The finite element method: its basis and fundamentals

O. C. Zienkiewicz, R. L. Taylor, J.Z. Zhu9780080472775, 9780750663205, 0750663200

The Sixth Edition of this influential best-selling book delivers the most up-to-date and comprehensive text and reference yet on the basis of the finite element method (FEM) for all engineers and mathematicians. Since the appearance of the first edition 38 years ago, The Finite Element Method provides arguably the most authoritative introductory text to the method, covering the latest developments and approaches in this dynamic subject, and is amply supplemented by exercises, worked solutions and computer algorithms. . The classic FEM text, written by the subject’s leading authors . Enhancements include more worked examples and exercises, plus a companion website with a solutions manual and downloadable algorithms . With a new chapter on automatic mesh generation and added materials on shape function development and the use of higher order elements in solving elasticity and field problems Active research has shaped The Finite Element Method into the pre-eminent tool for the modelling of physical systems. It maintains the comprehensive style of earlier editions, while presenting the systematic development for the solution of problems modelled by linear differential equations. Together with the second and third self-contained volumes (0750663219 and 0750663227), The Finite Element Method Set (0750664312) provides a formidable resource covering the theory and the application of FEM, including the basis of the method, its application to advanced solid and structural mechanics and to computational fluid dynamics. * The classic introduction to the finite element method, by two of the subject’s leading authors * Any professional or student of engineering involved in understanding the computational modelling of physical systems will inevitably use the techniques in this key text * Enhancements include more worked examples, exercises, plus a companion website with a worked solutions manual for tutors and downloadable algorithms

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Table of contents :
Front Matter……Page 1
Preface……Page 4
Table of Contents……Page 6
1.1 Introduction……Page 12
1.2 The Structural Element and the Structural System……Page 14
1.3 Assembly and Analysis of a Structure……Page 16
1.4 The Boundary Conditions……Page 17
1.5 Electrical and Fluid Networks……Page 18
1.6 The General Pattern……Page 20
1.7 The Standard Discrete System……Page 21
1.8 Transformation of Coordinates……Page 22
1.9 Problems……Page 24
References……Page 28
2.1 Introduction……Page 30
2.2 Direct Formulation of Finite Element Characteristics……Page 31
2.2.1 Displacement Function……Page 32
2.2.1.1 Triangle with 3 Nodes……Page 33
2.2.1.2 Rectangle with 4 Nodes……Page 34
2.2.2 Strains……Page 35
2.2.3 Stresses……Page 36
2.2.4 Equivalent Nodal Forces……Page 37
2.3 Generalization to the Whole Region – Internal Nodal Force Concept Abandoned……Page 42
2.4 Displacement Approach as a Minimization of Total Potential Energy……Page 45
2.4.1 Bound on Strain Energy in a Displacement Formulation……Page 46
2.4.2 Direct Minimization……Page 47
2.5 Convergence Criteria……Page 48
2.6 Discretization Error and Convergence Rate……Page 49
2.7 Displacement Functions with Discontinuity between Elements – Non-Conforming Elements and the Patch Test……Page 50
2.9.1 Problems for Accuracy Assessment……Page 51
2.9.2.1 Stress Flow around a Reinforced Opening……Page 56
2.10 Concluding Remarks……Page 57
2.11 Problems……Page 58
References……Page 63
3.1 Introduction……Page 65
3.2 Integral or ‘Weak’ Statements Equivalent to the Differential Equations……Page 68
3.3 Approximation to Integral Formulations: The Weighted Residual-Galerkin Method……Page 71
3.4 Virtual Work as the ‘Weak Form’ of Equilibrium Equations for Analysis of Solids or Fluids……Page 80
3.5 Partial Discretization……Page 82
3.6 Convergence……Page 85
3.7 What are ‘Variational Principles’?……Page 87
3.8.1 Euler Equations……Page 89
3.8.2 Relation of the Galerkin Method to Approximation via Variational Principles……Page 91
3.9 Establishment of Natural Variational Principles for Linear, Self-Adjoint, Differential Equations……Page 92
3.10 Maximum, Minimum, or a Saddle Point?……Page 94
3.11.1 Lagrange Multipliers……Page 95
3.11.2 Identification of Lagrange Multipliers. Forced Boundary Conditions and Modified Variational Principles……Page 98
3.12.1 Penalty Functions……Page 99
3.12.2 Perturbed Lagrangian……Page 100
3.13 Least Squares Approximations……Page 103
3.13.1 Galerkin Least Squares, Stabilization……Page 105
3.14 Concluding Remarks – Finite Difference and Boundary Methods……Page 106
3.15 Problems……Page 108
References……Page 111
4.1 Introduction……Page 114
4.2 Standard and Hierarchical Concepts……Page 115
4.3 Rectangular Elements – Some Preliminary Considerations……Page 118
4.4 Completeness of Polynomials……Page 120
4.5 Rectangular Elements – Lagrange Family……Page 121
4.6 Rectangular Elements – ‘Serendipity’ Family……Page 123
4.6.2 ‘Cubic’ Element……Page 124
4.7 Triangular Element Family……Page 127
4.7.1 Area Coordinates……Page 128
4.7.2 Shape Functions……Page 129
4.8 Line Elements……Page 130
4.9 Rectangular Prisms – Lagrange Family……Page 131
4.10.2 ‘Quadratic’ Element (20 Nodes)……Page 132
4.11.1 Volume Coordinates……Page 133
4.11.2.2 ‘Cubic’ Tetrahedron……Page 135
4.13 Hierarchic Polynomials in One Dimension……Page 136
4.15 Triangle and Tetrahedron Family……Page 139
4.16 Improvement of Conditioning with Hierarchical Forms……Page 141
4.17 Global and Local Finite Element Approximation……Page 142
4.18 Elimination of Internal Parameters before Assembly – Substructures……Page 143
4.20 Problems……Page 145
References……Page 147
5.1 Introduction……Page 149
5.2 Use of ‘Shape Functions’ in the Establishment of Coordinate Transformations……Page 150
5.4 Variation of the Unknown Function within Distorted, Curvilinear Elements. Continuity Requirements……Page 154
5.5 Evaluation of Element Matrices. Transformation in xi, eta, zeta Coordinates……Page 156
5.5.1 Computation of Global Derivatives……Page 157
5.5.2 Volume Integrals……Page 158
5.6 Evaluation of Element Matrices. Transformation in Area and Volume Coordinates……Page 159
5.7 Order of Convergence for Mapped Elements……Page 162
5.8 Shape Functions by Degeneration……Page 164
5.8.1 Higher Order Degenerate Elements……Page 166
5.9.2 Gauss Quadrature……Page 171
5.10 Numerical Integration – Rectangular (2D) or Brick Regions (3D)……Page 173
5.12 Required Order of Numerical Integration……Page 175
5.12.1 Minimum Order of Integration for Convergence……Page 176
5.12.2 Order of Integration for No Loss of Convergence Rate……Page 177
5.12.3 Matrix Singularity due to Numerical Integration……Page 178
5.13 Generation of Finite Element Meshes by Mapping. Blending Functions……Page 180
5.14.1 Introduction……Page 181
5.14.2 The Mapping Function……Page 183
5.15 Singular Elements by Mapping – Use in Fracture Mechanics, Etc…….Page 187
5.16 Computational Advantage of Numerically Integrated Finite Elements……Page 188
5.17 Problems……Page 189
References……Page 195
6.1 Introduction……Page 198
6.2.1 Displacement Function……Page 199
6.2.2 Strain Matrix……Page 200
6.2.3 Equilibrium Equations……Page 201
6.2.4 Boundary Conditions……Page 202
6.2.4.2 Symmetry and Repeatability……Page 203
6.2.4.3 Normal Pressure Loading……Page 204
6.2.5 Transformation of Stress and Strain……Page 205
6.2.6.1 Isotropic Materials……Page 206
6.2.6.2 Anisotropic Materials……Page 208
6.2.6.3 Initial Strain – Thermal Effects……Page 211
6.3 Finite Element Approximation……Page 212
6.3.1 Displacement and Strain Approximation……Page 213
6.3.2 Stiffness and Load Matrices……Page 216
6.4 Reporting of Results: Displacements, Strains and Stresses……Page 218
6.5 Numerical Examples……Page 220
6.5.1 A Dam Subject to External and Internal Water Pressures……Page 222
6.5.2 Rotating Disc……Page 224
6.5.3 Conical Water Tank……Page 226
6.5.5 Arch Dam in a Rigid Valley……Page 227
6.6 Problems……Page 228
References……Page 238
7.1 Introduction……Page 240
7.2.1 Governing Equations……Page 241
7.2.2 Anisotropic and Isotropic Forms for k……Page 242
7.3.1 Finite Element Discretization……Page 244
7.3.2 Two-Dimensional Plane and Axisymmetric Problem……Page 246
7.4 Partial Discretization – Transient Problems……Page 248
7.4.1 Finite Element Discretizations……Page 249
7.5 Numerical Examples – An Assessment of Accuracy……Page 250
7.5.1 Torsion of Prismatic Bars……Page 251
7.5.2 Transient Heat Conduction……Page 253
7.5.3 Anisotropic Seepage……Page 255
7.5.4 Electrostatic and Magnetostatic Problems……Page 256
7.5.6 Irrotational and Free Surface Flows……Page 262
7.7 Problems……Page 264
References……Page 272
8.1 Introduction……Page 275
8.2.1 Geometrical Characteristics of the Mesh……Page 277
8.2.2.1 Boundary Curve Representation……Page 278
8.2.3 Triangular Mesh Generation……Page 281
8.2.3.2 Boundary Node Generation……Page 282
8.2.3.4 Element Generation……Page 288
8.2.4.1 Mesh Smoothing……Page 291
8.2.4.2 Mesh Modification……Page 292
8.2.5 Higher Order Elements……Page 294
8.2.6 Remarks……Page 296
8.3 Surface Mesh Generation……Page 297
8.3.1 Geometrical Representation……Page 298
8.3.1.2 Surface Representation……Page 299
8.3.2.1 Mesh Control Function in Three Dimensions……Page 301
8.3.2.2 Mesh Parameters in Parametric Plane……Page 304
8.3.3.1 Node Generation on the Curves……Page 308
8.3.4 Element Generation in Parametric Plane……Page 310
8.3.6 Remarks……Page 312
8.4 Three-Dimensional Mesh Generation – Delaunay Triangulation……Page 314
8.4.1 Voronoi Diagram and Delaunay Triangulation……Page 315
8.4.2 Three-Dimensional Mesh Generation by Delaunay Triangulation……Page 317
8.4.2.1 Delaunay Triangulation Algorithm……Page 319
8.4.2.2 Automatic Node Generation……Page 322
8.4.2.3 Surface Mesh Recovery……Page 323
8.4.3.1 Element Transformation……Page 327
8.4.3.2 Node Addition and Node Elimination……Page 330
8.4.3.3 Mesh Smoothing……Page 331
8.4.6 Remarks……Page 332
8.6 Problems……Page 334
References……Page 335
9.1 Introduction……Page 340
9.2 Convergence Requirements……Page 341
9.3 The Simple Patch Test (Tests A and B) – A Necessary Condition for Convergence……Page 343
9.4 Generalized Patch Test (Test C) and the Single-Element Test……Page 345
9.6 Higher Order Patch Tests……Page 347
9.7 Application of the Patch Test to Plane Elasticity Elements with ‘Standard’ and ‘Reduced’ Quadrature……Page 348
9.8 Application of the Patch Test to an Incompatible Element……Page 354
9.10 Concluding Remarks……Page 358
9.11 Problems……Page 361
References……Page 365
10.1 Introduction……Page 367
10.2 Discretization of Mixed Forms – Some General Remarks……Page 369
10.3.1 Solvability Requirement……Page 371
10.3.2 Locking……Page 372
10.3.3 The Patch Test……Page 373
10.4.1 General……Page 374
10.4.2 The u-sigma Mixed Form……Page 375
10.4.3 Stability of Two-Field Approximation in Elasticity (u-sigma)……Page 376
10.5.1 The u-sigma-epsilon Mixed Form……Page 381
10.5.2 Stability Condition of Three-Field Approximation (u-sigma-epsilon)……Page 382
10.5.3 The u-sigma-epsilon_en Form. Enhanced Strain Formulation……Page 383
10.5.3.1 Remarks……Page 385
10.6.1 General Forms……Page 386
10.6.1.1 The Complementary Heat Transfer Problem……Page 387
10.6.1.2 The Complementary Elastic Energy Principle……Page 388
10.6.2 Solution Using Auxiliary Functions……Page 389
10.8 Problems……Page 390
References……Page 391
11.2 Deviatoric Stress and Strain, Pressure and Volume Change……Page 394
11.3 Two-Field Incompressible Elasticity (u-p Form)……Page 395
11.4 Three-Field nearly Incompressible Elasticity (u-p-epsilon_nu Form)……Page 404
11.4.1 The B-Bar Method for Nearly Incompressible Problems……Page 408
11.5 Reduced and Selective Integration and Its Equivalence to Penalized Mixed Problems……Page 409
11.6.1 General……Page 415
11.6.2 Iterative Solution for Incompressible Elasticity……Page 416
11.7 Stabilized Methods for Some Mixed Elements Failing the Incompressibility Patch Test……Page 418
11.7.1 Laplacian Pressure Stabilization……Page 419
11.7.2 Galerkin Least Squares Method……Page 420
11.7.3 Direct Pressure Stabilization……Page 421
11.7.4 Incompressibility by Time Stepping……Page 424
11.7.5 Numerical Comparisons……Page 427
11.8 Concluding Remarks……Page 432
11.9 Problems……Page 433
References……Page 436
12.1 Introduction……Page 440
12.2.1 Linking Subdomains for Quasi-Harmonic Equations……Page 441
12.2.1.2 Mortar and Dual Mortar Methods……Page 443
12.2.2 Linking Subdomains for Elasticity Equations……Page 445
12.3 Linking of Two or More Subdomains by Perturbed Lagrangian and Penalty Methods……Page 447
12.3.1 Nitsche Method and Discontinuous Galerkin Approximation……Page 449
12.3.1.1 Multiple Subdomain Problems……Page 451
12.4.1 General Remarks……Page 453
12.4.2 Linking Displacement Frame on Equilibrating Form Subdomains……Page 455
12.5 Linking of Boundary (or Trefftz)-Type Solution by the ‘Frame’ of Specified Displacements……Page 456
12.8 Problems……Page 462
References……Page 464
13.1 Definition of Errors……Page 467
13.1.1 Norms of Errors……Page 468
13.1.1.1 Effect of a Singularity……Page 469
13.2 Superconvergence and Optimal Sampling Points……Page 470
13.2.1 A One-Dimensional Example……Page 471
13.2.2 The Herrmann Theorem and Optimal Sampling Points……Page 473
13.3 Recovery of Gradients and Stresses……Page 476
13.4.1 Recovery for Gradients and Stresses……Page 478
13.5 Recovery by Equilibration of Patches – REP……Page 485
13.6 Error Estimates by Recovery……Page 487
13.7 Residual-Based Methods……Page 489
13.7.1 Explicit Residual Error Estimator……Page 490
13.7.2 Implicit Residual Error Estimators……Page 494
13.8 Asymptotic Behaviour and Robustness of Error Estimators – The Babuška Patch Test……Page 499
13.9 Bounds on Quantities of Interest……Page 501
13.10 Which Errors Should Concern Us?……Page 505
13.11 Problems……Page 506
References……Page 507
14.1 Introduction……Page 511
14.2.1 Predicting the Required Element Size in h Adaptivity……Page 514
14.2.2 Numerical Examples……Page 516
14.3 p-Refinement and hp-Refinement……Page 525
14.4 Concluding Remarks……Page 529
14.5 Problems……Page 531
References……Page 533
15.1 Introduction……Page 536
15.2.1 Least Squares Fit……Page 538
15.2.2 Weighted Least Squares Fit……Page 540
15.2.3 Interpolation Domains and Shape Functions……Page 541
15.3 Moving Least Squares Approximations – Restoration of Continuity of Approximation……Page 544
15.4 Hierarchical Enhancement of Moving Least Squares Expansions……Page 549
15.4.1.1 Shepard Interpolation……Page 550
15.5 Point Collocation – Finite Point Methods……Page 551
15.6.1 Introduction……Page 557
15.6.3 Galerkin Methods – Diffuse Elements……Page 558
15.7.1 Introduction……Page 560
15.7.2 Polynomial Hierarchical Method……Page 563
15.7.3 Application to Linear Elasticity……Page 564
15.7.4 Solution of Forms with Linearly Dependent Equations……Page 568
15.9 Problems……Page 569
References……Page 570
16.2.1 The ‘Quasi-Harmonic’ Equation with Time Differential……Page 574
16.2.2 Dynamic Behaviour of Elastic Structures with Linear Damping……Page 576
16.2.3 ‘Mass’ or ‘Damping’ Matrices for Some Typical Elements……Page 578
16.2.4 Mass ‘Lumping’ or Diagonalization……Page 579
16.3 General Classification……Page 581
16.4.1 Free Dynamic Vibration – Real Eigenvalues……Page 582
16.4.2 Determination of Eigenvalues……Page 583
16.4.3 Free Vibration with the Singular K Matrix……Page 584
16.4.5 Some Examples……Page 585
16.5 Free Response – Eigenvalues for First-Order Problems and Heat Conduction, Etc…….Page 587
16.6 Free Response – Damped Dynamic Eigenvalues……Page 589
16.8.2 Frequency Response Procedures……Page 590
16.8.3 Modal Decomposition Analysis……Page 591
16.9 Symmetry and Repeatability……Page 594
16.10 Problems……Page 595
References……Page 597
17.1 Introduction……Page 600
17.2.1 Weighted Residual Finite Element Approach……Page 601
17.2.2 Taylor Series Collocation……Page 604
17.2.4 Consistency and Approximation Error……Page 605
17.2.5 Stability……Page 607
17.2.6 Some Further Remarks. Initial Conditions and Examples……Page 610
17.3.1 Introduction……Page 611
17.3.2 The Weighted Residual Finite Element Form SSpj……Page 612
17.3.3 Truncated Taylor Series Collocation Algorithm GNpj……Page 617
17.3.3.1 The Newmark Algorithm (GN22)……Page 619
17.4 Stability of General Algorithms……Page 620
17.4.1 Stability of SS22/SS21 Algorithms……Page 623
17.4.2 Stability of Various Higher Order Schemes and Equivalence with Some Known Alternatives……Page 624
17.5.2 The Approximation Procedure for a General Multistep Algorithm……Page 626
17.6 Some Remarks on General Performance of Numerical Algorithms……Page 629
17.7 Time Discontinuous Galerkin Approximation……Page 630
17.8 Concluding Remarks……Page 635
17.9 Problems……Page 637
References……Page 639
18.1 Coupled Problems – Definition and Classification……Page 642
18.2.1 General Remarks and Fluid Behaviour Equations……Page 645
18.2.2 Boundary Conditions for the Fluid. Coupling and Radiation……Page 646
18.2.2.3 Radiation Boundary……Page 647
18.2.3 Weak Form for Coupled Systems……Page 648
18.2.4 The Discrete Coupled System……Page 649
18.2.6 Forced Vibrations and Transient Step-by-Step Algorithms……Page 650
18.2.6.1 Stability of the Fluid-Structure Time-Stepping Scheme……Page 653
18.2.7 Special Case of Incompressible Fluids……Page 655
18.3.1 The Problem and the Governing Equations. Discretization……Page 656
18.3.3 Transient Step-by-Step Algorithm……Page 659
18.3.4 Special Cases and Robustness Requirements……Page 660
18.3.5 Examples – Soil Liquefaction……Page 661
18.3.6 Biomechanics, Oil Recovery and Other Applications……Page 663
18.4 Partitioned Single-Phase Systems – Implicit-Explicit Partitions (Class I Problems)……Page 664
18.4.1.1 Implicit-Explicit Solution – Element Partition……Page 665
18.5.2 Staggered Process of Solution in Single-Phase Systems……Page 666
18.5.3 Staggered Schemes in Fluid-Structure Systems and Stabilization Processes……Page 669
References……Page 671
19.2 Pre-Processing Module: Mesh Creation……Page 675
19.2.1 Element Library……Page 676
19.4 Post-Processor Module……Page 677
References……Page 678
A.1 Definition of a Matrix……Page 679
A.2 Matrix Addition or Subtraction……Page 680
A.4 Inverse of a Matrix……Page 681
A.7 Symmetric Matrices……Page 682
A.9 The Eigenvalue Problem……Page 683
B.2 Indicial Notation: Summation Convention……Page 685
B.3 Derivatives and Tensorial Relations……Page 687
B.4 Coordinate Transformation……Page 688
B.5 Equilibrium and Energy……Page 689
B.6 Elastic Constitutive Equations……Page 690
B.7 Finite Element Approximation……Page 691
References……Page 693
C.1 Direct Solution……Page 694
C.2 Iterative Solution……Page 699
References……Page 702
Appendix D: Some Integration Formulae for a Triangle……Page 703
Appendix E: Some Integration Formulae for a Tetrahedron……Page 704
F.1 Addition and Subtraction……Page 705
F.3 Length of Vector……Page 706
F.5 ‘Vector’ or Cross Product……Page 707
F.6 Elements of Area and Volume……Page 708
Appendix G: Integration by Parts in Two or Three Dimensions (Green’s Theorem)……Page 710
Appendix H: Solutions Exact at Nodes……Page 712
References……Page 714
Appendix I: Matrix Diagonalization or Lumping……Page 715
A……Page 721
B……Page 722
C……Page 724
D……Page 726
E……Page 727
F……Page 728
G……Page 729
H……Page 730
I……Page 732
K……Page 733
L……Page 734
M……Page 736
N……Page 738
O……Page 739
P……Page 740
R……Page 741
S……Page 743
T……Page 745
V……Page 747
W……Page 748
Y……Page 749
Z……Page 750
A……Page 752
B……Page 754
C……Page 755
D……Page 758
E……Page 761
F……Page 766
G……Page 769
H……Page 770
I……Page 772
K……Page 774
L……Page 775
M……Page 776
N……Page 782
O……Page 783
P……Page 784
R……Page 787
S……Page 789
T……Page 794
U……Page 799
V……Page 800
W……Page 801
X……Page 802