Mathematical Methods in Elasticity Imaging

Habib Ammari, Elie Bretin, Josselin Garnier, Hyeonbae Kang, Hyundae Lee, Abdul Wahab

• Publisher: Princeton University Press
• Year: 2015
• Print ISBN: 9780691165318
• eBook ISBN: 9781400866625
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

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.

Chapter One

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Layer Potential Techniques

The asymptotic theory for elasticity imaging described in this book relies on layer potential techniques. In this chapter we prepare the way by reviewing a number of basic facts and preliminary results regarding the layer potentials associated with both the static and time-harmonic elasticity systems. The most important results in this chapter are on one hand the decomposition formulas for the solutions to transmission problems in elasticity and characterization of eigenvalues of the elasticity system as characteristic values of layer potentials and on the other hand, the Helmholtz-Kirchhoff identities. As will be shown later, the Helmholtz-Kirchhoff identities play a key role in the analysis of resolution in elastic wave imaging. We also note that when dealing with exterior problems for harmonic elasticity, one should introduce a radiation condition, known as the Sommerfeld radiation condition, in order to select the physical solution to the problem.

This chapter is organized as follows. In Section 1.1 we first review commonly used function spaces. Then we introduce in Section 1.2 equations of linear elasticity and use the Helmholtz decomposition theorem to decompose the displacement field into the sum of an irrotational (curl-free) and a solenoidal (divergence-free) field. Section 1.3 is devoted to the radiation condition for the time-harmonic elastic waves, which is used to select the physic...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Contents
• Introduction
• 1 Layer Potential Techniques
• 2 Elasticity Equations with High Contrast Parameters
• 3 Small-Volume Expansions of the Displacement Fields
• 4 Boundary Perturbations due to the Presence of Small Cracks
• 5 Backpropagation and Multiple Signal Classification Imaging of Small Inclusions
• 6 Topological Derivative Based Imaging of Small Inclusions in the Time-Harmonic Regime
• 7 Stability of Topological Derivative Based Imaging Functionals
• 8 Time-Reversal Imaging of Extended Source Terms
• 9 Optimal Control Imaging of Extended Inclusions
• 10 Imaging from Internal Data
• 11 Vibration Testing
• A Introduction to Random Processes
• B Asymptotics of the Attenuation Operator
• C The Generalized Argument Principle and Rouché’s Theorem
• References
• Index