Nous étudions l’homogénéisation d’une équation de transport dans un milieu périodique de période $\u03f5$. Cette équation est un problème aux valeurs propres qui modélise l’équilibre d’une densité de particules réagissant avec un milieu sous-jacent. Le libre parcours moyen des particules est supposé être aussi de taille $\u03f5$, ce qui entraîne que le modèle limite est une équation de diffusion. Lorsque les coefficients sont purement périodiques, on obtient une équation homogénéisée posée dans tout le domaine,...

This paper is concerned with optimal design problems with a
special assumption on the coefficients of the state equation.
Namely we assume that the variations of these coefficients
have a small amplitude. Then, making an asymptotic expansion
up to second order with respect to the aspect ratio of the
coefficients allows us to greatly simplify the optimal design
problem. By using the notion of -measures we are able to
prove general existence theorems for small amplitude
optimal design and to provide...

We consider the homogenization of both the parabolic and eigenvalue problems for a singularly perturbed
convection-diffusion equation in a periodic medium. All coefficients of the equation may vary both on the
macroscopic scale and on the periodic microscopic scale. Denoting by the period, the potential or zero-order
term is scaled as ${\epsilon}^{-2}$ and the drift or first-order term is scaled as ${\epsilon}^{-1}$. Under a structural
hypothesis on the first cell eigenvalue, which is assumed to admit a unique minimum in the...

This paper focus on the properties of boundary layers
in periodic homogenization in rectangular domains which are either
fixed or have an oscillating boundary. Such boundary layers are
highly oscillating near the boundary and decay exponentially fast
in the interior to a non-zero limit that we call boundary layer
tail. The influence of these boundary layer tails on interior
error estimates is emphasized. They mainly have
two effects (at first order with respect to the period ε): first,
they add...

We address the homogenization of an eigenvalue problem for the neutron transport
equation
in a periodic heterogeneous domain, modeling the criticality study of nuclear
reactor cores.
We prove that the neutron flux, corresponding to the first and unique positive
eigenvector,
can be factorized in the product of two terms, up to a remainder which goes
strongly to zero
with the period. One term is the first eigenvector of the transport equation in the
periodicity cell. The other term is the...

In the present work we investigate the numerical simulation of liquid-vapor phase change
in compressible flows. Each phase is modeled as a compressible fluid equipped with its own
equation of state (EOS). We suppose that inter-phase equilibrium processes in the medium
operate at a short time-scale compared to the other physical phenomena such as convection
or thermal diffusion. This assumption provides an implicit definition of an equilibrium
EOS...

In this paper, we study the homogenization and localization of a spectral transport equation posed in a locally periodic heterogeneous domain. This equation models the equilibrium of particles interacting with an underlying medium in the presence of a creation mechanism such as, for instance, neutrons in nuclear reactors. The physical coefficients of the domain are $\epsilon $-periodic functions modulated by a macroscopic variable, where $\epsilon $ is a small parameter. The mean free path of the particles is also...

This paper is a set of lecture notes for a short introductory course on homogenization.
It covers the basic tools of periodic homogenization (two-scale asymptotic expansions, the
oscillating test function method and two-scale convergence) and briefly describes the main
results of the more general theory of − or
−convergence. Several applications of the method are given: derivation
of Darcy’s law for flows in porous media, derivation of the porosity...

In this paper, we study the homogenization and localization of a
spectral transport equation posed in a locally periodic
heterogeneous domain. This equation models the equilibrium of
particles interacting with an underlying medium in the presence of a
creation mechanism such as, for instance, neutrons in nuclear
reactors. The physical coefficients of the domain are
-periodic functions modulated by a macroscopic variable, where
is a small parameter. The mean free path of the particles is
...

In the present work we investigate the numerical simulation of liquid-vapor phase change
in compressible flows. Each phase is modeled as a compressible fluid equipped with its own
equation of state (EOS). We suppose that inter-phase equilibrium processes in the medium
operate at a short time-scale compared to the other physical phenomena such as convection
or thermal diffusion. This assumption provides an implicit definition of an equilibrium
EOS...

The goal of this paper is to study the so-called worst-case or robust optimal design problem for minimal compliance. In the context of linear elasticity we seek an optimal shape which minimizes the largest, or worst, compliance when the loads are subject to some unknown perturbations. We first prove that, for a fixed shape, there exists indeed a worst perturbation (possibly non unique) that we characterize as the maximizer of a nonlinear energy. We also propose a stable algorithm to compute it....

The goal of this paper is to study the so-called worst-case or robust
optimal design problem for minimal compliance. In the context of linear
elasticity we seek an optimal shape which minimizes the largest, or worst,
compliance when the loads are subject to some unknown perturbations.
We first prove that, for a fixed shape, there exists indeed a worst
perturbation (possibly non unique) that we characterize as the maximizer
of a nonlinear energy. We also propose a stable algorithm to
compute...

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