Finite Element Method - The Basis (Volume 1)

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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
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Page 5: The Finite ElementMethodFifth editi
Page 10 and 11: Contentsix7.7 Concluding remarks 16
Page 12 and 13: Contentsxi12.7 Stabilized methods f
Page 15: Volume 2: Solid and structural mech
Page 18 and 19: 1Some preliminaries: the standarddi
Page 20 and 21: Table 1.1ENGINEERINGMATHEMATICSTria
Page 22 and 23: and for the corresponding nodal dis
Page 24 and 25: componentsf e " 08>:U iV iU nV nFin
Page 26 and 27: The boundary conditions 9components
Page 28 and 29: Electrical and ¯uid networks 11jP
Page 30 and 31: The general pattern 131 2(a)2 313 4
Page 32 and 33: Transformation of coordinates 15Onl
Page 34 and 35: 10. R.W. Clough. The ®nite element
Page 36 and 37: Direct formulation of ®nite elemen
Page 44 and 45: Generalization to the whole region
Page 46 and 47: Displacement approach as a minimiza
Page 48 and 49: Convergence criteria 312.5 Converge
Page 50 and 51: Displacement functions with discont
Page 52 and 53: The above energy expression is alwa
Page 54 and 55: and the sti€ness matrices for the
Page 56 and 57: 3Generalization of the ®nite eleme
Page 58 and 59:
Introduction 41then the approximati
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If the boundary conditions (3.12) a
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Integral or `weak' statements equiv
Page 64 and 65:
Approximation to integral formulati
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Page 68 and 69:
Page 70 and 71:
Virtual work as the `weak form' of
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In the ®rst set of bracketed terms
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Here the expression for K ij is ide
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Convergence 59In other examples of
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What are `variational principles'?
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`Natural' variational principles an
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`Natural' variational principles an
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Establishment of natural variationa
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To test for symmetry with any two f
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Constrained variational principles.
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This minimum is obviously zero at t
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Table 3.2 Finite element approximat
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6. Also attributed to Bubnov, 1913:
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4Plane stress and plane strain4.1 I
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Element characteristics 89in whicha
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and the initial strains as8>:" x0"
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Element characteristics 93for plane
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Element characteristics 95To obtain
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Examples ± an assessment of perfor
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Examples ± an assessment of perfor
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y521004910047CBA48100xRestrained in
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Some practical applications 103012-
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-300+200Arrow indicatestension-9220
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Some practical applications 107Temp
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yx250000 lb/ft 2Indicates tensionFi
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References 1114.6 Concluding remark
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Element characteristics 113z (v)mij
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Element characteristics 115With the
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Element characteristics 117Writing
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Element characteristics 119or notin
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Some illustrative examples 1215.2.9
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Early practical applications 123In
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Axisymmetry ± plane strain and pla
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6Three-dimensional stress analysis6
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Tetrahedral element characteristics
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in which23@N i@x ; 0; 0@N0; i@y ; 0
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Tetrahedral element characteristics
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Examples and concluding remarks 135
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Prestressing Concretesystem E = 5 x
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References 139generated mesh with a
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7.2 The general quasi-harmonic equa
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Finite element discretization 143Eq
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Some economic specializations 145Wi
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Examples ± an assessment of accura
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in which is the stress function, G
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Some practical applications 151H =
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Some practical applications 1531234
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Some practical applications 15590ea
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Some practical applications 157we
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Some practical applications 1592.62
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References 161The free surface, bei
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32. J.H. Argyris, G. Mareczek, and
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Standard and hierarchical concepts
Page 184 and 185:
Standard and hierarchical concepts
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Rectangular elements ± some prelim
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Completeness of polynomials 171yaa
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Rectangular elements ± Lagrange fa
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Rectangular elements ± `serendipit
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Elimination of internal variables b
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Triangular element family 179Fig. 8
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that contours of L 1 are equally pl
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The last shape again is a `bubble'
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Rectangular prisms ± `serendipity'
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Tetrahedral elements 18711(a) 4 nod
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ζ = 1123(a) 6 nodesζ4651121137102
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Hierarchic polynomials in one dimen
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Triangle and tetrahedron family 16;
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Triangle and tetrahedron family 16;
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Improvement of conditioning with hi
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15. J.G. Ergatoudis, B.M. Irons, an
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yη(x, y)(-1, 1) (1, 1)ηξξ(-1, -
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Use of `shape functions' in the est
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Use of `shape functions' in the est
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Variation of the unknown function w
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Consider,for instance,the set of lo
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Element matrices. Area and volume c
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Convergence of elements in curvilin
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Convergence of elements in curvilin
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Numerical integration ± one-dimens
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Numerical integration ± rectangula
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Numerical integration ± triangular
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Required order of numerical integra
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Required order of numerical integra
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Generation of ®nite element meshes
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In®nite domains and in®nite eleme
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Singular elements by mapping for fr
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Some practical examples of two-dime
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Three-dimensional stress analysis 2
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Symmetry and repeatability 245Analy
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References 24726. J. Peraire,M. Vah
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77. O.C. Zienkiewicz. Isoparametric
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Convergence requirements 251σ cons
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The simple patch test (tests Aand B
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Generalized patch test (test C) and
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10.5 The generality of a numerical
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Application of the patch test to pl
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Application of the patch test to pl
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(b)(a)(c)Fig. 10.9 Peculiar respons
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Application of the patch test to an
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Application of the patch test to an
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When a patch of elements is subject
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Table 10.3 Exact solution for patch
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Conclusion 2730-1Element size ln (h
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6. E.R. de Arantes Oliveira. The pa
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Introduction 277Clearly elimination
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Discretization of mixed forms ± so
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Stability of mixed approximation. T
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Stability of mixed approximation. T
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Two-®eld mixed formulation in elas
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Two-®eld mixed formulation in elas
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Here the stress interpolation is re
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Three-®eld mixed formulations in e
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functions yield a form8>:@N j@x@N j
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where…C ˆAt convergence the solu
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Complementary forms with direct con
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11.7.2 Solution using auxiliary fun
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8. R.S. Dunham and K.S. Pister. A
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12Incompressible materials,mixed me
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Two-®eld incompressible elasticity
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and is necessary for prevention of
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Two-®eld incompressible elasticity
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Three-®eld nearlyincompressible el
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Three-®eld nearlyincompressible el
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Reduced and selective integration a
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3 x 3 Gauss point integration2 x 2
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A simple iterative solution process
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A simple iterative solution process
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Stabilized methods for some mixed e
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Since the approximations for " v an
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Page 352 and 353:
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Page 356 and 357:
Page 358 and 359:
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perform well from those that do not
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References 34539. J.C. Simo and R.L
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Interface traction link of two (or
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Interface traction link of two or m
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which can be used directly in coupl
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We note that in the present de®nit
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Linking of boundary (or Trefftz)-ty
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13.7 Lagrange variables or disconti
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25. J.A. Ligget and P.L-F. Liu. The
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14Errors, recovery processes anderr
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De®nition of errors 367Similarly,
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5000300020001000500300200100NDF(0)4
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Superconvergence and optimal sampli
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which at an absolute minimum gives
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Recovery of gradients and stresses
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Superconvergent patch recovery ± S
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Page 400 and 401:
Recovery by equilibration of patche
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Error estimates by recovery 385and
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Other error estimators ± residual
Page 406 and 407:
Integrating by parts, we can write
Page 408 and 409:
Other error estimators ± residual
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Asymptotic behaviour and robustness
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Page 414 and 415:
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11. O.C. Zienkiewicz and J.Z. Zhu.
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15Adaptive ®nite element re®nemen
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Introduction 403(a) Original mesh(b
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here we have used 36 jjejj 2 ˆjjuj
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Number of degrees of freedom10 30 5
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Some examples of adaptive h-re®nem
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p-re®nement and hp-re®nement 415E
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p-re®nement and hp-re®nement 417y
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p-re®nement and hp-re®nement 419P
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p-re®nement and hp-re®nement 421P
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p-re®nement and hp-re®nement 423y
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p-re®nement and hp-re®nement 425(
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12. L. Demkowiez, J.T. Oden, W. Rac
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16Point-based approximations;elemen
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Function approximation 431approxima
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Function approximation 433If we con
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Function approximation 435Table 16.
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Function approximation 4376644220(x
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Moving least square approximations
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whereandH…x† ˆXnk ˆ 1w k …x
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For example, the ®rst derivatives
Page 462 and 463:
Hierarchical enhancement of moving
Page 464 and 465:
Point collocation ± ®nite point m
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andX ni ˆ 1N p i …x n†u i ˆ g
Page 468 and 469:
Galerkin weighting and ®nite volum
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Page 472 and 473:
Page 474 and 475:
Use of hierarchic and special funct
Page 476 and 477:
where L i are the area coordinates
Page 478 and 479:
Page 480 and 481:
Page 482 and 483:
6. J. Krok and J. Orkisz. A uni®ed
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55. W.K. Liu, S. Jun, S. Li, J. Ade
Page 486 and 487:
Direct formulation of time-dependen
Page 488 and 489:
Page 490 and 491:
Page 492 and 493:
Page 494 and 495:
form given byFree response ± eigen
Page 496 and 497:
e developed. The discussion of such
Page 498 and 499:
Free response ± eigenvalues for se
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Free response ± eigenvalues for se
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Forced periodic response 485and wri
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Transient response by analytical pr
Page 506 and 507:
as for true eigenvectors a ia T i M
Page 508 and 509:
References 491References1. S. Crand
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18The time dimension ± discreteapp
Page 512 and 513:
Simple time-step algorithms for the
Page 514 and 515:
Page 516 and 517:
Page 518 and 519:
Page 520 and 521:
Page 522 and 523:
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10501000700Simple time-step algorit
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General single-step algorithms for
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Page 530 and 531:
Page 532 and 533:
Page 534 and 535:
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Page 538 and 539:
Page 540 and 541:
Multistep recurrence algorithms 523
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Two point interpolation: p ˆ 1Eval
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Using the parameters q ˆ„ 10 W q
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Table 18.5 Identities between SSp2a
Page 548 and 549:
Some remarks on general performance
Page 550 and 551:
Page 552 and 553:
Page 554 and 555:
The discrete form of the governing
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3. P. Henrici. Discrete Variable Me
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51. G.G. Dahlquist. A special stabi
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Coupled problems ± de®nition and
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Fluid±structure interaction (Class
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and an additional variable q such t
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Page 572 and 573:
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Now, however, the term involving th
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in which r 0 n ‡ 1 is evaluated u
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Soil±pore ¯uid interaction (Class
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Partitioned single-phase systems ±
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Staggered solution processes 567thu
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23 2K 11 0 0 0 K 12 K 1kK 21 K 22
Page 588 and 589:
Staggered solution processes 571hav
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15. R. Ohayon. True symmetric formu
Page 592 and 593:
58. A.W. Craig and O.C. Zienkiewicz
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Introduction 577StartData Input Mod
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Data input module 579The notation u
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Data input module 581y13 14 15 167
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Data input module 583While the abov
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Data input module 585(a) Material p
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Data input module 587Table 20.4 Com
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Memory management for array storage
Page 608 and 609:
Since the problem given by Eq. (20.
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Solution module ± the command prog
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Solution module ± the command prog
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Computation of ®nite element solut
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Page 626 and 627:
Solution of simultaneous linear alg
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Page 634 and 635:
Page 636 and 637:
13. M. Adams. Heuristics for automa
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Matrix addition or subtraction 621T
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Symmetric matrices 623A sum of prod
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The eigenvalue problem 625The eigen
Page 644 and 645:
Indicial notation: summation conven
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Coordinate transformation 629The st
Page 648 and 649:
we can write the static equilibrium
Page 650 and 651:
Relation between indicial and matri
Page 652 and 653:
Appendix CBasic equations of displa
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Appendix ESome integration formulae
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Length of vector 639Addition and su
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Elements of area and volume 641we h
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Appendix GIntegration by parts in t
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Appendix HSolutions exact at nodesT
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Appendix H 647Thus, for linear shap
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Appendix I 649N iN iii(a)(b)Fig. I.
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Appendix I 65113 M113 M 3p = 1MO (h
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and has the ®rst variation… 2 ˆ
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656 Author indexBorouchaki, H. 229,
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658 Author indexJirousek, J. 356, 3
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660 Author indexSabin, M.A. 405, 42
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Subject indexa posteriori error est
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Subject index 665Boundary:¯ux, 74i
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Subject index 667D: elasticity matr
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Subject index 669incompatible, 264l
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Subject index 671Field approximatio
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Subject index 673Heat ¯ow, 41axisy
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Subject index 675L 2 projection of
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Subject index 677Mesh re®nement:ad
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Subject index 679Operator:linear, 6
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Subject index 681contrived variatio
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Subject index 683Reynolds equation,
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Subject index 685Sparse coe cient a
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Subject index 687Temperature, 41, 1
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Subject index 689Water/oil mixtures
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