Homogenization of a conductive and radiative heat transfer problem

Grégoire Allaire (1, 2) , K. El Ganaoui

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 DeFI - Shape reconstruction and identification

This paper is devoted to the homogenization of a heat conductionproblem in a periodically perforated domain with a nonlinear andnonlocal boundary condition modeling radiative heat transfer inthe perforations. Because of the critical scaling considered it isessential to use a method of two-scale asymptotic expansionsinside the variational formulation of the problem. We obtain anonlinear homogenized problem of heat conduction with effectivecoefficients which are computed via a cell problem featuring aradiative heat transfer boundary condition. We rigorously justifythis homogenization process for the linearized problem by usingtwo-scale convergence. We perform numerical simulations in twodimensions: we reconstruct an approximate temperature field byadding to the homogenized temperature a corrector term. Thecomputed numerical errors agree with the theoretical predictederrors and prove the effectiveness of our method for multiscalesimulation of conductive and radiative heat transfer problems inperiodically perforated domains.

Mots clés: homogenization; two-scale convergence; radiative transfer; heat conduction

Domaines: Analyse numérique [cs.NA]

Grégoire Allaire, K. El Ganaoui. Homogenization of a conductive and radiative heat transfer problem. Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, 2009, 7 (3), pp.1148-1170. ⟨10.1137/080714737⟩. ⟨hal-00784061⟩