Non-destructive testing of anisotropic materials
Date
2015
Authors
Journal Title
Journal ISSN
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Publisher
University of Delaware
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.