## Section: New Results

### Wave propagation in heterogeneous media

#### Homogenization of layered media

Participant : Jean-François Mercier.

Metamaterials have revived interest in the theory of homogenization techniques because some standard techniques, based on the Ross Nicholson-Weir method, can lead to unphysical effective parameters, since depending on the incident wave. In collaboration with Agnès Maurel and Abdelkader Ourir from the Langevin Institut and Simon Felix from the LAUM, we have proposed more suitable homogenization methods to describe wave propagation in artificial environments, by considering homogenization of sliced media. When the medium is structured at a sub-wavelength scale, it can be described as a simpler equivalent medium, homogeneous and anisotropic, with a tensor mass density and an effective modulus of elasticity. We considered two cases:

- for a propagating incident wave, we obtained the diffusion properties of the medium and we have shown that the effective medium correctly captures the acoustic properties of the real medium.

- however, in the real problem, evanescent waves are generated and if one of them is resonant, the properties of transmission and reflection of the incident wave are changed: this happens for the electromagnetic waves (Wood anomalies, "spoof plasmon"). To capture these resonance effects, we have considered evanescent incident waves. We then showed that the homogenization predicts the dispersion curves of the resonant waves: in the homogenized problem, they correspond to guided waves by the anisotropic layer.

#### High order transmission conditions between homogeneous and homogenized periodic half-spaces

Participants : Sonia Fliss, Valentin Vinoles.

This work is a part of the PhD of Valentin Vinoles, and is done in collaboration with Xavier Claeys (LJLL, Paris VI). It is motivated by the fact that classical homogenization theory poorly takes into account interfaces, which is particularly unfortunate when considering negative materials, because important phenomena arise precisely at their surface (plasmonic waves for instance). To overcome this limitation, we want to construct high order transmission conditions. Using matched asymptotics, we have treated the case of a plane interface between a homogeneous and a homogenized periodic half space. The analysis is based on an original combination of Floquet-Bloch transform and a periodic version of Kondratiev techniques. The obtained conditions involve Laplace- Beltrami operators at the interface and require to solve *cell problems* in infinite strips. The numerical computations are based on specific transparent conditions for periodic media. The error analysis and the numerical study are on-going works.

#### Scattering by small heterogeneities

Participants : Patrick Joly, Simon Marmorat.

Simon Marmorat has defended his thesis, done in collaboration with the CEA-LIST and with Xavier Claeys (LJLL, Paris VI). The goal was to develop an efficient numerical approach to simulate the propagation of waves in concrete, which is modelled as a smooth background medium, with many small embedded heterogeneities. To do so, one has proposed two reduced models relying on the asymptotic analysis of the problem with respect to the (small) size of the heterogeneities. The first model looks like a fictitious domain method in which the analysis of the near field (closed to the heterogeneities) is exploited. The second one is a method of auxiliary sources, based on the analysis of the far field (far from the heterogeneities). Rigorous error estimates have been established. From the numerical point of view, some points, related to the Galerkin enrichment of standard finite element methods, still need to be completed.

#### Effective boundary conditions for strongly heterogeneous thin layers

Participants : Mathieu Chamaillard, Patrick Joly.

This topic is the object of the PhD of Mathieu Chamaillard, done in collaboration with Houssem Haddar (Inria, Defi). We are interested in the construction of effective boundary conditions for the diffraction of waves by an obstacle covered with a thin coating whose physical characteristics vary “periodically'’. The width of the coating and the period are both proportional to the same small parameter $\delta $.

The results obtained previously on scalar propagation models have been extended to 3D Maxwell’s equations resulting in the construction of an effective condition of the form $E\times n=\delta ik{\mathcal{Z}}_{\Gamma}\left(n\times (H\times n)\right)$ where the impedance operator ${\mathcal{Z}}_{\Gamma}$, a second order tangential differential operator along $\Gamma $, depends on the geometry of the obstacle and of the material properties of the coating. The analysis, which is much more involved than in the scalar case (in particular in what concerns the stability analysis), provides error estimates in $O\left({\delta}^{2}\right)$.

The thesis will be defended in the end of January 2016.