Equivalent Conditions for an Elasto-Acoustic Problem with a Thin Layer

Abstract : Equivalent boundary Conditions have become a classic notion in the mathematical modeling of wave propagation phenomena. They are used for instance for scattering problems from thin obstacles. The general idea consist to replace an exact model inside the obstacle by approximate boundary conditions. This idea is pertinent if the Equivalent Condition (EC) can be easily handled numerically, for instance when it is local. In this talk, we consider the diffraction problem of elasto-acoustic waves in a solid medium surrounded by a thin layer of fluid medium. This problem is well suited for the notion of EC: due to the small thickness of the layer with respect to the wavelength, the effect of the fluid medium on the solid is as a first approximation local. We present asymptotic models in which the acoustic waves propagating in the fluid are represented by an EC. This approach leads us to solve only elastodynamic equations. To construct ECs, we derive a two-scale asymptotic expansion in power series of the thickness of the fluid layer. We study the mathematical models to ensure that the new conditions does not induce numerical instabilities. Questions regarding the implementation of the new conditions are addressed carefully in a joint work with Julien Diaz.