Comptes Rendus
no. 7
Homogenisation in domains with evolving microstructure
[Homogénéisation dans des domaines avec évolution de la micro-structure]

Présenté par : Évariste Sanchez-Palencia

Malte A. Peter1
1 Centre for Industrial Mathematics, FB 3, University of Bremen, Postfach 330 440, 28334 Bremen, Germany
Comptes Rendus. Mécanique, Volume 335 (2007) no. 7, pp. 357-362.

On propose une méthode pour inclure une évolution de la micro-structure en problème d'homogénéisation. L'idée de la méthode est assez générale et peut être employée pour une diversité de problèmes différents. Elle utilise la transformation d'un problème substitué homogénéisé dans un domaine fixe périodique.

We propose a method which accounts for an evolving microstructure in homogenisation problems. The concept of the method is quite general and can be applied to a number of different problems. It makes use of a transformation to a homogenisable substitute problem on a fixed periodic domain.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2007.05.024
Keywords: Homogenization, Multiscale approach, Porous media, Reaction–diffusion, Interfacial exchange
Mot clés : Homogénéisation, Approche multiéchelle, Milieux poreux, Reaction et diffusion, Transport de masse à travers une surface de séparation
Malte A. Peter 1

1 Centre for Industrial Mathematics, FB 3, University of Bremen, Postfach 330 440, 28334 Bremen, Germany 
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Malte A. Peter. Homogenisation in domains with evolving microstructure. Comptes Rendus. Mécanique, Volume 335 (2007) no. 7, pp. 357-362. doi : 10.1016/j.crme.2007.05.024. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2007.05.024/

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[2] E. Sanchez-Palencia Non-Homogeneous Media and Vibration Theory, Springer, 1980

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[6] R.E. Showalter Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, American Mathematical Society, 1997

[7] G. Nguetseng A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., Volume 20 (1989) no. 3, pp. 608-629

[8] G. Allaire Homogenization and two-scale convergence, SIAM J. Math. Anal., Volume 23 (1992) no. 6, pp. 1482-1518

[9] S.A. Meier; M. Böhm A two-scale reaction–diffusion system with time-dependent microstructure, Proc. Appl. Math. Mech., Volume 6 (2006) no. 1, pp. 647-648

[10] T. Arbogast; J. Douglas; U. Hornung Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., Volume 21 (1990) no. 4, pp. 823-836

[11] U. Hornung; W. Jäger Diffusion, convection, adsorption, and reaction of chemicals in porous media, J. Differential Equations, Volume 92 (1991), pp. 199-225

[12] (U. Hornung, ed.), Homogenization and Porous Media, Springer, 1997

[13] M.A. Peter, M. Böhm, Scalings in homogenisation of reaction, diffusion and interfacial exchange in a two-phase medium, in: Proc. Equadiff-11 (in press)

[14] U. Hornung Applications of the homogenization method to flow and transport in porous media, Beijing, China, 8–26 August 1988 (X. Shutie, ed.), World Scientific (1992), pp. 167-222

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