Deterministic non-periodic systems

Stochastic homogenization

The project-team has pursued its efforts in the field of stochastic homogenization of elliptic equations, aiming at designing numerical approaches that are practically relevant and keep the computational workload limited.

In addition, a question of interest is to describe how the oscillatory solution $u_{ϵ}$ fluctuates around its effective behavior (which is given by the homogenized limit $u^{*}$). This question is investigated in the PhD thesis of P.-L. Rothé. Results have been obtained for a weakly stochastic framework (with a periodic coefficient and a small random perturbation). It has been shown that, at the first order, the fluctuations are at the scale ${ϵ}^{-\frac{d}{2}}$. Furthermore when $ϵ$ is small, the localized fluctuations (characterized by a test function $g$) of $u_{ϵ}$ are Gaussian. The corresponding variance depends on the localization function $g$ and on a fourth order tensor $Q$. A numerical approach has been designed to approximate $Q$ and its convergence has been proven. Numerical experiments in more general settings (full stochastic case) following the same approach have been performed. The results are promising.

Multiscale Finite Element approaches

From a numerical perspective, the Multiscale Finite Element Method (MsFEM) is a classical strategy to address the situation when the homogenized problem is not known (e.g. in difficult nonlinear cases), or when the scale of the heterogeneities, although small, is not considered to be zero (and hence the homogenized problem cannot be considered as a sufficiently accurate approximation).

The MsFEM has been introduced almost 20 years ago. However, even in simple deterministic cases, there are still some open questions, for instance concerning multiscale advection-diffusion equations. Such problems are possibly advection dominated and a stabilization procedure is therefore required. How stabilization interplays with the multiscale character of the equation is an unsolved mathematical question worth considering for numerical purposes.

Dislocations

In the context of the PhD thesis of M. Josien, some results have been obtained regarding the modeling and numerical simulation of dislocations. Plastic properties of crystals are due to dislocations, which are thus objects of paramount importance in materials science. The geometrical shape of dislocations may be described by (possibly time-dependent) nonlinear integro-differential equations (e.g. the Weertman equation and the dynamical Peierls-Nabarro equation), involving non-local operators.