Boundary layer correctors and generalized polarization tensor for periodic rough thin layers. A review for the conductivity problem

Abstract

We study the behaviour of the steady-state voltage potential in a material composed of a two-dimensional object surrounded by a rough thin layer and embedded in an ambient medium. The roughness of the layer is supposed to be ε α-periodic, ε being the magnitude of the mean thickness of the layer, and α a positive parameter describing the degree of roughness. For ε tending to zero, we determine the appropriate boundary layer correctors which lead to approximate transmission conditions equivalent to the effect of the rough thin layer. We also provide an explicit characterization of the polarization tensor as defined by Capdeboscq and Vogelius in [9]. The present paper revisits the previous works of the author [11, 13, 16, 17], and it also provides new results for the very rough case α > 1. Résumé. Dans cet article, nous considérons le problème de conduction dans un domaine bidimensionnel composé d'une fine membrane rugueuse entourant un domaine conducteur, le tout plongé dans un milieu ambiant de conductivité différente. La rugosité de la membrane est supposée ε α-périodique, εétant l'épaisseur moyenne de la membrane, et α un paramètre positif décrivant le degré de rugosité. Nous déterminons des correcteurs de couche limite conduisantà la construction de conditions de transmission approchées lorsque le paramètre ε tend vers zero. Nous donnons aussi une caractérisation explicite du tenseur de polarisation défini par Capdeboscq and Vogelius dans [9]. Cet article revisite des résultats précédents de l'auteur obtenus dans [11,13,16,17], et présente de nouveaux résultats pour le cas très rugueux α > 1.

Derivation and Application of Effective Interface Conditions for Continuum Mechanical Models of Cell Invasion through Thin Membranes

“…Nonlinear partial differential equations describing reaction-diffusion processes and transport phenomena in spatial domains that comprise different parts separated by thin layers (i.e. films or membranes) arise in the mathematical modelling of various chemical, physical and biological systems [1,2,4,8,13,14,15,16,20,21,27,34,36,43,49,57,60,64,65,68,71,72]. Due to the analytical and numerical challenges posed by the presence of such layers [12], it is often convenient to approximate the original problem by an equivalent transmission problem whereby each thin layer is replaced by an effective interface.…”

Improved impedance conditions for a thin layer problem in a nonsmooth domain

“…On the other hand, the case of a random thickness has been investigated; see Basson and Gérard‐Varet in the context of rough surfaces, and Dambrine et al for a practical application of approximate boundary conditions to compute moments of solutions of boundary value problems inside random domains. Let us mention the works on polarization tensor for thin inclusions of rough layers.…”

“…We restrict here ourselves to layers of uniform thickness. For a review on problems with rapidly oscillating layers, see Poignard . See also Delourme et al for an example of mixing homogenization and matched asymptotic expansions.…”

“…Let us mention Abdelkader and Moussa, 22 Bellieud et al, 23 and Bonnet et al 24 for mechanical applications; Perrussel and Poignard, 6 Bendali and Poirier, 25 32 for a practical application of approximate boundary conditions to compute moments of solutions of boundary value problems inside random domains. Let us mention the works [33][34][35] on polarization tensor for thin inclusions of rough layers.…”

Asymptotic expansions and effective boundary conditions: a short review for smooth and nonsmooth geometries with thin layers

“…On the other hand, the case of a random thickness has been investigated, see [9] in the context of rough surfaces, and [23] for a practical application of approximate boundary conditions to compute moments of solutions of boundary value problems inside random domains. Let us mention the works [4,19,49] on polarization tensor for thin inclusions of rough layers.…”