Author: Harris, Isaac
Citable URI: http://udspace.udel.edu/handle/19716/17658
Advisor: Cakoni, Fioralba
Department: University of Delaware, Department of Mathematical Sciences
Publisher: University of Delaware
Date Issued: 2015
Abstract: In the thesis we consider the inverse problem of detecting the shape, size, position and some information about the material properties of a (possibly anisotropic) penetrable defective region in a known anisotropic material. First we consider the problem of detecting defects by using the measured transmission eigenvalues, which are related to non-scattering frequencies. In particular we consider the transmission eigenvalue problem corresponding to the scattering problem for an anisotropic magnetic material with voids, i.e. subregions with refractive index the same as the background, restricting ourselves to the scalar case of TE-polarization for electromagnetic waves or acoustic waves. Under appropriate assumptions on the material properties, we show that the transmission eigenvalues can be determined from the far field measurements, and we prove the existence of at least one real transmission eigenvalue for sufficiently small voids. We also show that the first transmission eigenvalue can be used to provide qualitative information about the size of the void. Even though the transmission eigenvalues can be used to determine the size of a defective region, to reconstruct the shape and position of the defect we need to use different techniques. To this end, we develop the Factorization Method (FM) which provides an indicator function for the defective region. The FM connects the support of the defective region to the range of a compact operator which is known from physical experiments. Hence, evaluating the indicator function derived from the FM amounts to applying Picard's criteria, which only requires the singular values and vectors of a known compact operator. Since evaluating the indicator function needs the far-field pattern of the background Green's function, we prove a mixed reciprocity result connecting the far-field pattern of the Greens function to the total field of the unperturbed material. We then consider the inverse scattering problem for an anisotropic media with small homogeneous penetrable defects. We considered the transmission eigenvalue problem for the perturbed media as well as derive a MUSIC algorithm to reconstruct the locations of the small defects. For the corresponding transmission eigenvalue problem we study the convergence and convergence rate of the transmission eigenvalues and construct appropriate corrector terms for the transmission eigenvalues as the size of the defects tends to zero. Using the corrector and the knowledge of the location of the inclusions one can derive an algorithm to reconstruct the constitutive parameters of the inclusions. In the same spirit as in using the transmission eigenvalues to determine information about a defective region, in the next project we use the transmission eigenvalues for parameter identification. In particular we consider the interior transmission problem associated with the scattering by an inhomogeneous (possibly anisotropic) highly oscillating periodic media. We show that, under appropriate assumptions, the solution of the interior transmission problem converges to the solution of a homogenized problem as the period goes to zero. Furthermore, we prove that the associated real transmission eigenvalues converge to transmission eigenvalues of the homogenized problem. In our investigation of the convergence, we construct boundary corrections for the anisotropic case, which are used to determine the convergence rate for the interior transmission problem. Finally we show how to use the first transmission eigenvalue of the period media, to obtain information about constant effective material properties of the periodic media.
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