PhD Student in Team MAGIQUE-3D, University of Pau and Pays de l’Adour, Inria Bordeaux Sud-Ouest, LMAP, UMR CNRS 5142
October 2016 / September 2019 under the supervision of Sébastien Tordeux and Victor Péron
email : justine.labat@inria.fr
Tel : +33 5 40 17 51 55
Context
In the context of non-destructive testing in medical imaging or civil engineering, difficulties to detect small bodies in three dimensional domains justify the interest of reduced models. The complexity for solving numerically direct problems both in terms of computation time and memory cost is due to the small size of obstacles in comparison with the incident wavelength and the large size of the domain of interest. Then the fine mesh size makes unsuitable or too expensive the use of classical numerical methods type continuous and discontinuous finite element methods or boundary element methods.
The method of Matched Asymptotic Expansions
The use of reduced models is used to get an approximation of the exact solution at a fixed precision with lower cost. It is based on a parametrization of the domain with respect to a small parameter, here the obstacle size. Then it allows the use of uniform meshes on the reduced models. We develop a Matched Asymptotic Expansions (MAE) method to solve the time-harmonic electromagnetic scattering problem by small obstacles.
This method allows to replace the scatterers by equivalent asymptotic point sources. In practice, it consists on defining an approximate solution using multi-scale expansions over far and near fields, related in a matching area. The approximation is valid for a range of small parameters and the smaller the parameter, the more accurate the method.
In a first time, we study the scattering problem by one small sphere perfectly conducting and we make explicit the asymptotic expansions until the second order, relatively to the sphere radius. Numerical results make evident convergence rate with respect to the sphere radius. Reference solutions are analytical solutions computed thanks to Montjoie.
Application to the Foldy-Lax model for the multiple scattering (In the pipeline)
In the other hand, we are performing the application to the Foldy-Lax model in order to extend our results for the multiple scattering problem by a finite number of small spheres. Meshes are performed with Gmsh.
Near perspectives
We want to study the electromagnetic scattering by small obstacles with general shape. The treatment of small bounded defects suggests the use of a boundary element method in electromagnetism.
