Handbook of numerical analysis: Volume 4 Finite element methods (part 2), numerical methods for solids (part 2)
Gespeichert in:
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| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Amsterdam
North-Holland
1996
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| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007111949&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Umfang: | X, 974 Seiten Illustrationen |
| ISBN: | 0444817948 |
Internformat
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| 245 | 1 | 0 | |a Handbook of numerical analysis |n Volume 4 |p Finite element methods (part 2), numerical methods for solids (part 2) |c general editor: P. G. Ciarlet (Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie), J. L. Lions |
| 264 | 1 | |a Amsterdam |b North-Holland |c 1996 | |
| 300 | |a X, 974 Seiten |b Illustrationen | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
| 700 | 1 | |a Ciarlet, Philippe G. |d 1938- |e Sonstige |0 (DE-588)143368362 |4 oth | |
| 700 | 1 | |a Lions, Jacques-Louis |d 1928-2001 |e Sonstige |0 (DE-588)124055397 |4 oth | |
| 700 | 1 | |a Du, Qiang |d 1964- |0 (DE-588)1188249320 |4 edt | |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-007111949 | |
Datensatz im Suchindex
| DE-BY-TUM_call_number | 0001 91 A 464-4 |
|---|---|
| DE-BY-TUM_katkey | 776259 |
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| adam_text | Contents
Chapter I. Origins, Milestones and Directions of the Finite Element Method—
A Personal View 7
1. Introduction 7
2. The origins 8
3. The variational approaches via extremum principles 9
4. Some early applications of the finite element method and alternatives 12
5. Virtual work and weighted residual approaches. Generalised finite element method
and other approximations 17
6. Non-conforming approximation and the patch test as a necessary and sufficient
condition of finite element method convergence 23
6.1. Plate bending—Conforming and non-conforming variants 23
6.2. The patch test 24
6.3. Diffuse element approximation 28
7. Higher-order elements and isoparametric mapping. Three-dimensional analysis,
plates, shells and reduced integration 30
7.1. The need governs development—Isoparametric mapping 30
7.2. Physics govern theory—The essential development for plates and shells 31
8. Adaptivity and error estimation 37
9. Fluid mechanics and non-self-adjoint problems 43
10. Epilogue 53
Appendix 55
References 59
Subject Index 67
Contents
Preface 73
Chapter I. General Definitions and Problem 75
Introduction 75
1. Basic ideas of finite element methods 75
2. Mesh definition 78
3. Mesh contents 84
4. Mesh structure 87
5. Mesh prescriptions 89
6. Methodology 92
7. Data structure 93
8. Properties and formula about triangles and tetrahedra 94
Chapter II. Review of Mesh Generation Methods 97
Introduction 97
9. Manual methods 99
10. Product methods 100
11. Special application methods 103
11.1. Algebraic methods 103
11.2. PDE methods 106
12. Semi-automatic methods 111
13. Automatic methods 114
13.1. Quadtree-octree type methods 114
13.2. Advancing-front methods 117
13.3. Voronol type methods 120
14. Surfaces 122
Chapter HI. Automatic Method (1): Advancing-front Type Mesh Generation 127
Introduction 127
15. General principles 128
16. Creation and insertion of an optimal point 130
16.1. Position of a point with respect to the domain 130
16.2. Finding an optimal location 131
16.3. Insertion 134
16.4. Updating the front 135
16.5. Control space 135
16.6. Neighborhood space 137
16.7. Mesh point and element smoothing 139
17. Scheme for the mesh generator 140
18. Applications in the case of a given control 141
19. Application examples 144
71
72 P-L. George
20. Structures and algorithms 145
20.1. Basic structure 146
20.2. Background structures 146
20.3. Basic algorithms 147
Chapter IV. Automatic Method (2): Voronoi Type Mesh Generation 149
Introduction 149
21. General principles 150
22. Connection from points to points 152
22.1. Original insertion point method 153
22.2. Derived insertion point method 156
23. Boundary integrity 158
23.1. The two-dimensional case 158
23.1.1. Creation of the midpoint of every missing boundary edge 158
23.1.2. Local diagonal swapping procedures 160
23.2. The three-dimensional case 162
23.2.1. Creation of the midpoint of all missing edges and creation of a
point at the centroid of all missing faces 162
23.2.2. Local modifications coupled with possible creation of internal points 163
23.3. Definition of the inside of the domain 166
24. Creation of the field points 168
24.1. Creation and insertion of the internal points 168
24.2. First type method 168
24.3. Second type method 169
24.4. Mesh point and element smoothing 171
25. Scheme for the mesh generator 172
26. Applications in the case of a given control 172
27. Application examples 175
28. Structures and algorithms 179
28.1. Basic structure 179
28.2. Background structures 179
28.3. Basic algorithms 181
References 183
Subject Index 189
Contents
Preface 197
Chapter I. Introduction 199
1. The physical problem 199
2. The mathematical model 202
3. Remarks on the development of limit analysis 206
Chapter II. The Duality Problem 209
4. Why so different from elasticity? 209
5. Existence of the collapse solution 213
5.1. The stress space X 216
5.2. The flow space U 217
5.3. The duality of limit analysis 219
5.4. The reduced problem 225
6. Generalizations and variations 227
6.1. Boundary flow constrained to a subspace 227
6.2. Presence of a pre-load 228
6.3. Bounded yield set 228
6.4. Yield conditions for granular materials 229
7. Green s formula in the collapse state 230
Chapter III. Discretization by Finite Elements 235
8. Formulation of the discrete problem 235
9. Elements for divergence-free flow 244
10. Bounds and convergence 247
Chapter IV. Solution of the Discrete Problem 255
11. General remarks 255
12. Linear programming methods 259
12.1. Solution with the simplex method 262
12.2. Solution with the primal affine scaling algorithm 265
12.3. Solution with the dual affine scaling algorithm 267
13. Convex programming methods 272
14. Divergence-free elements 276
14.1. Solution of the primal problem 277
14.2. Solution of the dual problem 278
15. Concluding remarks 282
15.1. On the solution to the test problem 284
196 E. Christiansen
Chapter V. Limit Analysis for Plates 285
16. The continuous problem 285
17. Duality of limit analysis 288
18. Discretization of the plate problem 291
19. Solution of the discrete problem 295
References 303
List of Symbols 307
Subject Index 311
Contents
Preface 317
Chapter I. Variational Inequalities of the Elliptic Type. Dual and Mixed Variational
Approach 319
1. Examples of unilateral boundary value problems 319
2. Preliminaries concerning the approximation of variational inequalities 325
3. Elliptic variational inequalities. Different variational formulations. Existence results 327
Bibliography and comments to Chapter I 347
Chapter II. Approximate Solution of Variational Inequalities 349
4. Approximation of the elliptic inequalities 349
4.1. Approximation of the primal formulation 349
4.2. Approximation of mixed formulations 353
Bibliography and comments to Chapter II 361
Chapter HI. Contact Problems in Elasticity 363
5. Formulation of contact problems 363
5.1. Problems with a bounded zone of contact 365
5.2. Problems with increasing zone of contact 367
5.3. Variational formulations 368
6. Existence and uniqueness of solution 369
6.1. Problem with a bounded zone of contact 370
6.2. Problem with increasing zone of contact 374
7. Contact problems with friction 377
7.1. Definitions and preliminary results 377
7.2. Contact problem with Coulomb s law 380
8. Approximation of contact problems by finite elements. Frictionless case 382
8.1. Approximation of the problem with a bounded zone of contact 383
8.2. Approximation of the problem with an increasing zone of contact 388
9. Approximation of the contact problem with given friction 391
10. Approximation of the Signorini problem obeying Coulomb s law of friction 404
11. Numerical realization of contact problems 411
Chapter IV. Contact of Elasto-Plastic Bodies 417
12. Deformation theory 417
12.1. Formulation of Signorini s contact problem in the deformation theory of
plasticity 417
12.2. Abstract convergence theorem for the secant modules method 418
12.3. Application to Signorini s contact problems 419
13. Contact of perfectly elasto-plastic bodies 420
13.1. Bounded zone of contact 421
316 J. Haslinger, I. Hlavacek and J. Necas
13.2. Increasing zone of contact 424
13.3. Solution of approximate problems 427
Chapter V. Problems of the Theory of Plasticity 431
14. Prandtl-Reuss model of plastic flow 431
15. Plastic flow with hardening 435
16. Solution of the Prandtl-Reuss model by the equilibrium finite-element method 440
16.1. A priori error estimates 442
17. Solution of isotropic hardening using finite elements 445
17.1. A priori error estimates 448
17.2. Convergence in the case of a nonregular solution 454
18. Bibliography and comments to Chapter V 457
Chapter VI. Unilateral Problems for Elastic Plates 459
19. Introduction. Statement of the basic problems 459
20. Unilateral displacement problem 461
20.1. Primal approach 461
20.2. A mixed variational formulation for a semicoercive problem 465
21. Unilateral rotation problem 466
22. Inner obstacle problem 468
22.1. A primal approach 468
22.2. A dual approach 469
22.3. A dual approach combined with the penalty method 470
22.4. A mixed formulation by factorization 471
Bibliography and comments to Chapter VI 472
References 473
List of Symbols 479
Subject Index 483
Contents
Preface 493
Introduction 497
Mathematical modelling of elastic plates and shells: Classical and asymptotic methods 497
Mathematical modelling of elastic rods 499
Mathematical modelling of elastic rods: Asymptotic methods 502
Overview of contents 507
Chapter I. Asymptotic Expansion Method for a Linearly Elastic Clamped Rod 511
1. The three-dimensional equations of a linearly elastic clamped rod 512
2. The fundamental scalings on the unknowns and assumptions on the data: The
displacement approach 515
3. The asymptotic expansion method 519
4. Cancellation of the factors of e?, —4 ^ q ^ —1. Some properties of «° and u2.
Boundary layer phenomenon 521
5. The weakly clamping conditions. The asymptotic expansion method in the
displacement approach 530
6. The asymptotic expansion method in the displacement-stress approach 535
7. Characterization of the zeroth order stress term and second order displacement
fields: Statement of the theorem 542
8. Characterization of the zeroth order stress and second order displacement fields:
Proof of the theorem 554
9. Characterization of the second order stress and fourth order displacement fields 565
10. Convergence of the scaled displacements and stresses 575
Chapter II. Linear Asymptotic Models. Comparison with Classical Theories 587
11. Asymptotic first order one-dimensional equations of a linearly elastic clamped
rod. Comparison with Bernoulli-Navier s theory 587
12. Asymptotic second order general model of a linearly elastic clamped rod 592
13. Comparison with Vlassov s beam theory for thick rods 601
14. Comparison with Saint Venant s pure torsion theory 603
15. Comparison with Timoshenko s beam theory. New Timoshenko constants 606
Chapter III. Asymptotic Modelling in Thin-Walled Rods - An Introduction 621
16. Formal asymptotic approach of bending and torsion properties for rods with a
thin rectangular cross section 621
17. Convergence and limit problem for Poisson s equation in a thin rectangle with
Dirichlet or Neumann boundary conditions 632
18. Convergence of torsion and warping functions for thin rectangular cross sections 638
19. Convergence of Timoshenko s functions and constants for thin rectangular cross
sections 641
490 L. Trabucho and J.M. Viano
20. Critical rectangular cross sections in Timoshenko s beam theory 650
21. Asymptotic problems in thin-walled beams 652
Chapter IV. Asymptotic Expansion Method for Nonhomogeneous Anisotropic Rods 657
22. The asymptotic expansion method for anisotropic and nonhomogeneous elastic
rods. Applied forces at one end 658
23. First order coupled bending-stretching model for anisotropic and
nonhomogeneous elastic rods. Convergence results. Fibre-reinforced beams 665
24. Higher order models for transversely nonhomogeneous isotropic rods 682
25. A general asymptotic model for homogeneous anisotropic rods 697
Chapter V. Asymptotic Modelling of Rods in Linearized Thermoelasticity 715
26. The asymptotic expansion method for stationary thermoelastic rods 716
27. Asymptotic justification of an evolution linear thermoelastic model for rods 734
Chapter VI. Asymptotic Method for Contact Problems in Rods and Rods with a
Variable Cross Section 763
28. Asymptotic derivation of unilateral contact models for rods on a rigid or elastic
foundation 764
29. The asymptotic expansion method for rods with a variable cross section 778
Chapter VII. Rods with a Multicellular Cross Section. Some Homogenization and
Asymptotic Results 801
30. The asymptotic model for rods with a multicellular cross section 802
31. Homogenization of the generalized beam theories 812
32. Generalized beam theory of the homogenized material 833
Chapter VIII. A Galerkin Type Method for Linear Elastic Rods 847
33. A Galerkin type approximation in linearized beam theory 847
34. A Galerkin type approximation for the multicellular case 862
35. Homogenization of the basis functions 872
36. A Galerkin approximation of the homogenized three-dimensional equations 879
37. The three-dimensional equations for a linearly elastic curved rod 888
38. The curved rod problem posed in a fixed domain 890
39. The Galerkin method for curved rods 898
39.1. The Galerkin method for the straight beam h^ 898
39.2. The Galerkin method for the beam he 901
Chapter IX. Asymptotic Method for Nonlinear Elastic Rods. Stationary Case 909
40. The three-dimensional equations of a nonlinearly elastic clamped rod 909
41. The fundamental scalings on the unknowns and assumptions on the data: the
displacement approach 912
42. Cancellation of the factors of sq, — 4 q ^ 0, in the scaled three-dimensional
problem 916
43. Identification of the leading term u° of the formal expansion 924
44. Existence of the leading term »° in the displacement approach 927
45. The displacement-stress approach for the nonlinear case 930
Chapter X. Asymptotic Method for Nonlinear Elastic Rods. Dynamic Case 941
46. The three-dimensional equations for the nonlinear elastodynamic case 942
Contents 491
47. The asymptotic expansion method for the nonlinear elastodynamic case 944
48. Computation of the first term in the expansion for the nonlinear elastodynamic
case 948
49. The limit problem for the nonlinear elastodynamic case 952
50. The general nonlinear elastodynamic case 954
References 959
Subject Index 971
|
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| indexdate | 2024-12-20T09:58:26Z |
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| isbn | 0444817948 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007111949 |
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| spellingShingle | Handbook of numerical analysis |
| title | Handbook of numerical analysis |
| title_auth | Handbook of numerical analysis |
| title_exact_search | Handbook of numerical analysis |
| title_full | Handbook of numerical analysis Volume 4 Finite element methods (part 2), numerical methods for solids (part 2) general editor: P. G. Ciarlet (Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie), J. L. Lions |
| title_fullStr | Handbook of numerical analysis Volume 4 Finite element methods (part 2), numerical methods for solids (part 2) general editor: P. G. Ciarlet (Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie), J. L. Lions |
| title_full_unstemmed | Handbook of numerical analysis Volume 4 Finite element methods (part 2), numerical methods for solids (part 2) general editor: P. G. Ciarlet (Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie), J. L. Lions |
| title_short | Handbook of numerical analysis |
| title_sort | handbook of numerical analysis finite element methods part 2 numerical methods for solids part 2 |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007111949&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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