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]

A variational method for second order shape derivatives

We consider shape functionals obtained as minima on Sobolev spaces of classical integrals having smooth and convex densities, under mixed Dirichlet-Neumann boundary conditions. We propose a new approach for the computation of the second order shape derivative of such functionals, yielding a general existence and representation theorem. In particular, we consider the p-torsional rigidity functional for p grater than or equal to 2.Comment: Submitted paper. 29 page

Plasmonic waves allow perfect transmission through sub-wavelength metallic gratings

We perform a mathematical analysis of the transmission properties of a metallic layer with narrow slits. Our analysis is inspired by recent measurements and numerical calculations that report an unexpected high transmission coefficient of such a structure in a subwavelength regime. We analyze the time harmonic Maxwell’s equations in the H-parallel case for a fixed incident wavelength. Denoting by ? the typical size of the grated structure, we analyze the limit n -> 0 and derive effective equations that take into account the role of plasmonic waves. We obtain a formula for the effective transmission coefficient

Mean field theory for a general class of short-range interaction functionals

In models of $N$ interacting particles in $\R^d$ as in Density Functional Theory or crowd motion, the repulsive cost is usually described by a two-point function c_\e(x,y) =\ell\Big(\frac{|x-y|}{\e}\Big) where $\ell: \R_+ \to [0,\infty]$ is decreasing to zero at infinity and parameter \e>0 scales the interaction distance. In this paper we identify the mean-field energy of such a model in the short-range regime \e\ll 1 under the sole assumption that $\exists r_0>0 \ : \ \int_{r_0}^\infty \ell(r) r^{d-1}\, dr <+\infty$. This extends recent results \cite{hardin2021, HardSerfLebl, Lewin} obtained in the homogeneous case $\ell(r) = r^{-s}$ where $s>d$

Homogenization of Maxwell’s equations with split rings

We analyze the time harmonic Maxwell’s equations in a complex geometry. The scatterer Omega subset R^3 contains a periodic pattern of small wire structures of high conductivity, the single element has the shape of a split ring. We rigorously derive effective equations for the scatterer and provide formulas for the effective permittivity and permeability. The latter turns out to be frequency dependent and has a negative real part for appropriate parameter values. This magnetic activity is the key feature of a left-handed meta-material

On the forces that cable webs under tension can support and how to design cable webs to channel stresses

In many applications of Structural Engineering the following question arises: given a set of forces $\mathbf{f}_1,\mathbf{f}_2,\dots,\mathbf{f}_N$ applied at prescribed points $\mathbf{x}_1,\mathbf{x}_2,\dots,\mathbf{x}_N$, under what constraints on the forces does there exist a truss structure (or wire web) with all elements under tension that supports these forces? Here we provide answer to such a question for any configuration of the terminal points $\mathbf{x}_1,\mathbf{x}_2,\dots,\mathbf{x}_N$ in the two- and three-dimensional case. Specifically, the existence of a web is guaranteed by a necessary and sufficient condition on the loading which corresponds to a finite dimensional linear programming problem. In two-dimensions we show that any such web can be replaced by one in which there are at most $P$ elementary loops, where elementary means the loop cannot be subdivided into subloops, and where $P$ is the number of forces $\mathbf{f}_1,\mathbf{f}_2,\dots,\mathbf{f}_N$ applied at points strictly within the convex hull of $\mathbf{x}_1,\mathbf{x}_2,\dots,\mathbf{x}_N$. In three-dimensions we show that, by slightly perturbing $\mathbf{f}_1,\mathbf{f}_2,\dots,\mathbf{f}_N$, there exists a uniloadable web supporting this loading. Uniloadable means it supports this loading and all positive multiples of it, but not any other loading. Uniloadable webs provide a mechanism for distributing stress in desired ways.Comment: 18 pages, 8 figure

Analyse limite de la diffraction d'ondes électromagnétiques par une structure mince

International audienceno abstrac

A complete-damage problem at small strains

The complete damage of a linearly-responding material that can thus completely disintegrate is addressed at small strains under time-varying Dirichlet boundary conditions as a rate-independent evolution problem in multidimensional situations. The stored energy involves the gradient of the damage variable. This variable as well as the stress and energies are shown to be well defined even under complete damage, in contrast to displacement and strain. Existence of an energetic solution is proved, in particular, by detailed investigating the $\Gamma$-limit of the stored energy and its dependence on boundary conditions. Eventually, the theory is illustrated on a one-dimensional example