P- and hp-finite element methods: theory and applications in solid and fluid mechanics
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford
Clarendon Press
1998
|
| Series: | Numerical mathematics and scientific computation
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. [364] - 371 |
| Physical Description: | XII, 374 S. graph. Darst. |
| ISBN: | 0198503903 |
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| 300 | |a XII, 374 S. |b graph. Darst. | ||
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| 650 | 4 | |a Differential equations |x Numerical solutions | |
| 650 | 4 | |a Finite element method | |
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Record in the Search Index
| _version_ | 1804135907481092096 |
|---|---|
| adam_text | CONTENTS
1 Variational formulation of boundary value problems 1
1.1 Model problems 1
1.1.1 Axially loaded, elastically supported bar 1
1.1.2 Membrane problem 4
1.2 Generalized solutions 7
1.2.1 Classical solution 7
1.2.2 Generalized formulations 7
1.2.3 Generalized formulations of the bar problem 9
1.2.4 Essential and natural boundary conditions 13
1.2.5 Generalized formulations of the membrane problem 13
1.3 Existence of weak solutions. Inf sup condition 15
1.3.1 Theory 15
1.3.2 Examples 21
1.4 Inh. essen. bound, conditions. Space HOq 27
1.4.1 The trace space Hl 2{T) 27
1.4.2 The problem of discontinuous Dirichlet data 28
1.4.3 The space ffo^2(rW) 31
1.5 Examples for generalized formulations 32
1.5.1 Mixed formulation of the bar problem I 32
1.5.2 Mixed formulation of the bar problem II 35
1.5.3 An initial value problem 36
1.5.4 The heat equation 37
1.5.5 Mixed formulation of the membrane problem 40
1.6 Bibliographical remarks 42
2 The Finite Element Method (FEM): Definition, Basic
Properties 43
2.1 Approximate solutions 43
2.2 Acceptance criteria and refinements 46
2.3 Stability, consistency and convergence 47
2.4 Variational crimes/nonconforming FEM 54
3 ftp Finite Elements in one dimension 57
3.1 The finite element spaces Sp l(tt, T) 57
3.1.1 Meshes 57
3.1.2 Element mappings 57
3.1.3 Definition of 5^(fi,T) 58
3.1.4 Basis of Sp 58
3.1.5 Element shape functions 60
3.1.6 Basis functions of Sp t(il, T) 60
X
3.2 Algorithmic pattern of the ftp FEM 62
3.3 The approximation properties of Sp (fi,T) 69
3.3.1 Basic approximation properties of Sp on 0 = (—1,1) 70
3.3.2 Rate of convergence of the ftp FEM with quasiuni
form meshes 75
3.3.3 Analytic solutions 78
3.3.4 Singular solutions, ft version FEM 80
3.3.5 Singular solutions, p version FEM 83
3.3.6 Singular solutions. hp FEM 89
3.3.7 Mesh optimization for ft version FEM 96
3.4 Problems with boundary layers. Robustness 100
3.4.1 Reaction diffusion problem 100
3.4.2 Regularity of the solution 102
3.4.3 The finite element method 105
3.4.4 Robustness 106
3.4.5 p approximation results 108
3.4.6 ftp approximation results 118
3.4.7 Numerical results 124
3.4.8 Convection diffusion problem 130
3.4.9 Variational formulation 132
3.4.10 ftp finite element discretization 135
3.5 A posteriori error estimate 142
3.5.1 Abstract residual error estimate 142
3.5.2 Application to hp FEM [6] 143
3.6 Inverse inequalities 148
3.6.1 Basic inverse inequality 148
3.6.2 Weighted inverse estimates 149
3.7 Bibliographical remarks 152
4 hp F mite Elements in two dimensions 154
4.1 Model problem 154
4.2 Regularity of the solution 1 156
4.3 Linear finite elements with mesh refinement 158
4.3.1 Meshes consisting of triangles 158
4.3.2 The finite element spaces Sp *(tt, T) 160
4.3.3 Approximation properties of 51 1(f2,T) 161
4.3.4 Extension to meshes containing quadrilaterals 167
4.4 hp FEM in two dimensions. Space Sp *(tt, T) 169
4.4.1 The geometric mesh families 169
4.4.2 Construction of 5p £(fi,T) 172
4.5 The rate of convergence of the hp FEM. 179
4.5.1 Regularity of the solution 179
4.5.2 Basic approximation results for hp FEM 180
4.5.3 ftp Approximation of u G i?| on a polygon 198
4.5.4 ftp FEM on quasiuniform meshes 200
xi
4.6 Inverse inequalities and trace liftings 207
4.6.1 Basic inverse inequalities 208
4.6.2 Inequalities in H1/2^!,!) 210
4.6.3 Polynomial trace liftings 213
4.7 /ip Preconditioning 214
4.7.1 Preliminary remarks 214
4.7.2 Preconditioning hp FEM by low order elements 215
4.8 Bibliographical remarks and further results 221
5 Finite Element Analysis of Saddle Point Problems. Mixed
hp FEM. in incompressible fluid flow 223
5.1 Introduction. Navier Stokes equations 223
5.2 Abstract saddle point problems. Inf sup condition 226
5.2.1 Minimization problems with constraints 226
5.2.2 Abstract saddle point problems 227
5.3 Existence for the Stokes problem 233
5.4 FE discretization of saddle point problems 234
5.5 Finite elements for the Stokes problem 240
5.5.1 General results 240
5.5.2 The Qi Pq element and spurious pressure modes 242
5.5.3 Verifying the inf sup conditions: Fortin s lemma 245
5.5.4 An approximation result: Clement interpolant 246
5.5.5 Some stable low order elements 247
5.5.6 Bubble stabilization of Stokes elements 252
5.5.7 Macroelement techniques 254
5.5.8 p and hp FEM for the Stokes Problem 259
5.6 Further results and bibliographical remarks 273
6 hp FEM in the Theory of Elasticity 275
6.1 A brief synopsis of the theory of elasticity 275
6.1.1 Kinematics 275
6.1.2 Equilibrium 277
6.1.3 Piola transform 278
6.1.4 Constitutive laws 279
6.1.5 Small strains. St. Venant Kirchhoff material 281
6.1.6 Hyper elasticity 282
6.1.7 Linearized three dimensional elasticity 284
6.2 Membranes, plates and shells 292
6.2.1 Membranes and plates 292
6.2.2 Shells 297
6.2.3 Koiter s shell model 301
6.2.4 Naghdi s shell model 306
6.2.5 Asymptotics of Koiter shells as e — ¦ 0. Bending and
membrane dominated shells 310
6.2.6 The case of a plate 314
xii
6.2.7 Membrane theories of shells of revolution 316
6.3 ft.p FEM for problems with constraints. Locking. 318
6.3.1 The problem of locking in FEM 319
6.3.2 Analysis of reduced constraint FEM 326
6.3.3 Application to Naghdi shell models 329
6.3.4 Reduced constraint hp FEM 331
A Sobolev Spaces 339
A.I Domains and boundaries 339
A.I.I 339
A.1.2 Lipschitz boundary 339
A. 1.3 Partition of unity 340
A.1.4 Ck boundary 340
A.1.5 Surface integral and normal vector 340
A.2 Spaces LS(Q), Ls(dn). 341
A.3 Spaces of continuous functions 342
A.4 Weak derivatives 343
A.5 Definition of the Sobolev spaces Wk s(n) 344
A.6 Density of smooth functions 346
A.7 Traces. Sobolev spaces on dQ, 346
A.8 Embedding theorems 349
A.9 Poincare inequalities 350
A.10 Green s formulas 351
A.ll Negative order spaces 352
A. 12 The space H(div, fi) and its properties 354
A. 13 Exercises 355
B Interpolation spaces 356
B.I Relation between Sobolev spaces 356
B.2 The real method of interpolation 357
B.3 Interpolation of Sobolev spaces 358
C Orthogonal polynomials 359
C.I General definitions 359
C.2 Legendre polynomials 359
C.3 Jacobi polynomials 361
C.4 Some special cases 363
References 364
Index 371
|
| adam_txt |
CONTENTS
1 Variational formulation of boundary value problems 1
1.1 Model problems 1
1.1.1 Axially loaded, elastically supported bar 1
1.1.2 Membrane problem 4
1.2 Generalized solutions 7
1.2.1 Classical solution 7
1.2.2 Generalized formulations 7
1.2.3 Generalized formulations of the bar problem 9
1.2.4 Essential and natural boundary conditions 13
1.2.5 Generalized formulations of the membrane problem 13
1.3 Existence of weak solutions. Inf sup condition 15
1.3.1 Theory 15
1.3.2 Examples 21
1.4 Inh. essen. bound, conditions. Space HOq 27
1.4.1 The trace space Hl'2{T) 27
1.4.2 The problem of discontinuous Dirichlet data 28
1.4.3 The space ffo^2(rW) 31
1.5 Examples for generalized formulations 32
1.5.1 Mixed formulation of the bar problem I 32
1.5.2 Mixed formulation of the bar problem II 35
1.5.3 An initial value problem 36
1.5.4 The heat equation 37
1.5.5 Mixed formulation of the membrane problem 40
1.6 Bibliographical remarks 42
2 The Finite Element Method (FEM): Definition, Basic
Properties 43
2.1 Approximate solutions 43
2.2 Acceptance criteria and refinements 46
2.3 Stability, consistency and convergence 47
2.4 Variational crimes/nonconforming FEM 54
3 ftp Finite Elements in one dimension 57
3.1 The finite element spaces Sp'l(tt, T) 57
3.1.1 Meshes 57
3.1.2 Element mappings 57
3.1.3 Definition of 5^(fi,T) 58
3.1.4 Basis of Sp 58
3.1.5 Element shape functions 60
3.1.6 Basis functions of Sp t(il, T) 60
X
3.2 Algorithmic pattern of the ftp FEM 62
3.3 The approximation properties of Sp''(fi,T) 69
3.3.1 Basic approximation properties of Sp on 0 = (—1,1) 70
3.3.2 Rate of convergence of the ftp FEM with quasiuni
form meshes 75
3.3.3 Analytic solutions 78
3.3.4 Singular solutions, ft version FEM 80
3.3.5 Singular solutions, p version FEM 83
3.3.6 Singular solutions. hp FEM 89
3.3.7 Mesh optimization for ft version FEM 96
3.4 Problems with boundary layers. Robustness 100
3.4.1 Reaction diffusion problem 100
3.4.2 Regularity of the solution 102
3.4.3 The finite element method 105
3.4.4 Robustness 106
3.4.5 p approximation results 108
3.4.6 ftp approximation results 118
3.4.7 Numerical results 124
3.4.8 Convection diffusion problem 130
3.4.9 Variational formulation 132
3.4.10 ftp finite element discretization 135
3.5 A posteriori error estimate 142
3.5.1 Abstract residual error estimate 142
3.5.2 Application to hp FEM [6] 143
3.6 Inverse inequalities 148
3.6.1 Basic inverse inequality 148
3.6.2 Weighted inverse estimates 149
3.7 Bibliographical remarks 152
4 hp F'mite Elements in two dimensions 154
4.1 Model problem 154
4.2 Regularity of the solution 1 156
4.3 Linear finite elements with mesh refinement 158
4.3.1 Meshes consisting of triangles 158
4.3.2 The finite element spaces Sp'*(tt, T) 160
4.3.3 Approximation properties of 51'1(f2,T) 161
4.3.4 Extension to meshes containing quadrilaterals 167
4.4 hp FEM in two dimensions. Space Sp'*(tt, T) 169
4.4.1 The geometric mesh families 169
4.4.2 Construction of 5p'£(fi,T) 172
4.5 The rate of convergence of the hp FEM. 179
4.5.1 Regularity of the solution 179
4.5.2 Basic approximation results for hp FEM 180
4.5.3 ftp Approximation of u G i?| on a polygon 198
4.5.4 ftp FEM on quasiuniform meshes 200
xi
4.6 Inverse inequalities and trace liftings 207
4.6.1 Basic inverse inequalities 208
4.6.2 Inequalities in H1/2^!,!) 210
4.6.3 Polynomial trace liftings 213
4.7 /ip Preconditioning 214
4.7.1 Preliminary remarks 214
4.7.2 Preconditioning hp FEM by low order elements 215
4.8 Bibliographical remarks and further results 221
5 Finite Element Analysis of Saddle Point Problems. Mixed
hp FEM. in incompressible fluid flow 223
5.1 Introduction. Navier Stokes equations 223
5.2 Abstract saddle point problems. Inf sup condition 226
5.2.1 Minimization problems with constraints 226
5.2.2 Abstract saddle point problems 227
5.3 Existence for the Stokes problem 233
5.4 FE discretization of saddle point problems 234
5.5 Finite elements for the Stokes problem 240
5.5.1 General results 240
5.5.2 The Qi Pq element and spurious pressure modes 242
5.5.3 Verifying the inf sup conditions: Fortin's lemma 245
5.5.4 An approximation result: Clement interpolant 246
5.5.5 Some stable low order elements 247
5.5.6 Bubble stabilization of Stokes elements 252
5.5.7 Macroelement techniques 254
5.5.8 p and hp FEM for the Stokes Problem 259
5.6 Further results and bibliographical remarks 273
6 hp FEM in the Theory of Elasticity 275
6.1 A brief synopsis of the theory of elasticity 275
6.1.1 Kinematics 275
6.1.2 Equilibrium 277
6.1.3 Piola transform 278
6.1.4 Constitutive laws 279
6.1.5 Small strains. St. Venant Kirchhoff material 281
6.1.6 Hyper elasticity 282
6.1.7 Linearized three dimensional elasticity 284
6.2 Membranes, plates and shells 292
6.2.1 Membranes and plates 292
6.2.2 Shells 297
6.2.3 Koiter's shell model 301
6.2.4 Naghdi's shell model 306
6.2.5 Asymptotics of Koiter shells as e — ¦ 0. Bending and
membrane dominated shells 310
6.2.6 The case of a plate 314
xii
6.2.7 Membrane theories of shells of revolution 316
6.3 ft.p FEM for problems with constraints. Locking. 318
6.3.1 The problem of locking in FEM 319
6.3.2 Analysis of reduced constraint FEM 326
6.3.3 Application to Naghdi shell models 329
6.3.4 Reduced constraint hp FEM 331
A Sobolev Spaces 339
A.I Domains and boundaries 339
A.I.I 339
A.1.2 Lipschitz boundary 339
A. 1.3 Partition of unity 340
A.1.4 Ck boundary 340
A.1.5 Surface integral and normal vector 340
A.2 Spaces LS(Q), Ls(dn). 341
A.3 Spaces of continuous functions 342
A.4 Weak derivatives 343
A.5 Definition of the Sobolev spaces Wk s(n) 344
A.6 Density of smooth functions 346
A.7 Traces. Sobolev spaces on dQ, 346
A.8 Embedding theorems 349
A.9 Poincare inequalities 350
A.10 Green's formulas 351
A.ll Negative order spaces 352
A. 12 The space H(div, fi) and its properties 354
A. 13 Exercises 355
B Interpolation spaces 356
B.I Relation between Sobolev spaces 356
B.2 The real method of interpolation 357
B.3 Interpolation of Sobolev spaces 358
C Orthogonal polynomials 359
C.I General definitions 359
C.2 Legendre polynomials 359
C.3 Jacobi polynomials 361
C.4 Some special cases 363
References 364
Index 371 |
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| id | DE-604.BV021945205 |
| illustrated | Illustrated |
| index_date | 2024-07-02T16:07:16Z |
| indexdate | 2024-07-09T20:47:58Z |
| institution | BVB |
| isbn | 0198503903 |
| language | English |
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| physical | XII, 374 S. graph. Darst. |
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| series2 | Numerical mathematics and scientific computation |
| spelling | Schwab, Christoph Verfasser aut P- and hp-finite element methods theory and applications in solid and fluid mechanics Oxford Clarendon Press 1998 XII, 374 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Numerical mathematics and scientific computation Literaturverz. S. [364] - 371 ELEMENTOS FINITOS (ESTRUTURA) larpcal Mathematik Differential equations Numerical solutions Finite element method Fluid dynamics Mathematics Solids Mathematics Finite-Elemente-Methode (DE-588)4017233-8 gnd rswk-swf Mechanik (DE-588)4038168-7 gnd rswk-swf Mechanik (DE-588)4038168-7 s Finite-Elemente-Methode (DE-588)4017233-8 s 1\p DE-604 DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015160355&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 | Schwab, Christoph P- and hp-finite element methods theory and applications in solid and fluid mechanics ELEMENTOS FINITOS (ESTRUTURA) larpcal Mathematik Differential equations Numerical solutions Finite element method Fluid dynamics Mathematics Solids Mathematics Finite-Elemente-Methode (DE-588)4017233-8 gnd Mechanik (DE-588)4038168-7 gnd |
| subject_GND | (DE-588)4017233-8 (DE-588)4038168-7 |
| title | P- and hp-finite element methods theory and applications in solid and fluid mechanics |
| title_auth | P- and hp-finite element methods theory and applications in solid and fluid mechanics |
| title_exact_search | P- and hp-finite element methods theory and applications in solid and fluid mechanics |
| title_exact_search_txtP | P- and hp-finite element methods theory and applications in solid and fluid mechanics |
| title_full | P- and hp-finite element methods theory and applications in solid and fluid mechanics |
| title_fullStr | P- and hp-finite element methods theory and applications in solid and fluid mechanics |
| title_full_unstemmed | P- and hp-finite element methods theory and applications in solid and fluid mechanics |
| title_short | P- and hp-finite element methods |
| title_sort | p and hp finite element methods theory and applications in solid and fluid mechanics |
| title_sub | theory and applications in solid and fluid mechanics |
| topic | ELEMENTOS FINITOS (ESTRUTURA) larpcal Mathematik Differential equations Numerical solutions Finite element method Fluid dynamics Mathematics Solids Mathematics Finite-Elemente-Methode (DE-588)4017233-8 gnd Mechanik (DE-588)4038168-7 gnd |
| topic_facet | ELEMENTOS FINITOS (ESTRUTURA) Mathematik Differential equations Numerical solutions Finite element method Fluid dynamics Mathematics Solids Mathematics Finite-Elemente-Methode Mechanik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015160355&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT schwabchristoph pandhpfiniteelementmethodstheoryandapplicationsinsolidandfluidmechanics |