Multiscale Problems

The following sections are included:

• Preface

• Contents

The purpose of this series of lectures is to give a short introduction to the theory of Homogenization in the case of periodic problems. The main questions addressed by periodic homogenization are presented and the three “classical” tools are briefly explained. They serve as a general introduction for the various lectures given during the ISFMA Symposium “Multiscale Problems: Theory, Numerical Approximation and Applications”.

These notes give a short presentation of the Periodic Unfolding Method and how it applies to the standard model problem of periodic homogenization. The method, originally introduced in 2002 in [24] is based on two distinct ideas, each leading to a new ingredient.

The first idea is a change of scale, which is embodied in the unfolding operator. At the expense of doubling the dimension, this allows to use standard weak or strong convergence theorems in L^p spaces. Here, it very easy follows that Two-scale Convergence is merely the weak convergence of the unfolding.

The second idea is the separation of scales, which is implemented as a macro-micro decomposition of functions and is especially suited for the weakly convergent sequences of Sobolev spaces.

The application to the standard homogenization problem is easily performed (as it actually follows the same lines as the application of Two-scale Convergence, but in a more explicit and elementary fashion).

The unfolding is particularly well-suited for multiscale problems (a simple backward iteration argument suffices) and for precise corrector results without extra regularity on the data.

These notes are a simplified version of the joint paper with the same title which appeared in SIAM J. Math. Anal. [26].

Deterministic homogenization of stationary Navier-Stokes type equations is studied, on one hand in fixed domains, on the other hand, on variable domains representing porous media. In the first case the macroscopic homogenized equations are still of Navier-Stokes type whereas in the second case the main homogenization result is Darcy's law. Our main tools are the so-called homogenization algebras and the sigma-convergence theory generalizing two-scale convergence.

We describe here the homogenization of the stationnary heat equation in a two-component composite with an ɛ-periodic interface. The heat flow through the interface is proportional by a function of order ɛ^γ to the jump of the temperature field, γ being a real parameter. This condition models an imperfect contact on the interface. We study the asymptotic behaviour of the problem as ɛ tends to zero and obtain different limit problems depending on the value of γ. To do that, we introduce a suitable class of test functions also having a jump on the interface and adapted to the problem.

In this paper, we introduce the decompositions of the displacements of curved rods and shells. We also give sharp estimates of the terms of these decompositions.

In this paper, we introduce a decomposition of the deformations of straight rods and we apply this tool to study the asymptotic behavior of an elastic thin rod in the framework of nonlinear elasticity.

In this paper, we investigate the asymptotic behavior of a multi-structure made of a periodic family of rods in junction with a plate. The model is linearized elasticity. We first consider the case where the plate has a thickness of order 1 with respect to the periodicity ɛ of the rods. The case of a thin plate, with a thickness of order ɛ^2/3, which corresponds to a critical case, is studied.

Mixing materials which present periodic heterogeneous micro structures (inclusions, fibers, lamina) is a classical way to obtain “new materials”. This has been called “the mixture theory”. By using the homogenization method, we rigorously justify the limit models obtained (when the size of the micro structures goes to zero) for composites with very promising properties. More precisely, we apply the method to three different examples of electromagnetic, piezoelectric and elastic materials to illustrate its power. The first example is related to bianisotropic electromagnetic material with memory as a first attempt for modeling photonic crystals. The second one describes a new bio-compatible piezoelectric/living cells material designed to improve bone regeneration. And the last one allows to understand acoustic band-gaps in the case of a strongly heterogeneous elastic composite. Numerical simulations are performed in order to illustrate the influence of the change of intrinsic parameters such as shape and physical characteristics of the micro structures.

Numerical methods coupling macroscopic and microscopic solvers for the efficient solution of partial differential equations with multiple scales have attracted much attention these last few years. Efficient implementations of such methods need a thorough understanding of the subtle interplay between the macroscopic scale (often the scale of interest) and the microscopic scale taking into account the fine-scale behavior of the problem of interest. This paper is concerned with the error analysis of a class of numerical homogenization methods, the so-called heterogeneous multiscale finite element methods. We discuss recent results obtained for the a priori and the a posteriori error analysis of these numerical methods and give a general framework to perform such analyses.