Analyse mathématique et numérique de problèmes de propagation des ondes dans des milieux périodiques infinis localement perturbés

Sonia Fliss 1, 2
2 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
Abstract : Periodic media play a major role in many applications, in particular in optics formicro and nanotechnology or in mechanics with composite materials. One of themain interesting features is the possibility offered by such media to select ranges of frequencies for which waves can or cannot propagate. In real applications, the media are not really periodic but differ from periodic media only in bounded regions (small with respect to the total size of the propagation domain). In such applications, there is a need for efficient numerical methods for computing the propagation of waves inside such structures. To reach this goal, the classical idea is to reduce the pure numerical computations to these regions and to try to take advantage of the periodic structure of the outside problem to construct artificial (but exact) boundary conditions. That is why we investigate the generalization to periodic media of the Neumann-to-Dirichlet approach which has already been well developed in homogeneous media. The new difficulty is that this NtD operator can no longer be determined explicitly and has to be computed numerically. We consider successively three specific situations of increasing complexity : the one-dimensional case, the case of a locally perturbed periodic waveguide and the more complicated case of a locally perturbed periodic 2D plane. For each situation, the approach is the same : first, we look for the solution of the problem in absorbing media and second we try to solve the problem in non absorbing media using the limiting absorption principle. We show then that the DtN operator can be characterized through the solution of local PDE cell problems, the use of analytical tools such as the Floquet-Bloch transformand the solution of operator-valued quadratic or linear equations.
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Submitted on : Tuesday, April 13, 2010 - 8:00:00 AM
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Sonia Fliss. Analyse mathématique et numérique de problèmes de propagation des ondes dans des milieux périodiques infinis localement perturbés. Mathématiques [math]. Ecole Polytechnique X, 2009. Français. ⟨pastel-00005464⟩

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