The Finite Element Method Set, Sixth Edition

Volume 1
Front Cover......Page 1
The Finite Element Method for Solid and Structural Mechanics......Page 4
Copyright Page......Page 5
Contents......Page 8
Preface......Page 14
1.1 Introduction......Page 18
1.2 Small deformation solid mechanics problems......Page 21
1.3 Variational forms for non-linear elasticity......Page 29
1.4 Weak forms of governing equations......Page 31
References......Page 32
2.2 Finite element approximation – Galerkin method......Page 34
2.3 Numerical integration – quadrature......Page 39
2.4 Non-linear transient and steady-state problems......Page 41
2.5 Boundary conditions: non-linear problems......Page 45
2.6 Mixed or irreducible forms......Page 50
2.7 Non-linear quasi-harmonic field problems......Page 54
2.8 Typical examples of transient non-linear calculations......Page 55
2.9 Concluding remarks......Page 60
References......Page 61
3.1 Introduction......Page 63
3.2 Iterative techniques......Page 64
3.3 General remarks – incremental and rate methods......Page 75
References......Page 77
4.1 Introduction......Page 79
4.2 Viscoelasticity – history dependence of deformation......Page 80
4.3 Classical time-independent plasticity theory......Page 89
4.4 Computation of stress increments......Page 97
4.5 Isotropic plasticity models......Page 102
4.6 Generalized plasticity......Page 109
4.7 Some examples of plastic computation......Page 112
4.8 Basic formulation of creep problems......Page 117
4.9 Viscoplasticity – a generalization......Page 119
4.10 Some special problems of brittle materials......Page 124
4.11 Non-uniqueness and localization in elasto-plastic deformations......Page 129
4.12 Non-linear quasi-harmonic field problems......Page 133
4.13 Concluding remarks......Page 135
References......Page 137
5.1 Introduction......Page 144
5.2 Governing equations......Page 145
5.3 Variational description for finite deformation......Page 152
5.4 Two-dimensional forms......Page 160
5.5 A three-field, mixed finite deformation formulation......Page 162
5.6 A mixed–enhanced finite deformation formulation......Page 167
5.7 Forces dependent on deformation– pressure loads......Page 171
5.8 Concluding remarks......Page 172
References......Page 173
6.2 Isotropic elasticity......Page 175
6.3 Isotropic viscoelasticity......Page 189
6.4 Plasticity models......Page 190
6.5 Incremental formulations......Page 191
6.6 Rate constitutive models......Page 193
6.7 Numerical examples......Page 195
6.8 Concluding remarks......Page 202
References......Page 206
7.1 Introduction......Page 208
7.2 Node–node contact: Hertzian contact......Page 210
7.3 Tied interfaces......Page 214
7.4 Node–surface contact......Page 217
7.5 Surface–surface contact......Page 235
7.6 Numerical examples......Page 236
References......Page 241
8.2 Pseudo-rigid motions......Page 245
8.3 Rigid motions......Page 247
8.4 Connecting a rigid body to a flexible body......Page 251
8.5 Multibody coupling by joints......Page 254
8.6 Numerical examples......Page 257
References......Page 259
9.1 Introduction......Page 262
9.2 Early DEM formulations......Page 264
9.3 Contact detection......Page 267
9.4 Contact constraints and boundary conditions......Page 273
9.5 Block deformability......Page 277
9.6 Time integration for discrete element methods......Page 284
9.7 Associated discontinuous modelling methodologies......Page 287
9.8 Unifying aspects of discrete element methods......Page 288
9.9 Concluding remarks......Page 289
References......Page 290
10.1 Introduction......Page 295
10.2 Governing equations......Page 296
10.3 Weak (Galerkin) forms for rods......Page 302
10.4 Finite element solution: Euler–Bernoulli rods......Page 307
10.5 Finite element solution: Timoshenko rods......Page 322
10.6 Forms without rotation parameters......Page 334
10.7 Moment resisting frames......Page 336
References......Page 337
11.1 Introduction......Page 340
11.2 The plate problem: thick and thin formulations......Page 342
11.3 Rectangular element with corner nodes (12 degrees of freedom)......Page 353
11.5 Triangular element with corner nodes (9 degrees of freedom)......Page 357
11.6 Triangular element of the simplest form (6 degrees of freedom)......Page 362
11.7 The patch test– an analytical requirement......Page 363
11.8 Numerical examples......Page 365
11.10 Singular shape functions for the simple triangular element......Page 374
11.11 An 18 degree-of-freedom triangular element with conforming shape functions......Page 377
11.12 Compatible quadrilateral elements......Page 378
11.13 Quasi-conforming elements......Page 379
11.14 Hermitian rectangle shape function......Page 380
11.15 The 21 and 18 degree-of-freedom triangle......Page 381
11.16 Mixed formulations – general remarks......Page 383
11.17 Hybrid plate elements......Page 385
11.18 Discrete Kirchhoff constraints......Page 386
11.19 Rotation-free elements......Page 388
11.20 Inelastic material behaviour......Page 391
References......Page 393
12.1 Introduction......Page 399
12.2 The irreducible formulation– reduced integration......Page 402
12.3 Mixed formulation for thick plates......Page 407
12.4 The patch test for plate bending elements......Page 409
12.5 Elements with discrete collocation constraints......Page 414
12.6 Elements with rotational bubble or enhanced modes......Page 422
12.7 Linked interpolation– an improvement of accuracy......Page 425
12.8 Discrete 'exact' thin plate limit......Page 430
12.9 Performance of various 'thick' plate elements – limitations of thin plate theory......Page 432
12.10 Inelastic material behaviour......Page 436
12.11 Concluding remarks – adaptive refinement......Page 437
References......Page 438
13.1 Introduction......Page 443
13.2 Stiffness of a plane element in local coordinates......Page 445
13.3 Transformation to global coordinates and assembly of elements......Page 446
13.4 Local direction cosines......Page 448
13.5 'Drilling' rotational stiffness – 6 degree-of-freedom assembly......Page 452
13.7 Choice of element......Page 457
13.8 Practical examples......Page 458
References......Page 467
14.2 Straight element......Page 471
14.3 Curved elements......Page 478
14.4 Independent slope–displacement interpolation with penalty functions (thick or thin shell formulations)......Page 485
References......Page 490
15.2 Shell element with displacement and rotation parameters......Page 492
15.3 Special case of axisymmetric, curved, thick shells......Page 501
15.5 Convergence......Page 504
15.7 Some shell examples......Page 505
15.8 Concluding remarks......Page 510
References......Page 512
16.1 Introduction......Page 515
16.2 Prismatic bar......Page 518
16.3 Thin membrane box structures......Page 521
16.4 Plates and boxes with flexure......Page 522
16.5 Axisymmetric solids with non-symmetrical load......Page 524
16.6 Axisymmetric shells with non-symmetrical load......Page 527
16.7 Concluding remarks......Page 531
References......Page 532
17.2 Large displacement theory of beams......Page 534
17.3 Elastic stability – energy interpretation......Page 540
17.4 Large displacement theory of thick plates......Page 543
17.5 Large displacement theory of thin plates......Page 549
17.6 Solution of large deflection problems......Page 551
17.7 Shells......Page 554
17.8 Concluding remarks......Page 559
References......Page 560
18.1 Introduction......Page 564
18.2 Asymptotic analysis......Page 566
18.3 Statement of the problem and assumptions......Page 567
18.4 Formalism of the homogenization procedure......Page 569
18.5 Global solution......Page 570
18.6 Local approximation of the stress vector......Page 571
18.7 Finite element analysis applied to the local problem......Page 572
18.8 The non-linear case and bridging over several scales......Page 577
18.9 Asymptotic homogenization at three levels: micro, meso and macro......Page 578
18.10 Recovery of the micro description of the variables of the problem......Page 579
18.11 Material characteristics and homogenization results......Page 582
18.12 Multilevel procedures which use homogenization as an ingredient......Page 584
18.13 General first-order and second-order procedures......Page 587
18.14 Discrete-to-continuum linkage......Page 589
18.16 Homogenization procedure – definition of successive yield surfaces......Page 595
18.17 Numerically developed global self-consistent elastic–plastic constitutive law......Page 597
18.18 Global solution and stress-recovery procedure......Page 598
18.19 Concluding remarks......Page 603
References......Page 604
19.1 Introduction......Page 607
19.2 Solution of non-linear problems......Page 608
19.3 Eigensolutions......Page 609
19.4 Restart option......Page 611
References......Page 612
Appendix A. Isoparametric finite element approximations......Page 614
Appendix B. Invariants of second-order tensors......Page 621
Author index......Page 626
Subject index......Page 636
Colour Plates......Page 650
Volume 2
Front Cover......Page 654
The Finite Element Method: Its Basis and Fundamentals......Page 657
Copyright Page......Page 658
Contents......Page 661
Preface......Page 667
1.1 Introduction......Page 669
1.2 The structural element and the structural system......Page 671
1.3 Assembly and analysis of a structure......Page 673
1.4 The boundary conditions......Page 674
1.5 Electrical and fluid networks......Page 675
1.6 The general pattern......Page 677
1.7 The standard discrete system......Page 678
1.8 Transformation of coordinates......Page 679
1.9 Problems......Page 681
2.1 Introduction......Page 687
2.2 Direct formulation of finite element characteristics......Page 688
2.3 Generalization to the whole region– internal nodal force concept abandoned......Page 699
2.4 Displacement approach as a minimization of total potential energy......Page 702
2.5 Convergence criteria......Page 705
2.6 Discretization error and convergence rate......Page 706
2.7 Displacement functions with discontinuity between elements – non-conforming elements and the patch test......Page 707
2.9 Numerical examples......Page 708
2.10 Concluding remarks......Page 714
2.11 Problems......Page 715
3.1 Introduction......Page 722
3.2 Integral or 'weak' statements equivalent to the differential equations......Page 725
3.3 Approximation to integral formulations: the weighted residual-Galerkin method......Page 728
3.4 Virtual work as the 'weak form' of equilibrium equations for analysis of solids or fluids......Page 737
3.5 Partial discretization......Page 739
3.6 Convergence......Page 742
3.7 What are 'variational principles'?......Page 744
3.8 'Natural' variational principles and their relation to governing differential equations......Page 746
3.9 Establishment of natural variational principles for linear, self-adjoint, differential equations......Page 749
3.10 Maximum, minimum, or a saddle point?......Page 751
3.11 Constrained variational principles. Lagrange multipliers......Page 752
3.12 Constrained variational principles. Penalty function and perturbed lagrangian methods......Page 756
3.13 Least squares approximations......Page 760
3.14 Concluding remarks – finite difference and boundary methods......Page 763
3.15 Problems......Page 765
4.1 Introduction......Page 771
4.2 Standard and hierarchical concepts......Page 772
4.3 Rectangular elements– some preliminary considerations......Page 775
4.4 Completeness of polynomials......Page 777
4.5 Rectangular elements– Lagrange family......Page 778
4.6 Rectangular elements– 'serendipity' family......Page 780
4.7 Triangular element family......Page 784
4.8 Line elements......Page 787
4.9 Rectangular prisms – Lagrange family......Page 788
4.10 Rectangular prisms – 'serendipity' family......Page 789
4.11 Tetrahedral elements......Page 790
4.13 Hierarchic polynomials in one dimension......Page 793
4.15 Triangle and tetrahedron family......Page 796
4.16 Improvement of conditioning with hierarchical forms......Page 798
4.17 Global and local finite element approximation......Page 799
4.18 Elimination of internal parameters before assembly – substructures......Page 800
4.20 Problems......Page 802
5.1 Introduction......Page 806
5.2 Use of 'shape functions' in the establishment of coordinate transformations......Page 807
5.4 Variation of the unknown function within distorted, curvilinear elements. Continuity requirements......Page 811
5.5 Evaluation of element matrices. Transformation in ξ,η,ζ coordinates......Page 813
5.6 Evaluation of element matrices. Transformation in area and volume coordinates......Page 816
5.7 Order of convergence for mapped elements......Page 819
5.8 Shape functions by degeneration......Page 821
5.9 Numerical integration– one dimensional......Page 828
5.10 Numerical integration– rectangular (2D) or brick regions (3D)......Page 830
5.12 Required order of numerical integration......Page 832
5.13 Generation of finite element meshes by mapping. Blending functions......Page 837
5.14 Infinite domains and infinite elements......Page 838
5.15 Singular elements by mapping – use in fracture mechanics, etc.......Page 844
5.16 Computational advantage of numerically integrated finite elements......Page 845
5.17 Problems......Page 846
6.1 Introduction......Page 855
6.2 Governing equations......Page 856
6.3 Finite element approximation......Page 869
6.4 Reporting of results: displacements, strains and stresses......Page 875
6.5 Numerical examples......Page 877
6.6 Problems......Page 885
7.1 Introduction......Page 897
7.2 General quasi-harmonic equation......Page 898
7.3 Finite element solution process......Page 901
7.4 Partial discretization– transient problems......Page 905
7.5 Numerical examples– an assessment of accuracy......Page 907
7.7 Problems......Page 921
8.1 Introduction......Page 932
8.2 Two-dimensional mesh generation– advancing front method......Page 934
8.3 Surface mesh generation......Page 954
8.4 Three-dimensional mesh generation– Delaunay triangulation......Page 971
8.6 Problems......Page 991
9.1 Introduction......Page 997
9.2 Convergence requirements......Page 998
9.3 The simple patch test (tests A and B) – a necessary condition for convergence......Page 1000
9.4 Generalized patch test (test C) and the single-element test......Page 1002
9.6 Higher order patch tests......Page 1004
9.7 Application of the patch test to plane elasticity elements with 'standard' and 'reduced' quadrature......Page 1005
9.8 Application of the patch test to an incompatible element......Page 1011
9.10 Concluding remarks......Page 1015
9.11 Problems......Page 1018
10.1 Introduction......Page 1024
10.2 Discretization of mixed forms – some general remarks......Page 1026
10.3 Stability of mixed approximation. The patch test......Page 1028
10.4 Two-field mixed formulation in elasticity......Page 1031
10.5 Three-field mixed formulations in elasticity......Page 1038
10.6 Complementary forms with direct constraint......Page 1043
10.8 Problems......Page 1047
11.2 Deviatoric stress and strain, pressure and volume change......Page 1051
11.3 Two-field incompressible elasticity (u–p form)......Page 1052
11.4 Three-field nearly incompressible elasticity (u–p–εv form)......Page 1061
11.5 Reduced and selective integration and its equivalence to penalized mixed problems......Page 1066
11.6 A simple iterative solution process for mixed problems: Uzawa method......Page 1072
11.7 Stabilized methods for some mixed elements failing the incompressibility patch test......Page 1075
11.8 Concluding remarks......Page 1089
11.9 Problems......Page 1090
12.1 Introduction......Page 1097
12.2 Linking of two or more subdomains by Lagrange multipliers......Page 1098
12.3 Linking of two or more subdomains by perturbed lagrangian and penalty methods......Page 1104
12.4 Interface displacement 'frame'......Page 1110
12.5 Linking of boundary (or Trefftz)-type solution by the 'frame' of specified displacements......Page 1113
12.8 Problems......Page 1119
13.1 Definition of errors......Page 1124
13.2 Superconvergence and optimal sampling points......Page 1127
13.3 Recovery of gradients and stresses......Page 1133
13.4 Superconvergent patch recovery – SPR......Page 1135
13.5 Recovery by equilibration of patches – REP......Page 1142
13.6 Error estimates by recovery......Page 1144
13.7 Residual-based methods......Page 1146
13.8 Asymptotic behaviour and robustness of error estimators – the Babuška patch test......Page 1156
13.9 Bounds on quantities of interest......Page 1158
13.10 Which errors should concern us?......Page 1162
13.11 Problems......Page 1163
14.1 Introduction......Page 1168
14.2 Adaptive h-refinement......Page 1171
14.3 p-refinement and hp-refinement......Page 1182
14.4 Concluding remarks......Page 1186
14.5 Problems......Page 1188
15.1 Introduction......Page 1193
15.2 Function approximation......Page 1195
15.3 Moving least squares approximations – restoration of continuity of approximation......Page 1201
15.4 Hierarchical enhancement of moving least squares expansions......Page 1206
15.5 Point collocation– finite point methods......Page 1208
15.6 Galerkin weighting and finite volume methods......Page 1214
15.7 Use of hierarchic and special functions based on standard finite elements satisfying the partition of unity requirement......Page 1217
15.9 Problems......Page 1226
16.2 Direct formulation of time-dependent problems with spatial finite element subdivision......Page 1231
16.3 General classification......Page 1238
16.4 Free response – eigenvalues for second-order problems and dynamic vibration......Page 1239
16.5 Free response – eigenvalues for first-order problems and heat conduction, etc.......Page 1244
16.6 Free response– damped dynamic eigenvalues......Page 1246
16.8 Transient response by analytical procedures......Page 1247
16.9 Symmetry and repeatability......Page 1251
16.10 Problems......Page 1252
17.1 Introduction......Page 1257
17.2 Simple time-step algorithms for the first-order equation......Page 1258
17.3 General single-step algorithms for first- and second-order equations......Page 1268
17.4 Stability of general algorithms......Page 1277
17.5 Multistep recurrence algorithms......Page 1283
17.6 Some remarks on general performance of numerical algorithms......Page 1286
17.7 Time discontinuous Galerkin approximation......Page 1287
17.8 Concluding remarks......Page 1292
17.9 Problems......Page 1294
18.1 Coupled problems– definition and classification......Page 1299
18.2 Fluid-structure interaction (Class I problems)......Page 1302
18.3 Soil-pore fluid interaction (Class II problems)......Page 1313
18.4 Partitioned single-phase systems – implicit-explicit partitions (Class I problems)......Page 1321
18.5 Staggered solution processes......Page 1323
18.6 Concluding remarks......Page 1328
19.2 Pre-processing module: mesh creation......Page 1332
19.4 Post-processor module......Page 1334
19.5 User modules......Page 1335
Appendix A: Matrix algebra......Page 1336
Appendix B: Tensor-indicial notation in the approximation of elasticity problems......Page 1342
Appendix C: Solution of simultaneous linear algebraic equations......Page 1351
Appendix D: Some integration formulae for a triangle......Page 1360
Appendix E: Some integration formulae for a tetrahedron......Page 1361
Appendix F: Some vector algebra......Page 1362
Appendix G: Integration by parts in two or three dimensions (Green's theorem)......Page 1367
Appendix H: Solutions exact at nodes......Page 1369
Appendix I: Matrix diagonalization or lumping......Page 1372
Author index......Page 1379
Subject index......Page 1387
Color Plate Section......Page 1403
Volume 3
Front Cover......Page 1407
The Finite Element Method for Fluid Dynamics......Page 1410
Copyright Page......Page 1411
Contents......Page 1414
Preface......Page 1418
Acknowledgements......Page 1420
1.1 General remarks and classification of fluid dynamics problems discussed in this book......Page 1422
1.2 The governing equations of fluid dynamics......Page 1425
1.3 Inviscid, incompressible flow......Page 1432
1.4 Incompressible (or nearly incompressible) flows......Page 1434
1.5 Numerical solutions: weak forms, weighted residual and finite element approximation......Page 1435
1.6 Concluding remarks......Page 1447
References......Page 1448
2.1 Introduction......Page 1449
2.2 The steady-state problem in one dimension......Page 1452
2.3 The steady-state problem in two (or three) dimensions......Page 1466
2.4 Steady state – concluding remarks......Page 1470
2.5 Transients – introductory remarks......Page 1471
2.6 Characteristic-based methods......Page 1474
2.7 Taylor–Galerkin procedures for scalar variables......Page 1486
2.9 Non-linear waves and shocks......Page 1487
2.10 Treatment of pure convection......Page 1491
2.11 Boundary conditions for convection–diffusion......Page 1493
2.12 Summary and concluding remarks......Page 1494
References......Page 1495
3.1 Introduction......Page 1500
3.2 Non-dimensional form of the governing equations......Page 1502
3.3 Characteristic-based split (CBS) algorithm......Page 1503
3.4 Explicit, semi-implicit and nearly implicit forms......Page 1513
3.5 Artificial compressibility and dual time stepping......Page 1516
3.6 'Circumvention' of the Babuška–Brezzi (BB) restrictions......Page 1518
3.7 A single-step version......Page 1519
3.8 Boundary conditions......Page 1521
3.9 The performance of two-step and one-step algorithms on an inviscid problem......Page 1524
3.10 Concluding remarks......Page 1525
References......Page 1526
4.1 Introduction and the basic equations......Page 1531
4.2 Use of the CBS algorithm for incompressible flows......Page 1533
4.3 Adaptive mesh refinement......Page 1544
4.5 Slow flows – mixed and penalty formulations......Page 1552
References......Page 1557
5.2 Non-Newtonian flows – metal and polymer forming......Page 1562
5.3 Viscoelastic flows......Page 1575
5.4 Direct displacement approach to transient metal forming......Page 1584
5.5 Concluding remarks......Page 1586
References......Page 1587
6.2 Free surface flows......Page 1591
6.3 Buoyancy driven flows......Page 1610
6.4 Concluding remarks......Page 1612
References......Page 1614
7.1 Introduction......Page 1618
7.2 The governing equations......Page 1619
7.3 Boundary conditions – subsonic and supersonic flow......Page 1620
7.4 Numerical approximations and the CBS algorithm......Page 1623
7.5 Shock capture......Page 1624
7.6 Variable smoothing......Page 1626
7.7 Some preliminary examples for the Euler equation......Page 1627
7.8 Adaptive refinement and shock capture in Euler problems......Page 1633
7.9 Three-dimensional inviscid examples in steady state......Page 1638
7.10 Transient two- and three-dimensional problems......Page 1647
7.11 Viscous problems in two dimensions......Page 1648
7.12 Three-dimensional viscous problems......Page 1661
7.13 Boundary layer-inviscid Euler solution coupling......Page 1662
References......Page 1663
8.1 Introduction......Page 1669
8.2 Treatment of incompressible turbulent flows......Page 1672
8.3 Treatment of compressible flows......Page 1685
8.4 Large eddy simulation......Page 1688
8.6 Direct Numerical Simulation (DNS)......Page 1691
References......Page 1692
9.1 Introduction......Page 1695
9.2 A generalized porous medium flow approach......Page 1696
9.3 Discretization procedure......Page 1700
9.5 Forced convection......Page 1703
9.6 Natural convection......Page 1705
9.7 Concluding remarks......Page 1709
References......Page 1710
10.1 Introduction......Page 1713
10.2 The basis of the shallow water equations......Page 1714
10.3 Numerical approximation......Page 1718
10.4 Examples of application......Page 1719
10.5 Drying areas......Page 1731
10.6 Shallow water transport......Page 1732
10.7 Concluding remarks......Page 1734
References......Page 1735
11.1 Introduction and equations......Page 1738
11.2 Waves in closed domains – finite element models......Page 1739
11.5 The short-wave problem......Page 1741
11.6 Waves in unbounded domains (exterior surface wave problems)......Page 1742
11.8 Local Non-Reflecting Boundary Conditions (NRBCs)......Page 1745
11.10 Mapped periodic (unconjugated) infinite elements......Page 1748
11.11 Ellipsoidal type infinite elements of Burnett and Holford......Page 1749
11.12 Wave envelope (or conjugated) infinite elements......Page 1751
11.14 Trefftz type infinite elements......Page 1753
11.15 Convection and wave refraction......Page 1754
11.16 Transient problems......Page 1756
11.17 Linking to exterior solutions (or DtN mapping)......Page 1757
11.18 Three-dimensional effects in surface waves......Page 1759
References......Page 1765
12.2 Background......Page 1770
12.4 Recent developments in short wave modelling......Page 1772
12.6 Finite elements incorporating wave shapes......Page 1773
12.7 Refraction......Page 1785
12.8 Spectral finite elements for waves......Page 1793
12.9 Discontinuous Galerkin finite elements (DGFE)......Page 1795
References......Page 1799
13.1 Introduction......Page 1803
13.2 The data input module......Page 1804
13.3 Solution module......Page 1805
References......Page 1808
Appendix A. Non-conservative form of Navier–Stokes equations......Page 1810
Appendix B. Self-adjoint differential equations......Page 1812
Appendix C. Postprocessing......Page 1813
Appendix D. Integration formulae......Page 1816
Appendix E. Convection–diffusion equations: vector-valued variables......Page 1818
Appendix F. Edge-based finite element formulation......Page 1826
Appendix G. Multigrid method......Page 1828
Appendix H. Boundary layer–inviscid flow coupling......Page 1830
Appendix I. Mass-weighted averaged turbulence transport equations......Page 1834
Author index......Page 1838
Subject index......Page 1848
Plates......Page 1858