Finite Element Method - The Basis (Volume 1)

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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 Òn 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, uÃ :8u û 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.
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Page 5: The Finite ElementMethodFifth editi
Page 8 and 9: ContentsPrefacexv1. Some preliminar
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 38 and 39: 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
Page 60 and 61: If the boundary conditions (3.12) a
Page 62 and 63: Integral or `weak' statements equiv
Page 64 and 65: Approximation to integral formulati
Page 70 and 71: Virtual work as the `weak form' of
Page 72 and 73: In the ®rst set of bracketed terms
Page 74 and 75: Here the expression for K ij is ide
Page 76 and 77: Convergence 59In other examples of
Page 78 and 79: What are `variational principles'?
Page 80 and 81: `Natural' variational principles an
Page 82 and 83: `Natural' variational principles an
Page 84 and 85: Establishment of natural variationa
Page 86 and 87:
To test for symmetry with any two f
Page 88 and 89:
Constrained variational principles.
Page 90 and 91:
Page 92 and 93:
Page 94 and 95:
Page 96 and 97:
This minimum is obviously zero at t
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Page 100 and 101:
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
Page 146 and 147:
Tetrahedral element characteristics
Page 148 and 149:
in which23@N i@x ; 0; 0@N0; i@y ; 0
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Tetrahedral element characteristics
Page 152 and 153:
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
Page 180 and 181:
32. J.H. Argyris, G. Mareczek, and
Page 182 and 183:
Standard and hierarchical concepts
Page 184 and 185:
Standard and hierarchical concepts
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Rectangular elements ± some prelim
Page 188 and 189:
Completeness of polynomials 171yaa
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Rectangular elements ± Lagrange fa
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Rectangular elements ± `serendipit
Page 194 and 195:
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
Page 226 and 227:
Consider,for instance,the set of lo
Page 228 and 229:
Element matrices. Area and volume c
Page 230 and 231:
Convergence of elements in curvilin
Page 232 and 233:
Convergence of elements in curvilin
Page 234 and 235:
Numerical integration ± one-dimens
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Numerical integration ± rectangula
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Numerical integration ± triangular
Page 240 and 241:
Required order of numerical integra
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Required order of numerical integra
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Generation of ®nite element meshes
Page 246 and 247:
In®nite domains and in®nite eleme
Page 248 and 249:
Page 250 and 251:
Page 252 and 253:
Singular elements by mapping for fr
Page 254 and 255:
Some practical examples of two-dime
Page 256 and 257:
Three-dimensional stress analysis 2
Page 258 and 259:
Page 260 and 261:
Page 262 and 263:
Symmetry and repeatability 245Analy
Page 264 and 265:
References 24726. J. Peraire,M. Vah
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77. O.C. Zienkiewicz. Isoparametric
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Convergence requirements 251σ cons
Page 270 and 271:
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
Page 298 and 299:
Stability of mixed approximation. T
Page 300 and 301:
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
Page 308 and 309:
Three-®eld mixed formulations in e
Page 310 and 311:
Page 312 and 313:
functions yield a form8>:@N j@x@N j
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Page 316 and 317:
where…C Ât convergence the solu
Page 318 and 319:
Complementary forms with direct con
Page 320 and 321:
11.7.2 Solution using auxiliary fun
Page 322 and 323:
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
Page 330 and 331:
Two-®eld incompressible elasticity
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Three-®eld nearlyincompressible el
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Three-®eld nearlyincompressible el
Page 336 and 337:
Reduced and selective integration a
Page 338 and 339:
3 x 3 Gauss point integration2 x 2
Page 340 and 341:
A simple iterative solution process
Page 342 and 343:
A simple iterative solution process
Page 344 and 345:
Stabilized methods for some mixed e
Page 346 and 347:
Page 348 and 349:
Since the approximations for " v an
Page 350 and 351:
Page 352 and 353:
Page 354 and 355:
Page 356 and 357:
Page 358 and 359:
Page 360 and 361:
perform well from those that do not
Page 362 and 363:
References 34539. J.C. Simo and R.L
Page 364 and 365:
Interface traction link of two (or
Page 366 and 367:
Interface traction link of two or m
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which can be used directly in coupl
Page 370 and 371:
We note that in the present de®nit
Page 372 and 373:
Linking of boundary (or Trefftz)-ty
Page 374 and 375:
Page 376 and 377:
Page 378 and 379:
13.7 Lagrange variables or disconti
Page 380 and 381:
25. J.A. Ligget and P.L-F. Liu. The
Page 382 and 383:
14Errors, recovery processes anderr
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De®nition of errors 367Similarly,
Page 386 and 387:
5000300020001000500300200100NDF(0)4
Page 388 and 389:
Superconvergence and optimal sampli
Page 390 and 391:
which at an absolute minimum gives
Page 392 and 393:
Recovery of gradients and stresses
Page 394 and 395:
Superconvergent patch recovery ± S
Page 396 and 397:
Page 398 and 399:
Page 400 and 401:
Recovery by equilibration of patche
Page 402 and 403:
Error estimates by recovery 385and
Page 404 and 405:
Other error estimators ± residual
Page 406 and 407:
Integrating by parts, we can write
Page 408 and 409:
Other error estimators ± residual
Page 410 and 411:
Asymptotic behaviour and robustness
Page 412 and 413:
Page 414 and 415:
Page 416 and 417:
11. O.C. Zienkiewicz and J.Z. Zhu.
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15Adaptive ®nite element re®nemen
Page 420 and 421:
Introduction 403(a) Original mesh(b
Page 422 and 423:
here we have used 36 jjejj 2 ˆjjuj
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Number of degrees of freedom10 30 5
Page 426 and 427:
Some examples of adaptive h-re®nem
Page 428 and 429:
Page 430 and 431:
Page 432 and 433:
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
Page 438 and 439:
p-re®nement and hp-re®nement 421P
Page 440 and 441:
p-re®nement and hp-re®nement 423y
Page 442 and 443:
p-re®nement and hp-re®nement 425(
Page 444 and 445:
12. L. Demkowiez, J.T. Oden, W. Rac
Page 446 and 447:
16Point-based approximations;elemen
Page 448 and 449:
Function approximation 431approxima
Page 450 and 451:
Function approximation 433If we con
Page 452 and 453:
Function approximation 435Table 16.
Page 454 and 455:
Function approximation 4376644220(x
Page 456 and 457:
Moving least square approximations
Page 458 and 459:
whereandH…x† ˆXnk ˆ 1w k …x
Page 460 and 461:
For example, the ®rst derivatives
Page 462 and 463:
Hierarchical enhancement of moving
Page 464 and 465:
Point collocation ± ®nite point m
Page 466 and 467:
andX ni ˆ 1N p i …x n†u i ˆ g
Page 468 and 469:
Galerkin weighting and ®nite volum
Page 470 and 471:
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
Page 484 and 485:
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
Page 500 and 501:
Free response ± eigenvalues for se
Page 502 and 503:
Forced periodic response 485and wri
Page 504 and 505:
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
Page 510 and 511:
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:
Page 524 and 525:
10501000700Simple time-step algorit
Page 526 and 527:
General single-step algorithms for
Page 528 and 529:
Page 530 and 531:
Page 532 and 533:
Page 534 and 535:
Page 536 and 537:
Page 538 and 539:
Page 540 and 541:
Multistep recurrence algorithms 523
Page 542 and 543:
Two point interpolation: p ˆ 1Eval
Page 544 and 545:
Using the parameters q ˆ„ 10 W q
Page 546 and 547:
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
Page 556 and 557:
3. P. Henrici. Discrete Variable Me
Page 558 and 559:
51. G.G. Dahlquist. A special stabi
Page 560 and 561:
Coupled problems ± de®nition and
Page 562 and 563:
Fluid±structure interaction (Class
Page 564 and 565:
Page 566 and 567:
Page 568 and 569:
and an additional variable q such t
Page 570 and 571:
Page 572 and 573:
Page 574 and 575:
Page 576 and 577:
Now, however, the term involving th
Page 578 and 579:
in which r 0 n ‡ 1 is evaluated u
Page 580 and 581:
Soil±pore ¯uid interaction (Class
Page 582 and 583:
Partitioned single-phase systems ±
Page 584 and 585:
Staggered solution processes 567thu
Page 586 and 587:
23 2K 11 0 0 0 K 12 K 1kK 21 K 22
Page 588 and 589:
Staggered solution processes 571hav
Page 590 and 591:
15. R. Ohayon. True symmetric formu
Page 592 and 593:
58. A.W. Craig and O.C. Zienkiewicz
Page 594 and 595:
Introduction 577StartData Input Mod
Page 596 and 597:
Data input module 579The notation u
Page 598 and 599:
Data input module 581y13 14 15 167
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Data input module 583While the abov
Page 602 and 603:
Data input module 585(a) Material p
Page 604 and 605:
Data input module 587Table 20.4 Com
Page 606 and 607:
Memory management for array storage
Page 608 and 609:
Since the problem given by Eq. (20.
Page 610 and 611:
Solution module ± the command prog
Page 612 and 613:
Solution module ± the command prog
Page 614 and 615:
Computation of ®nite element solut
Page 616 and 617:
Page 618 and 619:
Page 620 and 621:
Page 622 and 623:
Page 624 and 625:
Page 626 and 627:
Solution of simultaneous linear alg
Page 628 and 629:
Page 630 and 631:
Page 632 and 633:
Page 634 and 635:
Page 636 and 637:
13. M. Adams. Heuristics for automa
Page 638 and 639:
Matrix addition or subtraction 621T
Page 640 and 641:
Symmetric matrices 623A sum of prod
Page 642 and 643:
The eigenvalue problem 625The eigen
Page 644 and 645:
Indicial notation: summation conven
Page 646 and 647:
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
Page 654 and 655:
Appendix ESome integration formulae
Page 656 and 657:
Length of vector 639Addition and su
Page 658 and 659:
Elements of area and volume 641we h
Page 660 and 661:
Appendix GIntegration by parts in t
Page 662 and 663:
Appendix HSolutions exact at nodesT
Page 664 and 665:
Appendix H 647Thus, for linear shap
Page 666 and 667:
Appendix I 649N iN iii(a)(b)Fig. I.
Page 668 and 669:
Appendix I 65113 M113 M 3p = 1MO (h
Page 670:
and has the ®rst variation… 2 ˆ
Page 673 and 674:
656 Author indexBorouchaki, H. 229,
Page 675 and 676:
658 Author indexJirousek, J. 356, 3
Page 677 and 678:
660 Author indexSabin, M.A. 405, 42
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Subject indexa posteriori error est
Page 682 and 683:
Subject index 665Boundary:¯ux, 74i
Page 684 and 685:
Subject index 667D: elasticity matr
Page 686 and 687:
Subject index 669incompatible, 264l
Page 688 and 689:
Subject index 671Field approximatio
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Subject index 673Heat ¯ow, 41axisy
Page 692 and 693:
Subject index 675L 2 projection of
Page 694 and 695:
Subject index 677Mesh re®nement:ad
Page 696 and 697:
Subject index 679Operator:linear, 6
Page 698 and 699:
Subject index 681contrived variatio
Page 700 and 701:
Subject index 683Reynolds equation,
Page 702 and 703:
Subject index 685Sparse coe cient a
Page 704 and 705:
Subject index 687Temperature, 41, 1
Page 706:
Subject index 689Water/oil mixtures
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