Mathematical Methods in Elasticity Imaging

by Habib Ammari, Elie Bretin, Josselin Garnier, Hyeonbae Kang, Hyundae Lee

Overview

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.

Product Details

ISBN-13:	9780691165318
Publisher:	Princeton University Press
Publication date:	04/06/2015
Series:	Princeton Series in Applied Mathematics , #52
Pages:	240
Product dimensions:	6.80(w) x 10.10(h) x 0.90(d)

About the Author

Habib Ammari is director of research at the French National Center for Scientific Research and professor of mathematics at the École Normale Superiéure. Elie Bretin is a postdoctoral researcher in mathematics at the École Polytechnique. Josselin Garnier is professor of mathematics at Université Paris VII. Hyeonbae Kang is the Jungseok Chair Professor of Mathematics at Inha University in South Korea. Hyundae Lee is assistant professor of mathematics at Inha University. Abdul Wahab is a postdoctoral researcher in mathematics at Université Paris VII.

Table of Contents

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 Lamé 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-TimeMigration 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.1 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

B Asymptotics of the Attenuation Operator 210

B.1 Stationary Phase Theorem 210

B.2 Derivation of the Asymptotics 211

C The Generalized Argument Principle and Rouché’s Theorem 213

C.1 Notation and Definitions 213

C.2 Generalized Argument Principle 214

C.3 Generalization of Rouché’s Theorem 214

References 217

Index 229

What People are Saying About This

From the Publisher

"This book covers recent mathematical, numerical, and statistical approaches for elasticity imaging of inclusions and cracks. A precise and timely book, it is easy to follow and will interest readers."—Yanyan Li, Rutgers University

"This book is the first to provide readers with a comprehensive overview of mathematical and computational studies of elasticity imaging. It is a useful resource for scientists and engineers in the relevant fields and a solid reference for any newcomers to elasticity imaging."—Gang Bao, Zhejiang University

"This well-organized book explains recent mathematical, numerical, and statistical approaches for multistatic imaging of targets with waves at single or multiple frequencies. It makes an important contribution to the understanding of trade-offs between data size, computational complexity, signal-to-noise ratio, and resolution in multistatic imaging."—Jin Keun Seo, Yonsei University

Editorial Reviews

"This timely book that is devoted to a topic of paramount importance is very well written and it contains useful and interesting material. I strongly believe that it will be a valuable resource for researchers in elasticity imaging and related areas of nondestructive testing and inverse problems."—-Akhtar A. Khan, MathSciNet

From the Publisher