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Homogénéisation et limite de diffusion pour une équation de transport

Grégoire Allaire

Séminaire Équations aux dérivées partielles

Nous étudions l’homogénéisation d’une équation de transport dans un milieu périodique de période ϵ . 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 ϵ , 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,...

Homogenization of the criticality spectral equation in neutron transport

Grégoire AllaireGuillaume Bal — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

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...

Optimal design in small amplitude homogenization

Grégoire AllaireSergio Gutiérrez — 2007

ESAIM: Mathematical Modelling and Numerical Analysis

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...

Homogenization of periodic non self-adjoint problems with large drift and potential

Grégoire AllaireRafael Orive — 2007

ESAIM: Control, Optimisation and Calculus of Variations

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 ε - 2 and the drift or first-order term is scaled as ε - 1 . Under a structural hypothesis on the first cell eigenvalue, which is assumed to admit a unique minimum in the...

Boundary layer tails in periodic homogenization

Grégoire AllaireMicol Amar — 2010

ESAIM: Control, Optimisation and Calculus of Variations

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...

Modelling and simulation of liquid-vapor phase transition in compressible flows based on thermodynamical equilibrium

Gloria FaccanoniSamuel KokhGrégoire Allaire — 2012

ESAIM: Mathematical Modelling and Numerical Analysis

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...

Homogenization and localization in locally periodic transport

Grégoire AllaireGuillaume BalVincent Siess — 2002

ESAIM: Control, Optimisation and Calculus of Variations

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 ε -periodic functions modulated by a macroscopic variable, where ε is a small parameter. The mean free path of the particles is also...

A brief introduction to homogenization and miscellaneous applications

Grégoire Allaire — 2012

ESAIM: Proceedings

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...

Homogenization and localization in locally periodic transport

Grégoire AllaireGuillaume BalVincent Siess — 2010

ESAIM: Control, Optimisation and Calculus of Variations

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 ...

Modelling and simulation of liquid-vapor phase transition in compressible flows based on thermodynamical equilibrium

Gloria FaccanoniSamuel KokhGrégoire Allaire — 2012

ESAIM: Mathematical Modelling and Numerical Analysis

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...

Shape and topology optimization of the robust compliance via the level set method

François JouveGrégoire AllaireFrédéric de Gournay — 2008

ESAIM: Control, Optimisation and Calculus of Variations

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....

Shape and topology optimization of the robust compliance the level set method

Frédéric de GournayGrégoire AllaireFrançois Jouve — 2010

ESAIM: Control, Optimisation and Calculus of Variations

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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