Special Session 35: Direct and inverse problems in wave propagation
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The mathematical modelling of electromagnetic metamaterials and the homogenization theory are intimately related because metamaterials are precisely constructed by a periodic assembly of small resonating micro-structures involving dielectric materials presenting a high contrast with respect to a reference medium. We wish to look carefully at the treatment of boundaries and interfaces that are generally poorly taken into account by the first order homogenization.
This question is already relevant for standard homogenization (ie without high contrast) for which taking into account the presence of a boundary induces a loss of accuracy due to the inadequateness of the standard homogenization approach to take into account the boundary layers induced by the boundary. The objective of this work is to construct approximate effective boundary conditions that would restore the desired accuracy.
We have first considered a plane interface between a homogeneous and a periodic media in the standard case without high-contrast. We obtain high order transmission conditions between the homogeneous media and the periodic media. The technique we use involves matched asymptotic expansions combined with standard homogenization ansatz. Those conditions are non standard : they involve Laplace-Beltrami operators at the interface and requires to solve cell problems in infinite periodic waveguides. The analysis is based on a original combination of Floquet-Bloch and a periodic version of Kondratiev technique.
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