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On the use of sampling methods and spectral signatures to identify defects in inhomogeneous media

Abstract : This thesis is a contribution to inverse scattering theory. We are more specifically interested in the non-destructive testing of heterogeneous materials such as composite materials by using acoustic waves. Monitoring this type of materials in an industrial environment is of major importance, but their complex structure makes this task difficult. The so-called sampling methods seem very promising to address this issue. We develop these techniques to detect the appearance of defects from far field data. The defects considered are impenetrable Neumann obstacles. We distinguish two categories of them, each requiring a specific treatment: cracks and obstacles with non empty interior.Thanks to the two complementary factorizations of the far field operator that we establish, we show that it is possible to approach the solution of the Interior Transmission Problem (ITP) from the data. The ITP is a system of partial differential equations that takes into account the physical parameters of the material being surveyed. We show that it is then possible to detect an anomaly by comparing the solutions of two different ITPs, one associated with measurements made before the defect appeared and the other one associated with measurements made after. The validity of the described method requires avoiding particular frequencies, which are the elements of the ITP spectrum for which this problem is not well posed. We show that this spectrum is an infinite set, countable and without finite accumulation points.In the last chapter, we use the recent notion of artificial backgrounds to image crack networks embedded in a homogeneous background. This approach allows us to design a transmission problem with the choice of the artificial background, for instance made of an obstacle. The associated spectrum is then sensitive to the presence of cracks inside the artificial obstacle. This allows to quantify locally the crack density. However, the computation of the spectrum requires data at several frequencies and is expensive in terms of calculations. We propose an alternative method using only data at fixed frequency and which consists in working with the solutions of the ITP instead of it's spectrum.
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Kevish Napal. On the use of sampling methods and spectral signatures to identify defects in inhomogeneous media. Analysis of PDEs [math.AP]. Université Paris Saclay (COmUE), 2019. English. ⟨NNT : 2019SACLX102⟩. ⟨tel-02885422⟩

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