Section: New Results

Homogenization

Participants : Michaël Bertin, Ludovic Chamoin, Virginie Ehrlacher, Thomas Hudson, Marc Josien, Claude Le Bris, Frédéric Legoll, François Madiot, Pierre-Loïk Rothé.

Deterministic non-periodic systems

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 (Paris 7) and P.-L. Lions (Collège de France), C. Le Bris has introduced a possible theory, giving rise to a numerical approach, for the simulation of multiscale non-periodic systems. In former publications, several theoretical aspects have been considered, for the case of linear elliptic equations in divergence form. In the context of the PhD thesis of M. Josien, new issues are being explored, including the rate of convergence of the approximation, along with the convergence of the Green functions associated to the problems under consideration. The studies are motivated by several practically relevant problems, in particular the problem of defects in periodic structures and the "twin boundaries" problem in materials science. Also, some other equations than linear elliptic equations in divergence form have been considered lately. The case of advection-diffusion equations is currently examined. In addition, one ongoing work, in collaboration with P. Souganidis (University of Chicago) and P. Cardaliaguet (Université Paris-Dauphine), considers the non-periodic setting for Hamilton-Jacobi type equations.

Stochastic homogenization

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

Using 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.

Over the past years, the project-team has proposed several variance reduction techniques, which have been reviewed and compared to one another in [9], [20]. In particular, in [23], C. Le Bris, F. Legoll and W. Minvielle have investigated the possibility to use a variance reduction technique based on computing the corrector equation only for selected environments. These environments are chosen based on the fact that their statistics in the finite supercell matches the statistics of the materials in the infinite supercell. The efficiency of the approach has been demonstrated for various types of random materials, including composite materials with randomly located inclusions.

Besides the (averaged) behavior of the oscillatory solution $u_{\epsilon }$ on large space scales (which is given by the homogenized limit $u_{*}$ of $u_{\epsilon }$), another question of interest is to understand how much $u_{\epsilon }$ fluctuates around its coarse approximation $u_{*}$. This question will be explored in the PhD thesis of P.-L. Rothé, which started in October 2016.

Still another question investigated in the project-team is to find an alternative to standard homogenization techniques when the latter are difficult to use in practice, because not all the information on the microscopic medium is available. Following an interaction with A. Cohen (Paris 6), C. Le Bris, F. Legoll and S. Lemaire (post-doc in the project-team until 2015), have shown that the constant matrix that “best” (in a sense made precise in [44]) represents the oscillatory matrix describing the medium converges to the homogenized matrix in the limit of infinitely rapidly oscillatory coefficients. Furthermore, the corresponding optimization problem can be efficiently solved using standard algorithms and yield accurate approximation of the homogenized matrix. It has also been shown that it is possible to construct, in a second stage, approximations to the correctors, in order to recover an approximation of the gradient of the solution. The details are now available in [44].

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.

In the context of the PhD thesis of F. Madiot, current efforts are focused on the study of an advection-diffusion equation with a dominating convection in a perforated domain. The multiscale character of the problem stems here from the geometry of the domain. On the boundary of the perforations, we set either homogeneous Dirichlet or homogeneous Neumann conditions. In the spirit of the work [21], the purpose of our ongoing work is to investigate, on perforated domains, the behavior of several variants of the Multiscale Finite Element method, specifically designed to address multiscale advection-diffusion problems in the convection-dominated regime. Generally speaking, the idea of the MsFEM is to perform a Galerkin approximation of the problem using specific basis functions that are precomputed (in an offline stage) and adapted to the problem considered. All the variants considered are based upon local functions satisfying weak continuity conditions in the Crouzeix-Raviart sense on the boundary of mesh elements. Several possibilities for the basis functions have been examined (for instance, they may or may not encode the convection field). Depending on how basis functions are defined, stabilization techniques (such as SUPG) may be required. The type of boundary conditions on the perforations (either homogeneous Dirichlet or homogeneous Neumann boundary conditions) drastically affects the nature of the flow, and therefore the conclusions regarding which numerical approach is best to adopt. In short, homogeneous Dirichlet boundary conditions on the perforations damp the effect of advection, making the flow more stable than it would be in the absence of perforations, while this is not the case for homogeneous Neumann boundary conditions. This intuitive fact is investigated thoroughly at the numerical level, and particularly well exemplified, at the theoretical level, by the comparison of the respective homogenization limits.

Advection-diffusion equations that are both non-coercive and advection-dominated have also been considered (in a single-scale framework). Many numerical approaches have been proposed in the literature to address such difficult cases. C. Le Bris, F. Legoll and F. Madiot have proposed an approach based on the invariant measure associated to the original equation. The approach has been summarized in [22], and extensively described, analyzed and numerically tested in [45]. It is shown there that this approach allows for an unconditionally well-posed finite element approximation, and that it can be stable, as accurate as, and more robust than classical stabilization approaches.

Most of the numerical analysis studies of the MsFEM are focused on obtaining a priori error bounds. In collaboration with L. Chamoin, who was on leave in the project-team (from ENS Cachan, from September 2014 to August 2016), members of the project-team have been working on a posteriori error analysis for MsFEM approaches, with the aim of developing error estimation and adaptation tools. They have extended to the MsFEM case an approach that is classical in the computational mechanics community for single scale problems, and which is based on the so-called Constitutive Relation Error (CRE). Once a numerical solution $u_h$ has been obtained, the approach needs additional computations in order to determine a divergence-free field as close as possible to the exact flux $k\nabla u$. In the context of the MsFEM, it is important to be able to perform all expensive computations in an offline stage, independently of the right-hand side. The standard CRE approach thus needs to be adapted to that context. The proposed approach yields very interesting results, and provides an accurate and robust estimation of the global error. The approach has also been adapted towards the design of adaptive algorithms for specific quantities of interest (in the so-called “goal-oriented” setting), and towards the design of model reduction approaches (such as the Proper Generalized Decomposition (PGD)) in the specific context of multiscale problems. The work will be reported on in a forthcoming publication in preparation.

Discrete systems and their thermodynamic limit

In collaboration with X. Blanc (Paris 7), M. Josien has studied the macroscopic limit of a chain of atoms governed by Newton's equations. It is known from the works of X. Blanc (Paris 7), C. Le Bris and P.-L. Lions (Collège de France) that this limit is the solution of a nonlinear wave equation, as long as the solution remains smooth. For a large class of interaction potentials, X. Blanc and M. Josien have shown in [34], theoretically and numerically, that, if the distance between particles remains bounded, the above description in terms of a non-linear wave equation equation no longer holds when there are shocks. Indeed, the system of particles produces dispersive waves that are not predicted by the nonlinear wave equation.

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. Weertman's equation and the dynamical Peierls-Nabarro equation), involving non-local operators. In collaboration with Y.-P. Pellegrini (CEA), M. Josien has first focused on the steady state regime (where the equation of interest is the Weertman equation), and has designed an efficient numerical method for approximating its solution. The approach is based on a splitting strategy between the nonlinear local terms (which are integrated in real space) and the linear nonlocal terms (which are integrated in Fourier space). Current efforts are devoted to the simulation of physically relevant test-cases, with the aim of comparing the obtained numerical results with results of the physics literature. The work will be reported on in a forthcoming publication in preparation.