From periodic to random media


We study in this research axis wave propagation in heterogeneous media when the characteristic size of the heterogeneities is of the same order as the wavelength. In this regime, asymptotic techniques can not be applied and the heterogeneous media have to be considered as such. We have developed a longstanding expertise on periodic media, more precisely on spectral theory of differential operators with periodic coefficients and/or periodic geometry and the mathematical and numerical analysis of scattering problems in unbounded periodic media. More recently, we have also been interested in quasi-periodic media, which are ordered media without being periodic (see Figure 2) and have worked on randomly perturbed random media. Below are a few highlights on our works in this theme.

Spectral theory in periodic media

In some frequency ranges, periodic structures behave as insulators or filters: the corresponding monochromatic waves, also called Floquet modes, cannot propagate in the bulk. The study of these modes is, from a mathematical point of view, related to the spectrum of the underlying operator that presents so-called band structures: the spectrum may contain some forbidden frequency intervals, called band gaps. In the presence of a boundary, an interface or more generally a lineic perturbation in a periodic medium, energy localization and transport can be created. Such phenomena can be exploited in quantum, electronic or photonic device design. The questions here are (1) to find conditions on the periodic media or the perturbation to ensure energy transport (2) to propose a numerical method to simulate this phenomenon.

Wave propagation in periodic media

One of the difficulties when we study time harmonic wave propagation in unbounded domains is that the associated problem is not always well-posed in a classical framework. One has to impose a behaviour at infinity called the radiation condition or restrict the domain of interest to a bounded domain and impose transparent boundary conditions at its boundary. These difficulties are rarely addressed in the literature when the medium of propagation is heterogeneous.

For a few years, we have been studying the case of periodic media in different configurations (wave-guide, half-space, periodic medium unbounded in 2 directions...). We have proposed radiation conditions for wave-guides and worked extensively on the derivation of a transparent boundary condition that can be numerically implemented. The method works as follows : (1) we add dissipation (i.e. an imaginary part to the frequency) to our problem to fall into a classical framework, (2) we build a Dirichlet-to-Neumann operator based on the structure of the medium, (3) we let the dissipation parameter goes to 0.

Wave propagation in quasi-periodic media

PDEs with quasi-periodic coefficients have been the subject of a number of theoretical studies (see in particular the work of RM Levitan, VV Zhikov) but it seems that apart in the context of homogenization, there has been much less work on the numerical resolution of these equations. The overall objective is to develop original numerical methods for the solution of the time-harmonic wave equation in quasi-periodic media, in the spirit of the methods that we have developed previously for periodic media. The idea is to use that the study of an elliptic PDE (in the sense that the principal part of the operator is elliptic) with quasi-periodic coefficients comes down to the study of a non-elliptic PDE in higher dimension but whose coefficients are periodic.

Wave propagation in random media

The mathematical and the numerical analysis of the time harmonic wave equation in an unbounded random media is a challenging question. The difficulties concern the definition of radiation condition or equivalently the construction of transparent boundary conditions. We have studied for now this problem in simplified "asymptotic" configuration. More precisely, we have considered a periodic medium with weak random perturbations with two types of weak perturbations: (1) the case of stationary, ergodic and oscillating coefficients, the typical size of the oscillations being small compared to the wavelength (the periodi being comparable with the wavelength) and (2) the case of rare random perturbations of the medium, where each period has a small probability to have its coefficients modified, independently of the other periods. We have derived an asymptotic approximation of the solution with respect to the small parameter to construct absorbing boundary conditions for such media.