Bloch Approximation in Homogenization and Applications

The classical problem of homogenization of elliptic operators with periodically oscillating coefficients is revisited in this paper. As is well known, the homogenization process in a classical framework is concerned with the study of asymptotic behavior of solutions $u^\varepsilon$ of boundary value problems associated with such operators when the period $\varepsilon>0$ of the coefficients is small. In a previous work by C. Conca and M. Vanninathan [SIAM J. Appl. Math., 57 (1997), pp. 1639--1659], a new proof of weak convergence as $\varepsilon\to 0$ towards the homogenized solution was furnished using Bloch wave decomposition. Following the same approach here, we go further and introduce what we call Bloch approximation, which will provide energy norm approximation for the solution $u^\varepsilon$. We develop several of its main features. As a simple application of this new object, we show that it contains both the first and second order correctors. Necessarily, the Bloch approximation will have to capture the oscillations of the solution in a sharper way. The present approach sheds new light and offers an alternative for viewing classical results

Modulational stability of cellular flows

We present here the homogenization of the equations for the initial modulational (large-scale) perturbations of stationary solutions of the two-dimensional Navier–Stokes equations with a time-independent periodic rapidly oscillating forcing. The stationary solutions are cellular flows and they are determined by the stream function phi = sinx1/epsilonsinx2/epsilon+δ cosx1/epsiloncosx2/epsilon, 0 ≤ δ ≤ 1. Two results are given here. For any Reynolds number we prove the homogenization of the linearized equations. For small Reynolds number we prove the homogenization for the fully nonlinear problem. These results show that the modulational stability of cellular flows is determined by the stability of the effective (homogenized) equations

Fourier approach to homogenization problems

This article is divided into two chapters. The classical problem of homogenization of elliptic operators with periodically oscillating coefficients is revisited in the first chapter. Following a Fourier approach, we discuss some of the basic issues of the subject: main convergence theorem, Bloch approximation, estimates on second order derivatives, correctors for the medium, and so on. The second chapter is devoted to the discussion of some non-classical behaviour of vibration problems of periodic structures

Bloch wave homogenization of linear elasticity system

In this article, the homogenization process of periodic structures is analyzed using Bloch waves in the case of system of linear elasticity in three dimensions. The Bloch wave method for homogenization relies on the regularity of the lower Bloch spectrum. For the three dimensional linear elasticity system, the first eigenvalue is degenerate of multiplicity three and hence existence of such a regular Bloch spectrum is not guaranteed. The aim here is to develop all necessary spectral tools to overcome these difficulties. The existence of a directionally regular Bloch spectrum is proved and is used in the homogenization. As a consequence an interesting relation between homogenization process and wave propagation in the homogenized medium is obtained. Existence of a spectral gap for the directionally regular Bloch spectrum is established and as a consequence it is proved that higher modes apart from the first three do not contribute to the homogenization process

Optimization of dispersive coefficients in the homogenization of the wave equation in periodic structures

International audienceWe study dispersive effects of wave propagation in periodic media, which can be modelled by adding a fourth-order term in the homogenized equation. The corresponding fourth-order dispersive tensor is called Burnett tensor and we numerically optimize its values in order to minimize or maximize dispersion. More precisely, we consider the case of a two-phase composite medium with an 8-fold symmetry assumption of the periodicity cell in two space dimensions. We obtain upper and lower bound for the dispersive properties, along with optimal microgeometries