Two-scale convergence in thin domains with locally periodic rapidly oscillating boundary

The aim of this paper is to adapt the notion of two-scale convergence in $L^p$ to the case of a measure converging to a singular one. We present a specific case when a thin cylinder with locally periodic rapidly oscillating boundary shrinks to a segment, and the corresponding measure charging the cylinder converges to a one-dimensional Lebegues measure of an interval. The method is then applied to the asymptotic analysis of linear elliptic operators with locally periodic coefficients in a thin cylinder with locally periodic rapidly varying thickness.Comment: 13 pages, 1 figur

Homogenization of very thin elastic reticulated structures

This work is devoted to the homogenization of the anisotropic, linearized elasticity system posed on thin reticulated structures involving several parameters. We show that the result depends on the relative size of the parameters. In every case, we obtain a limit problem where both the microscopic and macroscopic scales appear together. From this problem, we get an asymptotic development which gives an approximation in L2 of the displacements and the linearized strain tensor.Ministerio de Ciencia y Tecnologí

Homogenization of random degenerated nonlinear monotone operators

This paper deals with homogenization of random nonlinear monotone operators in divergence form. We assume that the structure conditions (strict monotonicity and continuity conditions) degenerate and are given in terms of a weight function. Under proper integrability assumptions on the weight function we construct the effective operator and prove the homogenization result

Optimization of light structures: the vanishing mass conjecture

International audienceWe consider the shape optimization problem which consists in placing a given mass $m$ of elastic material in a design region so that the compliance is minimal. Having in mind optimal light structures, Our purpose is to show that the problem of finding thestiffest shape configuration simplifies as the total mass $m$ tends to zero: we propose an explicit relaxed formulation where the complianceappears after rescaling as a convex functional of the relative density of mass. This allows us to write necessary and sufficient optimality conditions for light structures following the Monge-Kantorovich approach developed recently in [5]

Homogenization of networks in domains with oscillating boundaries

We consider the asymptotic behaviour of integral energies with convex integrands defined on one-dimensional networks contained in a region of the three-dimensional space with a fast-oscillating boundary as the period of the oscillation tends to zero, keeping the oscillation themselves of fixed size. The limit energy, obtained as a $\Gamma$-limit with respect to an appropriate convergence, is defined in a `stratified' Sobolev space and is written as an integral functional depending on all, two or just one derivative, depending on the connectedness properties of the sublevels of the function describing the profile of the oscillations. In the three cases, the energy function is characterized through an usual homogenization formula for $p$-connected networks, a homogenization formula for thin-film networks and a homogenization formula for thin-rod networks, respectivel

Localization of eigenfunctions in a thin cylinder with a locally periodic oscillating boundary

We study a Dirichlet spectral problem for a second-order elliptic operator with locally periodic coefficients in a thin cylinder. The lateral boundary of the cylinder is assumed to be locally periodic. When the thickness of the cylinder ε tends to zero, the eigenvalues are of order ε−2 and described in terms of the first eigenvalue μ(x1) of an auxiliary spectral cell problem parametrized by x1, while the eigenfunctions localize with rate ε