Ammari / Kang Polarization and Moment Tensors

With Applications to Inverse Problems and Effective Medium Theory
1. Auflage 2007
ISBN: 978-0-387-71566-7
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 162, 314 Seiten

Reihe: Applied Mathematical Sciences

This book presents important recent developments in mathematical and computational methods used in impedance imaging and the theory of composite materials. By augmenting the theory with interesting practical examples and numerical illustrations, the exposition brings simplicity to the advanced material. An introductory chapter covers the necessary basics. An extensive bibliography and open problems at the end of each chapter enhance the text.

Autoren/Hrsg.

Ammari, Habib

Kang, Hyeonbae

Inhaltsverzeichnis

1;Preface;6
2;Contents;7
3;1 Introduction;11
4;2 Layer Potentials and Transmission Problems;17
4.1;Introduction;17
4.2;2.1 Notation and Preliminaries;17
4.3;2.2 Layer Potentials on Smooth Domains;24
4.4;2.3 Layer Potentials on Lipschitz Domains;34
4.5;2.4 Neumann and Dirichlet Functions;49
4.6;2.5 Representation Formula;55
4.7;2.6 Energy Identities;60
4.8;2.7 Anisotropic Transmission Problem;61
4.9;2.8 Periodic Isotropic Transmission Problem;62
4.10;2.9 Periodic Anisotropic Transmission Problem;69
4.11;2.10 Further Results and Open Problems;76
5;3 Uniqueness for Inverse Conductivity Problems;77
5.1;Introduction;77
5.2;3.1 Uniqueness With Many Measurements;78
5.3;3.2 Uniqueness With One Measurement;81
5.4;3.3 Further Results and Open Problems;84
6;4 Generalized Isotropic and Anisotropic Polarization Tensors;85
6.1;Introduction;85
6.2;4.1 Definition;86
6.3;4.2 Explicit Formulae;91
6.4;4.3 Extreme Conductivity Cases;98
6.5;4.4 Uniqueness Result;100
6.6;4.5 Symmetry and Positivity of GPTs;101
6.7;4.6 Estimates of the Harmonic Moments;104
6.8;4.7 Optimal Bounds for the Polarization Tensor;107
6.9;4.8 Monotonocity;113
6.10;4.9 Estimates of the Center of Mass;114
6.11;4.10 Polarization Tensors of Multiple Inclusions;116
6.12;4.11 Explicit Formulae for the Polarization Tensor of Multiple Disks;122
6.13;4.12 Anisotropic Polarization Tensors;129
6.14;4.13 Further Results and Open Problems;137
7;5 Full Asymptotic Formula for the Potentials;138
7.1;Introduction;138
7.2;5.1 Energy Estimates;140
7.3;5.2 Asymptotic Expansion;144
7.4;5.3 Derivation of the Asymptotic Formula for Closely Spaced Small Inclusions;149
7.5;5.4 Derivation of the Asymptotic Formula for Anisotropic Inclusions;151
7.6;5.5 Further Results and Open Problems;152
8;6 Near-Boundary Conductivity Inclusions;154
8.1;Introduction;154
8.2;6.1 Optimal Gradient Estimates;155
8.3;6.2 Asymptotic Expansions;159
8.4;6.3 Further Results and Open Problems;169
9;7 Impedance Imaging of Conductivity Inclusions;170
9.1;Introduction;170
9.2;7.1 Preliminary;171
9.3;7.2 Projection Algorithm — Reconstruction of a Single Inclusion;172
9.4;7.3 Quadratic Algorithm — Detection of Closely Spaced Inclusions;178
9.5;7.4 Simple Pole Method;181
9.6;7.5 Least-Squares Algorithm;182
9.7;7.6 Variational Algorithm;183
9.8;7.7 Linear Sampling Method;185
9.9;7.8 Lipschitz-Continuous Dependence and Moment Estimations;191
9.10;7.9 Detection of Anisotropic Inclusions;195
9.11;7.10 Further Results and Open Problems;201
10;8 Effective Properties of Electrical Composites;204
10.1;Introduction;204
10.2;8.1 Computation of Effective Conductivity;206
10.3;8.2 Anisotropic Composites;214
10.4;8.3 Further Results and Open Problems;219
11;9 Transmission Problem for Elastostatics;220
11.1;Introduction;220
11.2;9.1 Layer Potentials for the Lame System;220
11.3;9.2 Kelvin Matrix Under Unitary Transformations;224
11.4;9.3 Transmission Problem;227
11.5;9.4 Complex Representation of the Displacement Field;234
11.6;9.5 Periodic Green’s Function;239
11.7;9.6 Further Results and Open Problems;244
12;10 Elastic Moment Tensor;245
12.1;Introduction;245
12.2;10.1 Asymptotic Expansion in Free Space;245
12.3;10.2 Properties of EMTs;249
12.4;10.3 EMTs Under Linear Transformations;256
12.5;10.4 EMTs for Ellipses;259
12.6;10.5 EMTs for Elliptic Holes and Hard Ellipses;264
12.7;10.6 Further Results and Open Problems;267
13;11 Full Asymptotic Expansions of the Displacement Field;268
13.1;Introduction;268
13.2;11.1 Full Asymptotic Expansions;268
13.3;11.2 Further Results and Open Problems;274
14;12 Imaging of Elastic Inclusions;276
14.1;Introduction;276
14.2;12.1 Detection of EMTs;276
14.3;12.2 Representation of the EMTs by Ellipses;280
14.4;12.3 Detection of the Location;282
14.5;12.4 Numerical Results;284
14.6;12.5 Further Results and Open Problems;290
15;13 Effective Properties of Elastic Composites;292
15.1;Introduction;292
15.2;13.1 Derivation of the Effective Elastic Properties;293
15.3;13.2 Further Results and Open Problems;296
16;A Appendices;297
16.1;Introduction;297
16.2;A.1 Compact Operators;297
16.3;A.2 Theorem of Coifman, McIntosh, and Meyer;298
16.4;A.3 Continuity Method;299
17;References;301
18;Index;317