Multiscale analysis in nonlinear thermal diffusion problems in composite structures

Abstract

Abstract:The aim of this paper is to analyze the asymptotic behavior of the solution of a nonlinear problem arising in the modelling of thermal diffusion in a two-component composite material. We consider, at the microscale, a periodic structure formed by two materials with different thermal properties. We assume that we have nonlinear sources and that at the interface between the two materials the flux is continuous and depends in a dynamical nonlinear way on the jump of the temperature field. We shall be interested in describing the asymptotic behavior of the temperature field in the periodic composite as the small parameter which characterizes the sizes of our two regions tends to zero. We prove that the effective behavior of the solution of this system is governed by a new system, similar to Barenblatt's model, with additional terms capturing the effect of the interfacial barrier, of the dynamical boundary condition, and of the nonlinear sources. PACS

Multiscale analysis of diffusion processes in composite media

“…As u and v are uniquely determined (see [5,10]), the whole sequences of microscopic solutions converge to a solution of the unfolded limit problem and this completes the proof of Theorem 1.…”

“…Our approach, as already mentioned, is based on a different method, the periodic unfolding method, which allows us to avoid the use of extension operators and, hence, to deal with more general media (see Section 2 for a brief discussion concerning the advantages offered by the use of homogenization techniques based on this general method). The results presented in this paper constitute also a generalization of those obtained in [3,4,10,11]. As a matter of fact, some of them were announced, without detailed proofs, in [11].…”

“…In view of the above convergence results, it is obvious how to pass to the limit in the linear terms of (9) defined on Ω × Y 1 and, respectively, Ω × Y 2 (see, for details, [1,[10][11][12]). …”

“…There exists a unique weak solution (u ε , v ε ) of (5), with u ε ∈ L 2 (0, T ; H 1 ∂Ω (Ω ε 1 )), v ε ∈ L 2 (0, T ; H 1 (Ω ε 2 )) (see, for instance, [4,5,9,10,17]). Under the above hypotheses on the data, using Cauchy-Schwartz, Poincaré's, Young's and Gronwall's inequalities, we can obtain suitable energy estimates, independent of ε, for our solution (see [5,9,10,12]). More precisely, if we multiply the first equation in (1) by u ε , the second one by v ε and we integrate formally by parts, we obtain, for 0 < t < T ,…”

“…For results concerning the well posedness of problem (1), we refer to [5,9,10,13]. Unfortunately, it is impossible to solve a complicated model such as (1) that takes into account the fine-structure details of the geometry of the composite medium.…”