Section: New Results

Homogenization and related topics

Participants : Virginie Ehrlacher, Claude Le Bris, Frédéric Legoll, François Madiot, William Minvielle.

The homogenization of (deterministic) non periodic systems is a well known topic. Although well explored theoretically by many authors, it has been less investigated from the standpoint of numerical approaches (except in the random setting). In collaboration with X. Blanc and P.-L. Lions, C. Le Bris has introduced a possible theory, giving rise to a numerical approach, for the simulation of multiscale nonperiodic systems. The theoretical considerations are based on earlier works by the same authors (derivation of an algebra of functions appropriate to formalize a theory of homogenization). The numerical endeavour is completely new. Promising results have been obtained on a simple case of a periodic system perturbed by a localized defect. Ongoing works consider other configurations, such as for instance an interface between two different crystalline phases.

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

Using the standard homogenization theory, one knows that the homogenized tensor, which is a deterministic matrix, depends on the solution of a stochastic equation, the so-called corrector problem, which is posed on the whole space ${ℝ}^d$. This equation is therefore delicate and expensive to solve. In practice, the space ${ℝ}^d$ is truncated to some bounded domain, on which the corrector problem is numerically solved. In turn, this yields a converging approximation of the homogenized tensor, which happens to be a random matrix. For a given truncation of ${ℝ}^d$, the team has previously shown that the variance of this matrix can be reduced using the technique of antithetic variables. In [47] , F. Legoll and W. Minvielle have extended this technique to nonlinear, convex homogenization problems.

In addition, F. Legoll and W. Minvielle have investigated the possibility to use other variance reduction approaches, such as control variate techniques. A promising idea is to use the weakly stochastic model previously introduced by A. Anantharaman and C. Le Bris (in which a periodic model is perturbed by a rare stochastic perturbation) to build a control variate model. The preliminary results that have already been obtained are very encouraging.

Yet another approach to reduce the variance is the so-called Multi Level Monte Carlo (MLMC) approach, which is based on using a surrogate model for the quantity of interest. The MLMC approach consists in using many realizations of the surrogate model (which is cheap to evaluate) and few realizations of the reference model (which is more expensive to evaluate). In collaboration with Y. Efendiev and C. Kronsbein, F. Legoll has explored in [41] how this approach can be used in random homogenization.

We have discussed above approaches to efficiently compute the homogenized coefficient, assuming we have a complete knowledge of the microstructure of the material. We have recently started to consider a related inverse problem, and more precisely a parameter fitting problem. Knowing the homogenized quantities, is it possible to recover some features of the microstructure properties? Obviously, since homogenization is an averaging procedure, not everything can be recovered from macroscopic quantities. A realistic situation is the case when we assume a functional form of the distribution of the microscopic properties, but with some unknown parameters that we would like to determine. In collaboration with A. Obliger and M. Simon, F. Legoll and W. Minvielle have started to address that problem, determining the unknown parameters of the microscopic distribution on the basis of macroscopic (e.g. homogenized) quantities. The preliminary results that have been obtained are very encouraging.

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 an accurate enough approximation).

The MsFEM has been introduced more than 10 years ago. However, even in simple deterministic cases, there is actually still room for improvement in many different directions. In collaboration with A. Lozinski (University of Besançon), F. Legoll and C. Le Bris have introduced and studied a variant of MsFEM that considers Crouzeix-Raviart type elements on each mesh element. The continuity across edges (or facets) of the (multiscale) finite element basis set functions is enforced only weakly, using fluxes rather than point values. The approach has been analyzed (combining classical arguments from homogenization theory and finite element theory) and tested on simple, but highly convincing cases [27] . In particular, an elliptic problem set on a domain with a huge number of perforations has been considered in [37] . The variant developed outperforms all existing variants of MsFEM.

A follow up on this work, in collaboration with U. Hetmaniuk (University of Washington in Seattle) and A. Lozinski (University of Besançon), consists in the study of multiscale advection-diffusion problems. Such problems are possibly advection dominated and a stabilization procedure is therefore required. How stabilization interferes with the multiscale character of the equation is an unsolved mathematical question worth considering for numerical purposes. This is the aim of the PhD thesis of François Madiot, which began in October 2013.

Still another question related to homogenization theory that is investigated in the group is the following. Consider an elliptic equation, say in divergence form, with a highly oscillatory matrix coefficient. Is it possible to approximate the boundary value problem for different right hand sides using a similar problem with a constant matrix coefficient? How can this “best” constant matrix approximating the oscillatory problem be constructed in an efficient manner? We have addressed some of these questions in [25] , where we have in particular shown that this best constant matrix converges to the homogenized matrix, in the limit of infinitely rapidly oscillatory coefficients. Our approach can therefore be considered as an alternative way to compute the homogenized matrix. This is particularly interesting in random cases, where the standard approach is very expensive. Current work is directed towards extending the approach, in order to compute other quantities of interest than the homogenized coefficient.

To conclude this section, we mention the project undertaken by V. Ehrlacher during her six months postdoctoral position in the Cluster of Excellence Engineering of Advanced Materials (Erlangen University). This project, in collaboration with C. Le Bris, F. Legoll, G. Leugering and M. Stingl, aims at optimizing the shape of some materials (modelled as structurally graded linear elastic materials) in order to achieve the best mechanical response at the minimal cost. As often the case in shape optimization, the solution tends to be highly oscillatory, thus the need of homogenization techniques. We thus consider an initial microstructured material composed of steel and void and whose microstructure pattern is periodic (think e.g. of a periodic honeycomb structure). We next consider materials which are obtained from this initial material through a macroscopic deformation, and look for the optimal deformation achieving the best mechanical response. Encouraging first results have been obtained.