Inverse Scattering from Two-Dimensional Thin Inclusions and Cracks.

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Park, Won-Kwang (2009) Inverse Scattering from Two-Dimensional Thin Inclusions and Cracks. Doctorat, Centre de Mathématiques Appliquées, EP/X p.178.

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Résumé

Non-Destructive testing to retrieve a penetrable or impenetrable crack appears to be one of the interesting inverse problems which is arising in physics, medical science, material engineering, and so on, highly related with human life.



The aim of this thesis is to develop suitable reconstruction methods in order to apply them to various kinds of crack problems. First, we propose a non-iterative algorithm for retrieving the end points of conducting cracks based on an appropriate asymptotic formula and an identification method for simple poles and residues of a meromorphic function. Second, a non-iterative MUSIC(MUltiple SIgnal Classification)-type algorithm is considered to image a penetrable or impenetrable crack from scattered field data which can be represented via a rigorous asymptotic formulation. Third, a level-set technique is proposed to reconstruct a penetrable crack. Two level-set functions are used to express the crack because it cannot be easily described by traditional level-set methods due to the small thickness. Finally, this thesis deals with the reconstruction of small and extended cracks with Dirichlet boundary conditions. Based on the asymptotic expansion formula, we develop a MUSIC-type algorithm for retrieving small cracks and an optimization algorithm for reconstructing extended cracks. Comprehensive numerical simulations illustrate the performances of the proposed reconstruction methods.

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