Introduction .................................................... 1
1 Layer Potential Techniques ................................... 4
1.1 Sobolev Spaces .......................................... 4
1.2 Elasticity Equations .................................... 6
1.3 Radiation Condition .................................... 10
1.4 Integral Representation of Solutions to the Lamй
System ................................................. 11
1.5 Helmholtz-Kirchhoff Identities ......................... 21
1.6 Eigenvalue Characterizations and Neumann and
Dirichlet Functions .................................... 27
1.7 A Regularity Result .................................... 32
2 Elasticity Equations with High Contrast Parameters .......... 33
2.1 Problem Setting ........................................ 34
2.2 Incompressible Limit ................................... 34
2.3 Limiting Cases of Holes and Hard Inclusions ............ 36
2.4 Energy Estimates ....................................... 38
2.5 Convergence of Potentials and Solutions ................ 42
2.6 Boundary Value Problems ................................ 45
3 Small-Volume Expansions of the Displacement Fields .......... 48
3.1 Elastic Moment Tensor .................................. 48
3.2 Small-Volume Expansions ................................ 55
4 Boundary Perturbations due to the Presence of Small Cracks .. 66
4.1 A Representation Formula ............................... 66
4.2 Derivation of an Explicit Integral Equation ............ 69
4.3 Asymptotic Expansion ................................... 71
4.4 Topological Derivative of the Potential Energy ......... 75
4.5 Derivation of the Representation Formula ............... 76
4.6 Time-Harmonic Regime ................................... 79
5 Backpropagation and Multiple Signal Classification Imaging
of Small Inclusions ......................................... 80
5.1 A Newton-Type Search Method ............................ 80
5.2 A MUSIC-Type Method in the Static Regime ............... 82
5.3 A MUSIC-Type Method in the Time-Harmonic Regime ........ 82
5.4 Reverse-Time Migration and Kirchhoff Imaging in the
Time-Harmonic Regime ................................... 84
5.5 Numerical Illustrations ................................ 86
6 Topological Derivative Based Imaging of Small Inclusions
in the Time-Harmonic Regime ................................. 91
6.1 Topological Derivative Based Imaging ................... 91
6.2 Modified Imaging Framework ............................ 102
7 Stability of Topological Derivative Based Imaging
Functionals ................................................ 112
7.1 Statistical Stability with Measurement Noise .......... 112
7.2 Statistical Stability with Medium Noise ............... 118
8 Time-Reversal Imaging of Extended Source Terms ............. 125
8.1 Analysis of the Time-Reversal Imaging Functionals ..... 127
8.2 Time-Reversal Algorithm for Viscoelastic Media ........ 129
8.3 Numerical Illustrations ............................... 137
9 Optimal Control Imaging of Extended Inclusions ............. 148
9.1 Imaging of Shape Perturbations ........................ 149
9.2 Imaging of an Extended Inclusion ...................... 152
10 Imaging from Internal Data ................................. 160
10.1 Inclusion Model Problem ............................... 160
10.2 Binary Level Set Algorithm ............................ 162
10.3 Imaging Shear Modulus Distributions ................... 164
10.4 Numerical Illustrations ............................... 165
11 Vibration Testing .......................................... 168
11.1 Small-Volume Expansions of the Perturbations in the
Eigenvalues ........................................... 169
11.2 Eigenvalue Perturbations due to Shape Deformations .... 181
11.3 Splitting of Multiple Eigenvalues ..................... 192
11.4 Reconstruction of Inclusions .......................... 193
11.5 Numerical Illustrations ............................... 195
A Introduction to Random Processes ........................... 201
A.l Random Variables ...................................... 201
A.2 Random Vectors ........................................ 202
A.3 Gaussian Random Vectors ............................... 203
A.4 Conditioning .......................................... 204
A.5 Random Processes ...................................... 205
A.6 Gaussian Processes .................................... 206
A.7 Stationary Gaussian Random Processes .................. 208
A.8 Multi-valued Gaussian Processes ....................... 208
В Asymptotics of the Attenuation Operator .................... 210
B.l Stationary Phase Theorem .............................. 210
B.2 Derivation of the Asymptotics ......................... 211
С The Generalized Argument Principle and Rouchй's Theorem .... 213
C.l Notation and Definitions .............................. 213
C.2 Generalized Argument Principle ........................ 214
References .................................................... 217
Index ......................................................... 229
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