Asymptotic methods in inverse scattering for inhomogeneous media
Date
2017
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Publisher
University of Delaware
Abstract
In this thesis we study three different problems associated to the detection of two types of material defects: interfacial cracks and delaminations.
In Chapter 2 we address the problem of interfacial crack detection in layered isotropic elastic media. In the first part, a well-posedness result is established, and we use this result in the second part of the chapter to adapt the Factorization Method (FM) in order to propose a reconstruction algorithm.
In Chapter 3 we consider the problem of detecting if two materials that should be in contact have separated or delaminated. The goal is to find an acoustic technique to detect the delamination. We model the delamination as a thin opening between two materials of different acoustic properties, and using asymptotic techniques we derive an asymptotic model where the delaminated region is replaced by jump conditions on the acoustic field and flux. The asymptotic model has potential singularities due to the edges of the delaminated region, and we show that the forward problem is well posed for a large class of possible delaminations. We then design a special Linear Sampling Method (LSM) for detecting the shape of the delamination assuming that the background, undamaged, state is known.
In Chapter 4 we consider the problem of detecting planar delaminated regions of constant thickness. Here we aim to develop an electromagnetic technique to detect the delamination. Again, we derive a asymptotic model where the delaminated region is replaced by jump conditions on the electric and magnetic fields. We show that the forward problem is well posed under some assumptions on the material properties, and finally adapt again a LSM to detect the shape of the delamination assuming that the background state is known. In all three chapters we show, by numerical experiments, that our nondestructive testing (NDT) methods can indeed be used to determine the shape of the corresponding defect.
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Keywords
Applied sciences, Asymptotic methods, Crack, Delamination, Inverse scattering, Linear sampling method, Nondestructive testing