Homogenization and localization in locally periodic transport

Grégoire Allaire; Guillaume Bal; Vincent Siess

ESAIM: Control, Optimisation and Calculus of Variations (2002)

  • Volume: 8, page 1-30
  • ISSN: 1292-8119

Abstract

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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 of order ε . We assume that the leading eigenvalue of the periodicity cell problem admits a unique minimum in the domain at a point x 0 where its hessian matrix is positive definite. This assumption yields a concentration phenomenon around x 0 , as ε goes to 0 , at a new scale of the order of ε which is superimposed with the usual ε oscillations of the homogenized limit. More precisely, we prove that the particle density is asymptotically the product of two terms. The first one is the leading eigenvector of a cell transport equation with periodic boundary conditions. The second term is the first eigenvector of a homogenized diffusion equation in the whole space with quadratic potential, rescaled by a factor ε , i.e., of the form exp - 1 2 ε M ( x - x 0 ) · ( x - x 0 ) , where M is a positive definite matrix. Furthermore, the eigenvalue corresponding to this second term gives a first-order correction to the eigenvalue of the heterogeneous spectral transport problem.

How to cite

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Allaire, Grégoire, Bal, Guillaume, and Siess, Vincent. "Homogenization and localization in locally periodic transport." ESAIM: Control, Optimisation and Calculus of Variations 8 (2002): 1-30. <http://eudml.org/doc/244643>.

@article{Allaire2002,
abstract = {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 $\{\varepsilon \}$-periodic functions modulated by a macroscopic variable, where $\{\varepsilon \}$ is a small parameter. The mean free path of the particles is also of order $\{\varepsilon \}$. We assume that the leading eigenvalue of the periodicity cell problem admits a unique minimum in the domain at a point $x_0$ where its hessian matrix is positive definite. This assumption yields a concentration phenomenon around $x_0$, as $\{\varepsilon \}$ goes to $0$, at a new scale of the order of $\sqrt\{\{\varepsilon \}\}$ which is superimposed with the usual $\{\varepsilon \}$ oscillations of the homogenized limit. More precisely, we prove that the particle density is asymptotically the product of two terms. The first one is the leading eigenvector of a cell transport equation with periodic boundary conditions. The second term is the first eigenvector of a homogenized diffusion equation in the whole space with quadratic potential, rescaled by a factor $\sqrt\{\{\varepsilon \}\}$, i.e., of the form $\exp \left(- \frac\{1\}\{2 \{\varepsilon \}\} M (x-x_0)\cdot (x-x_0) \right)$, where $M$ is a positive definite matrix. Furthermore, the eigenvalue corresponding to this second term gives a first-order correction to the eigenvalue of the heterogeneous spectral transport problem.},
author = {Allaire, Grégoire, Bal, Guillaume, Siess, Vincent},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {homogenization; localization; transport; spectral transport equation; concentration phenomenon},
language = {eng},
pages = {1-30},
publisher = {EDP-Sciences},
title = {Homogenization and localization in locally periodic transport},
url = {http://eudml.org/doc/244643},
volume = {8},
year = {2002},
}

TY - JOUR
AU - Allaire, Grégoire
AU - Bal, Guillaume
AU - Siess, Vincent
TI - Homogenization and localization in locally periodic transport
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2002
PB - EDP-Sciences
VL - 8
SP - 1
EP - 30
AB - 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 ${\varepsilon }$-periodic functions modulated by a macroscopic variable, where ${\varepsilon }$ is a small parameter. The mean free path of the particles is also of order ${\varepsilon }$. We assume that the leading eigenvalue of the periodicity cell problem admits a unique minimum in the domain at a point $x_0$ where its hessian matrix is positive definite. This assumption yields a concentration phenomenon around $x_0$, as ${\varepsilon }$ goes to $0$, at a new scale of the order of $\sqrt{{\varepsilon }}$ which is superimposed with the usual ${\varepsilon }$ oscillations of the homogenized limit. More precisely, we prove that the particle density is asymptotically the product of two terms. The first one is the leading eigenvector of a cell transport equation with periodic boundary conditions. The second term is the first eigenvector of a homogenized diffusion equation in the whole space with quadratic potential, rescaled by a factor $\sqrt{{\varepsilon }}$, i.e., of the form $\exp \left(- \frac{1}{2 {\varepsilon }} M (x-x_0)\cdot (x-x_0) \right)$, where $M$ is a positive definite matrix. Furthermore, the eigenvalue corresponding to this second term gives a first-order correction to the eigenvalue of the heterogeneous spectral transport problem.
LA - eng
KW - homogenization; localization; transport; spectral transport equation; concentration phenomenon
UR - http://eudml.org/doc/244643
ER -

References

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