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Analyse mathématique et numérique de quelques problèmes d'ondes en milieu périodique

Abstract : The modeling of many interesting physical problems leads to partial differential equations, in a domain whose geometry and coefficients are functions periodic outside some regions, called scatterers, which are small with respect to the full domain of interest. The caracteristics of these problems often prevent us from applying classical homogeneization techniques, that is why we have developped new methods to restrict the computational domain to bounded domains. We have generalized the Lippmann-Schwinger equation approach, which allows us to treat bounded and structured unbounded scatterers, the main issue being that for a generic periodic media there is no analytic representation of the solution in the case without scatterers (i.e the Green function is unknown). Dirichlet-to-Neumann maps for periodic strips infinite in one direction play a key role in our approach. We treat two kinds of problems : time harmonic problems, for which the DtN maps for strip problems are known, and evolution problems, for which we present a method of derivation of these operators. In these two cases, we first treat the case of one bounded or unbounded scatterer, then we generalize the multiple scattering methods for homogeneous media to the case of periodic media, which allow us to handle several scatterers as well.
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https://pastel.archives-ouvertes.fr/pastel-00649212
Contributor : Julien Coatléven <>
Submitted on : Wednesday, December 7, 2011 - 1:10:20 PM
Last modification on : Wednesday, July 3, 2019 - 10:48:04 AM
Long-term archiving on: : Thursday, March 30, 2017 - 8:26:22 PM

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  • HAL Id : pastel-00649212, version 1

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Julien Coatléven. Analyse mathématique et numérique de quelques problèmes d'ondes en milieu périodique. Equations aux dérivées partielles [math.AP]. Ecole Polytechnique X, 2011. Français. ⟨pastel-00649212⟩

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