Séance Séminaire

Séminaire des Doctorant·e·s

Asymptotic behavior of the systematic error in stochastic homogenization.

In the physics of materials, we usually assume that every material is homogeneous. However, in several cases, this is not completely true: there are usually small parts of other materials inside (whose average size $s$ is very small compared to the characteristic size of the "main" material). This implies a variation of some physical properties. The goal of the homogenization theory is to derive, when $s$ tends to zero, a limit model corresponding to some homogeneous material, which accurately describes the physical properties of our "main" material. More precisely, the model concerns the electrical conductivity of a material with randomly distributed heterogeneities. The conductivity satisfies an elliptic differential equation whose numerical analysis is difficult due to the parameter $s$. This is why we are interested by the asymptotic when $s$ tends to zero, by answering the following questions : 1) What is the limit of the solution when $s$ goes to 0? Can we identify the problem solved by the limit? 2) Which kind of materiel do we get at the limit? Firstly, we investigate those issues in the particular case of a one-dimensional material. In this case, the elliptic equation is reduced to a second order ODE and the limit can be obtained by a direct computation. Secondly, we give a generalization of those results in higher dimensions. Finally, I will talk about the representative volume element approximation, which is a useful tool to get the solution of the asymptotic problem.