The finite element method: its basis and fundamentals
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Amsterdam [u.a.]
Elsevier Butterworth-Heinemann
2005
|
| Edition: | 6. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | 5. Aufl. mehrbd. erschienen Hier auch später erschienene, unveränderte Nachdrucke "In the present edition we have decided not to pursue the course of having three contiguous volumes but rather we treat the whole work as an assembly of three separate works, each one capable of being used without the others ... The two further volumes form again separate books ... The first of these is entitled The Finite Element Method in Solid and Structural Mechanics and the second is a text entitled The Finite Element Method in Fluid Dynamics." |
| Physical Description: | XIV, 733 S. zahlr. graph. Darst. |
| ISBN: | 0750663200 |
Staff View
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| 245 | 1 | 0 | |a The finite element method |b its basis and fundamentals |c O. C. Zienkiewicz ; R. L. Taylor ; J. Z. Zhu |
| 250 | |a 6. ed. | ||
| 264 | 1 | |a Amsterdam [u.a.] |b Elsevier Butterworth-Heinemann |c 2005 | |
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Record in the Search Index
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| adam_text | Contents
Preface xiii
1 The standard discrete system and origins of the finite element method 1
1.1 Introduction 1
1.2 The structural element and the structural system 3
1.3 Assembly and analysis of a structure 5
1.4 The boundary conditions 6
1.5 Electrical and fluid networks 7
1.6 The general pattern 9
1.7 The standard discrete system 10
1.8 Transformation of coordinates II
1.9 Problems 13
2 A direct physical approach to problems in elasticity: plane stress 19
2.1 Introduction 19
2.2 Direct formulation of finite element characteristics 20
2.3 Generalization to the whole region - internal nodal force concept
abandoned 31
2.4 Displacement approach as a minimization of total potential energy 34
2.5 Convergence criteria 37
2.6 Discretization error and convergence rate 3K
2.7 Displacement functions with discontinuity between elements -
non-conforming elements and the patch test 39
2.8 Finite element solution process 40
2.9 Numerical examples 40
2.10 Concluding remarks 46
2.11 Problems 47
3 Generalization of the finite element concepts. Galerkin-weighted residual and
variational approaches 54
3.1 Introduction 54
3.2 Integral or weak statements equivalent to the differential equations 57
3.3 Approximation to integral formulations: the weighted residual-
Galerkin method 60
viii Contents
3.4 Virtual work as the weak form of equilibrium equations for analysis
of solids or fluids 69
3.5 Partial discretization 71
3.6 Convergence 74
3.7 What are variational principles ? 76
3.8 Natural variational principles and their relation to governing
differential equations 78
3.9 Establishment of natural variational principles for linear, self-adjoint,
differential equations 81
3.10 Maximum, minimum, or a saddle point? 83
3.11 Constrained variational principles. Lagrange multipliers 84
3.12 Constrained variational principles. Penalty function and perturbed
lagrangian methods 88
3.13 Least squares approximations 92
3.14 Concluding remarks - finite difference and boundary methods 95
3.15 Problems 97
4 Standard and hierarchical element shape functions: some general families
of Co continuity 103
4.1 Introduction 103
4.2 Standard and hierarchical concepts 104
4.3 Rectangular elements - some preliminary considerations 107
4.4 Completeness of polynomials 109
4.5 Rectangular elements - Lagrange family 110
4.6 Rectangular elements - serendipity family 112
4.7 Triangular element family 116
4.8 Line elements 119
4.9 Rectangular prisms - Lagrange family 120
4.10 Rectangular prisms - serendipity family 121
4.11 Tetrahedral elements 122
4.12 Other simple three-dimensional elements 125
4.13 Hierarchic polynomials in one dimension 125
4.14 Two- and three-dimensional, hierarchical elements of the rectangle
or brick type 128
4.15 Triangle and tetrahedron family 128
4.16 Improvement of conditioning with hierarchical forms 130
4.17 Global and local finite element approximation 131
4.18 Elimination of internal parameters before assembly - substructures 132
4.19 Concluding remarks 134
4.20 Problems 134
5 Mapped elements and numerical integration - infinite and
singularity elements 138
5.1 Introduction 138
5.2 Use of shape functions in the establishment of coordinate
transformations 139
5.3 Geometrical conformity of elements 143
5.4 Variation of the unknown function within distorted, curvilinear
elements. Continuity requirements 143
Conten
5.5 Evaluation of element matrices. Transformation in £, r , £ coordinates 145
5.6 Evaluation of element matrices. Transformation in area and volume
coordinates 148
5.7 Order of convergence for mapped elements 151
5.8 Shape functions by degeneration 153
5.9 Numerical integration - one dimensional 160
5.10 Numerical integration - rectangular (2D) or brick regions (3D) 162
5.11 Numerical integration - triangular or tetrahedral regions 164
5.12 Required order of numerical integration 164
5.13 Generation of finite element meshes by mapping. Blending functions 169
5.14 Infinite domains and infinite elements 170
5.15 Singular elements by mapping - use in fracture mechanics, etc. 176
5.16 Computational advantage of numerically integrated finite elements 177
5.17 Problems 178
6 Problems in linear elasticity 187
6.1 Introduction 187
6.2 Governing equations 188
6.3 Finite element approximation 201
6.4 Reporting of results: displacements, strains and stresses 207
6.5 Numerical examples 209
6.6 Problems 217
7 Field problems - heat conduction, electric and magnetic potential
and fluid flow 229
7.1 Introduction 229
7.2 General quasi-harmonic equation 230
7.3 Finite element solution process 233
7.4 Partial discretization - transient problems 237
7.5 Numerical examples - an assessment of accuracy 239
7.6 Concluding remarks 253
7.7 Problems 253
8 Automatic mesh generation 264
8.1 Introduction 264
8.2 Two-dimensional mesh generation - advancing front method 266
8.3 Surface mesh generation 286
8.4 Three-dimensional mesh generation - Delaunay triangulation 303
8.5 Concluding remarks 323
8.6 Problems 323
9 The patch test, reduced integration, and non-conforming elements 329
9.1 Introduction 329
9.2 Convergence requirements 330
9.3 The simple patch test (tests A and B) - a necessary condition for
convergence 332
9.4 Generalized patch test (test C) and the single-element test 334
9.5 The generality of a numerical patch test 336
9.6 Higher order patch tests 336
x Contents
9.7 Application of the patch test to plane elasticity elements with
standard and reduced quadrature 337
9.8 Application of the patch test to an incompatible element 343
9.9 Higher order patch test - assessment of robustness 347
9.10 Concluding remarks 347
9.11 Problems 350
10 Mixed formulation and constraints - complete field methods 356
10.1 Introduction 356
10.2 Discretization of mixed forms - some general remarks 358
10.3 Stability of mixed approximation. The patch test 360
10.4 Two-field mixed formulation in elasticity 363
10.5 Three-field mixed formulations in elasticity 370
10.6 Complementary forms with direct constraint 375
10.7 Concluding remarks - mixed formulation or a test of element
robustness 379
10.8 Problems 379
11 Incompressible problems, mixed methods and other procedures of solution 383
11.1 Introduction 383
11.2 Deviatoric stress and strain, pressure and volume change 383
11.3 Two-field incompressible elasticity (u-p form) 384
11.4 Three-field nearly incompressible elasticity (n-p~ev form) 393
11.5 Reduced and selective integration and its equivalence to penalized
mixed problems 398
11.6 A simple iterative solution process for mixed problems: Uzawa
method 404
11.7 Stabilized methods for some mixed elements failing the
incompressibility patch test 407
11.8 Concluding remarks 421
11.9 Problems 422
12 Multidomain mixed approximations - domain decomposition and frame
methods 429
12.1 Introduction 429
12.2 Linking of two or more subdomains by Lagrange multipliers 430
12.3 Linking of two or more subdomains by perturbed lagrangian and
penalty methods 436
12.4 Interface displacement frame 442
12.5 Linking of boundary (or Trefftz)-type solution by the frame of
specified displacements 445
12.6 Subdomains with standard elements and global functions 451
12.7 Concluding remarks 451
12.8 Problems 451
13 Errors, recovery processes and error estimates 456
13.1 Definition of errors 456
13.2 Superconvergence and optimal sampling points 459
13.3 Recovery of gradients and stresses 465
Conten
13.4 Superconvergent patch recovery - SPR 467
13.5 Recovery by equilibration of patches - REP 474
13.6 Error estimates by recovery 476
13.7 Residual-based methods 478
13.8 Asymptotic behaviour and robustness of error estimators - the
Babuska patch test 488
13.9 Bounds on quantities of interest 490
13.10 Which errors should concern us? 494
13.11 Problems 495
14 Adaptive finite element refinement 500
14.1 Introduction 500
14.2 Adaptive ^-refinement 503
14.3 /^-refinement and /^-refinement 514
14.4 Concluding remarks 518
14.5 Problems 520
15 Point-based and partition of unity approximations. Extended finite
element methods 525
15.1 Introduction 525
15.2 Function approximation 527
15.3 Moving least squares approximations - restoration of continuity of
approximation 533
15.4 Hierarchical enhancement of moving least squares expansions 538
15.5 Point collocation - finite point methods 540
15.6 Galerkin weighting and finite volume methods 546
15.7 Use of hierarchic and special functions based on standard finite
elements satisfying the partition of unity requirement 549
15.8 Concluding remarks 558
15.9 Problems 558
16 The time dimension - semi-discretization of field and dynamic problems
and analytical solution procedures 563
16.1 Introduction 563
16.2 Direct formulation of time-dependent problems with spatial finite
element subdivision 563
16.3 General classification 570
16.4 Free response - eigenvalues for second-order problems and dynamic
vibration 571
16.5 Free response - eigenvalues for first-order problems and heat
conduction, etc. 576
16.6 Free response-damped dynamic eigenvalues 578
16.7 Forced periodic response 579
16.8 Transient response by analytical procedures 579
16.9 Symmetry and repeatability 583
16.10 Problems 584
17 The time dimension - discrete approximation in time 589
17.1 Introduction 589
xii Contents
17.2 Simple time-step algorithms for the first-order equation 590
17.3 General single-step algorithms for first- and second-order equations 600
17.4 Stability of general algorithms 609
17.5 Multistep recurrence algorithms 615
17.6 Some remarks on general performance of numerical algorithms 618
17.7 Time discontinuous Galerkin approximation 619
17.8 Concluding remarks 624
17.9 Problems 626
18 Coupled systems 631
18.1 Coupled problems - definition and classification 631
18.2 Fluid-structure interaction (Class I problems) 634
18.3 Soil-pore fluid interaction (Class II problems) 645
18.4 Partitioned single-phase systems - implicit-explicit partitions
(Class I problems) 653
18.5 Staggered solution processes 655
18.6 Concluding remarks 660
19 Computer procedures for finite element analysis 664
19.1 Introduction 664
19.2 Pre-processing module: mesh creation 664
19.3 Solution module 666
19.4 Post-processor module 666
19.5 User modules 667
Appendix A: Matrix algebra 668
Appendix B: Tensor-indicial notation in the approximation of elasticity problems 674
Appendix C: Solution of simultaneous linear algebraic equations 683
Appendix D: Some integration formulae for a triangle 692
Appendix E: Some integration formulae for a tetrahedron 693
Appendix F: Some vector algebra 694
Appendix G: Integration by parts in two or three dimensions (Green s theorem) 699
Appendix H: Solutions exact at nodes 701
Appendix I: Matrix diagonalization or lumping 704
Author index 711
Subject index 719
|
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| discipline | Physik Mathematik |
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| spelling | Zienkiewicz, Olgierd C. 1921-2009 Verfasser (DE-588)108459373 aut The finite element method its basis and fundamentals O. C. Zienkiewicz ; R. L. Taylor ; J. Z. Zhu 6. ed. Amsterdam [u.a.] Elsevier Butterworth-Heinemann 2005 XIV, 733 S. zahlr. graph. Darst. txt rdacontent n rdamedia nc rdacarrier 5. Aufl. mehrbd. erschienen Hier auch später erschienene, unveränderte Nachdrucke "In the present edition we have decided not to pursue the course of having three contiguous volumes but rather we treat the whole work as an assembly of three separate works, each one capable of being used without the others ... The two further volumes form again separate books ... The first of these is entitled The Finite Element Method in Solid and Structural Mechanics and the second is a text entitled The Finite Element Method in Fluid Dynamics." Engineering mathematics Finite element method Finite-Elemente-Methode (DE-588)4017233-8 gnd rswk-swf Approximation (DE-588)4002498-2 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Finite-Elemente-Methode (DE-588)4017233-8 s Approximation (DE-588)4002498-2 s 1\p DE-604 Taylor, Robert L. 1934- Verfasser (DE-588)122486536 aut Zhu, Jianzhong 1955- Verfasser (DE-588)13327540X aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013196574&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Zienkiewicz, Olgierd C. 1921-2009 Taylor, Robert L. 1934- Zhu, Jianzhong 1955- The finite element method its basis and fundamentals Engineering mathematics Finite element method Finite-Elemente-Methode (DE-588)4017233-8 gnd Approximation (DE-588)4002498-2 gnd |
| subject_GND | (DE-588)4017233-8 (DE-588)4002498-2 (DE-588)4123623-3 |
| title | The finite element method its basis and fundamentals |
| title_auth | The finite element method its basis and fundamentals |
| title_exact_search | The finite element method its basis and fundamentals |
| title_full | The finite element method its basis and fundamentals O. C. Zienkiewicz ; R. L. Taylor ; J. Z. Zhu |
| title_fullStr | The finite element method its basis and fundamentals O. C. Zienkiewicz ; R. L. Taylor ; J. Z. Zhu |
| title_full_unstemmed | The finite element method its basis and fundamentals O. C. Zienkiewicz ; R. L. Taylor ; J. Z. Zhu |
| title_short | The finite element method |
| title_sort | the finite element method its basis and fundamentals |
| title_sub | its basis and fundamentals |
| topic | Engineering mathematics Finite element method Finite-Elemente-Methode (DE-588)4017233-8 gnd Approximation (DE-588)4002498-2 gnd |
| topic_facet | Engineering mathematics Finite element method Finite-Elemente-Methode Approximation Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013196574&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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