Asymptotic analysis of fields in multi-structures:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford
Clarendon Press
1999
|
| Edition: | 1. publ. |
| Series: | Oxford mathematical monographs
Oxford science publications |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 282 S. graph. Darst. |
| ISBN: | 0198514956 |
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MARC
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| 100 | 1 | |a Kozlov, Vladimir |d 1954- |e Verfasser |0 (DE-588)103679014 |4 aut | |
| 245 | 1 | 0 | |a Asymptotic analysis of fields in multi-structures |c Vladimir Kozlov ; Vladimir Maz'ya ; Alexander Movchan |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Oxford |b Clarendon Press |c 1999 | |
| 300 | |a XV, 282 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Oxford mathematical monographs | |
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| 700 | 1 | |a Mazʹja, Vladimir Gilelevič |d 1937- |e Verfasser |0 (DE-588)121490602 |4 aut | |
| 700 | 1 | |a Movchan, Alexander B. |e Verfasser |4 aut | |
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Record in the Search Index
| _version_ | 1804127469099286528 |
|---|---|
| adam_text | CONTENTS
List of symbols xiv
1 Introduction to compound asymptotic expansions 1
1.1 Elementary examples of perturbation problems for ordi¬
nary differential equations 1
1.2 A one dimensional singularly perturbed problem 5
1.3 Neumann boundary value problem in a domain with small
cavity 10
1.3.1 Formulation of the problem 10
1.3.2 The leading order approximation 12
1.3.3 Remainder estimate 13
1.3.4 Complete asymptotic expansion 16
1.3.5 Asymptotic formula for the energy 20
1.4 Dirichlet boundary value problem in a domain with small
inclusion 22
1.4.1 The leading order approximation 22
1.4.2 The next approximation 24
1.4.3 The complete asymptotic expansion 25
1.5 Mixed boundary value problem for the Laplacian in a thin
rectangle 30
1.5.1 Formulation of the boundary value problem 30
1.5.2 Two term approximation 31
1.5.3 The next approximation 34
1.5.4 Higher order approximation 36
1.6 Problem of junction between thin bodies 38
1.6.1 Model problems 40
1.6.2 The leading order approximation 47
1.6.3 The next order approximation 49
1.6.4 The complete asymptotic expansion 51
1.6.5 The remainder estimate 53
2 A boundary value problem for the Laplacian in a multi
structure 56
2.1 Formulation of the problem 58
2.2 Model problems 59
2.2.1 Limit domains 59
2.2.2 Model problem in fi 60
2.2.3 Model problem for the junction region 63
2.2.4 Junction layer 70
x CONTENTS
2.2.5 Model problem for the bottom region 71
2.2.6 Two model problems for a thin cylinder 75
2.2.7 Algebraic system for the skeleton 76
2.3 Right hand sides 77
2.3.1 Local coordinates and limit domains 78
2.3.2 Cut off functions 78
2.3.3 Asymptotic representations of the right hand
sides 79
2.4 The leading term of the asymptotic solution 81
2.4.1 Domain ft 81
2.4.2 Junction layer 83
2.4.3 Thin cylinder 85
2.4.4 Bottom layer 87
2.4.5 Evaluation ot W0(i) 88
2.4.6 Evaluation of the constants T0(j) and Co 88
2.4.7 Concluding remarks on formal algorithm 89
2.5 Complete asymptotic expansion 89
2.5.1 Structure of the asymptotic expansion 89
2.5.2 The asymptotic algorithm 91
2.6 Justification of the asymptotic expansion 92
2.6.1 Auxiliary estimates for functions in H1 (H£) 92
2.6.2 Estimate for solutions 94
2.6.3 Estimate for the remainder term 97
2.7 A constant right hand side 98
2.8 Application to the asymptotics of the energy
integral 100
2.8.1 The case of the right hand sides concentra¬
ted in H 101
2.8.2 The case of the Dirichlet data at the bases
of thin cylinders 103
2.9 On a general 1D 3D multi structure 105
2.10 A multi structure with a thin walled tube 108
3 Auxiliary facts from mathematical elasticity 115
3.1 Basic formulae of linear elasticity 115
3.1.1 Stress and strain 115
3.1.2 Equations of equilibrium and boundary
conditions 116
3.2 Two dimensional problems of linear elasticity 118
3.2.1 Plane strain 118
3.2.2 Anti plane shear 119
3.3 Differential equations for engineering models of elastic
rods 120
CONTENTS xi
3.4 Classical solutions of linear elasticity for a half space 121
3.4.1 Boussinesq Cerruti s solution 121
3.4.2 Mindlin s solution 123
3.4.3 Connection between the Boussinesq Cerruti
and Mindlin solutions 124
3.5 Special solutions for a bounded two dimensional
domain 126
3.5.1 The torsion potential 127
3.5.2 The bending potentials 128
3.5.3 Example 128
3.6 Special solutions of linear elasticity for an infinite
cylinder 129
3.6.1 Representation of differential operators 129
3.6.2 The spectral problem 130
3.6.3 .^ polynomial solutions 132
3.6.4 Biorthogonality conditions 133
3.6.5 The normalised stiffness coefficients 135
3.6.6 Biorthogonality relations for eigenvectors and
generalised eigenvectors 136
3.6.7 There are no other polynomials 138
3.7 Green s matrix in Q 141
3.7.1 Definition 141
3.7.2 Asymptotics 141
3.8 Korn s inequalities 143
3.8.1 The case of bounded Lipschitz domains 143
3.8.2 Half space and a cylinder 146
3.9 Asymptotics at infinity for solutions to the traction
problem for a half cylinder 151
4 Elastic multi structure 155
4.1 Multi structure and boundary value problem 156
4.2 Model problems 158
4.2.1 Limit domains 158
4.2.2 Model problem for the body Q 158
4.2.3 Junction layer 160
4.2.4 Model problem for the bottom layer 167
4.2.5 Model problem for a bounded two dimensional
domain 171
4.2.6 Model problems on the axis of an elastic rod 172
4.2.7 Model matrices and the pile structure 174
4.2.8 Special cases 179
4.3 Asymptotic expansion of the solution 180
xii CONTENTS
4.3.1 Asymptotic representation of the right hand
sides 180
4.3.2 Description of the asymptotic series for the
solution 182
4.3.3 Auxiliary solutions of the Lame system in a
thin elastic rod 184
4.3.4 Expansions for displacement in a thin rod 187
4.3.5 Junction layer 189
4.3.6 Displacement in fi 190
4.3.7 Bottom layer 191
4.3.8 Functions v{™ j) 193
4.3 9 The recurrent procedure for the asymptotic
expansion 197
4.4 Justification of the asymptotic expansion 198
4.4.1 Korn s inequality in ile 198
4.4.2 An estimate for the solution 199
4.5 The leading order approximation 203
4.5.1 The term u^1) 203
4.5.2 The term u 0 204
4.6 Physical interpretation of the results 208
4.6.1 The case M31 + |FX | + |F21 ^ 0 208
4.6.2 The case Fi = F2 = M3 = 0 210
5 Non degenerate elastic multi structures 213
5.1 Pile structure model 214
5.1.1 Skeleton of the multi structure 214
5.1.2 The pile structure 216
5.1.3 Mathematical model of the pile structure 216
5.1.4 Solution of the pile structure equations 217
5.1.5 Algebraic system for the pile structure model 219
5.1.6 Non degenerate and degenerate pile structures 221
5.1.7 Examples 222
5.2 Multi structure and the boundary value problem 224
5.2.1 Description of the multi structure 224
5.2.2 Formulation of the boundary value problem 226
5.3 Model problems 227
5.3.1 Junction layer 227
5.3.2 Remaining model problems 230
5.4 Asymptotic expansion of the solution 231
5.4.1 Cut off functions 231
5.4.2 Asymptotic representation of the right hand sides
for the case of a non degenerate multi structure 233
CONTENTS xiii
5.4.3 Structure of the asymptotic series for the
displacement field in fle 233
5.4.4 The junction layer 235
5.4.5 Displacement in fl 237
5.4.6 The bottom layer 238
5.4.7 Functions v mJ) 239
5.4.8 Evaluation of the lock forces and moments at
junction points 241
5.4.9 Algebraic system for a(m), /5(m 242
5.4.10 The recurrent procedure for the asymptotic
expansion 243
5.5 Estimate for the remainder of the asymptotic expansion 244
5.6 Analysis of the leading term 245
5.7 Physical interpretation 246
6 Spectral analysis for 3D 1D multi structures 248
6.1 An abstract scheme for the asymptotics of eigenvalues 249
6.2 Spectral problem for the Laplacian 253
6.2.1 The first eigenvalue 253
6.2.2 The first eigenfunction 255
6.3 Asymptotics of first eigenvalues of the Lame operator 257
6.3.1 Spectral problem 258
6.3.2 Korn type inequalities 258
6.3.3 Spaces Xo and Ho 263
6.3.4 Asymptotic formula for the eigenvalues 263
6.4 Spectral problem for an inhomogeneous elastic multi
structure 267
6.4.1 The spectral problem 267
6.4.2 Asymptotic formulae for the eigenvalues 268
Bibliographical remarks 274
Bibliography 276
Index 281
|
| any_adam_object | 1 |
| author | Kozlov, Vladimir 1954- Mazʹja, Vladimir Gilelevič 1937- Movchan, Alexander B. |
| author_GND | (DE-588)103679014 (DE-588)121490602 |
| author_facet | Kozlov, Vladimir 1954- Mazʹja, Vladimir Gilelevič 1937- Movchan, Alexander B. |
| author_role | aut aut aut |
| author_sort | Kozlov, Vladimir 1954- |
| author_variant | v k vk v g m vg vgm a b m ab abm |
| building | Verbundindex |
| bvnumber | BV012798634 |
| classification_rvk | SK 540 |
| ctrlnum | (OCoLC)472302823 (DE-599)BVBBV012798634 |
| discipline | Mathematik |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV012798634 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:33:50Z |
| institution | BVB |
| isbn | 0198514956 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008703754 |
| oclc_num | 472302823 |
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| owner | DE-703 DE-11 |
| owner_facet | DE-703 DE-11 |
| physical | XV, 282 S. graph. Darst. |
| publishDate | 1999 |
| publishDateSearch | 1999 |
| publishDateSort | 1999 |
| publisher | Clarendon Press |
| record_format | marc |
| series2 | Oxford mathematical monographs Oxford science publications |
| spelling | Kozlov, Vladimir 1954- Verfasser (DE-588)103679014 aut Asymptotic analysis of fields in multi-structures Vladimir Kozlov ; Vladimir Maz'ya ; Alexander Movchan 1. publ. Oxford Clarendon Press 1999 XV, 282 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Oxford mathematical monographs Oxford science publications Asymptotische Entwicklung (DE-588)4112609-9 gnd rswk-swf Randwertproblem (DE-588)4048395-2 gnd rswk-swf Randwertproblem (DE-588)4048395-2 s Asymptotische Entwicklung (DE-588)4112609-9 s DE-604 Mazʹja, Vladimir Gilelevič 1937- Verfasser (DE-588)121490602 aut Movchan, Alexander B. Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008703754&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kozlov, Vladimir 1954- Mazʹja, Vladimir Gilelevič 1937- Movchan, Alexander B. Asymptotic analysis of fields in multi-structures Asymptotische Entwicklung (DE-588)4112609-9 gnd Randwertproblem (DE-588)4048395-2 gnd |
| subject_GND | (DE-588)4112609-9 (DE-588)4048395-2 |
| title | Asymptotic analysis of fields in multi-structures |
| title_auth | Asymptotic analysis of fields in multi-structures |
| title_exact_search | Asymptotic analysis of fields in multi-structures |
| title_full | Asymptotic analysis of fields in multi-structures Vladimir Kozlov ; Vladimir Maz'ya ; Alexander Movchan |
| title_fullStr | Asymptotic analysis of fields in multi-structures Vladimir Kozlov ; Vladimir Maz'ya ; Alexander Movchan |
| title_full_unstemmed | Asymptotic analysis of fields in multi-structures Vladimir Kozlov ; Vladimir Maz'ya ; Alexander Movchan |
| title_short | Asymptotic analysis of fields in multi-structures |
| title_sort | asymptotic analysis of fields in multi structures |
| topic | Asymptotische Entwicklung (DE-588)4112609-9 gnd Randwertproblem (DE-588)4048395-2 gnd |
| topic_facet | Asymptotische Entwicklung Randwertproblem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008703754&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT kozlovvladimir asymptoticanalysisoffieldsinmultistructures AT mazʹjavladimirgilelevic asymptoticanalysisoffieldsinmultistructures AT movchanalexanderb asymptoticanalysisoffieldsinmultistructures |