.6 Optimal superconvergent sampling and minimum integration points for some C 0 elements.The extension of the idea of superconvergent points from one-dimensionalelements to two-dimensional rectangles was fairly obvious. However, the full superconvergenceis lost when isoparametric distortion occurs. We have shown, however,that results at the pth-order Gauss±Legendre points still remain excellent and wesuggest that superconvergent properties of the integration points continue to beused for sampling.In all of the above discussion we have assumed that the weighting matrix A isdiagonal. But if such diagonality does not exist then the existence of superconvergentpoints is questionable. However excellent results are still available through thesampling points de®ned as above.Finally, we refer readers to references 5±9 for surveys on the superconvergencephenomenon and its detailed analyses.14.3 Recovery of gradients and stressesIn the previous section we have shown that sampling of the gradients and stresses atsome particular points is generally optimal and possesses a higher order accuracywhen such points are superconvergent. However, we would also like to have similarlyaccurate quantities elsewhere within each element for general analysis purposes, andin particular we need such highly accurate gradients and stresses when the energynorm or other similar norms have to be evaluated in error estimates. We have already

376 Errors, recovery processes and error estimates40302010Exact averageshear stressNodal values extrapolatedfrom Gauss points0–10Gauss pointvaluesLand 0.24 per unit2402 x 2 Gauss pointsFig. 14.7 Cantilever beam with four quadratic (Q8) elements. Stress sampling at cubic order …2 2† Gausspoints with extrapolation to nodes.shown how with some elements very large errors exist beyond the superconvergentpoint and attempts have been made from the earliest days to obtain a completepicture of stresses which is more accurate overall. Here attempts are generallymade to recover the nodal values of stresses and gradients from those sampledinternally and then to assume that throughout the element the recovered stresses r are obtained by interpolation in the same manner as the displacementsr ˆ N u ~r …14:27†We have already suggested a process used almost from the beginning of ®nite elementcalculations for triangular elements, where elements are sampled at the centroid(assuming linear shape functions have been used) and then the stresses are averagedat nodes. We have referred to such recovery in Chapter 4. However this is not thebest for triangles and for higher order elements such averaging is inadequate. Hereother procedures were necessary, for instance Hinton and Campbell 4 suggested a procedurein which stresses at all nodes were calculated by extrapolating the Gauss pointvalues. A further improvement of a similar kind was suggested by Brauchli and Oden 10who used the stresses in the manner given by Eq. (14.27) and assumed that these stressesshould represent in a least square sense the actual ®nite element stresses, therefore an L 2

Superconvergent patch recovery ± SPR 377projection. Though this has a similarity with the ideas contained in the Herrmanntheorem it reverses the order of least square application and has not proved to bealways stable and accurate, especially for even order elements. We have alreadydescribed this procedure in the chapter on mixed elements (see Sec. 11.6) and notedthat to obtain results it is necessary to invert a `mass' type matrix. This can only beachieved without high cost if the mass matrix is diagonal. However, in the followingpresentation we will show that highly improved results can be obtained by direct polynomial`smoothing' of the superconvergent values. Here the ®rst method of importanceis called superconvergent patch recovery. 11ÿ1314.4 Superconvergent patch recovery ± SPR14.4.1 Recovery for gradients and stressesWe have already noted that the stresses sampled at certain points in an elementpossess the superconvergent property (i.e., converge at the same rate as displacement)and have errors of order O…h p ‡ 1 †. A fairly obvious procedure for utilizing suchsampled values seems to the authors to be that of involving a smoothing of suchvalues by a polynomial of order p within a patch of elements for which the numberof sampling points can be taken as greater than the number of parameters in thepolynomial. In Fig. 14.8 we show several such patches each assembled around acentral corner node. The ®rst four represent rectangular elements where the superconvergentpoints are well de®ned. The last two give patches of triangles where the bestsampling points are used which are not superconvergent.If we accept the superconvergence of ^r at certain points s in each element then it is asimple matter (which also turns out computationally much less expensive than the L 2projection) to compute r which is superconvergent at all points within the element. Theprocedure is illustrated for two dimensions in Fig. 14.8, where we shall considerinterior patches (assembling all elements at interior nodes) as shown.At the superconvergent point the values of ^r are accurate to order p ‡ 1 (not p as istrue elsewhere). However, we can easily obtain an approximation given by a polynomialof degree p, with identical order to these occurring in the shape function fordisplacement, which has superconvergent accuracy everywhere if this polynomial ismade to ®t the superconvergent points in a least square manner.Thus we proceed for each component ^ i of ^r as follows: Writing the recoveredsolution as i ˆ pa ˆ ‰ 1; x; y; ; y p Šaa ˆ ‰ a 1 ; a 2 ; ; a m Š T …14:28†we minimize, for an element patch with total n sampling points, ˆ Xnk ˆ 1p k ˆ p…x k ; y k †‰ ^ i …x k ; y k †ÿp k aŠ 2…14:29†

Nodal values determinedfrom the patchPatch assembly pointSuperconvergent samplingpointsElement patches Ω s4-node elementsElementpatchesΩ s8-node elements9-node elements12- and 16-node elements3-node elements(linear)6-node elements(quadratic)Fig. 14.8 Interior superconvergent patches for quadrilateral elements (linear, quadratic, and cubic) and triangles (linear and quadratic).

Superconvergent patch recovery ± SPR 379InterfaceMaterial IMaterial IIPatch assembly node for boundary interfaceRecovered boundary and interface valuesFig. 14.9 Recovery of boundary or interface gradients.[(x k ; y k ) corresponding to coordinates of superconvergent points] obtaining immediatelythe coe cient a aswhereA ˆ Xnk ˆ 1a ˆ A ÿ1 bp T k p k and b ˆ Xnk ˆ 1p T k ^ i …x k ; y k †…14:30†…14:31†The availability of r allows the superconvergent values of ~r to be determined at allnodes. As some nodes belong to more than one patch, average values of ~r are bestobtained. The superconvergence of r throughout each element is achieved withEq. (14.27).It should be noted that on external boundaries and indeed on interfaces wherestresses are discontinuous the nodal values should be calculated from interior patchesin the manner shown in Fig. 14.9.In Fig. 14.10 we show in a one-dimensional example how the superconvergentpatch recovery reproduces exactly the stress (gradient) solutions of order p ‡ 1 forlinear or quadratic elements. Following the arguments of Chapter 10 on the patchtest it is evident that superconvergent recovery is now achieved at all points.Indeed, the same ®gure shows why averaging (or L 2 projection) is inferior (particularlyon boundaries).Figure 14.11 shows experimentally determined convergence rates for a onedimensionalproblem (stress distribution in a bar of length L ˆ 1; 0 4 x 4 1 andprescribed body forces). A uniform subdivision is used here to form the elements,and the convergence rates for the stress error at x ˆ 0:5 are shown using the directstress approximation ^, the L 2 recovery L and obtained by the SPR procedureusing linear, quadratic and cubic elements. It is immediately evident that issuperconvergent with a rate of convergence being at least one order higher thanthat of ^. However, as anticipated, the L 2 recovery gives much inferior answers, showingsuperconvergence only for odd values of p and almost no improvement for even

380 Errors, recovery processes and error estimatesσExact σ and σ*σ AVσ hInterior patchBoundary(a)xSuperconvergent valuesNodal SPR valuesσσ hQuadratic exact solution– recovered σ*hBoundary(b)xFig. 14.10 Recovery of exact of degree p by linear elements (p ˆ 1) and quadratic elements ( p ˆ 2).values of p, while shows a two-order increase of convergence rate for even orderelements (tests on higher order polynomials are reported in reference 14). This ultraconvergence has been veri®ed mathematically. 15 Although it is not observed whenelements of varying size are used, the important tests shown in Figs 14.12 and14.13 indicate how well the recovery process works.In the ®rst of these, Fig. 14.12, a ®eld problem is solved in two dimensions using avery irregular mesh for which the existence of superconvergent points is only inferredheuristically. The very small error in x is compared with the error of ^ x and theimprovement is obvious. Here x ˆ @u=@x where u is the ¯uid variable.In the second, i.e., Fig. 14.13, a problem of stress analysis, for which an exactsolution is known, is solved using three di€erent recovery methods. Once again therecovered solution (SPR) shows the much improved values compared with Land it is clear that the SPR process should be included in all codes if simply to presentimproved stress values.

Superconvergent patch recovery ± SPR 38120p = 11/h128 32 8 2p = 21/h128 32 8 2p = 31/h128 32 8 2log |e σ |log |e σ |–211–4–62–81–10–4 –3 –2 –1 0 1log h1/h0–2–4–6–8128 32 8p = 4142–106–121–14–4 –3 –2 –1 0 1log h1241–4 –3 –2 –1 0 1log h1/h128 32 8 2p = 51561–4 –3 –2 –1 0 1log h1341–4 –3 –2 –1 0 1log h1/h128 32 8 2p = 61681–4 –3 –2 –1 0 1log hFig. 14.11 Problem of a stressed bar. Rates of convergence (error) of stress, where x ˆ 0:5 (04 x 4 1).(^ ÐÐ; L ; ----)The SPR procedure which we have just outlined has proved to be a very powerfultool leading to superconvergent results on regular meshes and much improved results(nearly superconvergent) on irregular meshes. It has been shown numerically that itproduces superconvergent recovery even for triangular elements which do not havesuperconvergent points within the element. A recent mathematical proof con®rmsthis capability of SPR. 6 The procedure was introduced by Zienkiewicz and Zhu in1992 11ÿ13 and we still recommend it as the best procedure which is simple to use. However,many investigators have modi®ed the procedure by increasing the functional where(a) Arbitrary mesh (b) Error of σ x * (c) Error of σˆxFig. 14.12 Poisson equation in two dimensions solved using arbitrary shaped quadratic quadrilaterals.

382 Errors, recovery processes and error estimatesσ = 1.0yσ = 1.0BAECDx(a)P P P(b)Mesh 1 Mesh 2 Mesh 31.22401.20001.17601.1520x component –1.30–1.40–1.50–1.60–1.70–1.80–1.90y component–0.2050–0.2075–0.2100–0.2125–0.2150–0.2175–0.2200xy component–2.00–0.22251.1280–2.10–0.22500 1 2 3 4 0 1 2 3 4 0 1 2 3 4MeshMeshMeshSPR: σ* FEM: σˆL 2 : σ L(c) (d) (e)Fig. 14.13 Plane stress analysis of stresses around a circular hole in a uniaxial ®eld.the least square ®t is performed to include satisfaction of discrete equilibrium equationsor boundary conditions, etc. While the satisfaction of known boundary tractions can onoccasion be useful most of these additional constraints introduced have a€ected thesuperconvergent properties adversely and in general the modi®ed versions of SPR byWiberg et al. 17 and by Blacker and Belytschko 18 have not proved to be fully e€ective.

Recovery by equilibration of patches ± REP 38314.4.2 SPR for displacementsThe superconvergent patch recovery can be extended to produce superconvergentdisplacements. The procedure for the displacements is quite simple if we assumethe superconvergent points to be at nodes of the patch. However, as we have alreadyobserved it is always necessary to have more data than the number of coe cients inthe particular polynomial to be able to execute a least square minimization. Here ofcourse we occasionally need a patch which extends further than before, particularlysince the displacements will be given by a polynomial one order higher than thatused for the shape functions. In Fig. 14.8 however we show for most assembliesthat a similar patch as given before can be again applied producing a good approximationfor u within its interior. Larger element patches have also been suggested inreference 19.The recovered solution u has on occasion been used in dynamic problems (e.g.,Wiberg 19;20 ), because in dynamic problems the displacements themselves are oftenimportant. We shall ®nd such recovery useful in some problems of ¯uid dynamicsin Volume 3.The SPR recovery technique described in this section takes advantage of the superconvergenceproperty of the ®nite element solutions and the availability of theoptimal sampling points. Very recently a new method of recovery which does notneed such information has been devised and will be discussed in the next section.14.5 Recovery by equilibration of patches ± REPAlthough SPR has proved to work well generally, the reason behind its capability ofproducing an accurate recovered solution even when superconvergent points do notin fact exist remains an open question. We have therefore sought to determine viablerecovery alternatives. One of these, known by the acronym REP (recovery by equilibriumof patches), will be described next. This procedure was ®rst presented inreference 21 and later improved in reference 22.To some extent the motivation is similar to that of LadeveÁ ze et al. 23;24 who soughtto establish (for somewhat di€erent reasons) a fully equilibrating stress ®eld whichcan replace that of the ®nite element approximation. However we believe that theprocess derived in reference 21 is simpler though equilibration is only approximate.The starting point is the governing equilibrium equationS T r ‡ b ˆ 0In the ®nite element approximation this becomes…B T^r ……d ÿ N T b d ÿ N T t dÿ ˆ 0 p p ÿ p…14:32†…14:33†where ^r are the stresses from the ®nite element solution. In the above p is the domainof the patch and the last term comes from the tractions on the boundary of the patchdomain ÿ p . These can, of course, represent the whole of the problem, an elementpatch or only a single element.

384 Errors, recovery processes and error estimatesAs is well known the stresses ^r which result from the ®nite element analysis will ingeneral be discontinuous and we shall seek to replace them in every element patch by arecovered system which is smooth and continuous.To achieve the recovery we proceed in an exactly analogous way to that used in theSPR procedure, ®rst approximating the stress in each patch by a polynomial ofappropriate order r , second using this approximation to obtain nodal values of ~r and ®nally interpolating these values by standard shape functions.The stress r is taken as a vector of appropriate components, which for conveniencewe write as:8 9>< 1 >=r ˆ 2…14:34†>: >;The above notation is general with, for instance, 1 ˆ x , 2 ˆ y and 3 ˆ xy intwo-dimensional plane elastic analysis.We shall write each component of the above as a polynomial expansion of the form: i ˆ ‰ 1; x; y; Ša i ˆ p…x; y†a i …14:35†where p is a vector of polynomials and a i is a set of unknown coe cients for the ithcomponent of stress.For equilibrium we shall always attempt to ensure that the total smoothed stress r satis®es in the least square sense the same patch equilibrium conditions as the ®niteelement solution. Accordingly,…B T^r d p…B T r d p…14:36†where2 389p 0 0 >< a 1 >=r 6 7ˆ Pa ˆ 4 0 p 05a 2…14:37†>: >;0 0 pwritten here again for the case of three stress components. Obvious modi®cations aremade for more or less components.It has been found in practice that the constraints provided by Eq. (14.36) are notsu cient to always produce non-singular least square minimization. Accordingly,the equilibrium constraints are split into an alternative form in which each componentof stress is subjected to equilibrium requirements. This may be achieved by expressingthe stress asr ˆ X1 i i ˆ Xr i…14:38†ii^r ˆ X1 i ^ i ˆ X ^r i …14:39†wherei 3ia 31 1 ˆ ‰ 1; 0; 0 Š T …14:40†1 2 ˆ ‰ 0; 1; 0 Š T etc: …14:41†

Error estimates by recovery 385and imposing the set of constraints………B T^r i d pB T r i d ˆ pB T 1 i p da i p…14:42†The imposition of the approximate equation (14.42) allows each set of coe cientsa i to be solved independently reducing considerably the solution cost and here repeatinga procedure used with success in SPR.A least square minimization of Eq. (14.42) is expressed as ˆ … H i a i ÿ f p i † T … H i a i ÿ f p i † …14:43†where…H i ˆ B T 1 i p d p…14:44†and…f p i ˆ B T^r i d p…14:45†The minimization condition results ina i ˆ H T ÿ1H Ti H i i f p i …14:46†For patches in some problems Eq. (14.43) may be unstable. Generally, this may beeliminated by modifying the patch requirement to the minimization of ˆ … H i a i ÿ f p i† T … H i a i ÿ f p i †‡ X e…H e i a i ÿ f e i † T … H e i a i ÿ f e i † …14:47†where the added terms represent modi®cation on individual elements and is aparameter. Minimization now givesa i ˆ H T i H i ‡ X ÿ1 H e;Ti H e i H T i f p i ‡ X H e;Ti f e i …14:48†eeThe REP procedure follows precisely the details of SPR near boundaries and givesoverall an approximation which does not require knowledge of any superconvergentpoints. The accuracy of both processes is comparable and we are of the opinion thatmany other alternative recovery procedures are still possible.14.6Error estimates by recoveryOne of the most important applications of the recovery methods is its use in thecomputation of the a posteriori error estimators. With the recovered solutionsavailable, we can now evaluate errors simply by replacing the exact values of quantitiessuch as u, r, etc., which are in general unknown, in Eqs (14.1)±(14.3), by therecovered values which are much more accurate than the direct ®nite elementsolution. We write the error estimators in various norms such asjjejj jjejj ˆ jju ÿ ^ujj…14:49†

386 Errors, recovery processes and error estimatesjjejj L2 jjejj L2 ˆjju ÿ ^ujj L2 …14:50†jje jj L2 jje jj L2 ˆjjr ÿ ^rjj L2 …14:51†For example, the energy norm error estimator for elasticity problems has the form of …jjejj ˆ … r ÿ ^r † T D ÿ1 r 2… ÿ ^r †d1…14:52†Similarly, estimates of the RMS error in displacement and stress can be obtainedthrough Eqs (14.12) and (14.13). Error estimators formulated by replacing theexact solution with the recovered solution are sometimes called recovery basederror estimators. This type of error estimator was ®rst introduced by Zienkiewiczand Zhu. 25The accuracy or the quality of the error estimator is measured by the e€ectivityindex , which is de®ned as ˆ jjejjjjejj…14:53†A theorem proposed by Zienkiewicz and Zhu 12 shows that for all estimators basedon recovery we can establish the following bounds for the e€ectivity index:1 ÿ jje jjjjejj 4 4 1 ‡ jje jjjjejj…14:54†where e is the actual error and e is the error of the recovered solution, e.g.jje jj ˆ jju ÿ u jjThe proof of the above theorem is straightforward if we write Eq. (14.52) asjjejj ˆ jju ÿ ^ujj ˆ jj…u ÿ ^u †ÿ… u ÿ u †jj ˆ jje ÿ e jj …14:55†Using now the triangle inequality we havejjejj ÿ jje jj 4 jjejj 4 jjejj ‡ jje jj…14:56†from which the inequality (14.54) follows after division by jjejj. Obviously, thetheorem is also true for error estimators of other norms. Two important conclusionsfollow:1. any recovery process which results in reduced error will give a reasonable errorestimator and, more importantly,2. if the recovered solution converges at a higher rate than the ®nite element solutionwe shall always have asymptotically exact estimation.To prove the second point we consider a typical ®nite element solution with shapefunctions of order p where we know that the error (in the energy norm) is:jjejj ˆ O…h p †…14:57†If the recovered solution gives an error of a higher order, e.g.,jje jj ˆ O…h p ‡ † >0 …14:58†

Other error estimators ± residual based methods 387then the bounds of the e€ectivity index are:1 ÿ O…h † 4 4 1 ‡ O…h † …14:59†and the error estimator is asymptotically exact, that is ! 1 as h ! 0 …14:60†This means that the error estimator converges to the true error. This is a very importantproperty of error estimators based on recovery not generally shared by residualbased estimators which we shall discuss in the next section.14.7 Other error estimators ± residual based methodsOther methods to obtain error estimators have been proposed by many investigatorsworking in the ®eld. 26ÿ34 Most of these make use of the residuals of the ®nite elementapproximation, either explicitly or implicitly. Error estimators based on thesemethods are often called residual error estimators. Those using residuals explicitlyare termed explicit residual error estimators; the others are called implicit residualerror estimators.In this section we are mainly concerned with implicit residual error estimators, inparticular, the equilibrated element residual estimator which has been shown to bethe most robust among all the residual error estimators. 35ÿ37Here we consider the heat conduction problem in a two-dimensional domain as anexample. The di€erential equation is given bywith boundary conditionsÿ r T … k r† ˆ Q in …14:61† ˆ on ÿ q T n ˆ q n ˆ q on ÿ qIn the aboveq ˆÿk ris the heat ¯ux, n is the outward normal to the boundary ÿ and q n is the ¯ux normal tothe boundary (see Chapters 3 and 7).The error of the ®nite element solution ise ˆ ÿ ^and for element i the energy norm error is written as …1jjejj ˆ …r† T 2k r d…14:62† iIn what follows we shall construct the equilibrated residual error estimator for thisproblem. The procedure of constructing an estimator for other problems, such aselasticity problems, is analogous.We start by considering an interior element i. Substitute the ®nite element solution^ into Eq. (14.61). Subtracting the resulting equation from Eq. (14.61) gives an

388 Errors, recovery processes and error estimateselement boundary value problem for error e given byÿ r T … k re† ˆ r i in i …14:63†with boundary conditionÿ…k re† T n ˆ q n ÿ ^q n on ÿ iHerer i ˆ r T … k r†‡Qis the residual in the ®nite element and^q n ˆ ^q T nis the ®nite element normal ¯ux.We notice immediately that Eq. (14.63) is not solvable because the exact normal¯ux on the element boundary is in general unknown. A natural strategy to overcomethis di culty is to replace the exact normal ¯ux by a recovered solution q n which canbe computed from the ®nite element ¯ux in element i and its surrounding elements.We can now write the boundary value problem of the element error asÿ r T … k re† ˆ r i in i …14:64†with boundary conditionÿ… k re† T n ˆ q n ÿ ^q n on ÿ iThe approximate solution of the above equations e in the energy norm, jjejj, isde®ned as the element residual error estimator.Various recovery techniques can be used to recover the normal ¯ux q n. 30;31However, the Neumann problem of Eq. (14.64) will guarantee to have a solution ifq n is computed such that the residuals satisfy……N j r i d ‡ N j … q n ÿ ^q n †dÿ ˆ 0 …14:65† i ÿ iwhere N j is the shape function for node j of element i. Although N j can be a shapefunction of any order, a linear shape function seems to be the most practical in thefollowing computation.The residuals which satisfy Eq. (14.65) are said to be equilibrated, thus therecovered solution q n satisfying Eq. (14.65) is called the equilibrated ¯ux. An errorestimator which uses the solution of the element error problem of Eq. (14.64) withthe equilibrated ¯ux q n is termed an equilibrated residual error estimator. This typeof residual error estimator was ®rst introduced by Bank and Weiser 30 and laterpursued by Ainsworth and Oden. 34It is apparent that the most important step in the computation of the equilibratedresidual error estimator is to achieve the recovered normal ¯ux q n which satis®es Eq.(14.65). Once q n is determined, the error problem Eq. (14.64) can be readily solved,over an element, following the standard ®nite element procedure. Therefore weshall focus on the recovery process.The technique of recovering normal ¯ux by equilibrated residuals was ®rstproposed by Ladeve ze et al. 23 A di€erent version of this technique was later usedby Ainsworth and Oden. 34

Integrating by parts, we can write Eq. (14.65) in a computationally more convenientform:…… N j Q d ÿ r T k r ^ …d ‡ N j q n dÿ ˆ 0 …14:66† i i ÿ iLet the recovered element boundary normal ¯ux, for each edge of the element, havethe formq n ˆ 12… ^q i ‡ ^q k † T n s ‡ Z s …14:67†where the ®rst term on the right-hand side is the average of the normal ¯ux of the®nite element solution from element i and its neighbour element k; n s is the outwardnormal on the edge s of element i; and Z s is a linear function de®ned on the edge s,shared by elements i and k, with end nodes l and r andZ s ˆ L l a s l ‡ L r a s r…14:68†withL l ˆ 2 … 2Nl s ÿ Nrs † Ljh s jr ˆ 2 … 2Nr s ÿ Nls † …14:69†jh s jwhere Nl s and Nr s are linear shape functions de®ned over edge s and h s is the length ofedge s. The unknown parameters a s l and a s r are to be determined from the residualequilibrium equation (14.66).It is easy to verify that…NmL s n dÿ ˆ mn…14:70†swhere mn is the Kronecker delta, is given by: ij ˆ 1; i ˆ j; ij ˆ 0; i 6ˆ j …14:71†Let X n denote a typical interior vertex node. Choose N j ˆ N n in Eq. (14.66) and considerthe element patch associated with the linear shape function N n as shown in Fig.14.14. A local numbering for the elements and edges connected to node X n in thepatch is given. The edge normals shown here are the results of a global edge orientation.Assume X n be the end node l of all the edges connected with X n . For element e 1 inthe patch, substituting Eq. (14.67) into Eq. (14.66) for each edge and observing thatN n is non-zero only on s 1 and s 2 and at the directions of the edge normals, we have………N n Q d ÿ … rN n † T …k r ^† 1d ÿ2 N n…^q e1 ‡ ^q e5 † T n s1 dÿ i is 1………1‡2 N n…^q e1 ‡ ^q e2 † T n s2 dÿ ÿ N n Z s1 dÿ dÿ ÿ N n Z s2 dÿ ˆ 0 …14:72†s 2 s 1 s 2where the boundary integral takes a negative sign if the edge normal shown inFig. 14.15 is inward for the element.Let f e1 denote the ®rst four, computable, terms of the above equation and noticethat [using Eq. (14.70)]……s 1N n Z s1 dÿ ˆOther error estimators ± residual based methods 389s 1N n …L xn a s 1x n‡ L r a s 1r † dÿ ˆ a s 1x n…14:73†

390 Errors, recovery processes and error estimatesS 2 S 3S 4e 2e 1e 3X nS 1e 5e 4S 5Fig. 14.14 Typical patch with interior vertex node X n showing a local numbering of elements e i and edges S i .and……ÿN n Z s2 dÿ ˆ N n L xn a s 2x n‡ L r a s 2r dÿ ˆ as 2x n…14:74†s 2 s 2Equation (14.71) now becomesÿa s 1x n‡ a s 2x n ˆÿf e1…14:75†kn sslriFig. 14.15 Element interface for equilibrated ¯ux recovery

Other error estimators ± residual based methods 391Similarly for element e 2 to e 5 we haveÿa s 2x n‡ a s 3x n ˆÿf e2or in matrix formwhereÿa s 3x nÿ a s 4x n ˆÿf e3‡a s 4x n‡ a s 5x n ˆÿf e4ÿa s 5x n‡ a s 1x n ˆÿf e5…14:76†Aa ˆ b23ÿ1 1 0 0 00 ÿ1 1 0 0A ˆ0 0 ÿ1 ÿ1 0674 0 0 0 1 151 0 0 0 ÿ1…14:77†…14:78†a ˆ a s 1x n; a s 2x n; a s 3x n; a s 4x n; a s 5 T…14:79†x nand b ˆ ÿf e1 ; ÿf e2 ; ÿf e3 ; ÿf e4 ; ÿf Te5…14:80†It is easy to verify that these equations are linearly dependent but have solutionsdetermined up to an arbitrary constant. A procedure to obtain an optimal particularsolution is described as follows. 23;30;38 First, a particular solution a 0 isfound by choosing, for example, a s 5x n ˆ 0. Secondly, the corresponding homogeneousequationAb ˆ 0…14:81†with b ˆ ‰ b 1 ; b 2 ; b 3 ; b 4 ; b 5 Š T is solved for a non-zero particular solution with the choiceof, corresponding a s 5x n, b 5 ˆ 1. It is easy to verify that b i is either 1 or ÿ1 due to thestructure of A. In the element patch considered here b ˆ ‰ 1; 1; ÿ1; ÿ1; 1Š T .The ®nal particular solution of Eq. (14.77) takes the forma ˆ a 0 ‡ b…14:82†where the constant is determined by the minimization ofThe minimization condition gives ˆ a T a…14:83† ˆÿ bT a 0b T …14:84†bThe solution gives the nodal value a s ix nfor each edge connected to node X n in theelement patch.

392 Errors, recovery processes and error estimatesBoundary nodes and their related element patches can be considered in the samefashion except that we can take q n ˆ q n , the known ¯ux, for the element edge beingpart of ÿ q : For edges coincident with ÿ , we let the ®rst term on the right-hand sideof Eq. (14.67) be zero. By considering each vertex node of the mesh and itsassociated element patch, we will be able to determine a s l and a s r for every edge,thus the recovered normal ¯ux q n on the element boundary is achieved. Theprocedure described above for recovering the normal ¯ux is a recovery by elementresidual.We note that the non-uniqueness of the solution of Eq. (14.77) represents the nonuniquenessof the equilibrium status of the element residuals. The choice of thearbitrary constant in solving Eq. (14.77) will certainly a€ect the accuracy of therecovered solution q n, and therefore the accuracy of the error estimator.The local error problem Eq. (14.64) is usually solved by a higher order (e.g., p ‡ 1or even p ‡ 2) approximation. The solution of the problem is then employed in theelement equilibrated error estimator jjejj i . The global error estimator jjejj isobtained through Eq. (14.15). The global error estimator has been shown to bean upper bound of the exact error, 34 although it is not a trivial task to prove itsconvergence.We have shown here that the recovery method is the key to the computation ofimplicit residual error estimators. It can be shown that using a properly designedrecovery method some of the explicit residual error estimators or their equivalentcan, in fact, be directly derived from recovery based error estimators. 39;40 Numericalperformance of residual based error estimators was tested by BabusÏ ka et al. 35ÿ37 andcompared with that of recovery based error estimators.14.8 Asymptotic behaviour and robustness of errorestimators ± the BabusÏka patch testIt is well known that elements in which polynomials of order p are used to representthe unknown u will reproduce exactly any problem for which the exact solution is alsode®ned by such a polynomial. Indeed the veri®cation of this behaviour is an essentialpart of the `patch test' which has to be satis®ed by all elements to ensure convergence,as we have discussed in Chapter 10.Thus if we are attempting to determine the error in a general smooth solution wewill ®nd that this error is dominated by terms of order p ‡ 1. The response of anypatch to an exact solution of order p ‡ 1 will therefore determine the asymptoticbehaviour when both the size of the patch and of all the elements tends to zero. Ifthe patch is assumed to be one of a repeatable kind, its behaviour when subjectedto an exact solution of order p ‡ 1 will give the exact asymptotic error of the ®niteelement solution. Thus, any estimator can be compared with this exact value andthe asymptotic e€ectivity index can be established. Figure 14.16 shows such arepeatable patch of quadrilateral elements which evaluate the performance forquite irregular meshes.We have indeed shown how true superconvergent behaviour reproduces exactlysuch higher order solutions and thus leads to an e€ectivity index of unity in the

Asymptotic behaviour and robustness of error estimators ± the BabusÏka patch test 393Fig. 14.16 Repeating patch of irregular and quadrilateral elements.asymptotic limit. In the papers presented by BabusÏ ka et al. 35ÿ37;41 the procedure ofdealing with such repeatable patches for various patterns of two-dimensionalelements is developed. Thus, if we are interested in solving the di€erential equationL…u†‡f ˆ 0…14:85†where L is a linear di€erential operator of order 2p, we consider exact solutions(harmonic solutions) to the homogeneous equation ( f ˆ 0) of the formu ex ˆ Xa m X m Y n ˆ P…x; y†a; n ˆ p ‡ 1 ÿ m …14:86†mThe boundary conditions are taken asu ex j x ‡ Lxˆ u ex j x and u ex j y ‡ Lyˆ u ex j y …14:87†where L x and L y are periods in the x and y directions, respectively (viz. repeatabilitySection 9.18). In general, the individual terms of Eq. (14.86) do not satisfy thedi€erential equation and it is necessary to consider linear combinations in terms ofthe parameters in L asa 0 ˆ Ta…14:88†This solution serves as the basis for conducting a patch test in which the boundaryconditions are assigned to be periodic and to prevent constant changes to u.y Thecorrect constant value may be computed from……… N~u h ‡ C†d ˆ u ex d…14:89†patchTo compute upper and lower bounds ( U and L ) on the possible e€ectivity indices,all possible combinations of the harmonic solution must be considered. This may beachieved by constructing an error norm of the solutions, for example the L 2 norm ofthe ¯ux (or stress)…jje q jj 2 L 2 ˆpatchpatch… q ex ÿ q h † T ÿ… q ex ÿ q h †d ˆ a 0 TT T E ex Ta 0…14:90†y For elasticity type problems the periodic boundary conditions prevent rigid rotations.

394 Errors, recovery processes and error estimatesTable 14.1 Robustness index for the equilibratedresiduals (ERpB) and SPR (ZZ-discrete) estimatorsfor a variety of anisotropic situations andelement patterns, p ˆ 2EstimatorRobustness indexERpB 10.21SPR (ZZ-discrete) 0.02and…jje q jj 2 L 2 ˆpatch… q re ÿ q h † T ÿ… q re ÿ q h †d ˆ a 0 TT T E re Ta 0…14:91†and solving the eigenproblemT T E re Ta 0 ˆ 2 T T E ex Ta 0…14:92†to determine the minimum (lower bound) and maximum (upper bound) e€ectivityindices. Further details of the process summarized here are given in Boroomandand Zienkiewicz 21;22 and by Zienkiewicz et al. 42These bounds on the e€ectivity index are very useful for comparing various errorestimators and their behaviour for di€erent mesh and element patterns. However, asingle parameter called the robustness index has also been devised 35 and is useful asa guide to the robustness of any particular estimatorR ˆ max j1 ÿ L j‡j1 ÿ U j; j1 ÿ 1 j‡j1 ÿ 1 j …14:93† L UA large value of this index obviously indicates a poor performance. Conversely thebest behaviour is that in which L ˆ U ˆ 1…14:94†and this givesR ˆ 0…14:95†In the series of tests reported in references 35±41 various estimators have beencompared. Table 14.1 shows the highest robustness index value of an equilibratingresidual based error estimator and the SPR recovery error estimator for a set ofparticular patches of triangular elements. 37This performance comparison is quite remarkable and it seems that in all the testsquoted by BabusÏ ka et al. 35ÿ41 the SPR recovery estimator performs best. Indeed weshall observe that in many cases of regular subdivision, when full superconvergenceoccurs the ideal, asymptotically exact solution characterized by R ˆ 0 will beobtained.In Table 14.2 we show some results obtained for regular meshes of triangles andrectangles with linear and quadratic elements. In the rectangular elements used forproblems of heat conduction type, superconvergent points are exact and the idealresult is obtained for both linear and quadratic elements. It is surprising that this

Asymptotic behaviour and robustness of error estimators ± the BabusÏka patch test 395Table 14.2 E€ectivity bounds and robustness of SPR and REP recovery estimator for regular meshes oftriangles and rectangles with linear and quadratic shape function (applied to heat conduction and elasticityproblems). Aspect ratio ˆ length(L)/height(H) of elements in patch testedLinear triangles and rectangles (heat conduction/elasticity)SPRREPAspect ratio L=H L U R L U R1/1 1.0000 1.0000 0.0000 1.0000 1.0000 0.00001/2 1.0000 1.0000 0.0000 1.0000 1.0000 0.00001/4 1.0000 1.0000 0.0000 1.0000 1.0000 0.00001/8 1.0000 1.0000 0.0000 1.0000 1.0000 0.00001/16 1.0000 1.0000 0.0000 1.0000 1.0000 0.00001/32 1.0000 1.0000 0.0000 1.0000 1.0000 0.00001/64 1.0000 1.0000 0.0000 1.0000 1.0000 0.0000Quadratic rectangles (heat conduction) L U R L U R1/1 1.0000 1.0000 0.0000 1.0000 1.0000 0.00001/2 1.0000 1.0000 0.0000 1.0000 1.0000 0.00001/4 1.0000 1.0000 0.0000 1.0000 1.0000 0.00001/8 1.0000 1.0000 0.0000 1.0000 1.0000 0.00001/16 1.0000 1.0000 0.0000 1.0000 1.0000 0.00001/32 1.0000 1.0000 0.0000 1.0000 1.0000 0.00001/64 1.0000 1.0000 0.0000 1.0000 1.0000 0.0000Quadratic rectangles (elasticity) L U R L U R1/1 1.0000 1.0000 0.0000 0.9991 1.0102 0.01111/2 1.0000 1.0000 0.0000 0.9991 1.0181 0.01891/4 1.0000 1.0000 0.0000 0.9991 1.0136 0.01451/8 1.0000 1.0000 0.0000 0.9991 1.0030 0.00391/16 1.0000 1.0000 0.0000 0.9968 1.0001 0.00331/32 1.0000 1.0000 0.0000 0.9950 1.0000 0.00501/64 1.0000 1.0000 0.0000 0.9945 1.0000 0.0055Quadratic triangles (elasticity) L U R L U R1/1 0.9966 1.0929 0.0963 0.9562 1.0503 0.09401/2 0.9966 1.0931 0.0965 0.9559 1.0481 0.09231/4 0.9967 1.0937 0.0970 0.9535 1.0455 0.09241/8 0.9967 1.0943 0.0976 0.9522 1.0603 0.10811/16 0.9966 1.0946 0.0980 0.9518 1.0666 0.11481/32 0.9966 1.0947 0.0981 0.9517 1.0684 0.11671/64 0.9965 1.0947 0.0982 0.9516 1.0688 0.1172also occurs in elasticity where the proof of superconvergent points is lacking (for>0). Further, the REP procedure also seems to yield superconvergence exceptfor elasticity with quadratic elements.For regular meshes of quadratic triangles generally superconvergence is notexpected and it does not occur for either heat conduction or elasticity problems.However, the robustness index has very small values (R < 0:10 for SPR andR < 0:12 for REP) and these estimators are therefore very good.

396 Errors, recovery processes and error estimates(a)(e)(b)(f)(c)(g)(d)(h)Fig. 14.17.

Asymptotic behaviour and robustness of error estimators ± the BabusÏka patch test 397In Fig. 14.17 and Table 14.3 very irregular meshes of triangular and quadrilateralelements are analysed in repeatable patterns. It is of course not possible to presenthere all tests conducted by the e€ectivity patch test. The results shown are, however,typical ± others are given in reference 21. It is interesting to observe that theTable 14.3 E€ectivity bounds and robustness of SPR and REP recovery estimator for irregular meshes oftriangles (a, b, c, d) and quadrilaterals (e, f, g, h)Linear element (heat conduction)SPRREPMesh pattern L U R L U Ra 0.9626 1.0054 0.0442 0.9709 1.0145 0.0443b 0.9715 1.0156 0.0447 0.9838 1.0167 0.0329c 0.9228 1.4417 0.5189 0.8938 1.8235 0.9297d 0.8341 1.2027 0.3685 0.9463 1.9272 0.9810e 0.9943 1.0175 0.0232 0.9800 1.0589 0.0789f 0.9969 1.0152 0.0183 0.9849 1.0582 0.0733g 0.9987 1.0175 0.0188 0.9987 1.0175 0.0188h 0.9991 1.0068 0.0077 0.9979 1.0062 0.0083Linear elements (elasticity)SPRREP L U R L U Ra 0.9404 1.0109 0.0741 0.9468 1.0148 0.0707b 0.8869 1.0250 0.1520 0.9392 1.0275 0.0915c 0.8550 1.6966 0.8415 0.8037 2.0522 1.2486d 0.7945 1.2734 0.4788 0.7576 1.9416 1.1840e 0.9946 1.0247 0.0301 0.9579 1.0508 0.0928f 1.0038 1.0281 0.0318 0.9612 1.0467 0.0855g 0.9959 1.0300 0.0341 0.9960 1.0298 0.0338h 0.9972 1.0139 0.0168 0.9965 1.0122 0.0157Quadratic elements (heat conduction) L U R L U Ra 0.9443 1.0295 0.0877 0.9339 1.0098 0.0805b 0.8146 1.0037 0.2313 0.9256 1.0028 0.0832c 0.7640 1.0486 0.3000 0.9559 1.2229 0.2670d 0.8140 1.0141 0.2423 0.9091 1.2808 0.3717e 0.9762 1.0053 0.0296 0.9901 1.0177 0.0276f 0.9691 1.0045 0.0363 0.9901 1.0322 0.0421g 0.9692 1.0004 0.0322 0.9833 1.0024 0.0195h 0.9906 1.0113 0.0207 1.0045 1.0261 0.0307Quadratic elements (elasticity) L U R L U Ra 0.9144 1.0353 0.1277 0.9197 1.0244 0.1111b 0.7302 1.0355 0.4038 0.8643 1.0346 0.1905c 0.7556 1.1024 0.4163 0.8387 1.2422 0.4035d 0.7624 1.0323 0.3430 0.8244 1.2632 0.4388e 0.9702 1.0102 0.0408 0.9682 1.0058 0.0386f 0.9651 1.0085 0.0446 0.9749 1.0286 0.0537g 0.9457 1.0115 0.0688 0.9807 1.0125 0.0321h 0.9852 1.0141 0.0290 0.9996 1.0522 0.0526

398 Errors, recovery processes and error estimatesperformance measured by the robustness index on quadrilateral elements is alwayssuperior to that measured on triangles.The results in a recent paper of BabusÏ ka et al. 41 show that alternative versions ofSPR (such as references 17, 18, 43) generally give much worse robustness indexperformance than the original version, especially on irregular elements assemblednear boundaries.14.9 Which errors should concern us?In this chapter we have shown how various recovery procedures can accuratelyestimate the overall error of the ®nite element approximation and thus provide avery accurate error estimating method. We have also shown how superior are estimatorsbased on SPR recovery to those based on residual computation. The errorestimation discussed here concerns however only the original solution and if theuser takes advantage of the recovered values a much better solution is already available.In the next chapter we shall be concerned with adaptivity processes aiming atreduction of the original ®nite element error for which a vast body of literaturealready exists. Here again we shall show the excellent values of the e€ectivityindex which can be obtained with SPR type methods on examples for which an`exact' solution is available from very ®ne mesh computations. What perhaps weshould also be concerned with are the errors remaining in the recovered solutions,if indeed these are to be made use of. This problem is still unsolved and at themoment all the adaptive methods simply aim at the reduction of various normsof error in the ®nite element solution directly provided.References1. J. Barlow. Optimal stress locations in ®nite element models. Internat. J. Num. Meth. Eng.,10, 243±51, 1976.2. L.R. Herrmann. Interpretation of ®nite element procedures in stress error minimization.Proc. Am. Soc. Civ. Eng., 98(EM5), 1331±36, 1972.3. T. Moan. Orthogonal polynomials and `best' numerical integration formulas on a triangle.ZAMM, 54, 501±8, 1974.4. E. Hinton and J. Campbell. Local and global smoothing of discontinuous ®nite elementfunction using a least squares method. Internat. J. Num. Meth. Eng., 8, 461±80, 1974.5. M. Krizek and P. Neitaanmaki. On superconvergence techniques. Acta. Appl. Math., 9,75±198, 1987.6. Q.D. Zhu and Q. Lin. Superconvergence Theory of the Finite Element Methods. HunanScience and Technology Press, Hunan, China, 1989.7. L.B. Wahlbin. Superconvergence in Galerkin Finite Element Methods. Lectures Notes inMathematics, Vol. 1605. Springer, Berlin, 1995.8. C.M. Chen and Y. Huang. High Accuracy Theory of Finite Element Methods. HunanScience and Technology Press, Hunan, China, 1995.9. Q. Lin and N. Yan. Construction and Analyses of Highly E€ective Finite Elements. HebeiUniversity Press, Hebei, China, 1996.10. H.J. Brauchli and J.T. Oden. On the calculation of consistent stress distributions in ®niteelement applications. Internat. J. Num. Meth. Eng., 3, 317±25, 1971.

11. O.C. Zienkiewicz and J.Z. Zhu. Superconvergent patch recovery and a posteriori error estimationin the ®nite element method, Part I: A general superconvergent recovery technique.Internat. J. Num. Meth. Eng., 33, 1331±64, 1992.12. O.C. Zienkiewicz and J.Z. Zhu. The superconvergent patch recovery (SPR) and a posteriorierror estimates. Part 2: Error estimates and adaptivity. Internat. J. Num. Meth. Eng., 33,1365±82, 1992.13. O.C. Zienkiewicz and J.Z. Zhu. The superconvergent patch recovery (SPR) and adaptive®nite element re®nement. Comp. Meth. Appl. Mech. Eng., 101, 207±24, 1992.14. O.C. Zienkiewicz, J.Z. Zhu, and J. Wu. Superconvergent recovery techniques ± somefurther tests. Commun. Num. Meth. Eng., 9, 251±58, 1993.15. Z. Zhang. Ultraconvergence of the patch recovery technique. Math. Comput., 65, 1431±37,1996.16. B. Li and Z. Zhang. Analysis of a class of superconvergence patch recovery techniques forlinear and bilinear ®nite elements. Num. Meth. Partial Di€. Eq., 15, 151±67, 1999.17. N.-E. Wiberg, F. Abdulwahab, and S. Ziukas. Enhanced superconvergent patch recoveryincorporating equilibrium and boundary conditions. Internat. J. Num. Meth. Eng., 37,3417±40, 1994.18. T.D. Blacker and T. Belytschko. Superconvergent patch recovery with equilibrium andconjoint interpolation enhancements. Internat. J. Num. Meth. Eng., 37, 517±36, 1994.19. N.-E. Wiberg and X.D. Li. Superconvergent patch recovery of ®nite element solutions anda posterior l 2 norm error estimate. Commun. Num. Meth. Eng., 10, 313±20, 1994.20. X.D. Li and N.-E. Wiberg. A posteriori error estimate by element patch postprocessing,adaptive analysis in energy and L 2 norm. Comp. Struct., 53, 907±19, 1994.21. B. Boroomand and O.C. Zienkiewicz. Recovery by equilibrium patches (REP). Internat. J.Num. Meth. Eng., 40, 137±54, 1997.22. B. Boroomand and O.C. Zienkiewicz. An improved REP recovery and the e€ectivityrobustness test. Internat. J. Num. Meth. Eng., 40, 3247±77, 1997.23. P. LadeveÁ ze and D. Leguillon. Error estimate procedure in the ®nite element method andapplications. SIAMJ. Num. Anal., 20(3), 485±509, 1983.24. P. LadeveÁ ze, G. Co gnal, and J.P. Pelle. Accuracy of elastoplastic and dynamicanalysis. In I. BabusÏ ka, O.C. Zienkiewicz, J. Gago, and E.R. de A. Oliviera, editors.Accuracy Estimates and Adaptive Re®nements in Finite Element Computations, chapter11, 1986.25. O.C. Zienkiewicz and J.Z. Zhu. A simple error estimator and adaptive procedure forpractical engineering analysis. Internat. J. Num. Meth. Eng., 24, 337±57, 1987.26. I. BabusÏ ka and C. Rheinboldt. A-posteriori error estimates for the ®nite element method.Internat. J. Num. Meth. Eng., 12, 1597±1615, 1978.27. I. BabusÏ ka and W.C. Rheinboldt. Analysis of optimal ®nite element meshes in r 1 . Math.Comp., 33, 435±63, 1979.28. O.C. Zienkiewicz, J.P. De S.R. Gago, and D.W. Kelly. The hierarchical concept in ®niteelement analysis. Comp. Struct., 16(53-65), 53±65, 1983.29. D.W. Kelly, J.P. De S.R. Gago, O.C. Zienkiewicz, and I. BabusÏ ka. A posteriori erroranalysis and adaptive processes in the ®nite element method: Part 1 ± Error analysis.Internat. J. Num. Meth. Eng., 19, 1593±1619, 1983.30. R.E. Bank and A. Weiser. Some a posteriori error estimators for elliptic partial di€erentialequations. Math. Comput., 44, 283±301, 1985.31. J.T. Oden, L. Demkowicz, W. Rachowicz, and Westermann T, A. Toward a universal h-padaptive ®nite element strategy. Part 2: A posteriori error estimation. Comp. Meth. Appl.Mech. Eng., 77, 113±80, 1989.32. R. Verfurth. A posteriori error estimators for the stokes equations. Numer. Math., 55,309±25, 1989.References 399

400 Errors, recovery processes and error estimates33. C. Johnson and P. Hansbo. Adaptive ®nite element methods in computational mechanics.Comp. Meth. Appl. Mech. Eng., 101, 143±81, 1992.34. M. Ainsworth and J.T. Oden. A uni®ed approach to a posteriors error estimation usingelement residual methods. Numerische Mathematik, 65, 23±50, 1993.35. I. BabusÏ ka, T. Strouboulis, and C.S. Upadhyay. A model study of the quality of aposteriori error estimators for linear elliptic problems. Error estimation in the interior ofpatchwise uniform grids of triangles. Comp. Meth. Appl. Mech. Eng., 114, 307±78, 1994.36. I. BabusÏ ka, T. Strouboulis, C.S. Upadhyay, S.K. Gangaraj, and K. Copps. Validation ofa posteriori error estimators by numerical approach. Internat. J. Num. Meth. Eng., 37,1073±1123, 1994.37. I. BabusÏ ka, T. Strouboulis, C.S. Upadhyay, S.K. Gangaraj, and K. Copps. An objectivecriterion for assessing the reliability of a posteriori error estimators in ®nite elementcomputations. U.S.A.C.M. Bulletin, No. 7, 4±16, 1994.38. P. Ladeve ze, J.P. Pelle, and P. Rougeot. Error estimation and mesh optimization forclassical ®nite elements. Engng. Comput., 8, 69±80, 1991.39. J.Z. Zhu and O.C. Zienkiewicz. Superconvergence recovery technique and a posteriorierror estimators. Internat. J. Num. Meth. Eng., 30, 1321±39, 1990.40. J.Z. Zhu. A posteriori error estimation ± the relationship between di€erent procedures.Comp. Meth. Appl. Mech. Eng., 150, 411±22, 1997.41. I. BabusÏ ka, T. Strouboulis, and C.S. Upadhyay. A model study of the quality of a posteriorierror estimators for ®nite element solutions of linear elliptic problems, with particularreference to the behavior near the boundary. Internat. J. Num. Meth. Eng., 40, 2521±77,1997.42. O.C. Zienkiewicz, B. Boroomand, and J.Z. Zhu. Recovery procedures in error estimationand adaptivity: Adaptivity in linear problems. In P. LadeveÁ ze and J.T. Oden, editors,Advances in Adaptive Computational Mechanics in Mechanics, pages 3±23. Elsevier ScienceLtd., 1998.43. N.-E. Wiberg and F. Abdulwahab. Patch recovery based on superconvergent derivativesand equilibrium. Internat. J. Num. Meth. Eng., 36, 2703±24, 1993.

15Adaptive ®nite element re®nement15.1 IntroductionIn the previous chapter we have discussed at some length various methods of recoveryby which the ®nite element solution results could be made more accurate and this ledus to devise various procedures for error estimation. In this chapter we shall beconcerned with methods which can be used to reduce the errors generally once a®nite element solution has been obtained. As the process depends on previous resultsat all stages it is called adaptive. Such adaptive methods were ®rst introduced to ®niteelement calculations by BabusÏ ka and Rheinbolt in the late 1970s. 1;2 Before proceedingfurther it is necessary to clarify the objectives of re®nement and specify`permissible error magnitudes' and here the engineer or user must have very clearaims. For instance the naive requirement that all displacements or all stressesshould be given within a speci®ed tolerance is not always acceptable. The reasonsfor this are obvious as at singularities, for example, stresses will always be in®niteand therefore no ®nite tolerance could be speci®ed. The same di culty is true fordisplacements if point or knife edge loads are considered. The most common criterionin general engineering use is that of prescribing a total limit of the error computed inthe energy norm. Often this error is required not to exceed a speci®ed percentage ofthe total energy norm of the solution and in the many examples presented later weshall use this criterion. However, using a recovery type of error estimator it is possibleto adaptively re®ne the mesh so that the accuracy of a certain quantity of interest,such as the RMS error in displacement and/or RMS error in stress (see Chapter14, Eqs. (14.10a) and (14.10b)), satisfy some user-speci®ed criterion. We shouldrecognize that mesh re®nement based on reducing the RMS error in displacementis in e€ect reducing the average displacement error in each element; similarly meshre®nement based on reducing the RMS error in stress is the same as reducing the averagestress error in each element. Here we could, for instance, specify directly the permissibleerror in stresses or displacements at any location. Some investigators (e.g.,Zienkiewicz and Zhu 3 ) have used RMS error in stress in the adaptive mesh re®nementto obtain more accurate stress solutions. Others (e.g., OnÄ ate and Bugeda 4 ) have usedthe requirement of constant energy norm density in the adaptive analysis, which is infact equivalent to specifying a uniform distribution of RMS error in stress in eachelement. We note that the recovery type of error estimators are particularly useful

402 Adaptive ®nite element re®nementand convenient in designing adaptive analysis procedures for the quantities ofinterest.As we have already remarked in the previous chapter we will at all times consider theerror in the actual ®nite element solution rather than the error in the recovered solution.It may indeed be possible in special problems for the error in the recovered solution tobe zero, even if the error in the ®nite element solution itself is quite substantial. (Considerhere for instance a problem with a linear stress distribution being solved by linearelements which result in constant element stresses. Obviously the element error will bequite large. But if recovered stresses are used, exact results can be obtained and noerrors will exist.) The problem of which of the errors to consider still needs to beanswered. At the present time we shall consider the question of recovery as that of providinga very substantial margin of safety in the de®nition of errors.Various procedures exist for the re®nement of ®nite element solutions. Broadlythese fall into two categories:1. The h-re®nement in which the same class of elements continue to be used but arechanged in size, in some locations made larger and in others made smaller, toprovide maximum economy in reaching the desired solution.2. The p-re®nement in which we continue to use the same element size and simplyincrease, generally hierarchically, the order of the polynomial used in theirde®nition.It is occasionally useful to divide the above categories into subclasses, as the h-re®nement can be applied and thought of in di€erent ways. In Fig. 15.1 we illustratethree typical methods of h-re®nement.1. The ®rst of these h-re®nement methods is element subdivision (enrichment)[Fig. 15.1(b)]. Here re®nement can be conveniently implemented and existing elements,if they show too much error, are simply divided into smaller ones keepingthe original element boundaries intact. Such a process is cumbersome as manyhanging points are created where an element with mid-side nodes is joined to alinear element with no such nodes. On such occasions it is necessary to providelocal constraints at the hanging points and the calculations become more involved.In addition, the implementation of de-re®nement requires rather complex datamanagement which may reduce the e ciency of the method. Nevertheless, themethod of element subdivision is quite widely used.2. The second method is that of a complete mesh regeneration or remeshing[Fig. 15.1(c)]. Here, on the basis of a given solution, a new element size is predictedin all the domain and a totally new mesh is generated. Thus a re®nement and dere®nementare simultaneously allowed. This of course can be expensive, especiallyin three dimensions where mesh generation is di cult for certain types of elements,and it also presents a problem of transferring data from one mesh to another.However, the results are generally much superior and this method will be usedin most of the examples shown in this chapter. For many practical engineeringproblems, particularly of those for which the element shape will be severely distortedduring the analysis, adaptive mesh regeneration is a natural choice.3. The ®nal method, sometimes known as r-re®nement [Fig. 15.1(d)], keeps the totalnumber of nodes constant and adjusts their position to obtain an optimal

Introduction 403(a) Original mesh(b) Mesh enhancement by subdivision (enrichment)(c) Mesh enhancement by remeshing(d) r-refinement of original mesh by reposition of nodesFig. 15.1 Various procedures by h-re®nement.

404 Adaptive ®nite element re®nementapproximation. 5ÿ7 While this procedure is theoretically of interest it is di cult touse in practice and there is little to recommend it. Further it is not a true re®nementprocedure as a prespeci®ed accuracy cannot generally be reached.We shall see that with energy norms speci®ed as the criterion, it is a fairly simplematter to predict the element size required for a given degree of approximation.Thus very few re-solutions are generally necessary to reach the objective.With p-re®nement the situation is di€erent. Here two subclasses exist:1. One in which the polynomial order is increased uniformly throughout the wholedomain;2. One in which the polynomial order is increased locally using hierarchical re®nement.In neither of these has a direct procedure been developed which allows the predictionof the best re®nement to be used to obtain a given error. Here the procedures generallyrequire more resolutions and tend to be more costly. However, the convergencefor a given number of variables is more rapid with p-re®nement and it has much torecommend it.On occasion it is possible to combine e ciently the h- and p-re®nements and call itthe hp-re®nement. In this procedure both the size of elements h and their degree ofpolynomial p are altered. Much work has been reported in the literature by BabusÏ ka,Oden and others and the interested reader is referred to the references. 8ÿ18 In the nexttwo sections, Sec. 15.2 and 15.3, we shall discuss both the h- and the p-re®nements. InSec. 15.3 we also include some details of the very simple and yet e cient hp-re®nementprocess introduced by Zienkiewicz, Zhu and Gong. 1915.2 Some examples of adaptive h-re®nement15.2.1 Mesh regeneration proceduresIn the introduction to this chapter we have mentioned several alternative processes of h-adaptivity and we suggested that the process in which the complete mesh is regeneratedis in general the most e cient. Such a procedure allows elements to be de-re®ned (orenlarged) as well as re®ned (made smaller) and invariably starts at each stage of the analysisfrom a speci®cation of the mesh size de®ned at each nodal point of the previousmesh. Standard interpolation is used to ®nd the size of elements required at anypoint in the domain. This interpolation helps in the re®nement subsequently. Indeedat the starting point an initial mesh need not include the boundaries of the problemas it will be used only to interpolate the sizes required in the domain during the processof mesh generation. However, after this ®rst stage of analysis as the re®nement proceedsthe mesh sizes will be speci®ed at the nodes of the last mesh.In Chapter 9of this book, where we discussed mapping, we also discussed variouspossible mesh generators. These did not allow a mesh size variation of the re®ned kindto be speci®ed. In adaptivity it is very important to be able to de®ne quite precisely theelement size or density of mesh so that a minimum number of elements can be used.The generators which can do this have been developed since the mid-1980s. The®rst of these by Peraire et al. 20 was applied to aerospace engineering and ¯uid

here we have used 36 jjejj 2 ˆjjujj 2 ÿjj^ujj 2 …15:3†Some examples of adaptive h-re®nement 405mechanics calculations. Its basis is the frontal method of mesh generation developedoriginally by Cavendish 21 and Lo 22 and the original generator was made availableonly for triangular elements. Later such generators were generalized to include tetrahedralelements in three-dimensional space. 23 Today both triangular and tetrahedralgenerators form the basis of most adaptive codes.Extension to quadrilateral and hexahedral elements is by no means easy. First,procedures for generating quadrilateral elements in two dimensions have beendevised. The work of Zhu and Zienkiewicz 24;25 and Rank et al. 26;27 has to be noted.The procedures are based on the joining of two triangles into a quadrilateral atdi€erent stages of the mesh generation process.However, so far no extension of such methodologies to hexahedral elements inspace have been made. To the knowledge of the authors no e cient hexahedralmesh generators exist for adaptivity, though very many attempts have been reportedin the literature. 28ÿ33In the more recent mesh generators used for both triangles and tetrahedra thefrontal procedure has been largely replaced by Delauney triangulation and thereader is well advised to consult the following references and texts. 34;3515.2.2 Predicting the required element size in h adaptivityThe error estimators discussed in the previous chapter allow the global energy (orsimilar) norm of the error to be determined and the errors occurring locally (at theelement level) are usually also well represented. If these errors are within the limitsprescribed by the analyst then clearly the work is completed. More frequently theselimits are exceeded and re®nement is necessary. The question which this sectionaddresses is how best to e€ect this re®nement. Here obviously many strategies arepossible and much depends on the objectives to be achieved.In the simplest case we shall seek, for instance, to make the relative energy normpercentage error less than some speci®ed value (say 5% in many engineeringapplications). Thus 4 …15:1†is to be achieved.In an `optimal mesh' it is desirable that the distribution of energy norm error (i.e.,jjejj k ) should be equal for all elements. Thus if the total permissible error is determined(assuming that it is given by the result of the approximate analysis) asÿPermissible error jjujj jj^ujj 2 ‡jjejj 2 1=2…15:2†We could pose a requirement that the error in any element k should be jj^ujj 2 ‡jjejj 2 1=2jjejj k < emmwhere m is the number of elements involved.…15:4†

406 Adaptive ®nite element re®nementElements in which the above is not satis®ed are obvious candidates for re®nement.Thus if we de®ne the ratiojjejj kˆ e k …15:5†mwe shall re®ne whenevery k > 1…15:6† k can be approximated, of course, by replacing the true error in Eqs (15.4) and (15.5)with the error estimators.The re®nement could be carried out progressively by re®ning only a certain numberof elements in which is higher than a certain limit and at each time of re®ning halvethe size of such elements. This type of element subdivision process is also known asmesh enrichment. This process of re®nement though ultimately leading to a satisfactorysolution being obtained with a relatively small number of total degrees offreedom, is in general not economical as the total number of trial solutions may beexcessive.It is more e cient to try to design a completely new mesh which satis®es therequirement that k 4 1…15:7†One possibility here is to invoke the asymptotic convergence rate criteria at theelement level (although we have seen that these are not realistic in the presence ofsingularities) and to predict the element size distribution. For instance, if we assumejjejj k / h p k…15:8†where h k is the current element size and p the polynomial order of approximation,then to satisfy the requirement of Eq. (15.4) the new generated element size shouldbe no larger thanh new ˆ ÿ1=pkh k …15:9†Mesh generation programs in which the local element size can be speci®ed areavailable now as we have already stated and these can be used to design a newmesh for which the re-analysis is carried out. 20;24 In the ®gures we show how startingfrom a relatively coarse solution a single mesh prediction often allows a solution(almost) satisfying the speci®ed accuracy requirement to be achieved.The reason for the success of the mesh regeneration based on the simple assumptionof asymptotic convergence rate implied in Eq. (15.8) is the fact that with re®nementthe mesh tends to be `optimal' and the localized singularity in¯uence no longer a€ectsthe overall convergence.Of course the e€ects of singularity will still remain present in the elements adjacentto it and improved mesh subdivision can be obtained if in such elements we use theappropriate convergence and write, if in Eqs (15.8) and (15.9) p is replaced by ,see Chapter 14, Eq. (14.17)h new ˆ ÿ1=kh k …15:10†y We can indeed `de-re®ne' or use a larger element spacing where k < 1 if computational economy isdesired.

Number of degrees of freedom10 30 50 100 250 500 1000 2500 5000 10 000 30 000log ||e||–0.2–0.4–0.6–0.8–1.0–1.21h = 26 node elementsAdaptive refinement1.0141.0y16x6 node elementsUniform refinement18 110 1121.00.356 = λ/2N ≈ 3360Aim504030201054Relative error (%)–1.4–1.6Solutions reached in a single adaptivemesh regeneration aiming at 5% accuracy321.01.52.02.5log N3.03.54.04.5Fig. 15.2 The in¯uence of initial mesh to convergence rates in h version. Adaptive re®nement using quadratic triangular elements. Problem of Fig. 15.3. Note that if initialmesh is ®ner than h ˆ 1=8 adaptive re®nement reduces the number of equations.

408 Adaptive ®nite element re®nementin which is the singularity strength. A convenient number to use here is ˆ 0:5 asmost singularity parameters lie in the range 0:5±1:0. With this procedure, added tothe re®nement strategy, we frequently achieve accuracies better than the prescribedlimit in one remeshing.In the examples which follow we will show in general a process of re®nement inwhich the total number of degrees of freedom increases with each stage, eventhough the mesh is redesigned. This need not necessarily be the case as a ®ne butbadly structured mesh can show much greater error than a near-optimal one. Toillustrate this point we show in Fig. 15.2 the one stage re®nement designed to reachPoisson's ratio, ν = 0.3Thickness, t = 1.0yPlane strain conditionsp = 1.01.01.0 xMesh 1Mesh 2Mesh 3Fig. 15.3 Short cantilever beam and adaptive meshes of linear triangular elements.

Some examples of adaptive h-re®nement 409Poisson’s ratio, ν = 0.3Plane strain conditionsP = 1.01.01.0Mesh 1 (40 d.o.f) η = 27.0%Mesh 2 (228 d.o.f) η = 7.0%Mesh 3 (286 d.o.f) η = 4.0%Fig. 15.4 Adaptive mesh of quadratic triangular elements for short cantilever beam.5% accuracy in one step starting from uniform mesh subdivisions. The problem hereis the same as illustrated in Figs 15.3, 15.4, and 15.5 and in the re®nement process weuse both the mesh criteria of Eqs (15.9) and (15.10). 37 This problem refers to a shortstubby cantilever beam in which two very high singularities exist at the cornersattached to a rigid wall. In Fig. 15.4 we show three stages of an adaptive solutionand in Fig. 15.5 we indicate how rapidly this converges although all uniform re®nementsconverge at a very slow rate (due to the singularities).We note that now, in at least one re®nement, a decrease of total error occurs with areduction of total degrees of freedom (starting from a uniform 8 8 subdivision withNDF ˆ 544 and ˆ 9:8% to ˆ 3:1% with NDF ˆ 460).

410 Adaptive ®nite element re®nement103050Number of degrees of freedom100 250 500 1000 250050001000030000Log ||e||–0.2–0.4–0.6–1.0–1.2–1.4–1.66-node elementAdaptive refinement1.01.0Average slope1.02.526-node elementUniform refinement1.09-node elementUniform refinement0.356 = λ/250403020105432Relative error (%)1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5Log Nλ/2 = 0.356, theoretical rate of convergence for uniform refinementP/2 = 1.0, maximum rate of convergenceFig. 15.5 Experimental rates of convergence for short cantilever beam.The same problem is also solved by both mesh enrichment and mesh regenerationusing linear quadrilateral elements to achieve 5% accuracy. The prescribed accuracyis obtained with optimal rate of convergence being reached by both adaptive re®nementprocesses (Fig. 15.6). However, the mesh enrichment method requires seven0.5Number degrees of freedom10 30 100 300 1000 3000Log ||e||0–0.5–1.0–1.50.51.0Adaptive mesh enrichmentAdaptive mesh regeneration7040201053Relative error (%)–2.01.0 2.0 3.0 4.0Log N1P/2 = 0.5, optimal rate of convergenceFig. 15.6 Short cantilever beam. Mesh enrichment versus mesh regeneration using linear quadrilateralelements.

Some examples of adaptive h-re®nement 411yp = 1.01.01.0Mesh 1 η = 43.20%xMesh 2 η = 28.20% Mesh 3 η = 17.55% Mesh 4 η = 11.29%Mesh 5 η = 7.53% Mesh 6 η = 5.65% Mesh 7 η = 4.87%Fig. 15.7 Short cantilever solved by mesh enrichment. Linear quadrilateral elements.re®nements, as shown in Fig. 15.7, while mesh regeneration requires only three (seeFig. 15.8). Here the re®nement criterion, Eq. (15.8), is used for the mesh enrichmentprocess.As we mentioned earlier, the value of the energy norm error is not necessarily thebest criterion for practical re®nement. Limits on the local stress error can be usede€ectively. Such errors are quite simply obtained by the recovery processes describedin the previous chapter (SPR in Section 14.4 and REP in Section 14.5). In Fig. 15.9weshow a simple exercise recently conducted by OnÄ ate and Bugeda 4 in which a re®nementof a stressed cylinder is made using various criteria as described in the captionof Fig. 15.9. It will be observed that the stress tolerance method generally needs amuch ®ner mesh.Mesh 1 η = 43.20% Mesh 2 η = 9.60% Mesh 3 η = 5.00%Fig. 15.8 Short cantilever solved by mesh regeneration. Linear quadrilateral elements.

412 Adaptive ®nite element re®nementd.o.f.:236 236 236 236d.o.f.: 2348321634941732d.o.f.: 1980 7034 6944 1984d.o.f.: 2224737673982318(a) (b) (c) (d)Fig. 15.9 Sequence of adaptive mesh re®nement strategies based on (a) equal distribution of the global energeticerror between all the elements, (b) equal distribution of the density of energetic error, (c) equaldistribution of the maximum error in stresses at each point, and (d) equal distribution of the maximum percentageof the error in stresses at each point. All ®nal meshes have less than 5% energy norm error.15.2.3 Some further examplesWe shall now present further typical examples of h-re®nement with mesh adaptivity.In all of these, full mesh regeneration is used at every step.Example 1. A Poisson equation in a square domain This example is fairly straightforwardand starts from a simple square domain in which suitable loading termsexist in a Poisson equation to give the solution shown in Fig. 15.10. In Fig. 15.11we show the ®rst subdivision of this domain into regular linear and quadraticelements and the subsequent re®nements. The elements are of both triangular and

Some examples of adaptive h-re®nement 413(a)Base contour value = –0.365100Maximum contour value = 0.852200Contour interval = 0.060865(b)Base contour value = –0.365100Maximum contour value = 0.852200Contour interval = 0.060865Fig. 15.10 Poisson equation `exact' solutions. (a) @u=@x contours; (b) @u=@y contours.quadrilateral shape and for the linear ones a target error of 10% in total energy hasbeen set, while for quadratic elements the target error is 1% of total energy. Inpractically all cases three re®nements su ce to reach a very accurate solution satisfyingthe requirements despite the fact that the original mesh cannot capture in anyway the high intensity region illustrated in the previous ®gure. It is of interest tonote that the e€ectivity indices in all cases are very good ± this is true even for theoriginal re®nement. Figure 15.12 shows the convergence for various elements withthe error plotted against the total number of degrees of freedom. The readershould note that the asymptotic rate of convergence is exceeded when the re®nementgets closer to its ®nal objective.Example 2. An L-shaped domain It is of interest to note the results in Fig. 15.13which come from an analysis of a re-entrant corner using isoparametric quadraticquadrilaterals. Here a single re®nement is shown together with the convergence ofthe solution.Example 3. A machine part For this machine part problem plane strain conditions areassumed. A prescribed accuracy of 5% relative error is achieved in one adaptivere®nement (see Fig. 15.14) with linear quadrilateral elements. The convergence ofthe shear stress xy is shown in Fig. 15.15.Example 4. A perforated gravity dam The ®nal example of this section shows a morepractical engineering problem of a perforated dam. This dam was analysed in the late1960s during its construction. More recently, the problem was given to a youngengineer to choose a suitable mesh of quadratic triangles. Figure 15.16(a) showsthe mesh chosen. Despite the high order of elements the error is quite high, beingaround 17%. One stage of re®nement with a speci®ed value of 5% error in energynorm reaches this in a single operation. As we have seen in previous examples suchconvergence is not always possible but it is achieved here. We believe this typical

414 Adaptive ®nite element re®nement(a)Mesh 132 elements (9 d.o.f.)η = 83.320% θ* = 0.431θ L = 0.383Mesh 2742 elements (346 d.o.f.)η = 20.694% θ* = 0.943θ L = 0.864Mesh 32710 elements (1313 d.o.f.)η = 8.621% θ* = 1.036θ L = 0.971(b)Mesh 116 elements (9 d.o.f.)η = 57.652% θ* = 1.064θ L = 0.803Mesh 2451 elements (414 d.o.f.)η = 13.728% θ* = 1.155θ L = 0.994Mesh 31178 elements (1155 d.o.f.)η = 9.897% θ* = 1.024θ L = 0.941(c)Mesh 132 elements (49 d.o.f.)η = 55.742% θ* = 0.506θ L = 0.378Mesh 2744 elements (1421 d.o.f.)η = 3.243% θ* = 1.063θ L = 0.604Mesh 31406 elements (2765 d.o.f.)η = 0.782% θ* = 1.044θ L = 0.633(d)Mesh 116 elements (49 d.o.f.)η = 30.061% θ* = 0.988θ L = 0.464Mesh 2331 elements (1265 d.o.f.)η = 2.043% θ* = 1.178θ L = 0.634Mesh 3623 elements (2427 d.o.f.)η = 0.857% θ* = 1.032θ L = 0.527Fig. 15.11 Poisson problem of Fig. 15.10. Adaptive solutions for: (a) linear triangles; (b) linear quadrilaterals;(c) quadratic triangles; (d) quadratic quadrilaterals. based on SPR, L based on L 2 projection. Target error10% for linear elements and 1% for quadratic elements.

p-re®nement and hp-re®nement 415Error (%)100(1.064)10(0.436)(0.988)Theoreticalconvergencep = 2(0.506)11Theoreticalconvergencep = 11(0.943)(1.155)(1.178)(1.078)(1.024)(1.036)(1.063)110 100 1000 (1.032) 10000d.o.f.0.5(1.044)Linear triangleLinear quadsQuadratic triangleQuadratic quadsAim 10%Aim 1%Bracketed numbers showeffectivity index achievedFig. 15.12 Adaptive re®nement for Poisson problem of Fig. 15.10.example shows the advantages of adaptivity and the ease with which a ®nal goodmesh can be arrived at automatically.15.3 p-re®nement and hp-re®nementThe use of non-uniform p-re®nement is of course possible if done hierarchically andmany attempts have been made to do this e ciently. Some of this was done as early as1983. 38;39 However, the general process is di cult and necessitates many assumptionsabout the decrease of error. Certainly, the desired accuracy can seldom be obtained ina single step and most of the work on this requires a sequence of steps. We illustratesuch a re®nement process in Fig. 15.17 for the perforated dam problem presented inthe previous section.The same applies to hp-processes in which much work has been done during the lastdecade. 8ÿ18 We shall quote here only one particular attempt at hp-re®nement whichseems to be particularly e cient and where the number of resolutions is quitesmall. The methodology was introduced by Zienkiewicz et al. in 1989 19 and weshall quote here some of the procedures suggested.The ®rst procedure is that of pursuing an h-re®nement with lower order elements(e.g., linear or quadratic elements) to obtain, say, a 5% accuracy, at which stagethe energy norm error is nearly uniformly distributed throughout all elements.From there a p-re®nement is applied in a uniform manner (i.e., the same p is usedin all elements). This has very substantial computational advantages as programmingis easy and can be readily accomplished, especially if hierarchical functions are used.

416 Adaptive ®nite element re®nementp = 1.0505050 50Mesh 1: 27 elements (252 DOF)η = 8.3%, θ* = 1.110Mesh 2: 101 elements (876 DOF)η = 3.1%, θ* = 1.057Fig. 15.13 Adaptive re®nement of an L-shaped domain in plane stress with prescribed error of 1%.The uniform p-re®nement also allows the global energy norm error to be approximatelyextrapolated by three consecutive solutions. 40The convergence of the p-re®nement ®nite element solution can be written as 41jjejj 4 CN ÿ…15:11†

p-re®nement and hp-re®nement 417yxp = 1(a) Mesh 1 (565 d.o.f.) η = 9.75%(b) Mesh 2 (3155 d.o.f.) η = 4.85%Fig. 15.14 Adaptive re®nement of machine part using linear quadrilateral elements. Target error 5%.Base contour value = –1.833000Maximum contour value = 0.586500Contour interval = 0.163095Base contour value = –1.833000Maximum contour value = 0.586500Contour interval = 0.163095Fig. 15.15 Adaptive re®nement of machine part. Contours of shear stress for original and ®nal mesh.

418 Adaptive ®nite element re®nementMesh 1 (η = 16.5%, θ = 1.05, 728 DOF)(a)Mesh 2 (η = 4.9%, θ = 1.06, 1764 DOF)(b)Fig. 15.16 Quadratic triangle. Automatic mesh generation to achieve 5% accuracy. Plane strain analysis of adam with perforation, water loading only. (a) Original mesh. (b) Re®ned mesh.where C and are positive constants depending on the solution of the problem and Nis the number of degrees of freedom. We assume that for each re®nement the error is,observing Eq. (15.3),jjujj 2 ÿjj^u q jj 2 ˆ CNqÿ2…15:12†with q ˆ p ÿ 2; p ÿ 1; p for the three solutions. Eliminating the two constants C and from the above three equations, jjujj 2 can be solved byjjujj 2 ÿjju p jj 2 jjujj 2jjujj 2 ÿjj^u p ÿ 1 jj 2 ˆ ÿjju p ÿ 1 jj 2 log…N p ÿ 1 =N p†log…N p ÿ 2 =N p ÿ 1 †jjujj 2 ÿjj^u p ÿ 2 jj 2…15:13†

p-re®nement and hp-re®nement 419Percentage error intotal energyε = 6.0 ExactQuadratic d.o.f. (x direction)Cubic d.o.f. (x direction)ε = 3.6ε* = 7.8By estimatesBy ‘corrective’estimatesPercentage error intotal energyε = 3.0 Exactε = 2.9 By estimatesε* = 4.6By ‘corrective’estimatesOxOxOxOxOxOyOy OyOyOyCubic d.o.f. (x direction)Quartic d.o.f. (x direction)Ox Centre quadratic d.o.f. (x direction)≡ OxyQuadratics on element boundariesnot representedOxOyOxOxOyFig. 15.17 Adaptive solution of perforated dam by p-re®nement. (a) Stage three, 206 d.o.f.; (b) stage ®ve,365d.o.f.The global energy norm error for the ®nal solution and indeed the error at any stageof the p-re®nement can be determined usingjjejj 2 ˆjjujj 2 ÿjj^u q jj 2…15:14†q ˆ 1; 2; ...; p.

420 Adaptive ®nite element re®nementP = 1.0Poisson's ratio, ν = 0.3Plane stress conditions505050 50Mesh 1 (120 d.o.f.) η = 15% p = 2(a) Original meshMesh 2 (385 d.o.f.) η = 4.67% p = 2(1322 d.o.f.) η = 0.97% p = 4(b) Quadratic triangles for 5% errorLog ||e||–0.8–1.0–1.2–1.4–1.6–1.8–2.0–2.2–2.4Number of degrees of freedom30 50 250 1000 5000100 500 2500 100006-node elementAdaptiveh-refinementp=21.01.0p-refinement6-node elementUniform refinementp=31.00.2729-node elementUniform refinementp=41.5 2.0 2.5 3.0 3.5 4.0Log N(c) h-p refinement. 1% accuracy reached with 1322 d.o.f.Fig. 15.18 Solution of L-shaped domain by h-p re®nement (as de®ned in Example 2 of previous section)using procedure one of reference 19.302010543210.5Relative error (%)Generally the high accuracy is gained rapidly by re®nement, at least from examplesperformed to date. In Figs 15.18 and 15.19we show two examples for which we havepreviously used an h-re®nement. The ®rst illustrates an L-shaped domain with onesingularity and the second a short cantilever beam with two strong singularities. Inthe ®rst both problems are solved using h-re®nement and target 5% accuracy isreached using quadratic triangles. At this stage the p is increased to third andfourth order so that three solutions are available and when that is reached theerror is less than 1%.

p-re®nement and hp-re®nement 421Poisson's ratio, ν = 0.3Plain strain conditionsP = 1.01.01.0Mesh 1 (40 d.o.f.)η = 27.0% p = 2Mesh 2 (228 d.o.f.)η = 7.0% p = 2Mesh 3 (286 d.o.f.)η = 4.0% p = 2(1104 d.o.f.)η = 8.85% p = 4(a) Original mesh (b) First refinement (c) Second h-refinementLog ||e||–0.2–0.4–0.8–1.0–1.2–1.4–1.6–1.8–2.010Number of degrees of freedom30 100 500 2500 1000050 250 1000 5000 300006-nodeelementAdaptiveh-refinement1.01.0p-refinement6-node elementUniform refinementp=2p=3p=41.00.3569-node elementUniform refinement1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5Log N(d) p-refinement. 1% accuracy reached with 1104 d.o.f.Fig. 15.19 Solution of short cantilever by h-p adaptive re®nement using procedure one of reference 19.5040302010543210.5Relative error (%)In the same paper 19 an alternative procedure is suggested. This uses a very coarsemesh at the outset followed by p-re®nement. In this case the error at the element levelis estimated at the last stage of the p-re®nement as the di€erence between the last twore®nements (e.g., the third and fourth order). The global error estimator is calculatedby the extrapolation procedure used in the previous example. The element errorestimator is for order p ÿ 1 rather than the highest order p. It is, however, veryaccurate. The element error estimator is subsequently used to compute the optimalmesh size as described in Sec. 15.2.2. Nearly optimal rate of convergence isexpected to be achieved because the optimal mesh is designed for p ÿ 1 order

422 Adaptive ®nite element re®nement50p = 1.05050 50Mesh 1, for p = 4 d.o.f. = 430 η = 6.72%(a)Mesh 2, for p = 4 d.o.f. = 846 η = 0.76%(b)Log ||e||(c)–0.4–0.6–0.8–1.0–1.2–1.4–1.6–1.8–2.0–2.2–2.410Number of degrees of freedom30 100 500 2500 1000050 250 1000 5000 30000h-refinementp=1 p=16-node elementUniform refinementh-p refinementInitial mesh2Regenerated mesh2 3 1.040.2721.621.01.0 0.5441.031.9341.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5Log NNote: 1% accuracy reached with 846 d.o.f.60.050.040.030.020.010.05.04.03.02.0Fig. 15.20 Solution of L-shaped domain by h-p adaptive re®nement using alternative procedure of reference19.1.00.70.5Relative error (%)elements. Details of this process will be found again in the reference and will not bediscussed further.At no stage of the hp-re®nements have we used here any of the estimators quoted inthe previous chapter. However, their use would make the optimal mesh design atorder p possible, because the element error can be accurately estimated at order p.It will result in an optimal hp-re®nement.The two examples we have quoted above are re-analysed using the alternativeprocess described above and presented in Figs 15.20 and 15.21. In both cases

p-re®nement and hp-re®nement 423y p = 1.0A1.01.0Mesh 1, for p = 4 d.o.f. = 144η = 11.8%(a)xBABMesh 2, for p = 4 d.o.f. = 572η = 0.81%(b)10Number of degrees of freedom30 10050 250 500 1000 2500 5000 10000 30000Log ||e||(c)0–0.2–0.4–0.6–0.8–1.0–1.2–1.4–1.6–1.8–2.0–2.2p=1p=123421.41.03h-p refinementInitial meshRegenerated mesh41.01.81h-refinement6-node elementsUniform refinement1.00.7111.00.3651.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5Log NNote: 1% accuracy reached with 572 d.o.f.60.050.040.030.020.010.05.04.03.02.01.00.70.50.4Relative error (%)Fig. 15.21 Solution of short cantilever by h-p adaptive re®nement using alternative procedure of reference19.the ®nal accuracy shows an error of less than 1% but it is noteworthy thatthe total number of degrees of freedom used with the second method is considerablyless than that in the ®rst and still achieves a nearly optimal rate of convergence.We can conclude this section on hp-re®nement with a ®nal example where a highlysingular crack domain is studied. Once again the second procedure is used showing inFig. 15.22 a remarkable rate of convergence.

424 Adaptive ®nite element re®nementyOn all boundariesprescribed traction= analytic solutionyax710aa68 91151 2 3 412xa(a)aaaMesh 1, for p = 4 d.o.f. = 217 η = 9.17%(b)ABABCrackMesh 2, for p = 4 d.o.f. = 1245 η = 0.89%(c)Log ||e||–0.4–0.6–0.8–1.0–1.2–1.4–1.6–1.8–2.0–2.2–2.410Number of degrees of freedom30 100 500 2500 1000050 250 1000 5000 30000p=12p=134h-p refinementInitial meshRegeneratedmeshh-refinement6-node elementUniform refinement1.00.2521.501.01.01.66360.050.040.030.020.010.05.04.03.02.01.00.70.51.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5Log NNote: 1% accuracy reached with 1245 d.o.f.(d)Fig. 15.22 Adaptive h-p re®nement for a singular crack using alternative procedure of reference 19.4Relative error (%)

p-re®nement and hp-re®nement 425(a) Local mesh(b) Pressure coefficientsFig. 15.23 Directional mesh re®nement. Gas ¯ow past a circular cylinder ± Mach number 3. Third re®nementmesh 709 nodes (1348 elements).

426 Adaptive ®nite element re®nement15.4 Concluding remarksThe methods of estimating errors and adaptive re®nement which are described in thisand the previous chapter constitute a very important tool for practical application of®nite element methods. The range of applications is large and we have only touchedhere upon the relatively simple range of linear elasticity and similar self-adjointproblems. A recent survey shows many more areas of application 42 and the readeris referred to this publication for interesting details. At this stage we would like toreiterate that many di€erent norms or measures of error can be used and that forsome problems the energy norm is not in fact `natural'. A good example of this isgiven by problems of high-speed gas ¯ow, where very steep gradients (shocks) candevelop. The formulation of such problems is complex, but this is not necessary forthe present argument.For problems in ¯uid mechanics which we will discuss in Volume 3 and similarly forproblems of strain localization in plastic softening discussed in Volume 2 no globalnorms can be used e€ectively. We shall therefore base our re®nement on the valueof the maximum curvatures developed by the solution of u. On occasion an elongationof the elements will be used to re®ne the mesh appropriately. Figure 15.23shows a typical problem of shock capture solved adaptively.References1. I. BabusÏ ka and C. Rheinboldt. A-posteriori error estimates for the ®nite element method.Internat. J. Num. Meth. Eng., 12, 1597±615, 1978.2. I. BabusÏ ka and C. Rheinboldt. Adaptive approaches and reliability estimates in ®niteelement analysis. Comp. Meth. Appl. Mech. Eng., 17/18, 519±40, 1979.3. O.C. Zienkiewicz and J.Z. Zhu. A simple error estimator and adaptive procedure forpractical engineering analysis. Internat. J. Num. Meth. Eng., 24, 337±57, 1987.4. E. OnÄ ate and G. Bugeda. A study of mesh optimality criteria in adaptive ®nite elementanalysis. Eng. Comp., 10, 307±21, 1993.5. E.R. de Arantes e Oliveira. Theoretical foundations of the ®nite element method. Internat.J. Solids Struct., 4, 929±52, 1968.6. E.R. de Arantes e Oliveira. Optimization of ®nite element solutions. In Proc. 3rd Conf.Matrix Methods in Structural Mechanics, volume AFFDL-TR-71±160, 423±446,Wright-Patterson Air Force Base, Ohio, 1972.7. R.L. Taylor and R. Iding. Applications of extended variational principles to ®nite elementanalysis. In Proc. of the International Conference on Variational Methods in Engineering,volume II, pages 2/54±2/67. Southampton University Press, 1973.8. W. Gui and I. BabusÏ ka. The h, p and h-p version of the ®nite element method in 1 dimension.Part 1: The error analysis of the p-version. Part 2: The error analysis of the h- and h-pversion. Part 3: The adaptive h-p version. Numerische Math., 48, 557±683, 1986.9. B. Guo and I. BabusÏ ka. The h-p version of the ®nite element method. Part 1: The basicapproximation results. Part 2: General results and applications. Comp. Mech., 1, 21±41,203±26, 1986.10. I. BabusÏ ka and B. Guo. The h-p version of the ®nite element method for domains withcurved boundaries. SIAM J. Numer. Anal., 25, 837±61, 1988.11. I. BabusÏ ka and B.Q. Guo. Approximation properties of the hp version of the ®nite elementmethod. Comp. Meth. Appl. Mech. Eng., 133, 319±49, 1996.

12. L. Demkowiez, J.T. Oden, W. Rachowiez, and O. Hardy. Toward a universal h-p adaptive®nite element strategy. Part 1: Constrained approximation and data structure. Comp.Meth. Appl. Mech. Eng., 77, 79±112, 1989.13. W. Rachowicz, T.J. Oden, and L. Demkowicz. Toward a universal h-p adaptive ®nite elementstrategy. Part 3: Design of h-p meshes. Comp. Meth. Appl. Mech. Eng., 77, 181±211, 1989.14. K.S. Bey and J.T. Oden. hp-version discontinuous Galerkin methods for hyperbolicconservation laws. Comp. Meth. Appl. Mech. Eng., 133, 259±86, 1996.15. C.E. Baumann and J.T. Oden. A discontinuous hp ®nite element method for convectiondi€usionproblems. Comp. Meth. Appl. Mech. Eng., 175, 311±41, 1999.16. P. Monk. On the p and hp extension of Nedelecs curl-conforming elements. J. Comput.Appl. Math., 53, 117±37, 1994.17. L.K. Chilton and M. Suri. On the selection of a locking-free hp element for elasticityproblems. Internat. J. Num. Meth. Eng., 40, 2045±62, 1997.18. L. Vardapetyan and L. Demkowicz. hp-Adaptive ®nite elements in electromagnetics.Comp. Meth. Appl. Mech. Eng., 169, 331±44, 1999.19. O.C. Zienkiewicz, J.Z. Zhu, and N.G. Gong. E€ective and practical h-p-version adaptiveanalysis procedures for the ®nite element method. Internat. J. Num. Meth. Eng., 28,879±91, 1989.20. J. Peraire, M. Vahdati, K. Morgan, and O.C. Zienkiewicz. Adaptive remeshing for compressible¯ow computations. J. Comp. Phys., 72, 449±66, 1987.21. J.C. Cavendish. Automatic triangulation of arbitrary planar domains for the ®nite elementmethod. Internat. J. Num. Meth. Eng., 8, 679±96, 1974.22. S.H. Lo. A new mesh generation scheme for arbitrary planar domains. Internat. J. Num.Meth. Eng., 21, 1403±26, 1985.23. J. Peraire, K. Morgan, M. Vahdati, and O.C. Zienkiewicz. Finite element Euler computationsin 3-d. Internat. J. Num. Meth. Eng., 26, 2135±59, 1988.24. J.Z. Zhu, O.C. Zienkiewicz, E. Hinton, and J. Wu. A new approach to the developmentof automatic quadrilateral mesh generation. Internat. J. Num. Meth. Eng., 32, 849±66, 1991.25. J.Z. Zhu, E. Hinton, and O.C. Zienkiewicz. Mesh enrichment against mesh regenerationusing quadrilateral elements. Comm. Num. Meth. Eng., 9, 547±54, 1993.26. E. Rank and O.C. Zienkiewicz. A simple error estimator for the ®nite element method.Comp. Meth. Appl. Mech. Eng., 3, 243±50, 1987.27. E. Rank, M. Schweingruber, and M. Sommer. Adaptive mesh generation. Comm. Numer.Methods Engrg., 9, 121±29, 1993.28. T.D. Blacker, M.B. Stephenson, and S. Canann. Analysis automation with paving: A newquadrilateral meshing technique. Adv. Eng. Software, Elsevier, 56(13), 332±37, 1991.29. T.D. Blacker and M.B. Stephenson. Paving: A new approach to automated quadrilateralmesh generation. Internat. J. Num. Meth. Eng., 32, 811±47, 1991.30. M.A. Price, C.G. Armstrong, and M.A. Sabin. Hexahedral mesh generation by medial axissubdivision, I: solids with convex edges. Internat. J. Num. Meth. Eng., 38, 3335±59, 1995.31. T.J. Tautges, T.D. Blacker, and S.A. Mitchell. The whisker weaving algorithm: A connectivity-basedmethod for constructing all-hexahedral ®nite element meshes. Internat. J.Num. Meth. Eng., 39, 3327±49, 1996.32. M.A. Price, C.G. Armstrong, and M.A. Sabin. Hexahedral mesh generation by medialaxis subdivision, II, solids with ¯at and concave edges. Internat. J. Num. Meth. Eng.,40, 111±36, 1997.33. N. Chiba, I. Nishigaki, Y. Yamashita, C. Takizawa, and K. Fujishiro. A ¯exible automatichexahedral mesh generation by boundary-®t method. Comp. Meth. Appl. Mech. Eng., 161,145±54, 1998.34. N.P. Weatherill, P.R. Eiseman, J. Hause, and J.F. Thompson. Numerical Grid Generationin Computational Fluid Dynamics and Related Fields. Pineridge Press, Swansea, 1994.References 427

428 Adaptive ®nite element re®nement35. J.F. Thompson, B.K. Soni, and N.P. Weatherill, editors. Handbook of Grid Generation.CRC Press, January 1999.36. P.G. Ciarlet. The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam,1978.37. J.Z. Zhu and O.C. Zienkiewicz. Adaptive techniques in the ®nite element method. Comm.Appl. Num. Math., 4, 197±204, 1988.38. D.W. Kelly, J.P. De S.R. Gago, O.C. Zienkiewicz, and I. BabusÏ ka. A posteriori erroranalysis and adaptive processes in the ®nite element method: Part I ± Error analysis.Internat. J. Num. Meth. Eng., 19, 1593±619, 1983.39. J.P. De S.R. Gago, D.W. Kelly, O.C. Zienkiewicz, and I. BabusÏ ka. A posteriori erroranalysis and adaptive processes in the ®nite element method: Part II ± Adaptive meshre®nement. Internat. J. Num. Meth. Eng., 19, 1621±56, 1983.40. B.A. Szabo. Mesh design for the p version of the ®nite element. Comp. Meth. Appl. Mech.Eng., 55, 181±97, 1986.41. I. BabusÏ ka, B.A. Szabo, and I.N. Katz. The p version of the ®nite element method. SIAMJ. Numer. Anal., 18, 512±45, 1981.42. P. LadeveÁze and J.T. Oden (Editors), editors. Advances in Adaptive Computational Methodsin Mechanics Studies in Applied Mechanics 47. Elsevier, 1998.

16Point-based approximations;element-free Galerkin ± and othermeshless methods16.1 IntroductionIn all of the preceding chapters, the ®nite element method was characterized by thesubdivision of the total domain of the problem into a set of subdomains calledelements. The union of such elements gave the total domain. The subdivision ofthe domain into such components is of course laborious and di cult necessitatingcomplex mesh generation. Further if adaptivity processes are used, generally largeareas of the problem have to be remeshed. For this reason, much attention hasbeen given to devising approximation methods which are based on points withoutnecessity of forming elements.When we discussed the matter of generalized ®nite element processes in Chapter 3,we noted that point collocation or in general ®nite di€erences did in fact satisfy therequirement of the pointwise de®nition. However the early ®nite di€erences werealways based on a regular arrangement of nodes which severely limited their applications.To overcome this di culty, since the late 1960s the proponents of the ®nitedi€erence method have worked on establishing the possibility of ®nite di€erencecalculus being based on an arbitrary disposition of collocation points. Here thework of Girault, 1 Pavlin and Perrone, 2 and Snell et al. 3 should be mentioned. Howevera full realization of the possibilities was ®nally o€ered by Liszka and Orkisz, 4;5and Krok and Orkisz 6 who introduced the use of least square methods to determinethe appropriate shape functions.At this stage Orkisz and coworkers realized not only that collocation methodscould be used but also the full ®nite element, weak formulation could be adoptedby performing integration. Questions of course arose as to what areas such integrationshould be applied. Liszka and Orkisz 4 suggested determining a `tributary area'to each node providing these nodes were triangulated as shown in Fig. 16.1(a). Onthe other hand in a somewhat di€erent context Nay and Utku 7 also used the leastsquare approximation including triangular vertices and points of other trianglesplaced outside a triangular element thus simply returning to the ®nite elementconcept. We show this kind of approximation in Fig. 16.1(b). Whichever form oftributary area was used the direct least square approximation centred at each nodewill lead to discontinuities of the function between the chosen integration areas and

430 Point-based approximations(a)(b)Fig. 16.1 Patches of triangular elements and tributary areas.thus will violate the rules which we have imposed on the ®nite element method.However it turns out that such rules could be violated and here the patch test willshow that convergence is still preserved.However the possibility of determining a completely compatible form of approximationexisted. This compatible form in which continuity of the function and of its slope ifrequired and even higher derivatives could be accomplished by the use of so-calledmoving least square methods. Such methods were originated in another context(Shepard, 8 Lancaster and Salkauskas, 9;10 ). The use of such interpolation in the meshlessapproximation was ®rst suggested by Nayroles et al. 11ÿ13 This formulation wasnamed by the authors as the di€use ®nite element method.Belytschko and coworkers 14;15 quickly realized the advantages o€ered by such anapproach especially when dealing with the development of cracks and other problemsfor which standard elements presented di culties. His so-called `element-freeGalerkin' method led to many seminal publications which have been extensivelyused since.An alternative use of moving least square procedures was suggested by Duarte andOden. 16;17 They introduced at the same time a concept of hierarchical forms by notingthat all shape functions derived by least squares possess the partition of unityproperty (viz. Chapter 8). Thus higher order interpolations could be added at eachnode rather than each element, and the procedures of element-free Galerkin or ofthe di€use element method could be extended.The use of all the above methods still, however, necessitates integration. Now,however, this integration need not be carried out over complex areas. A backgroundgrid for integration purposes has to be introduced though internal boundaries wereno longer required. Thus such numerical integration on regular grids is currentlybeing used by Belytschko 18;19 and other approaches are being explored. Howeveran interesting possibility was suggested by BabusÏ ka and Melenk. 20;21BabusÏ ka and Melenk use a partition of unity but now the ®rst set of basic shapefunctions is derived on the simplest element, say the linear triangle. Most of the

Function approximation 431approximations then arise through addition of hierarchical variables centred atnodes. We feel that this kind of approach which necessitates very few elements forintegration purposes combines well the methodologies of `element free' and `standardelement' approximation procedures. We shall demonstrate a few examples later onthe application of such methods which seem to present a very useful extension ofthe hierarchical approach.Incidentally the procedures based on local elements also have the additionaladvantage that global functions can be introduced in addition to the basic ones torepresent special phenomena, for instance the presence of a singularity or waves.Both of these are important and the idea presented by this can be exploited. InVolume 3, we shall show the application of this to certain wave phenomena, seeChapter 8, Volume 3.This chapter will conclude with reference to other similar procedures which we donot have time to discuss. We shall refer to such procedures in the closure of this chapter.16.2 Function approximationWe consider here a local set of n points in two (or three) dimensions de®ned by thecoordinates x k ; y k ; z k ; k ˆ 1; 2; ...; n or simply x k ˆ‰x k ; y k ; z k Š at which a set ofdata values of the unknown function ~u k are given. It is desired to ®t a speci®edfunction form to the data points. In order to make a ®t it is necessary to:1. Specify the form of the functions, p…x†, to be used for the approximation. Here asin the standard ®nite element method, it is essential to include low order polynomialsnecessary to model the highest derivatives contained in the di€erentialequation or in the weak form approximation being used. Certainly a completelinear and sometimes quadratic polynomial will always be necessary.2. De®ne the procedure for establishing the ®t.Here we will consider some least square ®t methods as the basis for performing the®t. The functions will mostly be assumed to be polynomials, however, in additionother functions can be considered if these are known to model well the solutionexpected (e.g., see Chapter 8, Volume 3 on use of `wave' functions).16.2.1 Least square ®tWe shall ®rst consider a least square ®t scheme which minimizes the square of thedistance between n data values ~u k de®ned at the points x k and an approximatingfunction evaluated at the same points ^u…x k †. We assume the approximation functionis given by a set of monomials p j^u…x† ˆXnj ˆ 1p j …x† j p…x†a…16:1†in which p is a set of linearly independent polynomial functions and a is a set ofparameters to be determined. A least square scheme is introduced to perform the

432 Point-based approximations®t and this is written as (see Chapter 14 for similar operations): MinimizeJ ˆ 1X n2k ˆ 1… ^u…x k †ÿ~u k † 2ˆ min …16:2†where the minimization is to be performed with respect to the values of a. Substitutingthe values of ^u at the points x k we obtainwhere@J@ jˆ Xnk ˆ 1@ ^u k … ^u…x@ k †ÿ~u k † ˆ 0; j ˆ 1; 2; ...; n …16:3†j^u k ˆ Xp j …x k † jThis set of equations may be written in a compact matrix form asj@J@a ˆ Xnp T k … p k a ÿ ~u k † ˆ 0 …16:4†k ˆ 1where p k ˆ p…x k †. We can de®ne the result of the sums asH ˆ Xnp T k p k ˆ P T P…16:5†k ˆ 1g ˆ Xnk ˆ 1p T k ~u k ˆ P T ~u…16:6†in which2 38 9p 1~u 1P ˆ 6 p >< >=2 74 5 and ~u ˆ~u 2>:>;p n~u nThe above process yields the set of linear algebraic equationsHa ˆ g ˆ P T ~uwhich, provided H is non-singular, has the solutiona ˆ H ÿ1 g ˆ H ÿ1 P T ~u…16:7†We can now write the approximation for the function as^u ˆ p…x† H ÿ1 P T ~u ˆ N…x†~uwhere N…x† are the appropriate shape or basis functions. In general N i …x i † is not unityas it always has been in standard ®nite element shape functions. However, the partitionof unity [viz. Eq. (8.4)] is always preserved provided p…x† contains a constant.Example: Fit of a linear polynomial To make the process clear we ®rst consider adataset, ~u k , de®ned at four points, x k , to which we desire to ®t an approximationgiven by a linear polynomial^u…x† ˆ 1 ‡ x 2 ‡ y 3 ˆ p…x†a

Function approximation 433If we consider the set of data de®ned byx k ˆ ‰ ÿ4:0 ÿ1:0 0:0 6:0 Šy k ˆ ‰ 5:0 ÿ5:0 0:0 3:0 Š~u k ˆ ‰ ÿ1:5 5:1 3:5 4:3 Šwe can write the arrays as238 91 ÿ4 5ÿ1:51 ÿ1 ÿ5>=P ˆ 67 and ~u ˆ4 1 0 053:5>: >;1 6 34:3Using Eq. (16.5) we obtain the values2 38 94 1 3< 11:4 =H ˆ P T P ˆ 4 1 53 35 and g ˆ P T ~u ˆ 26:7: ;3 3 59ÿ20:1which from Eq. (16.7) has the solution8 9>< 3:1241 >=a ˆ 0:4745>: >;ÿ0:5237Thus, the values for the least square ®t at the data points are8 9ÿ1:5194>< 5:0698>=^u ˆ2:9676>: >;4:2820The least square ®t for these data points is shown in Fig. 16.2 and the di€erencebetween the data points and the values of the ®t at x k is given in Table 16.1.16.2.2 Weighted least square ®tLet us now assume that the point at the origin, x 0 ˆ 0, is the point about which we aremaking the expansion and, therefore, the one where we would like to have the bestaccuracy. Based on the linear approximation above we observe that the direct leastsquare ®t yields at the point in question the largest discrepancy. In order to improvethe ®t we can modify our least square ®t for weighting the data in a way thatemphasizes the e€ect of distance from a chosen point. We can write such a weightedleast square ®t as the minimization ofJ ˆ 1X n2k ˆ 1w…x k ÿ x 0 †… ^u…x k †ÿ~u k † 2ˆ min …16:8†where w is the weighting function. Many choices may be made for the shape of thefunction w. If we assume that the weight function depends on a radial distance, r,

434 Point-based approximations86u values~420–2–50–5(a)55086u values~420–2–50–5(b)550Fig. 16.2 Least square ®t: (a) four data points; (b) ®t of linear function on the four data points.

Function approximation 435Table 16.1 Di€erence between least square ®t and datax k ÿ4 ÿ1 0 6y k 5 ÿ5 0 3~u k ÿ1.500 5.100 3.500 4.300^u k ÿ1.392 5.268 3.124 4.400Di€erence ÿ0.108 ÿ0.168 0.376 ÿ0.100from the chosen point we havew ˆ w…r†; r 2 ˆ…x ÿ x 0 † …x ÿ x 0 †One functional form for w…r† is the exponential Gauss function:w…r† ˆexp…ÿcr 2 †; c > 0 and r 5 0 …16:9†For c ˆ 0:125 this function has the shape shown in Fig. 16.3 and when used with thepreviously given four data points yields the linear ®t shown in Table 16.2.16.2.3 Interpolation domains and shape functionsIn what follows we shall invariably use the least square procedure to interpolate theunknown function in the vicinity of a particular node i. The ®rst problem is thatwhen approximating to the function it is necessary to include a number of nodesequal at least to the number of parameters of a sought to represent a given polynomial.This number, for instance, in two dimensions is three for linear polynomials and six forquadratic ones. As always the number of nodal points has to be greater than or equal tothe bare minimum which is the number of parameters required. We should note inpassing that it is always possible to develop a singularity in the equation used forsolving a, i.e. Eq. (16.7) if the data points lie for instance on a straight line in two orthree dimensions. However in general we shall try to avoid such di culties by reasonablespacing of nodes. The domain of in¯uence can well be de®ned by making sure thatthe weighting function is limited in extent so that any point lying beyond a certaindistance r m are weighted by zero and therefore are not taken into account. Commonlyused weighting functions are, for instance, in direction r, givenby8< exp…ÿcr 2 †ÿexp…ÿcr 2 m†w…r† ˆ 1 ÿ exp…ÿcr 2 ; c > 0 and 0 4 r 4 r mm†…16:10†:0 ; r > r mwhich represents a truncated Gauss function. Another alternative is to use aHermitian interpolation function as employed for the beam example in Sec. 2.10:8 r 2 r 3 r m

436 Point-based approximations1.00.8Weight w (r)0.60.40.200 1 2 3 4 5 6 7 8r distanceFig. 16.3 Weighting function for Eq. (16.9): c ˆ 0:125.or alternatively the function8 " #r2n>:0 ; r > r mand n 5 2…16:12†is simple and has been e€ectively used. For circular domains, or spherical ones inthree dimensions, a simple limitation of r m su ces as shown in Fig. 16.4(a). Howeveroccasionally use of rectangular or hexahedral subdomains is useful as also shown inthat ®gure and now of course the weighting function takes on a di€erent form:w…x; y† ˆ Xi…x†Y j …y†; 0 4 x 4 x m ; 0 4 y 4 y m ; and i; j 5 2…16:13†0 ; x > x m ; y > y mwith" #x2i" #y2jX i …x† ˆ 1 ÿ ; Yx j …y† ˆ 1 ÿm y mTable 16.2 Di€erence between weighted least square ®tand datax k ÿ4 ÿ1 0 6y k 5 ÿ5 0 3~u k ÿ1.500 5.100 3.500 4.300^u k ÿ0.880 5.247 3.4872 5.246Error ÿ0.620 ÿ0.148 0.013 ÿ0.946

Function approximation 4376644220(x i , y i )0(x i , y i )–2–2–4–4–6–6 –4 –2 0 2 4 6(a)–6–6 –4 –2 0 2 4 6(b)Fig. 16.4 Two-dimensional interpolation domains: (a) circular; (b) rectangular.The above two possibilities are shown in Fig. 16.4. Extensions to three dimensionsusing these methods is straightforward.Clearly the domains de®ned by the weighting functions will overlap and it isnecessary if any of the integral procedures are used such as the Galerkin method toavoid such an overlap by de®ning the areas of integration. We have suggested acouple of possible ideas in Fig. 16.1 but other limitations are clearly possible. InFig. 16.5, we show an approximation to a series of points sampled in one dimension.The weighting function here always embraces three or four nodes. Limiting howeverthe domains of their validity to a distance which is close to each of the points providesa unique de®nition of interpolation. The reader will observe that this interpolation isPiecewise least square approximationExactJumpFig. 16.5 A one-dimensional approximation to a set of data points using parabolic interpolation and directleast square ®t to adjacent points.

438 Point-based approximationsdiscontinuous. We have already pointed out such a discontinuity in Chapter 3, but ifstrictly ®nite di€erence approximations are used this does not matter. It can howeverhave serious consequences if integral procedures are used and for this reason it isconvenient to introduce a modi®cation to the de®nition of weighting and methodof calculation of the shape function which is given in the next section.16.3Moving least square approximations ± restorationof continuity of approximationThe method of moving least squares was introduced in the late 1960s by Shepard 8 as ameans of generating a smooth surface interpolating between various speci®ed pointvalues. The procedure was later extended for the same reasons by Lancaster andSalkauskas 9;10 to deal with very general surface generation problems but again itwas not at that time considered of importance in ®nite elements. Clearly in the presentcontext the method of moving least squares could be used to replace the local leastsquares we have so far considered and make the approximation fully continuous.In moving least square methods, the weighted least square approximation isapplied in exactly the same manner as we have discussed in the preceding sectionbut is established for every point at which the interpolation is to be evaluated. Theresult of course completely smooths the weighting functions used and it also presentssmooth derivatives noting of course that such derivatives will depend on the locallyspeci®ed polynomial.To describe the method, we again consider the problem of ®tting an approximationto a set of data items ~u i ; i ˆ 1; ...; n de®ned at the n points x i . We again assume theapproximating function is described by the relationu…x† ^u…x† ˆXmj ˆ 1p j …x† j ˆ p…x†a…16:14†where p j are a set of linearly independent (polynomial) functions and i are unknownquantities to be determined by the ®t algorithm. A generalization to the weighted leastsquare ®t given by Eq. (16.8) may be de®ned for each point x in the domain by solvingthe problemJ…x† ˆ1X n2k ˆ 1w x …x k ÿ x†‰~u k ÿ p…x k †aŠ 2 ˆ min…16:15†In this form the weighting function is de®ned for every point in the domain and thuscan be considered as translating or moving as shown in Fig. 16.6. This produces acontinuous interpolation throughout the whole domain.Figure 16.7 illustrates the problem previously presented in Fig. 16.5 now showingcontinuous interpolation. We should note that it is now no longer necessary to specify`domains of in¯uence' as the shape functions are de®ned in the whole domain.The main di culty with this form is the generation of a moving weight functionwhich can change size continuously to match any given distribution of points x kwith a limited number of points entering each calculation. One expedient method

Moving least square approximations ± restoration of continuity of approximation 439x1.5uu i+1i–1u iu i+2u (x) and w (x)1.00.5u i–2w x (x i –x)0–2 –1 0 1 2 3x i coordinatesFig. 16.6 Moving weighting function approximation in MLS.to accomplish this is to assume the function is symmetric so thatw x …x k ÿ x† ˆw x …x ÿ x k †and use a weighting function associated with each data point x k asw x …x k ÿ x† ˆw k …x ÿ x k †Piecewise least square approximationExactMoving least squareapproximation(no jump)JumpFig. 16.7 The problem of Fig. 16.5 with moving least square interpolation.

440 Point-based approximationsx1.5uu i+1i–1u iu i+2u (x) and w (x)1.00.5u i–2w i (x – x i )0–2 –1 0 1 2 3x i coordinatesFig. 16.8 A `®xed' weighting function approximation to the MLS method.The function to be minimized now becomesJ…x† ˆ1X n2k ˆ 1w k …x ÿ x k †‰~u k ÿ p…x k †aŠ 2 ˆ min…16:16†In this form the weighting function is ®xed at a data point x k and evaluated at thepoint x as shown in Fig. 16.8. Each weighting function may be de®ned such thatw x …r† ˆ fk…r†; if jrj < r k…16:17†0; otherwiseand the terms in the sum are zero whenever r 2 ˆ…x ÿ x k † T …x ÿ x k † and jrj > r k . Theparameter r k de®nes the radius of a ball around each point, x k ; inside the ball theweighting function is non-zero while outside the radius it is zero. Each point mayhave a di€erent weighting function and/or radius of the ball around its de®ningpoint. The weighting function should be de®ned such that it is zero on the boundaryof the ball. This class of function may be denoted as Cq…r 0 k †, where the superscriptdenotes the boundary value and the subscript the highest derivative for which C 0continuity is achieved. Other options for de®ning the weighting function are availableas discussed in the previous section. The solution to the least square problem nowleads toa…x† ˆH ÿ1 …x† Xnj ˆ 1g j …x†~u j ˆ H ÿ1 …x†g…x†~u j…16:18†

whereandH…x† ˆXnk ˆ 1w k …x ÿ x k †p…x k † T p…x k †g j …x† ˆw j …x ÿ x j †p…x j † TIn matrix form the arrays H…x† and g…x† may be written asin whichH…x† ˆP T w…x†Pg…x† ˆw…x†P23w 1 …x ÿ x 1 † 0 0 w 2 …x ÿ x 2 † 0 w…x† ˆ . . . . . . 6745 0 w n …x ÿ x n †…16:19†…16:20†…16:21†…16:22†The moving least square algorithm produces solutions for a which depend continuouslyon the point selected for each ®t. The approximation for the function u…x† nowmay be written as^u…x† ˆXnN j …x†~u j…16:23†whereMoving least square approximations ± restoration of continuity of approximation 441j ˆ 1N j …x† ˆp…x†H ÿ1 …x†g j …x†…16:24†de®ne interpolation functions for each data item ~u j . We note that in general these`shape functions' do not possess the Kronecker delta property which we notedpreviously for ®nite element methods ± that isN j …x i †6ˆ ji…16:25†It must be emphasized that all least square approximations generally have values atthe de®ning points x j in which~u j 6ˆ ^u…x j † …16:26†i.e., the local values of the approximating function do not ®t the nodal unknownvalues (e.g., Fig. 16.2). Indeed ^u will be the approximation used in seeking solutionsto di€erential equations and boundary conditions and ~u j are simply the unknownparameters de®ning this approximation.The main drawback of the least square approach is that the approximation rapidlydeteriorates if the number of points used, n, largely exceeds that of the m polynomialterms in p. This is reasonable since a least square ®t usually does not match the datapoints exactly.A moving least square interpolation as de®ned by Eq. (16.23) can approximateglobally all the functions used to de®ne p…x†. To show this we consider the set of

442 Point-based approximationsapproximationsU ˆ Xnj ˆ 1N j …x† ~U j…16:27†whereU ˆ ‰ ^u 1 …x† ^u 2 …x† ... ^u n …x† Š T …16:28†and ~U j ˆ ~u j1 ~u j2 ... ~u Tjn…16:29†Next, assign to each ~u jk the value of the polynomial p k …x j † (i.e., the kth entry in p) sothatU j ˆ p…x j †…16:30†Using the de®nition of the interpolation functions given by Eqs (16.23) and (16.24) wehaveU ˆ XnN j …x†p…x †ˆXnj p…x†H ÿ1 …x†g j …x†p…x j †j ˆ 1j ˆ 1…16:31†which after substitution of the de®nition of g j …x† yieldsU ˆ Xnj ˆ 1p…x†H ÿ1 …x†w j …x ÿ x j †p…x j † T p…x j †Xnÿ1ˆ p…x†Hj ˆ 1w j …x ÿ x j †p…x j † T p…x j †ˆ p…x†H ÿ1 H…x† ˆp…x†…16:32†Equation (16.32) shows that a moving least square form can exactly interpolate anyfunction included as part of the de®nition of p…x†. If polynomials are used to de®ne thefunctions, the interpolation always includes exact representations for each includedpolynomial. Inclusion of the zero-order polynomial (i.e., 1), implies thatX nj ˆ 1N j …x† ˆ1…16:33†This is called a partition of unity (provided it is true for all points, x, in the domain). 22It is easy to recognize that this is the same requirement as applies to standard ®niteelement shape functions.Derivatives of moving least square interpolation functions may be constructedfrom the representationN j …x† ˆp…x†v j …x†…16:34†whereH…x†v j …x† ˆg j …x†…16:35†

For example, the ®rst derivatives with respect to x is given byandwhereHierarchical enhancement of moving least square expansions 443@H@x ˆ Xn@N j@x ˆ @p@x v j ‡ p @v j@xH @v j@x ‡ @H@x v j ˆ @g j@xk ˆ 1@w k …x ÿ x k †p…x@xk † T p…x k †…16:36†…16:37†…16:38†and@g j@x ˆ @w j…x ÿ x j †p…x@x j †…16:39†Higher derivatives may be computed by repeating the above process to de®ne thehigher derivatives of v j . An important ®nding from higher derivatives is the orderat which the interpolation becomes discontinuous between the interpolation subdomains.This will be controlled by the continuity of the weight function only. Forweight functions which are Cq 0 continuous in each subdomain the interpolation willbe continuous for all derivatives up to order q. For the truncated Gauss functiongiven by Eq. (16.10) only the approximated function will be continuous in thedomain, no matter how high the order used for the p basis functions. On the otherhand, use of the Hermitian interpolation given by Eq. (16.11) produces C 1 continuousinterpolation and use of Eq. (16.12) produces C n continuous interpolation. Thisgenerality can be utilized to construct approximations for high order di€erentialequations.Nayroles et al. suggest that approximations ignoring the derivatives of a may beused to de®ne the derivatives of the interpolation functions. 11ÿ13 While this approximationsimpli®es the construction of derivatives as it is no longer necessary tocompute the derivatives for H and g j , there is little additional e€ort required tocompute the derivatives of the weighting function. Furthermore, for a constant in pno derivatives are available. Consequently, there is little to recommend the use ofthis approximation.16.4 Hierarchical enhancement of moving least squareexpansionsThe moving least square approximation of the function u…x† was given in the previoussection as^u…x† ˆXnN j …x†~u j…16:40†j ˆ 1where N j …x† de®ned the interpolation or shape functions based on linearly independentfunctions prescribed by p…x† as given by Eq. (16.24). Here we shall restrict

444 Point-based approximationsattention to one-dimensional forms and employ polynomial functions to describep…x† up to degree k. Accordingly, we havep…x† ˆ 1 x x 2 ... x k…16:41†For this case we will denote the resulting interpolation functions using the notationNj k …x†, where j is associated with the location of the point where the parameter ~u jis given and k denotes the order of the polynomial approximating functions.Duarte and Oden suggest using Legendre polynomials instead of the form givenabove; 16 however, conceptually the two are equivalent and we use the above formfor simplicity. A hierarchical construction based on Nj k …x† can be established whichincreases the order of the complete polynomial to degree p. The hierarchical interpolationis written as08 91~b j1^u…x† ˆXnNj k …x†~u j ‡ Nj k >< ~b>=…x† x k ‡ 1 x k ‡ 2 ... x p j2Bj ˆ 1.C@. A>: >;ˆ Xnj ˆ 1( )Nj k ÿ X n~u j…x† ~u j ‡ q…x†~b j ˆ Nj k …x†‰ 1 q…x† Šj ˆ 1~b j~b jq…16:42†where q ˆ p ÿ k and ~ b jm ; m ˆ 1; ...; q; are additional parameters for the approximation.Derivatives of the interpolation function may be constructed using the methoddescribed by Eqs (16.34)±(16.39).The advantage of the above method lies in the reduced cost of computing theinterpolation function N k j …x† compared to that required to compute the p-orderinterpolations N p j …x†.Shepard interpolationFor example, use of the functions N 0 j …x†, which are called Shepard interpolations, 8leads to a scalar matrix H which is trivial to invert to de®ne the N 0 j . Speci®cally,the Shepard interpolations areN 0 j …x† ˆH ÿ1 …x†g j …x†…16:43†whereandH…x† ˆXnw k …x ÿ x k †k ˆ 1g j …x† ˆw j …x ÿ x j †…16:44†…16:45†The fact that the hierarchical interpolations include polynomials up to order p iseasy to demonstrate. Based on previous results from standard moving least squaresthe interpolation with ~b j ˆ 0 contains all the polynomials up to degree k. Higher

Hierarchical enhancement of moving least square expansions 445degree polynomials may be constructed from08~b j1~b jq91^u…x† ˆXnNj k …x†~u j ‡ Nj k >< ~b>=…x† x k ‡ 1 x k ‡ 2 ... x p j2.Bj ˆ 1@.CA>: >;…16:46†by setting all ~u j to zero and for each interpolation term setting one of the ~ b jk to unitywith the remaining values set to zero. For example, setting ~ b j1 to unity results in theexpansion^u…x† ˆXnNj k …x†x k ‡ 1 ˆ x k ‡ 1j ˆ 1This result requires only the partition of unity propertyX nj ˆ 1N k j …x† ˆ1…16:47†…16:48†The remaining polynomials are obtained by setting the other values of b ~ jk to unity oneat a time. We note further that the same order approximation is obtained usingk ˆ 0; 1orp. 16The above hierarchical form has parameters which do not relate to approximatevalues of the interpolation function. For the case where k ˆ 0 (i.e., Shepard interpolation),BabusÏ ka and Melenk 23 suggest an alternate expression be used in whichq in Eq. (16.42) is taken as 1 x x 2 ... x p and the interpolation written as!^u…x† ˆXn X pNj 0 …x† l p k …x†~u jk…16:49†j ˆ 1In this form the l p k…x† are Lagrange interpolation polynomials (e.g., see Sec. 8.5) and~u jk are parameters with dimensions of u for the jth term at point x k of the Lagrangeinterpolation. The above result follows since Lagrange interpolation polynomialshave the property 1; if k ˆ i;l k …x i †ˆ ki ˆ …16:50†0; otherwiseWe should also note that options other than polynomials may be used for the q…x†.Thus, for any function q i …x† we can set the associated b ~ ji to unity (with all others and~u j set to zero) and obtain^u…x† ˆXnNJ k …x†q i …x† ˆq i …x†…16:51†Again the only requirement is thatj ˆ 1X nj ˆ 1k ˆ 0N k j …x† ˆ1…16:52†

446 Point-based approximationsThus, for any basic functions satisfying the partition of unity a hierarchical enrichmentmay be added using any type of functions. For example, if one knows thestructure of the solution involves exponential functions in x it is possible to includethem as members of the q…x† functions and thus capture the essential part of thesolution with just a few terms. This is especially important for problems which involvesolutions with di€erent length scales. A large length scale can be included in the basicfunctions, Nj k …x†, while other smaller length scales may be included in the functionsq…x†. This will be illustrated further in Volume 3 in the chapter dealing with waves.The above discussion has been limited to functions in one space variable, however,extensions to two and three dimensions can be easily constructed. In the process ofthis extension we shall encounter some di culties which we address in more detailin the section on partition-of-unity ®nite element methods. Before doing this weexplore in the next section the direct use of least square methods to solve di€erentialequations using collocation methods.16.5 Point collocation ± ®nite point methodsFinite di€erence methods based on Taylor formula expansions on regular grids can, asexplained in Chapter 3, Sec. 3.13, always be considered as point collocation methodsapplied to the di€erential equation. They have been used to solve partial di€erentialequations for many decades. 24ÿ26 Classical ®nite di€erence methods commonly restrictapplications to regular grids. This limits their use in obtaining accurate solutions togeneral engineering problems which have curved (irregular) boundaries and/or multiplematerial interfaces. To overcome the boundary approximation and interface problemcurvilinear mapping may be used to de®ne the ®nite di€erence operators. 27The extension of the ®nite di€erence methods from regular grids to generalarbitrary and irregular grids or sets of point has received considerable attention(Girault, 1 Pavlin and Perrone, 2 Snell et al. 3 ). An excellent summary of the currentstate of the art may be found in a recent paper by Orkisz 27 who himself hascontributed very much to the subject since the late 1970s (Liszka and Orkisz 4 ).More recently such ®nite di€erence approximations on irregular grids have beenproposed by Batina 28 in the context of aerodynamics and by OnÄ ate et al. 29ÿ31 whointroduced the name `®nite point method'. Here both elasticity and ¯uid mechanicsproblems have been addressed.In point collocation methods the set of di€erential equations, which here is taken inthe form described in Sec. 3.1, is used directly without the need to construct a weakform or perform domain integrals. Accordingly, we considerA…u† ˆ0…16:53†as a set of governing di€erential equations in a domain subject to boundaryconditionsB…u† ˆ0…16:54†applied on the boundaries ÿ. An approximation to the dependent variable u may beconstructed using either a weighted or moving least square approximation since ateach collocation point the methods become identical. In this we must ®rst describe

Point collocation ± ®nite point methods 447the (collocation) points and the weighting function. The approximation is thenconstructed from Eq. (16.23) by assuming a su cient order polynomial for p inEq. (16.14) such that all derivatives appearing in Eqs (16.53) and (16.54) may becomputed. Generally, it is advantageous to use the same order of interpolation toapproximate both the di€erential and boundary conditions. 27 The resulting discreteform for the di€erential equations at each collocation point becomesA…N…x i †~u i †ˆ0; i ˆ 1; 2; ...; n e …16:55†and the discrete form for each boundary condition isB…N…x i †~u i †ˆ0; i ˆ 1; 2; ...; n b …16:56†The total number of equations must equal the number of collocation points selected.Accordingly,n e ‡ n b ˆ n…16:57†It would appear that little di€erence will exist between continuous approximationsinvolving moving least squares and discontinuous ones as in both locally the samepolynomial will be used. This may well account for the convergence of standardleast square approximations which we have observed in Chapter 3 for discontinuousleast square forms but in view of our previous remarks about di€erentiation, a slightdi€erence will in fact exist if moving least squares are used and in the work of OnÄ ateet al. 29ÿ31 which we mentioned before such moving least squares are adopted.In addition to the choice for p…x†, a key step in the approximation is the choice ofthe weighting function for the least square method and the domain over which theweighting function is applied. In the work of Orkisz 32 and Liszka 33 two methodsare used:1. A `cross' criterion in which the domain at a point is divided into quadrants in acartesian coordinate system originating at the `point' where the equation is to beevaluated. The domain is selected such that each quadrant contains a ®xednumber of points, n q . The product of n q and the number of quadrants, q, mustequal or exceed the number of polynomial terms in p less one (the central nodepoint). An example is shown in Fig. 16.9(a) for a two-dimensional problem(q ˆ 4 quadrants) and n q ˆ 2.2. A `Voronoi neighbour' criterion in which the closest nodes are selected as shownfor a two-dimensional example in Fig. 16.9(b).There are advantages and disadvantages to both approaches ± namely, the crosscriterion leads to dependence on the orientation of the global coordinate axes whilethe Voronoi method gives results which are sometimes too few in number to getappropriate order approximations. The Voronoi method is, however, e€ective foruse in Galerkin solution methods or ®nite volume (subdomain collocation) methodsin which only ®rst derivatives are needed.The interested reader can consult reference 27 for examples of solutions obtainedby this approach. Additional results for ®nite point solutions may be found inwork by OnÄ ate et al. 29 and Batina. 28One advantage of considering moving least square approximations instead ofsimple ®xed point weighted least squares is that approximations at points other

448 Point-based approximations151050–5–10–15–15 –10 –5 0 5 10 15(a)151050–5–10–15–15 –10 –5 0 5 10 15(b)Fig. 16.9 Methods for selecting points: (a) cross; (b) Voronoi.than those used to write the di€erential equations and boundary conditions are alsocontinuously available. Thus, it is possible to perform a full post-processing to obtainthe contours of the solution and its derivatives.In the next part of this section we consider the application of the moving leastsquare method to solve a second-order ordinary di€erential equation using pointcollocation.Example: Collocation (point) solution of ordinary di€erential equations We considerthe solution of ordinary di€erential equations using a point collocation method.The di€erential equation in our examples is taken asÿa d2 udx 2 ‡ b du ‡ cu ÿ f …x† ˆ0dx …16:58†on the domain 0 < x < L with constant coe cients a, b, c, subject to the boundary conditionsu…0† ˆg 1 and u…L† ˆg 2 . The domain is divided into an equally spaced set ofpoints located at x i ; i ˆ 1; ...; n. The moving least square approximation describedin Sec. 16.3 is used to write di€erence equations at each of the interior points (i.e.,i ˆ 2; ...; n ÿ 1). The boundary conditions are also written in terms of discrete approximationsusing the moving least square approximation. Accordingly, for the approximatesolution using p-order polynomials to de®ne the p…x† in the interpolations^u…x† ˆXnN p i …x†~u i…16:59†j ˆ 1we have the set of n equations in n unknowns:X ni ˆ 1ÿa d2 N p idx 2‡ b d2 N p idx 2‡ cNp iX ni ˆ 1!N p i …x 1†~u i ˆ g 1…16:60†x ˆ x j~u i ÿ f …x j †ˆ0; j ˆ 2; ...; n ÿ 1 …16:61†

andX ni ˆ 1N p i …x n†u i ˆ g 2Point collocation ± ®nite point methods 449…16:62†The above equations may be written compactly as:Ku ‡ f ˆ 0…16:63†where K is a square coe cient matrix, f is a load vector consisting of the entries fromg i and f …x j †, and u is the vector of unknown parameters de®ning the approximatesolution ^u…x†. A unique solution to this set of equations requires K to be non-singular(i.e., rank…K† ˆn). The rank of K depends both on the weighting function used toconstruct the least square approximation as well as the number of functions usedto de®ne the polynomials p. In order to keep the least square matrices as well conditionedas possible, a di€erent approximation is used at each node withp … j† …x† ˆ‰1 x ÿ x j …x ÿ x j † 2 ... …x ÿ x j † p Š …16:64†de®ning the interpolations associated with N p j …x†. The matrix K will be of correct rankprovided the weighting function can generate linearly independent equations.The accurate approximation of second derivatives in the di€erential equationrequires the use of quadratic or higher order polynomials in p…x†. 27 In addition, thespan of the weighting function must be su cient to keep the least squares matrixH non-singular at every collocation point. Thus, the minimum span needed tode®ne quadratic interpolations of p…x† (i.e., p ˆ k ˆ 2) must include at least threemesh points with non-zero contributions. At the problem boundaries only half ofthe weighting function span will be used (e.g., the right half at the left boundary).Consequently, for weighting functions which go smoothly to zero at their boundary,a span larger than four mesh spaces is required. The span should not be made toolarge, however, since the sparse structure of K will then be lost and overdi€usesolutions may result.Use of hierarchical interpolations reduces the required span of the weightingfunction. For example, use of interpolations with k ˆ 0 requires only a span ateach point for which the domain is just covered (since any span will include itsde®ning point, x k , the H matrix will always be non-singular). For a uniformlyspaced set of points this is any span greater than one mesh spacing.For the example we use the weighting function described by Eq. (16.12) with aweight span 4.4 (r m ˆ 2:2h) times the largest adjacent mesh space for the quadraticinterpolations with k ˆ p ˆ 2 and a weight 2.01 times the mesh space for thehierarchical quadratic interpolations with k ˆ 0, p ˆ 2.We consider the example of a string on an elastic foundation with the di€erentialequationÿa d2 u‡ cu ‡ f ˆ 0; 0 < x < 1 …16:65†2dxwith the boundary conditions u…0† ˆu…1† ˆ0. This is a special form of Eq. (16.58).The parameters for solution are selected asa ˆ 0:01 c ˆ 1 f ˆÿ1 …16:66†

450 Point-based approximationsCollocation moving least square ODE solution0.90.80.7u (x)0.60.50.4ExactApproximateNodeCol. pt.0.30.20.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0x coordinateFig. 16.10 String on elastic foundation solution using MLS form based on nodes: 27 points, k ˆ 0, p ˆ 2.Collocation moving least square ODE solution0.90.80.7u (x)0.60.50.4ExactApproximateNodeCol. pt.0.30.20.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0x coordinateFig. 16.11 String on elastic foundation hierarchic solution: 9 nodal points, k ˆ 0, p ˆ 2.

Galerkin weighting and ®nite volume methods 451Collocation moving least square ODE solution0.90.80.7u (x)0.60.50.4ExactApproximateNodeCol. pt.0.30.20.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0x coordinateFig. 16.12 String on elastic foundation hierarchic solution: 2 points, k ˆ 0, p ˆ 3.The exact solution is given byu…x† ˆ1 ÿ cosh…mx†‡…1 ÿ cosh…m†† sinh…mx† csinh…m† ; m ˆ …16:67†aThe problem is solved using 27 points and k ˆ p ˆ 2 producing the results shown inFig. 16.10.The process was repeated using the hierarchical interpolations with k ˆ 0 and p ˆ 2using nine points (which results in 27 parameters, the same as for the ®rst case). Theresults are shown in Fig. 16.11.The hierarchical interpolation permits the solution to be obtained using as few astwo points. A solution with two points and interpolations with k ˆ 0 and p ˆ 3 and 5is shown in Figs 16.12 and 16.13, respectively. Note however that with the hierarchicalform additional collocation points have to be introduced to achieve a su cientnumber of equations. We show such collocation points in Fig. 16.11. 1=216.6 Galerkin weighting and ®nite volume methods16.6.1 IntroductionPoint collocation methods are straightforward and quite easy to implement, the maintask being only the selection of the subdomain on which to perform the ®t of the

452 Point-based approximationsCollocation moving least square ODE solution0.90.80.7u (x)0.60.50.4ExactApproximateNodeCol. pt.0.30.20.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0x coordinateFig. 16.13 String on elastic foundation solution: 2 points, k ˆ 0, p ˆ 5.function from which the derivatives are computed. Disadvantages arise, however, inthe need to use high order interpolations such that accurate derivatives of the order ofthe di€erential equation may be computed. Further the treatment of boundaries andmaterial interfaces present di culties.An alternative, as we have discussed in Chapter 3, is the use of `weak' or `variational'forms which are equivalent to the di€erential equation. Approximationsthen require functions which have lower order than in the di€erential equation. Inaddition, boundary conditions often appear as `natural' conditions in the weakform ± especially for ¯ux (derivative or Neuman) type boundary conditions. Thisadvantage now is balanced by a need to perform integration over the whole domain.Here, we consider problems of the form given by (see Sec. 3.2)y……C…v† T D…u† d ‡ E…v† T F…u† dÿ ˆ 0…16:68†in which the operators C, D, E and F contain lower derivatives than those occurring inoperators A and B given in Eqs (16.55) and (16.56), respectively. For example, thesolution of second-order di€erential equations (such as those occurring in thequasi-harmonic or linear elasticity equation) have di€erential operators for C to Fwith derivatives no higher than ®rst order.ÿy We assume that the boundary terms are described such that v ˆ v.

Galerkin weighting and ®nite volume methods 453The approximate solution to forms given by Eq. (16.68) may be achievedusing moving least squares and alternative methods for performing the domainintegrals.16.6.2 Subdomain collocation ± ®nite volume methodA simple extension of the point collocation method is to use subdomains (elements)de®ned by the Voronoi neighbour criterion. The integrals for each subdomain areapproximated as a constant evaluated at the originating point asX n dC…v i † T D…u i † i ‡ Xn bE…v i † T F…u i †ÿ i ˆ 0…16:69†iiwhere n d ‡ n b ˆ n, the total number of unknown parameters appearing in theapproximations of u and v.The validity of the above approximation form can be established using patch tests(see Chapter 10). This approach is often called subdomain collocation or the ®nitevolume method. This approach has been used extensively in constructing approximationsfor ¯uid ¯ow problems. 34ÿ40 It has also been employed with some success in thesolution of problems in structural mechanics. 4116.6.3 Galerkin methods ± diffuse elementsMoving least square approximations have been used with weak forms to constructGalerkin type approximations. The origin of this approach can be traced to thework of Liszka 33 and Orkisz. 27 Additional work, originally called the di€use elementapproximation, was presented in the early 1990s by Nayroles et al. 11ÿ13 Beginning inthe mid-1990s the method has been extensively developed and improved byBelytschko and coauthors under the name element-free Galerkin. 14;15;42;43 A similarprocedure, call `hp-clouds', was also presented by Oden and Duarte. 16;17;44 Each ofthe methods is also said to be `meshless', however, in order to implement a trueGalerkin process it is necessary to carry out integrations over the domain. Whatdistinguishes each of the above processes is the manner in which these integrationsare carried out. In the element-free Galerkin method a background `grid' is oftenused to de®ne the integrals whereas in the hp cloud method circular subdomainsare employed. Di€ering weights are also used as means to generate the movingleast square approximation. The interested reader is referred to the appropriateliterature for more details. Another source to consult for implementation of theEFG method is reference 19. Here we present only a simple implementation forsolution of an ordinary di€erential equation.Example: Galerkin solution of ordinary di€erential equations The moving least squareapproximation described in Sec. 16.3; is now used as a Galerkin method to solve asecond-order ordinary di€erential equation. For an arbitrary function W…x†, aweak form for the di€erential equation may be deduced using the procedures

454 Point-based approximationspresented in Chapter 3. Accordingly, we obtaindWdx a du dx ‡ W bdu ‡ cu ÿ f …x†dx… L0dx ˆ 0…16:70†subject to the boundary conditions u…0† ˆg 1 and u…L† ˆg 2 . Using a hierarchicalmoving least square form a p-order polynomial approximation to the dependentvariable may be written as^u…x† ˆXnNj 0 …x†q jp …x†~u p j…16:71†wherej ˆ 1q jp ˆ‰1 x ÿ x j …x ÿ x j † 2 ... …x ÿ x j † p Š …16:72†Note that in the above form we have used the representationq jp …x†~u p j ˆ ~u j ‡ q…x†~b jThe approximation to the weight function is similarly taken as^W…x† ˆXnNj 0 …x†q jp …x† ~W p jj ˆ 1…16:73†in which W p j are arbitrary parameters satisfying W…0† ˆW…L† ˆ0. The approximationyields the discrete problem(X n… "… ~W p Xn Ld…q T ipNi 0 †i †T a d…q jpNj 0 †‡ q Ti ˆ 1 j ˆ 10 dx dxipNi 0 b d…q !#)jpNj 0 †‡ cqdx jp Nj0 dx ~u j… Li ˆ 1…W p i †T q T ipNi 0 f …x† dx…16:74†0Since W p i is arbitrary, the solution to the approximate weak form yields the set ofequations( "X n d…q T ipNi 0 †a d…q jpNj 0 †‡ q Tdx dxipNi 0 b d…q !#)jpNj 0 †‡ cqdx jp Nj0 dx ~u jˆ Xnj ˆ 1… L0ˆ… L0q T ipN 0 i f …x† dx; i ˆ 1; 2; ...; n …16:75†The set of equations only needs to be modi®ed to satisfy the essential boundaryequations. This is accomplished by replacing the equations corresponding toW 1 ˆ W n ˆ 0by~u 1 ˆ g 1 and ~u n ˆ g 2 .The Galerkin form requires only ®rst derivatives of the approximating functions asopposed to the second derivatives required for the point collocation method. Thisreduction, however, is accompanied by a need to perform integrals over thedomain. For weighting functions given by Eq. (16.12) all functions entering theapproximation are polynomial and rational polynomial expressions, thus, a closedform evaluation is impractical. Accordingly, we evaluate integrals using Gauss and

Galerkin weighting and ®nite volume methods 455Galerkin moving least square ODE solution0.90.80.7u (x)0.60.5ExactApproximateNode0.40.30.20.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0x coordinateFig. 16.14 String on elastic foundation solution: 3-point Gauss quadrature.Galerkin moving least square ODE solution0.90.80.7u (x)0.60.5ExactApproximateNode0.40.30.20.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0x coordinateFig. 16.15 String on elastic foundation solution: 4-point Gauss quadrature.

456 Point-based approximationsGalerkin moving least square ODE solution0.90.80.7u (x)0.60.5ExactApproximateNode0.40.30.20.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0x coordinateFig. 16.16 String on elastic foundation solution: 4-point Gauss±Lobatto quadrature.Galerkin moving least square ODE solution0.90.80.7u (x)0.60.5ExactApproximateNode0.40.30.20.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0x coordinateFig. 16.17 String on elastic foundation solution: 5-point Gauss±Lobatto quadrature.

Use of hierarchic and special functions based on standard ®nite elements 457Gauss±Lobatto quadrature over each interval generated by the basis points in themoving least square representation (i.e., x j for j ˆ 1; 2; ...; n). As an example of thetype of solutions possible we consider the string on elastic foundation problemgiven in the previous section. For the parameters a ˆ 0:004, c ˆ 1 with loadingf ˆÿ1 and zero boundary conditions a Galerkin solution using 3 and 4 pointGauss quadrature and 4 and 5 point Gauss±Lobatto quadrature is shown in Figs16.14±16.17. A mesh consisting of nine equally spaced points is used to de®ne theintervals for the solution and quadrature. The weight function is generated fork ˆ 0, p ˆ 2 with a span of 2.1 mesh points.Based upon this elementary example it is evident that the answers for a nine-pointmesh depend on accurate evaluation of integrals to produce high-quality answers.16.7 Use of hierarchic and special functions based onstandard ®nite elements satisfying the partition ofunity requirement16.7.1 IntroductionIn Sec. 16.4, we discussed the possibility of introducing hierarchical variables to shapefunctions based on moving least square interpolations. However a simpler approachto hierarchical forms and indeed to extensions by other functions can be based onsimple ®nite element shape functions.One important application of the partition of unity method starts from a set of®nite element basis functions, N i …x†. An approximation to u…x† is now given byu…x† ^u…x† ˆXN i …x† ~u i ‡ X q …i† …x†b i…16:76†iwhere N i …x† is the conventional (possibly isoparametric) ®nite element shape functionat node i, q …i† are global functions associated with node i, and~u i , and b i are parametersassociated with the added global hierarchical functions. We must note that asbefore ~u i will not represent a local value of the function unless the function q ibecome zero at the node i.Here we assume that conventional shape functions which satisfy the partition ofunity conditionXN i ˆ 1…16:77†iare used. Thus, the above form is a hierarchic ®nite element method based on thepartition of unity. 21;45;46 We note in particular that the function q …i† may be di€erentfor each node and thus the form may be e€ectively used in an adaptive ®nite elementprocedure as described in Chapter 15.Equation (16.76) provides options for a wide choice of functions for q …i† :1. Polynomial functions. In this case the method becomes an alternative hierarchicalscheme to that presented in Part 2 of Chapter 8.

458 Point-based approximations2. Harmonic `wave' functions. This is a multiscale method and will be discussed indetail in Volume 3.3. Singular functions. These can be used to introduce re-entrant corner or singularload e€ects in elliptic problems (e.g., heat conduction or elasticity forms).Derivatives of Eq. (16.76) are computed directly as@^uˆ X " #@x ki@N i@x k~u i ‡ X @N iq …i†@x k!q …i†‡ N i b@x ik…16:78†The reader will note that the narrow band structure of the standard ®nite elementmethod will always be maintained as it is determined by the connectivity of N i . Notealso that the standard element on which the shape functions N i were generated can beused for all subsequent integrations. Such a formulation is very easy to ®t into any®nite element program.16.7.2 Polynomial hierarchical methodTo give more details of the above hierarchical ®nite element method we ®rst considerthe one-dimensional approximation in a two-noded element wherein which^u ˆ N 1 ‰~u 1 ‡ q …1† b 1 Š‡N 2 ‰~u 2 ‡ q …2† b 2 Š…16:79†N 1 ˆ x2 ÿ x; Nx 2 ÿ x 2 ˆ x ÿ x 11 x 2 ÿ x 1andq …1† ˆ q …2† ˆ‰x k ; x k ‡ 1 ; Š…16:80†b i ˆ ‰ b i1 ; b i2 ; Š TWe recall that N 1 ‡ N 2 ˆ 1 and N 1 x 1 ‡ N 2 x 2 ˆ x.Investigation of the term x k in the approximation^u ˆ N 1 …x† ~u 1 ‡ x j a k1 ‡ N2 …x† ~u 2 ‡ x j a k2…16:81†we observe that a linear dependence with the usual ®nite element approximationoccurs when ~u i ˆ x i b0 ~ and k ˆ 1withb 11 ˆ b 12 ˆ ~b 1 . In this case Eq. (16.81) becomes^u ˆ ‰ N 1 x 1 ‡ N 2 x 2 Š b ~ 0 ‡ ‰ N 1 ‡ N 2 Šx b ~ 1ˆ x b ~ 0 ‡ x b ~ 1…16:82†In one dimension linear dependence can be avoided by setting k to 2 in Eqs (16.79)and (16.80). However, in two and three dimensional problems the linear dependencecannot be completely avoided, and we address this next. 45;47An approximation over two-dimensional triangles may be expressed asu…x; y† ^u…x; y† ˆX3i ˆ 1L i ‰~u i ‡ q …i† b i Š…16:83†

where L i are the area coordinates de®ned in Chapter 8. We consider the case wherecomplete quadratic functions are added asq …i† ˆ x 2 ; xy; y 2…16:84†to give a complete second-order polynomial approximation for u. Although this givesa complete second-order polynomial approximation there are two ways in which thecubic term x 2 y can be obtained.1. The ®rst setsgiving~u i ˆ b i1 ˆ b i3 ˆ 0 and b i2 ˆ x i ~^u ˆ X3i ˆ 1L i ‰xyŠ x i ~ ˆ x 2 y~2. The second alternative to compute the same term setsgivingUse of hierarchic and special functions based on standard ®nite elements 459~u i ˆ b i2 ˆ b i3 ˆ 0 and b i1 ˆ y i ~^u ˆ X3i ˆ 1L i ‰x 2 Š y i ~ ˆ x 2 y ~A similar construction may be made for the polynomial term xy 2 .An alternative is to construct the interpolation to depend on each node asq …i† ˆ …x ÿ x i † 2 …x ÿ x i †…y ÿ y i † …y ÿ y i † 2…16:85†This form, while conceptually the same as the original formulation, appears to bebetter conditioned and also avoids some of the problems of linear dependency. 47 InSec. 16.7.4 we will discuss in more detail a methodology to deal with the problemof linear dependency, however, before doing so we illustrate the use of the hierarchical®nite element method by an application to two-dimensional problems in linearelasticity.16.7.3 Application to linear elasticityIn the previous section the form for polynomial interpolation in two dimensionswas given. Here we consider the use of the interpolation to model the behaviourof problems in linear elasticity. For simplicity only the displacement model forplane strain as discussed in Chapters 2 and 4 is considered; however, the use ofthe hierarchic interpolations can easily be extended to other forms and to mixedmodels.For a displacement model the ®nite element arrays may be computed usingEq. (2.24). For two-dimensional plane strain problems, the strain±displacement

460 Point-based approximationsrelations may be written in matrix form as2@u3@x@ve ˆ6 @y4 @u@y ‡ @v75@x…16:86†Inserting the interpolations for u and v given by Eq. (16.76) and using Eq. (16.78) tocompute derivatives, the strain±displacement relations become2 3@Nik 0@x" #e ˆ XN@Nik ~u i0@yi ˆ 1~v i64 @Nik @Nik 75@y @x2!3@Nik@x qk i ‡ N k @q k ii0@x!" #‡ XN@Nik 0@x qk i ‡ Nik @q k i~b u i…16:87†@yi ˆ 1~b v i!!64 @Nik@x qk i ‡ Nik @q k i @Nik@y @x qk i ‡ Nik @q k 7i 5@xThe ®rst term is identical to the usual ®nite element strain±displacement matrices [seeEq. (4.10b)] and the second term has identical structure to the usual arrays. Thus, thedevelopment of all element arrays follows standard procedures.A quadratic triangular elementFor a triangular element with linear interpolation the shape functions and quadraticpolynomial hierarchic terms are given by N i ˆ L i and Eq. (16.85), respectively. Usingisoparametric concepts the coordinates are given byx ˆ X3i ˆ 1N 1 i ~x i ˆ X3i ˆ 1L i ~x i…16:88†and are used to construct all polynomials appearing in hierarchical form (16.85).A set of patch tests is ®rst performed to assess the stability and consistency of theabove hierarchic form. The set consists of one, two, four, and eight element patches asshown in Fig. 16.18. First, we perform a stability assessment by determining thenumber of zero eigenvalues for each patch. The results for hierarchical interpolationare shown in Table 16.3.The eigenproblem assessment reveals that the hierarchic interpolation has excesszero eigenvalues (i.e., spurious zero energy modes) only for meshes consisting of

Use of hierarchic and special functions based on standard ®nite elements 461(a) One element(b) Two element, a(c) Two element, b(d) Two element, c(e) Four element(f) Eight elementFig. 16.18 Patches for eigenproblem assessment.one or two elements. Furthermore, only two element meshes in which one side is astraight line through both elements have excess zero values. Once the mesh has nostraight intersections the number of zero modes becomes correct (e.g., contain onlythe three rigid body modes).Consistency tests verify that all meshes contain terms of up to quadratic polynomialorder ± thus also validating the correctness of the coding.As a simple test problem using the hierarchical ®nite element method we consider a®nite width strip containing a circular hole with diameter half the width of the strip.The strip is subjected to axial extension in the vertical direction and, due to symmetryTable 16.3 Triangle element patch tests: Number of zeroeigenvalues, minimum non-zero value, and maximum value…k ˆ 2† ± quadratic hierarchical termsMesh No. zero Min. value Max. value1 7 4.7340E ‡ 01 2.0560E ‡ 062a 5 4.0689E ‡ 01 2.1543E ‡ 052b 5 4.1971E ‡ 02 2.2648E ‡ 052c 3 1.5728E ‡ 02 2.3883E ‡ 064 3 1.0446E ‡ 02 2.9027E ‡ 058 3 9.5560E ‡ 01 3.4813E ‡ 05

462 Point-based approximations(a) 28 elements(b) 112 elementsFig. 16.19 Hierarchic elements: tension strip.(a) 28 elements(b) 112 elementsFig. 16.20 Isoparametric six-noded elements: tension strip.

Use of hierarchic and special functions based on standard ®nite elements 463Table 16.4 Hierarchical element. Boundary segmentsstraightNodes Elements Equations Energy30 28 156 131.708885 112 537 127.8260279 448 1971 126.76411003 1792 7527 126.5908of the loading and geometry, only one quadrant is discretized as shown in Figs 16.19and 16.20.The meshes in Fig. 16.19 employ the hierarchical interpolation considered above;whereas those in Fig. 16.20 use standard six-node isoparametric quadratic triangleswith two degrees of freedom per node (i.e., u and v). The material is taken as linearelastic with E ˆ 1000 and ˆ 0:25. The half-width of the strip is 10 units and thehalf-height is 18 units. The hole has radius 5.The problem size and computed energy (which indicates solution accuracy) areshown in Table 16.4 for the hierarchical method, in Table 16.5 for the six-node isoparametricformulation and in Table 16.6 for three-node linear triangular elements.The six-node isoparametric method gives overall the best accuracy; however, thehierarchical element is considerably better than the three-node triangular elementand o€ers great advantages when used in adaptive analysis. 4716.7.4 Solution of forms with linearly dependent equationsA typical problem for a steady-state analysis in which the algebraic equations aregenerated from the hierarchical ®nite element form described above, such as givenTable 16.5 Isoparametric element. Boundary segmentshave curved sidesNodes Elements Equations Energy30 28 129 127.3350279 112 483 126.64831003 448 1863 126.56613795 1792 7311 126.5593Table 16.6 Linear triangular elementNodes Elements Equations Energy30 28 36 137.65285 112 129 131.065279 448 483 128.0081003 1792 1863 126.9583795 7168 7311 126.662

464 Point-based approximationsby Eqs (16.83) and (16.84), produces algebraic equations in the standard form, i.e.,K~a ‡ f ˆ 0…16:89†where the parameters ~a include both nodal ~u i and hierarchical parameters b i .Weassume that occasionally the `sti€ness matrix' K and `force' vector f include equationswhich are linearly dependent with other equations in the system and, thus, K can besingular.If the system is solved by a direct elimination scheme (e.g., as described in Chapter 2or in books on linear algebra such as references 48 or 49) it is possible to set atolerance for the pivot below which an equation is assumed to be linearly dependentand can be omitted from the calculations (e.g., see reference 50, 51).An alternative to the above is to perturb Eq. (16.89) to‰ K ‡ "D K Š~a k ˆ f ÿ K~a k …16:90†where D K are diagonal entries of K, " is a speci®ed value anda k ‡ 1 ˆ a k ‡ a k…16:91†is used to de®ne an iterative strategy. An initial guess of zero may be used to start thesolution process. Certainly a choice of a small value for " (e.g., 10 ÿ6 ) leads to rapidconvergence. 4716.8 ClosureIn this chapter we have considered a number of methods which eliminate or reduceour dependence on meshing the total domain. There are a number of otherapproaches having the same aim which have been pursued with success. These includethe smooth particle hydrodynamics method (SPH) (Lucy, 52 Gingold and Monaghan, 53Benz 54 ) and the reproducing kernel method (RPK) (Liu et al. 55;56 ) applied to problemsin solid and ¯uid mechanics. Bonet and coworkers 57;58 improve the method of SPHand show its possibilities. Another approach has recently been introduced byYagawa. 59;60 These are not described here and the reader is referred to the literaturefor details.References1. V. Girault. Theory of a ®nite di€erence method on irregular networks. SIAMJ. Num.Anal., 11, 260±82, 1974.2. V. Pavlin and N. Perrone. Finite di€erence energy techniques for arbitrary meshes. Comp.Struct., 5, 45±58, 1975.3. C. Snell, D.G. Vesey, and P. Mullord. The application of a general ®nite di€erence methodto some boundary value problems. Comp. Struct., 13, 547±52, 1981.4. T. Liszka and J. Orkisz. Finite di€erence methods of arbitrary irregular meshes in nonlinearproblems of applied mechanics. In Proc. 4th Int. Conference on Structural Mechanicsin Reactor Technology, San Francisco, California, 1977.5. T. Liszka and J. Orkisz. The ®nite di€erence method at arbitrary irregular grids and itsapplications in applied mechanics. Comp. Struct., 11, 83±95, 1980.

6. J. Krok and J. Orkisz. A uni®ed approach to the FE generalized variational FD method innonlinear mechanics. Concept and numerical approach. In Discretization methods in structuralmechanics, pages 353±362. Springer-Verlag, Berlin-Heidelberg, 1990. IUTAM/IACMSymposium, Vienna, 1989.7. R. A. Nay and S. Utku. An alternative for the ®nite element method. Variat. Meth. Engng,1, 1972.8. D. Shepard. A two-dimensional function for irregularly spaced data. In ACMNationalConference, pages 517±24, 1968.9. P. Lancaster and K. Salkauskas. Surfaces generated by moving least squares methods.Math. Comput., 37, 141±58, 1981.10. P. Lancaster and K. Salkauskas. Curve and Surface Fitting. Academic Press, 1990.11. B. Nayroles, G. Touzot, and P. Villon. La me thode des e le ments di€use. C.R. Acad. Sci.Paris, 313, 133±38, 1991.12. B. Nayroles, G. Touzot, and P. Villon. L'approximation di€use. C.R. Acad. Sci. Paris, 313,293±96, 1991.13. B. Nayroles, G. Touzot, and P. Villon. Generalizing the FEM: di€use approximation anddi€use elements. Comput. Mech., 10, 307±18, 1992.14. T. Belytschko, Y. Lu, and L. Gu. Element free Galerkin methods. Intern. J. Num. Meth.Engng, 37, 397±414, 1994.15. T. Belytschko, Y. Lu, and L. Gu. Crack propagation by element-free Galerkin methods.Engng. Fracture Mech., 51, 295±315, 1995.16. A. Duarte and J.T. Oden. hp clouds ± A meshless method to solve boundary-valueproblems. Technical Report TICAM Report 95±05, The University of Texas, May1995.17. C.A. Duarte and J.T. Oden. An h ÿ p adaptive method using clouds. Comput. Meth. Appl.Mech. Engng, 139(1-4), 237±62, 1996.18. T. Belytschko, J. Fish, and A. Bayless. The spectral overlay on ®nite elements for problemswith high gradients. Comput. Meth. Appl. Mech. Engng, 81, 71±89, 1990.19. J. Dolbow and T. Belytschko. An introduction to programming the meshless element freeGalerkin method. Arch. Comput. Meth. Engng, 5(3), 207±41, 1998.20. I. BabusÏ ka and J.M. Melenk. The partition of unity ®nite element method. TechnicalReport Technical Note BN-1185, Institute for Physical Science and Technology, Universityof Maryland, April 1995.21. J.M. Melenk and I. BabusÏ ka. The partition of unity ®nite element method: Basic theoryand applications. Comput. Meth. Appl. Mech. Engng, 139, 289±314, 1996.22. W. Rudin. Principles of Mathematical Analysis. McGraw-Hill, 3rd edition, 1976.23. I. BabusÏ ka and J.M. Melenk. The partition of unity method. Intern. J. Num. Meth. Engng,40, 727±58, 1997.24. L. Collatz. The Numerical Treatment of Di€erential Equations. Springer, Berlin, 1966.25. G.E. Forsythe and W.R. Wasow. Finite Di€erence Methods for Partial Di€erentialEquations. John Wiley & Sons, New York, 1960.26. R. D. Richtmyer and K.W. Morton. Di€erence Methods for Initial Value Problems. Wiley(Interscience), New York, 1967.27. J. Orkisz. Finite di€erence method. In M. Kleiber, editor, Handbook of ComputationalSolid Mechanics. Springer-Verlag, Berlin, 1998.28. J. Batina. A gridless Euler/Navier-Stokes solution algorithm for complex aircraft applications.In AIAA 93±0333, Reno, NV, January 1993.29. E. OnÄ ate, S. Idelsohn, and O.C. Zienkiewicz. Finite point methods in computationalmechanics. Technical Report CIMNE Report 67, Int. Center for Num. Meth. Engr.,Barcelona, July 1995.30. E. OnÄ ate, S.R. Idelsohn, O.C. Zienkiewicz, R.L. Taylor, and C. Sacco. A stabilized ®niteReferences 465

466 Point-based approximationspoint method for analysis of ¯uid mechanics problems. Comput. Meth. Appl. Mech. Engng,139, 315±46, 1996.31. E. OnÄ ate, S.R. Idelsohn, O.C. Zienkiewicz, and R.L. Taylor. A ®nite point method incomputational mechanics. Applications to convective transport and ¯uid ¯ow. Intern. J.Num. Meth. Engng, 39, 3839±66, 1996.32. J. Orkisz. Computer approach to the ®nite di€erence method (in polish). Mechanika iKomputer, 2, 7±69, 1979.33. T. Liszka. An interpolation method for an irregular net of nodes. Intern. J. Num. Meth.Engng, 20, 1599±1612, 1984.34. R.B. Pelz and A. Jameson. Transonic ¯ow calculations using triangular ®nite elements.AIAA J., 23, 569±76, 1985.35. J.T. Batina. Vortex-dominated conical-¯ow computations using unstructured adaptivelyre®nedmeshes. AIAA J., 28(1l), 1925±32, 1990.36. J.T. Batina. Unsteady Euler airfoil solutions using unstructured dynamic meshes. AIAA J.,28(8), 1381±88, 1990.37. D. J. Mavriplis and A. Jameson. Multigrid solution of the navier-stokes equations ontriangular meshes. AIAA J., 28, 1415±25, 1990.38. R.D. Rausch, J.T. Batina, and H.T.Y. Yang. Spatial adaptation of unstructured meshesfor unsteady aerodynamic ¯ow computations. AIAA J., 30(5), 1243±51, 1992.39. R.D. Rausch, J.T. Batina, and H.T.Y. Yang. Three-dimensional time-marching aeroelasticanalyses using an unstructured-grid euler method. AIAA J., 31(9), 1626±33, 1993.40. K. Xu, L. Martinelli, and A. Jameson. Gas-kinetic ®nite volume methods. In S.M.Deshpande, S.S. Desai, and R. Narasimha (eds), Proc. 14th Int. Conf. Num. Meth. FluidDynamics, pages 106±111, 1995.41. E. OnÄ ate, F. Zarate, and F. Flores. A simple triangular element for thick and thin plate andshell analysis. Intern. J. Num. Meth. Engng, 37, 2569±82, 1994.42. Y. Lu, T. Belytschko, and L. Gu. A new implementation of the element-free. Comput.Meth. Appl. Mech. Engng, 113, 397±414, 1994.43. M. Tabbara, T. Blacker, and T. Belytschko. Finite element derivative recovery by movingleast square interpolates. Comput. Meth. Appl. Mech. Engng, 117, 211±23, 1994.44. C.A. Duarte. A review of some meshless methods to solve partial di€erential equations.Technical Report TICAM Report 95-06, The University of Texas, May 1995.45. R.L. Taylor, O.C. Zienkiewicz, and E. OnÄ ate. A hierarchical ®nite element method basedon the partition of unity. Comput. Meth. Appl. Mech. Engng, 152, 73±84, 1998.46. J.T. Oden, C.A. Duarte, and O.C. Zienkiewicz. A new cloud-based hp ®nite elementmethod. Comp. Meth. App. Mech. Eng., 152, 73±84, 1998.47. C.A. Duarte, I. BabusÏ ka, and J.T. Oden. Generalized ®nite element methods for threedimensional structural mechanics problems, in preparation.48. G. Strang. Linear Algebra and its Application. Academic Press, New York, 1976.49. J. Demmel. Applied Numerical Linear Algebra. Society for Industrial and AppliedMathematics, 1997.50. I.S. Du€ and J.K. Reid. Exploiting zeros on the diagonal in the direct solution of inde®nitesparse linear systems. ACMTransactions on Mathematical Software, 22, 227±57, 1996.51. I.S. Du€ and J.A. Scott. Ma62 ± a frontal code for sparse positive-de®nite symmetricsystems from ®nite element applications. In M. Papadrakakis and B.H.V. Topping (eds),Innovative Computational Methods for Structural Mechanics, pages 1±25, June 1999.52. L.B. Lucy. A numerical approach to the testing of fusion process. The Astron. J., 88,1977.53. R.A. Gingold and J.J. Monaghan. Smoothed particle hydrodynamics: Theory and applicationto non-spherical stars. Monthly Notices of Royal Astron. Sci., 181, 1977.54. W. Benz. Smoothed particle hydrodynamics: A review. Preprint 2884, 1989.

55. W.K. Liu, S. Jun, S. Li, J. Adee, and T. Belytschko. Reproducing kernel particle methodsfor structural dynamics. Comput. Meth. Appl. Mech. Engng, 38, 1655±79, 1995.56. W.K. Liu, S. Jun, and Y.F. Zhang. Reproducing kernel particle methods. Intern. J. Num.Meth. Engng, 20, 1081±1106, 1995.57. J. Bonet, S. Kulasetaram, and T.-S.L. Lok. Corrected smooth particle hydrodynamicsmethods for ¯uid and solid mechanics application. In W. Wunderlich, editor, ProceedingsFirst European Conference on Computational Mechanics, August±September 1999. CD-ROM Version.58. J. Bonet and S. Kulasegaram. Correction and stabilization of smooth particle hydrodynamicsmethods with applications in metal forming simulations. Intern. J. Num. Meth.Engng, 47(to appear), 2000.59. G. Yagawa and T. Yamada. Free mesh method. a kind of meshless ®nite element method.Comput. Mech., 18, 383±86, 1996.60. G. Yagawa, T. Yamada, and T. Furukawa. Parallel computing with free mesh method:Virtually meshless fem. In H.A. Mang and F.G. Rammerstorfer (eds), IUTAMSym.,Solid Mechanics and its Applications, pages 165±172. Kluwer Acd. Pub., 1997.References 467

17The time dimension ± semidiscretizationof ®eld and dynamicproblems and analytical solutionprocedures17.1 IntroductionIn all the problems considered so far in this text conditions that do not vary with timewere generally assumed. There is little di culty in extending the ®nite elementidealization to situations that are time dependent.The range of practical problems in which the time dimension has to be considered isgreat. Transient heat conduction, wave transmission in ¯uids and dynamic behaviourof structures are typical examples. While it is usual to consider these various problemsseparately ±sometimes classifying them according to the mathematical structure ofthe governing equations as `parabolic' or `hyperbolic' 1 ±we shall group them intoone category to show that the formulation is identical.In the ®rst part of this chapter we shall formulate, by a simple extension of themethods used so far, matrix di€erential equations governing such problems for avariety of physical situations. Here a ®nite element discretization in the space dimensiononly will be used and a semi-discretization process followed (see Chapter 3). Inthe remainder of this chapter various analytical procedures of the solution for theresulting ordinary linear di€erential equation system will be dealt with. These formthe basic arsenal of steady-state and transient analysis.Chapter 18 will be devoted to the discretization of the time domain itself.17.2 Direct formulation of time-dependent problemswith spatial ®nite element subdivision17.2.1 The `quasi-harmonic' equation with time differentialIn many physical problems the quasi-harmonic equation takes the form in which timederivatives of the unknown function occur. In the three-dimensional case typically

Direct formulation of time-dependent problems with spatial ®nite element subdivision 469we might have@@xk @ @x‡ @ @y k @ ‡ @ @y @zk @ @z!‡ Q ÿ @@t ÿ @2 @t 2 ˆ 0…17:1†In the above, quite generally, all the parameters may be prescribed functions of time,or in non-linear cases of , as well as of space x, i.e.,k ˆ k…x;;t† Q ˆ Q…x;;t† etc: …17:2†If a situation at a particular instant of time is considered, the time derivatives of and all the parameters can be treated as prescribed functions of space coordinates.Thus, at that instant the problem is precisely identi®ed with those treated in Chapter7 if the whole of the quantity in the last parentheses of Eq. (17.1) is identi®ed as thesource term Q.The ®nite element discretization of this in terms of space elements has already beenfully discussed and we found that with the prescription ˆ X N i a i ˆ NaN ˆ N…x; y; z†a ˆ a…t†…17:3†for each element, the standard form of assembled equationsyKa ‡ f ˆ 0…17:4†was obtained. Element contributions to the above matrices are de®ned in Chapter 7and need not be repeated here except for that representing the `load' term due to Q.This is given by… f ˆÿ N T Q d…17:5†Replacing Q by the last bracketed term of Eq. (17.1) we have… f ˆÿ N T Q ÿ @ @t ÿ @2 @t 2 d…17:6†However, from Eq. (17.3) it is noted that is approximated in terms of the nodalparameters a. On substitution of this approximation we have… … dad f ˆÿ N T Q d ‡ N T N ddt…‡ N T 2 aN ddt 2 …17:7†and on expanding Eq. (17.4) in its ®nal assembled form we get the following matrixdi€erential equation:Ma ‡ C_a ‡ Ka ‡ f ˆ 0…17:8†_a dadta d2 adt 2…17:9†y We have replaced the matrix H of Chapter 7 by K to facilitate comparison with other transientequations.

470 The time dimension ± semi-discretization of ®eld and dynamic problemsin which all the matrices are assembled from element submatrices in the standardmanner with submatrices K e and f e still given by relations (7) in Chapter 7 and…Cij e ˆ N i N j d…17:10†…Mij e ˆ N i N j d…17:11†Once again these matrices are symmetric as seen from the above relations.Boundary conditions imposed at any time instant are treated in the standard manner.The variety of physical problems governed by Eq. (17.1) is so large that a comprehensivediscussion of them is beyond the scope of this book. A few typical exampleswill, however, be quoted.Equation (17.1) with ˆ 0This is the standard transient heat conduction equation 1;2 which has been discussed inthe ®nite element context by several authors. 3ÿ6 This same equation is applicable inother physical situations ±one of these being the soil consolidation equations 7associated with transient seepage forms. 8Equation (17.1) with ˆ 0Now the relationship becomes the famous Helmholz wave equation governing a widerange of physical phenomena. Electromagnetic waves, 9 ¯uid surface waves 10 andcompression waves 11 are but a few cases to which the ®nite element process hasbeen applied.Equation (17.1) with 6ˆ 6ˆ 0This damped wave equation is of yet more general applicability and has particularsigni®cance in ¯uid mechanics (wave) problems.The reader will recognize that what we have done here is simply an application ofthe process of partial discretization described in Sec. 3.5. It is convenient, however, toperform the operations in the manner suggested above as all the matrices and discretizationexpressions obtained from steady-state analysis are immediately available.17.2.2 Dynamic behaviour of elastic structures with lineardampingWhile in the previous section we have been concerned with, apparently, a purelymathematical problem, identical reasoning can be applied directly to the wide classof dynamic behaviour of elastic structures following precisely the general lines ofChapter 2.When displacements of an elastic body vary with time two sets of additional forcesare called into play. The ®rst is the inertia force, which for an acceleration characterizedby u can be replaced by its static equivalentÿu

Direct formulation of time-dependent problems with spatial ®nite element subdivision 471using the well-known d'Alembert principle. This is a force with components in directionsidentical to those of the displacement u and (generally) given per unit of volume.In this context is simply the mass per unit volume.The second force is that due to (frictional) resistances opposing the motion. Thesemay be due to microstructure movements, air resistance, etc., and are often related ina non-linear way to the velocity _u. For simplicity of treatment, however, only a linearviscous-type resistance will be considered, resulting again in unit volume forces in anequivalent static problem of magnitudeÿl_uIn the above l is a set of viscosity parameters which can presumably be given numericalvalues. 12The equivalent static problem, at any instant of time, is now discretized preciselyin the manner of Chapter 2, but replacing the distributed body force b by itsequivalentb ÿ u ÿ l_uThe element (nodal) forces given by Eq. (2.13) now become (excluding initial stressand strain contributions)…………f e ˆÿ N T b d ˆÿ N T b d ‡ N T u d ‡ N T l_u d …17:12† e e e ein which the ®rst force is that due to an external distributed body load and need not beconsidered further.Substituting Eq. (17.12) into the general equilibrium equations we obtain ®nally,on assembly, the following matrix di€erential equation:Ma ‡ C_a ‡ Ka ‡ f ˆ 0…17:13†in which K and f are assembled sti€ness and force matrices obtained by the usualaddition of element sti€ness coe cients and of element forces due to externalspeci®ed loads, initial stresses, etc., in the manner fully described before. The newmatrices C and M are assembled by the usual rule from element submatrices given byy…C e ij ˆ N T i lN j d…17:14† eand…M e ij ˆ N T i N j d…17:15† eThe matrix M e is known as the element mass matrix and the assembled matrix M asthe system mass matrix. Similarly, the matrix C e is known as the element dampingmatrix and the assembled matrix C as the system damping matrix.It is of interest to note that in early attempts to deal with dynamic problems of hisnature the mass of the elements was usually arbitrarily `lumped' at nodes, alwaysresulting in a diagonal matrix even if no actual concentrated masses existed. They For simplicity we shall only consider distributed inertia ±concentrated mass and damping forces being alimiting case.

472 The time dimension ± semi-discretization of ®eld and dynamic problemsfact that such a procedure was, in fact, unnecessary and apparently inconsistent wassimultaneously recognized by Archer 13 and independently by Leckie and Lindberg 14in 1963. The general presentation of the results given in Eq. (17.15) is due to Zienkiewiczand Cheung. 15 The name consistent mass matrix has been coined for the massmatrix de®ned here, a term which may be considered to be unnecessary since it isthe logical and natural consequence of the discretization process. By analogy thematrices C e and C may be called consistent damping matrices.For many computational processes the lumped mass matrix is, however, more convenientand economical. Many practitioners are today using such matrices exclusively±sometimes showing good accuracy. While with simple elements a physically obviousmethodology of lumping is easy to devise, this is not the case with higher orderelements and we shall return to the process of `lumping' later.Determination of the damping matrix C is in practice di cult as knowledge of theviscous matrix l is lacking. It is often assumed, therefore, that the damping matrix is alinear combination of sti€ness and mass matrices, i.e.,C ˆ M ‡ K…17:16†Here the parameters and are determined experimentally. 12;16 Such damping isknown as `Rayleigh damping' and has certain mathematical advantages which weshall discuss later. On occasion C may be completely speci®ed and such approximationdevices are not necessary.It is perhaps worth recognizing that on occasion di€erent shape functions need tobe used to describe the inertia forces from those specifying the displacements u. Forinstance, in beams (Chapter 2) (also for plates considered in Chapter 4 of Volume2) the full strain state is prescribed simply by de®ning w, the lateral displacement, asadditional bending assumptions are introduced. When considering the intertiaforces it may be desirable not only to include the simple lateral inertia forcegiven byÿA @2 w@t 2(in which A is now the mass per unit length of the beam) but also to consider rotaryinertia couples of the type ÿI @2 @w@t 2 @xin which I is the rotatory inertia. Now it will be necessary to describe a more generalizeddisplacement u:( )wu ˆ @w@xˆ Na ein which N will follow directly from the de®nition of N which speci®es only the wcomponent. Relations such as Eq. (17.15) are still valid, providing we replace N byN and put in place of the matrix A 00 I

Direct formulation of time-dependent problems with spatial ®nite element subdivision 47317.2.3`Mass' or `damping' matrices for some typical elementsIt is impractical to present in an explicit form all the mass matrices for the variouselements discussed in previous chapters. Some selected examples only will bediscussed here.Plane stress and plane strainUsing triangular elements discussed in Chapter 4 the matrix N e is de®ned asN e ˆ N i N j N kwhereN e i ˆ N i I etc:andI ˆ 1 0 0 1Equation (4.8) gives the shape functions asN i ˆ ai ‡ b i x ‡ c i y; etc:2where is the area of the triangular element.If the thickness of the element is h and this is assumed to be constant within theelement, we have, for the mass matrix, Eq. (17.15),……M e ˆ h N T N dx dyor……M e rs ˆ hI N r N s dx dyIf the relationships of Eq. (4.8) are substituted, it is easy to verify that……(112 when r 6ˆ sN r N s dx dx ˆ16 when r ˆ sThus taking the total mass of the element asm ˆ hthe mass matrix becomes232 0 1 0 1 00 2 0 1 0 1M e ˆ m1 0 2 0 1 0120 1 0 2 0 1674 1 0 1 0 2 050 1 0 1 0 2…17:17†…17:18†

474 The time dimension ± semi-discretization of ®eld and dynamic problemsIf the mass is lumped at the nodes in three equal parts the `lumped' mass matrixcontributed by the element is231 0 0 0 0 00 1 0 0 0 0M e ˆ m0 0 1 0 0 030 0 0 1 0 0…17:19†674 0 0 0 0 1 050 0 0 0 0 1Certainly both matrices di€er considerably and yet in applications the results of theanalysis are almost identical.17.2.4 Mass `lumping' or diagonalizationWe have referred to the computational convenience of lumping of mass matrices andpresenting these in diagonal form. On some occasions such lumping is physicallyobvious (see the linear triangle for instance), in others this is not the case and a`rational' procedure is required. For matrices of the type given in Eq. (17.15) severalalternative approximations have been developed as discussed in Appendix I. In all ofthese the essential requirement of mass preservation is satis®ed, i.e.,X…~M ii ˆ d…17:20†where ~M ii is the diagonal of the lumped mass matrix ~M.Three main procedures exist (see Fig. 17.1):1. the row sum method in which~M ii ˆ X2. diagonal scaling in whichijM ij~M ii ˆ aM iiwith a adjusted so that Eq. (17.20) is satis®ed, 17;18 and3. evaluation of M using a quadrature involving only the nodal points and thusautomatically yielding a diagonal matrix for standard ®nite element shapefunctions 19;20 in which N i ˆ 0 for x ˆ x j ; j 6ˆ i.It should be remarked that Eq. (17.20) does not hold for hierarchical shape functionswhere no lumping procedure appears satisfactory.The quadrature (numerical integration) process is mathematically most appealingbut frequently leads to negative or zero lumped masses. Such a loss of positivede®niteness is undesirable in some solution processes and cancels out the advantagesof lumping. In Fig. 17.1 we show the e€ect of various lumping procedures on

Direct formulation of time-dependent problems with spatial ®nite element subdivision 4751/4 1/4IIIIII1/4 1/41/3III III1/3 1/315/24 5/243/16 3/161/8 1/8I II III7/24 7/2445/16 5/163/8 3/80 3/5701/3 1/316/57 16/571/3 1/3I II III01/303/5716/573/57 01/301/368/361/36–1/121/3–1/121/364/361/368/36 II8/36II1/3 1/3II II4/36 16/36 4/36III1/36 1/368/36–1/12 –1/121/31/36 1/364/36IIIIIIRow sum procedureDiagonal scaling procedureQuadrature using nodal pointsFig. 17.1 Mass lumping for some two-dimensional elements.triangular and quadrilateral elements of linear and quadratic type. It is clear fromthese that the optimal choice to lump the mass is by no means unique.In general we would recommend the use of lumped matrices only as a convenientnumerical device generally paid for by some loss of accuracy. An exception to this isfor `explicit' time integration of dynamics problems where the considerable e ciencyof their use more than compensates for any loss in accuracy (see Chapter 18). In someproblems of ¯uid mechanics (Volume 3) we shall indeed use lumping for an intermediateiterative step in getting the consistent solution. However, we note that ithas occasionally been shown that lumping can improve accuracy of some problemby error cancellation. It can be shown that in the transient approximation thelumping process introduces additional dissipation of the `sti€ness matrix' form andthis can help in cancelling out numerical oscillation.

476 The time dimension ± semi-discretization of ®eld and dynamic problemsTo demonstrate the nature of lumped and consistent mass matrices it is convenientto consider a typical one-dimensional problem speci®ed by the equation@@t ÿ @ @x @ @ÿ @ @x @t @xk @ @xSemi-discretization here gives a typical nodal equation i asÿ M ij ‡ H ij _aj ‡ K ij a j ˆ 0where…M ij ˆ N i N j dx…dNH ij ˆ i dx dN jdx dx…dNK ij ˆ i dx k dN jdx dxand it is observed that H and K have identical structure. With linear elements ofconstant size h the approximating equation at a typical node i (and surroundingnodes i ÿ 1ori ‡ 1) can be written as follows (as the reader can readily verify).M ij _a j h …6 _a i ÿ 1 ‡ 4 _a i ‡ _a i ‡ 1 †H ij _a j …h ÿ _a i ÿ 1 ‡ 2 _a i ÿ _a i ‡ 1 †K ij a j k …h ÿa i ÿ 1 ‡ 2a i ÿ a i ‡ 1 †If a lumped approximation is used for M, that is ~M, we have, simply by addingcoe cients using the row sum method,~M ij _a j ˆ h _a iThe di€erence between the two expressions is~M ij _a j ÿ M ij _a j h …6 ÿ _a i ÿ 1 ‡ 2 _a i ÿ _a i ‡ 1 †and is clearly identical to that which would be obtained by increasing by h 2 =6. As in the above example can be considered as a viscous dissipation we note that the e€ectof using a lumped matrix is that of adding an extra amount of such viscosity and canoften result in smoother (though probably less accurate) solutions.Eigenvalues and analytical solution procedures17.3 General classi®cationWe have seen that as a result of semi-discretization many time-dependent problemscan be reduced to a system of ordinary di€erential equations of the characteristic

form given byFree response ± eigenvalues for second-order problems and dynamic vibration 477Ma ‡ C_a ‡ Ka ‡ f ˆ 0…17:21†In this, in general, all the matrices are symmetric (some cases involving nonsymmetricmatrices will be discussed in Volume 3, Chapter 2). This second-ordersystem often becomes ®rst order if M is zero as, for instance, in transient heat conductionproblems. We shall now discuss some methods of solution of such ordinarydi€erential equation systems. In general, the above equations can be non-linear (if,for instance, sti€ness matrices are dependent on non-linear material properties or iflarge deformations are involved) but here we shall concentrate on linear cases only.Systems of ordinary linear di€erential equations can always in principle be solvedanalytically without the introduction of additional approximations. The remainder ofthis chapter will be concerned with such analytical processes. While such solutions arepossible they may be so complex that further recourse has to be taken to the processof approximation; we shall deal with this matter in the next chapter. The analyticalapproach provides, however, an insight into the behaviour of the system which theauthors always ®nd helpful.Some of the matter in this chapter will be an extension of standard well-knownprocedures used for the solution of di€erential equations with constant coe cientsthat are encountered in most studies of dynamics or mathematics. In the followingwe shall deal successively with:1. determination of free response (f ˆ 0)2. determination of periodic response (f…t† periodic)3. determination of transient response (f…t† arbitrary).In the ®rst two, initial conditions of the system are of no importance and a generalsolution is simply sought. The last, most important, phase presents a problem towhich considerable attention will be devoted.17.4 Free response ± eigenvalues for second-orderproblems and dynamic vibration17.4.1 Free dynamic vibration ± real eigenvaluesIf no damping or forcing terms exist in the dynamic problem of Eq. (17.21) it reducestoMa ‡ Ka ˆ 0…17:22†A general solution of such an equation may be written asa ˆ a exp…i!t†the real part of which simply represents a harmonic response as exp…i!t† cos !t ‡ i sin !t. Then on substitution we ®nd that ! can be determined from…ÿ! 2 M ‡ K†a ˆ 0…17:23†

478 The time dimension ± semi-discretization of ®eld and dynamic problemsThis is a general linear eigenvalue or characteristic value problem and for non-zerosolutions the determinant of the above coe cient matrix must be zero:jÿ! 2 M ‡ Kj ˆ0…17:24†Such a determinant will in general give n values of ! 2 (or ! j ; j ˆ 1; 2; ...; n) when thesize of the matrices K and M is n n, providing the matrices K and M are symmetricpositive de®nite.yWhile the solution of Eq. (17.24) cannot determine the actual values of a we can®nd n vectors a j that give the proportions for the various terms. Such vectors areknown as the normal modes of the system or eigenvectors and are made unique bynormalizing so thata T j Ma j ˆ 1; j ˆ 1; 2; ...; n …17:25†At this stage it is useful to note the property of modal orthogonality, i.e., thata T i Ma j ˆ 0; …i 6ˆ j† …17:26†a T i Ka j ˆ 0; …i 6ˆ j† …17:27†The proof of the above statement is simple. As Eq. (17.23) is valid for any mode wecan write! 2 i Ma i ˆ Ka i! 2 j Ma j ˆ Ka jPremultiplying the ®rst by a T j and the second by a T i and noting the symmetry of M andK so thata T j Ma i ˆ a T i Ma ja T j Ka i ˆ a T i Ka jthe di€erence becomes…! 2 i ÿ ! 2 j †a T i Ma j ˆ 0and if ! i 6ˆ ! j z the orthogonality condition for the matrix M has been proved. Fromthis the orthogonality of the vectors with K follows immediately. The ®nal conditiona T i Ka i ˆ ! 2follows from Eq. (17.25) and a premultiplication of Eq. (17.23) for equation i by a i .17.4.2 Determination of eigenvaluesTo ®nd the actual eigenvalues it is seldom practicable to write the polynomialexpanding the determinant given in Eq. (17.24) and alternative techniques have toy A symmetric matrix is positive de®nite if all the diagonals of the triangular factors are positive, this is ausual case with structural problems ±all roots of Eq. (17.24) are real positive numbers (for a proof see reference1). These are known as the natural frequencies of the system. If only the M matrix is symmetric positivede®nite while K is symmetric positive semide®nite the roots are real and positive or zero.z For any case where repeated frequencies occur we merely enforce the orthogonality by construction.

e developed. The discussion of such techniques is best left to specialist texts andindeed many standard computer programs exist as library routines.Many extremely e cient procedures are available and the reader can ®nd someinteresting matter in references. 21ÿ27In some processes the starting point is the standard eigenvalue problem given byHx ˆ x…17:28†in which H is a symmetric matrix and hence has real eigenvalues. Equation (17.23) canbe written asM ÿ1 Ka ˆ ! 2 aon inverting M with ˆ ! 2 , but symmetry is in general lost.If, however, we write in triangular formM ˆ LL T and M ÿ1 ˆ L ÿT L ÿ1in which L is a lower triangular matrix (i.e., has all zero coediagonal), Eq. (17.26) may now be written asKa ˆ ! 2 LL T aCallingLa ˆ xand multiplying by L ÿ1 we have ®nallyin whichFree response ± eigenvalues for second-order problems and dynamic vibration 479Hx ˆ ! 2 x…17:29†cients above the…17:30†…17:31†H ˆ L ÿ1 KL ÿT…17:32†which is of the standard form of Eq. (17.30), as H is now symmetric.Having determined ! 2 (all, or only a few of the selected smallest values correspondingto fundamental periods) the modes of x are found, and hence by use of Eq. (17.30)the modes of a.If the matrix M is diagonal ±as it will be if the masses have been `lumped' ±theprocedure of deriving the standard eigenvalue problem is simpli®ed and here appearsthe ®rst advantage of the diagonalization, which we have discussed in Sec. 17.2.4.17.4.3Free vibration with the singular K matrixIn static problems we have always introduced a suitable number of support conditionsto allow the sti€ness matrix K to be inverted, or what is equivalent to solve the staticequations uniquely. If such `support' conditions are in fact not speci®ed, as may wellbe the case with a rocket travelling in space, the arbitrary ®xing of a minimum numberof support conditions allows a static solution to be obtained without a€ecting thestresses. In dynamic situations such a ®xing is not permissible and frequently one isfaced with the problem of a free oscillation for which K is singular and thereforedoes not possess unique triangular factors or an inverse.

480 The time dimension ± semi-discretization of ®eld and dynamic problemsTo preserve the applicability of methods which require an inverse (e.g., methodsbased on inverse power iteration 26 ) a simple arti®ce is possible. Equation (17.23) ismodi®ed to‰…K ‡ M†ÿ…! 2 ‡ †MŠa ˆ 0…17:33†in which is an arbitrary constant of the same order as the typical ! 2 sought. The newmatrix …K ‡ M† is no longer singular and can be factored (or inverted) for use in thestandard eigensolution procedure to ®nd …! 2 ‡ †.This simple but e€ective avoidance of an otherwise serious di culty was ®rstsuggested by Cox 28 and Jennings. 29 Alternative methods of dealing with the aboveproblem are given in references 30 and 31.17.4.4 Reduction of the eigenvalue systemIndependent of which technique is used to determine the eigenpairs of the system(17.23), the e€ort for n n matrices is at least one order greater than that involvedin an equivalent static situation. Further, while the number of eigenvalues of thereal system is in®nite, in practice, we are generally interested only in a relativelysmall number of the lower frequencies and it is possible to simplify the computationby reducing the size of the problem.To achieve a reduced problem we assume that the unknown a can be expressed interms of m ( n) vectors t 1 ; t 2 ; ...; t m and corresponding participating factors x i .Wenow writea ˆ t 1 x 1 ‡ t 2 x 2 ‡‡t m x m ˆ Tx…17:34†Inserting Eq. (17.34) into Eq. (17.23) and premultiplying by T T we have a reducedproblem with only m eigenpairs:…! † 2 M x ˆ K x…17:35†whereM ˆ T T MT K ˆ T T KTand ! are now eigenvalues of the reduced system, which for the appropriate choice ofthe t i vectors can be good approximations to the eigenvalues of the original system.If by good fortune the trial vectors were to be chosen as eigenvectors of the originalmatrix the system would become diagonal and all eigenvalues (i.e., in this case! ˆ !) could be determined by a trivial calculation. This indeed is what someiterative eigenproblem strategies attempt (e.g., subspace or Lanczos methods 26;32 ).It is also of course possible by physical insight to ®nd vectors t that correspond closelyto the principal modes of the movement (e.g., see reference 33).17.4.5 Some examplesThere are a variety of problems for which practical solutions exist, so only a fewsimple examples will be shown.

Free response ± eigenvalues for second-order problems and dynamic vibration 481(a)(b)Fig. 17.2 Simply supported beam: (a) ! 1 ˆ 3:8050; (b) ! 2 ˆ 59:2236; (c) ! 3 ˆ 290:0804.(c)Vibration of a simply supported beamFigure 17.2 shows the ®rst three vibration modes of a simply supported beam withlength 40 and rectangular cross-section of width 1 and depth 2 units. The elasticproperties are E ˆ 30 000, ˆ 0, and ˆ 0:1 units. The beam is modelled using 9-noded quadrilateral elements of lagrangian type with the central node at the leftend restrained in the x and y direction and the central node at the right end restrainedonly in the y direction. The problem is also solved using a mesh with 1000 two-nodedbeam elements which include e€ects of transverse shearing deformation. In Table 17.1we present the values for the ®rst three frequencies obtained from the ®nite elementanalysis and compare to the value obtained from an exact solution for the beam withoutshear deformation.Vibration of an earth damFigure 17.3 shows the vibration of a two-dimensional earth dam resting on a rigidfoundation. The earth dam is modelled by linear triangular elements and includesthe e€ects of di€erent material layers.The `wave' equation. Electromagnetic and ¯uid problemsThe basic dynamic equation (17.8) can be derived for a variety of non-structural problems.The eigenvalue problem once again occurs with `sti€ness' and `mass' matricesnow having alternate physical meanings.Table 17.1 Frequencies for a simply supported beam9-noded element 3.7785 59.2236 290.08042-noded element 3.7787 59.2338 290.1774Beam theory 3.8050 60.8807 308.2080

482 The time dimension ± semi-discretization of ®eld and dynamic problems(a)(b)(c)(d)Fig. 17.3 (a) Mesh showing layers considered; (b) earth dam, ! 1 ˆ 3:8050; (c) earth dam, ! 2 ˆ 59:2236;(d) earth dam, ! 3 ˆ 290:0804.A particular form of the more general equations discussed earlier is the well-knownHelmholz wave equation which, in two-dimensional form, is@ 2 @x 2 ‡ @2 @y 2 ‡ 1 @ 2 c 2 @t 2 ˆ 0…17:36†If the boundary conditions do not force a response, an eigenvalue problem resultswhich has signi®cance in several ®elds of physical science.The ®rst application is to electromagnetic ®elds. 9 Figure 17.4 shows a modal shapeof a ®eld for a waveguide problem. Simple linear triangular elements are used here.More complex three-dimensional oscillations are also discussed in reference 9.A similar equation also describes to a reasonable approximation the behaviour ofshallow water waves in a body of water:@@xh @ ‡ @ @x @y h @@y‡ 1 @ 2g @t 2 ˆ 0…17:37†

Free response ± eigenvalues for second-order problems and dynamic vibration 4831020sr0'θ080999693906070503040Fig. 17.4 A `lunar' waveguide; 9 mode of vibration for electromagnetic ®eld. Outer diameter ˆ d,OO 0 ˆ 0:13d, r ˆ 0:29d, S ˆ 0:055d, ˆ 228.Scale of feet0 1000 2000yHarbour oscillation30 in40 in50 in75 in60 inLine of equalhorizontal movementHarbour entrance50 in60 in75 inLimit of water surface40 in30 inxFig. 17.5 Oscillations of a natural harbour: contours of velocity amplitudes. 10

484 The time dimension ± semi-discretization of ®eld and dynamic problemsin which h is the average water depth, the surface elevation above average and gthe gravity acceleration. The formulation of the last two problems are discussed indetail in Volume 3, Chapter 8.Thus natural frequencies of bodies of water contained in harbours of varyingdepths may easily be found. 10 Figure 17.5 shows the modal shape for a particularharbour.17.5 Free response ± eigenvalues for ®rst-orderproblems and heat conduction, etc.If in Eq. (17.21) M ˆ 0, we have a form typical of the transient heat conductionequation [see Eq. (17.1)]. For free response we seek a solution of the homogeneousequationC_a ‡ Ka ˆ 0…17:38†Once again an exponential form can be used:a ˆ a exp…ÿt†Substituting we have… ÿC ‡ K†a ˆ 0 …17:39†which again gives an eigenvalue problem identical to that of Eq. (17.23). As C and Kare usually positive de®nite, will be positive and real. The solution thereforerepresents simply an exponential decay term and is not really steady state. Combinationof such terms, however, can be useful in the solution of initial value transientproblems but is of little value per se.17.6 Free response ± damped dynamic eigenvaluesWe shall now consider the full equation (17.21) for free response conditions. WritingMa ‡ C_a ‡ Ka ˆ 0…17:40†and substitutingwe have the characteristic equationa ˆ a exp…t†… 2 M ‡ C ‡ K†a ˆ 0…17:41†…17:42†where and a will in general be found to be complex. The real part of the solutionrepresents a decaying vibration.The eigenvalue problem involved in Eq. (17.41) is more di cult than that arising inthe previous sections. In solutions to date the problem is usually solved by splittingEq. (17.40) into two ®rst-order equations. This is accomplished by de®ning_a ˆ v

Forced periodic response 485and writing the split form asM 0 _v0 ÿM _a ‡ C K vˆ 0 …17:43†M 0 a 0Now substitutinga ˆ a exp…t† v ˆ v exp…t†gives the general linear eigenproblem M 0 ‡ C K vˆ 0 …17:44†0 ÿM M 0 a 0This form has been studied by Chen et al. 34ÿ36 Similar to the ®rst-order problem, nosteady-state solution exists and once more the concept of eigenvalues of the abovekind is generally of importance only in modal analysis, as we shall see later.17.7 Forced periodic responseIf the forcing term in Eq. (17.21) is periodic or, more generally, if we can express it asf ˆ f exp…t†…17:45†where is complex, i.e. ˆ 1 ‡ i 2…17:46†then a general solution can once more be written asa ˆ a exp…t†…17:47†Substituting the above in Eq. (17.21) givesÿ 2 M ‡ C ‡ K a Ka ˆÿ f …17:48†which is no longer an eigenvalue problem but can be solved formally by inverting thematrix K asa ˆÿK ÿ1 f…17:49†The solution is thus precisely of the same form as that used for static problems butnow, however, has to be determined in terms of complex quantities. Computerprograms are available for operation of complex numbers but computation can bearranged in real numbers directly, noting thatexp…t† ˆexp… 1 t† ‰ cos 2 t ‡ i sin 2 tŠ f ˆ f1 ‡ i f 2…17:50†a ˆ a 1 ‡ i a 2in which 1 ; 2 ; f 1 ; f 2 ; a 1 and a 2 are real quantities. Inserting the above into Eq. (17.48)we have" #… 2 1 ÿ 2 2†M ‡ 1 C ‡ K; ÿ2 1 2 M ÿ 2 C a1ÿ2 1 2 M ÿ 2 C; ÿ… 2 1 ÿ 2 ˆÿ f 12†M ÿ 1 C ÿ K a 2 ÿ …17:51†f 2

486 The time dimension ± semi-discretization of ®eld and dynamic problemsEquations (17.51) form a system in which all quantities are real and from whichthe response to any periodic input can be determined by direct solution. The systemis no longer positive de®nite although it has been written in a form which is stillsymmetric.With periodic input the solution after an initial transient is not sensitive to theinitial conditions and the above solution represents the ®nally established response.It is valid for problems of dynamic structural and ¯uid-structure responses as wellas for problems typical of heat conduction in which we simply put M ˆ 0.17.8 Transient response by analytical procedures17.8.1 GeneralIn the previous sections we have been concerned with steady-state general solutionswhich took no account of the initial conditions of the system or of the non-periodicform of the forcing terms. The response taking these features into account is essentialif we consider, for instance, the earthquake behaviour of structures or the transientbehaviour of the heat conduction problem. The solution of such general casesrequires either a full-time discretization, which we shall discuss in detail in the nextchapter, or the use of special analytical procedures. Here two broad possibilities exist:1. the frequency response procedure2. the modal analysis procedure.We shall discuss these brie¯y.17.8.2 Frequency response proceduresIn Sec. 17.7 we have shown how the response of the system to any forcing terms of thegeneral periodic type or in particular to a periodic forcing functionf ˆ f exp…i !t†…17:52†can be obtained by solving a simple equation system. As a completely arbitraryforcing function can be represented approximately by a Fourier series or in thelimit, exactly, as a Fourier integral, the response to such an input can be obtainedby a synthesis of a curve representing the response of any quantity of interest, e.g.,the displacement at a particular point, etc., to all frequencies ranging from zero toin®nity. In fact only a limited number of such forcing frequencies has to be consideredand a result can be synthesized e ciently by fast Fourier transform techniques. 37 Weshall not discuss the mathematical details for such procedures which can be found instandard texts on structural dynamics. 12;16The technique of frequency response is readily adapted to problems where thedamping matrix C is of an arbitrary speci®ed form. This is not the case with themore widely used modal decomposition procedures which are to be described inthe next section.

Transient response by analytical procedures 487By way of illustration we show in Fig. 17.6 the frequency response of an arti®cialharbour [see Eq. (17.37)] to an input of waves with di€erent frequencies and dampingdue to the radiation of re¯ected waves which imposes a very particular form on thedamping matrix. Details of this problem are given elsewhere 38;39 (see also Volume3). Similar techniques are frequently used in the analysis for the foundation responseof structures where radiation of energy occurs. 4017.8.3Modal decomposition analysisThis procedure is probably the most important and widely used in practice. Further, itprovides an insight into the behaviour of the whole system, which is of value wherestrictly numerical processes are used. We shall therefore describe it in detail in thecontext of the general problem of Eq. (17.21), i.e.,Ma ‡ C_a ‡ Ka ‡ f ˆ 0…17:53†where f is an arbitrary function of time.We have seen that the general solution for the free response is of the forma ˆ Xni ˆ 1a i exp… i t†…17:54†where i are the (complex) eigenvalues and a i are the (complex) eigenvectors (Sec.17.6). For forced response we shall assume that the solution can be written in alinear combination of modes asa ˆ Xni ˆ 1a i y i …t† ˆ‰ a 1 ; a 1 ; ... Šy…t† …17:55†where the scalar mode participation factor y i is now a function of time. This shows ina clear manner the proportions of each mode occurring. Such a decomposition of anarbitrary vector presents no restriction as all the modes are linearly independentvectors (with those for repeated frequencies being constructed to be linearly independentas mentioned in Sec. 17.4).If expression (17.55) is substituted into Eq. (17.53) and the result is premultiplied bythe complex conjugate transposed, a T i (i ˆ 1; ...; n), then the result is simply a set ofscalar, independent, equationswherem i y i ‡ c i _y i ‡ k i y i ‡ f i ˆ 0m i ˆ a T i Ma i…17:56†c i ˆ a T i Ca ik i ˆ a T i Ka i…17:57†f i ˆ a T i f

488 The time dimension ± semi-discretization of ®eld and dynamic problemsOuter domainstretches to infinitySeries solution‘Infinite’ elementsB'a = 888 ftAyBxA'Idealization usingtriangular linear elementIdealization using quadraticisoparametric elementsλ for ka = 5(shortest wavelength)(a) Geometric details and FEM idealization.(a) Wave forcing frequency ω = k gh = ka, h = depth of water7Amplitude magnification65432Chen and Mei 38Zienkiewicz and Bettess 3910 0 1 2 3 4 5 6Frequency ka(b) Amplitude magnification response of mean depth in harbour(b) for various frequenciesFig. 17.6 Frequency response of an arti®cial harbour to an input of periodic wave.

as for true eigenvectors a ia T i Ma j ˆ a T i Ca j ˆ a T i Ka j ˆ 0 …17:58†Transient response by analytical procedures 489when i 6ˆ j (this result was proved in Sec. 17.4 for real eigenpairs but is valid generallyfor complex pairs, as could be veri®ed by the reader).Each scalar equation of (17.56) can be solved by elementary procedures independentlyand the total vector of response obtained by superposition followingEq. (17.57). In the general case, as we have shown in Sec. 17.6, the eigenpairs arecomplex and their determination is not simple. 30 The more usual procedure is touse real eigenpairs corresponding to the solution of Eq. (17.22):Ka ˆ ! 2 Ma…17:59†Now repetition of procedures using the process described in Eqs (17.55)±(17.58) leadsto decoupled equations with real variables y only ifa T i Ca j ˆ 0; i 6ˆ j …17:60†which generally does not occur as the eigenvectors now guarantee only orthogonalitywith M and K and not of the damping matrix. However, if the damping matrix C is ofthe form of Eq. (17.16), i.e., a linear combination of M and K, such orthogonality willobviously occur. Unless the damping is of a de®nite form which requires specialtreatment, an assumption of orthogonality is made and Eq. (17.56) is assumedvalid in terms of such eigenvectors.From Eq. (17.59) we haveKa i ˆ ! 2 i Ma i…17:61†and on premultiplying by a T i we obtaink i ˆ ! 2 i m i…17:62†Writing the modal damping in the formc i ˆ 2! i…17:63†(where i represents the ratio of damping to its critical value) and assuming that themodes have been normalized so that m i ˆ 1 [see Eq. (17.25)], Eq. (17.56) can berewritten in standard second order form:y i ‡ 2! i i _y i ‡ ! 2 i y i ‡ f i ˆ 0…17:64†A general solution can then be obtained by writingy i ˆ exp…ÿ i ! i t† _y i0 ‡ i ! i y i0sin !!i t ‡ y i0 cos ! i t… texp…ÿ i ! i ‰t ÿ Š† sin ! i …t ÿ †f i …† d0‡ 1 !ipin which ! i ˆ ! i 1 ÿ i2 and y i0 , _y i0 are initial conditions computed fromy i0 ˆ a T i Ma…0†_y i0 ˆ a T i M_a…0†…17:65†…17:66†

490 The time dimension ± semi-discretization of ®eld and dynamic problemsThe solution of Eq. (17.65) can be carried out by assuming the forcing function isgiven by linear interpolation between discrete time points t k and then evaluating theresulting integrals exactly. Alternatively, a numerical solution can be carried outand the response obtained. In practice, often a single calculation is carried outfor each mode to determine the maximum responses and a suitable addition ofthese results is used. Such processes are described in standard texts and are usedas procedures to calculate the bounds on behaviour of structures subjected toseismic loading. 12;16;2717.8.4 Damping and participation of modesThe type of calculation implied in modal decomposition apparently necessitates thedetermination of all modes and eigenvalues, a task of considerable magnitude. Infact only a limited number of modes usually need to be taken into consideration asoften the response to higher frequency is critically damped and insigni®cant.To show that this is true consider the form of the damping matrices. In Sec. 17.2[Eq. (17.16)] we have indicated that the damping matrix is often assumed asC ˆ M ‡ K…17:67†Indeed a form of this type is necessary for the use of modal decomposition, althoughother generalizations are possible. 41;42 From the de®nition of i , the critical dampingratio in Eq. (17.63), we see that this can now be written as i ˆ 1 a T i … M ‡ K†a 2! i ˆ 1 ÿ ‡ ! 2 i…17:68†i 2! iThus if the coe cient is of greater importance, as is the case with most structuraldamping, i grows with ! i and at high frequency an overdamped condition willarise. 12 This is indeed fortunate as, in general, an in®nite number of high frequenciesexist which are not modelled by any ®nite element discretization.We shall see in the next chapter that in the step-by-step recurrence computation thehigh frequencies often control the problem, and this e€ect needs to be `®ltered out' forrealistic results.17.9 Symmetry and repeatabilityIn concluding this chapter it is worth remarking that in dynamic calculation we haveonce again encountered all the general principles of assembly, etc., that are applicableto static problems. However, some aspects of symmetry and repeatability which wereused previously (see Sec. 9.18) need amending. It is obviously possible for symmetricstructures to vibrate in an unsymmetrical manner, for instance, and similarly arepeatable structure contains modes which are themselves non-repeatable. However,even here considerable simpli®cation can still be made; details of this are discussed byWilliams, 43 Thomas 44 and Evensen. 45

References 491References1. S. Crandall. Engineering Analysis. McGraw-Hill, New York, 1956.2. H.S. Carslaw and J.C. Jaeger. Conduction of Heat in Solids. Clarendon Press, Oxford, 2ndedition, 1959.3. W. Visser. A ®nite element method for the determination of non-stationary temperaturedistribution and thermal deformation. In Proc. 1st Conf. Matrix Methods in StructuralMechanics, volume AFFDL-TR-66±80, Wright-Patterson Air force Base, Ohio, October1966.4. O.C. Zienkiewicz and Y.K. Cheung. The Finite Element Method in Structural Mechanics.McGraw-Hill, London, 1967.5. E.L. Wilson and R.E. Nickell. Application of ®nite element method to heat conductionanalysis. Nucl. Eng. Design, 4, 1±11, 1966.6. O.C. Zienkiewicz and C.J. Parekh. Transient ®eld problems ±two and three dimensionalanalysis by isoparametric ®nite elements. Internat. J. Num. Meth. Eng., 2, 61±71, 1970.7. K. Terzhagi and R.B. Peck. Soil Mechanics in Engineering. John Wiley & Sons, New York,1948.8. D.K. Todd. Ground Water Hydrology. John Wiley & Sons, New York, 1959.9. P.L. Arlett, A.K. Bahrani, and O.C. Zienkiewicz. Application of ®nite elements to thesolution of Helmholz's equation. Proc. IEE, 115, 1762±64, 1968.10. C. Taylor, B.S. Patil, and O.C. Zienkiewicz. Harbour oscillation: a numerical treatment forundamped natural modes. Proc. Inst. Civ. Eng., 43, 141±56, 1969.11. O.C. Zienkiewicz and R.E. Newton. Coupled vibration of a structure submerged in a compressible¯uid. In Proc. Int. Symp. on Finite Element Techniques, pages 1±15, Stuttgart, 1969.12. A.K. Chopra. Dynamics of Structures. Prentice-Hall, Upper Saddle River, NJ, 1995.13. J.S. Archer. Consistent mass matrix for distributed systems. Proc. Am. Soc. Civ. Eng.,89(ST4), 161, 1963.14. F.A. Leckie and G.M. Lindberg. The e€ect of lumped parameters on beam frequencies.Aero. Q., 14, 234, 1963.15. O.C. Zienkiewicz and Y.K. Cheung. The ®nite element method for analysis of elasticisotropic and orthotropic slabs. Proc. Inst. Civ. Eng., 28, 471±88, 1964.16. R.W. Clough and J. Penzien. Dynamics of Structures. McGraw-Hill, New York, 2ndedition, 1993.17. S.W. Key and Z.E. Beisinger. The transient dynamic analysis of thin shells in the ®nite elementmethod. In Proc. 1st Conf. Matrix Methods in Structural Mechanics, volumeAFFDL-TR-6680, pages 667±710, Wright-Patterson Air force Base, Ohio, October 1966.18. E. Hinton, T. Rock, and O.C. Zienkiewicz. A note on mass lumping and related processesin the ®nite element method. Earthquake Eng. Struct. Dyn., 4, 245±49, 1976.19. P. Tong, T.H.H. Pian, and L.L. Bociovelli. Mode shapes and frequencies by the ®niteelement method using consistent and lumped matrices. Comp. Struct., 1, 623±38, 1971.20. I. Fried and D.S. Malkus. Finite element mass matrix lumping by numerical integrationwith the convergence rate loss. Internat. J. Solids Struct., 11, 461±65, 1975.21. J.H. Wilkinson. The Algebraic Eigenvalue Problem. Clarendon Press, Oxford, 1965.22. I. Fried. Gradient methods for ®nite element eigen problems. AIAA J., 7, 739±41, 1969.23. J.H. Wilkinson and C. Reinsch. Linear Algebra. Handbook for Automatic Computation,volume II. Springer-Verlag, Berlin, 1971.24. K.K. Gupta. Solution of eigenvalue problems by Sturm sequence method. Internat. J.Num. Meth. Eng., 4, 379±404, 1972.25. A. Jennings. Mass condensation and similarity iterations for vibration problems. Internat.J. Num. Meth. Eng., 6, 543±52, 1973.

492 The time dimension ± semi-discretization of ®eld and dynamic problems26. B.N. Parlett. The Symmetric Eigenvalue Problem. Prentice-Hall, Englewood Cli€s, NJ,1980.27. K.J. Bathe. Finite Element Procedures. Prentice-Hall, Englewood Cli€s, NJ, 1996.28. H.L. Cox. Vibration of missiles. Aircraft Eng., 33, 2±7 and 48±55, 1961.29. A. Jennings. Natural vibration of a free structure. Aircraft Eng., 34, 8, 1962.30. W.C. Hurty and M.F. Rubinstein. Dynamics of Structures. Prentice-Hall, EnglewoodCli€s, NJ, 1974.31. A. Craig and M.C.C. Bampton. On the iterative solution of semi de®nite eigenvalueproblems. Aero. J., 75, 287±90, 1971.32. J. Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics,1997.33. O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method, Volume 2. McGraw-Hill,London, 4th edition, 1991.34. H.-C. Chen and R.L. Taylor. Using Lanczos vectors and Pitz vectors for computingdynamic responses. Engrg. Comp., 6, 151±57, 1989.35. A. Ibrabimbegovic, H.C. Chen, E.L. Wilson, and R.L. Taylor. Ritz method for dynamicanalysis of large discrete linear systems with non-proportional damping. EarthquakeEng. Struct. Dyn., 19, 877±89, 1990.36. H.-C. Chen and R.L. Taylor. Properties and solutions of the eigensystem of nonproportionallydamped linear dynamic systems. Technical Report UCB/SEMM-86/10,University of California, Berkeley, November 1986.37. E.O. Brigham. The Fast Fourier Transform. Prentice-Hall, Englewood Cli€s, NJ, 1974.38. H.S. Chen and C.C. Mei. Hybrid-element method for water waves. In Proc. ModellingTechniques Conf. (Modelling 1975), volume 1, pages 63±81, San Francisco, 1975.39. O.C. Zienkiewicz and P. Bettess. In®nite elements in the study of ¯uid-structure interactionproblems. In 2nd Int. Symp. on Computing Methods in Applied Science and Engineering,Versailles, France, December 1975.40. J. Penzien. Frequency domain analysis including radiation damping and water load coupling.In O.C. Zienkiewicz, R.W. Lewis, and K.G. Stagg, editors, Numerical Methods inO€shore Engineering. John Wiley & Sons, 1978.41. E.L. Wilson and J. Penzien. Evaluation of orthogonal damping matrices. Internat. J. Num.Meth. Eng., 4, 5±10, 1972.42. H.T. Thomson, T. Collins, and P. Caravani. A numerical study of damping. EarthquakeEng. Struct. Dyn., 3, 97±103, 1974.43. F.W. Williams. Natural frequencies of repetitive structures. Q. J. Mech. Appl. Math., 24,285±310, 1971.44. D.L. Thomas. Standing waves in rotationally periodic structures. J. Sound Vibr., 37, 288±90, 1974.45. D.A. Evensen. Vibration analysis of multi-symmetric structures. AIAA J., 14, 446±53,1976.

18The time dimension ± discreteapproximation in time18.1 IntroductionIn the last chapter we have shown how semi-discretization of dynamic or transient®eld problems leads in linear cases to sets of ordinary di€erential equations of theformMa ‡ C_a ‡ Ka ‡ f ˆ 0whereda _a; etc.dt …18:1†subject to initial conditionsa…0† ˆa 0 and _a…0† ˆ_a 0for dynamics orC_a ‡ Ka ‡ f ˆ 0…18:2†subject to the initial conditiona…0† ˆa 0for heat transfer or similar problems.In many practical situations non-linearities exist, typically altering the aboveequations by makingM ˆ M…a† C ˆ C…a† Ka ˆ P…a† …18:3†The analytical solutions previously discussed, while providing much insight into thebehaviour patterns (and indispensable in establishing such properties as naturalsystem frequencies), are in general not economical for the solution of transientproblems in linear cases and not applicable when non-linearity exists. In this chapterwe shall therefore revert to discretization processes applicable directly to the timedomain.For such discretization the ®nite element method, including in its de®nition the®nite di€erence approximation, is of course widely applicable and provides thegreatest possibilities, though much of the classical literature on the subject uses

494 The time dimension ± discrete approximation in timeonly the latter. 1ÿ6 We shall demonstrate here how the ®nite element method providesa useful generalization unifying many existing algorithms and providing a variety ofnew ones.As the time domain is in®nite we shall inevitably curtail it to a ®nite time incrementt and relate the initial conditions at t n (and sometimes before) to those at timet n ‡ 1 ˆ t n ‡ t, obtaining so-called recurrence relations. In all of this chapter, thestarting point will be that of the semi-discrete equations (18.1) or (18.2), though, ofcourse, the full space-time domain discretization could be considered simultaneously.This, however, usually o€ers no advantage, for, with the regularity of the timedomain, irregular space-time elements are not required. Indeed, if product-typeshape functions are chosen, the process will be identical to that obtained by using®rst semi-discretization in space followed by time discretization. An exception hereis provided in convection dominated problems where simultaneous discretizationmay be desirable, as we shall discuss in the Volume 3.The ®rst concepts of space-time elements were introduced in 1969±70 7ÿ10 and thedevelopment of processes involving semi-discretization is presented in references 11±20. Full space-time elements are described for convection-type equations in references21, 22 and 23 and for elastodynamics in references 24, 25 and 26.The presentation of this chapter will be divided into four parts. In the ®rst weshall derive a set of single-step recurrence relations for the linear ®rst- and secondorderproblems of Eqs (18.2) and (18.1). Such schemes have a very generalapplicability and are preferable to multistep schemes described in the second part asthe time step can be easily and adaptively varied. In the third part we brie¯ydescribe a discontinuous Galerkin scheme and show its application in some simpleproblems. In the ®nal part we shall deal with generalizations necessary for nonlinearproblems.When discussing stability problems we shall often revert to the concept of modallyuncoupled equations introduced in the previous chapter. Here we recall that theequation systems (18.1) and (18.2) can be written as a set of scalar equations:m i y i ‡ c i _y i ‡ k i y i ‡ f i ˆ 0…18:4†orc i _y i ‡ k i y i ‡ f i ˆ 0…18:5†in the respective eigenvalue participation factors y i . We shall ®nd that the stabilityrequirements here are dependent on the eigenvalues associated with such equations,! i . It turns out, however, fortunately, that it is never necessary to obtain thesystem eigenvalues or eigenvectors due to a powerful theorem ®rst stated for ®niteelement problems by Irons and Treharne. 27The theorem states simply that the system eigenvalues can be bounded by theeigenvalues of individual elements ! e . Thusmin…! j † 2 5 min…! e † 2jemaxj…! j † 2 4 max…! e † 2e…18:6†The stability limits can thus (as will be shown later) be related to Eqs (18.4) or (18.5)written for a single element.

Simple time-step algorithms for the ®rst-order equation 495Single-step algorithms18.2 Simple time-step algorithms for the ®rst-orderequation18.2.1 Weighted residual ®nite element approachWe shall now consider Eq. (18.2) which may represent a semi-discrete approximationto a particular physical problem or simply be itself a discrete system. The objective isto obtain an approximation for a n ‡ 1 given the value of a n and the forcing vector facting in the interval of time t. It is clear that in the ®rst interval a n is the initialcondition a 0 , thus we have an initial value problem. In subsequent time intervals a nwill always be a known quantity determined from the previous step.In each interval, in the manner used in all ®nite element approximations, we assumethat a varies as a polynomial and take here the lowest (linear) expansion as shown inFig. 18.1 writinga ^a…t† ˆa n ‡ t …a n ‡ 1 ÿ a n †with ˆ t ÿ t n .This can be translated to the standard ®nite element expansion giving^a…t† ˆX N i a i ˆ 1 ÿ at n ‡ at n ‡ 1…18:7†…18:8†in which the unknown parameter is a n ‡ 1 .The equation by which this unknown parameter is provided will be a weightedresidual approximation to Eq. (18.2). Accordingly, we write the variational problem… t0w…† T ‰C_a ‡ Ka ‡ fŠ d ˆ 0…18:9†in which w…† is an arbitrary weighting function We write the approximate formw…† ˆW…†a n ‡ 1…18:10†a n (known)a n + 1(to be determined)τt n t n + 1∆tFig. 18.1 Approximation to a in the time domain.

496 The time dimension ± discrete approximation in timein which a n ‡ 1 is an arbitrary parameter. With this approximation the weighted residualequation to be solved is given by… t0W…†‰C _^a ‡ K^a ‡ fŠ d ˆ 0…18:11†Introducing as a weighting parameter given by ˆ 1„ t0 W d„t t…18:12†0 W dwe can immediately write1t C…a n ‡ 1 ÿ a n †‡K‰a n ‡ …a n ‡ 1 ÿ a n †Š ‡ f ˆ 0 …18:13†where f represents an average value of f given by„ t f ˆ0 Wf d„ t…18:14†0 W dor f ˆ fn ‡ …f n ‡ 1 ÿ f n †…18:15†if a linear variation of f is assumed within the time increment.Equation (18.13) is in fact almost identical to a ®nite di€erence approximation tothe governing equation (18.2) at time t n ‡ t, and in this example little advantage isgained by introducing the ®nite element approximation. However, the averaging ofthe forcing term is important, as shown in Fig. 18.2, where a constant W (that is ˆ 1=2) is used and a ®nite di€erence approximation presents di culties.Figure 18.3 shows how di€erent weight functions can yield alternate values of theparameter . The solution of Eq. (18.13) yieldsa n ‡ 1 ˆ…C ‡ tK† ÿ1 ‰…C ÿ…1 ÿ †tK†a n ÿ t fŠ…18:16†13∆tf12 2n∆tn + 1n n + 1∆tW1 1f = 1.5f n +1/3 indeterminate2f = 3f n +1 + 0(a)(b)Fig. 18.2 `Averaging' of the forcing term in the ®nite-element-time approach.

Simple time-step algorithms for the ®rst-order equation 497N n N n +1n +1nζ = t/∆t∆tθ = 0W =(a)θ = 1 2(b)θ = 1(c)θ = 1 21(d)θ = 2 3(e)θ = 1 3(f)Fig. 18.3 Shape functions and weight functions for two-point recurrence formulae.and it is evident that in general at each step of the computation a full equationsystem needs to be solved though of course a single inversion is su cient forlinear problems in which the time increment t is held constant. Methods requiringsuch an inversion are called implicit. However, when ˆ 0 and the matrix C isapproximated by its lumped equivalent C L the solution is called explicit and isexceedingly cheap for each time interval. We shall show later that explicit algorithmsare conditionally stable (requiring the t to be less than some critical value t crit )whereas implicit methods may be made unconditionally stable for some choices ofthe parameters.18.2.2 Taylor series collocationA frequently used alternative to the algorithm presented above is obtained by approximatingseparately a n ‡ 1 and _a n ‡ 1 by truncated Taylor series. We can write, assuming

498 The time dimension ± discrete approximation in timethat a n and _a n are known:a n ‡ 1 a n ‡ t_a n ‡ t…_a n ‡ 1 ÿ _a n †…18:17†and use collocation to satisfy the governing equation at t n ‡ 1 [or alternatively usingthe weight function shown in Fig. 18.3(c)]C_a n ‡ 1 ‡ Ka n ‡ 1 ‡ f n ‡ 1 ˆ 0…18:18†In the above is a parameter, 0 4 4 1, such that the last term of Eq. (18.17)represents a suitable di€erence approximation to the truncated expansion.Substitution of Eq. (18.17) into Eq. (18.18) yields a recurrence relation for _a n ‡ 1 :_a n ‡ 1 ˆÿ…C ‡ tK† ÿ1 ‰K…a n ‡…1 ÿ †t_a n †‡f n ‡ 1 Š …18:19†where a n ‡ 1 is now computed by substitution of Eq. (18.19) into Eq. (18.17).We remark that:(a) the scheme is not self-startingy and requires the satisfaction of Eq. (18.2) at t ˆ 0;(b) the computation requires, with identi®cation of the parameters ˆ , an identicalequation-solving problem to that in the ®nite element scheme of Eq. (18.16) and,®nally, as we shall see later, stability considerations are identical.The procedure is introduced here as it has some advantages in non-linear computationswhich will be shown later.18.2.3 Other single-step proceduresAs an alternative to the weighted residual process other possibilities of deriving ®niteelement approximations exist, as discussed in Chapter 3. For instance, variationalprinciples in time could be established and used for the purpose. This was indeeddone in the early approaches to ®nite element approximation using Hamilton's orGurtin's variational principle. 28ÿ31 However, as expected, the ®nal algorithms turnout to be identical. A variant on the above procedures is the use of a least squareapproximation for minimization of the equation residual. 12;13 This is obtained byinsertion of the approximation (18.7) into Eq. (18.2). The reader can verify that therecurrence relation becomes1t CT C ‡ 1 ÿ2 KT C ‡ C T 1K ‡3 tKT K a n ‡ 1ÿ1 t CT C ‡ 1 ÿ2 KT C ÿ C T 1K ÿ6 tKT K‡ 1 … tt 2 CT f d ‡ 1 … tt KT f d00a n…18:20†requiring a more complex equation solution and always remaining `implicit'. For thisreason the algorithm is largely of purely theoretical interest, though as expected itsy By `self-starting' we mean an algorithm is directly applicable without solving any subsidiary equations.Other de®nitions are also in use.

Simple time-step algorithms for the ®rst-order equation 4991.00.80.6Legend ∆t 0.5 0.9Crank–NicolsonLeast squaresEulerGalerkinBackwarddifferenceθ = 1 2θ = 0θ = 2 3θ = 1aExact solution0.40.2001.0 2.0 3.0tFig. 18.4 Comparison of various time-stepping schemes on a ®rst-order initial value problem.accuracy is good, as shown in Fig. 18.4, in which a single degree of freedom equation(18.2) is used withK ! K ˆ 1 C ! C ˆ 1 f ! f ˆ 0with initial condition a 0 ˆ 1. Here, the various algorithms previously discussed arecompared. Now we see from this example that the ˆ 1=2algorithm performsalmost as well as the least squares one. It is popular for this reason and is knownas the Crank±Nicolson scheme after its originators. 3218.2.4 Consistency and approximation errorFor the convergence of any ®nite element approximation, it is necessary and su cientthat it be consistent and stable. We have discussed these two conditions in Chapter 10and introduced appropriate requirements for boundary value problems. In thetemporal approximation similar conditions apply though the stability problem ismore delicate.Clearly the function a itself and its derivatives occurring in the equation have to beapproximated with a truncation error of O…t †, where 5 1 is needed for consistencyto be satis®ed. For the ®rst-order equation (18.2) it is thus necessary to usean approximating polynomial of order p 5 1 which is capable of approximating _ato at least O…t†.The truncation error in the local approximation of a with such an approximation isO…t 2 † and all the algorithms we have presented here using the p ˆ 1 approximationof Eq. (18.7) will have at least that local accuracy, 33 as at a given time, t ˆ nt, the

500 The time dimension ± discrete approximation in timetotal error can be magni®ed n times and the ®nal accuracy at a given time for schemesdiscussed here is of order O…t† in general.We shall see later that the arguments used here lead to p 5 2for the second-orderequation (18.1) and that an increase of accuracy can generally be achieved by use ofhigher order approximating polynomials.It would of course be possible to apply such a polynomial increase to the approximatingfunction (18.7) by adding higher order degrees of freedom. For instance, wecould write in place of the original approximation a quadratic expansion:a ^a…† ˆa n ‡ …t a n ‡ 1 ÿ a n †‡ t1 ÿ ta n ‡ 1…18:21†where a is a hierarchic internal variable. Obviously now both a n ‡ 1 and a n ‡ 1 areunknowns and will have to be solved for simultaneously. This is accomplished byusing the weighting functionw ˆ W…†a n ‡ 1 ‡ W…†a n ‡ 1…18:22†where W…† and W…† are two independent weighting functions. This will obviouslyresult in an increased size of the problem.It is of interest to consider the ®rst of these obtained by using the weighting Walone in the manner of Eq. (18.11). The reader will easily verify that we now haveto add to Eq. (18.13) a term involving a n ‡ 1 which is1t …1 ÿ 2†C ‡… ÿ †K ~ a n ‡ 1…18:23†where~ ˆ 1„ t0 W 2 dt 2 „ t0 W dIt is clear that the choice of ˆ ~ ˆ 1=2eliminates the quadratic term and regainsthe previous scheme, thus showing that the values so obtained have a local truncationerror of O…t 3 †. This explains why the Crank±Nicolson scheme possesses higheraccuracy.In general the addition of higher order internal variables makes recurrence schemestoo expensive and we shall later show how an increase of accuracy can be moreeconomically achieved.In a later section of this chapter we shall refer to some currently popular schemes inwhich often sets of a's have to be solved for simultaneously. In such schemes adiscontinuity is assumed at the initial condition and additional parameters (~a) canbe introduced to keep the same linear conditions we assumed previously. In thiscase an additional equation appears as a weighted satisfaction of continuity in time.The procedure is therefore known as the discontinuous Galerkin process and wasintroduced initially by Lesaint and Raviart 34 to solve neutron transport problems.It has subsequently been applied to solve problems in ¯uid mechanics and heattransfer 22;35;36 and to problems in structural dynamics. 24ÿ26 As we have alreadystated, the introduction of additional variables is expensive, so somewhat limiteduse of the concept has so far been made. However, one interesting application is inerror estimation and adaptive time stepping. 37

Simple time-step algorithms for the ®rst-order equation 50118.2.5 StabilityIf we consider any of the recurrence algorithms so far derived, we note that for thehomogeneous form (i.e., with f ˆ 0) all can be written in the forma n ‡ 1 ˆ Aa n…18:24†where A is known as the ampli®cation matrix.The form of this matrix for the ®rst algorithm derived is, for instance, evident fromEq. (18.16) asA ˆ … C ‡ tK† ÿ1 … C ÿ…1 ÿ †tK† …18:25†Any errors present in the solution will of course be subject to ampli®cation by preciselythe same factor.A general solution of any recurrence scheme can be written asa n ‡ 1 ˆ a n…18:26†and by insertion into Eq. (18.24) we observe that is given by eigenvalues of thematrix as… A ÿ I†a n ˆ 0 …18:27†Clearly if any eigenvalue is such thatjj > 1…18:28†all initially small errors will increase without limit and the solution will be unstable. Inthe case of complex eigenvalues the above is modi®ed to the requirement that themodulus of satis®es Eq. (18.28).As the determination of system eigenvalues is a large undertaking it is useful toconsider only a scalar equation of the form (18.5) (representing, say, one-elementperformance). The bounding theorems of Irons and Treharne 27 will show why wedo so and the results will provide general stability bounds if maximums are used.Thus for the case of the algorithm discussed in Eq. (18.27) we have a scalar A, i.e.A ˆc ÿ…1 ÿ †tkc ‡ tkˆ1 ÿ…1 ÿ †!t1 ‡ !tˆ …18:29†where ! ˆ k=c and is evaluated from Eq. (18.27) simply as ˆ A to allow nontriviala n . (This is equivalent to making the determinant of A ÿ I zero in the moregeneral case.)In Fig. 18.5 we show how (or A) varies with !t for various values. We observeimmediately that:(a) for 5 1=2jj 4 1…18:30†and such algorithms are unconditionally stable;(b) for

502 The time dimension ± discrete approximation in timeAmplification A1.00.80.60.40.20–0.2–0.4–0.6–0.8θ = 1θ = 1θ = 2 3θ = 1 2θ = 0.878‘Exact’ e –ω∆t–1.0 0 1 2 3 4ω∆tFig. 18.5 The ampli®cation A for various versions of the algorithm.The critical value of t below which the scheme is stable with

Simple time-step algorithms for the ®rst-order equation 5031.0Exact0.8a0.60.4∆t = 1.50.20 0 1 2 3 4 5 6tOscillatory resulta1.00.80.60.40.2ExactUnstable result∆t = 2.50 0 1 2 3 4 5 6tθ = 0 Eulerθ = 1 2Crank–Nicolsonθ = 2 3Galerkinθ = 1 BackwarddifferenceFig. 18.6 Performance of some algorithms in the problem of Fig. 18.4 and larger time steps. Note oscillationand instability.Stability computations which were presented for the algorithm of Sec. 18.2.1 can ofcourse be repeated for the other algorithms which we have discussed.If identical procedures are used, for instance on the algorithm of Sec. 18.2.2, weshall ®nd that the stability conditions, based on the determinant of the ampli®cationmatrix …A ÿ I†, are identical with the previous one providing we set ˆ .Algorithms that give such identical determinants will be called similar in the followingpresentations.

504 The time dimension ± discrete approximation in timeIn general, it is possible for di€erent ampli®cation matrices A to have identicaldeterminants of …A ÿ I† and hence identical stability conditions, but di€er otherwise.If in addition the ampli®cation matrices are the same, the schemes are known asidentical. In the two cases described here such an identity can be shown to exist despitedi€erent derivations.18.2.6 Some further remarks. Initial conditions and examplesThe question of choosing an optimal value of is not always obvious fromtheoretical accuracy considerations. In particular with ˆ 1=2oscillations aresometimes present, 13 as we observe in Fig. 18.6 (t ˆ 2:5), and for this reasonsome prefer to use 38 ˆ 2=3, which is considerably `smoother' (and which incidentallycorresponds to a standard Galerkin approximation). In Table 18.1 we showthe results for a one-dimensional ®nite element problem where a bar at uniforminitial temperature is subject to zero temperatures applied suddenly at the ends.Here 10 linear elements are used in the space dimension with L ˆ 1. The oscillationerrors occurring with ˆ 1=2are much reduced for ˆ 2=3. The time step usedhere is much longer than that corresponding to the lowest eigenvalue period, butthe main cause of the oscillation is in the abrupt discontinuity of the temperaturechange.For similar reasons Liniger 39 derives which minimizes the error in the whole timedomain and gives ˆ 0:878 for the simple one-dimensional case. We observe inFig. 18.5 how well the ampli®cation factor ®ts the exact solution with these values.Again this value will smooth out many oscillations. However, most oscillations areintroduced by simply using a physically unrealistic initial condition.In part at least, the oscillations which for instance occur with ˆ 1=2and t ˆ 2:5(see Fig. 18.6) in the previous example are due to a sudden jump in the forcing termintroduced at the start of the computation. This jump is evident if we consider thissimple problem posed in the context of the whole time domain. We can take theproblem as implyingf …t† ˆÿ1 for t < 0Table 18.1 Percentage error for ®nite elements in time: ˆ 2=3 and ˆ 1=2(Crank±Nicolson) scheme;t ˆ 0:01t x ˆ 0:1 x ˆ 0:2 x ˆ 0:3 x ˆ 0:4 x ˆ 0:5 2/3 1/2 2/3 1/2 2/3 1/2 2/3 1/2 2/3 1/20.01 10.8 28.2 1.6 3.2 0.5 0.7 0.6 0.1 0.5 0.20.020.5 3.5 2.1 9.5 0.1 0.0 0.5 0.7 0.7 0.40.03 1.3 9.9 0.5 0.7 0.8 3.1 0.5 0.20.5 0.60.05 0.5 4.5 0.4 0.20.5 2.3 0.4 0.8 0.5 1.00.10 0.1 1.4 0.1 2.0 0.1 1.4 0.1 1.9 0.1 1.60.15 0.3 2.2 0.3 2.1 0.3 2.2 0.3 2.1 0.3 2.20.20 0.6 2.6 0.6 2.6 0.6 2.6 0.6 2.6 0.6 2.60.30 1.4 3.5 1.4 3.5 1.4 3.5 1.4 3.5 1.4 3.5

Simple time-step algorithms for the ®rst-order equation 505aa 0ABExacttfSmoothed (A)∆tStandard (B)∆tFig. 18.7 Importance of `smoothing' the force term in elimination of oscillations in the solution. t ˆ 2:5.giving the solution u ˆ 1 with a sudden change at t ˆ 0, resulting inf …t† ˆ0 for t 5 0As shown in Fig. 18.7 this represents a discontinuity of the loading function att ˆ 0.Although load discontinuities are permitted by the algorithm they lead to asudden discontinuity of _u and hence induce undesirable oscillations. If in place ofthis discontinuity we assume that f varies linearly in the ®rst time step t(ÿt=2 4 t 4 t=2) then smooth results are obtained with a much improvedphysical representation of the true solution, even for such a long time step ast ˆ 2:5, as shown in Fig. 18.7.Similar use of smoothing is illustrated in a multidegree of freedom system (therepresentation of heat conduction in a wall) which is solved using two-dimensional®nite elements 40 (Fig. 18.8).Here the problem corresponds to an instantaneous application of prescribed temperature(T ˆ 1) at the wall sides with zero initial conditions. Now again troublesome

506 The time dimension ± discrete approximation in timeDiscontinuousInterpolatedTemperature (φ)1.0Temperature (φ)1.000 10 20 30Time(a) Crank–Nicholson (θ = 0.5)00 10 20 30TimeTemperature (φ)1.0Temperature (φ)1.000 10Time 20 30(b) Galerkin (θ = 0.667)00 10 20 30TimeTemperature (φ)1.0Temperature (φ)1.000 10Time 20 30(c) Liniger (θ = 0.878)00 10 20 30TimeAt 2.810 quadratic elementsθ = 11x4∂φ∂x= 0Centre-lineExactNumber ∆t = 2Fig. 18.8 Transient heating of a bar; comparison of discontinuous and interpolated (smoothed) initialconditions for single-step schemes.oscillations are almost eliminated for ˆ 1=2and improved results are obtained forothervaluesof (2/3, 0.878) by assuming the step change to be replaced by acontinuous one. Such smoothing is always advisable and a continuous representationof the forcing term is important.We conclude this section by showing a typical example of temperature distributionin a practical example in which high-order elements are used (Fig. 18.9).

10501000700Simple time-step algorithms for the ®rst-order equation 50750045040035055026003504005004504550500145040034505005550600500750850800700650600550Elements used650700750800A(a) t = 0.5 sB75065070070065060060075070065060065070075080080085085090080075090095010009501000(b) t = 1.0 s800850800850850800850900950850950100010509001100(c) Steady-state solutionSpecific heat c = 0.11 cal/gm °CDensity ρ = 7.99 gm/cm 3Conductivity k = 0.05 cal/s cm °CGas temperature around blade = 1145 °CHeat transfer coefficient α varies from 0.390 to 0.056 on the outside surfaces ofthe blade (A–B)Holenumber12Cooling holetemperature545 °C587 °Cα around perimeterof each hole0.09800.0871Fig. 18.9 Temperature distribution in a cooled rotor blade, initially at zero temperature.

508 The time dimension ± discrete approximation in time18.3 General single-step algorithms for ®rst- and secondorderequations18.3.1 IntroductionWe shall introduce in this section two general single-step algorithms applicable toEq. (18.1):Ma ‡ C_a ‡ Ka ‡ f ˆ 0These algorithms will of course be applicable to the ®rst-order problem of Eq. (18.2)simply by putting M ˆ 0.An arbitrary degree polynomial p for approximating the unknown function a willbe used and we must note immediately that for the second-order equations p 5 2isrequired for consistency as second-order derivatives have to be approximated.The ®rst algorithm SSpj (single step with approximation of degree p for equationsof order j ˆ 1; 2) will be derived by use of the weighted residual process and we shall®nd that the algorithm of Sec. 18.2.1 is but a special case. The second algorithm GNpj(generalized Newmark 41 with degree p and order j) will follow the procedures using atruncated Taylor series approximation in a manner similar to that described inSec. 18.2.2.In what follows we shall assume that at the start of the interval, i.e., at t ˆ t n ,weknow the values of the unknown function a and its derivatives, that is a n , _a n , a n uptop ÿ a1 n and our objective will be to determine a n ‡ 1 , _a n ‡ 1 , a n ‡ 1 up top ÿ a1 n ‡ 1, where pis the order of the expansion used in the interval.This is indeed a rather strong presumption as for ®rst-order problems we havealready stated that only a single initial condition, a…0†, is given and for secondorderproblems two conditions, a…0† and _a…0†, are available (i.e., the initial displacementand velocity of the system). We can, however, argue that if the system startsfrom rest we could take a…0† top ÿ a1 …0† as equal to zero and, providing that suitablycontinuous forcing of the system occurs, the solution will remain smooth in thehigher derivatives. Alternatively, we can di€erentiate the di€erential equation toobtain the necessary starting values.18.3.2 The weighted residual ®nite element form SSpj 18;19The expansion of the unknown vector a will be taken as a polynomial of degree p.With the known values of a n , _a n , a n up top ÿ a1 n at the beginning of the time step t,we write, as in Sec. 18.2.1, ˆ t ÿ t n t ˆ t n ‡ 1 ÿ t n …18:34†and using a polynomial expansion of degree p,a ^a ˆ a n ‡ _a n ‡ 1 2! 2 1a n ‡‡…p ÿ 1†! p ÿ 1 p ÿ a1 n ‡ 1 p! p a p n…18:35†

General single-step algorithms for ®rst- and second-order equations 509α n2 ∆t 2 /2a• na n + a• n ∆t = aˆna nτ∆tt n t n + 1Fig. 18.10 A second-order time approximation.where the only unknown is the vector a p n,a p n a p ˆ dpdt p a…18:36†which represents some average value of the pth derivative occurring in the interval t.The approximation to a for the case of p ˆ 2is shown in Fig. 18.10.We recall that in order to obtain a consistent approximation to all the derivativesthat occur in the di€erential equations (18.1) and (18.2), p 5 2is necessary for the fulldynamic equation and p 5 1 is necessary for the ®rst-order equation. Indeed thelowest approximation, that is p ˆ 1, is the basis of the algorithm derived in theprevious section.The recurrence algorithm will now be obtained by inserting a, _a and a obtained bydi€erentiating Eq. (18.35) into Eq. (18.1) and satisfying the weighted residualequation with a single weighting function W…†. This gives… t W…† M a n ‡ :::1a n ‡‡0…p ÿ 2†! p ÿ 2 a p n1‡ C _a n ‡ a n ‡‡…p ÿ 1†! p ÿ 1 a p n‡ K a n ‡ _a n ‡‡ 1 p! p a p n ‡ f dt ˆ 0 …18:37†as the basic equation for determining a p n.

510 The time dimension ± discrete approximation in timeWithout specifying the weighting function used we can, as in Sec. 18.2.1, generalizeits e€ects by writing„ t0 W k d k ˆ „ tk ˆ 0; 1; ...; p0 W d„ …18:38†t f ˆ0 Wf d„ t0 W dwhere we note 0 is always unity. Equation (18.37) can now be written more compactlyasAa p n ‡ Ma n ‡ 1 ‡ C_a n ‡ 1 ‡ Ka n ‡ 1 ‡ f ˆ 0…18:39†whereA ˆ t p ÿ 2…p ÿ 2†! M ‡ t p ÿ 1…p ÿ 1†! C ‡ t pKp!Xp ÿ 1 q t qa n ‡ 1 ˆq ˆ 0q!a q nXp ÿ 1 q ÿ 1 t_a q ÿ 1n ‡ 1 ˆq ˆ 1…q ÿ 1†!Xp ÿ 1 q ÿ 2 ta q ÿ 2n ‡ 1 ˆq ˆ 2…q ÿ 2†!a q na q n…18:40†As a n ‡ 1 , _a n ‡ 1 and a n ‡ 1 can be computed directly from the initial values we can solveEq. (18.39) to obtaina p n ˆÿA ÿ1 Ma n ‡ 1 ‡ C_a n ‡ 1 ‡ Ka n ‡ 1 ‡ f…18:41†It is important to observe that a n ‡ 1 , _a n ‡ 1 and a n ‡ 1 here represent some meanpredicted values of a n ‡ 1 , _a n ‡ 1 and a n ‡ 1 in the interval and satisfy the governingEq. (18.1) in a weighted sense if a p n is chosen as zero.The procedure is now complete as knowledge of the vector a p n permits the evaluationof a n ‡ 1 to p ÿ a 1n ‡ 1 from the expansion originally used in Eq. (18.35) by putting ˆ t. This givesa n ‡ 1 ˆ a n ‡ t_a n ‡‡ t pa p n ˆ ^ap! n ‡ 1 ‡ t pa p np!_a n ‡ 1 ˆ _a n ‡ ta n ‡‡ t p ÿ 1..p ÿ 1an ‡ 1 ˆ p ÿ a1 n ‡ ta p n…p ÿ 1†! a p n ˆ _^a n ‡ 1 ‡ t p ÿ 1…p ÿ 1†! a p n…18:42†In the above ^a, _^a, etc., are again quantities that can be written down a priori(before solving for a p n). These represent predicted values at the end of the intervalwith a p n ˆ 0.

General single-step algorithms for ®rst- and second-order equations 511To summarize, the general algorithm necessitates the choice of values for 1 to pand requires(a) computation of a, _a and a using the de®nitions of Eqs (18.40);(b) computation of a p n by solution of Eq. (18.41);(c) computation of a n ‡ 1 top ÿ a1 n ‡ 1 by Eqs (18.42).After completion of stage (c) a new time step can be started. In ®rst-order problemsthe computation of a can obviously be omitted.If matrices C and M are diagonal the solution of Eq. (18.41) is trivial providing wechoose p ˆ 0…18:43†With this choice the algorithms are explicit but, as we shall ®nd later, only sometimesconditionally stable.When p 6ˆ 0, implicit algorithms of various kinds will be available and some ofthese will be found to be unconditionally stable. Indeed, it is such algorithms thatare of great practical use.Important special cases of the general algorithm are the SS11 and SS22 forms givenbelow.The SS11 algorithmIf we consider the ®rst-order equation (that is j ˆ 1) it is evident that only the value ofa n is necessarily speci®ed as the initial value for any computation. For this reason thechoice of a linear expansion in the time interval is natural (p ˆ 1) and the SS11algorithm is for that reason most widely used.Now the approximation of Eq. (18.35) is simplya ˆ a n ‡ a …a 1 n ˆ a ˆ _a† …18:44†and the approximation to the average satisfaction of Eq. (18.2) is simplyCa ‡ K… a n ‡ 1 ‡ ta†‡ f ˆ 0 …18:45†with a n ‡ 1 ˆ a n . Solution of Eq. (18.45) determines a asa ˆÿC … ‡ tK† ÿ1 ÿ f ‡ Kan…18:46†and ®nallya n ‡ 1 ˆ a n ‡ ta…18:47†The reader will verify that this process is identical to that developed in Eqs (18.7)±(18.13) and hence will not be further discussed except perhaps for noting the moreelegant computation form above.The SS22 algorithmWith Eq. (18.1) we considered a second-order system ( j ˆ 2) in which the necessaryinitial conditions require the speci®cation of two quantities, a n and _a n . The simplestand most natural choice here is to specify the minimum value of p, that is p ˆ 2,asthis does not require computation of additional derivatives at the start. Thisalgorithm, SS22, is thus basic for dynamic equations and we present it here in full.

512 The time dimension ± discrete approximation in timeFrom Eq. (18.35) the approximation is a quadratica ˆ a n ‡ _a n ‡ 1 2 2 a …a 2 n ˆ a ˆ a† …18:48†The approximate form of the `average' dynamic equation is nowÿ ÿMa ‡ C _a n ‡ 1 ‡ 1 ta ‡ K an ‡ 1 ‡ 1 2 2ta ‡ f ˆ 0 …18:49†with predicted `mean' valuesa n ‡ 1 ˆ a n ‡ 1 t_a n_a n ‡ 1 ˆ _a n…18:50†After evaluation of a from Eq. (18.49), the values of a n ‡ 1 are found by Eqs (18.42)which become simplya n ‡ 1 ˆ a n ‡ t_a n ‡ 1 2 t2 a…18:51†_a n ‡ 1 ˆ _a n ‡ taThis completes the algorithm which is of much practical value in the solution ofdynamics problems.In many respects it resembles the Newmark algorithm 41 which we shall discuss inthe next section and which is widely used in practice. Indeed, its stability propertiesturn out to be identical with the Newmark algorithm, i.e., 1 ˆ 2 ˆ 2…18:52† 1 5 2 5 1 2for unconditional stability. In the above and are conventionally used Newmarkparameters.For 2 ˆ 0 the algorithm is `explicit' (assuming both M and C to be diagonal) andcan be made conditionally stable if 1 5 1=2.The algorithm is clearly applicable to ®rst-order equations described as SS21 andwe shall ®nd that the stability conditions are identical. In this case, however, it isnecessary to identify an initial condition for _a 0 and_a 0 ˆÿC ÿ1 ÿKaa 0 ‡ f 0is one possibility.18.3.3 Truncated Taylor series collocation algorithm GNpjIt will be shown that again as in Sec. 18.2.2 a non-self-starting process is obtained,which in most cases, however, gives an algorithm similar to the SSpj one we havederived. The classical Newmark method 41 will be recognized as a particular casetogether with its derivation process in a form presented generally in existing texts. 42Because of this similarity we shall term the new algorithm generalized Newmark(GNpj).

General single-step algorithms for ®rst- and second-order equations 513In the derivation, we shall now consider the satisfaction of the governing equation(18.1) only at the end points of the interval t [collocation which results from theweighting function shown in Fig. 18.3(c)] and writeMa n ‡ 1 ‡ C_a n ‡ 1 ‡ Ka n ‡ 1 ‡ f n ‡ 1 ˆ 0…18:53†with appropriate approximations for the values of a n ‡ 1 , _a n ‡ 1 and a n ‡ 1 .If we consider a truncated Taylor series expansion similar to Eq. (18.17) for thefunction a and its derivatives, we can writea n ‡ 1 ˆ a n ‡ t_a n ‡‡ t pa p t p n ‡ p! p a p n ‡ 1 ÿ a p np!_a n ‡ 1 ˆ _a n ‡ ta n ‡‡ t p ÿ 1..p ÿ 1an ‡ 1 ˆ p ÿ a1 t p ÿ 1n ‡ t 1…p ÿ 1†!t p ÿ 1 …p ÿ 1†! ap n ‡ p ÿ 1 a p n ‡ 1 ÿ a p n…p ÿ 1†!a p n ‡ 1 ÿ a p n… 0 1†…18:54†In Eqs (18.44) we have e€ectively allowed for a polynomial of degree p (i.e., byincluding terms up to t p ) plus a Taylor series remainder term in each of the expansionsfor the function and its derivatives with a parameter j ; j ˆ 1; 2; ...; p, whichcan be chosen to give good approximation properties to the algorithm.Insertion of the ®rst three expressions of (18.54) into Eq. (18.53) gives a singleequation from which a p n ‡ 1 can be found. When this is determined, a n ‡ 1 to a p ÿ 1n ‡ 1can be evaluated using Eqs (18.54). Satisfying Eq. (18.53) is almost a `collocation'which could be obtained by inserting the expressions (18.54) into a weighted residualform (18.37) with W ˆ …t n ‡ 1 † (the Dirac delta function). However, the expansiondoes not correspond to a unique function a.In detail we can write the ®rst three expansions of Eqs (18.54) ast pa n ‡ 1 ˆ a n ‡ 1 ‡ p a p n ‡ 1p!_a n ‡ 1 ˆ _a n ‡ 1 ‡ p ÿ 1t p ÿ 1…p ÿ 1†! ap n ‡ 1a n ‡ 1 ˆ a n ‡ 1 ‡ p ÿ 2t p ÿ 2…p ÿ 2†! ap n ‡ 1…18:55†wherea n ‡ 1 ˆ a n ‡ t_a n ‡‡…1 ÿ p † t pa p n ‡p!_a n ‡ 1 ˆ _a n ‡ ta n ‡‡…1 ÿ p ÿ 1 † t p ÿ 1…p ÿ 1†! ap n ‡a n ‡ 1 ˆ a n ‡ t ::: a n ‡‡…1 ÿ p ÿ 2 † t p ÿ 2…p ÿ 2†! ap n ‡…18:56†

514 The time dimension ± discrete approximation in timeInserting the above into Eq. (18.53) givesa p n ‡ 1 ˆÿA ÿ1 Ma n ‡ 1 ‡ C_a n ‡ 1 ‡ Ka n ‡ 1 ‡ f n ‡ 1…18:57†whereA ˆ p ÿ 2t p ÿ 2…p ÿ 2†!M ‡ p ÿ 1t p ÿ 1…p ÿ 1†!Solving the above equation for a p n ‡ 1, we haveC ‡ pt pKp!a p n ‡ 1 ˆÿA‰M a n ‡ 1 ‡ C _ a n ‡ 1 ‡ Ka n ‡ 1 ‡ f n ‡ 1 Š…18:58†We note immediately that the above expression is formally identical to that of theSSpj algorithm, Eq. (18.41), if we make the substitutions p ˆ p p ÿ 1 ˆ p ÿ 1 p ÿ 2 ˆ p ÿ 2 …18:59†However, a n ‡ 1 , _a n ‡ 1 , etc., in the generalized Newmark, GNpj, are not identical toa n ‡ 1 , _a n ‡ 1 , etc., in the SSpj algorithms. In the SSpj algorithm these representpredicted mean values in the interval t while in the GNpj algorithms they representpredicted values at t n ‡ 1 .The computation procedure for the GN algorithms is very similar to that for the SSalgorithms, starting now with known values of a n to a p n. As before we have the giveninitial conditions and we can usually arrange to use the di€erential equation and itsderivatives to generate higher derivatives for a at t ˆ 0. However, the GN algorithmrequires more storage because of the necessity of retaining and using a p 0 in thecomputation of the next time step.An important member of this family is the GN22 algorithm. However, beforepresenting this in detail we consider another form of the truncated Taylor seriesexpansion which has found considerable use recently, especially in non-linearapplications.An alternative is to use a weighted residual approach with a collocation weightfunction placed at t ˆ t n ‡ on the governing equation. This gives a generalizationto Eq. (18.13) of1t M … _a n ‡ 1 ÿ _a n †‡C_a n ‡ ‡ Ka n ‡ ‡ f n ‡ ˆ 0 …18:60†where an interpolated value for a n ‡ and _a n ‡ may be written asa n ‡ ˆ a n ‡ … a n ‡ 1 ÿ a n †_a n ‡ ˆ _a n ‡ …_a n ‡ 1 ÿ _a n †…18:61†This form may be combined with a weighted residual approach as described inreference 16. A collocation algorithm for this form is generalized in references 43±46. An advantage of this latter form is an option which permits the generation ofenergy and momentum conserving properties in the discrete dynamic problem.These generalizations are similar to the GNpj algorithm described in this sectionalthough the optimal parameters are usually di€erent.

General single-step algorithms for ®rst- and second-order equations 515The Newmark algorithm (GN22)We have already mentioned the classical Newmark algorithm as it is one of the mostpopular for dynamic analysis. It is indeed a special case of the general algorithm of thepreceding section in which a quadratic (p ˆ 2) expansion is used, this being theminimum required for second-order problems. We describe here the details in viewof its widespread use.The expansion of Eq. (18.54) for p ˆ 2givesa n ‡ 1 ˆ a n ‡ t_a n ‡ 1 2 …1 ÿ 2†t 2 a n ‡ 1 2 2t 2 a n ‡ 1 ˆ _a n ‡ 1 ‡ 1 2 2t 2 a n ‡ 1_a n ‡ 1 ˆ _a n ‡…1 ÿ 1 †ta n ‡ 1 ta n ‡ 1 ˆ a n ‡ 1 ‡ 1 ta n ‡ 1…18:62†and this together with the dynamic equation (18.53),Ma n ‡ 1 ‡ C_a n ‡ 1 ‡ Ka n ‡ 1 ‡ f n ‡ 1 ˆ 0…18:63†allows the three unknowns a n ‡ 1 , _a n ‡ 1 and a n ‡ 1 to be determined.We now proceed as we have already indicated and solve ®rst for a n ‡ 1 by substituting(18.62) into (18.63). This yields as the ®rst stepwherea n ‡ 1 ˆÿA ÿ1 ff n ‡ 1 ‡ C _ a n ‡ 1 ‡ Ka n ‡ 1 gA ˆ M ‡ 1 tC ‡ 1 2 2t 2 K…18:64†…18:65†After this step the values of a n ‡ 1 and _a n ‡ 1 can be found using Eqs (18.62).As in the general case, 2 ˆ 0 produces an explicit algorithm whose solution is verysimple if M and C are assumed diagonal.It is of interest to remark that the accuracy can be slightly improved and yet theadvantages of the explicit form preserved for SS/GN algorithms by a simple iterativeprocess within each time increment. In this, for the GN algorithm, we predict a i n ‡ 1,_a i n ‡ 1 and a i n ‡ 1 using expressions (18.55) withsetting for i ˆ 1…a p n ‡ 1† i ÿ 1…a p n ‡ 1† 0 ˆ 0This is followed by rewriting the governing equation (18.57) as" #M a i n ÿ ‡ 11 ‡ 2t p ÿ 2…p ÿ 2†! ap in ‡ 1 ‡ C_a i n ÿ ‡ 11 ‡ Ka i n ÿ ‡ 11 ‡ f n ‡ 1 ˆ 0…18:66†and solving for a p in ‡ 1 .This predictor±corrector iteration has been successfully used for variousalgorithms, though of course the stability conditions remain unaltered from thoseof a simple explicit scheme. 47For implicit schemes we note that in the general case, Eqs (18.62) have scalarcoe cients while Eq. (18.63) has matrix coe cients. Thus, for the implicit casesome users prefer a slightly more complicated procedure than indicated above inwhich the ®rst unknown determined is a n ‡ 1 . This may be achieved by expressing

516 The time dimension ± discrete approximation in timeEqs (18.62) in terms of the a n ‡ 1 to obtainwherea n ‡ 1 ˆ ^a n ‡ 1 ‡ 2 2 t 2 a n ‡ 1_a n ‡ 1 ˆ _^a n ‡ 1 ‡ 2 1 2 t a n ‡ 1^a n ‡ 1 ˆÿ 2 2 t 2 a n ÿ 2 2 t _a n ÿ 1 ÿ 2_^a n ‡ 1 ˆÿ 2 1 2 t a n ‡ 1 ÿ 2 1 2These are now substituted into Eq. (18.63) to give the resultwhere now 2a na n ‡ 1 ˆÿA ÿ1 …f n ‡ 1 ‡ C _^a n ‡ 1 ‡ M ^a n ‡ 1 †…18:67†_a n ‡ 1 ÿ …18:68†1ta n2…18:69†A ˆ 2 2 t 2 M ‡ 2 1 2 t C ‡ Kwhich again on using Eqs (18.67) and (18.68) gives _a and a. The inversion is hereidentical to within a scalar multiplier but as mentioned before precludes use of theexplicit form where 2 is zero.18.3.4 Stability of general algorithmsConsistency of the general algorithms of SS and GN type is self-evident and assuredby their formulation.In a similar manner to that used in Sec. 18.2.5 we can conclude from this that thelocal truncation error is O…t p ‡ 1 † as the expansion contains all terms up to p . However,the total truncation error after n steps is only O…t p † for ®rst-order equationsystem and O…t p ÿ 1 † for the second-order one. Details of accuracy discussionsand reasons for this can be found in reference 6.The question of stability is paramount and in this section we shall discuss it in detailfor the SS type of algorithms. The establishment of similar conditions for the GNalgorithms follows precisely the same pattern and is left as an exercise to thereader. It is, however, important to remark here that it can be shown that(a) the SS and GN algorithms are generally similar in performance;(b) their stability conditions are identical when p p .The proof of the last statement requires some elaborate algebra and is given inreference 6.The determination of stability requirements follows precisely the pattern outlinedin Sec. 18.2.5. However for practical reasons we shall(a) avoid writing explicitly the ampli®cation matrix A;

General single-step algorithms for ®rst- and second-order equations 517(b) immediately consider the scalar equation system implying modal decompositionand no forcing, i.e.,ma ‡ c _a ‡ ka ˆ 0…18:70†Equations (18.39), (18.40) and (18.42) written in scalar terms de®ne the recurrencealgorithms. For the homogeneous case the general solution can be written down asa n ‡ 1 ˆ a n_a n ‡ 1 ˆ _a n..p ÿ 1an ‡ 1 ˆ p ÿ a1 n…18:71†and substitution of the above into the equations governing the recurrence can bewritten quite generally asSX n ˆ 0…18:72†where8a n9>=X n ˆ n.>:t p a p >;n…18:73†The matrix S is given below in a compact form which can be veri®ed by the reader:23b 0 b 1 b 2 b p ÿ 1 b p1 1 11 ÿ 1 2! …p ÿ 1†! p!1 1S ˆ0 1ÿ 1 …p ÿ 2†! …p ÿ 1†!…18:74†. . . . . .. . . . . .164 0 0 0 1752!0 0 0 1 ÿ 1whereb 0 ˆ 0 t 2 k; 0 ˆ 1b 1 ˆ 0 tc ‡ 1 t 2 kb q ˆ q ÿ 2…q ÿ 2†! m ‡ q ÿ 1t…q ÿ 1†! c ‡ qt 2k; q ˆ 2; 3; ...; pq!For non-trivial solutions for the vector X n to exist it is necessary for the determinantof S to be zero:det S ˆ 0…18:75†

518 The time dimension ± discrete approximation in timeµ I z IStableµ R z RStableµ plane(µ = µ R + iµ l )z plane(z = z R + iz l )Fig. 18.11 The ˆ…1 ‡ z†=…1 ÿ z† transformation.This provides a characteristic polynomial of order p for which yields the eigenvaluesof the ampli®cation matrix. For stability it is su cient and necessary that themoduli of all eigenvalues [see Eq. (18.28)] satisfyjj 4 1…18:76†We remark that in the case of repeated roots the equality sign does not apply. Thereader will have noticed that the direct derivation of the determinant of S is muchsimpler than writing down matrix A and ®nding the eigenvalues. The results are, ofcourse, identical.The calculation of stability limits, even with the scalar (modal) equation system, isnon-trivial. For this reason in what follows we shall only do it for p ˆ 2and p ˆ 3.However, two general procedures will be introduced here.The ®rst of these is the so-called z transformation. In this we use a change ofvariables in the polynomial putting ˆ 1 ‡ z…18:77†1 ÿ zwhere z as well as are in general complex numbers. It is easy to show that therequirement of Eq. (18.76) is identical to that demanding the real part of z to benegative (see Fig. 18.11).The second procedure introduced is the well-known Routh±Hurwitz condition48ÿ50 which states that for a polynomial with c 0 > 0c 0 z n ‡ c 1 z n ÿ 1 ‡‡c n ÿ 1 z ‡ c n ˆ 0…18:78†the real part of all roots will be negative if, for c 1 > 0,2 3 c 1 c 3 c 5c 1 c 36 7det > 0 det4c 0 c 2 c 4 5 > 0 …18:79†c 0 c 20 c 1 c 3

General single-step algorithms for ®rst- and second-order equations 519and generally23c 1 c 3 c 5 c 7 c 0 c 2 c 4 c 6 0 cdet 1 c 3 c 5 > 0 …18:80†0 0 c 2 c 4 6 ..4 ..7. 50 0 0 0 c nWith these tools in hand we can discuss in detail the stability of speci®c algorithms.18.3.5 Stability of SS22/SS21 algorithmsThe recurrence relations for the algorithm given in Eqs (18.49) and (18.51) can bewritten after insertinga n ‡ 1 ˆ a n _a n ‡ 1 ˆ _a n f ˆ 0 …18:81†asÿm ‡ c …_a n ‡ 1 t†‡ka n ‡ t _a n ‡ 1 2 2t 2 ˆ 0ÿa n ‡ a n ‡ ta n ‡ 1 2 2t 2 ˆ 0ÿ _a n ‡ _a n ‡ ta n ‡ 1 t ˆ 0…18:82†…18:83†Changing the variable according to Eq. (18.77) results in the characteristic polynomialwithc 0 z 2 ‡ c 1 z ‡ c 2 ˆ 0c 0 ˆ 4m ‡…4 1 ÿ 2†tc ‡ 2… 2 ÿ 1 †t 2 kc 1 ˆ 2tc ‡…2 1 ÿ 1†t 2 kc 2 ˆ t 2 k…18:84†The Routh±Hurwitz requirements for stability is simply that c 1 0c 0 > 0 c 1 5 0 det > 0c 0 c 2or simplyc 0 > 0 c 1 5 0 c 2 > 0 …18:85†These inequalities give for unconditional stability the condition that 2 5 1 5 1 2…18:86†This condition is also generally valid when m ˆ 0, i.e., for the SS21 algorithm (the®rst-order equation) though now 2 ˆ 1 must be excluded.

520 The time dimension ± discrete approximation in timeIt is possible to satisfy the inequalities (18.85) only at some values of t yieldingconditional stability. For the explicit process 2 ˆ 0 with SS22/SS21 algorithms theinequalities (18.85) demand that2m ‡…2 1 ÿ 1†tc ÿ 1 t 2 k 5 02c ‡…2 1 ÿ 1†tk 5 0…18:87†The second one is satis®ed whenever 1 5 1 2…18:88†and for 1 ˆ 1=2the ®rst supplies the requirement thatt 2 4 4m k…18:89†The last condition does not permit an explicit scheme for SS21, i.e., when m ˆ 0.Here, however, if we take 1 > 1=2we have from the ®rst equation of Eq. (18.87)t < 2 1 ÿ 1 1ck…18:90†It is of interest for problems of structural dynamics to consider the nature of thebounds in an elastic situation. Here we can use the same process as that describedin Sec. 18.2.5 for ®rst-order problems of heat conduction. Looking at a single elementwith a single degree of freedom and consistent mass yields in place of condition(18.89)t 4 p2 3h C ˆ t critwhere h is the element size andC ˆsEis the speed p of elastic wave propagation. For lumped mass matrices the factorbecomes 2 .Once again the ratio of the smallest element size over wave speed governs thestability but it is interesting to note that in problems of dynamics the critical timestep is proportional to h while, as shown in Eq. (18.33), for ®rst-order problems itis proportional to h 2 . Clearly for decreasing mesh size explicit schemes in dynamicsare more e cient than in thermal analysis and are exceedingly popular in certainclasses of problems.18.3.6 Stability of various higher order schemes and equivalencewith some known alternativesIdentical stability considerations as those described in previous sections can beapplied to SS32/SS31 and higher order approximations. We omit here the algebraand simply quote some results. 6

General single-step algorithms for ®rst- and second-order equations 521Table 18.2 SS21 equivalentsAlgorithmsTheta valuesZlamal 38 1 ˆ 56 , 2 ˆ 2Gear 52 1 ˆ 32 , 2 ˆ 2Liniger 53 1 ˆ 1:0848, 2 ˆ 1Liniger 53 1 ˆ 1:2184, 2 ˆ 1:292SS32/31. Here for zero damping (c ˆ 0) in SS32we require for unconditional stabilitythat 1 > 1 2 2 5 1 ‡ 1 6 2 5 1 4 3 5 3 23 1 2 ÿ 3 2 1 ‡ 1 5 3…18:91†For ®rst-order problems (m ˆ 0), i.e., SS31, the ®rst requirements are as indynamics but the last one becomes3 1 2 1 ÿ 3 2 ‡ 1 5 3 ÿ ‰6 1… 1 ÿ 1†‡1Š 2…18:92†9…2 1 ÿ 1†With 3 ˆ 0, i.e., an explicit scheme when c ˆ 0,t 2 4 12…2 1 ÿ 1†6 2 ÿ 1mk…18:93†and when m ˆ 0,t 4 2 ÿ 1 c…18:94†6 2 ÿ 1 kSS42/41. For this (and indeed higher orders) unconditional stability in dynamicsproblems m 6ˆ 0 does not exist. This is a consequence of a theorem by Dahlquist. 51The SS41 scheme can have unconditional stability but the general expressions forthis are cumbersome. We quote one example that is unconditionally stable: 1 ˆ 52 2 ˆ 35 6 3 ˆ 25 2 4 ˆ 24This set of values corresponds to a backward di€erence four-step algorithm ofGear. 52It is of general interest to remark that certain members of the SS (or GN) families ofalgorithms are similar in performance and identical in the stability (and hencerecurrence) properties to others published in the large literature on the subject.Each algorithm claims particular advantages and properties. In Tables 18.2±18.4we show some members of this family. 38ÿ57 Clearly many more algorithms that areTable 18.3 SS31 equivalentsAlgorithmsTheta valuesGear 52 1 ˆ 2, 2 ˆ 11 3 , 3 ˆ 6Liniger 53 1 ˆ 1:84, 2 ˆ 3:07, 3 ˆ 4:5Liniger 53 1 ˆ 0:8, 2 ˆ 1:03, 3 ˆ 1:29

522 The time dimension ± discrete approximation in timeTable 18.4 SS32equivalentsAlgorithmsTheta valuesHoubolt 54 1 ˆ 2, 2 ˆ 11 3 , 3 ˆ 6Wilson 55 1 ˆ , 2 ˆ 2 , 3 ˆ 3( ˆ 1:4 common)Bossak±Newmark 56 1 ˆ 1 ÿ B(ma ‡ ka ˆ 0, 2 ˆ 23 ÿ B ‡ 2 B B ˆ 12 ÿ B) 3 ˆ 6 BBossak±Newmark 56 1 ˆ 1 ÿ B(ma ‡ c _a ‡ ka ˆ 0, 2 ˆ 1 ÿ 2 B B ˆ 12 ÿ B, 3 ˆ 1 ÿ 3 B B ˆ 16 ÿ 1 2 B)Hilber±Hughes±Taylor 57 1 ˆ 1(ma ‡ ka ˆ 0, 2 ˆ 23 ‡ 2 H ÿ 2 2 H H ˆ 12 ÿ H) 3 ˆ 6 H …1 ‡ H †applicable are present in the general formulae and a study of their optimal parametersis yet to be performed.We remark here that identity of stability and recurrence always occurs with multistepalgorithms, which we shall discuss in the next section.Multistep methods18.4Multistep recurrence algorithms18.4.1 IntroductionIn the previous sections we have been concerned with recurrence algorithms validwithin a single time step and relating the values of a n ‡ 1 , _a n ‡ 1 , a n ‡ 1 to a n , _a n , a n ,etc. It is possible to derive, using very similar procedures to those previously introduced,multistep algorithms in which we relate a n ‡ 1 to the values a n , a n ÿ 1 , a n ÿ 2 ,etc., without explicitly introducing the derivatives. Much classical work on stabilityand accuracy has been introduced on such multistep algorithms and hence theydeserve mention here.We shall show in this section that a series of such algorithms may be simply derivedusing the weighted residual process and that, for constant time increments t, this setpossesses identical stability and accuracy properties to the SSpj procedures.18.4.2 The approximation procedure for a general multistepalgorithmAs in Sec. 18.3.2we shall approximate the function a of the second-order equationMa ‡ C_a ‡ Ka ‡ f ˆ 0…18:95†

Multistep recurrence algorithms 523by a polynomial expansion of the order p, now containing a single unknown a n ‡ 1 .This polynomial assumes knowledge of the value of a n , a n ÿ 1 , ..., a n ÿ p ‡ 1 at appropriatetimes t n , t n ÿ 1 , ..., t n ÿ p ‡ 1 (Fig. 18.12).We can write this polynomial asa…t† ˆX1j ˆ 1 ÿ pN j …t†a n ‡ j…18:96†where Lagrange interpolation in time is given by (see Chapter 8)N j …t† ˆY1k ˆ 1 ÿ pk 6ˆ jt ÿ t n ‡ kt n ‡ j ÿ t n ‡ kThe derivatives of the shape functions may be constructed by writingN j ˆ nj…t†n j …t n ‡ j †…18:97†…18:98†and di€erentiating the numerator. Accordingly writingthe derivative becomesdn jdt ˆn j …t† ˆX1m ˆ p ÿ 1m 6ˆ jY1k ˆ 1 ÿ pk 6ˆ jY 1k ˆ p ÿ 1k 6ˆ jk 6ˆ m… t ÿ t n ‡ k † …18:99†… t ÿ t n ‡ k † p 5 2 …18:100†NowdN jdt ˆ 1 dn jn j …t n ‡ j † dt ˆ N_j…18:101†These expressions can be substituted into Eq. (18.84) giving_a ˆa ˆX1j ˆ 1 ÿ pX1j ˆ 1 ÿ p_ N j …t†a n ‡ jN j …t†a n ‡ j…18:102†Insertion of a, _a and a into the weighted residual equation form of Eq. (18.83) yields… tn ‡ 1t nW…t† X1j ˆ 1 ÿ pÿ N j M ‡ N_j C ‡ N j K an ‡ j ‡ N j f n ‡ j dt ˆ 0 …18:103†with the forcing functions interpolated similarly from its nodal values.

a n – p +1a n – 1a na n +1t∆t∆tn – p +1 n – 2 n – 1 n n+1N n – 1 N n N n +1Lagrangeinterpolationpolynomials of order p(shape function)1ApproximatedomainFig. 18.12 Multistep polynomial approximation.

Two point interpolation: p ˆ 1Evaluating Eq. (18.85) we obtainN 1 ˆ t ÿ t n ‡ 1t n ‡ 1 ÿ t nˆ 1t …t ÿ t n ‡ 1† ˆ tN 0 ˆ t ÿ t n ‡ 1t n ÿ t n ‡ 1ˆ 1t …t n ‡ 1 ÿ t† ˆ1 ÿ twhere t ˆ t n ‡ 1 ÿ t n and ˆ t ÿ t n . Here the derivative is computed directly asdN 1dtˆÿ dN 0dtˆ 1tSecond derivatives are obviously zero, hence this form may only be used for ®rstorderequations.Three point interpolation: p ˆ 2Evaluating Eq. (18.85)…t ÿ tN 1 ˆn ÿ 1 †…t ÿ t n †…t n ‡ 1 ÿ t n ÿ 1 †…t n ‡ 1 ÿ t n †N 0 ˆ …t ÿ t n ÿ 1†…t ÿ t n ‡ 1 †…t n ÿ t n ÿ 1 †…t n ÿ t n ‡ 1 †N ÿ1 ˆ…t ÿ t n †…t ÿ t n ‡ 1 †…t n ÿ 1 ÿ t n †…t n ÿ 1 ÿ t n ‡ 1 †The derivatives follow immediately from Eqs (18.87) and (18.88) asdN 1dtdN 0dtdN ÿ1dtˆ …t ÿ t n†‡…t ÿ t n ÿ 1 †…t n ‡ 1 ÿ t n ÿ 1 †…t n ‡ 1 ÿ t n †ˆ …t ÿ t n ‡ 1†‡…t ÿ t n ÿ 1 †…t n ÿ t n ÿ 1 †…t n ÿ t n ‡ 1 †ˆ …t ÿ t n ‡ 1†‡…t ÿ t n †…t n ÿ 1 ÿ t n †…t n ÿ 1 ÿ t n ‡ 1 †This is the lowest order which can be used for second-order equations and has secondderivativesd 2 N 1dt 2d 2 N 0dt 2ˆˆ2…t n ‡ 1 ÿ t n ÿ 1 †…t n ‡ 1 ÿ t n †2…t n ÿ t n ÿ 1 †…t n ÿ t n ‡ 1 †d 2 N ÿ12dt 2 ˆ frac 2…t n ÿ 1 ÿ t n †…t n ÿ 1 ÿ t n ‡ 1 †Multistep recurrence algorithms 525

526 The time dimension ± discrete approximation in time18.4.3 Constant t formFor the remainder of our discussion here we shall assume a constant time incrementfor all steps is given by t. To develop the constant increment form we introduce thenatural coordinate de®ned as ˆ t ÿ t n; 1 ÿ p 4 4 1t…18:104†j ˆ 1 ÿ p; 2 ÿ p; ...; 0; 1and now assume the shape functions N j in Eq. (18.84) are functions of the naturalcoordinates and given byN j …† ˆnj…†…18:105†n j …j†wheren j …† ˆn j … j† ˆDerivatives with respect to are given byY1k ˆ 1 ÿ pk 6ˆ jY1k ˆ 1 ÿ pk 6ˆ j… ÿ k†… j ÿ k†n 0 j…† ˆX1k ˆ 1 ÿ pk 6ˆ jY 1l ˆ 1 ÿ pl 6ˆ kl 6ˆ j… ÿ l†…18:106†n 00j …† ˆX1k ˆ 1 ÿ pk 6ˆ jX 1l ˆ 1 ÿ pl 6ˆ kl 6ˆ jY 1m ˆ 1 ÿ pm 6ˆ km 6ˆ lm 6ˆ j… ÿ m†…18:107†Using the chain rule these derivatives give the time derivatives_a ˆ 1tX 1j ˆ 1 ÿ pa ˆ 1t 2 X 1j ˆ 1 ÿ pn 0 jn j … j† a n ‡ kn 00jn j … j† a n ‡ k…18:108†…18:109†The weighted residual equation may now be written as… 1W…† X1 ÿNj 00 M ‡ tNjC 0 ‡ t 2 N j K an ‡ j ‡ t 2 N j f n ‡ j d ˆ 0 …18:111†0j ˆ 1 ÿ p

Using the parameters q ˆ„ 10 W q d„ 10 W d ; q ˆ 1; 2; ...; p; 0 ˆ 1 …18:112†we now have an algorithm that enables us to compute a n ‡ 1 from known valuesa n ÿ p ‡ 1 , a n ÿ p ‡ 2 , ..., a n .[Note: so long as the limits of integration are the same inEqs (18.111) and (18.112) it makes no di€erence what we choose them to be.]Four-point interpolation: p ˆ 3For p ˆ 3, Eq. (18.93) givesÿ N ÿ2 …† ˆÿ 1 63 ÿ ÿN ÿ1 …† ˆ12 3 ‡ 2 …18:113†ÿ 2ÿN 0 …† ˆÿ 1 2 3 ‡ 2 2 ÿ ÿ 2ÿN 1 …† ˆ16 3 ‡ 3 2 …18:114†‡ 2Similarly, from Eqs (18.94) and (18.95),Nÿ2…† 0 ÿ ˆÿ 1 6 32 ÿ 1Nÿ1…† 0 ÿ …18:115†ˆ12 32 ‡ 2 ÿ 2N0…† 0 ÿˆÿ 1 2 32 ‡ 4 ÿ 1N1…† 0 ÿ …18:116†ˆ16 32 ‡ 6 ‡ 2andNÿ2…† 00 ˆÿ…18:117†Nÿ1…† 00 ˆ3 ‡ 1N0 00 …† ˆÿ3 ÿ 2…18:118†N1 00 …† ˆ ‡ 1We now have a three-step algorithm for the solution of Eq. (18.83) of the form(taking f ˆ 0)whereX 1j ˆÿ2 j ‡ 2 M ‡ j ‡ 2 tC ‡ j ‡ 2 t 2 K an ‡ j ˆ 0 …18:119† j ‡ 2 ˆ… 10W…†N 00j dMultistep recurrence algorithms 527 j ‡ 2 ˆ… 10W…†N 0 j d…18:120† j ‡ 2 ˆ… 10W…†N j d

528 The time dimension ± discrete approximation in timeAfter integration the above gives 0 ˆÿ 1 0 ˆÿ 1 6 …3 2 ÿ 1† 0 ˆÿ 1 6 … 3 ÿ 1 † 1 ˆ 3 1 ‡ 1 1 ˆ 12 …3 2 ‡ 2 1 ÿ 2† 1 ˆ 12 … 3 ‡ 2 ÿ 2 1 † 2 ˆÿ3 1 ÿ 2 2 ˆÿ 1 2 …3 2 ‡ 2 1 ÿ 1† 2 ˆÿ 1 2 … …18:121†3 ‡ 2 2 ÿ 1 ÿ 2† 3 ˆ 1 ‡ 1 3 ˆ 16 …3 2 ‡ 6 1 ‡ 2† 3 ˆ 16 … 3 ‡ 3 2 ‡ 2 1 †An algorithm of the form given in Eq. (18.119) is called a linear three-step method.The general p-step form isX 11 ÿ p j ‡ p ÿ 1 M ‡ j ‡ p ÿ 1 tC ‡ j ‡ p ÿ 1 t 2 K an ‡ j ˆ 0 …18:122†This is the form generally given in mathematics texts; it is an extension of the formgiven by Lambert 2 for C ˆ 0. The weighted residual approach described here derivesthe 's, 's and 's in terms of the parameters i ; i ˆ 0; 1; ...; p and thus ensuresconsistency.From Eq. (18.122) the unknown a n ‡ 1 is obtained in the forma n ‡ 1 ˆ 3 M ‡ 3 tC ‡ 3 t 2 ÿ1FK…18:123†where F is expressed in terms of known values. For example, for p ˆ 3 the matrix tobe inverted is… 1 ‡ 1†M ‡… 1 2 2 ‡ 1 ‡ 1 3 †tC ‡…1 6 3 ‡ 1 2 2 ‡ 1 3 1†t 2 KComparing this with the matrix to be inverted in the SSpj algorithm given inEq. (18.41) suggests a correspondence between SSpj and the p-step algorithmabove which we explore further in the next section.18.4.4 The relationship between SSpj and the weighted residualp-step algorithmFor simplicity we now consider the p-step algorithm described in the previous sectionapplied to the homogeneous scalar equationma ‡ c _a ‡ ka ˆ 0…18:124†As in previous stability considerations we can obtain the general solution of therecurrence relationX 1j ˆ 1 ÿ p j ‡ p ÿ 1 m ‡ j ‡ p ÿ 1 tc ‡ j ‡ p ÿ 1 t 2 k an ‡ j ˆ 0 …18:125†by putting a n ‡ j ˆ p ÿ 1 ‡ j where the values of are the roots k of the stabilitypolynomial of the p-step algorithm:X 1j ˆ 1 ÿ p j ‡ p ÿ 1 m ‡ j ‡ p ÿ 1 tc ‡ j ‡ p ÿ 1 t 2 k p ÿ 1 ‡ j ˆ 0 …18:126†

Table 18.5 Identities between SSp2and p-step algorithmsSS22/21Multistep recurrence algorithms 529SS32/31SS42/41 1 ˆ 1 ‡ 1 2 2 ˆ 2 ‡ 1 1 ˆ 1 ‡ 1 2 ˆ 2 ‡ 2 1 ‡ 2 3 3 ˆ 3 ‡ 3 2 ‡ 2 1 1 ˆ 1 ‡ 3 2 2 ˆ 2 ‡ 3 1 ‡ 11 6 3 ˆ 3 ‡ 9 2 2 ‡ 11 2 1 ‡ 3 2 4 ˆ 4 ‡ 6 3 ‡ 11 2 ‡ 6 1This stability polynomial can be quite generally identi®ed with the one resultingfrom the determinant of Eq. (18.74) as shown in reference 6, by using a suitable setof relations linking i and i . Thus, for instance, in the case of p ˆ 3 discussed weshall have the identity of stability and indeed of the algorithm when 1 ˆ 1 ‡ 1 2 ˆ 2 ‡ 2 1 ‡ 2 3…18:127† 3 ˆ 3 ‡ 3 2 ‡ 2 1Table 18.5 summarizes the identities of p ˆ 2, 3 and 4.Many results obtained previously with p-step methods 15;58 can be used to give theaccuracy and stability properties of the solution produced by the SSpj algorithms.Tables 18.6 and 18.7 give the accuracy of stable algorithms from the SSp1 andSSp2families respectively for p ˆ 2, 3, 4. Algorithms that are only conditionallystable (i.e., only stable for values of the time step less than some critical value) aremarked CS. Details are given in reference 2.We conclude this section by writing in full the second degree (two-step) algorithmthat corresponds precisely to SS22 and GN22 methods. Indeed, it is written below inthe form originally derived by Newmark 41 :‰ M ‡tC ‡ t 2 K a n ‡ 1‡ ÿ2M ‡…1 ÿ 2†tC ‡… 1 2 ÿ 2 ‡ †t2 K an‡ M ÿ…1 ÿ †tC ‡… 1 2 ‡ ÿ †t2 K an ÿ 1 ‡ t 2 f ˆ 0 …18:128†Table 18.6 Accuracy of SSp1 algorithmsMethod parameters errorSS11 1 O…t† 1 ˆ 12O…t 2 †SS21 1 ; 2 O…t 2 † 1 ÿ 2 ˆ 16 O…t 3 †CSSS31 1 ; 2 ; 3 O…t 3 † 1 ÿ 3 2 ; ‡2 3 ˆ 0 O…t 4 †CSSS41 1 ; 2 ; 3 ; 4 O…t 4 †

530 The time dimension ± discrete approximation in timeTable 18.7 Accuracy of SSp2algorithmsMethod Parameters ErrorC ˆ 0 C 6ˆ 0SS22 1 ; 2 O…t† O…t† 1 ; 2 ˆ 12O…t 2 † O…t 2 † 1 ˆ 12 ; 2 ˆ 16O…t 4 †CS O…t 2 †CSSS32 1 ; 2 ; 3 O…t 2 † O…t 2 † 2 ˆ 1 ÿ 1 6O…t 3 †CS O…t 3 †CS 3 ˆ 12 1 O…t 4 †CS ±SS42 1 ; 2 ; 3 ; 4 O…t 4 †CS O…t 4 †CSHere of course, we have the original Newmark parameters , , which can be changedto correspond with the SS22/GN22 form as follows: ˆ 1 ˆ 1 ˆ 12 2 ˆ 12 2The explicit form of this algorithm ( ˆ 2 ˆ 2 ˆ 0) is frequently used as analternative to the single-step explicit form. It is then known as the central di€erenceapproximation obtained by direct di€erencing. The reader can easily verify that thesimplest ®nite di€erence approximation of Eq. (18.1) in fact corresponds to theabove with ˆ 0 and ˆ 1=2.18.5 Some remarks on general performance of numericalalgorithmsIn Secs 18.2.5 and 18.3.3 we have considered the exact solution of the approximaterecurrence algorithm given in the forma n ‡ 1 ˆ a n ; etc: …18:129†for the modally decomposed, single degree of freedom systems typical of Eqs (18.4)and (18.5). The evaluation of was important to ensure that its modulus does notexceed unity and stability is preserved.However, analytical solution of the linear homogeneous di€erential equations isalso easy to obtain in the forma ˆ a e t…18:130†a n ‡ 1 ˆ a n e t…18:131†and comparison of with such a solution is always instructive to provide informationon the performance of algorithms in the particular range of eigenvalues.In Fig. 18.5 we have plotted the exact solution e ÿ!t and compared it with thevalues of for various algorithms approximating the ®rst-order equation, notingthat hereand is real. ˆÿ! ˆÿ k c…18:132†

Some remarks on general performance of numerical algorithms 531Immediately we see that there the performance error is very di€erent for variousvalues of t and obviously deteriorates at large values. Such values in a real multivariableproblem correspond of course to the `high-frequency' responses which areoften less important, and for smooth solutions we favour algorithms where tendsto values much less than unity for such problems. However, response through thewhole time range is important and attempts to choose an optimal value of forvarious time ranges has been performed by Liniger. 53 Table 18.1 of Sec. 18.2.6illustrates how an algorithm with ˆ 2=3 and a higher truncation error than that1.0θ 2 /2 = β ≥ 0.25, θ 1 = γ = 0.5|µ|0.9θ 2 /2 = β = 0.5, θ 1 = γ = 0.60.8θ 2 /2 = β = 0.3025, θ 1 = γ = 0.60.710 –2 10 –1 1∆t/T10 10 2 10 3(a) Spectral radius |µ|0.50.4θ 1 = γ = 0.6, θ 2 /2 = β = 0.5θ 1 = γ = 0.6, θ 2 /2 = β = 0.3025(T–T)/T0.30.20.1θ 1 = γ = 0.5, θ 2 /2 = β = 0.2500 0.1 0.2 0.3 0.4∆t/T(b) Relative period elongationFig. 18.13 SS22, GN22 (Newmark) or their two-step equivalent.

532 The time dimension ± discrete approximation in time1.11.00.9Hilber et al. 57α = 0.1|µ|0.80.7AWilson 55θ = 1.4Houbolt 420.60.510 –2 10 –1 1 10 10 2 10 3∆t/T(a) Spectral radius0.5Houbolt 54 Hilber et al. 570.40.3(T–T)/T0.20.1Wilson 55θ = 1.4α = 0.100 0.1 0.2 0.3 0.4∆t/T(b) Relative period elongationFig. 18.14 SS23, GN23 or their two-step equivalent.

Some remarks on general performance of numerical algorithms 5332SS22Displacement10–1Displacement–20 10 20 30 40 50 60 70 80 90 100Time210–1Newmark–20.20.10 10 20 30 40 50 60 70 80 90 100TimeSS22Error0–0.1–0.20.20.10 10 20 30 40 50 60 70 80 90 100TimeNewmarkError0–0.1–0.20 10 20 30 40 50 60 70 80 90 100TimeFig. 18.15 Comparison of the SS22 and GN22 (Newmark) algorithms: a single DOFdynamic equation withperiodic forcing term, 1 ˆ 1 ˆ 1=2, 2 ˆ 2 ˆ 0.

534 The time dimension ± discrete approximation in time0.30SS22Velocity0.150–0.15Velocity–0.300 10 20 30 40 50 60 70 80 90 100Time0.300.150–0.15Newmark–0.300.20.10 10 20 30 40 50 60 70 80 90 100TimeSS22Error0–0.1–0.20.20.10 10 20 30 40 50 60 70 80 90 100TimeNewmarkError0–0.1–0.20 10 20 30 40 50 60 70 80 90 100TimeFig. 18.15 Continued.

Some remarks on general performance of numerical algorithms 5350.2SS22Acceleration0.10–0.1SS22Exact–0.20 10 20 30 40 50 60 70 80 90 1000.2TimeNewmarkAcceleration0.10–0.1GN22Exact–0.20.20.10 10 20 30 40 50 60 70 80 90 100TimeSS22Error0Error inSS22–0.1–0.20.20.10 10 20 30 40 50 60 70 80 90 100TimeNewmarkError0Error inGN22–0.1–0.2Fig. 18.15 Continued.0 10 20 30 40 50 60 70 80 90 100Time

536 The time dimension ± discrete approximation in timeof ˆ 1=2can perform better in a multidimensional system because of suchproperties.Similar analysis can be applied to the second-order equation. Here, to simplifymatters, we consider only the homogeneous undamped equation in the formma ‡ ka ˆ 0…18:133†in which the value of is purely imaginary and corresponds to a simple oscillator. Byexamining we can ®nd not only the amplitude ratio (which for high accuracy shouldbe unity) but also the phase error.In Fig. 18.13(a) we show both the variation of the modulus (which is called thespectral radius) and in Fig. 18.13(b) that of the relative period for the SS22/GN22schemes, which of course are also applicable to the two-step equivalent. The resultsare plotted againsttTwhere T ˆ 2 ! ; !2 ˆ km…18:134†In Fig. 18.14(a) and (b) similar curves are given for the SS23 and GN23 schemesfrequently used in practice and discussed previously.Here as in the ®rst-order problem we often wish to suppress (or damp out) theresponse to frequencies in which t=T is large (say greater than 0.1) in multidegreeof freedom systems, as such a response will invariably be inaccurate. At the sametime below this limit it is desirable to have amplitude ratios as close to unity aspossible. It is clear that the stability limit with 1 ˆ 2 ˆ 1=2giving unit responseeverywhere is often undesirable (unless physical damping is su cient to damp highfrequency modes) and that some algorithmic damping is necessary in these cases.The various schemes shown in Figs 18.13 and 18.14 can be judged accordingly andprovide the reason for a search for an optimum algorithm.We have remarked frequently that although schemes can be identical with regard tostability their performances may di€er slightly. In Fig. 18.15 we illustrate the applicationof SS22 and GN22 to a single degree of freedom system showing results anderrors in each scheme.18.6 Time discontinuous Galerkin approximationA time discontinuous Galerkin formulation may be deduced from the ®nite element intime approximation procedure considered in this chapter. This is achieved by assumingthe weight function W and solution variables a are approximated within each timeinterval t asa ˆ a ‡ n ‡ a…t† t ÿ n 4 t < t ÿ n ‡ 1…18:135†W ˆ W ‡ n ‡ W…t† t ÿ n 4 t < t ÿ n ‡ 1where the time t ÿ n is the limit from times smaller than t n and t ‡ n is the limit from timeslarger than t n and, thus, admit a discontinuity in the approximation to occur at eachdiscrete time location. The functions a and W are de®ned to be zero at t n andcontinuous up to the time t ÿ n ‡ 1 where again a discontinuity can occur during thenext time interval.

The discrete form of the governing equations may be deduce starting from the timedependent partial di€erential equations where standard ®nite elements in space arecombined with the time discontinuous Galerkin approximation and de®ning aweak form in a space-time slab. Alternatively, we may begin with the semi-discreteform as done previously in this chapter for other ®nite element in time methods. Inthis second form, for the ®rst-order case, we writeI ˆ… tÿn ‡ 1t ÿ nW T … C_a ‡ Ka ‡ f†d ˆ 0 …18:136†Due to the discontinuity at t n it is necessary to split the integral intoI ˆwhich gives… t‡nt ÿ nW T … C_a ‡ Ka ‡ f†d ‡… tÿn ‡ 1…I ˆ…W ‡ n † T ÿCa ‡ n ÿ a ÿ tÿn ‡…W‡n † Tt ‡ nW T … C_a ‡ Ka ‡ f†d ˆ 0 …18:137†t ‡ nn ‡ 1… C_a ‡ Ka ‡ f†d… tÿn ‡ 1‡ …W† T … C_a ‡ Ka ‡ f†d ˆ 0 …18:138†t ‡ nin which now all integrals involve approximations to functions which are continuous.To apply the above process to a second-order equation it is necessary ®rst to reducethe equation to a pair of ®rst-order equations. This may be achieved by de®ning themomentap ˆ M_a…18:139†and then writing the pairTime discontinuous Galerkin approximation 537M_a ÿ p ˆ 0_p ‡ C_a ‡ Ka ‡ f ˆ 0…18:140†…18:141†The time discrete process may now be applied by introducing two weighting functionsas described in reference 37.Example: Solution of the scalar equation To illustrate the process we consider thesimple ®rst-order scalar equationc _u ‡ ku ‡ f ˆ 0…18:142†We consider the speci®c approximationsu…t† ˆu ‡ n ‡ u ÿ n ‡ 1W…t† ˆW ‡ n ‡ W ÿ n ‡ 1…18:143†where u ÿ n ‡ 1 ˆ u ÿ n ‡ 1 ÿ u ‡ n , etc., and ˆt ÿ t nˆ t ÿ t nt n ‡ 1 ÿ t n t

538 The time dimension ± discrete approximation in timede®nes the time interval 0

3. P. Henrici. Discrete Variable Methods in Ordinary Di€erential Equations. John Wiley &Sons, New York, 1962.4. F.B. Hildebrand. Finite Di€erence Equations and Simulations. Prentice-Hall, EnglewoodCli€s, N.J., 1968.5. G.W. Gear. Numerical Initial Value Problems in Ordinary Di€erential Equations. Prentice-Hall, Englewood Cli€s, N.J., 1971.6. W.L. Wood. Practical Time Stepping Schemes. Clarendon Press, Oxford, 1990.7. J.T. Oden. A general theory of ®nite elements. Part II. Applications. Internat. J. Num.Meth. Eng., 1, 247±54, 1969.8. I. Fried. Finite element analysis of time-dependent phenomena. AIAA J., 7, 1170±73,1969.9. J.H. Argyris and D.W. Scharpf. Finite elements in time and space. Nucl. Eng. Design, 10,456±69, 1969.10. O.C. Zienkiewicz and C.J. Parekh. Transient ®eld problems ± two and three dimensionalanalysis by isoparametric ®nite elements. Internat. J. Num. Meth. Eng., 2, 61±71,1970.11. O.C. Zienkiewicz. The Finite Element Method in Engineering Science. McGraw-Hill,London, 2nd edition, 1971.12. O.C. Zienkiewicz and R.W. Lewis. An analysis of various time stepping schemes for initialvalue problems. Earthquake Eng. Struct. Dyn., 1, 407±8, 1973.13. W.L. Wood and R.W. Lewis. A comparison of time marching schemes for the transientheat conduction equation. Internat. J. Num. Meth. Eng., 9, 679±89, 1975.14. O.C. Zienkiewicz. A new look at the Newmark, Houbolt and other time stepping formulas.A weighted residual approach. Earthquake Eng. Struct. Dyn., 5, 413±18, 1977.15. W.L. Wood. On the Zienkiewicz four-time-level scheme for numerical integration ofvibration problems. Internat. J. Num. Meth. Eng., 11, 1519±28, 1977.16. O.C. Zienkiewicz, W.L. Wood, and R.L. Taylor. An alternative single-step algorithm fordynamic problems. Earthquake Eng. Struct. Dyn., 8, 31±40, 1980.17. W.L. Wood. A further look at Newmark, Houbolt, etc. time-stepping formulae. Internat.J. Num. Meth. Eng., 20, 1009±17, 1984.18. O.C. Zienkiewicz, W.L. Wood, N.W. Hine, and R.L. Taylor. A uni®ed set of single-stepalgorithms. Part 1: general formulation and applications. Internat. J. Num. Meth. Eng.,20, 1529±52, 1984.19. W.L. Wood. A uni®ed set of single-step algorithms. Part 2: theory. Internat. J. Num. Meth.Eng., 20, 2302±09, 1984.20. M. Katona and O.C. Zienkiewicz. A uni®ed set of single-step algorithms. Part 3: the beta-mmethod, a generalization of the Newmark scheme. Internat. J. Num. Meth. Eng., 21, 1345±59, 1985.21. E. Varoglu and N.D.L. Finn. A ®nite element method for the di€usion convectionequations with concurrent coe cients. Adv. Water Resources, 1, 337±41, 1973.22. C. Johnson, U. NaÈ vert, and J. PitkaÈ ranta. Finite element methods for linear hyperbolicproblems. Comp. Meth. Appl. Mech. Engng, 45, 285±312, 1984.23. T.J.R. Hughes, L.P. Franca, and G.M. Hulbert. A new ®nite element formulation for computational¯uid dynamics: VIII. The Galerkin/least-squares method for advective-di€usiveequations. Com. Meth. Appl. Mech. Eng., 73, 173±89, 1989.24. T.J.R. Hughes and G.M. Hulbert. Space-time ®nite element methods in elastodynamics:Formulation and error estimates. Com. Meth. Appl. Mech. Eng., 66, 339±63, 1988.25. G.M. Hulbert and T.J.R. Hughes. Space-time ®nite element methods for second-orderhyperbolic equations. Com. Meth. Appl. Mech. Eng., 84, 327±48, 1990.26. G.M. Hulbert. Time ®nite element methods for structural dynamics. Internat. J. Num.Meth. Eng., 33, 307±31, 1992.References 539

540 The time dimension ± discrete approximation in time27. B.M. Irons and C. Treharne. A bound theorem for eigenvalues and its practical application.In Proc. 3rd Conf. Matrix Methods in Structural Mechanics, volume AFFDL-TR-71±160,pages 245±54, Wright-Patterson Air Force Base, Ohio, 1972.28. K. Washizu. Variational Methods in Elasticity and Plasticity. Pergamon Press, New York, 3edition, 1982.29. M. Gurtin. Variational principles for linear initial-value problems. Q. Appl. Math., 22,252±56, 1964.30. M. Gurtin. Variational principles for linear elastodynamics. Arch. Rat. Mech. Anal., 16,34±50, 1969.31. E.L. Wilson and R.E. Nickell. Application of ®nite element method to heat conductionanalysis. Nucl. Eng Design, 4, 1±11, 1966.32. J. Crank and P. Nicolson. A practical method for numerical integration of solutions ofpartial di€erential equations of heat conduction type. Proc. Camb. Phil. Soc., 43, 50, 1947.33. R.L. Taylor and O.C. Zienkiewicz. A note on the Order of Approximation. Internat. J.Solids Structures, 21, 793±838, 1985.34. P. Lesaint and P.-A. Raviart. On a ®nite element method for solving the neutron transportequation. In C. de Boor, editor, Mathematical Aspects of Finite Elements in Partial Di€erentialEquations. Academic Press, New York, 1974.35. C. Johnson. Numerical Solutions of Partial Di€erential Equations by the Finite ElementMethod. Cambridge University Press, Cambridge, 1987.36. K. Eriksson and C. Johnson. Adaptive ®nite element methods for parabolic problems I, Alinear model problem. SIAM J. Numer. Anal., 28, 43±77, 1991.37. X.D. Li and N.-E. Wiberg. Structural dynamic analysis by a time-discontinuous Galerkin®nite element method. Internat. J. Num. Meth. Eng., 39, 2131±52, 1996.38. M. Zlamal. Finite element methods in heat conduction problems. In J. Whiteman, editor,The Mathematics of Finite Elements and Applications, pages 85±104. Academic Press,London, 1977.39. W. Liniger. Optimisation of a numerical integration method for sti€ systems of ordinarydi€erential equations. Technical Report RC2198, IBM Research, 1968.40. J.M. Bettencourt, O.C. Zienkiewicz, and G. Cantin. Consistent use of ®nite elements intime and the performance of various recurrence schemes for heat di€usion equation.Internat. J. Num. Meth. Eng., 17, 931±38, 1981.41. N. Newmark. A method of computation for structural dynamics. J. Eng. Mech. Div., 85,67±94, 1959.42. T. Belytschko and T.J.R. Hughes, editors. Computational Methods for Transient Analysis.North-Holland, Amsterdam, 1983.43. J.C. Simo and K. Wong. Unconditionally stable algorithms for rigid body dynamics thatexactly conserve energy and momentum. Internat. J. Num. Meth. Eng., 31, 19±52, 1991.44. J.C. Simo and N. Tarnow. The discrete energy-momentum method. conserving algorithmfor nonlinear elastodynamics. ZAMP, 43, 757±93, 1992.45. J.C. Simo and N. Tarnow. Exact energy-momentum conserving algorithms and symplecticschemes for nonlinear dynamics. Com. Meth. Appl. Mech. Eng., 100, 63±116, 1992.46. O. Gonzalez. Design and analysis of conserving integrators for nonlinear Hamiltoniansystems with symmetry. Ph.d thesis, Stanford University, Stanford, California, 1996.47. I. Miranda, R.M. Ferencz, and T.J.R. Hughes. An improved implicit-explicit time integrationmethod for structural dynamics. Earthquake Eng. Struct. Dyn., 18, 643±55, 1989.48. E.J. Routh. A Treatise on the Stability of a Given State or Motion. Macmillan, London,1977.49. A. Hurwitz. Uber die Bedingungen, unter welchen eine Gleichung nur WuÈ rzeln mitnegatives reellen teilen besitzt. Math. Ann., 46, 273±84, 1895.50. F.R. Gantmacher. The Theory of Matrices. Chelsea, New York, 1959.

51. G.G. Dahlquist. A special stability problem for linear multistep methods. BIT, 3, 27±43,1963.52. C.W. Gear. The automatic integration of sti€ ordinary di€erential equations. In A.J.H.Morrell, editor, Information Processing 68. North Holland, Dordrecht, 1969.53. W. Liniger. Global accuracy and A-stability of one and two step integration formulae forsti€ ordinary di€erential equations. In Proc. Conf. on Numerical Solution of Di€erentialEquations, Dundee University, 1969.54. J.C. Houbolt. A recurrence matrix solution for dynamic response of elastic aircraft. J. Aero.Sci., 17, 540±50, 1950.55. K.J. Bathe and E.L. Wilson. Stability and accuracy analysis of direct integration methods.Earthquake Eng. Struct. Dyn., 1, 283±91, 1973.56. W. Wood, M. Bossak, and O.C. Zienkiewicz. An alpha modi®cation of Newmark'smethod. Internat. J. Num. Meth. Eng., 15, 1562±66, 1980.57. H. Hilber, T.J.R. Hughes, and R.L. Taylor. Improved numerical dissipation for the timeintegration algorithms in structural dynamics. Earthquake Eng. Struct. Dyn., 5, 283±92,1977.58. W.L. Wood. On the Zienkiewicz three- and four-time-level schemes applied to the numericalintegration of parabolic problems. Internat. J. Num. Meth. Eng., 12, 1717±26, 1978.References 541

19Coupled systems19.1 Coupled problems ± de®nition and classi®cationFrequently two or more physical systems interact with each other, with the independentsolution of any one system being impossible without simultaneous solution ofthe others. Such systems are known as coupled and of course such coupling maybe weak or strong depending on the degree of interaction.An obvious `coupled' problem is that of dynamic ¯uid±structure interaction. Hereneither the ¯uid nor the structural system can be solved independently of the otherdue to the unknown interface forces.A de®nition of coupled systems may be generalized to include a wide range ofproblems and their numerical discretization as: 1Coupled systems and formulations are those applicable to multiple domains and dependentvariables which usually (but not always) describe di€erent physical phenomena andin which(a) neither domain can be solved while separated from the other;(b) neither set of dependent variables can be explicitly eliminated at the di€erentialequation level.The reader may well contrast this with de®nitions of mixed and irreducibleformulations given in Chapter 11 and ®nd some similarities. Clearly `mixed' and`coupled' formulations are analogous, with the main di€erence being that in theformer elimination of some dependent variables is possible at the governing di€erentialequation level. In the coupled system a full analytical solution or inversion of a(discretized) single system is necessary before such elimination is possible.Indeed, a further distinction can be made. In coupled systems the solution of anysingle system is a well-posed problem and is possible when the variables correspondingto the other system are prescribed. This is not always the case in mixed formulations.It is convenient to classify coupled systems into two categories:Class I. This class contains problems in which coupling occurs on domain interfacesvia the boundary conditions imposed there. Generally the domains describedi€erent physical situations but it is possible to consider coupling between

Coupled problems ± de®nition and classi®cation 543domains that are physically similar in which di€erent discretization processeshave been used.Class II. This class contains problems in which the various domains overlap (totallyor partially). Here the coupling occurs through the governing di€erentialequations describing di€erent physical phenomena.Typical of the ®rst category are the problems of ¯uid±structure interactionillustrated in Fig. 19.1(a) where physically di€erent problems interact and alsoInterface(a) Fluid–structure interaction (physically different domains)Interface(b) Structure–structure interaction (physically identical domains)Fig. 19.1 Class I problems with coupling via interfaces (shown as thick line).

544 Coupled systemsImposed position(a) Seepage through a porous medium interacts with its dynamic,(a) structural behaviour(b) Problem of metal extrusion in which the plastic flow is coupled(b) with the thermal fieldFig. 19.2 Class II problems with coupling in overlapping domains.structure±structure interactions of Fig. 19.1(b) where the interface simply dividesarbitrarily chosen regions in which di€erent numerical discretizations are used.The need for the use of di€erent discretization may arise from di€erent causes. Herefor instance:1. Di€erent ®nite element meshes may be advantageous to describe the subdomains.2. Di€erent procedures such as the combination of boundary method and ®niteelements in respective regions may be computationally desirable.3. Domains may simply be divided by the choice of di€erent time-steppingprocedures, e.g. of an implicit and explicit kind.In the second category, typical problems are illustrated in Fig. 19.2. One of these isthat of metal extrusion where the plastic ¯ow is strongly coupled with the temperature®eld while at the same time the latter is in¯uenced by the heat generated in the plastic¯ow. This problem will be considered in more detail in Volume 2 but is included toillustrate a form of coupling that commonly occurs in analyses of solids. The otherproblem shown in Fig. 19.2 is that of soil dynamics (earthquake response of adam) in which the seepage ¯ow and pressures interact with the dynamic behaviourof the soil `skeleton'.

Fluid±structure interaction (Class I problem) 545We observe that, in the examples illustrated, motion invariably occurs. Indeed, thevast majority of coupled problems involve such transient behaviour and for thisreason the present chapter will only consider this area. It will thus follow andexpand the analysis techniques presented in Chapters 17 and 18.As the problems encountered in coupled analysis of various kinds are similar, weshall focus the presentation on three examples:1. ¯uid±structure interaction (con®ned to small amplitudes);2. soil±¯uid interaction;3. implicit±explicit dynamic analysis of a structure where the separation involves theprocess of temporal discretization.In these problems all the typical features of coupled analysis will be found andextension to others will normally follow similar lines. In Volume 2 we shall, forinstance, deal in more detail with the problem of coupled metal forming 2 and thereader will discover the similarities.As a ®nal remark, it is worthwhile mentioning that problems such as linear thermalstress analysis to which we have referred frequently in this volume are not coupled inthe terms de®ned here. In this the stress analysis problem requires a knowledge of thetemperature ®eld but the temperature problem can be solved independently of thestress ®eld.y Thus the problem decouples in one direction. Many examples of trulycoupled problems will be found in available books. 4ÿ619.2 Fluid±structure interaction (Class I problem)19.2.1 General remarks and ¯uid behaviour equationsThe problem of ¯uid±structure interaction is a wide one and covers many forms of¯uid which, as yet, we have not discussed in any detail. The consideration of problemsin which the ¯uid is in substantial motion is deferred until Volume 3 and, thus, weexclude at this stage such problems as ¯utter where movement of an aerofoilin¯uences the ¯ow pattern and forces around it leading to possible instability. Forthe same reason we also exclude here the `singing wire' problem in which the sheddingof vortices reacts with the motion of the wire.However, in a very considerable range of problems the ¯uid displacement remainssmall while interaction is substantial. In this category fall the ®rst two examples ofFig. 19.1 in which the structural motions in¯uence and react with the generation ofpressures in a reservoir or a container. A number of symposia have been entirelydevoted to this class of problems which is of considerable engineering interest, andhere fortunately considerable simpli®cations are possible in the description of the¯uid phase. References 7±22 give some typical studies.y In a general setting the temperature ®eld does depend upon the strain rate. However, these terms are notincluded in the form presented in this volume and in many instances produce insigni®cant changes to thesolution. 3

546 Coupled systemsIn such problems it is possible to write the dynamic equations of ¯uid behavioursimply as@…v† @v@t @t ˆ rp…19:1†where v is the ¯uid velocity, is the ¯uid density and p the pressure. In postulating theabove we have assumed1. that the density varies by a small amount only so may be considered constant;2. that velocities are small enough for convective e€ects to be omitted;3. that viscous e€ects by which deviatoric stresses are introduced can be neglected inthe ¯uid.The reader can in fact note that with the preceding assumption Eq. (19.1) is aspecial form of a more general relation (described in Chapter 1 of Volume 3).The continuity equation based on the same assumption is div v r T v ˆÿ @…19:2†@tand noting thatd ˆ K dp…19:3†where K is the bulk modulus, we can writer T v ˆÿ 1 @p…19:4†K @tElimination of v between (19.1) and (19.4) gives the well-known Helmholtzequation governing the pressure p:wherer 2 p ˆ 1 @ 2 pc 2 @t 2c ˆsK…19:5†…19:6†denotes the speed of sound in the ¯uid.The equations described above are the basis of acoustic problems.19.2.2 Boundary conditions for the ¯uid. Coupling and radiationIn Fig. 19.3 we focus on the Class I problem illustrated in Fig. 19.1(a) and on theboundary conditions possible for the ¯uid part described by the governing equation(19.5). As we know well, either normal gradients or values of p now need to bespeci®ed.

Fluid±structure interaction (Class I problem) 547Actual surfacezη 3Mean surface1 N f 4N sFig. 19.3 Boundary conditions for the ¯uid component of the ¯uid±structure interaction.2Interface with solidOn the boundaries 1 and 2 in Fig. 19.3 the normal velocities (or their timederivatives) are prescribed. Considering the pressure gradient in the normal directionto the face n we can thus write, by Eq. (19.1),@p@n ˆÿ _v n ˆÿn T _v…19:7†where n is the direction cosine vector for an outward pointing normal to the ¯uidregion and _v n is prescribed.Thus, for instance, on boundary 1 coupling with the motion of the structuredescribed by displacement u occurs. Here we put_v n ˆ u n ˆ n T u…19:8†while on boundary 2where only horizontal motion exists we have_v z ˆ 0Coupling with the structure motion occurs only via boundary 1 .Free surfaceOn the free surface (boundary 3…19:9†in Fig. 19.3) the simplest assumption is thatp ˆ 0…19:10†However, this does not allow for any possibility of surface gravity waves. These canbe approximated by assuming the actual surface to be at an elevation relative to themean surface. Nowp ˆ g…19:11†where g is the acceleration due to gravity. From Eq. (19.1) we have, on notingv z ˆ @=@t and assuming to be constant, @2 @t 2 ˆÿ@p…19:12†@z

548 Coupled systemsand on elimination of , using Eq. (19.11), we have a speci®ed normal gradientcondition@p@z ˆÿ1 @ 2 pg @t 2 ˆÿ1 g p…19:13†This allows for gravity waves to be approximately incorporated in the analysis and isknown as the linearized surface wave condition.Radiation boundaryBoundary 4 physically terminates an in®nite domain and some approximation toaccount for the e€ect of such a termination is necessary. The main dynamic e€ectis simply that the wave solution of the governing equation (19.5) must here becomposed of outgoing waves only as no input from the in®nite domain exists.If we consider only variations in x (the horizontal direction) we know that thegeneral solution of Eq. (19.5) can be written asp ˆ F…x ÿ ct†‡G…x ‡ ct†…19:14†where c is the wave velocity given by Eq. (19.6) and the two waves F and G travel inpositive and negative directions of x, respectively.The absence of the incoming wave G means that on boundary 4 we have onlyp ˆ F…x ÿ ct†…19:15†Thus@p@n @p@x ˆ F 0…19:16†and@p@t ˆÿcF 0…19:17†where F 0 denotes the derivative of F with respect to …x ÿ ct†. We can thereforeeliminate the unknown function F 0 and write@p@n ˆÿ1 c _p…19:18†which is a condition very similar to that of Eq. (19.13). This boundary condition was®rst presented in reference 7 for radiating boundaries and has an analogy with adamping element placed there.19.2.3 Weak form for coupled systemsA weak form for each part of the coupled system may be written as described inChapter 3. Accordingly, for the ¯uid we can write the di€erential equation as… f ˆ p 1 fc 2 p ÿ r2 p d ˆ 0…19:19†

Fluid±structure interaction (Class I problem) 549which after integration by parts and substitution of the boundary conditionsdescribed above yields… p 1 … fc 2 p ‡…r†T rp d ‡ pn T u dÿ ‡ pÿ 1…ÿ 1 …3g p dÿ ‡ p 1 _p dÿ ˆ 0ÿ 4c…19:20†where f is the ¯uid domain and ÿ i the integral over boundary part i .Similarly for the solid the weak form after integration by parts is given by……u s u ‡ S T DSu d ÿ u T t dÿ ˆ 0…19:21†ÿ twhere for pressure de®ned positive in compression the surface traction is de®ned ast ˆÿpn s ˆ pn…19:22†since the outward normal to the solid is n s ˆÿn. The traction integral in Eq. (19.21) isnow expressed as……u T t dÿ ˆ u T np dÿ…19:23†ÿ t ÿ t(1) In complex physical situations, the interaction between compressibility andinternal gravity waves (interaction between acoustic modes and sloshing modes)leads to a modi®ed Helmholz equation. The Eq. (19.5) should then be replaced bya more complex equation: in a strati®ed medium for instance, the irrotationalitycondition for the ¯uid is not totally veri®ed (the ¯uid is irrotational in a plane perpendicularto the strati®cation axis). 16(2) The variational formulation de®ned by Eq. (19.20) is valid in the static case providedthe following constraints conditions are added „ fp d ‡ c 2 „ @ fn T u dÿ ˆ 0for a compressible ¯uid ®lling a cavity, „ ÿ 1n T u dÿ ‡ „ ÿ 2p=g dÿ ˆ 0, for an incompressibleliquid with a free surface contained inside a reservoir. The static behaviouris important for the modal response of coupled systems when modal truncation needstatic corrections in order to accelerate the convergence of the method. This staticbehaviour is also of prime importance for the construction of reduced matrixmodels when using dynamic substructuring methods for ¯uid structure interactionproblems. 17;1819.2.4 The discrete coupled systemWe shall now consider the coupled problem discretized in the standard (displacement)manner with the displacement vector approximated asu ^u ˆ N u ~u…19:24†and the ¯uid similarly approximated byp ^p ˆ N p ~p…19:25†where ~u and ~p are the nodal parameters of each ®eld and N u and N p are appropriateshape functions.

550 Coupled systemsThe discrete structural problem thus becomesM ~u ‡ C ~u _ ‡ K~u ÿ Q~p ‡ f ˆ 0…19:26†where the coupling term arises due to the pressures (tractions) speci®ed on theboundary as……N T u t dÿ ˆ N T u nN p dÿ ˆ Q~p…19:27†ÿ t ÿ tThe terms of the other matrices are already well known to the reader as mass, damping,sti€ness and force.Standard Galerkin discretization applied to the weak form of the ¯uid equation(19.20) leads toS ~p ‡ C ~ ~p _ ‡ H~p ‡ Q T ~u ‡ q ˆ 0…19:28†where……S ˆ N T 1p c 2 N p d ‡ N T 1pÿ 3g N p dÿ…~C ˆ N T 1pÿ 4c 2 N p dÿ…19:29†…H ˆ …rN p † T rN p dand Q is identical to that of Eq. (19.27).19.2.5 Free vibrationsIf we consider free vibrations and omit all force and damping terms (noting that in the¯uid component the damping is strictly that due to radiation energy loss) we can writethe two equations (19.26) and (19.28) as a set: ( ) ~uM 0Q T S~p‡ K ÿQ ~u0 H ~pˆ 0…19:30†and attempt to proceed to establish the eigenvalues corresponding to naturalfrequencies. However, we note immediately that the system is not symmetric (norpositive de®nite) and that standard eigenvalue computation methods are not directlyapplicable. Physically it is, however, clear that the eigenvalues are real and that freevibration modes exist.The above problem is similar to that arising in vibration of rotating solids and specialsolution methods are available, though costly. 23 It is possible by various manipulationsto arrive at a symmetric form and reduce the problem to a standard eigenvalue one. 14ÿ26A simple method proposed by Ohayon proceeds to achieve the symmetrizationobjective by putting ~u ˆ u e i!t , ~p ˆ p e i!t and rewriting Eq. (19.30) asKu ÿ Qp ÿ ! 2 Mu ˆ 0…19:31†Hp ÿ ! 2 Sp ÿ ! 2 Qu ˆ 0

and an additional variable q such thatp ˆ ! 2 qFluid±structure interaction (Class I problem) 551…19:32†After some manipulation and substitution we can write the new system as823 23989>< K 0 0 M 0 Q >= >< u >=6 74 0 S 05 ÿ ! 2 674 0 0 S 5 p>:>; >: >; ˆ 0 …19:33†0 0 0 Q T S T H qwhich is a symmetric generalized eigenproblem. Further, the variable q can now beeliminated by static condensation and the ®nal system becomes symmetric and nowcontains only the basic variables. The system (19.32), with static corrections, maylead to convenient reduced matrix models through appropriate dynamic substructuringmethods. 19An alternative that has frequently been used is to introduce a new symmetrizingvariable at the governing equation level, but this is clearly not necessary. 14;15As an example of a simple problem in the present category we show an analysis of athree-dimensional ¯exible wall vibrating with a ¯uid encased in a `rigid' container 27(Fig. 19.4).19.2.6 Forced vibrations and transient step-by-step algorithmsThe reader can easily verify that the steady-state, linear response to periodic input canbe readily computed in the complex frequency domain by the procedures described inChapter 17. Here no di culties arise due to the non-symmetric nature of equationsand standard procedures can be applied. Chopra and co-workers have, for instance,done many studies of dam/reservoir interaction using such methods. 28;29 However,such methods are not generally economical for very large problems and fail in nonlinearresponse studies. Here time-stepping procedures are required in the mannerdiscussed in the previous chapter. However, simple application of methods developedthere leads to an unsymmetric problem for the combined system (with ~u and ~p asvariables) due to the form of the matrices appearing in (19.30) and a modi®edapproach is necessary. 30 In this each of the equations (19.26) and (19.28) is ®rstdiscretized in time separately using the general approaches of Chapter 18.Thus in the time interval t we can approximate ~u using, say, the general SS22procedure as follows. First we write~u ˆ ~u n ‡ ~u _ n ‡ a 2…19:34†2with a similar expression for p,~p ˆ ~p n ‡ ~p _ n ‡ b 2…19:35†2where ˆ t ÿ t n .Insertion of the above into Eqs (19.26) and (19.28) and weighting with two separateweighting functions results in two relations in which a and b are the unknowns. These

552 Coupled systemsMode 1Frequency: 9.8 Hz(a)Mode 2Frequency: 43.6 Hz(b)Mode 3Frequency: 55 Hz(c)Fig. 19.4 Body of ¯uid with a free surface oscillating with a wall. Circles show pressure amplitude andsquares indicate opposite signs. Three-dimensional approach using parabolic elements.

Fluid±structure interaction (Class I problem) 553areÿ ÿMa ‡ C _u n ‡ 1 ‡ 1 ta ‡ K un ‡ 1 ‡ 1 2 2t 2 aÿÿ Q p n ‡ 1 ‡ 1 2 2 t 2 b ‡ fn ‡ 1 ˆ 0 …19:36†andSb ‡ Q T ÿa ‡ H p n ‡ 1 ‡ 1 2 2t 2 b ‡ qn ‡ 1 ˆ 0 …19:37†whereu n ‡ 1 ˆ ~u n ‡ 1 t _ ~u n_u n ‡ 1 ˆ _~u n…19:38†p n ‡ 1 ˆ ~p n ‡ 1 t _ ~p nare the predictors for the n ‡ 1 time step. In the above the parameters i and i aresimilar to those of Eq. (18.49) and can be chosen by the user. It is interesting tonote that the equation system can be put in symmetric form as2364…M ‡ 1 tC ‡ 1 2 2t 2 K† ÿQ ÿQ T ÿ 2H ‡ 2 7 2 2 t 2 S5 a^b ˆF1F 2…19:39†where the second equation has been multiplied by ÿ1, the unknown b has beenreplaced by^b ˆ 12 2t 2 b …19:40†and the forces are given byF 1 ˆÿf n ‡ 1 ÿ C_u n ‡ 1 ÿ Ku n ‡ 1 ‡ Qp n ‡ 1F 2 ˆ q n ‡ 1 ‡ Hp n ‡ 1…19:41†It is not necessary to go into detail about the computation steps as these follow theusual patterns of determining a and b and then evaluation of the problem variables,that is ~u n ‡ 1 , ~p n ‡ 1 , _ ~u n ‡ 1 and _ ~p n ‡ 1 at t n ‡ 1 before proceeding with the next time step.Non-linearity of structural behaviour can readily be accommodated using proceduresdescribed in Volume 2.It is, however, important to consider the stability of the linear system which will, ofcourse, depend on the choice of i and i . Here we ®nd, by using procedures describedin Chapter 18, that unconditional stability is obtained when 2 5 1 1 5 1 2 2 5 2 1 5 1 2…19:42†It is instructive to note that precisely the same result would be obtained if GN22approximations were used in Eqs (19.34) and (19.35).The derivation of such stability conditions is straightforward and follows preciselythe lines of Sec. 18.3.4 of the previous chapter. However, the algebra is sometimes

554 Coupled systemstedious. Nevertheless, to allow the reader to repeat such calculations for any caseencountered we shall outline the calculations for the present example.Stability of the ¯uid±structure time-stepping scheme 30For stability evaluations it is always advisable to consider the modally decomposedsystem with scalar variables. We thus rewrite Eqs (19.36) and (19.37) omitting theforcing terms and putting i ˆ i asm ‡ c… _u n ‡ 1 t†‡k…u n ‡ 1 t _u n ‡ 1 2 2t 2 †ÿ q…p n ‡ 1 t _p n ‡ 1 2 2t 2 †ˆ0…19:43†ands ‡ q ‡ h…p n ‡ 1 t _p ‡ 1 2 2t 2 †ˆ0…19:44†To complete the recurrence relations we haveu n ‡ 1 ˆ u n ‡ t _u n ‡ 1 2 t2 _u n ‡ 1 ˆ _u n ‡ tp n ‡ 1 ˆ p n ‡ t _p n ‡ 1 2 t2 _p n ‡ 1 ˆ _p n ‡ t…19:45†The exact solution of the above system will always be of the formu n ‡ 1 ˆ u n_u n ‡ 1 ˆ _u np n ‡ 1 ˆ p n…19:46†_p n ‡ 1 ˆ _p nand immediately we put ˆ 1 ÿ z1 ‡ zknowing that for stability we require the real part of z to be negative.Eliminating all n ‡ 1 values from Eqs (19.45) and (19.46) leads to_u n ˆ 2zt u n ˆ_p n ˆ 2zt p n4z 2…1 ÿ z†t 2 u 4z 2n ˆ…1 ÿ z†t 2 p n…19:47†Inserting (19.47) into the system (19.43) and (19.44) gives…a 11 z 2 ‡ b 11 z ‡ k†u n ‡…a 12 z 2 ‡ b 12 z ÿ q†p n ˆ 04qz 2 u n ‡…a 22 z 2 ‡ b 22 z ‡ h 0 †p n ˆ 0…19:48†

Fluid±structure interaction (Class I problem) 555wherea 11 ˆ 4m 0 ÿ 2…1 ÿ 2 1 †c 0 ÿ 2k… 1 ÿ 2 †a 12 ˆ 2q… 1 ÿ 2 †a 22 ˆ 4s ÿ 2… 1 ÿ 2 †h 0b 11 ˆ 2c 0 ÿ k…1 ÿ 2 1 †…19:49†in whichb 12 ˆ…1 ÿ 2 1 †qb 22 ˆÿ…1 ÿ 2 1 †h 0m 0 ˆ mt 2c 0 ˆ cth 0 ˆ t 2 hFor non-trivial solutions to exist the determinant of Eq. (19.48) has to be zero. Thisdeterminant provides the characteristic equation for z which, in the present case, is apolynomial of fourth order of the forma 0 z 4 ‡ a 1 z 3 ‡ a 2 z 2 ‡ a 3 z ‡ a 4 ˆ 0Thus use of the Routh±Hurwitz conditions given in Sec. 18.3.4 ensures stabilityrequirements are satis®ed, i.e., that the roots of z have negative real parts. For thepresent case the requirements are the followingThe inequalitya 0 > 0 and a i 5 0; i ˆ 1; 2; 3; 4a 11 a 22 ÿ 8q 2 … 1 ÿ 2 † > 0…19:50†is satis®ed for m 0 , c 0 , k, s, h 0 5 0if 1 5 1 2 2 5 1The inequalitya 1 ˆ a 11 ÿh 0 …1 ÿ 2 1 † ‡ 2c 0 ÿ k…1 ÿ 2 1 † a22 5 0 …19:51†is also satis®ed if 1 5 1 2 2 5 1The inequalitiesa 2 ˆ a 11 h 0 ‡ b 11 b 22 ‡ a 22 k ‡ 4q 2 5 0a 3 ˆ b 11 h 0 ‡ b 22 k 5 0…19:52†…19:53†are satis®ed if (19.50) and (19.51) are satis®ed. The inequalitya 4 ˆ kh 0 5 0…19:54†is automatically satis®ed. Finally the two inequalitiesa 1 a 2 ÿ a 0 a 3 5 0a 1 a 2 a 3 ÿ a 0 a 2 3 ÿ a 4 a 2 1 5 0…19:55†…19:56†are also satis®ed if (19.50) and (19.51) are satis®ed.If all the equalities hold then m 0 s > 0 has to be satis®ed. In case m 0 s ˆ 0andc 0 ˆ 0then 2 > 1 must be enforced.

556 Coupled systems19.2.7 Special case of incompressible ¯uidsIf the ¯uid is incompressible as well as being inviscid, its behaviour is described by asimple laplacian equationr 2 p ˆ 0…19:58†obtained by putting c ˆ1in Eq. (19.5).In the absence of surface wave e€ects and of non-zero prescribed pressures thediscrete equation (19.28) becomes simplyH~p ˆÿQ T ~u…19:59†as wave radiation disappears. It is now simple to obtain~p ˆÿH ÿ1 Q T ~u …19:60†and substitution of the above into the structure equation (19.26) results inÿM ‡ QH ÿ1 Q T ~u ‡ C ~u _ ‡ K~u ‡ f ˆ 0 …19:61†This is now a standard structural system in which the mass matrix has beenaugmented by an added mass matrix asM u ˆ QH ÿ1 Q T…19:62†and its solution follows the standard procedures of previous chapters.We have to remark that1. In general the complete inverse of H is not required as pressures at interface nodesonly are needed.2. In general the question of when compressibility e€ects can be ignored is a di cultone and will depend much on the frequencies that have to be considered in theanalysis. For instance, in the analysis of the reservoir±dam interaction muchdebate on the subject has been recorded. 31 Here the fundamental compressibleperiod may be of order H=c where H is a typical dimension (such as height ofthe dam). If this period is of the same order as that of, say, earthquake forcingmotion then, of course, compressibility must be taken into account. If it is muchshorter then its neglect can be justi®ed.19.2.8 Cavitation effects in ¯uidsIn ¯uids such as water the linear behaviour under volumetric strain ceases whenpressures fall below a certain threshold. This is the vapour pressure limit. When thisis reached cavities or distributed bubbles form and the pressure remains almostconstant. To follow such behaviour a non-linear constitutive law has to be introduced.Although this volume is primarily devoted to linear problems we here indicate some ofthe steps which are necessary to extend analyses to account for non-linear behaviour.A convenient variable useful in cavitation analysis was de®ned by Newton 32s ˆ div…u† r T …u†…19:63†

Fluid±structure interaction (Class I problem) 557where u is the ¯uid displacement. The non-linearity now is such thatp ˆÿK div u ˆ c 2 s; if s < …p a ÿ p v †=c 2p ˆ p a ÿ p v ; if s > …p a ÿ p v †=c 2 …19:64†Here p a is the atmospheric pressure (at which u ˆ 0 is assumed), p v is the vapourpressure and c is the sound velocity in the ¯uid.Clearly monitoring strains is a di cult problem in the formulation using thevelocity and pressure variables [Eq. (19.1) and (19.5)]. Here it is convenient tointroduce a displacement potential such thatu ˆÿr…19:65†175.40186.00 m186 m0.00 m(a) Structure–fluid mesh (quadratic elements)–100.00 m0.15 s 0.20 s0.25 s 0.30 s0.35 s 0.40 s0.50 s 0.55 s(b) Zones in which cavitation develops0.60 s 0.65 sFig. 19.5 The Bhakra dam±reservoir system. 33 Interaction during the ®rst second of earthquake motionshowing the development of cavitation.

558 Coupled systemsFrom the momentum equation (19.1) we see thatand thusThe continuity equation (19.2) now givesu ˆÿr ˆÿrp ˆ p…19:66†s ˆ div u ˆÿr 2 ˆ 1c 2 p ˆ 1c 2 …19:67†in the linear case [with an appropriate change according to conditions (19.64) duringcavitation].Details of boundary conditions, discretization and coupling are fully described inreference 33 and follow the standard methodology previously given. Figure 19.5,taken from that reference, illustrates the results of a non-linear analysis showingthe development of cavity zones in a reservoir.19.3 Soil±pore ¯uid interaction (Class II problems)19.3.1 The problem and the governing equations. DiscretizationIt is well known that the behaviour of soils (and indeed other geomaterials) is stronglyin¯uenced by the pressures of the ¯uid present in the pores of the material. Indeed, theconcept of e€ective stress is here of paramount importance. Thus if r describes the totalstress (positive in tension) acting on the total area of the soil and the pores, and p is thepressure of the ¯uid (positive in compression) in the pores (generally of water), the e€ectivestress is de®ned asr 0 ˆ r ‡ mp…19:68†Here m T ˆ ‰ 1; 1; 1; 0; 0; 0Š if we use the notation in Chapter 12. Now it is well knownthat it is only the stress r 0 which is responsible for the deformations (or failure) of thesolid skeleton of the soil (excluding here a very small volumetric grain compressionwhich has to be included in some cases). Assuming for the development given herethat the soil can be represented by a linear elastic model we haver 0 ˆ De…19:69†Immediately the total discrete equilibrium equations for the soil±¯uid mixture can bewritten in exactly the same form as is done for all problems of solid mechanics:…M ~u ‡ C ~u _ ‡ B T r d ‡ f ˆ 0…19:70†where ~u are the displacement discretization parameters, i.e.u ^u ˆ N~u…19:71†B is the strain±displacement matrix and M, C, f have the usual meaning of mass,damping and force matrices, respectively.

Now, however, the term involving the stress must be split as………B T r d ˆ B T r 0 d ÿ B T mp d…19:72†to allow the direct relationship between e€ective stresses and strains (and hencedisplacements) to be incorporated. For a linear elastic soil skeleton we immediatelyhaveM ~u ‡ C ~u _ ‡ K~u ÿ Q~p ‡ f ˆ 0…19:73†where K is the standard sti€ness matrix written as……B T r 0 d ˆ B T DB d ~u ˆ K~u …19:74†and Q couples the ®eld of pressures in the equilibrium equations assuming these arediscretized asp ^p ˆ N p ~p…19:75†Thus…Q ˆSoil±pore ¯uid interaction (Class II problems) 559B T mN p d…19:76†In the above discretization conventionally the same element shapes are used for the~u and ~p variables, though not necessarily identical interpolations. With the dynamicequations coupled to the pressure ®eld an additional equation is clearly needed fromwhich the pressure ®eld can be derived. This is provided by the transient seepageequation of the formÿr T … krp†‡ 1 Q _p ‡ _" v ˆ 0 …19:77†where Q is related to the compressibility of the ¯uid, k is the permeability and " v is thevolumetric strain in the soil skeleton, which on discretization of displacements is givenby" v ˆ m T e ˆ m T B~u …19:78†The equation of seepage can now be discretized in the standard Galerkin manner asQ T _~u ‡ S ~p _ ‡ H~p ‡ q ˆ 0…19:79†where Q is precisely that of Eq. (19.76), and……S ˆ N T 1p Q N p d H ˆ …rN p † T krN p d …19:80†with q containing the forcing and boundary terms. The derivation of coupled ¯ow±soil equations was ®rst introduced by Biot 34 but the present formulation is elaboratedupon in references 30 to 37 where various approximations, as well as the e€ect ofvarious non-linear constitutive relations, are discussed.We shall not comment in detail on any of the boundary conditions as these are ofstandard type and are well documented in previous chapters.

560 Coupled systems19.3.2 The format of the coupled equationsThe solution of coupled equations often involves non-linear behaviour, as notedpreviously in the cavitation problem. However, it is instructive to consider thelinear version of Eqs (19.73) and (19.79). This can be written as ( )M 0 ~u0 0 ~p‡ C 0Q T S( )_~u_~p‡ K ÿQ ~u0 H ~p (ˆÿ~ )f~q…19:81†Once again, like in the ¯uid±structure interaction problem, overall asymmetryoccurs despite the inherent symmetry of the M, C, K, S and H matrices. As thefree vibration problem is of no great interest here, we shall not discuss its symmetrization.In the transient solution algorithm we shall proceed in a similar manner tothat described in Sec. 19.2.6 and again symmetry will be observed.19.3.3 Transient step-by-step algorithmTime-stepping procedures can be derived in a manner analogous to that presented inSec. 19.2.6. Here we choose to use the GNpj algorithm of lowest order to approximateeach variable.Thus for ~u we shall use GN22, writing~u n ‡ 1 ˆ ~u n ‡ t _ ~u n ‡ 1 2 t2 ~u n ‡ 1 2 2t 2 ~u n ‡ 1 ~u p n ‡ 1 ‡ 1 2 2t 2 ~u n ‡ 1_~u n ‡ 1 ˆ _~u n ‡ t ~u n ‡ 1 t ~u n ‡ 1…19:82† _ ~u p n ‡ 1 ‡ 1t ~u n ‡ 1For the variables p that occur in ®rst-order form we shall use GN11, as~p n ‡ 1 ˆ ~p n ‡ t _ ~u n ‡ t _ ~u n ‡ 1 ~p p n ‡ 1 ‡ t_ ~p n ‡ 1…19:83†In the above ~u p n ‡ 1, etc., denote values that can be `predicted' from knownparameters at time t n and ~u n ‡ 1 ˆ ~u n ‡ 1 ÿ ~u n _ ~p n ‡ 1 ˆ _~p n ‡ 1 ÿ _ ~p n …19:84†are the unknowns.To complete the recurrence algorithm it is necessary to insert the above into thecoupled governing equations [(19.70) and (19.79)] written at time t n ‡ 1 . Thus werequire the following equalities…M ~u n ‡ 1 ‡ C ~u _ n ‡ 1 ‡ B T r 0 n ‡ 1 ÿ Q~p n ‡ 1 ‡ f n ‡ 1 ˆ 0…19:85†Q T _~u n ‡ 1 ‡ S ~p _ n ‡ 1 ‡ H~p n ‡ 1 ‡ q n ‡ 1 ˆ 0

in which r 0 n ‡ 1 is evaluated using the constitutive equation (19.69) in incremental formand knowledge of r 0 n asr 0 n ‡ 1 ˆ r 0 n ‡ De n ‡ 1 ˆ r 0 n ‡ DB~u n ‡ 1…19:86†In general the above system is non-linear and indeed on many occasions the Hmatrix itself may be dependent on the values of u due to permeability variationswith strain. Solution methods of such non-linear systems will be discussed inVolume 2; however, it is of interest to look at the linear form as the non-linearsystem usually solves a similar form iteratively.Here insertion of Eqs (19.82), (19.83) and (19.86) into (19.85) results in the equationsystem2364…M ‡ 1 tC ‡ 1 2 2t 2 K†ÿQÿQ T ÿ H ‡ 1t SSoil±pore ¯uid interaction (Class II problems) 561(75 ) ~u n ‡ 1 p _ ˆn ‡ 1F1F 2…19:87†where F 1 and F 2 are vectors that can be evaluated from loads and solution values at t n .Symmetry in the above is obtained by multiplying Eq. (19.37) by ÿ1 and de®ning p _ n ‡ 1 ˆ 1 t ~p _ n ‡ 1…19:88†The solution of Eq. (19.87) and the use of Eqs (19.82) and (19.83) complete therecurrence relation.The stability of the linear scheme can be found by following identical procedures tothose used in Sec. 19.2.6 and the result is 25 that stability is unconditional when 2 5 1 1 5 1 2 5 1 2…19:89†19.3.4 Special cases and robustness requirementsFrequently the compressibility of the ¯uid phase, which forms the matrix S, is suchthatS 0compared with other terms. Further, the permeability k may on occasion also be verysmall (as, say, in clays) andH 0leading to so-called `undrained' behaviour.Now the coe cient matrix in (19.87) becomes of the lagrangian constrained form(see Chapter 11), i.e. ( ) A ÿQ ~u n ‡ 1ÿQ T 0 p _ ˆF1…19:90†n ‡ 1F 2and is solvable only ifn u 5 n pwhere n u and n p denote the number of ~u and ~p parameters, respectively.

562 Coupled systemsupFig. 19.6 `Robust' interpolations for the coupled soil±¯uid problem.The problem is indeed identical to that encountered in incompressible behaviourand the interpolations used for the u and p variables have to satisfy identical criteria.As C 0 interpolation for both variables is necessary for the general case, suitableelement forms are shown in Fig. 19.6 and can be used with con®dence.The formulation can of course be used for steady-state solutions but it must beremarked that in such cases an uncoupling occurs as the seepage equation can besolved independently.Finally, it is worth remarking that the formulation also solves the well-known soilconsolidation problem where the phenomena are so slow that the dynamic term M ~utends to 0. However, no special modi®cations are necessary and the algorithm form isagain applicable.19.3.5 Examples ± soil liquefactionAs we have already mentioned, the most interesting applications of the coupled soil±¯uid behaviour is when non-linear soil properties are taken into account. Inparticular, it is a well-known fact that repeated straining of a granular, soil-like materialin the absence of the pore ¯uid results in a decrease of volume (densi®cation) due toparticle rearrangement. In Volume 2 we present constitutive equations which includethis e€ect and here we only represent a typical result which they can achieve whenused in a coupled soil±¯uid solution. When a pore ¯uid is present, densi®cation will(via the coupling terms) tend to increase the ¯uid pressures and hence reduce the soilstrength. This, as is well known, decreases with the compressive mean e€ective stress.It is not surprising therefore that under dynamic action the soil frequently loses allof its strength (i.e., lique®es) and behaves almost like a ¯uid, leading occasionally tocatastrophic failures of structural foundations in earthquakes. The reproduction ofsuch phenomena with computational models is not easy as a complete constitutive

Soil±pore ¯uid interaction (Class II problems) 563501010019573811765+C1260T2+789DBase with prescribedearthquake movement(a) Outline of centrifuge model.A, D, C are location of pressure transducersfor which comparisons are shown,T is displacement transducer+A13 1001 Sand mixture2 Concrete dyke3 Retaining wall4 Oil overflow35 Oil sea6 Coarse sand6 drains6 4Rigid boundaryModel dimensions in mm(b) Finite element mesh and final permanentdistorted shape (x 10 magnification)Excess porepressure (kPa)70Experiment00 0.04 0.08 0.12 0.16Excess porepressure (kPa)70Computation00 0.04 0.08 0.12 0.16SecondsExcess pore pressure at DSecondsExcess porepressure (kPa)7000 0.04 0.08 0.12 0.16Excess porepressure (kPa)7000 0.04 0.08 0.12 0.16SecondsExcess pore pressure at ASecondsExcess porepressure (kPa)7000 0.04 0.08 0.12 0.16Excess porepressure (kPa)7000 0.04 0.08 0.12 0.16SecondsExcess pore pressure at CSecondsDisplacement (mm)0Displacement (mm)–2.5 –2.50 0.04 0.08 0.12 0.160 0.04 0.08 0.12 0.16SecondsSecondsVertical displacement of the dyke(c) Comparison of excess pore pressuresat transducer locations D, A, C,and displacement at T (top of dyke)0 < time < 0.16 seconds,earthquake stops at t = 0.12 sFig. 19.7 Soil±pressure water interaction. Computation and centrifuge model results compared on aproblem of a dyke foundation subject to a simulated earthquake.0

564 Coupled systemsExcess porepressure (kPa)Excess porepressure (kPa)Excess porepressure (kPa)Displacement (mm)Fig. 19.7 Continued.ExperimentDevice type: 6 pressure transducerDevice number: 233870behaviour description for soils is imperfect. However, much e€ort devoted to thesubject has produced good results 35ÿ42 and a reasonable con®dence in predictionsachieved by comparison with experimental studies exists. One such study is illustratedin Fig. 19.7 where a comparison with tests carried out in a centrifuge is made. 41;42 Inparticular the close correlation between computed pressure and displacement withexperiments should be noted.Excess porepressure (kPa)0 00 0.5 1.0 1.5 2.0 2.50 0.5 1.0 1.5 2.0 2.5SecondsSecondsExcess pore pressure at DDevice type: 6 pressure transducerDevice number: 28517070Excess porepressure (kPa)0 00 0.5 1.0 1.5 2.0 2.5 0 0.5 1.0 1.5 2.0 2.5SecondsSecondsExcess pore pressure at ADevice type: 6 pressure transducerDevice number: 28487070Excess porepressure (kPa)0 00 0.5 1.0 1.5 2.0 2.5 0 0.5 1.0 1.5 2.0 2.5SecondsSecondsExcess pore pressure at CDevice type: 6 pressure transducerDevice number: 87300–2.50 0.5 1.0 1.5 2.0 2.5–2.50 0.5 1.0 1.5 2.0 2.5SecondsSecondsVertical displacement of the dyke(d) Comparison of excess pore pressuresat transducer locations D, A, C,and displacement at T (top of dyke)0 < time < 2.5 seconds,Note consolidation processDisplacement (mm)70Computation19.3.6 Biomechanics, oil recovery and other applicationsThe interaction between a porous medium and interstitial ¯uid is not con®ned to soils.The same equations describe, for instance, the biomechanics problem of bone±¯uidinteraction in vivo. Applications in this ®eld have been documented. 43;44

Partitioned single-phase systems ± implicit±explicit partitions (Class I problems) 565On occasion two (or more) ¯uids are present in the pores and here similar equationscan again be written 45;46 to describe the interaction. Problems of ground settlement inoil ®elds due to oil extraction, or ¯ow of water/oil mixtures in oil recovery are goodexamples of application of techniques described here.19.4 Partitioned single-phase systems ± implicit±explicitpartitions (Class I problems)In Fig. 19.1(b), describing problems coupled by an interface, we have alreadyindicated the possibility of a structure being partitioned into substructures andlinked along an interface only. Here the substructures will in general be of a similarkind but may di€er in the manner (or simply size) of discretization used in each oreven in the transient recurrence algorithms employed. In Chapter 13 we havedescribed special kinds of mixed formulations allowing the linking of domains inwhich, say, boundary-type approximations are used in one and standard ®niteelements in the other. We shall not return to this phase and will simply assume thatthe total system can be described using such procedures by a single set of equationsin time. Here we shall only consider a ®rst-order problem (but a similar approachcan be extended to the second-order dynamic system):C_a ‡ Ka ‡ f ˆ 0…19:91†which can be partitioned into two (or more) components, writing C 11 C 12 _a1‡ K 11 K 12 a1‡ f 1ˆ 0 …19:92†C 21 C 22 _a 2 K 21 K 22 a 2 f 2 0Now for various reasons it may be desirable to use in each partition a di€erenttime-step algorithm. Here we shall assume the same structure of the algorithm(SS11) and the same time step (t) but simply a di€erent parameter in each.Proceeding thus as in the other coupled analyses we writea 1 ˆ a 1n ‡ a 1…19:93†a 2 ˆ a 2n ‡ a 2Inserting the above into each of the partitions and using di€erent weight functions, weobtainC 11 a 1 ‡ C 12 a 2 ‡ K 11 …a 1n ‡ ta 1 †‡K 12 …a 2n ‡ ta 2 †‡ f 1 ˆ 0 …19:94†C 21 a 1 ‡ C 22 a 2 ‡ K 21 …a 1n ‡ ta 1 †‡K 22 …a 2n ‡ ta 2 †‡ f 2 ˆ 0 …19:95†This system may be solved in the usual manner for a 1 and a 2 and recurrencerelations obtained even if and di€er. The remaining details of the time-stepcalculations follow the obvious pattern but the question of coupling stability mustbe addressed. Details of such stability evaluation in this case are given elsewhere 47but the result is interesting.1. Unconditional stability of the whole system occurs if 5 1 2 5 1 2

566 Coupled systems2. Conditional stability requires thatt 4 t critwhere the t crit condition is that pertaining to each partitioned system consideredwithout its coupling terms.Indeed, similar results will be obtained for the second-order systemsMa ‡ C_a ‡ Ka ‡ f ˆ 0…19:96†partitioned in a similar manner with SS22 or GN22 used in each.The reader may well ask why di€erent schemes should be used in each partition ofthe domain. The answer in the case of implicit±implicit schemes may be simply thedesire to introduce di€erent degrees of algorithmic damping. However, much moreimportant is the use of implicit±explicit partitions. As we have shown in both`thermal' and dynamic-type problems the critical time step is inversely proportionalto h 2 and h (the element size), respectively. Clearly if a single explicit scheme wereto be used with very small elements (or very large material property di€erences)occurring in one partition, this time step may become too short for economy to bepreserved in its use. In such cases it may be advantageous to use an explicit scheme(with ˆ 0 in ®rst-order problems, 2 ˆ 0 in dynamics) for a part of the domainwith larger elements while maintaining unconditional stability with the same timestep in the partition in which elements are small or otherwise very `sti€'. For thisreason such implicit±explicit partitions are frequently used in practice.Indeed, with a lumped representation of matrices C or M such schemes are in e€ectstaggered as the explicit part can be advanced independently of the implicit part andimmediately provides the boundary values for the implicit partition. We shall returnto such staggered solutions in the next section.The use of explicit±implicit partitions was ®rst recorded in 1978. 48ÿ50 In the ®rstreference the process is given in an identical manner as presented here; in thesecond, a di€erent algorithm is given based on an element split (instead of the impliednodal split above) as described next.Implicit±explicit solution ± element partitionWe again consider the ®rst-order problem given in Eq. (19.91) and split asC I _a I ‡ C E _a E ‡ K I a I ‡ K E a E ‡ f ˆ 0…19:97†where the subscript I denotes an implicit partition and subscript E an explicit one.The recurrence relation for a is now written using GN11 asa … j†n ‡ 1 ˆ a n ‡…1 ÿ †t_a n ‡ t_a … j†n ‡ 1…19:98†witha …0†n ‡ 1 ˆ a n ‡…1 ÿ †t_a nThe approximations for the split are now taken asa I ˆ a … j†n ‡ 1a E ˆ a … j ÿ 1†n ‡ 1_a I ˆ _a E ˆ _a … j†n ‡ 1…19:99†

Staggered solution processes 567thus yielding the system of equations at iteration j as…C ‡ tK I †_a … j†n ‡ 1 ‡ F… j† ˆ 0…19:100†where F … j† contains the loading terms which depend on known values at t n andprevious iterate values …j ÿ 1†. The above algorithm has stability properties whichdepend on the choice of . For a linear system with 5 0:5 the implicit part isunconditionally stable and stability depends on the t crit of the explicit elements. 49;50Performing only one iteration in each time step is permitted; however improvedaccuracy in the explicit partition can occur if additional iterations are used, althoughthe cost of each time step is obviously increased.19.5 Staggered solution processes19.5.1 General remarksWe have observed in the previous section that in the nodal based implicit±explicitpartitioning of time stepping it was possible to proceed in a staggered fashion,achieving a complete solution of the explicit scheme independently of the implicitone and then using the results to progress with the implicit partition. It is temptingto examine the possibility of such staggered procedures generally even if each usesan independent algorithm.In such procedures the ®rst equation would be solved with some assumed (predicted)values for the variable of the other. Once the solution for the ®rst systemwas obtained its values could be substituted in the second system, again allowingits independent treatment. If such procedures can be made stable and reasonablyaccurate many possibilities are immediately open, for instance:1. Completely di€erent methodologies could be used in each part of the coupledsystem.2. Independently developed codes dealing e ciently with single systems could becombined.3. Parallel computation with its inherent advantages could be used.4. Finally, in systems of the same physics, e cient iterative solvers could easily bedeveloped.The problems of such staggered solutions have been frequently discussed 33;51ÿ54and on occasion unconditional stability could not be achieved without substantialmodi®cation. In the following we shall indicate some options available.19.5.2 Staggered process of solution in single-phase systemsWe shall look at this possibility ®rst, having already mentioned it as a special formarising naturally in the implicit±explicit processes of Sec. 19.4. We return here to

568 Coupled systemsconsider the problem of Eq. (19.91) and the partitioning given in Eq. (19.92). Further,for simplicity we shall assume a diagonal form of the C matrix, i.e., that the problem isposed asC 11 0 _a10 C 22 _a 2 ‡ K 11 K 12 a1K 21 K 22 a 2‡ f 1f 2ˆ 0 0…19:101†As we have already remarked, the use of ˆ 0 in the ®rst equation and 5 0:5 inthe second [see Eqs (19.94) and (19.95)] allowed the explicit part to be solved independentlyof the implicit. Now, however, we shall use the same in both equations but inthe ®rst of the approximations, analogous to Eq. (19.94), we shall insert a predictedvalue for the second variable:a 2 ˆ a p 2 ˆ a 2n…19:102†This is similar to the treatment of the explicit part in the element split of the implicit±explicit scheme and gives in place of Eq. (19.94)C 11 a 1 ‡ K 11 …a 1n ‡ ta 1 †ˆÿf 1 ÿ K 12 a 2n…19:103†allowing direct solution for a 1 .Following this step, the second equation can be solved for a 2 with the previousvalue of a 1 inserted, i.e.C 22 a 2 ‡ K 22 …a 2n ‡ ta 2 †ˆÿf 2 ÿ K 21 …a 1n ‡ ta 1 †…19:104†This scheme is unconditionally stable if 5 0:5, i.e., the total system is stableprovided each stagger is unconditionally stable. A similar condition holds forlinear second-order dynamic problems.Obviously, however, some accuracy will be lost as the approximation of Eq.(19.103) is that of the explicit form in a 2 . The approximation is consistent andhence convergence will occur.The advantage of using the staggered process in the above is clear as the equationsolving, even though not explicit, is now con®ned to the magnitude of each partitionand computational economy occurs.Futher, it is obvious that precisely the same procedures can be used for anynumber of partitions and that again the same stability conditions will apply.De®ne the arrays2C 11 C 22 .. .C ˆ64C ii. . .C kk375…19:105†

23 2K 11 0 0 0 K 12 K 1kK 21 K 22 . .0 0 . .. .. . ... . .K ˆ‡K ii0 . . .6. .4. 0 7 6.5 4.K k1 K k;k ÿ 1 K kk 0 0ˆ K L ‡ K Uand consider the partitionC_a ‡ K L a ‡ K U a p ‡ f ˆ 0Introducing now the approximationa i ˆ a in ‡ a iand using Eq. (19.102) gives the discrete formStaggered solution processes 569K k ÿ 1;k375…19:106†…19:107†…19:108†Ca ‡ K L … a n ‡ ta†‡K U a n ‡ f ˆ 0… C ‡ K L ta†‡Ka n ‡ …19:109†f ˆ 0In approximating the ®rst equation set it is necessary to use predicted values for a 2 ,a 3 , , a k , writing in place of Eq. (19.103),C 11 a 1 ‡ K 11 …a 1n ‡ ta 1 †‡K 12 a 2n ‡ K 13 a 3n ‡‡f 1 ˆ 0 …19:110†and continue similarly to (19.104), with the predicted values now continually beingreplaced by better approximations as the solution progresses.The partitioning of Eq. (19.105) can be continued until only a single equation set isobtained. Then at each step the equation that requires solving for a i is of the form… C ii ‡ tK ii †a i ˆ F i …19:111†where F i contains the e€ects of the load and all the previously computed a i . Forpartitions where each submatrix is a scalar Eq. (19.111) is a scalar equation andcomputation is thus fully explicit and yet preserves unconditional stability for 5 0:5. This type of partitioning and the derivation of an unconditionally stableexplicit scheme was ®rst proposed by Zienkiewicz et. al. 55 An alternative andsomewhat more limited scheme of a similar kind was given by Trujillo. 56Clearly the error in the approximation in the time step decreases as the solutionsweeps through the partitions and hence it is advisable to alter the sweep directionsduring the computation. For instance, in Fig. 19.8 we show quite reasonable accuracyfor a one-dimensional heat-conduction problem in which the explicit-split process wasused with alternating direction of sweeps. Of course the accuracy is much inferior tothat exhibited by a standard implicit scheme with the same time step, though theprocess could be used quite e€ectively as an iteration to obtain steady-statesolutions. Here many other options are also possible.

570 Coupled systemsT c0.80.60.40.2Implicit θ = 1∆t/∆t crit = 4Exact‘Explicit’–split∆t/∆t crit = 400 20 40 60 80 100 120 140t/∆t critC LT = 1T o = 0∆t crit = critical time step for standard explicit formT c = temperature on centre-lineFig. 19.8 Accuracy of an explicit-split procedure compared with a standard implicit process for heat conductionof a bar.It is, for instance, of interest to consider the system given in Eq. (19.105) asoriginating from a simple ®nite di€erence approximation to, say, a heat-conductionequation on the rectangular mesh of Fig. 19.9.Here it is well known that the so-called alternating direction implicit (ADI)scheme 57 presents an e cient solution for both transient and steady-state problems.It is fairly obvious that the scheme simply represents the procedure just outlined withpartitions representing lines of nodes such as …1; 5; 9; 13†, …2; 6; 10; 14†, etc., of Fig 19.9alternating with partitions …1; 2; 3; 4†, …5; 6; 7; 8†, etc.Obviously the bigger the partition, the more accurate the scheme becomes, thoughof course at the expense of computational costs. The concept of the staggered partitionclearly allows easy adoption of such procedures in the ®nite element context.Here irregular partitions arbitrarily chosen could be made but so far applications1 2 3 45 6 7 89 10 11 1213 14 15 161 2 3 45 6 7 89 10 11 1213 14 15 16Fig. 19.9 Partitions corresponding to the well-known ADI (alternating direction implicit) ®nite differencescheme.

Staggered solution processes 571have only been recorded in regular mesh subdivisions. 57 The ®eld of possibilities isobviously large. Use in parallel computation is obvious for such procedures.A further possibility which has many advantages is to use hierarchical variablesbased on, say, linear, quadratic and higher expansions and to consider each set ofthese variables as a partition. 58 Such procedures are particularly e cient in iterationif coupled with suitable preconditioning 59 and form a basis of multigrid procedures.19.5.3 Staggered schemes in ¯uid±structure systems andstabilization processesThe application of staggered solution methods in coupled problems representingdi€erent phenomena is more obvious, though, as it turns out, more di cult.For instance, let us consider the linear discrete ¯uid±structure equations withdamping omitted, written as [see Eqs (19.26) and (19.28)] M 0 uQ T ‡ K ÿQ u‡ f ˆ 0 …19:112†S p 0 H p q 0where we have omitted the tilde superscript for simplicity.For illustration purposes we shall use the GN22 type of approximation for bothvariables and write using Eq. (19.82)u n ‡ 1 ˆ u p n ‡ 1 ‡ 1 2 2t 2 u n ‡ 1_u n ‡ 1 ˆ _u p n ‡ 1 ‡ 1tu n ‡ 1p n ‡ 1 ˆ p p n ‡ 1 ‡ 1 2 2 t 2 p n ‡ 1…19:113†_p n ‡ 1 ˆ _p p n ‡ 1 ‡ 1 tp n ‡ 1which together with Eq. (19.112) written at t ˆ t n ‡ 1 completes the system ofequations requiring simultaneous solution for u n ‡ 1 and p n ‡ 1 .Now a staggered solution of a fairly obvious kind would be to write the ®rst set ofequations (19.112) corresponding to the structural behaviour with a predicted(approximate) value of p n ‡ 1 ˆ p p n ‡ 1, as this would allow an independent solutionfor u n ‡ 1 writingMu n ‡ 1 ‡ Ku n ‡ 1 ˆÿf ‡ Qp p n ‡ 1…19:114†This would then be followed by the solution of the ¯uid problem for p n ‡ 1 writingSp n ‡ 1 ‡ Hu n ‡ 1 ˆÿq ÿ Q T u n ‡ 1…19:115†This scheme turns out, however, to be only conditionally stable, 47 even if i and i are chosen so that unconditional stability of a simultaneous solution is achieved.(The stability limit is indeed the same as if a fully explicit scheme were chosen forthe ¯uid phase.)Various stabilization schemes can be used here. 25;47 One of these is given below. Inthis Eq. (19.114) is augmented toÿMu n ‡ 1 ‡ K ‡ QS ÿ1 Q T un ‡ 1 ˆÿf ‡ Qp p n ‡ 1 ‡ QSÿ1 Q T u p n ‡ 1…19:116†

572 Coupled systemsbefore solving for u n ‡ 1 . It turns out that this scheme is now unconditionally stableprovided the usual conditions 2 5 1 1 5 1 2are satis®ed.Such stabilization involves the inverse of S but again it should be noted that thisneeds to be obtained only for the coupling nodes on the interface. Another stablescheme involves a similar inversion of H and is useful as incompressible behaviouris automatically given.Similar stabilization processes have been applied with success to the soil±¯uidsystem. 60;61References1. O.C. Zienkiewicz. Coupled problems and their numerical solution. In Numerical Methodsin Coupled Systems (Eds R.W. Lewis, P. Bettis and E. Hinton), pp. 65±68, John Wiley andSons, Chichester, 1984.2. O.C. Zienkiewicz, E. OnÄ ate, and J.C. Heinrich. A general formulation for coupled thermal¯ow of metals using ®nite elements. Internat. J. Num. Meth. Eng., 17, 1497±514, 1980.3. B.A. Boley and J.H. Weiner. Theory of Thermal Stresses. John Wiley & Sons, New York,1960. Reprinted by R.E. Krieger Publishing Co., Malabar Florida, 1985.4. R.W. Lewis, P. Bettess, and E. Hinton, editors. Numerical Methods in Coupled Systems,Chichester, 1984. John Wiley & Sons.5. R.W. Lewis, E. Hinton, P. Bettess, and B.A. Schre¯er, editors. Numerical Methods inCoupled Systems, Chichester, 1987. John Wiley & Sons.6. J.C. Simo and T.J.R. Hughes. Computational Inelasticity, volume 7 of InterdisciplinaryApplied Mathematics. Springer-Verlag, Berlin, 1998.7. O.C. Zienkiewicz and R.E. Newton. Coupled vibration of a structure submerged in a compressible¯uid. In Proc. Int. Symp. on Finite Element Techniques, pp. 1±15, Stuttgart, 1969.8. P. Bettess and O.C. Zienkiewicz. Di€raction and refraction of surface waves using ®niteand in®nite elements. Internat. J. Num. Meth. Eng., 11, 1271±90, 1977.9. O.C. Zienkiewicz, D.W. Kelly, and P. Bettess. The Sommer®eld (radiation) condition onin®nite domains and its modelling in numerical procedures. In Proc. IRIA 3rd Int. Symp.on Computing Methods in Applied Science and Engineering, Versailles, December 1977.10. O.C. Zienkiewicz, P. Bettess, and D.W. Kelly. The ®nite element method for determining¯uid loadings on rigid structures. Two- and three-dimensional formulations. In O.C.Zienkiewicz, R.W. Lewis, and K.G. Stagg, editors, Numerical Methods in O€shoreEngineering, pp. 141±183. John Wiley & Sons, Chichester, 1978.11. O.C. Zienkiewicz and P. Bettess. Dynamic ¯uid-structure interaction. Numerical modelling ofthe coupled problem. In O.C. Zienkiewicz, R.W. Lewis, and K.G. Stagg, editors, NumericalMethods in O€shore Engineering, pp. 185±193. John Wiley & Sons, Chichester, 1978.12. O.C. Zienkiewicz and P. Bettess. Fluid-structure dynamic interaction and wave forces. Anintroduction to numerical treatment. Internat. J. Num. Meth. Eng., 13, 1±16, 1978.13. O.C. Zienkiewicz and P. Bettess. Fluid-structure dynamic interaction and some `uni®ed'approximation processes. In Proc. 5th Int. Symp. on Uni®cation of Finite Elements, FiniteDi€erences and Calculus of Variations, University of Connecticut, May 1980.14. R. Ohayon. Symmetric variational formulations for harmonic vibration problems couplingprimal and dual variables ± applications to ¯uid-structure coupled systems. La RecherecheAerospatiale, 3, 69±77, 1979.

15. R. Ohayon. True symmetric formulation of free vibrations for ¯uid-structure interaction inbounded media. In Numerical Methods in Coupled Systems, Chichester, 1984. John Wiley &Sons.16. R. Ohayon. Fluid±structure interaction. Proc. of the ECCM'99 Conference, IACM/ECCM'99, 31 August±3 September 1999, Munich, Germany.17. H. Morand and R. Ohayon. Fluid-Structure Interaction, Wiley, 1995.18. M.P. Paidoussis and P.P. Friedmann (eds). 4th International Symposium on Fluid±StructureInteractions, Aeroelasticity, Flow-Induced Vibration and Noise, vol. 1, 2, 3, ASME/Winter Annual Meeting, 16±21 November 1997, Dallas, Texas, AD-vol. 52-3.19. T. Kvamsdal et al. (eds). Computational Methods for Fluid±Structure Interaction, TapirPublishers, Trondheim, 1999.20. R. Ohayon and C.A. Felippa (eds). Computational Methods for Fluid±Structure Interactionand Coupled Problems. Comp. Meth. in Appl. Mech. Eng., special issue, to appear2000.21. M. Geradin, G. Roberts, and J. Huck. Eigenvalue analysis and transient response of ¯uidstructure interaction problems. Eng. Comp., 1, 152±60, 1984.22. G. Sandberg and P. Gorensson. A symmetric ®nite element formation of acoustic ¯uidstructureinteraction analysis. J. Sound Vib., 123, 507±15, 1988.23. K.K. Gupta. On a numerical solution of the supersonic panel ¯utter eigenproblem.Internat. J. Num. Meth. Eng., 10, 637±45, 1976.24. B.M. Irons. The role of part inversion in ¯uid-structure problems with mixed variables. JAIAA, 7, 568, 1970.25. W.J.T. Daniel. Modal methods in ®nite element ¯uid-structure eigenvalue problems.Internat. J. Num. Meth. Eng., 15, 1161±75, 1980.26. C.A. Felippa. Symmetrization of coupled eigenproblems by eigenvector augmentation.Commun. Appl. Num. Meth., 4, 561±63, 1988.27. J. Holbeche. Ph. D. Thesis. PhD thesis, University of Wales, Swansea, 1971.28. A.K. Chopra and S. Gupta. Hydrodynamic and foundation interaction e€ects in earthquakeresponse of a concrete gravity dam. J. Struct. Div. Am. Soc. Civ. Eng., 578, 1399±412, 1981.29. J.F. Hall and A.K. Chopra. Hydrodynamic e€ects in the dynamic response of concretegravity dams. Earthquake Eng. Struct. Dyn., 10, 333±95, 1982.30. O.C. Zienkiewicz and R.L. Taylor. Coupled problems ± a simple time-stepping procedure.Comm. Appl. Num. Meth., 1, 233±39, 1985.31. O.C. Zienkiewicz, R.W. Clough, and H.B. Seed. Earthquake analysis procedures forconcrete and earth dams ± state of the art. Technical Report Bulletin 32, Int. Commissionon Large Dams, Paris, 1986.32. R.E. Newton. Finite element study of shock induced cavitation. In ASCE Spring Convention,Portland, Oregon, 1980.33. O.C. Zienkiewicz, D.K. Paul, and E. Hinton. Cavitation in ¯uid-structure response (withparticular reference to dams under earthquake loading). Earthquake Eng. Struct. Dyn., 11,463±81, 1983.34. M.A. Biot. Theory of propagation of elastic waves in a ¯uid saturated porous medium, PartI: low frequency range; Part II: high frequency range. J. Acoust. Soc. Am., 28, 168±91, 1956.35. O.C. Zienkiewicz, C.T. Chang, and E. Hinton. Non-linear seismic responses and liquefaction.Internat. J. Num. Anal. Meth. Geomech., 2, 381±404, 1978.36. O.C. Zienkiewicz and T. Shiomi. Dynamic behaviour of saturated porous media, thegeneralized Biot formulation and its numerical solution. Internat. J. Num. Anal. Meth.Geomech., 8, 71±96, 1984.37. O.C. Zienkiewicz, K.H. Leung, and M. Pastor. Simple model for transient soil loading inearthquake analysis: Part I ± basic model and its application. Internat. J. Num. Anal. Meth.Geomech., 9, 453±76, 1985.References 573

574 Coupled systems38. O.C. Zienkiewicz, K.H. Leung, and M. Pastor. Simple model for transient soil loading inearthquake analysis: Part II ± non-associative models for sands. Internat. J. Num. Anal.Meth. Geomech., 9, 477±98, 1985.39. O.C. Zienkiewicz, A.H.C. Chan, M. Pastor, and T. Shiomi. Computational approach tosoil dynamics. In A.S. Czamak, editor, Soil Dynamics and Liquefaction, volume Developmentsin Geotechnial Engineering 42. Elsevier, Amsterdam, 1987.40. O.C. Zienkiewicz, A.H.C. Chan, M. Pastor, D.K. Paul, and T. Shiomi. Static and dynamicbehaviour of soils: A rational approach to quantitative solutions, I. Proc. Roy. Soc.London, 429, 285±309, 1990.41. O.C. Zienkiewicz, Y.M. Xie, B.A. Schre¯er, A. Ledesma, and N. Bicanic. Static anddynamic behaviour of soils: A rational approach to quantitative solutions, II. Proc. Roy.Soc. London, 429, 311±21, 1990.42. O.C. Zienkiewicz, A.H. Chan, M. Pastor, B.A. Schre¯er and T. Shiomi. ComputationalGeomechanics with Special Reference to Earthquake Engineering, John Wiley and Sons,Chichester, 1999.43. B.R. Simon, J. S-S. Wu, M.W. Carlton, L.E. Kazarian, E/P. France, J.H. Evans, and O.C.Zienkiewicz. Poroelastic dynamic structural models of rhesus spinal motion segments.Spine, 10(6), 494±507, 1985.44. B.R. Simon, J. S-S. Wu, and O.C. Zienkiewicz. Higher order mixed and Hermitian ®niteelement procedures for dynamic analysis of saturated porous media. Internat. J. Num.Meth. Eng., 10, 483±99, 1986.45. R.W. Lewis and B.A. Schre¯er. The Finite Element method in the Deformation andConsolidation of Porous Media. John Wiley & Sons, Chichester, 1987.46. X.K. Li, O.C. Zienkiewicz, and Y.M. Xie. A numerical model for immiscible two-phase¯uid ¯ow in porous media and its time domain solution. Internat. J. Num. Meth. Eng.,30, 1195±212, 1990.47. O.C. Zienkiewicz and A.H.C. Chan. Coupled problems and their numerical solution. InAdvanced in Computational Non-linear Mechanics, chapter 3, pp. 109±176. Springer-Verlag, Berlin, 1988.48. T. Belytschko and R. Mullen. Stability of explicit-implicit time domain solution. Internat.J. Num. Meth. Eng., 12, 1575±86, 1978.49. T.J.R. Hughes and W.K. Liu. Implicit-explicit ®nite elements in transient analyses. Part Iand Part II. J. Appl. Mech., 45, 371±78, 1978.50. T. Belytschko and T.J.R. Hughes, editors. Computational Methods for Transient Analysis.North-Holland, Amsterdam, 1983.51. C.A. Felippa and K.C. Park. Staggered transient analysis procedures for coupled mechanicalsystems: formulation. Comp. Meth. Appl. Mech. Eng., 24, 61±111, 1980.52. K.C. Park. Partitioned transient analysis procedures for coupled ®eld problems: stabilityanalysis. J. Appl. Mech., 47, 370±76, 1980.53. K.C. Park and C.A. Felippa. Partitioned transient analysis procedures for coupled ®eldproblems: accuracy analysis. J. Appl. Mech., 47, 919±26, 1980.54. O.C. Zienkiewicz, E. Hinton, K.H. Leung, and R.L. Taylor. Staggered time marchingschemes in dynamic soil analysis and selective explicit extrapolation algorithms. In R.Shaw et al., editors, Proc. Conf. on Innovative Numerical Analysis for the EngineeringSciences, University of Virginia Press, 1980.55. O.C. Zienkiewicz, C.T. Chang, and P. Bettess. Drained, undrained, consolidating dynamicbehaviour assumptions in soils. Geotechnique, 30, 385±95, 1980.56. D.M. Trujillo. An unconditionally stable explicit scheme of structural dynamics. Internat.J. Num. Meth. Eng., 11, 1579±92, 1977.57. L.J. Hayes. Implementation of ®nite element alternating-direction methods on nonrectangularregions. Internat. J. Num. Meth. Eng., 16, 35±49, 1980.

58. A.W. Craig and O.C. Zienkiewicz. A multigrid algorithm using a hierarchical ®nite elementbasis. In D.J. Pedolon and H. Holstein, editors, Multigrid Methods in Integral and Di€erentialEquations, pp. 310±312. Clarendon Press, Oxford, 1985.59. I. BabusÏka, A.W. Craig, J. Mandel, and J. PitkaÈranta. E cient preconditioning for the p-inversion ®nite element method in two dimensions. SIAM J. Num. Anal, 28, 624±61, 1991.60. K.C. Park. Stabilization of partitioned solution procedures for pore ¯uid-soil interactionanalysis. Internat. J. Num. Meth. Eng., 19, 1669±73, 1983.61. O.C. Zienkiewicz, D.K. Paul, and A.H.C. Chan. Unconditionally stable staggered solutionprocedures for soil-pore ¯uid interaction problems. Internat. J. Num. Meth. Eng., 26,1039±55, 1988.References 575

20Computer procedures for ®niteelement analysis20.1 IntroductionIn this chapter we consider some of the steps that are involved in the development of a®nite element computer program to carry out analyses for the theory presented inprevious chapters. The computer program discussed here may be used to solve anyone-, two-, or three-dimensional linear steady-state or transient problem. The programmay also be used to solve non-linear problems as will be discussed in Volume 2.Source listings are not included in the book but may be obtained at no charge fromthe publisher's internet web site (http://www.bh.com/companions/fem). Any errorsreported by readers will be corrected frequently so that up-to-date versions will beavailable.The program is an extension of the work presented in the 4th edition. 1;2 The versiondiscussed here is called FEAPpv to distinguish the current program from that presentedearlier. The program name is an acronym for Finite Element Analysis Program ±personal version. It is intended mainly for use in learning ®nite element programmingmethodologies and in solving small to moderate size problems on single processorcomputers. A simple memory management scheme is employed to permit e cientuse of main memory with limited need to read and write information to disk.The current version of FEAPpv permits both `batch'and `interactive'problemsolution. The ®nite element model of the problem is given as an input ®le and maybe prepared using any text editor capable of writing ASCII ®le output. A simplegraphics capability is also included to display the mesh and results from one- andtwo-dimensional models in either their undeformed or reference con®guration. Theavailable versions for graphics is limited to X-window applications and compilerscompatible with the current Compac Fortran 95 compiler for Windows based systems.Experienced programmers should be able to easily adapt the routines to other systems.Finite element programs can be separated into three basic parts:1. data input module and preprocessor2. solution module3. results moduleFigure 20.1 shows a simpli®ed schematic for a typical ®nite element programsystem. Each of the modules can in practice be very complex. In the subsequent

Introduction 577StartData Input Module(Preprocessor)Solution andOutput Module(Postprocessor)StopFig. 20.1 Simpli®ed schematic of ®nite element program.sections we shall discuss in some detail the programming aspects for each of themodules. It is assumed that the reader is familiar with the ®nite element principlespresented in this book, linear algebra, and programming in either Fortran or C.Readers who merely intend to use the program may ®nd information in this chapteruseful for understanding the solution process; however, for this purpose it is onlynecessary to read the user instructions available from the web site where the programis downloaded.This chapter is divided into seven sections. Section 20.2 describes the procedureadopted for data input, necessary to de®ne a ®nite element problem and basic instructionsfor data ®le preparation. The data to be provided consists of nodal quantities(e.g., coordinates, boundary condition data, loading, etc.) and element quantities(e.g., connection data, material properties, etc.).Section 20.3 describes the memory management routines.Section 20.4 discusses solution algorithms for various classes of ®nite elementanalyses. In order to have a computer program that can solve many types of ®niteelement problems a command language strategy is adopted. The command languageis associated with a set of compact subprograms, each designed to compute one orat most a few basic steps in a ®nite element process. Examples in the language arecommands to form a global sti€ness matrix, as well as commands to solve equations,display results, enter graphics mode, etc. The command language concept permitsinclusion of a wide range of solution algorithms useful in solving steady-state andtransient problems in either a linear or non-linear mode.In Section 20.5 we discuss a methodology commonly used to develop elementarrays. In particular, numerical integration is used to derive element `sti€ness',`mass'and `residual'(load) arrays for problems in linear heat transfer and elasticity.The concept of using basic shape function routines is exploited in these developments(Chapters 8 and 9).In Section 20.6 we summarize methods for solving the large set of linear algebraicequations resulting from the ®nite element formulation. The methods may bedivided into direct and iterative categories. In a direct solution a variant of Gaussian

578 Computer procedures for ®nite element analysiselimination is used to factor the problem coe cient matrix (e.g., sti€ness matrix)into the product of a lower triangular, diagonal and upper triangular form. A solution(or indeed subsequent resolutions) may then be easily obtained. A directsolution has the advantage that an a priori calculation may be made on thenumber of numerical operations which need to be performed to obtain a solution.On the other hand, a direct solution results in ®ll-in of the initial, sparse ®niteelement coe cient array ± this is especially signi®cant in three-dimensional solutionsand results in very large storage and compute times. In the second category iterativestrategies are used to systematically reduce a residual equation to zero, and thusyield an acceptable solution to the set of linear algebraic equations. The schemediscussed in this chapter is limited to solution of symmetric equations by a preconditionedconjugate gradient method.20.2 Data input moduleThe data input module shown in Fig. 20.1 must obtain su cient information topermit the de®nition and solution of each problem by the other modules. In theprogram discussed in this book the data input module is used to read the necessarygeometric, material, and loading data from a ®le or from information speci®ed bythe user using the computer keyboard or mouse. In the program a set of dynamicallydimensioned arrays is established which store nodal coordinates, element connectionlists, material properties, boundary condition indicators, prescribed nodal forces anddisplacements, etc. Table 20.1 lists the names of variables which are used in assigningarray sizes for mesh data and Table 20.2 indicates some of the main arrays used tostore mesh data.Table 20.1 Control parametersVariable name Description DefaultNUMNP Number of nodal points in mesh 0NUMEL Number of elements in mesh 0NUMMAT Number of material sets in mesh 0NDM Spatial dimension of mesh noneNDF Number of degrees of freedom per node (maximum) noneNEN Number of nodes per element (maximum) noneNDD Number of material property values per set 200Table 20.2 Variable names used for data storageVariable name (dimension) Type DescriptionID(NDF,NUMNP,2) Integer (1) Boundary codes; (2) Equation numbersIE(NIE,NUMMAT) Integer Element pointers for degrees of freedom, historypointers, material set type, etc.IX(NEN1,NUMEL) Integer Element connections, set ¯ag, etc.D(NDD,NUMMAT) Real Material property data setsF(NDF,NUMNP,2) Real (1) Nodal forces; (2) and displacementsX(NDM,NUMNP) Real Nodal coordinates

Data input module 579The notation used for the arrays often di€ers from that used in the text. Forexample, in the text it was found convenient to refer to nodal coordinates as x i , y i ,z i , whereas in the program these are called X(1,i), X(2,i), X(3,i), respectively.This change is made so that all arrays used in the program can be dynamicallyallocated. Thus, if a two-dimensional problem is analysed, space will not be reservedfor the X(3,i) coordinates. Similarly the nodal displacements in the text werecommonly named a i ; in the program these are called U(1,i), U(2,i), etc., wherethe ®rst subscript refers to the degrees of freedom at a node (from 1 to NDF).20.2.1 Control data and storage allocationThe allocation of the major arrays for storage of mesh and solution variables isperformed in a control program as indicated in Fig. 20.2. Since a dynamicmemory allocation is used it is not possible to establish absolute values for themaximum number of nodes, elements or material sets. The value for the parameterNUM_MR de®nes the amount of memory available to solve a given problem and isassigned to the main program module; however, if this is not su cient an errormessage is given and the program stops execution.To facilitate the allocation of all the arrays data de®ning the size of the problem isinput by the control program as shown schematically in Fig. 20.2. The required data isshown in Table 20.1; however, the number of nodes, elements and material sets maybe omitted and FEAPpv.f will use the subsequent input data to determine the actualsize required. Using the size data the remaining mesh storage requirements aredetermined and allocated by the control program.20.2.2 Element and coordinate dataAfter a user has determined the mesh layout for a problem solution the data must betransmitted to the analysis program. As an example consider the speci®cation of thenodal coordinate and element connection data for the simple two-dimensional (NDM =2) rectangular region shown in Fig. 20.3, where a mesh of nine four-node rectangularelements (NUMEL = 9 and NEN =4) and 16 nodes (NUMNP = 16) has been indicated. Todescribe the nodal and element data, values must be assigned to each X(i,j) fori ˆ 1; 2 and j ˆ 1 to 16 and to each IX(k,n) for k ˆ 1 to 4 and n ˆ 1 to 9. In thede®nition of the coordinate array X, the subscript i indicates the coordinate directionand the subscript j the node number. Thus, the value of X(1,3) is the x coordinate fornode 3 and the value of X(2,3) is the y coordinate for node 3. Similarly for theelement connection array IX the subscript k is the local node number of the elementand n is the element number. The value of any IX(k,n) (for k less than or equal toNEN) is the number of a global node. Values of k larger than NEN are used to storeother data. The convention for the ®rst local node number is somewhat arbitrary.The local node number 1 for element 3 in Fig. 20.3 could be associated with globalnode 3, 4, 7, or 8. Once the ®rst local node is established the others must followaccording to the convention adopted for each particular element type. For example,

580 Computer procedures for ®nite element analysisStart1CCCC = FEAPT100FCC = MACRT200FData Input andEquation Profile Module100CC = STOP1FTStopSet PointersSolution andOutput ModulePMESH200PROFILPMACR11Fig. 20.2 Control program ¯ow chart.it is conventional to number the connections by a right-hand rule and the four-nodedquadrilateral element can be numbered according to that shown in Fig. 20.4. If weconsider once again element 3 from the mesh in Fig. 20.3 we have four possibilitiesfor specifying the IX(k,3) array as shown in Fig. 20.4. The computation of theelement arrays from any of the above descriptions must produce the same coe cientsfor the global arrays and is known as element invariance to data input.Data input modulesIn FEAPpv two subprograms PINPUT and TINPUT are available to perform datainput operations. For example, all the nodal coordinates may be input using thesubprogram

Data input module 581y13 14 15 167 8 99 10 11 123 x 3=94 5 65 6 7 81 2 31 2 3 43 x 3=9xNI= element number= global node numberFig. 20.3 Simple two-dimensional mesh.KL43N1I = IX (1, N)2JiIN= local node number= global node number= element numberOptionnumberabcdLocal node number1 2 3 43 4 8 74 8 7 38 7 3 47 3 4 8Fig. 20.4 Typical four-noded element and numbering options.

582 Computer procedures for ®nite element analysisSUBROUTINE XDATA(X,NDM,NUMNP)IMPLICIT NONELOGICAL ERRCK, PINPUTINTEGER NUMNP, NDM , NREAL*8 X(NDM,NUMNP)DO N = 1,NUMNPERRCK = PINPUT(X(1,N),NDM}IF(ERRCK) THENSTOP ` Coordinate error: Node:',NENDIFEND DO ! NENDThe above use of the PINPUT routine obtains NDM values from each record and assignsthem to the coordinate components of node N. The data input routines obtain theirinformation from the current input ®le speci®ed by a user. The routines are alsoused in cases where input is to be provided from the keyboard. These input all datain character mode, and parse the data for embedded function names or parameters(use of functions and parameters is described in the user manual). Users who areextending the capability of the program are encouraged to use the routines toavoid possible errors. The subprogram TINPUT permits character data to precedenumerical values use is given asERRCK = TINPUT(TEXT,M,DATA,N)in which TEXT is a CHARACTER*15 array of size M and DATA is a REAL*8 array of size N.For cases where integer information is to be input the information must be moved.For example, a simple input routine for the IX data isSUBROUTINE IXDATA(IX,NEN1,NUMEL)IMPLICIT NONELOGICAL ERRCK, PINPUTINTEGER NUMEL, NEN1 , N, IINTEGER IX(NEN1,NUMEL)REAL*8 RIX(16)DO N = 1,NUMELERRCK = PINPUT(RIX,NEN1}IF(ERRCK) THEN! Stop onerrorSTOP ` Connection error: ELEMENT:',NELSE! Move data to IXDO I = 1,NEN1IX(I,N) = NINT(RIX(I))END DO ! IENDIFEND DO ! NEND

Data input module 583While the above form is not optimal it is an expedient method to permit the arbitrarymixing of real and integer data on the same record. In the above two examples thenode and element numbers are associated with the record number read. The formused in the routines supplied with FEAPpv include the node and element numbersas part of the data record. In this form the inputs need not be sequential nor alldata input at one instance.For a very large problem the preparation of each node and element record for themesh data would be very tedious; consequently, some methods are provided inFEAPpv to generate missing data. These include simple interpolation between missingnumbers of nodes or elements, use of super-elements to perform generation of blocksof nodes and elements, and use of blending function methods. Even with these aidsthe preparation of the mesh data for nodes and coordinates can be time consumingand users should consider the use of mesh generation programs such as GiD 3 toassist in this task. Generally, however, the data input scheme included in the programis su cient to solve academic and test examples. Moreover the organization of themesh input module (subprogram PMESH) is data driven and permits users to interfacetheir own program directly if desired (see below for more information on addingfeatures). The data-driven format of the mesh input routine is controlled by keywordswhich direct the program to the speci®c segment of code to be used. In this form eachinput segment does not interact with any of the others as shown schematically in the¯ow chart in Fig. 20.5.20.2.3 Material property speci®cation ± multiple elementroutinesThe above discussion considered the data arrays for nodal coordinates and elementconnections. It is also necessary to specify the material properties associated witheach element, loadings, and the restraints to be applied to each node.Each element has associated property sets, for example in linear isotropic elasticmaterials Young's modulus E and Poisson's ratio describe the material parametersfor an isotropic state. In most situations several elements have the same propertysets and it is unnecessary to specify properties for each element individually. Inthe data structure used in FEAPpv an element is associated with a material set bya number on the data record for each element. The material properties are thengiven once for each number. For example, if the region shown in Fig. 20.3 is allthe same material, only one material set is required and each element would referencethis set. To accommodate the storage of the material set numbers the IXarray is increased in size to NEN1 entries and the material set number is stored inthe entry IX(NEN1,n) for element n. InFEAPpv the material properties are storedin the array D(NDD,NUMMAT), where NUMMAT is the number of di€erent materialsets and NDD is the number of allowable properties for each material set (the defaultfor NDD is 200).Each material set de®nes the element type to which the properties are to be assigned.In realistic engineering problems several element types may be needed to de®ne theproblem to be solved. A simple example involving di€erent element types is shown in

584 Computer procedures for ®nite element analysisStart10CC50n1, NWD10IF [CC.EQ.WD (n)]T100100501 2 NWDRead NodalCoord DataRead ElementDataReturnIF (NPR)TT10 IF (NPR)10FOutputNodal CoordOutput ElementConnections10 10Fig. 20.5 Flow chart for mesh data input.Fig. 1.4(a) in Chapter 1 where elements 1, 2, 4, and 5 are plane stress elastic elementsand element 3 is a truss element. In this case at least two di€erent types of elementsti€ness formulations must be computed. In FEAPpv it is possible to use ten di€erentuser provided element formulations in any analysis.y The program has been designed sothat all computations associated with each individual element are performed in oneelement subprogram called ELMTnn, wherenn is between 01 and 10 (see Sec. 20.5.3for a discussion on the organization of ELMTnn). Each element type to be used isspeci®ed as part of the material set data. Thus if element type 1, e.g., computationsperformed by ELMT01, is a plane linear elastic three- or four-noded element and elementtype 4 is a truss element, the data given for example Fig. 1.4(a) would be:y In addition, some standard element formulations are provided as described in the user instructions.

Data input module 585(a) Material propertiesMaterial Element Material propertyset number type data1 4 E 1 , A 12 1 E 2 , 2(b) Element connectionsElement Material set Connection1 2 1 3 42 2 1 4 23 1 2 54 2 36745 2 4785where E is Young's modulus, is Poisson's ratio and A is area. Thus, elements 1, 2, 4,and 5 have material property set 2 which is associated with element type 1 and element3 has a material property set 1 which is associated with element type 4. It will be seenlater that the above scheme leads to a simple organization of an element routine whichcan input material property sets and perform all the necessary computations for the®nite element arrays.More sophisticated schemes could be adopted; however, for the educational andresearch type of program described here this added complexity is not warranted.20.2.4 Boundary conditions ± equation numbersThe process of specifying the boundary conditions at nodes and the procedure forimposing speci®ed nodal displacements is closely associated with the method adoptedto store the global solution matrix, e.g., the sti€ness matrix. In FEAPpv the directsolution procedure included uses a variable band (pro®le) storage for the globalsolution matrix. Accordingly, only those coe cients within the non-zero pro®lesare stored.While the nodal displacements associated with boundary restraints may beimposed using the `penalty'method described in Chapter 1, a more e cient directsolution results if the rows and columns for these equations are deleted. As anexample consider the sti€ness matrix corresponding to the problem shown inFig. 1.1; storing all terms within the upper pro®le leads to the result shown inFig. 20.6(a) and requires 54 words, whereas if the equations corresponding to therestrained nodes 1 and 6 are deleted the pro®le shown in Fig. 20.6(b) results andrequires only 32 words. In addition to a reduction in storage requirements, thecomputer time to solve the equations is also reduced.To facilitate a compact storage operation in forming the global arrays, a boundarycondition array is used for each node. The array is named ID and is dimensioned asshown in Table 20.2. During input of data, degrees of freedom with known value orwhere no unknown exists have a non-zero value assigned to ID(i,j,1). All active

586 Computer procedures for ®nite element analysis1 2 3 4 5 6ProfileNodes123Profile storage = 54 words(a)4562 3 4 5Nodes23Profile storage = 32 words(b)45Fig. 20.6 Stiffness matrix: (a)total stiffness storage; (b)storage after deletion of boundary conditions.degrees of freedom have a zero value in the ID array. After the input phase the valuesin ID(i,j,2) are assigned values of the active equation numbers. Restrained DOFshave zero (or negative) values.Table 20.3 shows the ID values for the example shown in Fig. 1.1(a), where it isevident that nodes 1 and 6 are fully restrained.The numbers for the equations associated with unknowns are constructed fromTable 20.3 by replacing each non-zero value with a zero and each zero value bythe appropriate equation number. In FEAPpv this is performed by subprogramPROFIL starting with the degrees of freedom associated with node 1 followed bynode 2, etc. The result for the example leads to values shown in Table 20.4, andthis information is stored in ID(i,j,2). This information is used to assemble allthe global arrays.Table 20.3 Boundary restraint code valuesafter data input of problem in Fig. 1.1Degree of freedomNode 1 21 1 12 0 03 0 04 0 05 0 06 1 1

Data input module 587Table 20.4 Compacted equation numbersfor problem in Fig. 1.1Degree of freedomNode 1 21 0 02 1 23 3 44 5 65 7 86 0 0The above scheme may be modi®ed in a number of ways for either e ciency or toaccommodate more general problems. For problems in which the node numbers areinput in an order which creates a very large pro®le it is advisable to employ a programto renumber the nodes for better e ciency (often called bandwidth minimizationschemes). Using the renumbered node order the equation numbers may then beconstructed.The solution of mixed formulations which have matrices with zero diagonalsrequires special care in solving for the parameters. For example in the q, formulationdiscussed in Sec. 11.2 it is necessary to eliminate all ~q i parameters associatedwith each ~ i parameter when a direct method of solution without pivoting is used(e.g., those discussed in Sec. 20.6.1). This may be achieved by numbering theID(i,j,2) entries so that ~q i have smaller equation numbers than the one for theassociated ~ i .The equation number scheme may be further exploited to handle repeating boundaries(see Chapter 9, Sec. 9.18) where nodes on two boundaries are required to havethe same displacement but the value is unknown. This is accomplished by setting theequation numbers to the same value (and discard the unused ones). Similarly, regionsmay be joined by assigning nodes with the same coordinate values the same equationnumbers.All modi®cations of the above type must be performed prior to computing thepro®le of the global matrix.20.2.5 Loading ± nodal forces and displacementsIn FEAPpv the speci®ed nodal forces and displacements associated with each degreeof freedom are stored in the array F(NDF,NUMNP,2). The speci®ed force values fordegree of freedom i at node j are retained in F(i,j,1) and speci®ed values for thecorresponding speci®ed displacements in F(i,j,2). The actual value to be usedduring each phase of an analysis depends on the current value stored inID(i,j,1). Thus if the value of the ID(i,j,1) is zero a force value is taken fromF(i,j,1) whereas if the value is non-zero a displacement value is taken fromF(i,j,2). For the example of Fig. 1.1, an 0.01 settlement of the node 1 can beinput by setting F(1,2,2) = -0.01, where it is assumed that the second degree of

588 Computer procedures for ®nite element analysisfreedom is a displacement in the vertical direction. Similarly, a horizontal force atnode 4 can be speci®ed by setting F(1,4,1) = 5, (i.e., X 4 in the ®gure).In many problems the loading may be distributed and in these cases the loadingmust ®rst be converted to nodal forces. In FEAPpv there are some provisions includedto perform the computation automatically. Users may develop additional schemes fortheir own problems and add a new input command in the subprogram PMESH. Otheroptions could also be added to compute necessary nodal quantities.The necessary steps to add a feature in PMESH are:1. Increase the dimensioned size of the array WD which is a character array to store thecommand names.2. Set the value of LIST in the DATA statement to the new number of entries in WD.3. Add a new statement label entries to the GO TO statement.4. For each statement label entry add the program statements for the new feature.The speci®c instructions to prepare data for FEAPpv are contained in the usermanual available at the publisher's web site.20.2.6 Mesh data checkingOnce all the data for the geometric, material and loading conditions are suppliedFEAPpv is ready to initiate execution of the solution module; however prior to thisstep it is usually preferable to perform some checks on the input data (and anygenerated values).After the mesh is input the program will pass to solution mode. During solutionadditional arrays may be required which can also exceed the available space in theblank common. The most intensive storage requirement is for the global coe cientmatrix for the set of linear algebraic equations de®ning the nodal solution parameters.In direct solution mode a variable band, pro®le solution scheme is used for simplicity.The solver has the capability of solving both symmetric and unsymmetric coe cientarrays and this is generally adequate for one- and two-dimensional problems ofmoderate size. However, for three-dimensional applications the storage demandsfor the coe cient matrix can exceed the capabilities of even the largest computersavailable at the time of writing this volume. Thus, an alternative iterative scheme isincluded in FEAPpv using a simple preconditioned conjugate gradient solver.20.3 Memory management for array storageA single array is partitioned to store all the main data arrays, as well as other arraysneeded during the solution and output phases. This is accomplished using a datamanagement system which can de®ne, resize or destroy an integer or real array. Dependingon the computer system used real arrays may be de®ned in the main programmodule FEAPpv.F in either single precision or double precision form. Using the datamanagement system each array indicated in Table 20.2 is dynamically dimensioned tothe size and precision required for each problem. The result is a set of pointers de®ning

Memory management for array storage 589the location in a single array located in blank common. Blank common is de®ned asREAL*8 HRINTEGER MRCOMMON HR(1),MR(NUM_MR)and pointers are assigned into the array NP stored in the named common POINTERSgiven byINTEGERNPCOMMON /POINTERS/ NP(NUM_NP)The size of each array is de®ned by parameters NUM_MR and NUM_NP. While notstrictly de®ned by programming standards the above size for HR is not limited to 1.By working outside the array bound real arrays may be de®ned up to size NUM_MR/2for the double precision indicated. Using this arti®ce of pointers subroutines may becalled asCALL SUBX(MR(NP(5)), HR(NP(33)), ... )where the ®rst argument is integer and the second real. The subroutine would thenreadSUBROUTINE SUBX(I1, R1, .... )and real names associated with each array as determined by a programmer. At thisstage the missing ingredient is assignment of values to each speci®c pointer. InFEAPpv this is accomplished by the subprogram PALLOC. This logical functionsubprogram associates a number with a name for each variable to be de®ned,changed or deleted. Each programmer must use a listing of this routine to understandwhich variable is being de®ned and whether the variable is to be real or integer. Aspeci®cation of an array action is accomplished using the assignment statementSETVAL = PALLOC{ NUM , NAME , LENGTH , PRECISION )For example the statementSETVAL = PALLOC{ 43 , `X' , NDM*NUMNP , 2 )de®nes the real array for the nodal coordinates to have a size as indicated in Table20.2. Similarly, the statementSETVAL = PALLOC{ 33 , `IX' , NEN1*NUMEL , 1 )de®nes an integer array for the element connection array. Repeating the use of theallocation statement with a di€erent size (either larger or smaller) will rede®nethe size of the array. Similarly, use of the statement with a zero (0) size deletes thearray from the allocation table. Accordingly, use ofSETVAL = PALLOC{ 33 , `IX' , 0 , 1 )would destroy the storage (and values) for the connection data. Thus, using thememory management scheme above it is possible to rede®ne a mesh in an adaptivesolution scheme to add or delete speci®c element data. Alternatively, data may beused in a temporary manner by allocating and then deleting after use.

590 Computer procedures for ®nite element analysis20.4 Solution module ± the command programminglanguageAt the completion of data input and any checks on the mesh we are prepared toinitiate a problem solution. It is at this stage that the particular type of solutionmode must be available to the user. In many existing programs only a smallnumber of solution modes are generally included. For example, the program mayonly be able to solve linear steady-state problems, or in addition it may be able tosolve linear transient problems for a single method. In a research mode or indeedin practical engineering problems ®xed algorithm programs are often too restrictiveand continual modi®cation of the program is necessary to solve speci®c problemsthat arise ± often at the expense of features needed by another user. For thisreason it is desirable to have a program that has modules for various algorithmcapabilities and, if necessary, can be modi®ed without a€ecting other users'capabilities. The program form that we discuss here is basic and the reader canundoubtedly ®nd many ways to improve and extend the capabilities to be able tosolve other classes of problems.The command language concept described in this section has been used bythe authors for more than 20 years and, to date, has not inhibited our researchactivities by becoming outdated. Applications are routinely conducted on personalcomputers and workstations using an identical program except for graphical displaymodules.20.4.1 Linear steady-state problemsA basic aspect of the variable algorithm program FEAPpv is a command instructionlanguage which is associated with speci®c program solution modules for speci®calgorithms as needed. A user needs only to understand the association betweenspeci®c commands and the operations carried out by the associated solutionmodules.In a steady-state problem we are required to solve the problem given, for example,byr …k† ˆ f ÿ Ka …k†…20:1†where k is an index related to the solution iteration number. We call r …k† the residual ofthe problem for iteration k and note that a solution results when it is zero. In a datadrivensolution mode using the command language of FEAPpv the formulation ofEq. (20.1) is given by the command FORM, which is a mnemonic for form residual.In addition an incremental form of the solution of Eq. (20.1) is adopted inFEAPpv. Accordingly we letand solve the problema …k ‡ 1† ˆ a …k† ‡ a …k†r …k ‡ 1† ˆ r …k† ÿ Ka …k† ˆ 0…20:2†…20:3†

Since the problem given by Eq. (20.1) is linear this iterative form must converge in oneiteration. That is, if we solve the problem for k ˆ 0 for any speci®ed a …0† , the residualfor k ˆ 1 will be zero (to machine precision). The only exceptions to this will be: (a)animproperly formulated or implemented ®nite element formulation for the sti€nessand/or the residual; (b) an incorrect setting of the necessary boundary conditionsto avoid singularity of the resulting sti€ness matrix; or (c) the problem is so illposedthat round-o€ in computer arithmetic leads to signi®cant error in the resultingsolution.In FEAPpv the command language statement to form a symmetric sti€ness matrixis TANG, which is a mnemonic for tangent sti€ness. An unsymmetric sti€ness matrixcan be formed by specifying the command UTAN. By now the reader should haveobserved that commands for FEAPpv are given by four-character mnemonics. Ingeneral, users can use up to 14 characters to issue any command, however, onlythe ®rst four are interpreted by the program. Thus, if a user desires, the commandto form the tangent may be given as TANGENT. Finally, to solve the systems ofequations given by Eq. (20.3) the command SOLV is used. Thus to solve a steadystateproblem the three commands issued are:TANGentFORMSOLVeThe ®rst two commands can be reversed without a€ecting the algorithm.The basic structure for all command language statements is:COMMAND OPTION VALUE_1 VALUE_2 VALUE_3Since the above three statements occur so often in any ®nite element solution strategya shorthand command option is provided in FEAPpv asTANGent,,1where a comma is used to separate the ®elds and leave a blank option parameter. Anypositive non-zero number may be used for the VALUE_1 parameter.A user can check that the solution is correct by including another FORM commandafter the SOLV statement.After a solution has been performed for the steady-state problem it is necessary toissue additional commands in order to obtain the solution results. For example, thecommandsDISPlacement ALLSTREss ALLSolution module ± the command programming language 591will output all the nodal displacements and stresses in an output ®le speci®ed at theinitiation of running FEAPpv. Table 20.5 lists some of the commands available inthe program. A complete list is available in the user manual.The variable algorithm program described by a command language program canoften be extended as necessary without need to reprogram the modules. Additionaloptions are described in the user manual.

592 Computer procedures for ®nite element analysisTable 20.5 Partial list of solutions commandsCommand Option Value_1 Value_2 Value_3 DescriptionCHECkPerform check of mesh(ISW = 2) 1DISP ALL N1 N2 N3 Output displacement for nodesN1 to N2 at increments of N3ALL outputs allDT V1 Set time increment to V1FORMForm equation residual(ISW = 6)LOOP N Loop N times all instructionsto a matching NEXT commandMESHInput changes to meshNEXTEnd of LOOP instructionPLOT OPTION Enter graphical modeor perform command OPTIONREAC ALL N1 N2 N3 Output reactions at nodesN1 to N2 at increments of N3ALL outputs all(ISW = 6)SOLVSolve for new solutionincrement (after FORM)STRE ALL N1 N2 N3 Output element variablesN1 to N2 at increments of N3ALL outputs all(ISW = 4)TANGent N1 Form symmetric tangentSolve if N1 positive(ISW = 3)TIMEAdvance time by DT valueTOL V1 Set solution tolerance to V1UTAN N1 Form unsymmetric tangent(ISW = 3)20.4.2 Transient solution methodsThe integration of second-order di€erential equations of motion for time-dependentstructural systems can be treated using the command language program. The ®rstorderdi€erential equations resulting from the heat equation may also be similarlyintegrated. For the transient second-order case the residual equation is modi®ed tor …k† ˆ f ÿ Ka …k† ÿ C_a …k† ÿ Ma …k†…20:4†where C and M are damping and mass matrices, respectively, and _a and a are velocityand acceleration, respectively. To solve this problem it is necessary to:1. specify the time integration method to be used (see Chapter 18);2. specify the time increment for the integration;3. specify the number of time steps to perform;4. form the residual r …k† ;5. form the tangent matrix for the speci®c time integration method;6. solve the equation for each time step;7. report answers as needed.

Solution module ± the command programming language 593As an example we consider the Newmark method (GN22) as described in Chapter18, Sec. 18.33. Using Eq. (18.12) we can de®ne the updates at iteration k asa …k†n ‡ 1 ˆ a n ‡ 1 ‡ 1 2 2t 2 a …k†n ‡ 1…20:5†_a …k†n ‡ 1 ˆ _a n ‡ 1 ‡ 1 ta …k†n ‡ 1…20:6†where a n ‡ 1 and _a n ‡ 1 are expressed in terms of solution variables at time n. Theseequations may also be written in an incremental form as…k ‡ 1†an ‡ 1ˆ a …k†n ‡ 1 ‡ 1 2 2t 2 a …k†n ‡ 1…20:7†…k ‡ 1†_an ‡ 1ˆ _a …k†n ‡ 1 ‡ 1ta …k†n ‡ 1…20:8†Comparing Eq. (20.7) with Eq. (20.3) we obtaina …k†n ‡ 1 ˆ 12 2t 2 a …k†n ‡ 1…20:9†Similarly_a …k†n ‡ 1 ˆ 1ta …k†n ‡ 1…20:10†Thus, selecting the incremental nodal displacements as the primary unknown, theresidual equation for k ‡ 1 may be written asr …k ‡ 1† ˆ r …k† ÿ K a …k†n ‡ 1…20:11†whereK ˆ c 1 K ‡ c 2 C ‡ c 3 M…20:12†withc 1 ˆ 1c 2 ˆ 2 1 2 tc 3 ˆ 2 2 t 2…20:13†obtained from the relations between the incremental displacement, velocity andacceleration vectors. As we have noted in Chapter 18 the changing of the primaryunknown from displacement to acceleration or velocity or, indeed, changing theintegration algorithm from Newmark to any other method only changes the residualequation by the parameters c i which de®ne the tangent matrix K . The other changesfrom di€erent integration algorithms appear in the number of vectors required for thealgorithm and the way they are initialized and updated within each time increment.In program FEAPpv the parameters c i are passed to each element routine asCTAN(i) together with the values of the localized nodal displacement, velocity andacceleration vectors. This permits an element module to be programmed in a generalmanner without knowing which integration method will be used during the solutionspeci®ed in the command language instructions. In Sec. 20.5 we will discuss the stepsneeded to program the residual terms, as well as the sti€ness and mass terms needed toform the global tangent matrix.

594 Computer procedures for ®nite element analysisHere we note also that the steady-state algorithm discussed in the previous sectionmerely requires that the velocity and acceleration vectors and the parameters c 2 and c 3be set to zero before calling an element module. Similarly, for a ®rst-order system theacceleration vector and parameter c 3 are set to zero prior to entering the elementmodule.The command language instructions to solve a linear transient problem over 50time steps in which all results are reported at each time is given asTRANS,NEWMark ! Selects Newmark MethodDT,,0.024 ! Sets time increment to 0.024TANG! Form tangent matrixLOOP,time,50 ! Loop 50 times to NEXTFORM! Form residualSOLVe ! Solve equationsDISP,ALL ! Output nodal displacementsSTRE,ALL ! Output element variablesNEXT,time ! End of LOOPThe issuing of the instructions TRANsient causes the parameters c i to be set for theNewmark method. The default for the transient option is the steady-state solutionalgorithm with c 1 ˆ 1 and c 2 ˆ c 3 ˆ 0.20.4.3 Non-linear solutions: Newton's methodsThe command language programming instructions may also be used to solve nonlinearproblems. For example, the steady-state set of non-linear algebraic equationsgiven by the residual equationr …k† ˆ f ÿ P…a …k† †…20:14†in which P is a non-linear function of a is considered. A solution may be obtained bywriting a linear approximation for the residual at k ‡ 1asr …k ‡ 1† r …k† ÿ K …k†T a…k† ˆ 0…20:15†in which K T is some non-singular coe cient matrix used to obtain the increments…k ‡ 1†a …k† . Now the update for a …k ‡ 1† using Eq. (20.2) will not in general make rzero in one iteration.A common method to generate the coe cient matrix is Newton's method whereK …k†T ˆ @P@a j a ˆ a …20:16†…k†When properly implemented the norm of the residual should converge at a quadraticasymptotic rate. Thus if jjrjj is the norm of the residual then for an approximationclose to the solution the ratios for two successive iterations should bejjr …k† jjjjr …0† jj ˆ C 110 ÿq ;jjr …k ‡ 1† jjjjr …0† jjˆ C 2 10 ÿ2q…20:17†

Solution module ± the command programming language 595In general, this is the best one can obtain with the type of algorithm given by Eq.(20.15).In FEAPpv a norm of the solution is computed for each iteration and a check of thecurrent norm versus the initial value is performed as indicated in Eq. (20.17). Once thevalue of the ratio of the norm is below a speci®ed tolerance, convergence is assumed.The solution tolerance is set using the command language instruction TOL as indicatedin Table 20.5 (the default value for the norm is 10 ÿ12 ). The instructions to perform asolution using the algorithm indicated in Eq. (20.15) is given byLOOP,iteration,10TANG,,1NEXT,iteration! Perform a maximum of 10 iterations! Compute tangent, residual and solve! End for LOOP instructionOnce the ratio of the norms is reached, FEAPpv will exit the iteration loop andexecute the instruction following the NEXT statement. If the element module usedhas a tangent matrix computed using Eq. (20.16) the asymptotic behaviour ofNewton's method should be attained. Failure to achieve a quadratic rate ofconvergence during the last few iterations indicates an incorrect implementation inthe element module, a data input error, or extreme sensitivity in the formulationsuch that round-o€ prevents the asymptotic rate being reached. One can neverachieve convergence beyond that where the round-o€ limit is reached.An alternative to the above program is the modi®ed solution method in which thetangent is used from an earlier state. For example, the command language instructionsetTANGLOOP,iteration,10FORMSOLVeNEXT,iteration! Compute tangent! Perform a maximum of 10 iterations! Compute residual! Solve equations! End for LOOP instructionexecutes a modi®ed Newton's algorithm and, for general non-linear systems, resultsin less than a quadratic asymptotic rate of convergence (generally linear or less, sothat if iteration k gives a ratio of order 10 ÿp , iteration k ‡ 1 gives about 10 ÿ…p ‡ 1† ).The execution of each TANG, UTAN, FORM, etc. instruction uses the current problemtype and time increment to de®ne the parameters c i along with the current solutionvalues for a …k† , _a …k† and a …k† to calculate a tangent, residual, etc., respectively.Many additional solution algorithms may be established using the commandsavailable in the program. Some of these are discussed in the user manual wheretopics ranging from time-dependent loading to general transient, non-linear solutionstrategies included in FEAPpv are described. Authors may be found in Volume 2.20.4.4 Programming command language statementsThe command language module for FEAPpv is contained in a set of subprogramswhose names begin with PMAC. The routine PMACR calls the other routines andestablishes the limits on the number of commands available to the program. Included

596 Computer procedures for ®nite element analysisSUBROUTINE UMACR1(LCT,CTL,PRT)IMPLICIT NONEC Inputs:C LCT - Command character parametersC CTL(3) - Command numerical parametersC PRT - Flag, output if trueC Outputs:C N.B. Users are responsible for command actions.IMPLICIT NONELOGICAL PCOMP,PRTCHARACTER LCT*15REAL*8 CTL(3)CHARACTER UCT*4COMMON /UMAC1/ UCTC Set command word to user selected nameIF(PCOMP(UCT,'MAC1',4)) THENUCT = `xxxx'RETURNELSEC Implement user solution stepENDIFENDFig. 20.7 Structure of a user command subprogram.in the current command list is an option to access a set of user subprograms namedUMACRn where n ranges from 1 to 5. Each user subprogram has a structure as shown inFig. 20.7. A user is required to select a four character name for xxxx which does notalready exist in the command list in PMACR and to program the desired solution step.It should be noted that all arrays identi®ed in the subprogram PALLOC can beaccessed directly using the data management system described in Sec. 20.3. Inaddition data may be assigned to space in memory using the TEMPn array namesthat are also available in PALLOC. Thus it is not necessary to pass the names ofarrays through the argument list of the subprograms UMACRn. Quite general routinescan be created using these routines; however, if a more involved command is deemednecessary by a user the routines PMACRn may be modi®ed to add additional instructions.This is not an option which should be considered without a thorough studyof the new solution option needed, as well as, options already available in thecommands included.If it is decided to modify the PMACRn routines it is necessary to:1. Increase the size of the WD array in subprogram PMACR by the number of commandsto be added.

Computation of ®nite element solution modules 5972. Add the new command name to the list in the data statement for WD in subprogramPMACR noting which of the routines PMACRn will have the solution module added(the continue labels indicate the value of n).3. Increase the value of the variable NWDn in the data statement by the number ofcommands added for each n.4. Add the solution module to the subprograms PMACRn. This requires either amodi®cation of a GO TO or an IF-THEN-ELSE program form in addition toadding the statements.Again users are reminded that extreme care must be exercised when addingcommands in this way. Despite the fact that each command involves a speci®csolution step or steps there are some interactions between instructions that exist. Ifthese are changed in any way the program may not function properly after newcommands are added. This is particularly true for setting the parameters NWDn sinceif these are not correct transfer to incorrect locations in the list can occur.20.5 Computation of ®nite element solution modules20.5.1 Localization of element dataWhen we want to compute an element array, e.g., an element sti€ness matrix, S,oranelement load or residual vector, P, we only need those quantities associated with theone element in question. The nodal and material quantities that are required can bedetermined from the node and material set numbers stored in the IX array for eachelement. In the program FEAPpv the necessary values are moved from each globalarray to a set of local arrays before the appropriate element routine, ELMTnn, iscalled. The process will be called localization. The quantities that are localized are:1. nodal coordinates which are stored in the local array XL(NDM,NEN);2. nodal displacements, displacement increments, velocity and acceleration which arestored in the array UL(NDF,NEN,5);3. nodal T-variables which are stored in the array TL(NEN);4. equation numbers for assembly which are stored in the destination array LD(NEN).The LD array described in Step 4 above is used to map the element arrays to theglobal arrays. Accordingly, for the following element array:2323S…1; 1† S…1; 2† S…1; 3† P…1†6‰ LD…1† LD…2† LD…3† Š4S…2; 1† S…2; 2† 765 P…2† 74 5. ... ..the term S(i,j) would be assembled into the global coe cient array (e.g., sti€nessmatrix) in the position corresponding to row LD(i) and column LD(j). Similarly,P(i) would be assembled into the position corresponding to the LD(i) value. Thatis, the LD array contains the equation numbers of the global arrays. The LD(i) assignmentof the degrees of freedom for each node is made using the data stored in theID(j,k,2) array as shown in Table 20.2.

598 Computer procedures for ®nite element analysisThe localization process is the same for every type of ®nite element and is performedin the subprogram PFORM, which organizes all computations associated with elementsusing the connections given in the IX array. The maximum number of nodes actuallyconnected to an element is determined and assigned to the parameter NEL, whichmay be less than the maximum NEN, and is determined by ®nding the largest nonzeroentry in the IX array for each element number. Intermediate zero values areinterpreted as no node connected. In this way FEAPpv permits the mixing of elementswith di€erent numbers of connected nodes, e.g., three-noded triangles can be mixedwith four-noded quadrilaterals. Also di€erent types of elements can be mixed such astwo-noded shell elements with four-noded quadrilaterals.Since the current value of the nodal displacements and their increments, as well asthe nodal velocities and accelerations for transient problems, is localized for allelement computations, the program can be used to solve non-linear problems. Thisis, in fact, the only additional information required over that needed to solve linearproblems and will be discussed further in Volume 2.20.5.2 Element array computationsThe e cient computation of element arrays (in both programmer and computer time)is a crucial aspect of any ®nite element development. The development of subprogramsto evaluate element sti€ness and load arrays (or for non-linear problemstangent sti€ness and residual arrays) can be e ciently accomplished by a combinationof appropriate numerical methods. In order to illustrate a typical developmenta statement of the essential steps is ®rst given and then some details shown for thetwo-dimensional linear elastic problem.A ¯ow chart describing two alternative methods for computing a sti€ness matrix isshown in Fig. 20.8. Key steps in the computation are:1. use of appropriate numerical integration procedures;2. use of shape function subprograms (which are the same for all problems with thesame required continuity);3. e cient organization of numerical steps.Gauss±Legendre quadrature formulae are usually utilized to compute elementarrays since they provide the highest accuracy for a given number of integrationpoints (see Chapter 9). In some instances it is desirable to use other formulae. Forexample, if a quadrature formula which samples only at nodes is used, the evaluationof an inertial term leads to a diagonal mass matrix which is more e cient in explicitdynamics calculations.Shape function subprograms allow a programmer to develop elements for manyproblems quickly and reliably. A shape function subprogram should evaluate boththe shape functions and their derivatives with respect to the global coordinateframe. As an example consider the two-dimensional C 0 problem where we needonly ®rst derivatives of each shape function N i . For the four-noded isoparametricquadrilateral we haveN i ˆ 14… 1 ‡ i† … 1 ‡ i † …20:18†

Computation of ®nite element solution modules 599StartStartSet up DSet up CS 0100S 0L1, LINT100L1, LINTShape FunctionDeterminationShape FunctionDeterminationSet up BW ij∇N i ∇N jDBD • B100SS + B T DB100S ijC W ijReturnReturn(a)(b)Fig. 20.8 Element stiffness matrix computation: (a)general form; (b)form for constant material properties.where , are natural coordinates on the bi-unit square parent element and i , i theirvalues at the four nodes.Using the isoparametric concept we havex ˆ N i x iy ˆ N i y i…20:19†

600 Computer procedures for ®nite element analysiswith derivatives given by( ) " #( )N i;ˆ x; y ; Ni;xN i; x ; y ; N i;y( ) " #( )N i;xˆ 1 y ; ÿy ; Ni;N i;yJ ÿx ; x ; N i;…20:20†…20:21†where J is the jacobian determinant and … † ;x denotes the partial derivative@… †=@x, etc. The above relations de®ne the steps for the shape function subprogramgiven in Fig. 20.9 where it is assumed that the nodal coordinates have been transferredto the local coordinate array XL.This shape function routine can be used for all two-dimensional C 0 problems whichuse the four-noded element (e.g., two-dimensional plane and axisymmetric elasticity,heat conduction, ¯ow in porous media, ¯uid ¯ow, etc.). Shape function subprogramscan also be used for the generation of mesh data. 4 It is a simple task to extend theshape function routine to higher order elements (e.g., see the listing for subprogramSHAP2 in FEAPpv which includes options for up to nine-node quadrilaterals). Usingsuch routines permits the use of elements which have individual edges with eitherlinear or quadratic interpolation.The generation of the matrix products occurring in the sti€ness matrix of elasticityproblems deserves special attention since zeros often exist in the B and D matrices.Several methods can be used to reduce the number of operations performed. The®rst is to form explicitly the matrix products. While this involves extra hand computationsit is in fact elementary if performed on a nodal basis. For example, considerthe two-dimensional axisymmetric linear elastic problem where23N i;r 00 N i;zB i ˆ 67…20:22†4 cN i =r 0 5N i;zN i;rA two-dimensional plane problem may be considered by replacing r; z by x; y andsetting the constant c to zero. For axisymmetry the constant c is unity. For anisotropic linear elastic material the moduli are given by23D 11 D 12 D 12 0D 12 D 11 D 12 0D ˆ 67…20:23†4 D 12 D 12 D 11 0 50 0 0 D 33where D 33 is the shear modulus given by …D 11 ÿ D 12 †=2. Thus for a typical nodal pair iand j a contribution to the element sti€ness K ij may be computed usingQ j ˆ DB j…20:24†andK ij ˆ B T i Q j…20:25†

Computation of ®nite element solution modules 601CCCCSUBROUTINE SHAPE(SS,XL, J,SHP)Shape function routine for 4-node quadrilateralIMPLICIT NONEINTEGER II ,JJ ,KKREAL*8 SS(2),XL(2,4),J,SHP(3,4),SI(4),TI(4),XS(2,2),TEMPDATA SI / -0.5D0, 0.5D0, 0.5D0, -0.5D0/DATA TI / -0.5D0, -0.5D0, 0.5D0, 0.5D0/Compute shape functions and natural coordinate derivativesDO II = 1,4SHP(1,II) = SI(II)*(0.5D0 + TI(II)*SS(2))SHP(2,II) = TI(II)*(0.5D0 + SI(II)*SS(1))SHP(3,II) = (0.5D0 + SI(II)*SS(1))*(0.5D0 + TI(II)*SS(2))END DO ! IICompute Jacobian and Jacobian determinantDO II = 1,2DO JJ = 1,2XS(II,JJ) = 0.0D0DO KK = 1,4XS(II,JJ) = XS(II,JJ) + XL(II,KK)*SHP(JJ,KK)END DO ! KKEND DO ! JJEND DO ! IIJ = XS(1,1)*XS(2,2) - XS(1,2)*XS(2,1)Transform to X,Y derivativesDO II = 1,4TEMP = ( XS(2,2)*SHP(1,II) - XS(2,1)*SHP(2,II))/JSHP(2,II) = (-XS(1,2)*SHP(1,II) + XS(1,1)*SHP(2,II))/JSHP(1,II) = TEMPEND DO ! IIENDFig. 20.9 Shape function subprogram for four-noded element.Thus, using Eqs (20.22) and (20.23) and settingn j ˆ cr N jwe obtain2…D 11 N j;r ‡ D 12 n j †…D 12 N j;r ‡ D 12 n j †Q j ˆ 64 …D 12 N j;r ‡ D 11 n j †D 33 N j;z3D 12 N j;zD 22 N j;z7D 12 N j;z 5D 33 N j;r…20:26†…20:27†

602 Computer procedures for ®nite element analysisand ®nally the sti€ness asK ij ˆ …N i;rQ 11 ‡ n i Q 31 ‡ N i;z Q 41 † …N i;r Q 12 ‡ n i Q 32 ‡ N i;z Q 42 †…20:28†…N i;z Q 21 ‡ N i;r Q 41 † …N i;z Q 22 ‡ N i;r Q 42 †Accordingly, for each nodal pair it is required to perform 21 multiplications to form eachK ij , whereas formal multiplication of B T i DB j including all zero operations would require48 multiplications. When the element sti€ness matrix is symmetric it is only necessary toform the upper or lower triangular parts of K (the other half is formed from the symmetrycondition). A typical routine for the sti€ness computation is given in Figs 20.10and 20.11 where it is assumed that the quadrature points are available as SG(1,L)equal to L , SG(2,L) equal to L ,andSG(3,L) equal to the quadrature weight.The increments by NDF are to keep the sti€ness array stored in nodal order withNDFNDF submatrix blocks. This is required by FEAPpv to maintain proper compatibilitywith the routine used to assemble the global arrays.ccSUBROUTINE ELSTIF(D, XL, AXI, NDF,NDM,NST, S)IMPLICIT NONELOGICAL AXIINTEGER II,I1, JJ,J1, L, LINT, NDF,NDM,NSTREAL*8 DV, D11,D12,D33, J, RREAL*8 D(*), XL(NDM,4), S(NST,NST)REAL*8 SG(3,4), SHP(3,4), Q(4,2), N(4)CALL INT2D(2,LINT, SG) ! Set up 2x2 quadrature pointsDo numerical integrationDO L = 1,LINTCALL SHAPE(SG(1,L),XL, J,SHP)DV = J*SG(3,L) ! SG(3,L) is quadrature weightD11 = D(1)*DV ! D(1) is D_11 modulusD12 = D(2)*DV ! D(2) is D_12 modulusD33 = D(3)*DV ! D(3) is shear modulusCompute n_i = c*N_i/rR = 0.0D0! R is radiusDO II = 1,4R = R + SHP(3,II)*XL(1,II)END DO ! IIDO II = 1,4IF(AXI) THENN(II) = SHP(3,II)/RELSEN(II) = 0.0D0ENDIFEND DO ! IIFig. 20.10 Element stiffness calculation. Part 1.

Computation of ®nite element solution modules 603cCompute Q_j = D * B_jJ1 = 1DO JJ = 1,4Q(1,1) = D11*SHP(1,JJ) + D12*N(JJ)Q(2,1) = D12*SHP(1,JJ) + D12*N(JJ)Q(3,1) = D12*SHP(1,JJ) + D11*N(JJ)Q(4,1) = D33*SHP(2,JJ)Q(1,2) = D12*SHP(2,JJ)Q(2,2) = D11*SHP(2,JJ)Q(3,2) = D12*SHP(2,JJ)Q(4,2) = D33*SHP(1,JJ)cCompute stiffness term: k_ijI1 = 1DO II = 1,JJS(I1 ,J1 ) = S(I1 ,J1 ) + SHP(1,II)*Q(1,1)+N(II)*Q(3,1)&+ SHP(2,II)*Q(4,1)S(I1 ,J1+1) = S(I1 ,J1+1) + SHP(1,II)*Q(1,2)+N(II)*Q(3,2)&+ SHP(2,II)*Q(4,2)S(I1+1,J1 ) = S(I1+1,J1 ) + SHP(2,II)*Q(2,1)&+ SHP(1,II)*Q(4,1)S(I1+1,J1+1) = S(I1+1,J1+1) + SHP(2,II)*Q(2,2)&+ SHP(1,II)*Q(4,2)I1 = I1 + NDFEND DO ! IIJ1 = J1 + NDFEND DO ! JJEND DO ! LcCompute lower part by symmetryDO II = 1,NSTDO JJ = 1,IIS(II,JJ) = S(JJ,II)END DO ! JJEND DO ! IIENDFig. 20.11 Element stiffness calculation. Part 2.An extension to anisotropic problems can be made by replacing the isotropic Dmatrix by the appropriate anisotropic one and then recomputing the Q j matrix.The computation of element sti€ness matrices for two-dimensional plane andthree-dimensional problems which have constant material properties within anelement can be made more e cient than that given above. This is obtained by

604 Computer procedures for ®nite element analysisnoting from Appendix B that the internal energy may be written in indicial form asW…u† ˆ12 ~ui aD abcd…V eN i;b N j;d dVu j c …20:29†where a, b, c, d are indices from the elasticity equations and range over the spacedimension of the problem and i, j are nodal indices which range from 1 to NEL ineach element. The element sti€ness for the nodal pair i, j may be written asKac ij ˆ W ijbd D abcd…20:30†where…W ijbd ˆ N i;b N j;d dV…20:31†V eFor isotropic materialsD abcd ˆ ab cd ‡ … ac bd ‡ ad bc † …20:32†where and are the Lame elastic constants which are related to the usual elasticconstants E and asE ˆ…1 ‡ †…1 ÿ 2† ; ˆ E2…1 ‡ †Thus, the sti€ness matrix for an isotropic material is given asKac ij ˆ Wac ij ‡ …Wca ij ‡ ac W ijbb †…20:33†Using this approach the steps to compute the element sti€ness matrix for planeelasticity are given in Fig. 20.8(b). This procedure for computing sti€ness matriceswas noted in reference 5 and for plane problems results in about 25% fewer numericaloperations than the procedure shown in Fig. 20.8(a). In three dimensions the savingsare even greater.The computation of other element arrays can also be performed using a shapefunction routine. For example, the computation of the element consistent anddiagonal mass matrices by the row sum method (see Appendix I) for transient oreigenvalue computations can be easily constructed. The consistent mass matrix fortwo- and three-dimensional problems is obtained from…M ij ˆ I N i N j dV…20:34†V ewhereas the diagonal mass is computed from…M jj ˆ I N j dV…20:35†V eIn the above I is an identity matrix of size NDM and is the mass density. A set ofstatements to compute the mass matrix for these cases is shown in Fig. 20.12 wherethe element consistent mass is stored in the square matrix S and the diagonal massmatrix is stored in the rectangular array P.The shape function routine may also be used to compute strains, stresses andinternal forces in an element. The strains at each point in an element may be

Computation of ®nite element solution modules 605CCCCS(NST,NST) : Consistent mass arrayP(NDM,NEL) : Diagonal mass arrayNumerical integration loopDO L = 1,LINTCALL SHAPE(SG(1,L), XL, J, SHP)DMASS = RHO*J*SG(3,L)J1 = 1DO JJ = 1,NELJMASS = DMASS*SHP(3,JJ)P(1,JJ) = P(1,JJ) + JMASSI1 = 1DO II = 1,NELS(I1,J1) = S(I1,J1) + SHP((3,II)*JMASSI1 = I1 + NDFEND DO ! IIJ1 = J1 + NDFEND DO ! JJEND DO ! LCopy using identity matrixJ1 = 0DO JJ = 1,NELDO KK = 2,NDMP(KK,JJ) = P(1,JJ)END DO ! KKI1 = 0DO II = 1,NELDO KK = 2,NDMS(I1+KK,J1+KK) = S(I1+1,J1+1)END DO ! KKI1 = I1 + NDFEND DO ! IIJ1 = J1 + NDFEND DO ! JJFig. 20.12 Diagonal (lumped)and consistent mass matrix for an isoparametric element.computed frome ˆ B i …n† i ~u i…20:36†where n is the set of local natural coordinates and ~u i are the nodal displacements atnode i. A subprogram to compute the strains for the two-dimensional case givenby Eq. (20.22) is shown in Fig. 20.13. Stresses are now computed as usual fromr ˆ De…20:37†or any other relationship expressed in terms of the strains. The above form ismore general and e cient than saving the values in the Q i matrices during sti€ness

606 Computer procedures for ®nite element analysisCCCSUBROUTINE STRAIN(XL, UL, SHP, NDM,NDF,NEN,NEL, EPS,R, AXI)IMPLICIT NONELOGICAL AXIINTEGER NDM,NDF,NEN,NEL, IIREAL*8 XL(NDM,*),UL(NDF,NEN,*),SHP(3,*), EPS(4),RInitialize strains and radiusDO II = 1,4EPS(II) = 0.0D0END DO ! IIR = 0.0D0Sum strains from shape functions and nodal valuesDO II = 1,NELEPS(1) = EPS(1) + SHP(1,II)*UL(1,II,1)EPS(2) = EPS(2) + SHP(2,II)*UL(2,II,1)EPS(3) = EPS(3) + SHP(3,II)*UL(1,II,1)EPS(4) = EPS(4) + SHP(1,II)*UL(2,II,1) + SHP(2,II)*UL(1,II,1)R = R + SHP(3,II)*XL(1,II)END DO ! IIModify hoop strainif axisymmetric; zero for plane problemIF(AXI) THENEPS(3) = EPS(3)/RELSEEPS(3) = 0.0D0ENDIFENDFig. 20.13 Strain calculation for isoparametric element.evaluation and then computing the stresses fromr ˆ DB i ~u i ˆ Q i ~u i…20:38†This would require signi®cant additional storage or saving and retrieving the Q ifrom backing store as given in reference 6. Moreover, it is often desirable to computethe stresses at points other than those used to compute the sti€ness matrix asindicated in Chapter 14 for recovery processes. In non-linear problems the computationof strains and stresses must also be performed directly. Thus, for all the abovereasons it is desirable to compute strains as necessary using the technique given inFig. 20.13.In FEAPpv the stresses must also be determined to compute element residuals.One of the main terms in the element residual is the internal stress term and hereagain shape function routines are useful. The internal force term for problems inelasticity (and, as will be shown in the Volume 2, also for ®nite deformation inelastic

Computation of ®nite element solution modules 607CCCCCQuadrature loopDO L = 1,LINTCompute shape functionsCALL SHAPE(SG(1,L), XL, J, SHP)DV = J*SG(3,L)Compute strainsCALL STRAIN(XL, UL, SHP, NDM,NDF,NEN,NEL, EPS,R, AXI)DO II = 1,NELIF(AXI) THENN(II) = SHP(3,II)/RELSEN(II) = 0.0D0ENDIFEND DO ! IICompute stressesCALL STRESS(EPS, SIG)Compute internal forcesDO II = 1,NELP(1,II) = P(1,II) - (SHP(1,II)*SIG(1) + SHP(2,II)*SIG(4)&+ N(II)*SIG(3))*DVP(2,II) = P(2,II) - (SHP(2,II)*SIG(2) + SHP(1,II)*SIG(4))*DVEND DO ! IIEND DO ! LFig. 20.14 Internal force computation.problems) is given by…P i ˆÿ B T i r dV…20:39†V eThe programming steps to compute are given in Fig. 20.14.The generality of an isoparametric C 0 shape function routine can be exploited toprogram element routines for other problems. For example, Fig. 20.15 gives thenecessary program instructions to compute the `sti€ness'matrix for problems ofthe quasi-harmonic equation discussed in Chapters 3 and 7.20.5.3 Organization of element routinesThe previous discussion has focused on procedures for determining element arrays.The reader will note that the element square matrices for sti€ness and mass were

608 Computer procedures for ®nite element analysisCQuadrature loopDO L = 1,LINTCCompute shape functionsCALL SHAPE(SG(1,L), XL, J, SHP)DV = J*SG(3,L)KK = D(1)*DV ! Conductivity times volumeCFor each JJ-node compute the D*BDO JJ = 1,NELDO KK = 1,NDMQ(KK) = D1*SHP(KK,JJ)END DO ! KKCFor each II-node compute the coefficient matrixDO II = 1,JJDO KK = 1,NDMS(II,JJ) = S(II,JJ) + SHP(KK,II)*Q(KK)END DO ! KKEND DO ! IIEND DO ! JJEND DO ! LFig. 20.15 Coef®cient matrix for quasi-harmonic operator.both stored in the square array S while element vectors were stored in the rectangulararray P. This was intentional since all aspects of computing element arraysfor the program are to be consolidated into a single subprogram called the elementroutine. An element routine is called by the element library subprogram ELMLIB. Asgiven here, the element library provides space for ten element subprograms at anyone time, where as noted previously these are named ELMTnn with nn rangingfrom 01 to 10. This can easily be increased by modifying the subprogram ELMLIB.The subprogram ELMLIB is, in turn, called from the subprogram PFORM which isthe routine to loop through all elements and perform the localization step to setup local coordinates XL, displacements, etc., UL and equation numbers for globalassembly LD. The subprogram PFORM also uses subprogram DASBLE to assembleelement arrays into global arrays and uses subprogram MODIFY to perform appropriatemodi®cations for prescribed non-zero displacements. When an element routine isaccessed the value of a parameter ISW is given a value between 1 and 10. The parameterspeci®es what action is to be performed in the element routine. Each elementroutine must provide appropriate transfers for each value of ISW. A mock elementroutine for FEAPpv is shown in Fig. 20.16.

Solution of simultaneous linear algebraic equations 609CCCCCCCCCSUBROUTINE ELMTnn(D,UL,XL,IX,TL, S,P, NDF,NDM,NST, ISW)IMPLICIT NONEINTEGER NDF,NDM,NST, ISW, IX(NEN1,*)REAL*8 D(*),UL(NDF,NEN,*),XL(NDM,*),TL(*), S(NST,*),P(NDF,*)Input and output material set dataIF(ISW.EQ.1) THENUse D(*) to store input parametersCheck element for errorsELSEIF(ISW.EQ.2) THENCheck element for negative jacobians, etc.Form element coefficient matrix and residual vectorELSEIF(ISW.EQ.3 .OR. ISW.EQ.6) THENThe S(NST,NST) array stores coefficient matrixThe P(NDF,NEL) array stores residual vectorOutput element results (e.g., stress, strain, etc.)ELSEIF(ISW.EQ.4) THENCompute element mass arraysELSEIF(ISW.EQ.5) THENThe S(NST,NST) array stores consistent massThe P(NDF,NEL) array stores lumped massCompute element error estimatesELSEIF(ISW.EQ.7) THENProject element results to nodesELSEIF(ISW.EQ.8) THENProject element error estimatorELSEIF(ISW.EQ.9) THENAugmented lagragian updateELSEIF(ISW.EQ.10) THENENDIFENDFig. 20.16 Mock element routine functions.20.6 Solution of simultaneous linear algebraic equationsA ®nite element problem leads to a large set of simultaneous linear algebraicequations whose solution provides the nodal and element parameters in the formulation.For example, in the analysis of linear steady-state problems the direct assemblyof the element coe cient matrices and load vectors leads to a set of linear algebraicequations. In this section methods to solve the simultaneous algebraic equationsare summarized. We consider both direct methods where an a priori calculation of

610 Computer procedures for ®nite element analysisthe number of numerical operations can be made, and indirect or iterative methodswhere no such estimate can be made.20.6.1 Direct solutionConsider ®rst the general problem of direct solution of a set of algebraic equationsgiven byKa ˆ b…20:40†where K is a square coe cient matrix, a is a vector of unknown parameters and b is avector of known values. The reader can associate these with the quantities describedpreviously: namely, the sti€ness matrix, the nodal unknowns, and the speci®ed forcesor residuals.In the discussion to follow it is assumed that the coe cient matrix has propertiessuch that row and/or column interchanges are unnecessary to achieve an accuratesolution. This is true in cases where K is symmetric positive (or negative) de®nite.yPivoting may or may not be required with unsymmetric, or inde®nite, conditionswhich can occur when the ®nite element formulation is based on some weightedresidual methods. In these cases some checks or modi®cations may be necessary toensure that the equations can be solved accurately. 7ÿ9For the moment consider that the coe cient matrix can be written as the productof a lower triangular matrix with unit diagonals and an upper triangular matrix.Accordingly,whereandK ˆ LU21 0 30L 21 1 0L ˆ . . . . . 6745L n1 L n2 12U 11 U 12 3U 1n0 U 22 U 2nU ˆ . . . . . 67450 0 U nn…20:41†…20:42†…20:43†y For mixed methods which lead to forms of the type given in Eq. (11.14) the solution is given in terms of apositive de®nite part for ~q followed by a negative de®nite part for ~ f. Thus, interchanges are not neededproviding the ordering of the equation is de®ned as described in Sec. 20.2.4.

Solution of simultaneous linear algebraic equations 611This form is called a triangular decomposition of K. The solution to the equations cannow be obtained by solving the pair of equationsLy ˆ b…20:44†andUa ˆ y…20:45†where y is introduced to facilitate the separation, e.g., see references 7±11 foradditional details.The reader can easily observe that the solution to these equations is trivial. In termsof the individual equations the solution is given byy 1 ˆ b 1y i ˆ b i ÿ Xi ÿ 1L ij y jj ˆ 1i ˆ 2; 3; ...; n…20:46†anda n ˆ ynU nna i ˆ 1U iiy i ÿ Xnj ˆ i ‡ 1U ij a j!i ˆ n ÿ 2; n ÿ 3; ; 1…20:47†Equation (20.46) is commonly called forward elimination while Eq. (20.47) is calledback substitution.The problem remains to construct the triangular decomposition of the coe cientmatrix. This step is accomplished using variations on Gaussian elimination. Inpractice, the operations necessary for the triangular decomposition are performeddirectly in the coe cient array; however, to make the steps clear the basic steps areshown in Fig. 20.17 using separate arrays. The decomposition is performed in thesame way as that used in the subprogram DATRI contained in the FEAPpv program;thus, the reader can easily grasp the details of the subprograms included once thesteps in Fig. 20.17 are mastered. Additional details on this step may be found inreferences 9±11.In DATRI the Crout form of Gaussian elimination is used to successively reduce theoriginal coe cient array to upper triangular form. The lower portion of the array isused to store L ÿ I as shown in Fig. 20.17. With this form, the unit diagonals for L arenot stored.Based on the organization of Fig. 20.17 it is convenient to consider the coe cientarray to be divided into three parts: part one being the region that is fully reduced;part two the region which is currently being reduced (called the active zone); andpart three the region which contains the original unreduced coe cients. These regionsare shown in Fig. 20.18 where the jth column above the diagonal and the jth row tothe left of the diagonal constitute the active zone. The algorithm for the triangulardecomposition of an n n square matrix can be deduced from Fig. 20.17 and

612 Computer procedures for ®nite element analysisActive zoneL 11 = 1 K 11 K 12 K 13U 11 = K 11K 31 K 32 K 33K 21 K 22 K 23Step 1. Active zone. First row and column to principal diagonal.Reduced zoneActive zoneK 12 K 13 1 0U 11 U 12 = K 12K 31 K 32 K 33K 21 K 22 K 23 L 21 = K 21 /U 11 L 22 = 1 0 U 22 = K 22 – L 21 U 12Step 2. Active zone. Second row and column to principal diagonal. Use first row of K toeliminate L 21 U 11 . The active zone uses only values of K from the active zone andvalues of L and U which have already been computed in steps 1 and 2.Reduced zoneActive zoneU 11K 13 1 0 0 U 12K 31 K 32K 23K 33L 21L 311L 320L 33 = 100U 220U 13 = K 13U 23 = K 23 – L 21 U 13U 33 = K 33 – L 31 U 13 – L 32 U 23L 31 = K 31 /U 11L 32 = (K 32 – L 31 U 12 )/U 22Step 3. Active zone. Third row and column to principal diagonal. Use first row toeliminate L 31 U 11 ; use second row of reduced terms to eliminate L 32 U 22 (reducedcoefficient K 32 ). Reduce column 3 to reflect eliminations below diagonal.Fig. 20.17 Triangular decomposition of K.Ujth column active zoneK 1jLReducedzoneK 2j•••jth rowactive zoneK j1 , K j2•••K jjUnreduced zoneFig. 20.18 Reduced, active and unreduced parts.

Solution of simultaneous linear algebraic equations 613U 1iU 2iU 1jU 2j• • •• • •U i – 1, iU i – 1, jL i1 L i2 • • • L i, i – 1U iiK ijL j1 L j2 • • • L j, i – 1 K jiFig. 20.19 Terms used to construct U ij and L ji .Fig. 20.19 as follows:U 11 ˆ K 11 ; L 11 ˆ 1 …20:48†For each active zone j from 2 to n,and ®nallyL j1 ˆ Kj1; UU 1j ˆ K 1j …20:49†11L ji ˆ 1 KU ji ÿ Xi ÿ 1 L jm U miiim ˆ 1U ij ˆ K ij ÿ Xi ÿ 1L im U mj i ˆ 2; 3; ...; j ÿ 1m ˆ 1L jj ˆ 1U jj ˆ K jj ÿ Xj ÿ 1L jm U mjm ˆ 1…20:50†…20:51†The ordering of the reduction process and the terms used are shown in Fig. 20.19. Theresults from Fig. 20.17 and Eqs (20.48)±(20.51) can be veri®ed by the reader using thematrix given in the example shown in Table 20.6.Once the triangular decomposition of the coe cient matrix is computed, severalsolutions for di€erent right-hand sides b can be computed using Eqs (20.46) and(20.47). This process is often called a resolution since it is not necessary to recompute

614 Computer procedures for ®nite element analysisTable 20.6 Example: triangular decomposition of 3 3 matrix2K3 2L3 2U4 2 1 1464 2 4 2767 65 45 41 2 4375Step 1. L 11 ˆ 1, U 11 ˆ 42 12 4 2 1 2 410:5 14 23Step 2. L 21 ˆ 24 ˆ 0:5, U 12 ˆ 2, U 22 ˆ 1, U 22 ˆ 4 ÿ 0:5 2 ˆ 3114 2 120:5 13 1:5 1 2 4 0:25 0:5 1 3 Step 3. L 31 ˆ 14 ˆ 0:25, U 13 ˆ 1, L 32 ˆ 2 ÿ 0:25 2 ˆ 1:53 3 ˆ 0:5U 23 ˆ 2 ÿ 0:5 1 ˆ 1:5, L 33 ˆ 1, U 33 ˆ 4 ÿ 0:25 1 ÿ 0:5 1:5 ˆ 32323 2 314 2 1 4 2 164 0:5 1 76543 1:5 75 ˆ6 2 4 274 50:25 0:5 1 3 1 2 4Step 4. Checkthe L and U arrays. For large size coe cient matrices the triangular decompositionstep is very costly while a resolution is relatively cheap; consequently, a resolutioncapability is necessary in any ®nite element solution system using a direct method.The above discussion considered the general case of equation solving (without rowor column interchanges). In coe cient matrices resulting from a ®nite element formulationsome special properties are usually present. Often the coe cient matrix issymmetric (K ij ˆ K ji ) and it is easy to verify in this case thatU ij ˆ L ji U ii…20:52†For this problem class it is not necessary to store the entire coe cient matrix. It issu cient to store only the coe cients above (or below) the principal diagonal andthe diagonal coe cients. Equation (20.52) may be used to construct the missingpart. This reduces by almost half the required storage for the coe cient array aswell as the computational e€ort to compute the triangular decomposition.The required storage can be further reduced by storing only those rows andcolumns which lie within the region of non-zero entries of the coe cient array.Problems formulated by the ®nite element method and the Galerkin process normallyhave a symmetric pro®le which further simpli®es the storage form. Storing the upperand lower parts in separate arrays and the diagonal entries of U in a third array is usedin DATRI. Figure 20.20 shows a typical pro®le matrix and the storage order adopted

Solution of simultaneous linear algebraic equations 615Half band widthProfileHalf band widthK 11 K 12 K 13 K 14K 11 K 12 K 13 K 14K 22 K 23 K 24K 22 K 23 K 24K 33 K 34 K 35K 33 K 34 K 35K 44 K 45 K 46K 44 K 45 K 46SymmetricK 55 K 56K 55 K 56K 66 K 67 K 68K 66 K 67 K 68K 77 K 78K 77 K 78K 88K 88Banded storage arrayIAD iI AU i AL i J JD i1K 111 K 12 K 211023K 22K 3323K 13 K 31K 23 K 3223134567K 44K 55K 66K 77456K 14 K 41K 24 K 42K 34 K 4345676810118K 887K 35 K 53818Diagonals8K 45 K 54Profile910K 46 K 64K 56 K 65K 11 K 12 K 13 K 14 K 18K 22 K 23 K 24K 33 K 34 K 35K 44 K 45 K 46Symmetric K 55 K 56K 66 K 67K 28K 38K 48K 58K 68K 77 K 78K 8811 K 67 K 761218K 18 K 81••••••K 78 K 87Storage ofarraysFig. 20.20 Pro®le storage for coef®cient matrix.

616 Computer procedures for ®nite element analysisfor the upper array AU, the lower array AL and the diagonal array AD. An integer arrayJD is used to locate the start and end of entries in each column. With this scheme it isnecessary to store and compute only within the non-zero pro®le of the equations. Thisform of storage does not severely penalize the presence of a few large columns/rowsand is also an easy form to program a resolution process (e.g., see subprogram DASOLin FEAPpv and reference 10).The routines included in FEAPpv are restricted to problems for which thecoe cient matrix can ®t within the space allocated in the main storage array. Intwo-dimensional formulations, problems with several thousand degrees of freedomcan be solved on today's personal computers. In three-dimensional cases howeverproblems are restricted to a few thousand equations. To solve larger size problemsthere are several options. The ®rst is to retain only part of the coe cient matrix inthe main array with the rest saved on backing store (e.g., hard disk). This can bequite easily achieved but the size of problem is not greatly increased due to thevery large solve times required and the rapid growth in the size of the pro®le-storedcoe cient matrix in three-dimensional problems.A second option is to use sparse solution schemes. These lead to signi®cantprogram complexity over the procedure discussed above but can lead to signi®cantsavings in storage demands and compute time ± especially for problems in threedimensions. Nevertheless, capacity in terms of storage and compute time is againrapidly encountered and alternatives are needed.20.6.2 Iterative solutionOne of the main problems in direct solutions is that terms within the coe cient matrixwhich are zero from a ®nite element formulation become non-zero during thetriangular decomposition step. While sparse methods are better at limiting this ®llthan pro®le methods they still lead to a very large increase in the number of nonzeroterms in the factored coe cient matrix. To be more speci®c consider the caseof a three-dimensional linear elastic problem solved using eight-node isoparametrichexahedron elements. In a regular mesh each interior node is associated with 26other nodes, thus, the equation of such a node has 81 non-zero coe cients ± threefor each of the 27 associated nodes. On the other hand, for a rectangular block of elementswith n nodes on each of the sides the typical column height is approximatelyproportional to n 2 and the number of equations to n 3 . In Table 20.7 we show thesize and approximate number of non-zero terms in K from a ®nite formulation forlinear elasticity (i.e., with three degrees of freedom per node). The table also indicatesthe size growth with column height and storage requirements for a direct solutionbased on a pro®le solution method.From the table it can be observed that the demands for a direct solution aregrowing very rapidly (storage is approximately proportional to n 5 ) while at thesame time the demands for storing the non-zero terms in the sti€ness matrix growsproportional to the number of equations (i.e., proportional to n 3 for the block).Iterative solution methods use the terms in the sti€ness matrix directly and thus forlarge problems have the potential to be very e cient for large three-dimensionalproblems. On the other hand, iterative methods require the resolution of a set of

Solution of simultaneous linear algebraic equations 617Table 20.7SidenodesNumber ofequationsNon-zeros in KPro®le storage dataWords (10 ÿ6 ) Mbytes Col. Ht. Words (10 ÿ6 ) Mbytes5 375 0.02 0.12 90 0.03 0.2710 3000 0.12 0.96 330 0.99 7.9220 24000 0.96 7.68 1260 30.24 241.8240 192000 7.68 61.44 4920 944.64 7557.1280 1536000 61.44 491.52 18440 28323.84 226584.72equations until the residual of the linear equations, given byr …i† ˆ b ÿ Ka …i†…20:53†becomes less than a speci®ed tolerance.In order to be e€ective the number of iterations i to achieve a solution must be quitesmall ± generally no larger than a few hundred. Otherwise, excessive solution costswill result. At the time of writing this book the subject of iterative solution for general®nite element problems remains a topic of intense research. There are some impressiveresults available for the case where K is symmetric positive (or negative) de®nite;however, those for other classes (e.g., unsymmetric or inde®nite forms) are generallynot e cient enough for reliable use in the solution of general problems.For the symmetric positive de®nite case methods based on a preconditionedconjugate gradient method have been particularly e€ective. 12ÿ14 The convergenceof the method depends on the condition number of the matrix K ± the larger thecondition number, the slower the convergence (see reference 9 for more discussion).The condition number for a ®nite element problem with a symmetric positive de®nitesti€ness matrix K is de®ned as ˆ n…20:54† 1where 1 and n are the smallest and largest eigenvalue from the solution of theeigenproblem (viz. Chapter 17)K ˆ …20:55†in which is a diagonal matrix containing the individual eigenvalues i and thecolumns of are the eigenvectors i associated with each of the eigenvalues.Usually, the condition number for an elasticity problem modelled by the ®niteelement method is too large to achieve rapid convergence and a preconditionedconjugate gradient (PCG) is used. 12 A symmetric form of preconditioned system iswritten aswhereK p z ˆ PKP T z ˆ Pb…20:56†P T z ˆ a…20:57†Now the convergence of the PCG algorithm depends on the condition number ofK p . The problem remains to construct a preconditioner which adequately reduces

618 Computer procedures for ®nite element analysisthe condition number. In FEAPpv the diagonal of K is used, however, moree cient schemes incorporating also multigrid methods are discussed in references13 and 14.20.7 Extension and modi®cation of computer programFEAPpvThe previous sections brie¯y discussed the basis for the program FEAPpv which isavailable from the publishers web site at no cost. The capabilities of the programare quite signi®cant ± mainly due to the ¯exibility of the command language solutionstrategy. However, the program can be improved in many ways. Improvements toincrease the size of problems which can be solved have already been mentioned.Other improvements include additional command language statements to handlespecial needs of each user, preprocessors to assist in preparation of input data andpostprocessors to permit a wider range of graphical output options. In the lattertwo instances the program GiD 3 provides features which can greatly assist users inthe preparation of mesh data and the display of resultsy.In order to facilitate the addition of new input features and/or new commandlanguage statements the program FEAPpv includes a number of options for usersto add subprograms without the need to modify the PMESH or the PMACRn routines.References1. O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method, volume 1. McGraw-Hill,London, 4th edition, 1989.2. O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method, volume 2. McGraw-Hill,London, 4th edition, 1991.3. GiD ± The Personal Pre/Postprocesor (Version 5.0). Barcelona, Spain, 1999.4. O.C. Zienkiewicz. The Finite Element Method in Engineering Science. McGraw-Hill,London, 2nd edition, 1971.5. A.K. Gupta and B. Mohraz. A method of computing numerically integrated sti€nessmatrices. Internat. J. Num. Meth. Eng., 5, 83±9, 1972.6. E.L. Wilson. SAP ± a general structural analysis program for linear systems. Nucl. Engr.Des., 25, 257±74, 1973.7. A. Ralston. A First Course in Numerical Analysis. McGraw-Hill, New York, 1965.8. J.H. Wilkinson and C. Reinsch. Linear Algebra. Handbook for Automatic Computation,volume II. Springer-Verlag, Berlin, 1971.9. J. Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics,1997.10. R.L. Taylor. Solution of linear equations by a pro®le solver. Eng. Comp., 2, 344±50, 1985.11. G. Strang. Linear Algebra and its Application. Academic Press, New York, 1976.12. R.M. Ferencz. Element-by-element preconditioning techniques for large-scale, vectorized®nite element analysis in nonlinear solid and structural mechanics. Ph.D thesis, StanfordUniversity, Stanford, California, 1989.y Options to acquire GiD are also provided at the publishers web sit.

13. M. Adams. Heuristics for automatic construction of coarse grids in multigrid solvers for®nite element matrices. Technical Report UCB//CSD-98±994, University of California,Berkeley, 1998.14. M. Adams. Parallel muiltigrid algorithms for unstructured 3D large deformation elasticityand plasticity ®nite element problems. Technical Report UCB//CSD-99±1036, Universityof California, Berkeley, 1999.References 619

Appendix AMatrix algebraThe mystique surrounding matrix algebra is perhaps due to the texts on the subjectrequiring a student to `swallow too much' at one time. It will be found that inorder to follow the present text and carry out the necessary computation only alimited knowledge of a few basic de®nitions is required.De®nition of a matrixThe linear relationship between a set of variables x and ba 11 x 1 ‡ a 12 x 2 ‡ a 13 x 3 ‡ a 14 x 4 ˆ b 1a 21 x 1 ‡ a 22 x 2 ‡ a 23 x 3 ‡ a 24 x 4 ˆ b 2…A:1†a 31 x 1 ‡ a 32 x 2 ‡ a 33 x 3 ‡ a 34 x 4 ˆ b 3can be written, in a shorthand way, as‰ AŠfxg ˆ fbg…A:2†orAxˆ b…A:3†where23a 11 a 12 a 13 a 14A ‰ AŠ ˆ64 a 21 a 22 a 23 a724 5 …A:4†x b a 31 a 32 a 33 a 348 9fxg ˆfbg ˆ>:8>:x 1x 2x 3x 4b 1b 2b 3>=>;9>=>;

Matrix addition or subtraction 621The above notation contains within it both the de®nition of a matrix and of the processof multiplication of two matrices. Matrices are de®ned as `arrays of numbers' of thetype shown in Eq. (A.4). The particular form listing a single column of numbers is oftenreferred to as a vector or column matrix, whereas a matrix with multiple columns androws is called a rectangular matrix. The multiplication of a matrix by a column vector isde®ned by the equivalence of the left and right sides of Eqs (A.1) and (A.2).The useof boldcharacters tode®ne both vectorsand matrices will be followed throughoutthe text, generally lower case letters denoting vectors and capital letters matrices.If another relationship, using the same a-constants, but a di€erent set of x and b,exists and is written asa 11 x 0 1 ‡ a 12 x 0 2 ‡ a 13 x 0 3 ‡ a 14 x 0 4 ˆ b 0 1a 21 x 0 1 ‡ a 22 x 0 2 ‡ a 23 x 0 3 ‡ a 24 x 0 4 ˆ b 0 2…A:5†then we could writein whichX ‰ XŠ ˆa 31 x 0 1 ‡ a 32 x 0 2 ‡ a 33 x 0 3 ‡ a 34 x 0 4 ˆ b 0 3‰ AŠ‰ XŠ ˆ ‰ BŠ or AX ˆ B …A:6†2x 1 ; x 0 31x 2 ; x 0 26x 3 ; x 0 7435x 4 ; x 0 4B ‰ BŠ ˆ2 3b 1 ; b 0 16b 2 ; b 0 74 2 5 …A:7†b 3 ; b 0 3implying both the statements (A.1) and (A.5) arranged simultaneously as232 3a 11 x 1 ‡; a 11 x 0 1 ‡b 1 ; b 0 16a 21 x 1 ‡; a 21 x 0 741 ‡5 ˆ B ‰ BŠ 6ˆ b 2 ; b 0 74 2 5 …A:8†a 31 x 1 ‡; a 31 x 0 1 ‡b 3 ; b 0 3It is seen, incidentally, that matrices can be equal only if each of the individual terms isequal.The multiplication of full matrices is de®ned above, and it is obvious that it has ameaning only if the number of columns in A is equal to the number of rows in X for arelation of the type (A.6).One property that distinguishes matrix multiplication isthat, in general,AX 6ˆ XAi.e., multiplication of matrices is not commutative as in ordinary algebra.Matrix addition or subtractionIf relations of the form (A.1) and (A.5) are added then we havea 11 …x 1 ‡ x 0 1†‡a 12 …x 2 ‡ x 0 2†‡a 13 …x 3 ‡ x 0 3†‡a 14 …x 4 ‡ x 0 4†ˆb 1 ‡ b 0 1a 21 …x 1 ‡ x 0 1†‡a 22 …x 2 ‡ x 0 2†‡a 23 …x 3 ‡ x 0 3†‡a 24 …x 4 ‡ x 0 4†ˆb 2 ‡ b 0 2…A:9†a 31 …x 1 ‡ x 0 1†‡a 32 …x 2 ‡ x 0 2†‡a 33 …x 3 ‡ x 0 3†‡a 34 …x 4 ‡ x 0 4†ˆb 3 ‡ b 0 3

622 Appendix Awhich will also follow fromAx ‡ Ax 0 ˆ b ‡ b 0if we de®ne the addition of matrices by simple addition of the individual terms of thearray. Clearly this can be done only if the size of the matrices is identical, i.e., forexample,2 3 2 3 23a 11 a 12 b 11 b 12 a 11 ‡ b 11 a 12 ‡ b 1264 a 21 a722 5 ‡64 b 21 b722 5 ˆ6a 21 ‡ b 21 a 22 ‡ b7422 5a 31 a 32 b 31 b 32 a 31 ‡ b 31 a 32 ‡ b 32orA ‡ B ˆ C…A:10†implies that every term of C is equal to the sum of the appropriate terms of A and B.Subtraction obviously follows similar rules.Transpose of a matrixThis is simply a de®nition for reordering the terms in an array in the followingmanner:2 3" # a 11 a 21a 11 a 12 a 13Tˆ6a 12 a74 22 5 …A:11†a 21 a 22 a 23a 13 a 23and will be indicated by the symbol T as shown.Its use is not immediately obvious but will be indicated later and can be treated hereas a simple prescribed operation.Inverse of a matrixIf in the relationship (A.2) the matrix A is `square', i.e., it represents the coe cients ofsimultaneous equations of type (A.1) equal in number to the number of unknowns x,then in general it is possible to solve for the unknowns in terms of the knowncoe cients b. This solution can be written asx ˆ A ÿ1 b…A:12†in which the matrix A ÿ1 is known as the `inverse' of the square matrix A. Clearly A ÿ1is also square and of the same size as A.We could obtain (A.12) by multiplying both sides of (A.2) by A ÿ1 and henceA ÿ1 A ˆ I ˆ AA ÿ1…A:13†where I is an `identity' matrix having zero on all o€-diagonal positions and unity oneach of the diagonal positions.If the equations are `singular' and have no solution then clearly an inverse does notexist.

Symmetric matrices 623A sum of productsIn problems of mechanics we often encounter a number of quantities such as forcethat can be listed as a matrix `vector':8 9f 1>< f>= 2f ˆ. .…A:14†>: >;f nThese in turn are often associated with the same number of displacements given byanother vector, say,8 9a 1>=a ˆ 2.…A:15†. >: >;a nIt is known that the work is represented as a sum of products of force and displacementW ˆ Xnf k a kk ˆ 1Clearly the transpose becomes useful here as we can write, by the rule of matrixmultiplication,8 9a 1>=2W ˆ f 1 f 2 f n . ˆ f T a ˆ a T f…A:16†>: >;a nUse of this fact is made frequently in this book.Transpose of a productAn operation that sometimes occurs is that of taking the transpose of a matrixproduct. It can be left to the reader to prove from previous de®nitions that… AB† Tˆ B T A T …A:17†Symmetric matricesIn structural problems symmetric matrices are often encountered. If a term of amatrix A is de®ned as a ij , then for a symmetric matrixa ij ˆ a ji or A ˆ A T

624 Appendix AA symmetric matrix must be square. It can be shown that the inverse of a symmetricmatrix is also symmetricA ÿ1 ˆ … A ÿ1 † T A ÿTPartitioningIt is easy to verify that a matrix product AB in which for example23a 11 a 12 a 13 a 14 a 15A ˆ 6 a 21 a 22 a 23 a 24 a 425 75a 31 a 32 a 33 a 34 a 352b 113b 12b 21 b 22B ˆb 31 b 3264 b 41 b 742 5b 51 b 52could be obtained by dividing each matrix into submatrices, indicated by the lines,and applying the rules of matrix multiplication ®rst to each of such submatrices asif it were a scalar number and then carrying out further multiplication in the usualway. Thus, if we write" # " #A ˆ A11 A 12B ˆB1A 21 A 22B 2then"AB ˆ A11B#1 A 12 B 2A 21 B 1 A 22 B 2can be veri®ed as representing the complete product by further multiplication.The essential feature of partitioning is that the size of subdivisions has to be such asto make the products of the type A 11 B 1 meaningful, i.e., the number of columns in A 11must be equal to the number of rows in B 1 , etc. If the above de®nition holds, then allfurther operations can be conducted on partitioned matrices, treating each partitionas if it were a scalar.It should be noted that any matrix can be multiplied by a scalar (number). Here,obviously, the requirements of equality of appropriate rows and columns no longerapply.If a symmetric matrix is divided into an equal number of submatrices A ij in rowsand columns thenA ij ˆ A T ji

The eigenvalue problem 625The eigenvalue problemAn eigenvalue of a symmetric matrix A of size n n is a scalar i , which allows thesolution of… A ÿ i I†f i ˆ 0 and det jA ÿ i Ijˆ0 …A:18†where f i is called the eigenvector.There are, of course, n such eigenvalues i to each of which corresponds aneigenvector f i . Such vectors can be shown to be orthonormal and we write(f T i f j ˆ ij ˆ 1for i ˆ j0 for i 6ˆ jThe full set of eigenvalues and eigenvectors can be written as23 1 06 . ˆ .7 4 . 5 ˆ f 1 ; ... f n0 nUsing these the matrix A may be written in its spectral form by noting from the orthonormalityconditions on the eigenvectors that ÿ1 ˆ TThen fromA ˆ it follows immediately thatA ˆ T…A:19†The condition number (which is related to equation solution roundo€) is de®ned as ˆ j maxj…A:20†j min j

Appendix BTensor-indicial notation in theapproximation of elasticityproblemsIntroductionThe matrix type of notation used in this volume for the description of tensorquantities such as stresses and strains is compact and we believe easy to understand.However, in a computer program each quantity will often still have to be identi®ed byappropriate indices (see Chapter 20) and the conciseness of matrix notation does notalways carry over to the programming steps. Further, many readers are accustomedto the use of indicial-tensor notation which is a standard tool in the study of solidmechanics. For this reason we summarize here the formulation of ®nite elementarrays in an indicial form.Some advantages of this reformulation from the matrix setting become apparentwhen evaluation of sti€ness arrays for isotropic materials is considered. Here somemultiplication operations previously necessary become redundant and the elementmodule programs can be written more economically.When ®nite deformation problems in solid mechanics have to be considered the useof indicial notation is almost essential to form many of the arrays needed for theresidual and tangent terms.This appendix adds little new to the discretization ideas ± it merely repeats in adi€erent language the results already presented.Indicial notation: summation conventionA point P in three-dimensional space may be represented in terms of its cartesiancoordinates x a , a ˆ 1; 2; 3. The limits that a can take de®ne its range. To de®nethese components we must ®rst establish an oriented orthogonal set of coordinatedirections as shown in Fig. B.1. The distance from the origin of the coordinateaxes to the point de®ne a position vector x. If along each of the coordinate axeswe de®ne the set of unit orthonormal base vectors, i a , a ˆ 1; 2; 3; which have thepropertyi a i b ˆ ab ˆ 1 for a ˆ b…B:1†0for a 6ˆ b

Indicial notation: summation convention 627x 1 (P)Pxx 2 (P)i 2x 2x 1Fig. B.1 Orthogonal axes and a point: cartesian coordinates.i 1where … † … † denotes the vector dot product, the components of the positionvector are constructed from the vector dot productx a ˆ i a x; a ˆ 1; 2; 3 …B:2†From this construction it is easy to observe that the vector x may be represented asx ˆ X3a ˆ 1x a i a…B:3†In dealing with vectors, and later tensors, the form x is called the intrinsic notation ofthe coordinates and x a i a the indicial form. An intrinsic form is a physical entity whichis independent of the coordinate system selected, whereas an indicial form depends ona particular coordinate system.To simplify notation we adopt the common convention that any index which isrepeated in any given term implies a summation over the range of the index. Thus,our shorthand notation for Eq. (B.3) isx ˆ x a i a ˆ x 1 i 1 ‡ x 2 i 2 ‡ x 3 i 3…B:4†For two-dimensional problems unless otherwise stated it will be understood that therange of the index is two.Similarly, we can de®ne the components of the displacement vector u asu ˆ u a i a…B:5†Note that the components …u 1 ; u 2 ; u 3 † replace the components …u; v; w† used throughoutmost of this volume.To avoid confusion with nodal quantities to which we previously also attachedsubscripts we shall simply change their position to a superscript. Thusused previously, etc.u j 2 has the same meaning as v j…B:6†

628 Appendix BDerivatives and tensorial relationsIn indicial notation the derivative of any quantity with respect to a coordinatecomponent x a is written compactly as@… †@x ;a …B:7†aThus we can write the gradient of the displacement vector as@u a u@x a;b a; b ˆ 1; 2; 3 …B:8†bIn a cartesian coordinate system the base vectors do not change their magnitude ordirection along any coordinate direction. Accordingly their derivatives with respect toany coordinate is zero as indicated in Eq. (B.9):@i aˆ i@x a;b ˆ 0…B:9†bThus in Cartesian co-ordinates, the derivative of the intrinsic displacement u is given byu ;b ˆ u a;b i a ‡ u a i a;b ˆ u a;b i a…B:10†The collection of all the derivatives de®nes the displacement gradient which we write inintrinsic notation asru ˆ u a;b i a i b…B:11†The symbol denotes the tensor product between two base vectors and since only twovectors are involved the gradient of the displacement is called second rank. Thenotation used to de®ne a tensor product follows that used in reference 31.Any second rank intrinsic quantity can be split into symmetric and skew symmetric(antisymmetric) parts asA ˆ 1 ‰2 A ‡ AT Š‡ 1 ‰2 A ÿ AT Š ˆ A …s† ‡ A …a† …B:12†where A and its transpose have cartesian componentsA ˆ A ab i a i b ; A T ˆ A ba i a i b …B:13†The symmetric part of the displacement gradient de®nes the (small) strainye ˆ ru …s† ˆ 12ru ‡…ru†Tˆ 12 ‰ u a;b ‡ u b;a Ši a i bˆ " ab i a i b ˆ " ba i a i b…B:14†and the skew symmetric part gives the (small) rotationx ˆ ru …a† ˆ 12ru ÿ…ru†Tˆ 12 ‰ u a;b ÿ u b;a Ši a i bˆ ! ab i a i b ˆÿ! ba i a i b…B:15†y Note that this de®nition is slightly di€erent from that occurring in Chapters 2±6. Now " ab ˆ 1=2 ab wheni 6ˆ j.

Coordinate transformation 629The strain expression is analogous to Eq. (2.2). The components " ab and ! ab may berepresented by a matrix as23 23" 11 " 12 " 13 " 11 " 12 " 13" ab ˆ 64 " 21 " 22 " 2375 ˆ6" 12 " 22 " 23745 …B:16†" 31 " 32 " 33 " 13 " 23 " 3323 230 ! 12 ! 13 0 ! 12 ! 13! ab ˆ 64 ! 21 0 ! 2375 ˆ6ÿ! 12 0 ! 23745 …B:17†! 31 ! 32 0 ÿ! 13 ÿ! 23 0Coordinate transformationConsider now the representation of the intrinsic coordinates in a system which hasdi€erent orientation than that given in Fig. B.1. We represent the components inthe new system byx ˆ x 0 a 0 i0 a 0…B:18†Using Eq. (B.2) we can relate the components in the prime system to those in theoriginal system asx 0 a 0ˆ i0 a xˆ i 0 a i b x bˆ a 0 bx b…B:19†where a 0 b ˆ i 0 a 0 i b ˆ cos…x 0 a 0; x b†…B:20†de®ne the direction cosines of the coordinate in a manner similar to that of Eq. (1.25).Equation (B.19) de®nes how the cartesian coordinate components transform fromone coordinate frame to another. Recall that the summation convention impliesx 0 a 0 ˆ a 0 1x 1 ‡ a 0 2x 2 ‡ a 0 3x 3 a 0 ˆ 1; 2; 3 …B:21†In Eq. (B.19) a 0 is called a free index whereas b is called a dummy index since it may bereplaced by any other unique index without changing the meaning of the term (notethat the notation does not permit an index to appear more than twice in any term).The summation convention will be employed throughout the remainder of thisdiscussion and the reader should ensure that the concept is fully understood beforeproceeding. Some examples will be given occasionally to illustrate its use.Using the notion of the direction cosines, Eq. (B.19) may be used to transform anyvector with three components. Thus, transformation of the components of thedisplacement vector is given byu 0 a 0 ˆ a 0 bu b a 0 b ˆ 1; 2; 3 …B:22†

630 Appendix BIndeed we can also use the above to express the transformation for the base vectorssincei 0 a 0 ˆ ÿ i0 a 0 i b ib ˆ a 0 bi b …B:23†Similarly, by interchanging the role of the base vectors we obtainÿ i b ˆ i b i 0 a 0 i0a 0 ˆ a 0 bi 0 a 0 …B:24†which indicates that the inverse of the direction cosine coe cient array is the same asits transpose.The strain transformation follows from the intrinsic form written ase ˆ " 0 a 0 b 0i a 0 i b 0ˆ " cd i c i d…B:25†Substitution of the base vectors from Eq. (B.24) into Eq. (B.25) givese ˆ a 0 c" cd b 0 di a 0 i b 0…B:26†Comparing Eq. (B.26) with Eq. (B.25) the components of the strain transformaccording to the relation" 0 a 0 b 0 ˆ a 0 c" cd b 0 d …B:27†Variables that transform according to Eq. (B.22) are called ®rst rank cartesiantensors whereas quantities that transform according to Eq. (B.27) are called secondrank cartesian tensors. The use of indicial notation in the context of cartesiancoordinates will lead naturally to each mechanics variable being de®ned in terms ofa cartesian tensor of appropriate rank.Stress may be written in terms of its components ab which may be written in matrixform similar to Eq. (B.16)23 11 12 1367 ab ˆ 4 21 22 23 5 a; b ˆ 1; 2; 3 …B:28† 31 32 33In intrinsic form the stress is given byr ˆ ab i a i b…B:29†and, using similar logic as was used for strain, can be shown to transform as a secondrank cartesian tensor. The symmetry of the components of stress may be establishedby summing moments (angular momentum balance) about each of the coordinateaxes to obtain ab ˆ ba…B:30†Equilibrium and energyIntroducing a body force vectorb ˆ b a i a…B:31†

we can write the static equilibrium equations (linear momentum balance) for adi€erential element asÿ div r ‡ b ba;b ‡ b a ia ˆ 0 …B:32†where the repeated index again implies summation over the range of the index, i.e., ba;b X3b ˆ 1 ba;bˆ 1a;1 ‡ 2a;2 ‡ 3a;3Note that the free index a must appear in each term for the equation to be meaningful.As a further example of the summation convention consider an internal energy termW ˆ ab " ab…B:33†This expression implies a double summation; hence summing ®rst on a givesW ˆ 1b " 1b ‡ 2b " 2b ‡ 3b " 3band then summing on b gives ®nallyW ˆ 11 " 11 ‡ 12 " 12 ‡ 13 " 13‡ 21 " 21 ‡ 22 " 22 ‡ 23 " 23‡ 31 " 31 ‡ 32 " 32 ‡ 33 " 33Elastic constitutive equations 631We may use symmetry conditions on ab and " ab to reduce the nine terms to six terms.Accordingly,W ˆ 11 " 11 ‡ 22 " 22 ‡ 33 " 33‡ 2… 12 " 12 ‡ 23 " 23 ‡ 31 " 31 † …B:34†Following a similar expansion we can also show the result ab ! ab 0…B:35†Elastic constitutive equationsFor an elastic material the most general linear relationship we can write for componentsof the stress±strain characterization isÿ ab ˆ D abcd " cd ÿ " 0 cd ‡ 0ab …B:36†Equation (B.36) is the equivalent of Eq. (2.5) but now written in index notation. Wenote that the elastic moduli which appear in Eq. (B.36) are components of the fourthrank tensorD ˆ D abcd i a i b i c i d…B:37†The elastic moduli possess the following symmetry conditionsD abcd ˆ D bacd ˆ D abdc ˆ D cdab…B:38†the latter, arising from the existence of an internal energy density in the form 2W…e† ˆ12 " abD abcd " cd ‡ " ab 0 ab ÿ D abcd " 0 cd…B:39†

632 Appendix Bwhich yields the stress from ab ˆ @W…B:40†@" abBy writing the constitutive equation with respect to x 0 a0 and using properties of thebase vectors we can deduce the transformation equation for moduli asD 0 a 0 b 0 c 0 d 0 ˆ a 0 e b 0 f c 0 g d 0 hD efgh …B:41†A common notation for the intrinsic form of Eq. (B.36) isr ˆ D : … e ÿ e 0 †‡ r 0 …B:42†in which : denotes the double summation (contraction) between the elastic moduli andthe strains.The elastic moduli for an isotropic elastic material may be written in indicial form asD abcd ˆ ab cd ‡ … ac bd ‡ ad bc † …B:43†where , are the Lame constants. An isotropic linear elastic material is alwayscharacterized by two independent elastic constants. Instead of the Lame constantswe can use Young's modulus, E, and Poisson's ratio, , to characterize the material.The Lame constants may be deduced from ˆE2…1 ‡ †and ˆE…1 ‡ †…1 ÿ 2†…B:44†Finite element approximationIf we now introduce the ®nite element displacement approximation given by Eq. (2.1),using indicial notation we may write for a single elementu a ^u a ˆ N i ~u i a a ˆ 1; 2; 3; i ˆ 1; 2; ...; n …B:45†where n is the total number of nodes on an element. The strain approximation in eachelement is given by the de®nition of Eq. (B.14) as^" ab ˆ 12 N i;b~u i a ‡ N i;a ~u i b…B:46†The internal virtual work for an element is given as……U I ˆ e : r dV ˆ " ab ab dV…B:47†V e V eUsing Eqs (B.46) and (B.47) and noting the symmetries in D abcd we may write theinternal virtual work for a linear elastic material as…U I ˆ ~u i a N i;b D abcd N j;d dV ~u j cV e…ÿ ‡ ~u i a N i;b 0 ab ÿ D abcd " 0 cd dV …B:48†V ewhich replaces the terms obtained in Chapter 2 in indicial notation.In describing a sti€ness coe cient two subscripts have been used previously and thesubmatrix K ij implied 2 2or3 3 entries for the ij nodal pair, depending on

Relation between indicial and matrix notation 633whether two or three dimensional displacement components were involved. Now thescalar componentsK ijaba; b ˆ 1; 2; 3; i; j ˆ 1; 2; ...; n …B:49†de®ne completely the appropriate sti€ness coe cient with ab indicating the relativesubmatrix position (in this case for a three-dimensional displacement).Note that for a symmetric matrix we have previously required thatK ij ˆ K T ji…B:50†In indicial notation the same symmetry is implied ifK ijab ˆ Kji ba…B:51†The sti€ness tensor is now de®ned from Eq. (B.48) as…Kac ij ˆ N i;b D abcd N j;d dV…B:52†V eWhen the elastic properties are constant over the element we may separate theintegration from the material constants by de®ning…W ijbd ˆ N i;b N j;d dV…B:53†V eand then perform the summations with the material moduli asKac ij ˆ W ijbd D abcd…B:54†In the case of isotropy a particularly simple result is obtainedKac ij ˆ Wac ij ‡ ‰Wca ij ‡ ac W ijbb Š…B:55†which allows the construction of the sti€ness to be carried out using fewer arithmeticoperations as compared with the use of the matrix form. 3Using indicial notation the ®nal equilibrium equations of the system are written asKacu ij j c ‡ f i a ˆ 0 a ˆ 1; 2; 3 …B:56†and in this scalar form every coe cient is simply identi®ed. The reader can, as asimple exercise, complete the derivation of the force terms due to the initial strain" 0 ab, stress 0 ab,body force b a and external traction t a .Indicial notation is at times useful in clarifying individual terms, and this introductionshould be helpful as a key to reading some of the current literature.Relation between indicial and matrix notationThe matrix form used throughout most of this volume can be deduced from theindicial form by a simple transformation between the indices. The relationshipbetween the indices of the second rank tensors and their corresponding matrixform can be performed by an inspection of the ordering in the matrix for stressand its representation shown in Eq. (B.28). In the matrix form the stress was given

634 Appendix Bin Chapter 6 as Tr ˆ 11 22 33 12 23 31…B:57†This form includes the use of the symmetry of stress components. The mapping of theindices follows that shown in Table B.1.Table B.1 Mapping between matrix and tensor indices for second rank symmetric tensorsFormIndex numberMatrix 1 2 3 4 5 6Tensor 11 22 33 12 & 21 23 & 32 31 & 13xx yy zz xy & yx yz & zy zx & xzTable B.1 may also be used to perform the map of the material moduli by notingthat the components in the energy are associated with the index pairs from thestress and the strain. Accordingly, the moduli transform asD 1111 ! D 11 ; D 2233 ! D 23 ; D 1231 ! D 46 ; etc. …B:58†The symmetry of the stress and strain is embedded in Table B.1 and the existence ofan energy function yields the symmetry of the modulus matrix, i.e., D ij ˆ D ji .References1. P. Chadwick. Continuum Mechanics. John Wiley & Sons, New York, 1976.2. I.S. Sokolniko€. The Mathematical Theory of Elasticity. McGraw-Hill, New York, 2ndedition, 1956.3. A.K. Gupta and B. Mohraz. A method of computing numerically integrated sti€nessmatrices. Internat. J. Num. Meth. Eng., 5, 83±89, 1972.

Appendix CBasic equations of displacementanalysis (Chapter 2)Displacementu ^u ˆ X N i a i ˆ Na…C:1†Straine ˆ Sue ˆ X B i a i ˆ BaB i ˆ SN iB ˆ SN…C:2†…C:3†…C:4†Stress±strain constitutive relation of linear elasticityr ˆ D… e ÿ e 0 †‡ r 0 …C:5†Approximate equilibrium equationsr ˆ Ka ‡ f…K ij ˆ B T i DB j dVV…f i ˆÿV……N T i b dV ÿ N T i t dA ÿAVB T i … De 0 ‡ r 0 †dV…C:6†…C:7†

Appendix DSome integration formulae fora triangleLet a triangle be de®ned in the xy plane by three points (x i , y i ), (x j , y j ), (x m , y m )withthe origin of the coordinates taken at the centroid (or baricentre), i.e.,x i ‡ x j ‡ x m3ˆ yi ‡ y j ‡ y m3ˆ 0Then integrating over the triangle area we obtain:…1 x i y idx dy ˆ 11 x2 j y jˆ ˆ area of triangle 1 x m y m… …x dx dy ˆ y dx dy ˆ 0…x 2 dx dy ˆ 12 …x2 i ‡ x 2 j ‡ x 2 m†…y 2 dx dy ˆ 12 …y2 i ‡ y 2 j ‡ y 2 m†…xy dx dy ˆ 12 …x i y i ‡ x j y j ‡ x m y m †

Appendix ESome integration formulae fora tetrahedronLet a tetrahedron be de®ned in the xyz coordinate system by four points (x i , y i , z i ),(x j , y j , z j ), (x m , y m , z m ), (x p , y p , z p ) with the origin of the coordinates taken at thecentroid, i.e.,x i ‡ x j ‡ x m ‡ x p4ˆ yi ‡ y j ‡ y m ‡ y p4ˆ zi ‡ z j ‡ z m ‡ z p4Then integrating over the tetrahedron volume gives1 x i y i z i…dx dy dz ˆ 11 x j y j z jˆ V ˆ tetrahedron volume61 x m y m z m 1 x p y p z pProvided the order of numbering the nodes is as indicated on Fig. 6.1then also:………x dx dy dz ˆ y dx dy dz ˆ z dx dy dz ˆ 0…x 2 dx dy dz ˆ V20 …x2 i ‡ x 2 j ‡ x 2 m ‡ x 2 p†…y 2 dx dy dz ˆ V20 …y2 i ‡ y 2 j ‡ y 2 m ‡ y 2 p†…z 2 dx dy dz ˆ V20 …z2 i ‡ z 2 j ‡ z 2 m ‡ z 2 p†…xy dx dy dz ˆ V20 …x i y i ‡ x j y j ‡ x m y m ‡ x p y p †…yz dx dy dz ˆ V20 …y iz i ‡ y j z j ‡ y m z m ‡ y p z p †…zx dx dy dz ˆ V20 …z ix i ‡ z j x j ‡ z m x m ‡ z p x p †ˆ 0

Appendix FSome vector algebraSome knowledge and understanding of basic vector algebra is needed in dealing withcomplexities of elements oriented in space as occur in beams, shells, etc. Some of theoperations are summarized here.Vectors (in the geometric sense) can be described by their components along thedirections of the x, y, z axes.Thus, the vector V 01 shown in Fig. F.1 can be written asV 01 ˆ x 1 i ‡ y 1 j ‡ z 1 k…F:1†in which i, j, k are unit vectors in the direction of the x, y, z axes.Alternatively, the same vector could be written as8 9>< x 1 >=V 01 ˆ y 1…F:2†>: >;(now a `vector' in the matrix sense) in which the components are distinguished bypositions in the column.z 1z21V 21z z 1 V 202x 1y 2yx 2V 010y 1xFig. F.1 Vector addition.

Length of vector 639Addition and subtractionAddition and subtraction is de®ned by addition and subtraction of components.Thus, for example,V 02 ÿ V 01 ˆ…x 2 ÿ x 1 †i ‡…y 2 ÿ y 1 †j ‡…z 2 ÿ z 1 †k…F:3†The same result is achieved by the de®nitions of matrix algebra; thus8 9>< x 2 ÿ x 1 >=V 02 ÿ V 01 ˆ V 21 ˆ y 2 ÿ y 1>: >;z 2 ÿ z 1…F:4†'Scalar' productsThe scalar product of two vectors is de®ned asIfthenA B ˆ B A ˆ X3a k b kk ˆ 1A ˆ a x i ‡ a y j ‡ a z kB ˆ b x i ‡ b y j ‡ b z kA B ˆ a x b x ‡ a y b y ‡ a z b z…F:5†…F:6†…F:7†Using the matrix notation8>:a xa ya z9>=>;8>:b xb yb z9>=>;…F:8†the scalar product becomesA B ˆ A T B ˆ B T A…F:9†Length of vectorThe length of the vector V 21 is given, purely geometrically, asql 21 ˆ …x 2 ÿ x 1 † 2 ‡…y 2 ÿ y 1 † 2 ‡…z 2 ÿ z 1 † 2or in terms of matrix algebra aspl 21 ˆ V 21 V 21qˆ V T 21 V 21…F:10†…F:11†

640 Appendix FDirection cosinesDirection cosines of a vector are simply given from the de®nition of the projectedcomponent of lengths as (Fig. F.1)cos x ˆ vx ˆ x2 ÿ x 1l 21The scalar product may also be written as (Fig. F.2)A B ˆ B A ˆ l a l b cos ˆ V il 21; etc: …F:12†…F:13†where is the angle between the two vectors A and B and l a and l b are their lengths,respectively.'Vector' or cross productAnother product of vectors is de®ned as a vector oriented normally to the plane givenby two vectors and equal in magnitude to the product of the length of the two vectorsmultiplied by the sine of the angle between them. Further, the direction of the normalvector follows the right-hand rule as shown in Fig. F.2 in whichA B ˆ C…F:14†is shown.Thus, from the right-hand rule, we haveA B ˆÿB A…F:15†It is worth noting that the magnitude (or length) of C is equal to the area of theparallelogram shown in Fig. F.2.Using the de®nition of Eq. (F.6) and noting thati i ˆ j j ˆ k k ˆ 0i j ˆ k j k ˆ i k i ˆ j…F:16†CI c = I a , I b sin γBl bγl cAFig. F.2 Vector multiplication (cross product).

Elements of area and volume 641we havei j kA B ˆ deta x a y a z b x b y b zˆ…a y b z ÿ a z b y †i ‡…a z b x ÿ a x b z †j ‡…a x b y ÿ a y b x †kThere is no simple counterpart in matrix algebra but we can use the above to de®nethe vector C.y89>< a y b z ÿ a z b y >=C ˆ A B ˆ a z b x ÿ a x b z…F:17†>:>;a x b y ÿ a y b xThe vector product will be found particularly useful when the problem of erecting anormal direction to a surface is considered.Elements of area and volumeIf and are curvilinear coordinates, then the following vectors in the two-dimensionalplane8 98 9@x@x>< >=>=dn ˆ d dg ˆ d…F:18†@y@y>: >;>: >;@@de®ned from the relationship between the cartesian and curvilinear coordinates, arevectors directed tangentially to the and equal-constant contours, respectively.As the length of the vector resulting from a cross product of dn dg is equal tothe area of the elementary parallelogram we can write@x @x@ @d…area† ˆdet@y @yd d…F:19†@ @by Eq. (F.17).y If we rewrite A as a skew symmetric matrix20 ÿa z a y36 ^A ˆ 4 a z 0 ÿa7x 5ÿa y a x 0then an alternative representation of the vector product in matrix form is C ˆ ^AB.

642 Appendix FSimilarly, if we have three curvilinear coordinates , , in cartesian space, the`triple' or box product de®nes a di€erential volume@x @x @x@ @ @@y @y @yd…vol† ˆ…dn dg† df ˆ detd d d …F:20†@ @ @@z @z @z@ @ @This follows simply from the geometry. The bracketed product, by de®nition,forms avector whose length is equal to the parallelogram area with sides tangent to two of thecoordinates. The second scalar multiplication by a length and the cosine of the anglebetween that length and the normal to the parallelogram establishes a di€erentialvolume element.The above equations serve in changing the variables in surface and volumeintegrals.

Appendix GIntegration by parts in two or threedimensions (Green's theorem)Consider the integration by parts of the following two-dimensional expression…… @ dx dy@x …G:1†Integrating ®rst with respect to x and using the well-known relation for integration byparts in one-dimension… xRu dv ˆÿx L… xRv du ‡…uv† x ˆ xR ÿ…uv† x ˆ xLx L…G:2†we have, using the symbols of Fig. G.1…… @ @x……dx dy ˆÿ @dx dy ‡@x… yTy B … † x ˆ xR ÿ… dy …G:3†† x ˆ xLIf we now consider a direct segment of the boundary dÿ on the right-handboundary, we note thatdy ˆ n x dÿ…G:4†where n x is the direction cosine between the outward normal and the x direction.Similarly on the left-hand section we havedy ˆÿn x dÿ…G:5†The ®nal term of Eq. (G.3) can thus be expressed as the integral taken around ananticlockwise direction of the complete closed boundary:‡ n x dÿ…G:6†ÿIf several closed contours are encountered this integration has to be taken aroundeach such countour. The general expression in all cases is…… @‡ @x……dx dy ˆÿ @dx dy ‡ n@xx dÿ…G:7†ÿ

644 Appendix GFig. G.1 De®nitions for integrations in two-dimensions.Similarly, if di€erentiation in the y direction arises we can write…… @‡ @y……dx dy ˆÿ @dx dy ‡ n@yy dÿ…G:8†ÿwhere n y is the direction cosine between the outward normal and the y axis.In three dimensions by an identical procedure we can write……… @‡ @y………dx dy dz ˆÿ @dx dy dz ‡ n@yy dÿ …G:9†ÿwhere dÿ becomes the element of the surface area and the last integral is taken overthe whole surface. A similar expression holds for derivatives like Eq. (G.9) in x and z.

Appendix HSolutions exact at nodesThe ®nite element solution of ordinary di€erential equations may be made exact atthe interelement nodes by a proper choice of the weighting function in the weak(Galerkin) form. To be more speci®c, let us consider the set of ordinary di€erentialequations given byA…u†‡f…x† ˆ0…H:1†where u is the set of dependent variables which are functions of the single independentvariable `x' and f is a vector of speci®ed load functions. The weak form of this set ofdi€erential equations is given by… xRv T ‰A…u†‡f Š dx ˆ 0…H:2†x LThe weak form may be integrated by parts to remove all the derivatives from u andplace them on v. The result of this step may be expressed as… xR‰u T A …v†‡v T f Š dx ‡‰B …v†Š T B…u† x Rx Lˆ 0 …H:3†x Lwhere A …v† is the adjoint di€erential equation and B …v† and B…u† are terms on theboundary resulting from integration by parts.If we can ®nd the general integral to the homogeneous adjoint di€erential equationA …v† ˆ0…H:4†then the weak form of the problem reduces to… xRv T f dx ‡‰B …v†Š T B…u† x Rx Lˆ 0 …H:5†x LThe ®rst term is merely an expression to generate equivalent forces from the solutionto the adjoint equation and the last term is used to construct the residual equation forthe problem. If the di€erential equation is linear these lead to a residual whichdepends linearly on the values of u at the ends x L and x R . If we now let these bethe location of the end nodes of a typical element we immediately ®nd an expressionto generate a sti€ness matrix. Since in this process we have never had to construct anapproximation for the dependent variables u it is immediately evident that at the end

646 Appendix Hpoints the discrete values of the exact solution must coincide with any admissibleapproximation we choose. Thus, we always obtain exact solutions at these points.If we consider that all values of the forcing function are contained in f (i.e., no pointloads at nodes), the terms in B…u† must be continuous between adjacent elements. Atthe boundaries the terms in B…u† include a ¯ux term as well as displacements.As an example problem, consider the single di€erential equationwith the associated weak form… xRx Lvd 2 udx 2 ‡ P dudx ‡ f ˆ 0 d 2 udx 2 ‡ P dudx ‡ f dx ˆ 0…H:6†…H:7†After integration by parts the weak form becomes… xR d 2 vux Ldx 2 ÿ P dv du‡ vf dx ‡ vdxdx ‡ Pu ÿ dv xRdx u ˆ 0 …H:8†x LThe adjoint di€erential equation is given bya …v† ˆd2 vdx 2 ÿ P dvdx ˆ 0…H:9†and the boundary terms by( )vB …v† ˆÿ dv…H:10†dxand( )duB…u† ˆ dx ‡ Pu…H:11†uFor the above example two cases may be identi®ed:1. P zero, where the adjoint di€erential equation is identical to the homogeneousequation in which case the problem is called self-adjoint.2. P non-zero, where we then have the non-self-adjoint problem.The ®nite element solution for these two cases is often quite di€erent. In the ®rst casean equivalent variational theorem exists, whereas for the second case no such theoremexists.yIn the ®rst case the solution to the adjoint equation is given byv ˆ Ax ‡ B…H:12†which may be written as conventional linear shape functions in each element asN L ˆ xR ÿ xx R ÿ x LN R ˆ x ÿ x Lx R ÿ x L…H:13†y An integrating factor may often be introduced to make the weak form generate a self-adjoint problem;however, the approximation problem will remain the same. See Sec. 3.9.2.

Appendix H 647Thus, for linear shape functions in each element used as the weighting function theinterelement nodal displacements for u will always be exact (e.g., see Fig. 3.4)irrespective of the interpolation used for u.For the second case the exact solution to the adjoint equation isv ˆ Ae Px ‡ B ˆ Az ‡ B…H:14†This yields the shape functions for the weighting functionN L ˆ zR ÿ zNz R ÿ z R ˆ z ÿ z L…H:15†L z R ÿ z Lwhich when used in the weak form again yield exact answers at the interelementnodes.After constructing exact nodal solutions for u, exact solutions for the ¯ux at theinterelement nodes can also be obtained from the weak form for each element. Theabove process was ®rst given by Tong for self-adjoint di€erential equations.yy P. Tong. Exact solutions of certain problems by the ®nite element method. J. AIAA, 7, 179±80, 1969.

Appendix IMatrix diagonalization or lumpingSome of the algorithms discussed in this volume become more e cient if one of theglobal matrices can be diagonalized (also called `lumped' by many engineers). Forexample, the solution of some mixed and transient problems are more e cient if aglobal matrix to be inverted (or equations solved) is diagonal [see Chapter 12,Eq. (12.95) and Chapter 17, Sec. 17.2.4 and 17.4.2]. Engineers have persisted withpurely physical concepts of lumping; however, there is clearly a need for devising asystematic and mathematically acceptable procedure for such lumping.We shall de®ne the matrix to be considered as…A ˆ N T cN d…I:1†where c is a matrix with small dimension. Often c is a diagonal matrix (e.g., in mass orsimple least square problems c is an identity matrix times some scalar).When A is computedexactly it has full rank and is not diagonal ± this is called the consistent form of Asince it is computed consistently with the other terms in the ®nite element model. Thediagonalized form is de®ned with respect to `nodes' or the shape functions, e.g.,N i ˆ N i I; hence, the matrix will have small diagonal blocks, each with the maximumdimension of c. Onlywhenc is diagonal can the matrix A be completely diagonalized.Four basic lines of argument may be followed in constructing a diagonal form.The ®rst procedure is to use di€erent shape functions to approximate each term inthe ®nite element discretization. For the A matrix we use substitute shape functions N ifor the lumping process. No derivatives exist in the de®nition of A, hence, for thisterm the shape functions may be piecewise continuous within and between elementsand still lead to an acceptable approximation. If the shape functions used to de®ne Aare piecewise constants, such that N i in a certain part of the element surrounding thenode i and zero elsewhere, and such parts are not overlapping or disjoint, then clearlythe matrix of Eq. (I.1) becomes nodally diagonal as8 ……

Appendix I 649N iN iii(a)(b)Fig. I.1 (a) Linear and (b) piecewise constant shape functions for a triangle.this using a patch test to show that consistency is still maintained in the approximation.The functions selected need only satisfy the conditionXN i ˆ N i I with N i ˆ 1…I:3†for all points in the element and this also maintains a partition of unity property in allof . In Fig. I.1 we show the functions N i and N i for a triangular element.The second method to diagonalize a matrix is to note that condition (I.1) is simply arequirement that ensures conservation of the quantity c over the element. Forstructural dynamics applications this is the conservation of mass at the elementlevel. Accordingly, it has been noted that any lumping that preserves the integral ofc on the element will lead to convergent results, although the rate of convergencemay be lower than with use of a consistent A. Many alternatives have been proposedbased upon this method. The earliest procedures performed the diagonalization usingphysical intuition only. Later alternative algorithms were proposed. One suggestion,often called a `row sum' method, is to compute the diagonal matrix from8X…>< N T i cN k d i ˆ jA ij ˆ k i …I:4†>:0 i 6ˆ jThis simpli®es to8 …

650 Appendix IThe third procedure uses numerical integration to obtain a diagonal array withoutapparently introducing additional shape functions. Use of numerical integration toevaluate the A matrix of Eq. (I.1) yields a typical term in the summation form(following Chapter 9)…A ij ˆ N T i cN j d ˆ X…N T i cN j † nqJ q W q…I:8†where n q refers to the quadrature point at which the integrand is evaluated, J is thejacobian volume transformation at the same point and W q gives the appropriatequadrature weight.If the quadrature points for the numerical integration are located at nodes then (forstandard shape functions) by Eq. (I.3) the diagonal matrix isA ij ˆ c J iW i i ˆ j…I:9†0 i 6ˆ jwhere J i is the jacobian and W i is the quadrature weight at node i.Appropriate weighting values may be deduced by requiring the quadrature formulato exactly integrate particular polynomials in the natural coordinate system. Ingeneral the quadrature should integrate a polynomial of the highest complete orderin the shape functions. Thus, for four-noded quadrilateral elements, linear functionsshould be exactly integrated. Integrating additional terms may lead to improvedaccuracy but is not required. Indeed, only conservation of c is required.For low-order elements, symmetry arguments may be used to lump the matrix. It is,for instance, obvious that in a simple triangular element little improvement can beobtained by any other lumping than the simple one in which the total c is distributedin three equal parts. For an eight-noded two-dimensional isoparametric element nosuch obvious procedure is available. In Fig. I.2 we show the case of rectangularq1414136836136136436136836836436436144-nodeAll methods141368368-node(I.6)1361364369-nodeAll methods1361– 12131– 1213131– 12138-node(I.5) and (I.9)1– 12Fig. I.2 Diagonalization of rectangular elements by three methods.

Appendix I 65113 M113 M 3p = 1MO (h)0 Mp = 2O (h 2 )13 M0 M1– 60 M110 M1910 M20 M1– 60 Mp = 3O (h 5 )Fig. I.3 Diagonalization of triangular elements by quadrature.elements of four-, eight-, and nine-noded type and lumping by Eqs (I.5), (I.6) and(I.9).It is noted that for the eight-noded element some of the lumped quantities arenegative when Eq. (I.5) or Eq. (I.9) is used. These will have some adverse e€ects incertain algorithms (e.g., time-stepping schemes to integrate transient problems) andpreclude their use. In Fig. I.3 we show some lumped matrices for triangular elementscomputed by quadrature (i.e., Eq. (I.9)). It is noted here that the cubic element hasnegative terms while the quadratic element has zero terms. The zero terms areparticularly di cult to handle as the resulting diagonal matrix A no longer has fullrank and thus may not be inverted.Another aspect of lumping is the performance of the element when distorted fromits parent element shape. For example, as a rectangular element is distorted andapproaches a triangular shape it is desirable to have the limit triangular shape casebehave appropriately. In the case of a four-noded rectangular element the lumpedmatrix for all three procedures gives the same answer. However if the element isdistorted by a transformation de®ned by one parameter f as shown in Fig. I.4 thenthe three lumping procedures discussed so far give di€erent answers. The jacobiantransformation is given byJ ˆ ab…1 ÿ f †…I:10†and c is here taken as the identity matrix.The form (I.5) gives ab…1 ÿ f =3† at top nodesA ii ˆ …I:11†ab…1 ‡ f =3† at bottom nodesfa (1–f )aabxbFig. I.4 Distorted four-noded element.

652 Appendix Ithe form (I.6) gives ab…1 ÿ f =2† at top nodesA ii ˆ …I:12†ab…1 ‡ f =2† at bottom nodesand the quadrature form (I.9) yields ab…1 ÿ f † at top nodesA ii ˆ …I:13†ab…1 ‡ f † at bottom nodesThe four-noded element has the property that a triangle may be de®ned by coalescingtwo nodes and assigning them to the same global node in the mesh. Thus, thequadrilateral is identical to a three-noded triangle when the parameter f is unity.The limit value for the row sum method will give equal lumped terms at the threenodes while method (I.6) yields a lumped value for the coalesced node which istwo-thirds the value at the other nodes and the quadrature method (I.9) yields azero lumped value at the coalesced node. Thus, methods (I.6) and(I.9) give limitcases which depend on how the nodes are numbered to form each triangular element.This lack of invariance is not desirable in computer programs; hence for the fournodedquadrilateral, method (I.5) appears to be superior to the other two. On theother hand,we have observed above that the row sum method (I.5) leads to negativediagonal elements for the eight-noded element; hence there is no universal method fordiagonalizing a matrix.A fourth but not widely used method is available which may be explored to deducea consistent matrix that is diagonal. This consists of making a mixed representationfor the term creating the A matrix.Consider a functional given by… 1 ˆ 12u T cu d…I:14†The ®rst variation of 1 yields… 1 ˆu T cu dApproximation using the standard formu ^u ˆ N i ~u i ˆ N~uyields… 1 ˆ ~u T N T cN d ~u…I:15†…I:16†…I:17†This yields exactly the form for A given by Eq. (I.1).We can construct an alternative mixed form by introducing a momentum typevariable given byp ˆ cu…I:18†The Hellinger±Reissner type mixed form may then be expressed as…… 2 ˆ u T p d ÿ 1 2p T c ÿ1 p d…I:19†

and has the ®rst variation… 2 ˆ…ÿ u T p d ‡ p T u ÿ c ÿ1 p d …I:20†The term with variation on u will combine with other terms so is not set to zero;however the other term will not appear elsewhere so can be solved separately.If we now introduce an approximation for p asp ^p ˆ n j ~p j ˆ n~p…I:21†then the variational equation becomes…… 2 ˆ ~u T N T n d ~p ‡ ~p TIf we now de®ne the matrices…n T N d ~u ÿ n T c ÿ1 n d ~p…G ˆ N T n d…H ˆ n T c ÿ1 n dthen the weak form is GT ~u 2 ˆ‰~u T ~p T ŠG ÿ H ~pEliminating ~p using the second row of Eq. (I.24) gives ˆ0Appendix I 653…I:22†…I:23†…I:24†A ˆ G T H ÿ1 G…I:25†for which diagonal forms may now be sought. This form again has the same optionsas discussed above but, in addition, forms for the shape functions n can be soughtwhich also render the matrix diagonal.

Author indexPage numbers in bold refer to the list of references at the end of each chapter.Abdulwahab, F. 382, 398, 399, 400Abramowitz, N. 217, 219, 246Adams, M. 617, 618, 619Adee, J. 464, 467Ahmad, S. 237, 248; 252, 275Ainsworth, M. 387, 388, 392, 400Allen, D.N.de G. 1, 16; 84, 86; 146, 161Allwood, R.J. 355, 362Andel®nger, U. 295, 305Anderson, R.G. 222, 246Ando, Y. 74, 85; 361, 363Archer, J.S. 472, 491Argyris, J.H. 2, 3, 16; 127, 128, 139; 158, 159,162, 163; 172, 182, 183, 184, 198; 324, 344;494, 539Arlett, P.L. 140, 153, 155, 161; 470, 482, 483, 491Armstrong, C.G. 405, 427Arnold, D. 287, 305Arnold, D.N. 314, 320, 343, 344Arrow, K.J. 301, 306; 323, 344Arya, S.K. 79, 81, 85At-Abdulla, J. 355, 362Atluri, S.N. 276, 304Atluri, T.H. 355, 362BabusÆ ka, I. 251, 275; 276, 280, 304, 305; 361,363; 387, 392, 393, 394, 398, 399, 400; 401,404, 415, 416, 426, 427, 428; 430, 445, 457,458, 459, 463, 465, 466; 571, 575Bachrach, W.E. 226, 246; 260, 275Back, P.A.A. 153, 162Bahrani, A.K. 470, 482, 483, 491Bahrani, A.L. 140, 153, 155, 161Baiocchi, C. 161, 163Balestra, M. 326, 345Bampton, M.C.C. 480, 492Banerjee, P.K. 84, 86; 356, 362Bank, R.E. 387, 388, 391, 399Barlow, J. 371, 398Barsoum, R.S. 234, 248Bathe, K.J. 161, 163; 217, 246; 479, 490, 492;521, 541Batina, J. 446, 447, 453, 465, 466Baumann, C.E. 361, 363, 364; 404, 415, 427Bayless, A. 430, 465Baynham, J.A.W. 276, 280, 304; 311, 320, 343Bazeley, G.P. 31, 34, 38; 250, 274Becker, E.B. 82, 86Beckers, P. 251, 270, 274Beer, G. 84, 86; 229, 247Beisinger, Z.E. 474, 491Belytschko, T. 226, 246; 260, 275; 382, 398, 399;430, 453, 464, 465, 466, 467; 512, 521, 540;566, 574Benz, W. 464, 466Benzley, S.E. 234, 248Bercovier, H. 319, 344Beresford, P.J. 265, 268, 275Bettencourt, J.M. 505, 521, 540Bettess, P. 74, 85; 229, 233, 235, 247; 356, 363,487, 492; 545, 569, 572, 574Bey K.S. 404, 415, 427Bicà anicà , N. 261, 263, 275; 564, 574Biezeno, O.C. 2, 3, 17; 46, 84Bijlaard, P.P. 127, 139Biot, M.A. 559, 573Bischo€, M. 295, 305Blackburn, W.S. 234, 248Blacker, T.D. 382, 398, 399; 405, 427; 453, 466Bociovelli, L.L. 474, 491Bogner, F.K. 35, 38Boley, B.A. 545, 572Bonet, J. 464, 467Booker, J.F. 158, 162Boroomand, B. 97, 111; 251, 274; 383, 394, 397,399, 400

656 Author indexBorouchaki, H. 229, 247Bossak, M. 521, 522, 541Braess, D. 295, 305Brauchli, H.J. 298, 306; 376, 398Brebbia, C.A. 84, 86; 356, 361, 363Brezzi, F. 280, 287, 305; 314, 326, 333, 343, 345Brigham, E.O. 486, 492Brown, C.B. 161, 162Bruch, J.C. 161, 163Buck, K.E. 172, 184, 198Bugeda, G. 401, 411, 426Butter®eld, R. 356, 362Byskov, E. 234, 248Campbell, D.M. 285, 305Campbell, J. 77, 85; 176, 177, 185, 192, 198, 198,199; 237, 244, 248; 374, 376, 398Canann, S. 405, 427Cantin, G. 298, 302, 305; 505, 521, 540Caravani, P. 490, 492Carey, G.F. 82, 86Carlton, M.W. 564, 574Carpenter, C.J. 229, 247Carslaw, H.S. 470, 491Cassell, A.C. 153, 162Cavendish, J.C. 405, 427Chadwick, P. 628, 634Chan, A.H.C. 34, 38; 103, 111; 251, 256, 275;564, 565, 571, 572, 574, 575Chan, S.T.K. 159, 163Chang, C.T. 559, 564, 569, 573, 574Chari, M.V.K. 153, 162Chen, C.M. 375, 398Chen, D.P. 352, 362Chen, H.-C. 485, 492Chen, H.S. 74, 85; 487, 492Cheung, Y.K. 3, 17; 31, 34, 38; 99, 102, 111; 140,161; 250, 274; 470, 472, 491Chiba, N. 405, 427Chilton, L.K. 404, 415, 427Chopra, A.K. 471, 472, 486, 490, 491; 551, 573Chu, T.Y. 158, 162Ciarlet, P.G. 32, 38; 405, 428Clough, R.W. 2, 3, 16, 17; 19, 32, 37; 87,111;112, 126; 318, 344; 472, 486, 490, 491; 556,559, 573Codina, R. 326, 335, 345Co gnal, G. 383, 399Collatz, L. 446, 465Collins, T. 490, 492Comincioli, V. 161, 163Cook, R.D. 355, 362Coons, S.A.203 246Copps, K. 387, 392, 393, 394, 400Cornes, G.M.M. 355, 362Courant, R. 2, 3, 17; 19, 38; 81, 86; 326, 345Cowper, G.R. 222, 246Cox, H.L. 480, 492Craig, A. 480, 492; 571, 575Crandall, S.H. 1, 16; 46, 84; 468, 470, 491Crank, J. 499, 540Crochet, M. 217, 246Crouzcix, M. 314, 343Cruse, T.A. 234, 248Dahlquist, G.G. 521, 541Daniel, W.J.T. 550, 561, 571, 573Davies, A.J. 82, 86de Arrantes Oliveira, E.R. 32, 38; 251, 257, 275;404, 426de Vries, G. 159, 162, 163Demkowiecz, L. 387, 388, 399; 404, 415, 427Demmel, J. 464, 466; 480, 492; 610, 611, 618Desai, C.S. 160, 161, 163Dixon, J.R. 234, 247Doctors, L.J. 159, 163Doherty, W.P. 179, 198; 264, 266, 269, 275Dolbow, J. 430, 453, 465Douglas, J. 287, 305Duarte, A. 430, 444, 445, 453, 465Duarte, C.A. 430, 453, 457, 458, 459, 463, 464,465, 466Du€, I.S. 464, 466Dungar, R. 153, 162; 355, 362Dunham, R.S. 279, 305Dunne, P.C. 168, 169, 198Eiseman, P.R. 139, 139; 229, 247; 405, 427Elias, Z.M. 304, 306Ely, J.F. 149, 162Emson, C. 229, 233, 247Engelman, M.S. 319, 344Ergatoudis, J.G. 3, 128, 139; 169, 172, 174, 185,198, 198, 199; 237, 239, 248Eriksson, K. 500, 540Evans, J.H. 564, 574Evensen, D.A. 490, 492Farhat, Ch. 348, 362Felippa, C.A. 222, 246; 323, 344; 545, 550, 567,573, 574Ferencz, R.M. 514, 521, 540; 617, 618Finlayson, B.A. 1, 16; 46, 84Finn, N.D.L. 494, 539Fish, J. 430, 465Fix, G.J. 32, 34, 38; 206, 246; 251, 257, 275Fjeld, S. 128, 139Fletcher, C.A.T. 82, 86Flores, F. 453, 466Formaggia, L. 138, 139Forrest, A.R. 203, 246Forsythe, G.E. 446, 465

Author index 657Fortin, M. 314, 323, 324, 343, 344Fortin, N. 314, 343Fox, R.L. 35, 38Fraeijs de Veubeke, B. 34, 38; 183, 198; 251, 274;280, 304, 305, 306Franca, L.P. 81, 86; 326, 333, 338, 345; 494, 539France, E.P. 564, 574Fraser, G.A. 260, 275Frazer, R.A. 46, 84Freund, J. 82, 86Fried, I. 78, 82, 85, 86; 182, 198; 224, 246; 474,479, 491; 494, 539Friedmann, P.P. 545, 549, 550, 573Fujishiro, K. 405, 427Furukawa, T. 464, 467Gago, J.P. de S.R. 193, 197, 199; 387, 399; 415,428Galerkin, B.G. 2, 3, 17; 46, 47, 84Gallagher, R.H. 127, 139; 234, 247; 276, 304Gangaraj, S.K. 387, 392, 393, 394, 400Gantmacher, F.R. 518, 521, 540Gauss, C.F. 2, 3, 17Gear, C.W. 494, 521, 539, 541George, P.L. 229, 247Geradin, M. 545, 550, 573Ghaboussi, J. 264, 266, 269, 275GiD ± the Personal Pre/Postprocessor 139, 139;229, 247; 583, 618, 618Gingold, R.A. 464, 466Girault, V. 429, 446, 464Glowinski, R. 323, 324, 344Godbole, P.N. 318, 343Gong, N.G. 404, 415, 420, 421, 424, 427Gonzalez, O. 514, 521, 540Goodier, J.N. 22, 23, 38; 54, 85; 87, 99, 111; 114,126; 130, 139; 303, 306Gordon, W.J. 227, 246Gorensson, P. 545, 550, 573Gould, P.L. 355, 362Gourgeon, H. 356, 363Grammel, R. 46, 84Gresho, P.M. 319, 344Gri ths, A.A. 234, 247Gri ths, D. 84, 86Gri ths, R.E. 251, 275Gu, L. 430, 453, 465, 466Gui, W. 404, 415, 426Guo, B. 404, 415, 426Gupta, A.K. 604, 618; 633, 634Gupta, K.K. 479, 491; 550, 573Gupta, S. 551, 573Gurtin, M. 498, 540Guymon, G.L. 69, 85Hall, C.A. 227, 246Hall, J.F. 551, 573Hammer, P.C. 222, 246Hansbo, P. 387, 400Hardy, O. 404, 415, 427Hause, J. 139, 139; 229, 247; 405, 427Hayes, L.J. 570, 571, 574Hearmon, R.F.S. 91, 111Heinrich, J.C. 545, 572Hellan, K. 279, 304Hellen, T.K. 221, 234, 246, 248Hellinger, E. 285, 305Henrici, P. 494, 539Henshell, R.D. 234, 248; 355, 362Herrera, I. 356, 363Herrmann, L.R. 69, 85; 140, 159, 161, 163; 279,285, 304, 305; 309, 311, 343; 371, 373, 398Hestenes, M.R. 323, 344Hibbitt, H.D. 234, 248Hilber, H.M. 172, 184, 198; 521, 522, 532, 541Hildebrand, F.B. 60, 63, 84, 85, 86; 494, 539Hine, N.W. 494, 539Hinton, E. 78, 85; 261, 263, 275; 318, 344; 374,376, 398; 405, 406, 427; 474, 491; 545, 557,558, 559, 564, 567, 572, 573, 574Holbeche, J. 551, 573Hood, P. 314, 343Houbolt, J.C. 521, 522, 532, 541Hreniko€, A. 1, 3, 16Hsieh, M.S. 153, 162Huang, Y. 375, 398Huck, J. 545, 550, 573Huebner, K.H. 158, 162Hughes, T.J.R. 81, 86; 293, 305; 317, 318, 326,333, 338, 343, 344, 345; 494, 500, 512, 514,521, 522, 532, 539, 540, 541; 545, 566, 572,574Hulbert, G.M. 326, 333, 338, 345; 494, 500, 539Humpheson, C. 104, 111Hurty, W.C. 480, 489, 492Hurwicz, K.J. 301, 306Hurwicz, L. 301, 306, 323, 344Hurwitz, A. 518, 521, 540Ibrabimbegovic, A. 485, 492Idelsohn, S.R. 446, 447, 465, 466Iding, R. 404, 426Irons, B.M. 10, 17; 31, 34, 38; 118, 126; 128, 139;169, 172, 174, 176, 177, 181, 185, 192, 198,198, 199; 203, 221, 222, 236, 237, 239, 244,246, 248; 250, 252, 274; 494, 501, 540; 550,573Jaeger, J.C. 470, 491Jameson, A. 453, 466Javandel, I. 160, 163Jennings, A. 479, 480, 491, 492

658 Author indexJirousek, J. 356, 358, 361, 363Johnson, C. 387, 400, 494, 500, 539, 540Johnson, M.W. 32, 38Jones, W.P. 46, 84Jun, S. 464, 467Kamei, A. 234, 247Kantorovitch, L.V. 56, 85Kassos, T. 72, 85Katona, M. 494, 539Katz, I.N. 416, 428Kaupp, P. 229, 247Kazarian, L.E. 564, 574Kelly, D.W. 74, 85; 197, 199; 229, 235, 247; 356,363; 387, 399; 415, 428; 545, 572Key, S.W. 309, 343; 474, 491Kikichi, F. 74, 85, 361, 363Kikuchi, N. 161, 163; 283, 305Koch, J.J. 2, 3, 17Kong, D. 352, 362Koshgoftar, M. 161, 163Koslo€, D. 226, 246; 260, 275Krizek, M. 375, 398Krok, J. 429, 465Kron, G. 348, 362Krylov, V.I. 56, 85Kulasetaram, S. 464, 467Kvamsdal, T. 545, 550, 551, 573Kythe, P.K. 84, 86LadeveÁ ze, P. 383, 388, 391, 394, 399, 400; 426,428Ladkany, S.G. 355, 362Lambert, T.D. 494, 528, 529, 538Lan Guex 356, 363Lancaster, P. 430, 438, 465Larock, B.E. 159, 163Leckie, F.A. 472, 491Ledesma, A. 564, 574Lee, K.N. 79, 81, 86Lee, Nam-Sua 217, 246Lefebvre, D. 287, 292, 305Leguillon, D. 383, 388, 391, 399Lekhnitskii, S.G. 91, 92, 111Lesaint, P. 500, 540Leung, K.H. 559, 564, 567, 573, 574Levy, J.F. 318, 344Levy, N. 234, 236, 247Lewis, R.W. 104, 111; 494, 498, 504, 539; 545,565, 572, 574Li, B. 381, 399Li, G.C. 161, 163Li, S. 464, 467Li, X.D. 383, 399; 500, 537, 540Li, X.K. 565, 574Liebman, H. 2, 3, 17Ligget, J.A. 356, 363Lin, Q. 375, 381, 398Lindberg, G.M. 472, 491Liniger, W. 504, 521, 531, 540, 541Liszka, T. 429, 446, 447, 453, 464, 466Liu, P.L-F. 356, 363Liu, W.K. 464, 467; 566, 574Livesley, R.K. 14, 17Lo, S.H. 405, 427Lok, T.-S.L. 464, 467Lomacky, O. 234, 247Loubignac, C. 298, 302, 305Lowther, D.A. 229, 247Lu, Y. 430, 453, 465, 466Lucy, L.B. 464, 466Luenberger, D.G. 323, 344Luke, J.C. 161, 162Lyness, J.F. 154, 162Lynn, P.P. 79, 81, 85McDonald, B.H. 153, 162McHenry, D. 1, 3, 16MacKerle, J. 84, 86McLay, R.W. 32, 38Maione, V. 161, 163Makridakis, C.G. 361, 363Malkus, D.S. 314, 318, 343, 344; 474, 491Mallett, R.H. 35, 38Mandel, J. 571, 575MarcË al, P.V. 234, 234, 236, 247Mareczek, G. 159, 163; 172, 184, 198Marlowe, O.P. 222, 246Martin, H.C. 2, 3, 14, 16, 17; 87,111; 159, 162Martinelli, L. 453, 466Mavriplis, D.J. 453, 466Mayer, P. 140, 161Meek, J.L. 229, 247Mei, C.C. 74, 85; 487, 492Melenk, J.M. 430, 445, 457, 465Melosh, R.J. 127, 139Mikhlin, S.C. 32, 38; 47, 66, 85Minich, M.D. 35, 38Miranda, I. 514, 521, 540Mitchell, A.R. 84, 86; 251, 275Mitchell, S.A. 405, 427Moan, T. 374, 398Mohraz, B. 604, 618Monaghan, J.J. 464, 466Monk, P. 404, 415, 427Morand, H. 545, 549, 550, 573Morgan, K. 37, 38; 82, 86; 138, 139; 229, 247;326, 335, 345; 356, 363; 404, 406, 427Morton, K.W. 446, 465, 494, 538Mote, C.D. 196, 199Mowbray, D.F.234 248Mullen, R. 566, 574

Author index 659Mullord, P. 429, 446, 464Munro, E. 153, 162Murthy Krishna, A. 234, 248Nagtegaal, J.C. 320, 322, 344Nakazawa, S. 34, 38; 251, 275; 282, 299, 305,306; 318, 323, 324, 325, 344Nath, B. 149, 162NaÈ vert, U. 494, 500, 539Nay, R.A. 429, 465Naylor, D.J. 78, 85; 318, 343Nayroles, B. 430, 443, 453, 465Neitaanmaki, P. 375, 398Newmark, N.M. 1, 3, 16; 508, 512, 521, 522, 529,540, 541Newton, R.E. 153, 162; 470, 491; 545, 548, 572Nickell, R.E. 470, 491; 498, 540Nicolson, P. 499, 540Nishigaki, I. 405, 427Nithiarasu, P. 326, 335, 345Norrie, D.H. 159, 162, 163Oden, J.T. 66, 82, 85, 86; 283, 298, 305, 306; 314,343; 361, 363, 364; 376, 387, 388, 392, 398,399, 400; 404, 415, 426, 427, 428; 430, 444,445, 453, 457, 458, 459, 463, 464, 465, 466;494, 539Oglesby, J.J. 234, 247Oh, K.P. 158, 162Ohayon, R. 545, 549, 550, 551, 572, 573OnÄ ate, E. 326, 345; 401, 411, 426; 446, 447, 453,457, 465, 466Orkisz, J. 429, 446, 447, 449, 453, 464, 465Ortiz, P. 326, 335, 345Osborn, J.E. 276, 304Ostergren, W.J. 234, 236, 247Owen, D.R.J. 79, 81, 86; 154, 162Padlog, J. 127, 139Paidoussis, M.P. 545, 549, 550, 573Parekh, C.J. 470, 491; 494, 539Park, K.C. 567, 572, 574, 575Parks, D.M. 234, 248; 320, 322, 344Parlett, B.N. 479, 480, 492Pastor, M. 103, 111; 559, 564, 573, 574Patil, B.S. 470, 483, 484, 491Paul, D.K. 557, 558, 559, 564, 567, 572, 573, 574,575Pavlin, V. 429, 446, 464Pawsey, S.F. 318, 344Payne, N.A. 10, 17Peano, A.G. 192, 193, 199Peck, R.B. 470, 491Pedro, O. 135, 139Peiro, J. 138, 139Pelle, J.P. 383, 391, 394, 399, 400Pelz, R.B. 453, 466Penzien, J. 472, 486, 487, 490, 491, 492Peraire, J. 138, 139; 229, 247; 404, 406, 427Perrone, N. 429, 446, 464Phillips, D.V. 226, 246Pian, T.H.H. 32, 38; 234, 248; 290, 305; 352, 353,355, 362; 474, 491Pierre, R. 337, 338, 345Piltner, R. 356, 363Ping Tong 32, 38Pister, K.S. 279, 293, 305; 317, 343PitkaÈ ranta, J. 326, 333, 345; 494, 500, 539; 571,575Pook, L.P. 234, 247Powell, M.J.D. 323, 344Prager, W. 2, 3, 17; 19, 23, 38Price, M.A. 405, 427Przemieniecki, J.S. 14, 17Qu, S. 34, 38; 251, 275; 282, 305Rachowicz, W. 387, 388, 399; 404, 415, 427Radau 222, 246Raju, I.S. 234, 248Ralston, A. 252, 275; 610, 611, 618Ramm, E. 295, 305Randolf, M.F. 320, 344Rank, E. 405, 427Rao, A.K. 234, 248Rao, I.C. 157, 162Rashid, Y.R. 112, 126; 127, 139Rausch, R.D. 453, 466Raviart, P.A. 287, 305; 314, 343; 500, 539Rayleigh, Lord 2, 3, 17; 30, 38; 60, 85Razzaque, A. 34, 38; 250, 274Reddi, M.M. 158, 162Redshaw, J.C. 128, 139Reid, J.K. 464, 466Reinsch, C. 479, 491; 610, 611, 618Reissner, E. 285, 305Rheinboldt, C. 387, 399, 401, 426Rice, J.R. 234, 236, 247, 248; 320, 322, 344Richardson, L.F. 2, 3, 17, 33, 38Richtmyer, R.D. 446, 465; 494, 538Rifai, M.S. 294, 305Ritz, W. 2, 17; 30, 38; 60, 85Roberts, G. 545, 550, 573Robinson, J. 251, 275Rock, T. 474, 491Rockenhauser, W. 127, 139Rohde, S.M. 158, 162Rougeot, P. 391, 394, 400Routh, E.J. 518, 521, 540Roux, F.-X. 348, 362Rubinstein, M.F. 480, 489, 492Rudin, W. 166, 198; 442, 465

660 Author indexSabin, M.A. 405, 427Sabina, F.J. 356, 363Sabo, B.A. 72, 85Sacco, C. 446, 447, 465Salkauskas, K. 430, 438, 465Salonen, E-M. 82, 86Samuelsson, A. 268, 275Sandberg, G. 545, 550, 573Sander, G. 251, 270, 274Sani, R.L. 319, 344Satya Sai, B.V.K. 326, 335, 345Savin, G.N. 99, 111Scharpf, D.W. 158, 159, 162, 163; 172, 182, 184,198; 494, 539Schmit, L.A. 35, 38Schre¯er, B. 103, 111Schre¯er, B.A. 545, 564, 565, 572, 574Schweingruber, M. 405, 427Scott, F.C. 175, 176, 177, 185, 192, 198, 198, 199;237, 244, 245, 248, 249Scott, J.A. 464, 466Scott, V.H. 69, 85Seed, H.B. 556, 559, 573Sen, S.K. 355, 362Severn, R.T. 153, 162; 355, 362Shaw, K.G. 234, 248Shen, S.F. 229, 247Shepard, D. 430, 438, 444, 465Shiomi, T. 103, 111Silvester, P. 153, 162; 182, 198; 229, 247Simkin, J. 154, 162Simo, J.C. 34, 38; 251, 256, 275; 293, 294, 305;317, 325, 343, 345; 514, 521, 540Simon, B.R. 564, 574Sken, S.W. 46, 84Sloan, S.W. 320, 344Sloss, J.M. 161, 163Snell, C. 429, 446, 464Sokolniko€, I.S. 631, 634Sommer, M. 405, 427Soni, B.K. 139, 139; 229, 247; 405, 428Southwell, R.V. 1, 3, 8, 16, 17; 53, 84, 85, 86;304, 306Specht, B. 268, 275Stagg, K.G. 99, 102, 111Stakgold, I. 76, 85Stanton, E.L. 35, 38Stegun, I.A. 217, 219, 246Steinberg, S. 229, 247Stephenson, M.B. 405, 427Strang, G. 32, 34, 38; 206, 246; 251, 257, 275;464, 466; 611, 616, 618Strannigan, J.S. 234, 247Strouboulis, T. 251, 275; 387, 392, 393, 394, 398,400Stroud, A.H. 222, 246Strutt, J.W. 2, 3, 17; 30, 38; 60, 85Stummel, F. 251, 268, 275Sumihara, K. 290, 305Suri, M. 404, 415, 427Swedlow, J.L. 234, 247Synge, J.L. 2, 3, 17; 19, 38Szabo, B.A. 416, 428Szmelter, J. 19, 38Tabbara, M. 453, 466Taig, I.C. 203, 246Takizawa, C. 405, 427Tanesa, D.V. 157, 162Tarnow, N. 514, 521, 540Tautges, T.J. 405, 427Taylor, C. 314, 343; 470, 483, 484, 491Taylor, R.L. 34, 38; 161, 163; 173, 176, 177, 179,182, 198; 251, 256, 264, 265, 266, 268, 269,275; 276, 277, 279, 280, 282, 293, 304, 305;311, 317, 318, 320, 325, 343, 344; 404, 426;446, 447, 457, 465, 466; 480, 485, 492; 494,499, 521, 522, 532, 539, 540, 541; 551, 554,559, 573, 576, 611, 618Teodorescu, P. 356, 358, 363Terzhagi, K. 103, 111; 470, 491Thatcher, R.W. 229, 247Thomas, D.L. 490, 492Thomas, J.M. 287, 305Thomasset, F. 323, 344Thompson, J.F. 139, 139; 229, 247; 405, 427,428Thompson, J.P. 229, 247Thomson, H.T. 490, 492Tieu, A.K. 158, 162Timoshenko, S.P. 22, 38; 54, 76, 85; 87, 99, 111;114, 126; 130, 139; 303, 306Todd, D.K. 470, 491Tong, P. 50, 85; 234, 248; 353, 355, 362; 474, 491;647, 647Tonti, E. 66, 85Too, J. 318, 344Topp, L.J. 2, 3, 16; 87,111Touzot, C. 298, 302, 305Touzot, G. 430, 443, 453, 465Toyoshima, S. 299, 306, 323, 324, 325, 344Tracey D.M. 234, 236, 247, 248Tre€tz, E. 355, 362Treharne, C. 494, 501, 540Trowbridge, C.W. 154, 162Trujillo, D.M. 569, 574Turner, M.J. 2, 3, 16; 87,111Upadhyay, C.S. 251, 275; 387, 392, 393, 394, 398,400Utku, S. 429, 465Uzawa, H. 301, 306; 323, 344

Author index 661Vahdati, M. 138, 139; 229, 247; 404, 406, 427Vainberg, M.M. 66, 85Valliappan, S. 324, 344Vanburen, W. 234, 248Vardapetyan, L. 404, 415, 427Varga, R.S. 2, 3, 17; 53, 85Varoglu, E. 494, 539Vazquez, M. 326, 335, 345Verfurth, R. 387, 399Vesey, D.G. 429, 446, 464Villon, P. 430, 443, 453, 465Vilotte, J.P. 299, 306; 323, 324, 325, 344Visser, W. 140, 161; 470, 491Vogelius, M. 320, 344Wachspress, E.L. 217, 246; 262, 275Wahlbin, L.B. 375, 398Walker, S. 356, 361, 363Walsh, P.F. 234, 248Washizu, K. 30, 38; 73, 74, 85; 161, 162; 277,304; 498, 540Wasow, W.R. 446, 465Watson, J.O. 84, 86Watwood, V.B. 234, 247Weatherill, N.P. 139, 139; 229, 247; 405, 427, 428Weiner, J.H. 545, 572Weiser, A. 387, 388, 391, 399Westergaard, H.M. 149, 162Westermann, A. 387, 388, 399Wexler, A. 153, 162Wiberg, N.E. 348, 362; 382, 383, 398, 399, 400;500, 540Wilkinson, J.H. 479, 491; 610, 611, 618Williams, F.W. 490, 492Wilson, E.L. 124, 126; 179, 198; , 264, 265, 266,268, 269, 275; 470, 485, 490, 491, 492, 498,521, 522, 532, 540, 541, 606, 611, 618Winslow, A.M. 140, 153, 156, 161Witherspoon, P.A. 160, 163Wolf, J.P. 355, 361, 362Wong, K. 514, 521, 540Wood, W.L. 494, 498, 504, 516, 520, 521, 529,539, 541WroÂblewski, A. 356, 363Wu, J.S-S. 326, 333, 345; 405, 406, 427; 564, 574Wyatt, E.A. 229, 247Xi Kui Li 299, 306Xie, Y.M 564, 574Xu, K. 453, 466Yagawa, G. 464, 467Yamada, T. 464, 467Yamashita, Y. 405, 427Yan, N. 375, 398Yang, H.T.Y. 453, 466Yokobori, T. 234, 247Yoshida, Y. 355, 362Zarate, F. 453, 466Zhang, Y.F. 464, 467Zhang, Z. 381, 398, 399Zhong, W.X. 251, 275Zhu, J.Z. 97, 111; 377, 380, 381, 386, 392, 394,399, 400; 401, 404, 405, 406, 409, 415, 420,421, 424, 426, 427, 428Zhu, Q.D. 375, 381, 398Zielinski, A.P. 356, 361, 363Zienkiewicz, O.C. 3, 17; 31, 34, 37, 38; 74, 77, 78,81, 82, 85, 86; 97, 99, 102, 103, 104, 111;124, 126; 128, 138, 139; 140, 149, 153, 154,155, 161, 162; 169, 172, 174, 176, 177, 181,184, 185, 192, 197, 198, 198, 199; 226, 229,233, 235, 237, 239, 244, 245, 246, 247, 248,249; 250, 251, 256, 275; 276, 277, 279, 282,287, 292, 299, 304, 304, 305, 306; 311, 318,320, 323, 324, 325, 326, 334, 335, 343, 344,345, 356, 363; 377, 380, 381, 383, 386, 387,392, 393, 394, 397, 399, 400; 401, 404, 405,406, 409, 415, 420, 421, 424, 426, 427, 428;446, 447, 457, 458, 465, 466; 470, 472, 474,480, 482, 483, 484, 487, 491, 492; 494, 498,499, 505, 521, 539, 540, 541; 542, 545, 548,551, 554, 556, 557, 559, 564, 565, 567, 569,571, 572, 572, 573, 574, 575; 304, 576, 600,618Ziukas, S. 382, 398, 399Zlamal, M. 224, 246; 504, 521, 540

Subject indexa posteriori error estimator, 385Absciss\ae, integration, 220, 222, 223Accelerator matrix, convergence, 323, 324Accuracy, local, 499Acoustic problems, 546Adaptive analysis, 463Adaptive mesh re®nement, 401Adaptive methods, 398Adaptive time stepping, 500Adaptivity, h, 405Added mass, 153, 556ADI (Alternating Direction Implicit), 570Adjoint functions and operators, 75, 83, 645Aerofoil, 233, 545Aircraft, 138Airy stress function, 303Algebraic equations:linear, 577non-linear, 594simultaneous linear, 609Algorithm (see also algorithms, methods):Bossak±Newmark, 522CBS, 338, 341Gear, 521generalized Newmark, 508GiD, 583GN, 516GN11, 566GN22, 514, 515, 529, 531, 533, 553, 566, 571, 593GNpj, 508, 560Hilber±Hughes±Taylor, 522Houbolt, 522Liniger, 521Newmark (see SS22), 512, 515, 531, 533recurrence, 560SS (Single Step), 495±521SS11, 511, 565SS21, 519, 520SS22 (see Newmark), 511, 519, 520, 529, 530,531, 533, 566SS31, 520SS32, 520, 530SS42, 530SS42/41 algorithms, 529SSpj, 508, 514steady-stage, 594three-step, 527transient step-by-step, 560Uzawa, 301Wilson, 522Zlamal, 521Algorithmic damping, 536Algorithms:general single-step, 508GNpj, 514implicit, 511multi-step, 522multistep recurrence, 522recurrence, 522single-step, 495transient recurrence, 565transient step-by-step, 551Alternating direction implicit (ADI), 570Alternating direction of sweeps, 569Ampli®cation matrix, 501, 516Analysis procedure, modal, 486Anchorages, cable, 110Angular velocity, 119Anisotropic elasticity, 309Anisotropic materials, 91Anisotropic media, 144Anisotropic porous foundation, 149Anisotropic problems, 603Anisotropic seepage, 149Anisotropic valley, 102Anisotropy, strati®ed, 115Approximation:central di€erence, 530discontinuous stress, 286®nite element, 632

664 Subject indexApproximation ± cont.function, 431incomplete, 346least square, 78mixed, 309multi-step polynomial, 524partial ®eld, 346, 348point, 84point-based, 429three-®eld, 292, 329Approximation error, 499Approximation in time, 493Arbitrary tetrahedral meshes, 138Arbitrary weighting function, 495Arch dam in rigid valley, 239, 241, 242Area:elements of, 641tributary, 429Array computations, element, 598Area coordinates, 180±182, 186±190Array storage, 588Arrays:dynamically dimensioned, 578global, 585residual load, 577Arti®cial harbour, 487, 488ASCII ®le, 576Assembly equations, 13Assembly process, general, 9Asymptotic convergence rate, 59, 250, 257,406Atmospheric pressure, 556Augmented Lagrangian form, 324Automatic element subdivision, 226Auxiliary bending shape functions, 265Auxiliary functions, 303Average dynamic equation, 512Average stress error, 401Axes:orthogonal, 627principal, 149Axial co-ordinates, 112Axial tension, 7Axisymmetric elasticity matrix, D,116Axisymmetric heat ¯ow, 149Axisymmetric initial strain, 115Axisymmetric loading, 112Axisymmetric solids, 112Axisymmetric sti€ness matrix, 117Axisymmetric strain matrix, B, 115Axisymmetric stress analysis, 112Axisymmetric thermal strain, 115Axisymmetrical pressure vessel, 152Axisymmetry plane strain, 124Axisymmetry plane stress, 124B, strain matrix, 90±95, 115, 600B-bar method, 316BabusÆ ka patch test, 392BabusÆ ka±Brezzi conditions, 307, 326Back substitution, 611Background grid, 453Backward di€erence four-step algorithm, 521Backward di€erence implicit scheme, 538Band:variable, 588Bandwidth, 127, 337, 358Bandwidth minimization, 587Bar:pin-ended, 6, 15Barlow points, 371Bars:elastic, 2pin-jointed, 178Base polynomial solutions, 253, 258Basic shape function routines, 577Batch processing, 576Beam: 82cantilever, 376narrow, 171simply supported, 481Beam reactions, simple, 6Beam shape function, 36Beam vibration, 481Bearing pad, 157Bending moment, 7, 35Bending moments, internal, 26Bending of prismatic beams, 140Best ®t stresses, 98Bhakra dam±reservoir system, 557Bilinear mapping, 215Bilinear shape functions, 205Bimetallic shaft, hollow, 148Biomechanics, 244, 564Blade cascade, in®nite, 245Blade, cooled rotor, 507Blending functions, 226, 227Blocks of elements, 583Blocks of nodes, 583Body force components, 23Body force potential, 96Body force vectors, 208Body forces, 25, 54, 96, 166, 254, 4713D distributed, 132distributed, 23, 119Body of revolution, 118Bone-¯uid interaction, 564Bossak±Newmark algorithm, 522Boundaries:external, 379internal, 356repeating, 587

Subject index 665Boundary:¯ux, 74internal element, 256radiation, 548Boundary condition:forced, 44, 45, 66, 143natural, 44, 45, 66, 143, 278prescribed, 13radiation, 147traction, 356Boundary condition data, 577, 578Boundary elements, 356, 359Boundary methods, 82, 346, 355Boundary traction, 256Boundary Tre€tz-type elements, 357Boussinesq problem, 125, 135, 233Box product, 642Brick elements, hierarchical, 193Brick-type elements, 132Bubble function, 183, 311, 326cubic, 338hierarchical, 327Bubble mode, 329, 331, 334, 338Bubbles, distributed (cavitation), 556Bubnov±Galerkin method (see also Galerkin), 47Bulk modulus, 308, 325, 546Buttress dam, 102C programming language, 577C ÿ1 continuity, 279C 0 continuity, 43, 52, 164, 279C 0 continuous shape functions, 168±1981 dimensional, 183±1842 dimensional, 168±1773 dimensional, 184±198C 1 continuity, 29, 80, 268, 304, 443Cable anchorages, 110Cancellation of error, 223Cantilever, 374, 376Cartesian tensors:®rst rank, 630second rank, 630Cascade, in®nite blade, 245Cavitation, 556, 557, 560Cavities, 556Cavity problem:driven, 338, 340Cavity zones, 558CBS (Characteristic Based Split), algorithm, 326,335, 338, 341Central di€erence approximation, 530Centrifugal action, 238Centrifuge, 563, 564Chain rule, 526Characteristic Based Split (CBS), 326, 335, 338,341Characteristic equation, 484, 554Characteristic polynomial, 518Characteristic value, 478Checking, mesh data, 588Circular hole, 99Circular subdomains, 453Circumferential strain, 112Co-ordinates:axial, 112radial, 112Coalescing two nodes, 236Coe cient array, sparse, 578Coe cient matrix, global, 588Coe cient of thermal expansion, 94Cofactors, 129Collocation:point, 46, 83, 446, 447, 448, 451subdomain, 46, 83, 447, 453Taylor series, 497Command programming language, 577, 590Compact storage operation, 585Compatibility, 8, 19, 206, 250Complementary elastic energy principle, 302Complementary forms, 301Complementary heat transfer, 301Complete ®eld methods, 276Complete polynomial, 165, 171Completeness, 59Complex form, 12Complex impedance, 12Complex mesh generation, 429Complex pairs, 489Composite elements, 134Compressibility, 556, 559, 561Compressibility matrix, ®ctitious, 324Computer procedures, 576Concrete reactor pressure vessel, 123Condition: (see also boundary condition),constant gradient, 280constant strain, 31Routh±Hurwitz, 518, 519, 555stability, 252, 292, 294, 325Condition number, 197, 617Condition of stationarity, 60Conditional stability, 497, 501, 511, 566, 571Conditioning, 167Conditioning of hierarchical forms, 197Conditioning, matrix, 33Conditions:BabusÆ ka±Brezzi, 307, 326constraint, 270equilibrium, 19, 284, 302initial, 504, 505natural, 452stability, 359, 516support, 479

666 Subject indexConduction:heat, 41, 47, 123, 140, 149, 276, 279, 287, 370,395, 458, 486, 502, 505, 520transient heat, 57, 468, 477Conduction±convection, steady-state heat, 41Conductivity, 41, 149Conductors, electric, 156Conformability, geometrical, 206Conical water tank, 237Connection data, 577Connection:element, 578, 583, 585Conservation equation, 276Conservation, energy, 7Consistency, 252, 253, 273, 282, 325, 499,516Consistent damping matrix, 472Consistent mass matrix, 472, 476, 604Constant:radiation, 144spring, 268Constant t form, 526Constant energy norm density, 401Constant gradient condition, 280Constant jacobian, 269Constant strain, 250Constant strain condition, 31Constitutive equations, elastic, 631Constitutive relation, 284, 301Constrained functional, 78Constrained Lagragian forms, 83Constrained variational principle, 70, 76Constraint conditions, 270, 320Constraint variable, 280, 282, 285, 320Continuity: (see also shape functions)C ÿ1 , 279C 0 , 43, 52, 164, 279C 1 , 29, 80, 268, 304, 443displacement, 347excessive, 287interelement, 74stress ¯ux, 361subdomain, 349Continuity equation, 160, 546, 557Continuity of ¯ux, 283Continuous interpolation, 438Continuous systems, 1Continuum domain, 1Continuum, three-dimensional, 54Contravariant transformation, 15Contrived variational principle, 61Control data, 579Control program, 579Convective di€usion equation, 69Convergence, 58, 250, 252h, 59order of, 32, 59p, 59rate of, 223, 367, 381ultra, 380Convergence accelerator, 323, 324Convergence criteria, 31Convergence of elements, 213Convergence rate, 32asymptotic, 59, 250, 406Convergence requirements, 251Cooled rotor blade, 507Coordinate transformation, 629Coordinates, 577area, 180curvilinear, 213global, 208nodal, 578, 583parametric curvilinear, 203parent, 207transformation of, 15triangle normalized, 179volume, 186Corner shape functions, 176, 188, 190Corners, re-entrant, 365Coupled analyses, 542, 549, 560, 565Coupled metal forming, 545Coupled systems weak form, 548Couples, rotary inertia, 472Coupling, 351, 546Coupling forms, symmetric, 356Cracking, 106Crank±Nicolson scheme, 499, 500, 503, 504Crest gates (spillway), 110Crime, 255Critical damping ration, 490Cross criterion, 447Cross product, 640Crystal growth, 22, 94Cubic bubble, 338Cubic elements, 186Cubic tetrahedron, 188Cubic triangle, 182Cubic variation, 168Current, 11, 12Current balance, 11Current, electric, 155Curvature, 35Curved strata, 103Curvilinear coordinates, 213parametric, 203Curvilinear mapping, 446Cutouts, 196Cyclic permutation, 89, 327Cylinder, pressure loaded, 121D'Alembert principle, 471

Subject index 667D: elasticity matrix, 90, 116, 366, 600Dam, 151buttress, 102gravity, 110perforated, 419perforated gravity, 413Dam±reservoir system, Bhakra, 557Dam/reservoir interaction, 551Damped dynamic eigenvalues, 484Damped wave equation, 470Damping:algorithmic, 536linear, 470modal, 489Rayleigh, 472zero, 521Damping element, 548Damping matrix, 473, 487, 489, 490, 549, 592consistent, 472element, 471Damping ratio, critical, 490Damping terms, 550Darcy's law, 141Data:boundary condition, 577connection, 577control, 579element, 597Data checking, mesh, 588Data input module, 576, 577, 578Data input modules, 580De-re®ne (®nite element mesh), 406DEC Fortran 95, 576Decomposed, modally, 553Decomposition of matrix, 611Decomposition:modal, 486±490, 517, 530triangular, 611±614De®ciency, rank, 261De®nition of error, 365De®nition of matrix, 620De®nition, pointwise, 429De¯ection, 35lateral, 26Degenerate isoparametrics, 236Degree of freedom count, 311Degree of robustness, 252Degrees of freedom, 5Delauney triangulation, 405Delta function, Dirac, 513Delta:Kronecker, 170, 389, 441Densi®cation, 562Density:¯ux, 155mesh, 229Determinant, jacobian, 205Deviatoric projection matrix, 308Deviatoric strains, 307, 332, 335Deviatoric stress, 307, 308, 335Diagonal matrix, 578Diagonal scaling, 474Diagonal terms, zero, 73Diagonal:principal, 612zero, 294, 323Diagonality, 197Diagonalization, 474matrix, 648, 649Diagonals, unit, 611Di€erential equations, 1adjoint, 645Euler, 62non-linear, 66ordinary, 56self-adjoint, 66Di€erential volume, 642Di€erential, time, 468Di€use elements, 453Di€use ®nite element method, 430Di€usion, 141Di€usive term, 81Dilatation, thermal, 94Dimension:time, 468, 493Dimensioned arrays, dynamically, 578Dirac delta function, 513Direct elimination, 72Direct rank test, 311Direct solution, 578, 609, 610Direction cosine array inverse, 630Direction cosine array transpose, 630Direction cosines, 640Directional mesh re®nement, 425Disc:rotating, 196rotating, 237Discontinuities, load, 505Discontinuity, 279element, 33Discontinuous Galerkin method, 361, 494,500Discontinuous slope, 43Discontinuous stress approximation, 286Discrete analysis, 16Discrete coupled system, 549Discrete elements, 1, 18Discrete pressure equation, 336Discrete system: 1, 2, 14Discrete variables, 1Discretization, 1, 71®nite element, 143

668 Subject indexDiscretization ± cont.partial, 55Discretization error, 32Displacement, 635prescribed, 348radial, 112virtual, 24, 55Displacement compatibility, 8Displacement continuity, 347Displacement ®eld, 87Displacement formulation, 19Displacement frame, interface, 350Displacement function, 18, 87Displacement gradient, 628Displacement potential, 556Displacement shape functions, 348Displacements:nodal, 8, 88prescribed, 10rigid body, 31, 352SPR for, 383Distorted four-noded element, 651Distorted Lagrangian elements, 216Distorted serendipity elements, 216Distorted triangular prism, 213Distortion:gross, 204violent, 204Distributed body forces, 23, 119Distributed bubbles (cavitation), 556Distributed loads, nodal forces, 5Domain:analysis, 245continuum, 1L-shaped, 368, 413, 416, 420space-time, 494square, 412time, 495Domain element, polygonal, 356Domain integral, 355Domain of in¯uence, 435, 438Domains:in®nite, 229interpolation, 435multiple, 542overlapping, 544Domains overlap, 543Dome, hemispheric, 238Double summation (numerical integration),221Driven cavity problem, 338, 340Dummy index, 629Dyke foundation, 563Dynamic analysis, implicit±explicit, 545Dynamic eigenvalues, damped, 484Dynamic equation, average, 512Dynamic memory allocation, 579Dynamic problems, 468Dynamically dimensional arrays, 578Dynamics:explicit, 226soil, 544E: Young's Modulus of elasticity, 23, 35, 36,90±94, 116±117, 131±132, 583, 632Earthed trough, 155Earthquake, 562Earthquake forcing motion, 556Earthquake motion, 557Earthquake, simulated, 563E€ective stress, 103, 558E€ectivity index, 386, 392, 393, 413E€ects, surface wave, 555Eigenpairs, real, 489Eigenproblem, 336, 394, 485Eigenvalue computations, 604Eigenvalue participation factors, 494Eigenvalue problem, 625standard, 479Eigenvalues, 476, 478, 494, 518, 550damped dynamic, 484real, 477zero, 460Eigenvector, 478, 489, 494, 625Elastic bars, 2Elastic constants, 91Lame 604, 632Elastic constitutive equations, 631Elastic foundation, 268string on, 450, 455Elastic membrane, 268Elastic wave propagation, 520Elasticity, 284, 395, 458anisotropic, 309incompressible, 309, 323, 334linear, 4, 39, 459two-®eld incompressible, 308Elasticity matrix D, 90, 317, 366Elastodynamics, 494Electric conductors, 156Electric current, 155Electric potential, 140Electrical networks, 10Electrical resistance, 11Electromagnetics, 155, 482Electrostatic potential, 155Electrostatics, 153Element:damping, 548distorted four-noded, 651enhanced strain, 296general triangular, 182

Subject index 669incompatible, 264line, 183linear, 185non-robust, 252Pian±Sumihara, 343polygonal domain, 356quadratic, 185quadratic triangular, 460Simo±Rifai, 343tetrahedral, 127, 128triangular, 20, 87Element analysis, 576Element array computations, 598Element boundary, internal, 256Element connection, 578, 583, 585Element damping matrix, 471Element data, 597Element discontinuity, 33Element invariance, 580Element mass matrix, 471Element matrix, 192, 211singular, 225Element modes:singular, 256zero energy, 256Element partition, 566Element patch (see also SPR), 285, 384, 391Element residual, 606Element residual error estimator, 388Element residual, recovery by, 392Element robustness, 273, 304Element routines, 607multiple, 583Element shape functions, 165±198Element singularities, local, 226Element size, required, 405Element sti€ness calculation, 602Element sti€ness matrix, 5, 471, 604Element sti€ness matrix computation, 599Element subdivision, 59, 402automatic, 226Element test, single, 255Element-free Galerkin method, 429, 430, 453Elements of area, 641Elements of volume, 641Elements:blocks of, 583boundary, 356, 359boundary Tre€tz-type, 357brick-type, 132composite, 134convergence of, 213cubic, 186di€use, 453discrete, 1, 18distorted Lagrangian, 216distorted serendipity, 216exterior, 356®nite, not indexed because too pervasivehierarchical brick, 193hierarchical rectangle, 193hierarchical tetrahedron, 193hierarchical triangle, 193hybrid-stress, 355in®nite, 157, 229interior, 356irreducible, 282, 342isoparametric, 355linear, 183mapped, 200non-conforming, 33, 250one-dimensional, 183patch of, 377rectangular, 168singular, 235singularity, 200superparametric, 207suspect, 261tetrahedral, 186three-dimensional, 184Tre€tz-type, 356, 358, 359triangular prism, 189two-dimensional, 168Elimination:direct, 72forward, 611Gaussian, 577, 578Elimination of internal variables, 177Elimination of singularities, 226ELMLIB (Element Library), 608Elongation, 7relative period, 532Energy bound, 30, 34Energy conservation, 7Energy functional, 372Energy loss, radiation, 550Energy norm, 366, 386, 387, 405Energy principle, complementary elastic, 302Enhanced strain element, 293, 296Enhanced strain stabilization, 329, 338Equation:adjoint di€erential, 645average dynamic, 512characteristic, 484characteristic, 554conservation, 276continuity, 160continuity, 546, 557convective di€usion, 69damped wave, 470discrete pressure, 336equilibrium, 471

670 Subject indexEquation ± cont.®rst-order, 495heat, 592Helmholtz, 470, 482, 546hyperbolic, 468Laplace, 140±161, 333, 337, 555Maxwell's, 155parabolic, 468Poisson, 140±161, 333, 360, 381, 412quasi-harmonic, 140±161, 276Reynolds, 157second-order, 508wave, 481Equations:assembly, 13constraint, 320coupled, 560di€erential, 1elastic constitutive, 631equilibrium, 10, 55, 303, 332, 635Euler, 62, 63, 75linear algebraic, 577non-linear algebraic, 594non-linear di€erential, 66ordinary di€erential, 56quasi-harmonic, 360self-adjoint di€erential, 66singular, 77slope-de¯ection, 37symmetric, 278system, 14Equilibrated element residual estimator, 387, 388,394Equilibrating form subdomains, 353Equilibrating stresses, 354Equilibrium and energy, 630Equilibrium conditions, 19, 284, 302Equilibrium equations, 10, 55, 303, 332, 471,635Equilibrium equations weak form, 53Equilibrium, overall, 8Equivalent nodal forces, 23Error:approximation, 499average stress, 401cancellation of, 223de®nition of, 365discretization, 32energy norm, 387, 405local stress, 411local truncation, 516order of, 165RMS stress, 367total permissible, 405true, 406truncation, 499, 531Error estimates by recovery, 385Error estimation, 33, 251, 365, 401, 500Error estimator, 405a posteriori, 385element residual, 388energy norm, 386equilibrated residual, 388, 394explicit residual, 387global, 421implicit residual, 387residual, 387residual based, 365, 392Error magnitudes, permissible, 401Error norms, 365Escher modes, 263Euler equations, 62, 63, 75Exact integration, 320Exact nodal answers, 147Excessive continuity, 287Expansion:isotropic thermal, 131polynomial, 32quadratic, 500stress ®eld, 353thermal, 6three-dimensional thermal, 130Explicit dynamics, 226Explicit residual error estimator, 387Explicit scheme, 336, 497, 521Explicit-split process, 569Exponential Gauss function, 435Exterior elements, 356External boundaries, 379External loading, 25External nodal forces, 118External water pressure, 102External work, 24Extrapolation:Richardson, 33stress, 376Extrusion problem, 325Extrusion, metal, 544Factors, eigenvalue participation, 494Fast Fourier transform, 486FEAPpv (Finite Element Analysis Program), 576,579, 583, 618Fibreglass, 115Fick's law, 141Fictitious compressibility matrix, 324Field:displacement, 87incomplete, 346oil, 565thermal, 544uniform stress, 98

Subject index 671Field approximation:partial, 346±348Field expansion, stress, 353Field methods:complete, 276incomplete hybrid, 346Field problems, steady-state, 140Field strength, magnetic, 155File:ASCII, 576output, 591Fill-in (matrix), 578Film thickness, 157Films, squeeze, 158Finite di€erences, 3, 82, 84, 406, 429, 446, 493,570Finite element analysis program, 576Finite element approximation, 632Finite element discretization, 143Finite element program schematic, 577Finite element solution modules, 597Finite element not indexed because too pervasiveFinite increment calculus, 326Finite point method, 446Finite volume, 447, 451, 453First rank cartesian tensors, 630First-order equation, 495First-order problem, 565Fixed weighting function, 440Flexible wall, 551Flow, 12axisymmetric heat, 149¯uid, 140heat, 41ideal ¯uid, 160plastic, 544slow viscous, 320sphere heat, 122Stokes, 318, 335Flows:free surface, 159incompressible, 159irrotational, 159Fluid:interstitial, 564pore, 562Fluid ¯ow, 140ideal, 160Stokes, 318Fluid interaction, soil±pore, 558Fluid networks, 10Fluid oscillating with wall, 552Fluid phase, 561, 571Fluid problems, 481Fluid±structure interaction, 542, 543, 545, 547Fluid±structure systems, 571Fluid±structure time-stepping scheme, 553Fluids, incompressible, 555Flutter, 545Flux, 145, 146, 282, 388, 393, 452continuity of, 283normal, 301, 388Flux boundary, 74Flux continuity, stress, 361Flux density, 155Force:inertia, 470lateral inertia, 472Force matrix, 549Forced boundary condition, 44, 45, 66, 143Forced periodic response, 485, 551Forces:body, 25, 54, 96, 166, 254, 471distributed body, 23, 119equivalent nodal, 23external nodal, 118inertia, 470interelement, 28lateral inertia, 472nodal, 6, 95, 351, 471prescribed nodal, 578Forcing motion, earthquake, 556Forcing term, 406, 504, 506Forcing, continuous, 508Formula, two-point recurrence, 497(also see algorithms, methods, formulations)Formulation:displacement, 19enhanced strain, 293Galerkin, 326hybrid, 346incompatible, 264irreducible, 42, 276, 280, 542, 565, 587mixed, 42, 276, 304, 342, 346, 542stress function, 304two-®eld mixed, 284three-®eld mixed, 291Fortran, 577Fortran 95, DEC, 576Forward elimination, 611Foundation:anisotropic porous, 149dyke, 563elastic, 268rock, 102strati®ed, 150Foundation pile, 124Foundation response, 261Four-noded element, distorted, 651Four-point interpolation, 527Four-step algorithm, backward di€erence, 521Fourier series, 47, 167

672 Subject indexFourier transform, fast, 486Fourier's law, 141Fracture mechanics, 234Frame links, 353Frame of speci®ed displacements, 355Frame parameters, 351Frame, interface displacement, 350Free index, 629Free jet, 160Free response, 477, 484Free surface, 160, 356, 547Free surface ¯ows, 159Free vibration, 479, 550Frequencies, natural, 550Frequency response procedure, 486Frictional resistance, 471Frontal method, 405Fully reduced matrix, 611Function: see under names of functions, e.g.Green's, shapeFunctional, 1, 63, 76, 143constrained, 78energy, 372maximum of, 69minimum of, 69quadratic, 61variational, 40Galerkin approximation, time discontinuous, 536Galerkin form, 333, 645Galerkin formulation, 326Galerkin least square (GLS), 81, 82, 333, 338Galerkin method, 47, 64, 83, 453discontinuous, 361, 494, 500element-free, 429, 430, 453Galerkin method, 39, 46, 143, 284, 447, 451(see also Petrov±Galerkin, Bubnov±Galerkin)Gates, crest (spillway), 110Gauss function:exponential, 435truncated, 435Gauss points, 374, 376Gauss quadrature, 218, 219, 259, 371, 374, 598Gauss±Legendre points, 373Gauss±Lobatto quadrature, 457Gaussian elimination, 577, 578Gear algorithm, 521General assembly process, 9General single-step algorithms, 508General triangular element, 182General variational principle, 75Generalised patch test, 255Generalized Newmark algorithm, 508Generalized strain, 35Generators, hexahedral mesh, 405Geometrical conformability, 206GiD algorithm, program 583, 618Global arrays, 585Global coe cient matrix, 588Global coordinates, 208Global energy norm, 405Global energy norm error, 416Global error estimator, 421Global ®nite element approximation, 196Global functions, 360, 457Global matrix, 12Global shape functions, 50Global sti€ness matrix, 577Global tangent matrix, 593GLS (see Galerkin Least Squares), 338GN algorithm, 516GN11 algorithm, 566GN22 algorithm, 514, 515, 529, 531, 533, 553,566, 571, 593GNpj algorithm, 508, 512, 514, 560Gradient singularities, 365Gradients:interface, 379potential, 141recovery of, 375Gravity dam: 110perforated, 413Gravity loading, 102Gravity waves, surface, 547Green's functions, 356Green's theorem, 278, 301, 353, 643Grid:background, 453Grids, nesting, 229Gross distortion, 204Growth:crystal, 22, 94Gurtin's variational principle, 498h adaptivity, 405h convergence, 59h-re®nement, 402, 404, 412, 420Half-space, semi-in®nite, 229Hamilton's variational principle, 498Hanging points, 402Harbour:arti®cial, 487, 488Harmonic wave functions, 458Head:hydraulic, 12, 149Heat conduction, 47, 123, 140, 276, 279, 287,370, 395, 458, 486, 505, 520steady-state, 41, 149transient, 57, 468, 477weak form, 44Heat conduction±convection, steady-state, 41Heat equation, 592

Subject index 673Heat ¯ow, 41axisymmetric, 149sphere, 122Heat storage, 56Heat transfer problem, 303Heat transfer, complementary, 301Hellinger±Reissner variational principle, 285, 652Helmholtz equation, 470, 482, 546Hemispheric dome, 238Hermitian interpolation function, 435Herrmann theorem, 372, 377Hexahedral mesh generators, 405Hierarchical brick elements, 193Hierarchical bubble function, 327Hierarchical enhancement, 443Hierarchical forms, 167, 208Hierarchical interpolation, 444, 449, 463Hierarchical method, polynomial, 458Hierarchical moving least square, 454Hierarchical polynomials, 190Hierarchical rectangle elements, 193Hierarchical shape functions, 164, 166, 190, 194Hierarchical tetrahedron elements, 193Hierarchical triangle elements, 193Hierarchical variables, 431, 457, 571High Poisson's ratio, 78(see also almost incompressible, Poisson'sratio ˆ 1/2)High-frequency responses, 531Higher derivatives, 443Higher order patch test, 257, 261, 271Higher order stability, 520Hilber±Hughes±Taylor algorithm, 522Hinged member, 9Hole, circular, 99Hollow bimetallic shaft, 148Horace Walpole, 176Houbolt algorithm, 522Hourglass control, 226hp-clouds, 453hp-re®nement, 415, 422Hybrid:superelement, 350Hybrid form, 352Hybrid formulations, 346Hybrid-stress elements, 355Hydraulic head, 12, 149Hydrodynamic pressure, 149Hydrodynamics, smooth particle, 464Hydrostatic pressure, 103Hyperbolic equation, 468Ideal ¯uid ¯ow, 160Ideal ¯uid irrotational ¯ow, 140Impedance, complex, 12Impeller, 245Implicit algorithms, 511Implicit methods, 497Implicit residual error estimator, 387Implicit scheme:backward di€erence, 538Implicit schemes, 515Implicit±explicit dynamic analysis, 545Implicit±explicit partitions, 565, 566Implicit±explicit solution, 566Implicit±implicit schemes, 566Inclined pile wall, 150Incompatible element, 264, 268Incompatible element patch test, 267Incompatible formulation, 264Incompatible mode, 264, 269Incomplete approximation, 346Incomplete ®eld, 346Incomplete hybrid ®eld methods, 346Incompressibility, 226, 301, 311, 334, 335Incompressibility patch test, 326Incompressibility:near, 271, 293, 298, 307, 316, 320(see also large Poisson's ratio, Poisson'sratio ˆ 1/2)Incompressible constraint, 324Incompressible elasticity, 309, 323, 334three-®eld nearly 314two-®eld, 308Incompressible ¯ows, 159, 555Incompressible material, 110, 307, 308Independent linear relations, 225Independent strain relations, 226Index:dummy, 629e€ectivity, 386, 392free, 629robustness, 394, 395Indicial and matrix notation, 630, 633Indirect methods, 610Inertia couples, rotary, 472Inertia force, 470lateral, 472Inertia, rotatory, 472Inexact integration, 223In®nite blade cascade, 245In®nite domains, 229In®nite elements, 157, 229In®nite line, 231In®nitesimal patch, 254In®nity, 229, 356Initial conditions, 504, 505Initial strain, 25, 91, 94axi-symmetric, 115Initial strains, volumetric, 324Initial stress, 25Initial value problem, 495

674 Subject indexInput module:data, 576, 577, 578, 580mesh, 583Instability, 311, 503Insulator, porcelain, 155Integral:domain, 355volume, 117, 209surface, 210Integrating factor, 646Integrating points, 221Integration:exact, 320Gauss, 218gaussian, 219inexact, 223numerical, 117, 200, 217, 219, 224, 650order of, 224prism numerical, 219rectangular numerical, 219reduced, 250, 255, 258, 286, 318, 320selective, 255, 318, 319standard, 258tetrahedral numerical, 223triangle numerical, 222Integration by parts, 43, 143, 278, 351, 389, 549,643Integration formulae for tetrahedron, 637Integration formulae for triangle, 636Integration formulae:tetrahedra numerical, 223triangle numerical, 222Integration limits, 212Integration orders, low, 224Integration points, minimum, 375Integration procedures, numerical, 598Interaction:bone±¯uid, 564dam/reservoir, 551¯uid±structure, 542, 543, 545, 547reservoir±dam, 556soil±¯uid, 545soil±pore ¯uid, 558soil±pressure, 563structure±structure, 543Interelement continuity, 74Interelement forces, 28Interface displacement frame, 350Interface gradients, 379Interface line, 348Interface traction, 346, 349Interface traction link, 346, 349Interface with solid, 547Interior elements, 356Internal bending moments, 26Internal boundaries, 356Internal element boundary, 256Internal nodes, 177Internal parameters, nodeless, 177Internal variables, 265, 348Internal water pressure, 102Internal work, 24, 89Interpolation:continuous, 438domains, 435four-point, 527hierarchical, 444, 449, 463Lagrange, 228local, 361Shepard, 444three point, 525two point, 525Interpolation function, Hermitian, 435Interstitial ¯uid, 564Invariance, element, 580Inverse of matrix, 622Irreducible elements, 282, 342Irreducible form, 279, 304, 318, 353Irreducible form subdomains, 346Irreducible formulation, 42, 276, 280, 542Irregular mesh, 147Irregular partitions, 570Irrotational ¯ow: 159ideal ¯uid, 140Isoparametric element, 203±216, 355Isoparametrics, degenerate, 236Isotropic behaviour, 308Isotropic material, 90, 142transversely, 91Isotropic thermal expansion, 131Iteration, predictor±corrector, 515Iterative method, 12, 298, 323, 588, 610Iterative solution, 127, 616Iterative solvers, 567Jacobian, 210, 600, 650Jacobian determinant, 205Jacobian matrix, 209, 290transformation, 232Jacobian, constant, 269Jet over¯ow, 160Jet, free, 160Joints, rigid, 5Jump, 504K matrix, singular, 479k: permeability, 561Kernel method, reproducing, 464Kronecker delta, 170, 389, 441L-shaped domain, 368, 413, 416, 420L 2 norm, 366

Subject index 675L 2 projection of stress, 376Lagragian forms, constrained, 83Lagrange interpolation, 228Lagrange interpolation in time, 523Lagrange multiplier, 282, 346, 349, 361Lagrange polynomials 172, 445Lagrange shape functions 171Lagrangian elements, distorted, 216Lagrangian form, augmented, 324Laminar ¯ow regime, 12Laminations, 115Lame elastic constants 604, 632Lanczos methods 480Language statements, programming command, 595Language:C programming, 577command, 577command programming, 590Laplace equation, 140±161, 333, 337, 555Lateral de¯ection, 26Lateral inertia force, 472Lateral load, 6Law:Darcy's, 141Fick's, 141Fourier's, 141Least square, 440Least square approximation, 78Least square ®t, 372, 431Least square forms, 83Least square method, 76, 79Least square minimization, 385Least square smoothing, 298Least square:Galerkin (GLS), 338hierarchical moving, 454Legendre polynomials, 219, 444Length scale, 446Limit:stability, 571vapour pressure, 556Limitation:principle of, 280, 286, 304, 354Limits:integration, 212stability, 518Line:in®nite, 231interface, 348Line element, 183Linear algebraic equations, 577Linear damping, 470Linear elastic model, 558Linear elasticity, 4, 39, 459Linear element, 183, 185Linear operator, 67Linear polynomials, 88Linear relations, independent, 225Linear steady-state problems, 590Linearly varying stress ®eld, 98Liniger algorithm, 521Link:interface traction, 346, 349Links, frame, 353Liquefaction, soil, 562Listings, source, 576Load:lateral, 6Load arrays, residual, 577Load discontinuities, 505Load matrix, 3D, 132Loaded cylinder, pressure, 121Loaded sphere, pressure, 121Loaded string, 47Loading, 577, 587axisymmetric, 112external, 25gravity, 102non-symmetrical, 124Local accuracy, 499Local element singularities, 226Local ®nite element approximation, 196Local interpolations, 361Local stability, 538Local stress error, 411Local truncation error, 516Localization, 597Locally based functions, 50Locking, 78, 82, 258, 281, 298, 311, 314, 315,318Logarithmic terms, 118Low integration orders, 224Low reynolds number ¯ow, 322Lower bound solution, 34Lower triangular matrix, 578Lubrication, 157Lubrication of pad bearings, 140Lumped mass matrix, 471, 520, 605Lumped mass, negative, 474Lumping, 336, 648, 650Lumping procedures, 474Lumping:mass, 474, 475Lunar waveguide, 483Machine part, 413Magnet, 156Magnetic ®eld strength, 155Magnetic permeability, 155Magnetic potential, 140Magnetostatics, 153, 157Mapped elements, 200

676 Subject indexMapped mesh generation, 226Mapped shape functions, 230Mapping:bilinear, 215curvilinear, 446uniqueness of, 204, 205Mapping function, 212, 230, 254Mass:added, 153lumped, 471negative lumped, 474Mass array, 577Mass lumping, 474, 475Mass matrix, 473, 549, 592added, 556consistent, 472, 476, 604element, 471lumped, 520, 605Material:anisotropic, 91incompressible, 110, 307, 308isotropic, 90, 142strati®ed, 92transversely isotropic, 91Material properties, 577, 578, 585Material property speci®cation, 583Matrix:3D load, 1323D sti€ness, 1323D stress, 132added mass, 556ampli®cation, 501axi-symmetric sti€ness, 117B, 2D strain 90, 600B, 3D strain 130B, axi-symmetric strain, 115bubble, 329consistent damping, 472consistent mass, 472, 476, 604convergence accelerator, 323D, 2D elasticity 90, 600D, 3D elasticity 131D, axi-symmetric elasticity, 116damping, 473, 487, 489, 490, 549, 592decomposition, 611de®nition of, 620deviatoric projection, 308diagonal, 578element damping, 471element mass, 471element sti€ness, 5, 471, 604®ctitious compressibility, 324force, 549fully reduced, 611global, 12global coe cient, 588global sti€ness, 577global tangent, 593inverse of, 622jacobian, 209, 290lower triangular, 578lumped mass, 520, 605mass, 473, 549, 592non-singular, 252positive de®nite, 35pro®le, 614singular element, 225singular K, 479skew symmetric, 68, 628, 641sti€ness, 7, 12, 95, 208, 350, 549, 578strain-displacement, 558symmetric, 35, 61, 623tangent, 61three-dimensional strain, 130transformation jacobian, 232unsymmetric sti€ness, 591upper triangular, 578Matrix algebra, 620Matrix conditioning, 33Matrix diagonalization, 648, 649Matrix notation, 633Matrix singularity, 224, 225, 318Matrix subtraction, 621Maximum of functional, 69Maximum primary variables, 282Maximum principle, 76Maxwell's equation, 155Maxwell±Betti reciprocal theorem, 7Mechanics, fracture, 234Mechanisms, spurious, 252Media:anisotropic, 144non-homogeneous, 144porous, 103Medium:porous, 544, 564Member, hinged, 9Membrane, elastic, 268Memory allocation, dynamic, 579Memory management, 588Mesh:irregular, 147optimal, 405regular, 147Mesh data checking, 588Mesh density, 229Mesh enrichment, 406, 410Mesh generation:complex, 429mapped, 226Mesh generators, hexahedral, 405Mesh input module, 583

Subject index 677Mesh re®nement:adaptive, 401directional, 425Mesh regeneration, 402, 404, 410Meshes, arbitrary tetrahedral, 138Meshless methods, 429, 430, 453Metal extrusion, 544Metal forming, coupled, 545Method of weighted residuals, 46Method: (see also algorithm)B-bar, 316Bubnov±Galerkin, 47discontinuous Galerkin, 361, 494, 500element-free Galerkin, 429, 430, 453®nite point, 446®nite volume, 453frontal, 405Galerkin, 47, 64, 83, 451, 453iterative, 12, 298, 323, 588, 610iterative solution, 127least square, 76, 79Petrov±Galerkin, 47polynomial hierarchical, 458Rayleigh±Ritz, 60relaxation, 8reproducing kernel, 464row sum, 474, 649subspace, 480Uzawa, 323weighted residual, 42Methods:adaptive, 398boundary, 82, 346complete ®eld, 276explicit, 497®nite di€erence, 446®nite point, 446®nite volume, 451Galerkin solution, 447implicit, 497indirect, 610Lanczos, 480meshless, 429, 430, 453mixed, 307multistep, 522Newton's, 594stabilized, 326stress recovery, 298transient solution, 592Tre€tz, 346variational, 3Mid-edge nodes, 188Mid-face nodes, 188Mid-side shape functions, 176Mid-side shift, 234Middle half rule, 206Middle third rule, 206MINI, 338Minimization, 19bandwidth, 587least square, 385Minimum constraint variables, 282Minimum integration points, 375Minimum of functional, 69Minimum order of integration, 223Minimum principle, 76Minimum span, 449Miracles, 233Miscellaneous weight functions, 83Mixed approximation, 309Mixed form, 278, 284, 304, 319, 352, 353Mixed form subdomains, 349, 351Mixed formulation, 42, 276, 304, 342, 346,542two-®eld, 284Mixed formulations, 565, 587three-®eld, 291Mixed methods, 307Mixed patch test, 307, 325, 349Mixed problem, 299Mixtures, water/oil, 565Modal analysis, 485, 486Modal damping, 489Modal decomposition, 486, 487, 490, 517,530Modal orthogonality, 478Modally decomposed, 553Mode:bubble, 329, 331, 334, 338incompatible, 269spurious, 262zero-energy, 261Model:linear elastic, 558physical, 83Modes:Escher, 263normal, 478participation of, 490quadratic, 338rigid body, 348, 359singular element, 256vibration, 550wild, 263zero energy, 460Modi®ed variational principle, 74Module:data input, 576±578mesh input, 583results, 576solution, 576, 590Moli®cation, 43

678 Subject indexMoment:bending, 7, 35Moments, internal bending, 26Motion:earthquake, 557earthquake forcing, 556rigid body, 10, 226Moving least square approximation, 438, 443,457Moving weight function, 438, 439Multi-step algorithms, 522Multi-step polynomial approximation, 524Multigrid procedures, 571Multiple domains, 542Multiple element routines, 583Multiplication, vector, 640Multiplier:Lagrange, 70, 71, 73, 75, 76, 280, 282, 346,349, 361Multistep methods, 494, 522Multistep recurrence algorithms, 522Narrow beams, 171Natural boundary condition, 44, 45, 66, 143, 278Natural conditions, 452Natural frequencies, 550Natural variational principle, 61, 79, 80Near incompressibility, 271, 293, 298, 307, 316,320(see high Poisson's ration, Poisson's ratio ˆ 1/2)Negative lumped mass, 474Neighbour criterion, Voronoi, 447, 453Nesting grids, 229Networks:electrical, 10¯uid, 10Neutron transport, 500Newmark 531, 533, 593, 594Newmark algorithm 512Newmark algorithm (GN22), 515Newmark algorithm, generalized, 508Newton's methods 594Newton±Cotes quadrature 217Newton's laws of motion 54Nodal coordinates, 578, 583Nodal displacements, 8, 88, 587Nodal forces, 6, 95, 351, 471, 587equivalent, 23external, 118prescribed, 578Nodal forces distributed loads, 5Nodal point quadrature, 474Nodal points, 18Nodeless internal parameters, 177Nodeless variables, 177Nodes, 4blocks of, 583coalescing two, 236for corner, 188internal, 177mid-edge, 188mid-face, 188Non-conforming elements, 33, 250Non-homogeneous media, 144Non-homogeneous shaft, 148Non-linear algebraic equations, 594Non-linear di€erential equations, 66Non-linear problems, 494Non-robust element, 252Non-self adjoint operators, 81Non-self-adjoint problem, 646Non-singular matrix, 252Non-square integrable, 43Non-symmetrical loading, 124Non-trivial solutions, 554Non-unique mapping, 204Norm density, constant energy, 401Norm:energy, 366global energy, 405L2, 366Norm error, 365energy, 387energy, 405global energy, 416relative energy, 367Normal ¯ux, 301, 388Normal modes, 478Normalized coordinates, triangle, 179Notation:indicial, 626, 630Nuclear pressure vessel, 137, 149Number:condition, 197, 617penalty, 76, 317Numerical integration, 117, 200, 217, 224, 598,650(see also Gauss, integration, quadrature)Numerical integration procedures, 598Numerical integration:prism, 219rectangular, 219tetrahedral, 221triangle, 221Numerical oscillation, 475Numerical patch test, 257Numerically integrated ®nite elements, 236Oil ®eld, 565Oil recovery, 564One-dimensional elements, 183Opening, reinforced, 101

Subject index 679Operator:linear, 67non-self adjoint, 81self-adjoint, 67small strain, 55Optimal mesh, 405Optimal sampling points, 370, 371Optimal superconvergent sampling, 375Optimization, 323Order of convergence, 32, 59Order of error, 165Order of integration, 224Order stability, higher, 520Ordinary di€erential equations, 56Ori®ce, 320, 322Orthogonal axes, 627Orthogonal form, 191Orthogonal series, 167Orthogonality, 192, 489modal, 478Oscillation, 81, 503, 505numerical, 475Oscillations of natural harbour, 483Oscillations, three-dimensional, 482Oscillatory results, 307Outgoing waves, 548Output ®le, 591Output module, solution and, 577Overall equilibrium, 8Over¯ow, jet, 160Overlapping domains, 544Overspill, 204p convergence, 59p-re®nement, 402, 404, 415, 416, 421p-step algorithm, weighted residual, 528Pad:bearing, 157stepped (lubrication), 157Pairs, complex, 489PALLOC (memory management), 589Parabolic equation, 468Parallel computation, 567Parameter:frame, 351nodeless internal, 177penalty, 325system, 14undetermined, 30weighting, 406Parametric curvilinear coordinates, 203Parent coordinates, 207Partial discretization, 55Partial ®eld approximation, 346, 348Participation factors, eigenvalue, 494Participation of modes, 490Particle hydrodynamics, smooth, 464Partition of unity, 166, 430, 442, 445, 457Partition:element, 566Partitioned single-phase systems, 565Partitioning, 569, 624Partitions:implicit±explicit, 565implicit±explicit, 566irregular, 570Pascal triangle patch, 253Patch equilibrium, recovery by, 383Patch:element, 285, 384, 391in®nitesimal, 254superconvergent, 378Patch of elements, 377Patch recovery:superconvergent, 377, 383Patch test, 33, 250±270, 430Patch test A, 260Patch test B, 254, 260Patch test C, 260Patch test:BabusÆ ka, 392generalised, 255higher order, 257, 261, 271incompatible element, 267incompressibility, 326mixed, 307, 325, 349numerical, 257simple, 253weak, 267, 270Pathological arrangements of elements, 257PCG, 617Penalty function, 76, 77, 78, 83Penalty number, 76, 317Penalty parameter, 325Penalty term, 78Perforated dam, 413, 419Periodic response, 477forced, 485Permeability, 149, 559, 561Permeability, magnetic, 155Permissible error, 401, 405Permutation:cyclic, 89, 327Petrov±Galerkin method, 47Phase:¯uid, 561, 571Physical model, 83Pian±Sumihara, 287±297Piecewise constant shape functions, 649Piers, 110Pile in strati®ed soil, 125Pile wall, inclined, 150

680 Subject indexPile, foundation, 124Pin-ended bar, 6, 15Pin-jointed bars, 178PINPUT (program input command), 580Pipes, 12Pivot, 464Plane strain, 87axisymmetry, 124Plane stress, 20, 87axisymmetry, 124Plastic ¯ow, 544Plate, slopes of, 26PMACRn routines, 596PMESH, 583Point:collocation, 448quarter, 234saddle, 70superconvergent, 377Point approximation, 84Point-based approximation, 429Point collocation, 46, 83, 446, 451Pointers, 588, 589Points:Barlow, 371collocation, 447Gauss, 374, 376Gauss±Legendre, 373hanging, 402integrating, 221minimum integration, 375nodal, 18optimal sampling, 370, 371quadrature, 320sampling, 219superconvergent, 380Pointwise de®nition, 429Poisson equation, 140±161, 333, 360, 381, 412Poisson's ratio, 78, 272Poisson's ratio ˆ 1/2, 258, 271, 298, 320, 343Poisson's ratio, high, 78Polygonal domain element, 356Polynomial:characteristic, 518complete, 165stability, 528, 529Polynomial approximation, multi-step, 524Polynomial expansion, 32Polynomial hierarchical method, 458Polynomial order, 404, 406Polynomials:completeness of, 171hierarchical, 190Lagrange, 172, 445Legendre, 219, 444linear, 88Porcelain insulator, 155Pore ¯uid, 562Pore pressure, 102Porous foundation, anisotropic, 149Porous media, 103, 544, 564seepage through, 140Positive de®nite matrix, 35, 617Positive valued weighting function, 79Potential:body force, 96displacement, 556electric, 140electrostatic, 155magnetic, 140scalar, 156vector, 154Potential energy:total, 29, 349Potential gradients, 141Potential temperature, 276Power station:underground, 108, 110Preconditioned conjugate gradient method, 578,617Preconditioned conjugate gradient solver, 588Preconditioning, 571Predictor±corrector iteration, 515Preprocessor, 576Prescribed boundary condition, 13Prescribed displacement, 10, 348Prescribed nodal forces, 578Prescribed tractions, 352Pressure, 307, 549Pressure equation, discrete, 336Pressure limit, vapour, 556Pressure loaded cylinder, 121Pressure loaded sphere, 121Pressure stabilization, 332Pressure vessel, 243axisymmetrical, 152concrete reactor, 123nuclear, 122, 137, 149Pressure:atmospheric, 556external water, 102hydrodynamic, 149hydrostatic, 103internal water, 102pore, 102Primary variable, 280, 285, 320Primary variables, maximum, 282Princes of Serendip, 176Principal axes, 149Principal diagonal, 612Principle:constrained variational, 70, 76

Subject index 681contrived variational, 61d'Alembert, 471Galerkin, 143general variational, 75Gurtin's variational, 498Hamilton's variational, 498Hellinger±Reissner variational, 285, 652maximum, 76minimum, 76modi®ed variational, 74natural variational, 61, 79, 80variational, 60, 83, 143, 277, 291, 352, 353Principle of limitation, 280, 286, 304, 354Prism elements, triangular, 189Prism numerical integration, 219Prism:distorted triangular, 213rectangular, 184triangular, 190Problem:acoustic, 546anisotropic, 603Boussinesq, 125, 135, 233continuous, 1discrete, 1driven cavity, 338, 340dynamic, 468eigenvalue, 625extrusion, 325®rst-order, 565¯uid, 481heat transfer, 303initial value, 495linear steady-state, 590magnetostatic, 157mixed, 299non-linear, 494non-self-adjoint, 646saddle point, 76seepage, 159self-adjoint, 646slot, 342standard discrete, 2standard eigenvalue, 479steady-state ®eld, 140Stokes, 334three-dimensional, 576transient, 576two-dimensional, 145waveguide, 482Procedure: (see also methods, algorithms)analytical, 486computer, 576frequency response, 486lumping, 474modal analysis, 486multigrid, 571numerical integration, 598Rayleigh±Ritz, 19recovery, 374REP, 385single-step, 498SSpj, 522stabilized, 307weighted residual, 1Process:explicit-split, 569general assembly, 9recovery, 365stabilization, 571staggered solutions, 567weighted residual, 75Processing, batch, 576Product:box, 642cross, 640scalar, 639tensor, 628triple, 642vector, 210, 640Products:sum of, 623PROFIL (program command), 586Pro®le matrix, 614Pro®le solution scheme, 588Pro®le storage, 615Program:control, 579GiD, 618Programming command language statements,595Programming language:C, 577command, 590Projection matrix, deviatoric, 308Propagation, elastic wave, 520Properties:material, 577, 578, 585stability, 567Property speci®cation, material, 583Quadratic element, 185Quadratic expansion, 500Quadratic functional, 61Quadratic modes, 338Quadratic tetrahedron, 188Quadratic triangle, 182Quadratic triangular element, 460Quadrature: (see also integration)Gauss, 219, 371Gauss±Legendre, 219, 374, 598Gauss±Lobatto, 457

682 Subject indexQuadrature ± cont.gaussian, 259Newton±Cotes, 217nodal point, 474selective reduced, 270Quadrature points, 320Quadrature weight, 650Quadrilateral:Pian±Sumihara, 287, 290, 295Simo±Rifai, 294, 296Quarter point, 234Quasi-harmonic equation, 140, 142, 276, 360, 468r-re®nement, 404Radial co-ordinates, 112Radial displacement, 112Radiation, 546Radiation boundary, 548Radiation boundary condition, 147Radiation coe cient, 142Radiation constant, 144Radiation energy loss, 550Radiation of re¯ected waves, 487Radius:spectral, 532, 536Rank de®ciency, 261Rank test, direct, 311Rate:asymptotic convergence, 59, 250, 406convergence, 32Rate of convergence, 223, 367, 381Rayleigh damping, 472Rayleigh±Ritz method 19, 30, 60Re-entrant corners, 365Reactions, 259simple beam, 6Reciprocal theorem, Maxwell±Betti, 7Recovery:oil, 564superconvergent patch (SPR), 377, 383, 394Recovery based estimator, 365Recovery by element residual, 392Recovery by patch equilibrium, 383Recovery methods, stress, 298Recovery procedures, 298, 374, 375, 365, 374,375, 388Rectangle elements, hierarchical, 193Rectangular elements, 168Rectangular numerical integration, 219Rectangular prism, 184Rectangular prisms, serendipity family, 185Rectangular shaft torsion, 148Recurrence algorithm, 522, 560multistep, 522transient, 565Recurrence formulae, two-point, 497Recurrence relation, 229, 494, 498, 528single-step, 494Reduced integration, 250, 255, 258, 286, 318, 320Reduced quadrature, selective, 270Reduction of eigenvalue system, 480Re®nement, 406adaptive mesh, 401directional mesh, 425Regeneration:mesh, 402, 404, 410Regular mesh, 147Reinforced opening, 101Relative energy norm error, 367Relative period elongation, 532Relaxation method, 8Relaxation solution, 147Remainder, Taylor series, 513Remeshing, 402REP procedure, 385Repeatability, 244, 490Repeatability segments, 245Repeating boundaries, 587Reproducing kernel method, 464Required element size, 405Reservoir, 151, 558Reservoir bottom, 153Reservoir±dam interaction, 556Residual, 46, 590element, 606form, 590stresses 25tolerance on, 617weighted, 3, 40, 55Residual based error estimator, 365, 392Residual error estimator, 387Residual estimator, equilibrated element, 387Residual load arrays, 577Residual method:Galerkin weighted, 39, 46weighted, 42Residual procedures, weighted, 1, 75Resistance, 12electrical, 11frictional, 471Resolution (of matrix equations), 613Response:forced periodic, 485foundation, 261free, 477, 484high-frequency, 531periodic, 477transient, 477, 486Response procedure, frequency, 486Results module, 576Results, oscillatory, 307Revolution, body of, 118

Subject index 683Reynolds equation, 157Richardson extrapolation, 33Right-hand rule, 580Rigid body constraints, 350Rigid body displacements, 31, 352Rigid body modes, 348, 359Rigid body motion, 10, 226Rigid joints, 5Rigid valley and arch dam, 239, 241, 242RMS stress error robustness index, 394, 395Robustness:assessment of, 271degree of, 252element, 273, 304Robustness of error estimator, 392Robustness requirements, 561Rock foundation, 102Root mean square error (RMS), 366Rotary inertia couples, 472Rotating disc, 196, 237Rotating solids, 550Rotating sphere, 238, 240Rotational symmetry, 92Rotatory inertia, 472Rotor blade, cooled, 507Round-o€, 33, 591Routh±Hurwitz condition, 518, 519, 555Routines:element, 607multiple element, 583PMACRn, 596Row sum method, 474, 649Saddle point, 70, 76Safe zone for midpoint, 205Sampling points, 219optimal, 370, 371Sampling:optimal superconvergent, 375stress, 376Scalar potential, 156Scalar products, 639Scale, length, 446Scaling, diagonal, 474Scheme:Crank±Nicolson, 499, 500, 503explicit, 336, 521®nite di€erence, 570¯uid-structure time-stepping, 553implicit, 515implicit±implicit, 566multistep, 494pro®le solution, 588self-starting, 498sparse solution, 616staggered, 566, 571Second rank cartesian tensors, 630Second rank symmetric tensors, 634Second-order equation, 508Seepage, 140, 159, 160, 544, 559, 562Seepage, anisotropic, 149Selective integration, 255, 318, 319Selective reduced quadrature, 270Self-adjoint, 67, 68, 277Self-adjoint di€erential equations, 66Self-adjoint operator, 67Self-adjoint problem, 646Self-equilibrating, 25Self-starting scheme, 498Semi-discretization, 468, 494Semi-in®nite half-space, 229Serendip, Princes of, 176Serendipity, 190Serendipity cubic shape functions, 175Serendipity elements, distorted, 216Serendipity family rectangular prisms, 185Serendipity linear shape functions, 174Serendipity quadratic shape functions, 174Serendipity quartic shape functions, 175Serendipity shape functions, 174Series collocation:Taylor, 497truncated Taylor, 512Series remainder, Taylor, 513Series:Fourier, 47, 167orthogonal, 167Taylor, 32, 446Shaft torsion, rectangular, 148Shaft:hollow bimetallic, 148non-homogeneous, 148Shape function, 97Shape function subprograms, 598Shape function:beam, 36Shape functions, 21, 58, 140, 164, 357, 435auxillary bending, 265bilinear, 205C 0 continuous, 168corner, 176displacement, 348element, 165global, 50hierarchical, 164, 166, 190, 194incompatible, 268Lagrange, 171mapped, 230mid-side, 176piecewise constant, 649serendipity, 174serendipity cubic, 175

684 Subject indexShape functions ± cont.serendipity linear, 174serendipity quadratic, 174serendipity quartic, 175standard, 166, 384tetrahedron, 187tetrahedron cubic, 187tetrahedron linear, 187tetrahedron quadratic, 187triangle, 181Shear strain singularity, 236Shedding of vortices, 545Shells, 238Shepard interpolation shift, mid-side, 234Shrinkage, 22, 94Simo±Rifai element, 343Simo±Rifai quadrilateral, 294, 296Simple beam reactions, 6Simple patch test, 253Simply supported beam, 481Simpson one third rule, 218Simulated earthquake, 563Simultaneous linear algebraic equations,609Singing wire, 545Single element test, 255Single-phase systems, 567partitioned, 565Single-step algorithms, 495general, 508Single-step procedures, 498Single-step recurrence relations, 494Singular element matrix, 225Singular element modes, 256Singular elements, 235Singular elements by mapping, 234Singular equations, 77Singular functions, 458Singular k matrix, 479Singular solutions, 356Singularities, 59elimination of, 226gradient, 365local element, 226Singularity elements, 200Singularity, 356, 368, 370, 406, 420, 591avoidance of, 281matrix, 224matrix, 225matrix, 318shear strain, 236weak, 252Size, required element, 405Skeleton:soil, 544solid, 558Skew symmetric matrix, 68, 628, 641Slope:discontinuous, 43Slope-de¯ection equations, 37Slopes of plate, 26Slot problems, 342Slotted tension strip, 341Slow viscous ¯ow, 320Small strain operators, 55Smooth particle hydrodynamics, 464Smoothed stress, 298, 384Smoothing, 505least square, 298Soil dynamics, 544Soil liquefaction, 562Soil skeleton, 544Soil±¯uid interaction, 545Soil±¯uid system, 572Soil±pore ¯uid interaction, 558Soil±pressure interaction, 563Solid:elastic, 2interface with, 547Solids:axisymmetric, 112rotating, 550Solid skeleton, 558Solution:base, 253, 258boundary, 355direct, 578, 609, 610implicit±explicit, 566iterative, 616lower bound, 34relaxation, 147staggered, 571Tre€tz-type, 355Solution and output module, 577Solution iteration number, 590Solution method:iterative, 127Solution methods:Galerkin, 447transient, 592Solution module, 576, 590Solution modules, ®nite element, 597Solution scheme:pro®le, 588Solution schemes, sparse, 616Solutions exact at nodes, 645Solution processes, staggered, 567Solvers, iterative, 567Sound, speed of, 546Source listings, 576Space±time domain, 494Span, minimum, 449

Subject index 685Sparse coe cient array, 578Sparse solution schemes, 616Speci®cation, material property, 583Spectral form, 625Spectral radius, 532, 536Speed of sound, 546Sphere:pressure loaded, 121rotating, 238, 240Sphere heat ¯ow, 122SPR (Superconvergent Patch Recovery), 377±383SPR for displacements, 383Spring constant, 268Spurious mechanisms, 252Spurious mode, 262Spurious solutions, 333Square domain 412Squeeze ®lms, 158SS (Single Step), algorithms, 495±521SS11 algorithm, 511, 565SS21 algorithm, 519, 520SS22 algorithm, 511, 519, 520, 529, 530, 531, 533,566, 533, 566SS31 algorithm, 520SS32 algorithm, 520, 530SS42 algorithm, 530SS42/41 algorithms, 529SSpj algorithm 508, 514Stability, 282, 354, 499, 501, 519, 553, 555conditional, 497, 501, 566higher order, 520local, 538unconditional, 497, 501, 519, 521, 565, 567,569Stability check, 273Stability condition, 252, 292, 294, 325, 359, 516Stability criteria, 317, 352Stability limit, 518, 571Stability of general algorithms, 516Stability of mixed approximation, 280Stability polynomial, 528, 529Stability properties, 567Stability requirements, 337, 494Stabilization:enhanced strain, 329, 338pressure, 332Stabilized methods, 307, 326Stabilizing terms, 335Stable:conditionally, 511, 571unconditionally, 511Stagger, 568Staggered schemes, 566, 567, 571Standard discrete system, 2, 14Standard eigenvalue problem, 479Standard shape functions, 166, 384Stationary point of functional, 69Steady-stage algorithm, 594Steady-state ®eld problems, 140Steady-state heat conduction, 41, 149Steady-state heat conduction-convection, 41Steady-state problems, linear, 590Steady-state solutions, 562Step-by-step algorithm:transient, 551, 560Stepped pad bearing (lubrication), 157Sti€ness array, 577Sti€ness calculation, element, 602Sti€ness matrix, 7, 12, 95, 208, 350, 549, 5783D, 132axi-symmetric, 117element, 5, 471, 604global, 577unsymmetric, 591Sti€ness, tangent, 591Stokes ¯ow, 318, 335Storage:array, 588heat, 56pro®le, 615Storage allocation, 579Storage operation, compact, 585Strain, 18, 22, 635axi-symmetric initial, 115axi-symmetric thermal, 115circumferential, 112constant, 250deviatoric, 307generalized, 35initial, 25, 91, 94plane, 87thermal, 94volumetric, 308, 316, 317, 559Strain element, enhanced, 296Strain energy, 29Strain formulation, enhanced, 293Strain matrix, B, axi-symmetric 115Strain matrix, B, 2D 90, 600Strain matrix, B, 3D 130Strain operators, small, 55Strain relations, independent, 226Strain stabilization:enhanced, 329, 338Strain tensor, 22Strain-displacement, 317, 558Strains:deviatoric, 332, 335volumetric initial, 324Strata, 92curved, 103Strati®ed anisotropy, 115Strati®ed foundation, 150

686 Subject indexStrati®ed material, 92Strati®ed soil, pile in, 125Stream function, 160Streamline, 161Strength, magnetic ®eld, 155Stress:axisymmetry plane, 124deviatoric, 307, 308e€ective, 558initial, 25plane, 20, 87smoothed, 298, 384techtonic, 102total, 103Stress approximation, discontinuous, 286Stress components, 54Stress concentration, 99Stress error:average, 401local, 411RMS, 367Stress evaluation, 97Stress extrapolation, 376Stress ®eld expansion, 353Stress ®eld:linearly varying, 98uniform, 98Stress ¯ux continuity, 361Stress function, 359Stress function formulation, 304Stress function, Airy, 303Stress matrix, 3D, 132Stress recovery methods, 298Stress sampling, 376Stress tensor, symmetric cartesian, 54Stress±strain relation, 23, 635Stresses:best ®t, 98deviatoric, 335e€ective, 103equilibrating, 354recovery of, 375residual, 25tectonic, 25thermal, 108, 123String on elastic foundation, 450, 455String, loaded, 47Strip, slotted tension, 341Structure-structure interaction, 543Subdivision:automatic element, 226element, 59, 402Subdomain, 351, 355, 356Subdomain collocation, 46, 83, 447, 453Subdomain continuity, 349Subdomains with standard elements, 360Subdomains, 346, 348, 367circular, 453equilibrating form, 353irreducible form, 346mixed form, 349, 351Submatrix, 6Submerged surface, 149Subparametric, 207Subprograms, shape function, 598Subspace method, 480Substitution, back, 611Substructures, 177, 178Substructuring, 179Subtraction:matrix, 621vector, 639Su cient condition for convergence, 256Sum of products, 623Sumihara, Pian and, 291Summation convention, 626Summation, double (integration), 221Superconvergence, 370, 371, 374, 395Superconvergent patch, 378Superconvergent patch recovery (SPR), 377, 383Superconvergent point, 377, 380Superconvergent sampling, optimal, 375Superelement hybrid, 350Superelements, 360Superparametric elements, 207Support conditions, 479Surface:free, 160, 356, 547submerged, 149Surface ¯ows, free, 159Surface gravity waves, 547Surface integrals, 210Surface traction, 549Surface wave e€ects, 555Suspect elements, 261Symmetric cartesian stress tensor, 54Symmetric coupling forms, 356Symmetric equations, 278Symmetric matrix, 35, 61, 623skew, 628, 641Symmetric tensors, second rank, 634Symmetry, 67, 244, 490rotational, 92T elements (Tre€tz), 356Tangent matrix, 61global, 593Tangent sti€ness, 591Tank, conical water, 237Taylor series collocation, 497, 513Techniques, recovery, 388Techtonic stress, 25, 102

Subject index 687Temperature, 41, 141, 149, 282, 506Temperature change, 5, 22, 94Temperature, potential, 276Tension strip with slot, 338, 341Tension, axial, 7Tensor:strain, 22Tensor product, 628Tensor-indicial notation, 626Tensorial relations, 628Tetrahedra numerical integration formulae,223Tetrahedral element, 127, 128Tetrahedral elements, 186Tetrahedral meshes, arbitrary, 138Tetrahedral numerical integration, 221Tetrahedron:cubic, 188quadratic, 188Tetrahedron cubic shape functions, 187Tetrahedron elements, hierarchical, 193Tetrahedron linear shape functions, 187Tetrahedron quadratic shape functions, 187Tetrahedron shape functions, 187Theorem:Green's, 278, 301, 353, 643Herrmann, 372, 377Maxwell±Betti reciprocal, 7variational, 316Thermal conduction, 502Thermal dilatation, 94Thermal expansion, 6isotropic, 131three-dimensional, 130Thermal ®eld, 544Thermal strain, 94axi-symmetric, 115Thermal stresses, 108, 123Three point interpolation, 525Three-dimensional continuum, 54Three-dimensional elements, 184Three-dimensional oscillations, 482Three-dimensional problem, 576Three-dimensional strain matrix, 130Three-dimensional stress analysis, 127Three-dimensional thermal expansion, 130Three-dimensional transformer, 158Three-®eld approximation, 292, 329Three-®eld mixed formulations, 291Three-®eld nearly incompressible elasticity, 314Three-step algorithm, 527Time di€erential, 468Time discontinuous Galerkin approximation, 536Time domain, 495Time stepping, adaptive, 500Time stepping scheme, ¯uid-structure, 553TINPUT, 580, 582Tolerance on residual, 617Torsion of prismatic shafts, 140Torsion, rectangular shaft, 148Total permissible error, 405Total potential energy, 29, 349Total stress, 103Traction, 284boundary, 256interface, 349surface, 549Traction boundary condition, 356Traction link:interface, 346, 349Tractions, 347, 549interface, 346prescribed, 352Transfer coe cient, 142Transfer problem, heat, 303Transfer, complementary heat, 301Transform:fast Fourier, 486Transformation jacobian matrix, 232Transformation of coordinates, 15Transformation:contravariant, 15coordinate, 629z, 518Transformations, 208Transformer, 157three-dimensional, 158Transient computations, 604Transient heat conduction, 57, 468, 477Transient heat conduction equation, 470Transient heating of bar, 506Transient problem, 576Transient recurrence algorithms, 565Transient response, 477, 486Transient solution methods, 592Transient step-by-step algorithm, 560Transient step-by-step algorithms, 551Transpose of a matrix, 622Transpose of a product, 623Transversely isotropic material, 91Trapezoidal rule, 218Tre€tz methods, 346, 356, 358, 359Tre€tz-type elements, boundary, 357Tre€tz-type solution, 355Trial function, 3, 60Triangle:cubic, 182Pascal, 171, 173, 186, 215quadratic, 182Triangle elements, hierarchical, 193Triangle normalized coordinates, 179Triangle numerical integration, 221

688 Subject indexTriangle numerical integration formulae, 222Triangle shape functions, 181Triangular decomposition, 611, 612, 613,614Triangular element, 20, 87general, 182quadratic, 460Triangular matrix:lower, 578upper, 578Triangular prism elements, 189Triangular prism:distorted, 213Triangular prisms, 190Triangulation, Delauney, 405Tributary area, 429Triple product, 642Trough, earthed, 155True error, 406Truncated gauss function, 435Truncated taylor series collocation, 512Truncation error, 499, 531local, 516Two nodes, coalescing, 236Two point interpolation, 525Two-dimensional elements, 168Two-dimensional problem, 145Two-®eld incompressible elasticity, 308Two-®eld mixed formulation, 284Two-point recurrence formulae, 497Ultra convergence, 380Unconditional stability, 497, 501, 511, 519, 521,565, 567, 569Underground power station, 108, 110Undetermined parameters, 30Undrained behaviour, 561Uniform stress ®eld, 98Uniqueness of mapping, 205Unit diagonals, 611Unity:partition of, 166, 430, 442, 445, 457Universal shape function routines, 236Unreduced coe cient, 611Unsymmetric sti€ness matrix, 591Upper triangular matrix, 578User manual, 582Uzawa algorithm, 301Uzawa method, 323Valley, anisotropic, 102Vapour pressure limit, 556Variable band, 588Variational approaches, 39Variational forms, 452Variational functional, 40Variational methods, 3Variational principle, 60, 83, 143, 277, 291, 352,353complementary energy, 302constrained, 70, 76contrived, 61general, 75Gurtin's, 498Hamilton's, 498Hellinger±Reissner, 285, 652modi®ed, 74natural, 61, 79, 80Variational theorem, 316Vector addition, 638, 639Vector algebra, 638Vector multiplication, 640Vector potential, 154Vector product, 210, 640Vector subtraction, 639Vectors components, 638Velocity, 592angular, 119virtual, 55Vessel:axisymmetrical pressure, 152nuclear pressure, 137pressure, 243reactor pressure, 122Vibration modes, 550Vibration of earth dam, 481Vibration:beam, 481forced, 551free, 479, 550Violent distortion, 204Virtual displacement, 24, 55Virtual velocity, 55Virtual work, 53, 54, 83, 346, 351, 632Viscosity, 157, 471, 476Viscous ¯ow, slow, 320Voltage, 12Volume coordinates, 186±190Volume integral, 117, 209Volumetric initial strains, 324Volumetric strain, 308, 316, 317, 559Voronoi neighbour criterion 447, 453Vortices, shedding of, 545Wall:¯exible, 551inclined pile, 150Walpole, Horace, 176Water pressure:external, 102internal, 102Water tank, conical, 237

Subject index 689Water/oil mixtures, 565Wave e€ects, surface, 555Wave equation, 481damped, 470Helmholtz, 470, 482Wave functions, harmonic, 458Wave propagation, elastic, 520Wave transmission, 468Waveguide problem, 482Waveguide, lunar, 483Waves:outgoing, 548surface gravity, 547Weak form, 43, 142, 309, 346Weak form heat conduction, 44Weak form:coupled systems, 548equilibrium equations, 53Weak patch test, 267, 270Weak patch test satisfaction, 255Weak singularity, 252Weak statements, 42Weight coe cient, 220Weight function:moving, 438, 439Weight functions, miscellaneous, 83Weight:quadrature, 650Weighted least square ®t, 433Weighted residual, 55Weighted residual ®nite element, 495, 508Weighted residual method, 42Weighted residual p-step algorithm, 528Weighted residual procedures, 1Weighted residual process, 75Weighted residuals, 3, 40Weighting coe cient, 219Weighting function, 54, 278, 440, 447, 449, 500,510, 551arbitrary, 495®xed, 440positive valued, 79Weighting method, Galerkin, 451Weighting parameter, 406Weighting:collocation, 82Galerkin, 284Wild modes, 263Wilson algorithm 522Windows based systems 576Wire, singing, 545Work:external, 24internal, 24, 89virtual, 53, 54, 83, 346, 351, 632X-window applications 576Young's modulus, E, 23, 35, 36, 90±94, 116±117,131±132, 583, 632z transformation, 518Zero damping, 521Zero diagonal, 294, 323Zero diagonal terms, 73Zero eigenvalues, 460Zero energy element modes, 256Zero energy modes, 261, 460Zlamal algorithm 521Zone:active, 611:Poisson's ratio, 583, 632

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