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E-raamat: Mathematical Methods in Elasticity Imaging

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Mathematical Methods in Elasticity ImagingSuurem pilt
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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.


Arvustused

"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

Introduction 1(3)
1 Layer Potential Techniques 4(29)
1.1 Sobolev Spaces
4(2)
1.2 Elasticity Equations
6(4)
1.3 Radiation Condition
10(1)
1.4 Integral Representation of Solutions to the Lame System
11(10)
1.5 Helmholtz-Kirchhoff Identities
21(6)
1.6 Eigenvalue Characterizations and Neumann and Dirichlet Functions
27(5)
1.7 A Regularity Result
32(1)
2 Elasticity Equations with High Contrast Parameters 33(15)
2.1 Problem Setting
34(1)
2.2 Incompressible Limit
34(2)
2.3 Limiting Cases of Holes and Hard Inclusions
36(2)
2.4 Energy Estimates
38(4)
2.5 Convergence of Potentials and Solutions
42(3)
2.6 Boundary Value Problems
45(3)
3 Small-Volume Expansions of the Displacement Fields 48(18)
3.1 Elastic Moment Tensor
48(7)
3.2 Small-Volume Expansions
55(11)
4 Boundary Perturbations due to the Presence of Small Cracks 66(14)
4.1 A Representation Formula
66(3)
4.2 Derivation of an Explicit Integral Equation
69(2)
4.3 Asymptotic Expansion
71(4)
4.4 Topological Derivative of the Potential Energy
75(1)
4.5 Derivation of the Representation Formula
76(3)
4.6 Time-Harmonic Regime
79(1)
5 Backpropagation and Multiple Signal Classification Imaging of Small Inclusions 80(11)
5.1 A Newton-Type Search Method
80(2)
5.2 A MUSIC-Type Method in the Static Regime
82(1)
5.3 A MUSIC-Type Method in the Time-Harmonic Regime
82(2)
5.4 Reverse-Time Migration and Kirchhoff Imaging in the Time-Harmonic Regime
84(2)
5.5 Numerical Illustrations
86(5)
6 Topological Derivative Based Imaging of Small Inclusions in the Time-Harmonic Regime 91(21)
6.1 Topological Derivative Based Imaging
91(11)
6.2 Modified Imaging Framework
102(10)
7 Stability of Topological Derivative Based Imaging Functionals 112(13)
7.1 Statistical Stability with Measurement Noise
112(6)
7.2 Statistical Stability with Medium Noise
118(7)
8 Time-Reversal Imaging of Extended Source Terms 125(23)
8.1 Analysis of the Time-Reversal Imaging Functionals
127(2)
8.2 Time-Reversal Algorithm for Viscoelastic Media
129(8)
8.3 Numerical Illustrations
137(11)
9 Optimal Control Imaging of Extended Inclusions 148(12)
9.1 Imaging of Shape Perturbations
149(3)
9.2 Imaging of an Extended Inclusion
152(8)
10 Imaging from Internal Data 160(8)
10.1 Inclusion Model Problem
160(2)
10.2 Binary Level Set Algorithm
162(2)
10.3 Imaging Shear Modulus Distributions
164(1)
10.4 Numerical Illustrations
165(3)
11 Vibration Testing 168(33)
11.1 Small-Volume Expansions of the Perturbations in the Eigenvalues
169(12)
11.2 Eigenvalue Perturbations due to Shape Deformations
181(11)
11.3 Splitting of Multiple Eigenvalues
192(1)
11.4 Reconstruction of Inclusions
193(2)
11.5 Numerical Illustrations
195(6)
A Introduction to Random Processes 201(9)
A.1 Random Variables
201(1)
A.2 Random Vectors
202(1)
A.3 Gaussian Random Vectors
203(1)
A.4 Conditioning
204(1)
A.5 Random Processes
205(1)
A.6 Gaussian Processes
206(2)
A.7 Stationary Gaussian Random Processes
208(1)
A.8 Multi-valued Gaussian Processes
208(2)
B Asymptotics of the Attenuation Operator 210(3)
B.1 Stationary Phase Theorem
210(1)
B.2 Derivation of the Asymptotics
211(2)
C The Generalized Argument Principle and Rouche's Theorem 213(4)
C.1 Notation and Definitions
213(1)
C.2 Generalized Argument Principle
214(1)
C.3 Generalization of Rouche's Theorem
214(3)
References 217(12)
Index 229
Habib Ammari is director of research at the French National Center for Scientific Research and professor of mathematics at the Ecole Normale Superieure. Elie Bretin is a postdoctoral researcher in mathematics at the Ecole Polytechnique. Josselin Garnier is professor of mathematics at Universite 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 Universite Paris VII.
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