Topological derivatives in shape optimization:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin ; Heidelberg
Springer
[2013]
|
| Series: | Interaction of Mechanics and Mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXI, 412 Seiten Illustrationen, Diagramme |
| ISBN: | 9783642352447 9783642352454 |
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MARC
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| 245 | 1 | 0 | |a Topological derivatives in shape optimization |c Antonio André Novotny ; Jan Sokołowski |
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| 264 | 4 | |c © 2013 | |
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Record in the Search Index
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| adam_text | IMAGE 1
1 INTRODUCTION 1
1.1 THE TOPOLOGICAL DERIVATIVE CONCEPT 3
1.2 RELATIONSHIP BETWEEN SHAPE AND TOPOLOGICAL DERIVATIVES 10
1.2.1 THE TOPOLOGICAL-SHAPE SENSITIVITY METHOD 12
1.2.2 AN EXAMPLE O F TOPOLOGICAL DERIVATIVE EVALUATION 14
1.3 MONOGRAPH ORGANIZATION 2 0
1.4 EXERCISES 23
2 DOMAIN DERIVATION IN CONTINUUM MECHANICS 25
2.1 MATERIAL AND SPATIAL DESCRIPTIONS 26
2.1.1 GRADIENT O F SCALAR FIELDS 28
2.1.2 GRADIENT O F VECTOR FIELDS 28
2.1.3 SPATIAL DESCRIPTION O F VELOCITY FIELDS 29
2.2 MATERIAL DERIVATIVES OF SPATIAL FIELDS 30
2.2.1 DERIVATIVE O F THE GRADIENT O F A SCALAR FIELD 31
2.2.2 DERIVATIVE O F THE GRADIENT O F A VECTOR FIELD 32
2.3 MATERIAL DERIVATIVES O F INTEGRAL EXPRESSIONS 33
2.3.1 DOMAIN INTEGRAL 33
2.3.2 BOUNDARY INTEGRAL 35
2.4 SUMMARY OF THE DERIVED FORMULAE 38
2.5 THE ESHELBY ENERGY-MOMENTUM TENSOR 4 0
2.6 EXERCISES 43
3 MATERIAL AND SHAPE DERIVATIVES FOR BOUNDARY VALUE PROBLEMS 47 3.1
PRELIMINARIES 48
3.1.1 SOBOLEV-SLOBODETSKII SPACES 49
3.1.2 ELLIPTIC REGULARITY 50
3.1.3 ELLIPTIC PROBLEMS IN NONSMOOTH DOMAINS 51
3.1.4 SHAPE DERIVATIVES 52
HTTP://D-NB.INFO/1027239242
IMAGE 2
3.2 MATERIAL DERIVATIVES FOR SECOND ORDER ELLIPTIC EQUATIONS 55
3.2.1 WEAK MATERIAL DERIVATIVES FOR THE DIRICHLET PROBLEM 55
3.2.2 STRONG MATERIAL DERIVATIVES FOR THE DIRICHLET PROBLEM 6 0 3.2.3
MATERIAL DERIVATIVES FOR THE NEUMANN PROBLEM 62
3.3 SHAPE DERIVATIVES FOR SECOND ORDER ELLIPTIC EQUATIONS 64
3.3.1 SHAPE DERIVATIVES FOR THE DIRICHLET PROBLEM 64
3.3.2 SHAPE DERIVATIVES FOR THE NEUMANN PROBLEM 66
3.4 MATERIAL AND SHAPE DERIVATIVES FOR ELASTICITY PROBLEMS 68
3.4.1 PROBLEM FORMULATION 68
3.4.2 MATERIAL DERIVATIVES FOR ELASTICITY 71
3.4.3 SHAPE DERIVATIVES FOR ELASTICITY 73
3.4.4 SHAPE DERIVATIVES FOR INTERFACES 76
3.5 MATERIAL AND SHAPE DERIVATIVES FOR KIRCHHOFF PLATES 77
3.5.1 PROBLEM FORMULATION 78
3.5.2 MATERIAL DERIVATIVES FOR THE KIRCHHOFF PLATE 79
3.5.3 SHAPE DERIVATIVES FOR THE KIRCHHOFF PLATE 80
3.6 MATERIAL AND SHAPE DERIVATIVES IN FLUID MECHANICS 82
3.6.1 THE ADJUGATE MATRIX CONCEPT 82
3.6.2 SHAPE DERIVATIVES FOR THE STATIONARY, HOMOGENEOUS NAVIER-STOKES
PROBLEM 87
3.6.3 MATERIAL DERIVATIVES FOR THE STATIONARY, HOMOGENEOUS NAVIER-STOKES
PROBLEM 87
3.7 EXERCISES 89
SINGULAR PERTURBATIONS O F ENERGY FUNCTIONALS 91
4.1 SECOND ORDER ELLIPTIC EQUATION: THE POISSON PROBLEM 92
4.1.1 PROBLEM FORMULATION 92
4.1.2 SHAPE SENSITIVITY ANALYSIS 94
4.1.3 ASYMPTOTIC ANALYSIS O F THE SOLUTION 97
4.1.4 TOPOLOGICAL DERIVATIVE EVALUATION 100
4.1.5 EXAMPLES WITH EXPLICIT FORM O F TOPOLOGICAL DERIVATIVES . . . 106
4.1.6 ADDITIONAL COMMENTS AND SUMMARY O F THE RESULTS 110
4.2 SECOND ORDER ELLIPTIC SYSTEM: THE NAVIER PROBLEM 112
4.2.1 PROBLEM FORMULATION 112
4.2.2 SHAPE SENSITIVITY ANALYSIS 114
4.2.3 ASYMPTOTIC ANALYSIS O F THE SOLUTION 117
4.2.4 TOPOLOGICAL DERIVATIVE EVALUATION 120
4.3 FOURTH ORDER ELLIPTIC EQUATION: THE KIRCHHOFF PROBLEM 122
4.3.1 PROBLEM FORMULATION 123
4.3.2 SHAPE SENSITIVITY ANALYSIS 125
4.3.3 ASYMPTOTIC ANALYSIS O F THE SOLUTION 129
4.3.4 TOPOLOGICAL DERIVATIVE EVALUATION 132
4.4 EXERCISES 136
IMAGE 3
CONTENTS XIII
5 CONFIGURATIONAL PERTURBATIONS O F ENERGY FUNCTIONALS 137
5.1 SECOND ORDER ELLIPTIC EQUATION: THE LAPLACE PROBLEM 138
5.1.1 PROBLEM FORMULATION 138
5.1.2 SHAPE SENSITIVITY ANALYSIS 140
5.1.3 ASYMPTOTIC ANALYSIS O F THE SOLUTION 143
5.1.4 TOPOLOGICAL DERIVATIVE EVALUATION 145
5.1.5 NUMERICAL EXAMPLE 147
5.2 SECOND ORDER ELLIPTIC SYSTEM: THE NAVIER PROBLEM 148
5.2.1 PROBLEM FORMULATION 148
5.2.2 SHAPE SENSITIVITY ANALYSIS 151
5.2.3 ASYMPTOTIC ANALYSIS O F THE SOLUTION 153
5.2.4 TOPOLOGICAL DERIVATIVE EVALUATION 157
5.2.5 NUMERICAL EXAMPLE 159
5.3 FOURTH ORDER ELLIPTIC EQUATION: THE KIRCHHOFF PROBLEM 160
5.3.1 PROBLEM FORMULATION 161
5.3.2 SHAPE SENSITIVITY ANALYSIS 164
5.3.3 ASYMPTOTIC ANALYSIS O F THE SOLUTION 167
5.3.4 TOPOLOGICAL DERIVATIVE EVALUATION 172
5.3.5 NUMERICAL EXAMPLE 173
5.4 ESTIMATES FOR THE REMAINDERS 175
5.5 EXERCISES 180
6 TOPOLOGICAL DERIVATIVE EVALUATION WITH ADJOINT STATES 181
6.1 PROBLEM FORMULATION 181
6.2 SHAPE SENSITIVITY ANALYSIS 184
6.3 ASYMPTOTIC ANALYSIS O F THE SOLUTION 189
6.3.1 ASYMPTOTIC EXPANSION O F THE DIRECT STATE 190
6.3.2 ASYMPTOTIC EXPANSION O F THE ADJOINT STATE 191
6.4 TOPOLOGICAL DERIVATIVE EVALUATION 192
6.5 EXERCISES 194
7 TOPOLOGICAL DERIVATIVE FOR STEADY-STATE ORTHOTROPIC HEAT DIFFUSION
PROBLEMS 195
7.1 PROBLEM FORMULATION 196
7.2 SHAPE SENSITIVITY ANALYSIS 198
7.3 ASYMPTOTIC ANALYSIS O F THE SOLUTION 199
7.4 TOPOLOGICAL DERIVATIVE EVALUATION 201
8 TOPOLOGICAL DERIVATIVE FOR THREE-DIMENSIONAL LINEAR ELASTICITY
PROBLEMS 203
8.1 PROBLEM FORMULATION 203
8.2 SHAPE SENSITIVITY ANALYSIS 206
8.3 ASYMPTOTIC ANALYSIS O F THE SOLUTION 207
IMAGE 4
XIV
CONTENTS
8.4 TOPOLOGICAL DERIVATIVE EVALUATION 211
8.5 NUMERICAL EXAMPLE 213
8.6 MULTISCALE TOPOLOGICAL DERIVATIVES 217
8.6.1 MULTISCALE MODELING IN SOLID MECHANICS 217
8.6.2 THE HOMOGENIZED ELASTICITY TENSOR 219
8.6.3 SENSITIVITY OF THE MACROSCOPIC ELASTICITY TENSOR TO TOPOLOGICAL
MICROSTRUCTURAL CHANGES 221
9 COMPOUND ASYMPTOTIC EXPANSIONS FOR SPECTRAL PROBLEMS 225
9.1 PRELIMINARIES AND EXAMPLES 226
9.2 DIRICHLET LAPLACIAN IN DOMAINS WITH SMALL CAVITIES 228
9.2.1 FIRST ORDER ASYMPTOTIC EXPANSION 229
9.2.2 SECOND ORDER ASYMPTOTIC EXPANSION 232
9.2.3 COMPLETE ASYMPTOTIC EXPANSION 236
9.3 NEUMANN LAPLACIAN IN DOMAINS WITH SMALL CAVERNS 239
9.3.1 FIRST BOUNDARY LAYER CORRECTOR 242
9.3.2 SECOND BOUNDARY LAYER CORRECTOR 246
9.3.3 CORRECTION TERM O F REGULAR TYPE 250
9.3.4 MULTIPLE EIGENVALUES 255
9.4 CONFIGURATIONAL PERTURBATIONS O F SPECTRAL PROBLEMS IN ELASTICITY .
. . 256 9.4.1 ANISOTROPIC AND INHOMOGENEOUS ELASTIC BODY 257
9.4.2 VIBRATIONS O F ELASTIC BODIES 258
9.4.3 FORMAL CONSTRUCTION O F ASYMPTOTIC EXPANSIONS 261
9.4.4 POLARIZATION MATRICES 273
10 TOPOLOGICAL ASYMPTOTIC ANALYSIS FOR SEMILINEAR ELLIPTIC BOUNDARY
VALUE PROBLEMS 277
10.1 TOPOLOGICAL DERIVATIVES IN R 2 279
10.1.1 FORMAL ASYMPTOTIC ANALYSIS 280
10.1.2 FORMAL ASYMPTOTICS O F SHAPE FUNCTIONAL 284
10.2 TOPOLOGICAL DERIVATIVES IN K 3 287
10.2.1 ASYMPTOTIC APPROXIMATION O F SOLUTIONS 287
10.2.2 ASYMPTOTICS O F SHAPE FUNCTIONAL 292
10.3 EXERCISES 295
11 TOPOLOGICAL DERIVATIVES FOR UNILATERAL PROBLEMS 299
11.1 PRELIMINARIES 300
11.2 DOMAIN DECOMPOSITION AND THE STEKLOV-POINCARE OPERATOR 302 11.2.1
DOMAIN DECOMPOSITION TECHNIQUE 304
11.2.2 STEKLOV-POINCARE PSEUDODIFFERENTIAL BOUNDARY OPERATORS . . . 305
11.3 DOMAIN DECOMPOSITION METHOD FOR VARIATIONAL INEQUALITIES 308 11.3.1
PROBLEM FORMULATION 308
11.3.2 HADAMARD DIFFERENTIABILITY O F MINIMIZER 310
IMAGE 5
CONTENTS X V
11.3.3 TOPOLOGICAL DERIVATIVE EVALUATION 311
11.4 CRACKS ON BOUNDARIES OF RIGID INCLUSIONS 314
11.4.1 PROBLEM FORMULATION 315
11.4.2 APPROXIMATION O F A RIGID INCLUSION WITH CONTRAST PARAMETER 317
11.4.3 HADAMARD DIFFERENTIABILITY OF SOLUTIONS TO VARIATIONAL
INEQUALITIES 320
11.4.4 TOPOLOGICAL DERIVATIVE EVALUATION 321
11.5 EXERCISES 324
A AUXILIARY RESULTS FOR SPECTRAL PROBLEMS 325
A.L PRELIMINARIES RESULTS 325
A.2 LEMMA ON ALMOST EIGENVALUES AND EIGENVECTORS 329
B SPECTRAL PROBLEM FOR THE NEUMANN LAPLACIAN 331
B. 1 THE WEIGHTED POINCARE INEQUALITY 331
B.2 ASYMPTOTICS FOR SPECTRAL PROBLEM 335
C SPECTRAL PROBLEMS IN ELASTICITY 345
C.L JUSTIFICATION O F ASYMPTOTIC EXPANSIONS 345
C.2 PROOF OF THEOREM 9.2 352
D POLARIZATION TENSOR IN ELASTICITY 355
D . L ELASTICITY BOUNDARY VALUE PROBLEMS 355
D. 1.1 VOIGT NOTATION IN ELASTICITY 355
D.L.2 KORN INEQUALITY 356
D.2 POLARIZATION MATRICES IN ELASTICITY 357
D.3 THE POLARIZATION MATRICES FOR THREE-DIMENSIONAL ANISOTROPIC
ELASTICITY PROBLEMS 360
D.3.1 SOLVABILITY O F THE TRANSMISSION PROBLEMS 360
D.3.2 ASYMPTOTIC BEHAVIOR O F THE SOLUTIONS 364
D.3.3 THE POLARIZATION MATRIX AND ITS PROPERTIES 366
D.3.4 HOMOGENEOUS INCLUSION 368
E COMPOUND ASYMPTOTIC EXPANSIONS FOR SEMILINEAR PROBLEMS 371 E.L
LINEARIZED PROBLEM IN WEIGHTED HOLDER SPACES 372
E.2 ESTIMATES FOR THE REMAINDERS 374
F SENSITIVITY ANALYSIS FOR VARIATIONAL INEQUALITIES 381
F.L POLYHEDRAL CONVEX SETS IN SOBOLEV SPACES O F THE DIRICHLET TYPE . .
. 382 F.2 COMPACTNESS O F THE ASYMPTOTIC ENERGY EXPANSION 383
IMAGE 6
XVI CONTENTS
G TENSOR CALCULUS 389
G. 1 INNER, VECTOR AND TENSOR PRODUCTS 389
G.2 GRADIENT, DIVERGENCE AND CURL 390
G.3 INTEGRAL THEOREMS 391
G.4 SOME USEFUL DECOMPOSITIONS 392
G.5 POLAR AND SPHERICAL COORDINATE SYSTEMS 394
REFERENCES 397
INDEX 409
|
| any_adam_object | 1 |
| author | Novotný, Antonín 1959-2021 Sokołowski, Jan 1949- |
| author_GND | (DE-588)143304194 (DE-588)172379571 |
| author_facet | Novotný, Antonín 1959-2021 Sokołowski, Jan 1949- |
| author_role | aut aut |
| author_sort | Novotný, Antonín 1959-2021 |
| author_variant | a n an j s js |
| building | Verbundindex |
| bvnumber | BV040671095 |
| classification_rvk | SK 950 |
| ctrlnum | (OCoLC)826594214 (DE-599)DNB1027239242 |
| dewey-full | 515.642 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.642 |
| dewey-search | 515.642 |
| dewey-sort | 3515.642 |
| dewey-tens | 510 - Mathematics |
| discipline | Maschinenbau / Maschinenwesen Mathematik |
| format | Book |
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| id | DE-604.BV040671095 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:28:46Z |
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| isbn | 9783642352447 9783642352454 |
| language | English |
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| owner | DE-384 DE-19 DE-BY-UBM DE-11 DE-20 DE-83 |
| owner_facet | DE-384 DE-19 DE-BY-UBM DE-11 DE-20 DE-83 |
| physical | XXI, 412 Seiten Illustrationen, Diagramme |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | Springer |
| record_format | marc |
| series2 | Interaction of Mechanics and Mathematics |
| spelling | Novotný, Antonín 1959-2021 (DE-588)143304194 aut Topological derivatives in shape optimization Antonio André Novotny ; Jan Sokołowski Berlin ; Heidelberg Springer [2013] © 2013 XXI, 412 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Interaction of Mechanics and Mathematics Elliptisches Randwertproblem (DE-588)4193399-0 gnd rswk-swf Asymptotische Entwicklung (DE-588)4112609-9 gnd rswk-swf Topologieoptimierung (DE-588)7662388-9 gnd rswk-swf Singuläre Störung (DE-588)4055100-3 gnd rswk-swf Gestaltoptimierung (DE-588)4329076-0 gnd rswk-swf Gestaltoptimierung (DE-588)4329076-0 s Topologieoptimierung (DE-588)7662388-9 s Elliptisches Randwertproblem (DE-588)4193399-0 s Singuläre Störung (DE-588)4055100-3 s Asymptotische Entwicklung (DE-588)4112609-9 s DE-604 Sokołowski, Jan 1949- (DE-588)172379571 aut Erscheint auch als Online-Ausgabe 978-3-642-35245-4 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025497659&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Novotný, Antonín 1959-2021 Sokołowski, Jan 1949- Topological derivatives in shape optimization Elliptisches Randwertproblem (DE-588)4193399-0 gnd Asymptotische Entwicklung (DE-588)4112609-9 gnd Topologieoptimierung (DE-588)7662388-9 gnd Singuläre Störung (DE-588)4055100-3 gnd Gestaltoptimierung (DE-588)4329076-0 gnd |
| subject_GND | (DE-588)4193399-0 (DE-588)4112609-9 (DE-588)7662388-9 (DE-588)4055100-3 (DE-588)4329076-0 |
| title | Topological derivatives in shape optimization |
| title_auth | Topological derivatives in shape optimization |
| title_exact_search | Topological derivatives in shape optimization |
| title_full | Topological derivatives in shape optimization Antonio André Novotny ; Jan Sokołowski |
| title_fullStr | Topological derivatives in shape optimization Antonio André Novotny ; Jan Sokołowski |
| title_full_unstemmed | Topological derivatives in shape optimization Antonio André Novotny ; Jan Sokołowski |
| title_short | Topological derivatives in shape optimization |
| title_sort | topological derivatives in shape optimization |
| topic | Elliptisches Randwertproblem (DE-588)4193399-0 gnd Asymptotische Entwicklung (DE-588)4112609-9 gnd Topologieoptimierung (DE-588)7662388-9 gnd Singuläre Störung (DE-588)4055100-3 gnd Gestaltoptimierung (DE-588)4329076-0 gnd |
| topic_facet | Elliptisches Randwertproblem Asymptotische Entwicklung Topologieoptimierung Singuläre Störung Gestaltoptimierung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025497659&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT novotnyantonin topologicalderivativesinshapeoptimization AT sokołowskijan topologicalderivativesinshapeoptimization |