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Includes bibliographical references and index.

Cover -- Title -- Copyright -- Contents -- Introduction -- Chapter 1: Layer Potential Techniques -- 1.1 Sobolev Spaces -- 1.2 Elasticity Equations -- 1.3 Radiation Condition -- 1.4 Integral Representation of Solutions to the Lam�e System -- 1.5 Helmholtz-Kirchhoff Identities -- 1.6 Eigenvalue Characterizations and Neumann and Dirichlet Functions -- 1.7 A Regularity Result -- Chapter 2: Elasticity Equations with High Contrast Parameters -- 2.1 Problem Setting -- 2.2 Incompressible Limit -- 2.3 Limiting Cases of Holes and Hard Inclusions -- 2.4 Energy Estimates -- 2.5 Convergence of Potentials and Solutions -- 2.6 Boundary Value Problems -- Chapter 3: Small-Volume Expansions of the Displacement Fields -- 3.1 Elastic Moment Tensor -- 3.2 Small-Volume Expansions -- Chapter 4: Boundary Perturbations due to the Presence of Small Cracks -- 4.1 A Representation Formula -- 4.2 Derivation of an Explicit Integral Equation -- 4.3 Asymptotic Expansion -- 4.4 Topological Derivative of the Potential Energy -- 4.5 Derivation of the Representation Formula -- 4.6 Time-Harmonic Regime -- Chapter 5: Backpropagation and Multiple Signal Classification Imaging of Small Inclusions -- 5.1 A Newton-Type Search Method -- 5.2 A MUSIC-Type Method in the Static Regime -- 5.3 A MUSIC-Type Method in the Time-Harmonic Regime -- 5.4 Reverse-Time Migration and Kirchhoff Imaging in the Time-Harmonic Regime -- 5.5 Numerical Illustrations -- Chapter 6: Topological Derivative Based Imaging of Small Inclusions in the Time-Harmonic Regime -- 6.1 Topological Derivative Based Imaging -- 6.2 Modified Imaging Framework -- Chapter 7: Stability of Topological Derivative Based Imaging Functionals -- 7.1 Statistical Stability with Measurement Noise -- 7.2 Statistical Stability with Medium Noise -- Chapter 8: Time-Reversal Imaging of Extended Source Terms.

8.1 Analysis of the Time-Reversal Imaging Functionals -- 8.2 Time-Reversal Algorithm for Viscoelastic Media -- 8.3 Numerical Illustrations -- Chapter 9: Optimal Control Imaging of Extended Inclusions -- 9.1 Imaging of Shape Perturbations -- 9.2 Imaging of an Extended Inclusion -- Chapter 10: Imaging from Internal Data -- 10.1 Inclusion Model Problem -- 10.2 Binary Level Set Algorithm -- 10.3 Imaging Shear Modulus Distributions -- 10.4 Numerical Illustrations -- Chapter 11: Vibration Testing -- 11.1 Small-Volume Expansions of the Perturbations in the Eigenvalues -- 11.2 Eigenvalue Perturbations due to Shape Deformations -- 11.3 Splitting of Multiple Eigenvalues -- 11.4 Reconstruction of Inclusions -- 11.5 Numerical Illustrations -- Appendix A: Introduction to Random Processes -- A.1 Random Variables -- A.2 Random Vectors -- A.3 Gaussian Random Vectors -- A.4 Conditioning -- A.5 Random Processes -- A.6 Gaussian Processes -- A.7 Stationary Gaussian Random Processes -- A.8 Multi-valued Gaussian Processes -- Appendix B: Asymptotics of the Attenuation Operator -- B.1 Stationary Phase Theorem -- B.2 Derivation of the Asymptotics -- Appendix C: The Generalized Argument Principle and Rouch�e's Theorem -- C.1 Notation and Definitions -- C.2 Generalized Argument Principle -- C.3 Generalization of Rouch�e's Theorem -- References -- Index.

This book is the first to comprehensively explore elasticity imaging and examines recent, important developments in asymptotic imaging, modeling, and analysis of deterministic and stochastic elastic wave propagation phenomena. It derives the best possible functional images for small inclusions and cracks within the context of stability and resolution, and introduces a topological derivative-based imaging framework for detecting elastic inclusions in the time-harmonic regime. For imaging extended elastic inclusions, accurate optimal control methodologies are designed and the effects of uncertainties of the geometric or physical parameters on stability and resolution properties are evaluated. In particular, the book shows how localized damage to a mechanical structure affects its dynamic characteristics, and how measured eigenparameters are linked to elastic inclusion or crack location, orientation, and size. Demonstrating a novel method for identifying, locating, and estimating inclusions and cracks in elastic structures, the book opens possibilities for a mathematical and numerical framework for elasticity imaging of nanoparticles and cellular structures.

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