Qualitative methods for heterogeneous media

Abstract : This thesis focuses on non destructive testing of concrete using ultrasonic waves, and thus examines imaging in complex heterogeneous media. We assume that measurements are multistatic, which means that we record the total field on different points by using several sources. For this type of data we wish to build methods that are able to image inclusions or defects that contributed to the measured field. We focus in this work on the extension of so called sampling methods to deal with the over-mentioned application where the main additional difficulty is the lack of knowledge of the reference media (media without defects, also referred to as background media). The first part of this thesis consists of a new theoretical analysis of the Linear Sampling Method leading to new mathematically sound formulation of this method. Such analysis is done in the framework of regularization theory, and our main contribution is to provide and analyze a regularization term that ensures exact characterization of the shape in terms of measured data. We also prove that one is able to reconstruct from regularized solutions a sequence of functions that strongly converges to the solution of the so-called interior transmission problem. This result gives a central place to the interior transmission problem as it allows describing the asymptotic behaviour of our regularized problem. More importantly it also allows us to compare solutions coming from two different datasets. Based on the result of this comparison, we manage to produce an image of the connected components of the background that contain the defects appearing between two measurement campaigns and this is regardless of background “microstructure”. This strategy is well suited for applications to concrete- like backgrounds as shown on several numerical examples with realistic concrete-like microstructures. Finally, we extend our theoretical results to the case of limited aperture, anisotropic medium and elastic waves, which correspond to the real physics of the ultrasounds.
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Submitted on : Wednesday, September 23, 2015 - 2:59:02 PM
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  • HAL Id : tel-01203647, version 1

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Lorenzo Audibert. Qualitative methods for heterogeneous media. Analysis of PDEs [math.AP]. Ecole polytechnique X, 2015. English. ⟨tel-01203647⟩

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