Higher-order finite element methods:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton, FL [u.a.]
Chapman & Hall/CRC
2004
|
| Series: | Studies in advanced mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XXI, 382 S. Ill., graph. Darst. 1 CD-ROM (12 cm) |
| ISBN: | 158488438X |
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| 246 | 1 | 3 | |a Higher order finite element methods |
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| adam_text | Titel: Higher order finite element methods
Autor: Šolín, Pavel
Jahr: 2004
Higher-Order
Finite Element
Methods
PAVEL SOLIN, Rice University, Houston, Texas
KAREL SEGETH, Academy ofSciences ofthe Czech Republic, Prague
V
IVO DOLEZEL, Czech Technical University, Prague
·nCHAPMAN HALL/CRC
A CRC Press Company
Boca Raton London New York Washington, D.C.
Contents
1 Introduction 1
1.1 Finite elements 1
1.1.1 Function Spaces H1
, H(curl) and iJ(div) 1
1.1.2 Unisolvency of finite elements 2
1.1.3 Finite element mesh 7
1.1.4 Finite element interpolants and conformity 8
1.1.5 Reference domains and reference maps 16
1.1.6 Finite element discretization 17
1.1.7 Method of lines for evolutionary problems 19
1.2 Orthogonal polynomials 20
1.2.1 The family of Jacobi polynomials 20
1.2.2 Legendre polynomials 22
1.2.3 Lobatto shape functions 25
1.2.4 Kernel functions 27
1.2.5 Horner s algorithm for higher-order polynomials . . . 27
1.3 A one-dimensional example 28
1.3.1 Continuous and discrete problem 28
1.3.2 Transformation to reference domain 31
1.3.3 Higher-order shape functions 32
1.3.4 Design of basis functions 37
1.3.5 Sparsity structure and Connectivity 39
1.3.6 Assembling algorithm 41
1.3.7 Compressed representation of sparse matrices 42
2 Hierarchie master elements of arbitrary order 43
2.1 De Rham diagram 45
2.2 H1
-conforming approximations 46
2.2.1 One-dimensional master element K 47
2.2.2 Quadrilateral master element K}q 48
2.2.3 Triangulär master element K 55
2.2.4 Brick master element K}B 62
2.2.5 Tetrahedral master element K 68
2.2.6 Prismatic master element K}p 73
2.3 iJ(curl)-conforming approximations 78
2.3.1 De Rham diagram and finite elements in ff(curl) . . . 79
2.3.2 Quadrilateral master element Kc
TMx
80
xi
Xll
2.3.3 Triangulär master element /QurI
83
2.3.4 Brick master element /Cglrl
88
2.3.5 Tetrahedral master element /C^url
91
2.3.6 Prismatic master element JCc
prl
97
2.4 i/(div)-conforming approximations 105
2.4.1 De Rham diagram and finite elements in H(div) . . . 105
2.4.2 Quadrilateral master element /C^lv
106
2.4.3 Triangulär master element Kfv
108
2.4.4 Brick master element /C^iv
111
2.4.5 Tetrahedral master element JCT
[v
113
2.4.6 Prismatic master element ICplv
117
2.5 L2
-conforming approximations 121
2.5.1 De Rham diagram and finite elements in L2
121
2.5.2 Master elements for L2
-conforming approximations . . 121
3 Higher-order finite element discretization 125
3.1 Projection-based interpolation on reference domains 125
3.1.1 i/1
-conforming elements 126
3.1.2 H(curl)-conforming elements 136
3.1.3 i/(div)-conforming elements 141
3.2 Transfinite interpolation revisited 144
3.2.1 Projectors 145
3.2.2 Bipolynomial Lagrange interpolation 147
3.2.3 Transfinite bivariate Lagrange interpolation 147
3.3 Construction of reference maps 148
3.3.1 Mapping (curved) quad elements onto Kq 148
3.3.2 Mapping (curved) triangulär elements onto Kt . . . . 152
3.3.3 Mapping (curved) brick elements onto Kp 153
3.3.4 Mapping (curved) tetrahedral elements onto KT · · · 155
3.3.5 Mapping (curved) prismatic elements onto Kp . . . . 157
3.3.6 Isoparametric approximation of reference maps . . . . 158
3.3.7 Simplest case - lowest-order reference maps 159
3.3.8 Inversion of reference maps 160
3.4 Projection-based interpolation on physical mesh elements . . 161
3.5 Technology of discretization in two and three dimensions . . 163
3.5.1 Outline of the procedure 163
3.5.2 Orientation of master element edge and face functions 164
3.5.3 Transformation of master element polynomial Spaces . 172
3.5.4 Design of global basis functions 177
3.5.5 Minimum rules for higher-order FE discretizations . . 183
3.5.6 Enumeration of functions and Connectivity arrays . . . 184
3.5.7 Variational formulation on the reference domain . . . 185
3.5.8 Local and global assembling procedures 187
3.5.9 Static condensation of internal DOF 191
3.6 Constrained approximation 194
xiii
3.6.1 Continuous constrained approximation in 2D 194
3.6.2 Vector-valued constrained approximation in 2D . . . . 200
3.6.3 Continuous constrained approximation in 3D 203
3.6.4 Vector-valued constrained approximation in 3D . . . . 212
3.7 Selected software-technical aspects 213
3.7.1 Data structure for /ip-adaptivity 213
3.7.2 One-irregular mesh division algorithms 215
4 Higher-order numerical quadrature 217
4.1 One-dimensional reference domain Ka 218
4.1.1 Newton-Cotes quadrature 219
4.1.2 Chebyshev quadrature 222
4.1.3 Lobatto (Radau) quadrature 224
4.1.4 Gauss quadrature 227
4.2 Reference quadrilateral Kq 231
4.2.1 Composite Gauss quadrature 231
4.2.2 Economical Gauss quadrature 231
4.2.3 Tables of Gauss quadrature points and weights . . . . 232
4.3 Reference triangle Kt 234
4.3.1 Translation of quadrature to the ref. quadrilateral Kq 234
4.3.2 Newton-Cotes quadrature 235
4.3.3 Gauss quadrature 236
4.3.4 Tables of Gauss Integration points and weights . . . . 237
4.4 Reference brick KB 240
4.4.1 Composite Gauss quadrature 240
4.4.2 Economical Gauss quadrature 240
4.4.3 Tables of Gauss Integration points and weights . . . . 241
4.5 Reference tetrahedron KT 243
4.5.1 Translation of quadrature to the reference brick Kß · 243
4.5.2 Economical Gauss quadrature 244
4.5.3 Tables of Gauss Integration points and weights . . . . 246
4.6 Reference prism Kp 250
4.6.1 Composite Gauss quadrature 250
5 Numerical Solution of flnite element equations 251
5.1 Direct methods for linear algebraic equations 252
5.1.1 Gaussian elimination and matrix factorization 252
5.1.2 Banded Systems 256
5.1.3 General sparse Systems 257
5.1.4 Fast methods for special Systems 260
5.2 Iterative methods for linear algebraic equations 265
5.2.1 ORTHOMIN and steepest descent methods 265
5.2.2 Conjugate gradient and biconjugate gradient methods 269
5.2.3 MINRES and GMRES methods 272
5.2.4 Classical iterative methods and preconditioning . . . . 274
XIV
5.2.5 Block iterative methods 280
5.2.6 Multigrid methods 282
5.3 Choice of the method 288
5.4 Solving initial value problems for ordinary differential equations 290
5.4.1 Method of lines 291
5.4.2 Multistep methods 293
5.4.3 One-step methods 294
6 M e s h optimization, reference Solutions and /ip-adaptivity 297
6.1 Automatic mesh optimization in one dimension 298
6.1.1 Minimization of projection-based interpolation error . 299
6.1.2 Automatic mesh optimization algorithms 302
6.1.3 Automatic /i-adaptive mesh optimization 304
6.1.4 Automatic p-adaptive mesh optimization 310
6.1.5 Automatic /ip-adaptive mesh optimization 311
6.2 Adaptive strategies based on automatic mesh optimization . 314
6.2.1 Reference Solutions 315
6.2.2 A strategy based on automatic mesh optimization . . 316
6.2.3 Model problem 317
6.2.4 Automatic /i-adaptivity 318
6.2.5 Automatic p-adaptivity 320
6.2.6 Automatic /ip-adaptivity 321
6.3 Goal-oriented adaptivity 324
6.3.1 Quantities of interest 324
6.3.2 Formulation of the dual problem 325
6.3.3 Error control in quantity of interest 326
6.3.4 Selected nonlinear and unbounded functionals 327
6.4 Automatic goal-oriented h-, p- and /ip-adaptivity 329
6.4.1 Automatic goal-oriented adaptive strategies 330
6.4.2 Example: average of Solution over a subdomain . . . . 331
6.4.3 Goal-oriented and energy-driven /i-adaptivity 332
6.4.4 Goal-oriented and energy-driven /ip-adaptivity . . . . 335
6.5 Automatic goal-oriented /ip-adaptivity in two dimensions . . 337
6.5.1 Mesh optimization step in two dimensions 338
6.5.2 Example: Singular Solution in the L-shape domain . . 341
6.5.3 Goal-oriented and energy-driven /i-adaptivity 343
6.5.4 Goal-oriented and energy-driven /ip-adaptivity . . . . 348
6.5.5 Comparison of convergence in the quantity of interest 353
References 359
Author index 375
Subject index 379
|
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| spelling | Šolín, Pavel Verfasser aut Higher-order finite element methods Pavel Šolín, Karel Segeth, Ivo Doležel Higher order finite element methods Boca Raton, FL [u.a.] Chapman & Hall/CRC 2004 XXI, 382 S. Ill., graph. Darst. 1 CD-ROM (12 cm) txt rdacontent n rdamedia nc rdacarrier Studies in advanced mathematics Includes bibliographical references and index Método dos elementos finitos larpcal Éléments finis, Méthode des Finite element method Finite-Elemente-Methode (DE-588)4017233-8 gnd rswk-swf Finite-Elemente-Methode (DE-588)4017233-8 s DE-604 Segeth, Karel Verfasser aut Doležel, Ivo Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010321730&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Šolín, Pavel Segeth, Karel Doležel, Ivo Higher-order finite element methods Método dos elementos finitos larpcal Éléments finis, Méthode des Finite element method Finite-Elemente-Methode (DE-588)4017233-8 gnd |
| subject_GND | (DE-588)4017233-8 |
| title | Higher-order finite element methods |
| title_alt | Higher order finite element methods |
| title_auth | Higher-order finite element methods |
| title_exact_search | Higher-order finite element methods |
| title_full | Higher-order finite element methods Pavel Šolín, Karel Segeth, Ivo Doležel |
| title_fullStr | Higher-order finite element methods Pavel Šolín, Karel Segeth, Ivo Doležel |
| title_full_unstemmed | Higher-order finite element methods Pavel Šolín, Karel Segeth, Ivo Doležel |
| title_short | Higher-order finite element methods |
| title_sort | higher order finite element methods |
| topic | Método dos elementos finitos larpcal Éléments finis, Méthode des Finite element method Finite-Elemente-Methode (DE-588)4017233-8 gnd |
| topic_facet | Método dos elementos finitos Éléments finis, Méthode des Finite element method Finite-Elemente-Methode |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010321730&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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