406 research outputs found
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
Optimization of light structures: the vanishing mass conjecture
International audienceWe consider the shape optimization problem which consists in placing a given mass 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 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]
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 applied at
prescribed points , 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
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 elementary loops,
where elementary means the loop cannot be subdivided into subloops, and where
is the number of forces
applied at points strictly within the convex hull of
. In three-dimensions we show
that, by slightly perturbing ,
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
Homogenization of nonlocal wire metamaterial via a renormalization approach
It is well known that defining a local refractive index for a metamaterial
requires that the wavelength be large with respect to the scale of its
microscopic structure (generally the period). However, the converse does not
hold. There are simple structures, such as the infinite, perfectly conducting
wire medium, which remain non-local for arbitrarily large wavelength-to-period
ratios. In this work we extend these results to the more realistic and relevant
case of finite wire media with finite conductivity. In the quasi-static regime
the metamaterial is described by a non-local permittivity which is obtained
analytically using a two-scale renormalization approach. Its accuracy is tested
and confirmed numerically via full vector 3D finite element calculations.
Moreover, finite wire media exhibit large absorption with small reflection,
while their low fill factor allows considerable freedom to control other
characteristics of the metamaterial such as its mechanical, thermal or chemical
robustness.Comment: 8 pages on two columns, 7 figures, submitted to Phys. Rev.
Realizability of metamaterials with prescribed electric permittivity and magnetic permeability tensors
We show that any pair of real symmetric tensors \BGve and \BGm can be
realized as the effective electric permittivity and effective magnetic
permeability of a metamaterial at a given fixed frequency. The construction
starts with two extremely low loss metamaterials, with arbitrarily small
microstructure, whose existence is ensured by the work of Bouchitt{\'e} and
Bourel and Bouchitt\'e and Schweizer, one having at the given frequency a
permittivity tensor with exactly one negative eigenvalue, and a positive
permeability tensor, and the other having a positive permittivity tensor, and a
permeability tensor having exactly one negative eigenvalue. To achieve the
desired effective properties these materials are laminated together in a
hierarchical multiple rank laminate structure, with widely separated length
scales, and varying directions of lamination, but with the largest length scale
still much shorter than the wavelengths and attenuation lengths in the
macroscopic effective medium.Comment: 12 pages, no figure
Mean field theory for a general class of short-range interaction functionals
In models of interacting particles in 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 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
. This
extends recent results \cite{hardin2021, HardSerfLebl, Lewin} obtained in the
homogeneous case where
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