Section: New Results

Homogenization

Participants : Virginie Ehrlacher, Marc Josien, Claude Le Bris, Frédéric Legoll, Adrien Lesage, Pierre-Loïk Rothé.

Deterministic non-periodic systems

In homogenization theory, members of the project-team have pursued their ongoing systematic study of perturbations of periodic problems (by local and nonlocal defects). This has been done in two different directions.

For linear elliptic equations, C. Le Bris has written, in collaboration with X. Blanc (Paris Diderot, France) and P.-L. Lions (Collège de France, France), two manuscripts that present a more versatile proof of the existence of a corrector function for periodic problems with local defects, and also extend the results: the first manuscript [34] addresses the case of an equation (or a system) in divergence form, while the second manuscript [12] extends the analysis to advection-diffusion equations.

Second, they have also provided more details on the quality of approximation achieved by their theory. The fact that a corrector exists with suitable properties allows one to quantify the rate of convergence of the two-scale expansion using that corrector to the actual exact solution, as the small homogenization parameter $\epsilon$ vanishes. These works by C. Le Bris, in collaboration with X. Blanc and M. Josien (and in the context of the PhD thesis of the latter), will be presented in a series of manuscripts in preparation. The precise results have been announced in [11] and proven in [33]. A related study [47] has been performed by M. Josien and addresses issues regarding periodic Green functions.

Also in the context of homogenization theory, C. Le Bris and F. Legoll have initiated a collaboration with R. Cottereau (Ecole Centrale and now CNRS Marseille, France). The topic is in some sense a follow-up on both an earlier work of R. Cottereau and the series of works completed by C. Le Bris and F. Legoll in collaboration with K. Li and next S. Lemaire over the years. Schematically, the purpose of the work is to determine the homogenized coefficient for a medium without explicitly performing a homogenization approach nor using a MsFEM type approach. In earlier works, an approximation approach, somewhat engineering-style, was designed. The purpose now is to examine the performance of this approach in the context of the so-called Arlequin method, a very popular method in the mechanical engineering community. One couples a sub-region of the medium where a homogeneous model is employed, along with a complementary sub-region where the original multiscale model is solved explicitly. The coupling is performed using the Arlequin method. Then, one optimizes a suitable criterion so that optimization leads to an homogeneous sub-region indeed described by the homogenized coefficient seeked for. Some numerical analysis questions, together with practical perspectives for computational enhancements of the approach, are currently examined.

Finally, C. Le Bris has informally participated into the supervision of the master thesis of S. Wolf (Ecole Normale Supérieure, Paris, France), and in this context performed some works in interaction with the student and X. Blanc. The purpose is to investigate perturbations of periodic homogenization problems when the perturbation is geometric in nature. The test case considered is that of a domain perforated by holes the locations of which are not necessarily periodic, but only periodic up to a local perturbation. The results proven, on the prototypical Poisson equation, are natural extensions of the celebrated results by J.-L. Lions published in the late 1960s for the periodic case. This provides a proof of concept, showing that perturbations of a periodic geometry are also possible, a fact that will be more thoroughly investigated in the near future within the above mentioned collaboration.

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.

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. A standard approach consists in truncating the space ${ℝ}^d$ to some bounded domain, on which the corrector problem is numerically solved.

In collaboration with B. Stamm (Aachen University, Germany) and S. Xiang (now also at Aachen University, Germany), E. Cancès, V. Ehrlacher and F. Legoll have studied, both from a theoretical and a numerical standpoints, new alternatives for the approximation of the homogenized matrix. They all rely on the use of an embedded corrector problem, previously introduced by the authors, where a finite-size domain made of the highly oscillatory material is embedded in a homogeneous infinite medium whose diffusion coefficients have to be appropriately determined. In [40], they have shown that the different approximations introduced all converge to the homogenized matrix of the medium when the size of the embedded domain goes to infinity. In [41], they present an efficient algorithm for the resolution of such problems for particular heterogeneous materials, based on the reformulation of the embedded corrector problem as an integral equation, which is discretized using spherical harmonics and solved using the fast multipole method.

Besides the averaged behavior of the oscillatory solution $u_{\epsilon }$ on large space scales (which is given by its homogenized limit), a question of interest is to describe how $u_{\epsilon }$ fluctuates. This question is investigated in the PhD thesis of P.-L. Rothé, both from a theoretical and a numerical viewpoints. First, theoretical results have been obtained for a weakly stochastic setting (where the coefficient is the sum of a periodic coefficient and a small random perturbation). It has been shown that, at the first order and when $\epsilon$ is small, the localized fluctuations (characterized by a test function $g$) of $u_{\epsilon }$ are Gaussian. The corresponding variance depends on the localization function $g$, on the right-hand side $f$ of the problem satisfied by $u_{\epsilon }$, and on a fourth order tensor $Q$ which is defined in terms of the corrector. Since the corrector function is challenging to compute, so is $Q$. A numerical approach has hence been designed to approximate $Q$ and its convergence has been proven. Second, numerical experiments in more general settings (i.e. full stochastic case) following the same approach have been performed. The results are promising, and consistent with the theoretical results obtained in the weakly stochastic setting. These results are collected in a manuscript in preparation.

In collaboration with T. Hudson (University of Warwick, United Kingdom), F. Legoll and T. Lelièvre have considered in [46] a scalar viscoelastic model in which the constitutive law is random and varies on a lengthscale which is small relative to the overall size of the solid. Using stochastic two-scale convergence, they have obtained the homogenized limit of the evolution, and have demonstrated that, under certain hypotheses, the homogenized model exhibits hysteretic behaviour which persists under asymptotically slow loading. This work is motivated by rate-independent stress-strain hysteresis observed in filled rubber.

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).

During the year, several research tracks have been pursued in this general direction.

The final writing of the various works performed in the context of the PhD thesis of F. Madiot is still ongoing. The issues examined there are on the one hand the application (and adequate adjustment) of MsFEM approaches to the case of an advection-diffusion equation with a dominating convection term posed in a perforated domain, and on the other hand some more general study of a numerical approach based, again in the case of convection-dominated flows, on the introduction of the invariant measure associated to the problem. The final version of the two manuscripts describing the efforts in each of these directions should be completed in a near future.

The MsFEM approach uses a Galerkin approximation of the problem on a pre-computed basis, obtained by solving local problems mimicking the problem at hand at the scale of mesh elements, with carefully chosen right-hand sides and boundary conditions. The initially proposed version of MsFEM uses as basis functions the solutions to these local problems, posed on each mesh element, with null right-hand sides and with the coarse P1 elements as Dirichlet boundary conditions. Various improvements have next been proposed, such as the oversampling variant, which solves local problems on larger domains and restricts their solutions to the considered element. In collaboration with U. Hetmaniuk (University of Washington in Seattle, USA), C. Le Bris, F. Legoll and P.-L. Rothé have introduced and studied a MsFEM method improved differently. They have considered a variant of the classical MsFEM approach with enrichments based on Legendre polynomials, both in the bulk of the mesh elements and on their interfaces. A convergence analysis of this new variant has been performed. Promising numerical results have been obtained. These results are currently being collected in a manuscript in preparation.

One of the perspectives of the team, through the PhD thesis of A. Lesage, is the development of Multiscale Finite Element Methods for thin heterogeneous plates. The fact that one of the dimension of the domain of interest scales as the typical size of the heterogeneities within the material induces theoretical and practical difficulties that have to be carefully taken into account. The first steps of the work of V. Ehrlacher, F. Legoll and A. Lesage, in collaboration with A. Lebée (École des Ponts) have consisted in studying the homogenized limit (and the two-scale expansion) of problems posed on thin heterogeneous plates. The case of a diffusion equation has been first dealt with, while the more challenging case of elasticity is currently under study.