Etude mathématique et numérique de modèles homogénéisés de métamatériaux

Dans la première partie des études des problèmes de propagation d'ondes en présence de métamatériaux homogénéisés tels que les équations de Maxwell, le systèmes de l'acoustique ou de l'élasticité linéaire. Nous établissons des résultats d'existence et d'unicité pour ces systèmes sous des hypothèses phénoménologiques sur le métamatériaux en accord avec certains modèles de la littérature. Nous abordons ensuite leurs approximations numériques. Nous présentons des résultats concernant les éléments finis pour l'approximation de l'équation de Helmholtz qui montrent que ce schéma peut ne pas converger en présence de métamatériaux. On propose alors un schéma Galerkin Discontinu dont on montre numériquement sa convergence sur des exemples de métamatériauxIn the first part, we investigate wave propagation problems with homogenized metamaterials for Maxwell's equations and acoustics or linear elasticity systems. We establish that each of these systems is well-posed under assumptions that are relevant for some models already designed in the literature. We next tackle their numerical approximation. We give results showing that the finite element method for the approximation of Helmholtz equation, when metatmaterials are involved, may not converges. We propose then a numerical scheme, the EF-AL schemen which can be with metamaterials and we prove that it converges as soon as the considered problem is well-posed. We finish studying the discontinuous galerkin scheme. We show numerically its convergence for some examples of metamaterials

Approximation by multipoles of the multiple acoustic scattering by small obstacles and application to the Foldy theory of isotropic scattering.

50 (avec 1,5 interligne)International audienceThe asymptotic analysis, carried out in this paper, for the problem of a multiple scattering of a time-harmonic wave by obstacles whose size is small as compared with the wavelength establishes that the effect of the small bodies can be approximated at any order of accuracy by the field radiated by point sources. Among other issues, this asymptotic expansion of the wave furnishes a mathematical justification with optimal error estimates of Foldy's method that consists in approximating each small obstacle by a point isotropic scatterer. Finally, it is shown how this theory can be further improved by adequately locating the center of phase of the point scatterers and taking into account of self-interactions

Optimization of Bathymetry for Long Waves with Small Amplitude

This paper deals with bathymetry-oriented optimization in the case of long waves with small amplitude. Under these two assumptions, the free-surface incompressible Navier-Stokes system can be written as a wave equation where the bathymetry appears as a parameter in the spatial operator. Looking then for time-harmonic fields and writing the bottom topography as a perturbation of a flat bottom, we end up with a heterogeneous Helmholtz equation with impedance boundary condition. In this way, we study some PDE-constrained optimization problem for a Helmholtz equation in heterogeneous media whose coefficients are only bounded with bounded variation. We provide necessary condition for a general cost function to have at least one optimal solution. We also prove the convergence of a finite element approximation of the solution to the considered Helmholtz equation as well as the convergence of discrete optimum toward the continuous ones. We end this paper with some numerical experiments to illustrate the theoretical results and show that some of their assumptions could actually be removed

The Boussinesq system with non-smooth boundary conditions : existence, relaxation and topology optimization.

In this paper, we tackle a topology optimization problem which consists in finding the optimal shape of a solid located inside a fluid that minimizes a given cost function. The motion of the fluid is modeled thanks to the Boussinesq system which involves the unsteady Navier-Stokes equation coupled to a heat equation. In order to cover several models presented in the literature, we choose a non-smooth formulation for the outlet boundary conditions and an optimization parameter of bounded variations. This paper aims at proving existence of solutions to the resulting equations, along with the study of a relaxation scheme of the non-smooth conditions. A second part covers the topology optimization problem itself for which we proved the existence of optimal solutions and provides the definition of first order necessary optimality conditions

LiQuOFETI : a FETI-inspired method for elliptic quadratic optimal control problems

International audienc

A domain decomposition approach for a Topology Optimisation problem

International audienc

LiQuOFETI : a FETI-inspired method for elliptic quadratic optimal control problems

International audienc

A domain decomposition approach for a Topology Optimisation problem

International audienc