Section: New Results
Imaging and inverse problems
Participants : Laurent Bourgeois, Jérémi Dardé, Éric Lunéville.
We have continued our works on the method of quasi-reversibility to solve second order ill-posed Cauchy problems for an elliptic equation (as they appear in standard inverse problems). This constitutes part of the subject of the PhD thesis of J. Dardé. In particular, we have developped a new strategy to identify obstacles in a domain from partial Cauchy data on the boundary of such domain. This strategy uses a coupling between the method of quasi-reversibility and a level set method. Too types of level set method have been studied in the case of an obstacle which is characterized by a Dirichlet boundary condition u = 0 . The first one relies on the solution of an eikonal equation, while the second one relies on the solution of a simple Poisson equation. The second one turns out to be particularly easy to implement and very efficient. Some theoretical justifications have been provided for each method. We have also extended this strategy to the case of non standard boundary conditions such as |u| = 1 . Such condition arises in the field of mechanical engineering, precisely in the problem of identification of plastic zones from simultaneous measurements of displacements and forces on the boundary. A promising application is the Non Destructive Evaluation of cracks in elastoplastic media.
Linear sampling Method
Participants : Laurent Bourgeois, Frédérique Le Louër, Éric Lunéville.
We aim at extending some previous work concerning the Linear Sampling Method in an acoustic waveguide to the case of an elastic waveguide. This constitutes the subject of the Post Doc of F. Le Louër and we have begun the first investigations. The main challenge relies on the fact that we can no longer use a family of transverse modes in order to project the displacement field. One possibility consists in using a family of two vector fields formed with particular combinations of displacement and stress components, for which a bi-orthogonality relationship in the transverse section has been proved (Fraser relationship). We hence expect to use propagating modes based on these vector fields as incident waves and the corresponding scattered waves measured at long distance in order to retrieve unknown obstacles, in the framework of the Linear Sampling Method.
Identification of generalized boundary conditions
Participants : Laurent Bourgeois, Nicolas Chaulet, Houssem Haddar.
This work is a collaboration between POEMS and DEFI projects and constitutes the subject of the PhD thesis of N. Chaulet. In the context of acoustics in the harmonic regime, we have first considered the problem of identification of some Generalized Impedance Boundary Conditions (GIBC) at the boundary of an object (which is supposed to be known) from far field measurements associated with a single incident plane wave at a fixed frequency. The GIBCs are approximate models for thin coatings or corrugated surfaces. We have addressed the theoretical questions of uniqueness, stability, as well as the numerical reconstruction of these boundary parameters.