Section: New Results

Asymptotic methods and approximate models

Effective boundary conditions for thin periodic coatings

Participants : Mathieu Chamaillard, Patrick Joly.

This topic is developed in collaboration with H. Haddar (DEFI, INRIA Saclay) can be seen as a continuation of the PhD thesis of B. Delourme (see the activity report of last year) on effective transmission conditions for thin rough interfaces. On this last subject, the mathematical analysis of such transmission conditions in the (difficult) case of 3D Maxwell's equations has been completed and submitted for publication.

We are now coming back to the more traditional issue of effective or approximate boundary conditions for simulating thin periodic coatings at the surface of a diffractive obstacle. This subject has already been more widely investigated, in particular in France (see the works by Achdou, Ammari and their collaborators for instance). However, we attack, with the PhD thesis of M. Chamaillard, supported by a DGA/INRIA scholarship, various new aspects of the problem, namely :

• the treatment of surfaces of general geometry,

• the use of non standard materials, such as ferromagnetic materials, for the coating,

• the construction of higher order impedance conditions.

This is motivated by various recent progress in the domain of stealth technology and we hope to develop a collaboration in this domain with CEA-DAM (CESTA and Le Ripault).

Elastic wave propagation in strongly heterogeneous media

Participants : Patrick Joly, Simon Marmorat.

This subject enters our long term collaboration with CEA-LIST on the development on numerical methods for time-domain non destructive testing experiments using ultra-sounds. This is also the subject of the PhD thesis of Simon Marmorat. Our objective is to develop an efficient numerical approach for the propagation of elastic waves in a medium which is made of many small inclusions / heterogeneities embedded in a smooth (or piecewise smooth) background medium, without any particular assumption (such as periodicity) on the spatial distribution of these heterogeneities. Our idea is to exploit the smallness of the inclusions (with respect to the wavelength in the background medium) to derive a simplified approximate model in which each inclusion would de described by very few parameters (functions of time) coupled to the displacement field in background medium for which we could use a computational mesh that ignores the presence of the heterogeneities. For deriving such a model. we intend to use and adapt the asymptotic methods previously developed at Poems (such as matched asymptotic expansions).

Approximate models in aeroacoustics

Participants : Anne-Sophie Bonnet-Ben Dhia, Patrick Joly, Guillaume Legendre, Ricardo Weder.

This topic concerns the 2D acoustic propagation in presence of a mean flow, modeled for instance by the Galbrun equation. We had previously derived effective boundary conditions taking into account the boundary layers of the mean flow near a rigid or treated boundary. These boundary conditions are in general non local with respect to the normal coordinate inside the boundary layer. However when the Mach profile in the shear layer is piecewise linear, the condition can be replaced by a system of 1D advection equations, which are coupled with the Galbrun equation in the 2D domain. We have derived a variational formulation for this model, in time-harmonic regime and for the case of a rigid boundary. This formulation has been implemented in the Melina code: the first results are promising and the validation, by comparison to the solution obtained by a full discretization of the shear layer, is in progress.