Séance Séminaire

Séminaire ACSIOM

Asymptotic models for geophysical applications

This work enters into the scope of the FP7 project "High Performance Computing for Geophysics Applications". In this project, one application concerns the mathematical modeling of Earthquakes. In this context, we are interesting with numerical models on the coupling of seismic waves (elastic and acoustic waves) which propagate in complex media (the Earth's subsurface, the ocean, the atmosphere). In this talk, we present Equivalent boundary Conditions and asymptotic models for diffraction problems of elasto-acoustic waves set in a domain with a thin layer. Equivalent boundary conditions have become a classical notion in the mathematical modeling of wave propagation phenomena. They are used for instance for scattering problems from thin obstacles. The general idea is to replace an exact model inside the obstacle by approximate boundary conditions. This idea is pertinent if the boundary condition can be easily handled numerically, for instance when it is local. First, we consider a model problem set in a domain made of a solid obstacle coated with a thin layer of fluid medium. This problem is well suited for the notion of equivalent conditions since the small thickness of the coating (with respect to the wavelength) ensures that the effect of the layer on the solid medium is as a first approximation local. We present equivalent models for the elasto-acoustic problem : acoustic waves propagating in water are represented by an equivalent boundary condition. This approach leads us to solve only elastic equations. We present elements of derivation together with mathematical justications for equivalent boundary conditions. We illustrate these models with numerical simulations (from a Discontinuous Galerkin Method) and analytical solutions. Then, we discuss the influence of a layer with a variable thickness. Finally, we consider the case of a solid obstacle coated with a thin layer of fluid medium, and surrounded by a third medium. Keywords: asymptotic analysis, impedance conditions, elastic waves, acoustic waves