Section: New Results

Wave propagation in heterogeneous media

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 requires to solve cell problems in infinite strips.

Multiple scattering by small homogeneities

Participants : Patrick Joly, Simon Marmorat.

This is the topic of the PhD of Simon Marmorat, done in collaboration with the CEA-LIST and with Xavier Claeys (LJLL, Paris VI). We aim at developing 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. This kind of problem is very costly to handle with classical numerical methods, due the refined meshes needed around the inclusions. To overcome these issues, two models have been developed, which rely on the asymptotic analysis of the problem: each of them can be interpreted as a full space wave equation, which can be discretized using a defects-free mesh, coupled to some auxiliary unknowns accounting for the presence of the inclusions. While the first model is established by using a special Galerkin approximation in the vicinity of the inclusions, the second model only focuses on the far field. The challenge is then to simulate source points coupled to the incident field and this is achieved thanks to the introduction of a special relaxed version of the Dirac mass. Rigorous error estimates as well as some numerical tests have been established, highlighting the efficiency of the two methods.

Finite Element Heterogeneous Multiscale Method for Maxwell's Equations

Participants : Patrick Ciarlet, Sonia Fliss, Christian Stohrer.

This work is the subject of the post-doc of Christian Stohrer. The standard Finite Element Heterogeneous Multiscale Method (FE-HMM) can be used to approximate the effective behavior of solutions to the classical Helmholtz equation in highly oscillatory media. Using a novel combination of well-known results about FE-HMM and the notion of T-coercivity, we derive an a priori error bound. Numerical experiments corroborate the analytical findings. We work now on the application of HMM in presence of interfaces, for Maxwell's equations and finally in presence of high contrast materials.

Effective boundary conditions for strongly heterogeneous thin layers

Participants : Matthieu Chamaillard, Patrick Joly.

This topic is the object of the PhD of Matthieu Chamaillard, done in collaboration with Houssem Haddar (CMAP École Polytechnique). 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 $\delta$. In the scalar case, we proved that the error between the exact model (with the thin coat) and the one with the effective boundary condition of order $n$ for $n\in \{1,2\}$ is of the order $𝒪\left({\delta }^{n+1}\right)$. This has been checked numerically for some two dimensional configurations. Recently, we also succeeded to extend our theoretical work to Maxwell equations. We found a first order boundary condition of the form $E\times n=\delta ik{𝒵}_{\Gamma }\left(n\times (H\times n)\right)$ where $n$ is the unit outward normal to the boundary $\Gamma$ and ${𝒵}_{\Gamma }$ is a second order tangential differential operator along $\Gamma$. The coefficients of this operator depend only on the deformation mapping ${\psi }_{\Gamma }$ and the material properties of the coating, through the resolution of particular unbounded cell problems in the flat reference configuration. When the coating is homogeneous, one recovers the well known first order thin layer condition. We have moreover proven that this effective condition provides an error of the order $𝒪\left({\delta }^2\right)$.