Asymptotic Analysis of Solutions to Transmission Problems in Solids with Many Inclusions

Abstract

We construct an asymptotic approximation to the solution of a transmission problem for a body containing a region occupied by many small inclusions. The cluster of inclusions is characterised by two small parameters that determine the nominal diameter of individual inclusions and their separation within the cluster. These small parameters can be comparable to each other. Remainder estimates of the asymptotic approximation are rigorously justified. Numerical illustrations demonstrate the efficiency of the asymptotic approach when compared with benchmark finite element algorithms.

Series expansion for the effective conductivity of a periodic dilute composite with thermal resistance at the two-phase interface

“…Concerning asymptotic methods for general elliptic problems we mention, e.g., Maz'ya, Nazarov, and Plamenewskij [37,38] and Maz'ya, Movchan, and Nieves [36]. In particular, a uniform asymptotic approximation of Green's kernel for the transmission problem for domains with small inclusions has been obtained in Maz'ya, Movchan, and Nieves [35] and Nieves [45]. Boundary value problems in domains with small holes have been also analyzed with the method of multiscale asymptotic expansions (cf., e.g., Bonnaillie-Noël, Dambrine, Tordeux, and Vial [13] and Bonnaillie-Noël, Dambrine, and Lacave [12]).…”

Interaction of scales for a singularly perturbed degenerating nonlinear Robin problem

“…As examples, we mention the method of matching outer and inner asymptotic expansions of Il'in [19] and the compound asymptotic expansion method of Maz'ya et al [20,21], which allows the treatment of general Douglis-Nirenberg elliptic boundary value problems in domains with perforations and corners. More recently, Maz'ya et al [22] provided asymptotic analysis of Green's kernels in domains with small cavities by applying the method of mesoscale asymptotic approximations (see also the papers [23][24][25][26][27]). Moreover, we refer to Ammari and Kang [28] for several applications to inverse problems and Novotny and Sokołowski [29] for applications to topological optimization.…”

Asymptotic analysis of in-plane dynamic problems for elastic media with rigid clusters of small inclusions

“…Meso-scale approximations have also been shown to be effective in modelling the response of elastic solids with rigid and free clouds of impurities [48,49], low frequency acoustic problems involving rigid defect clusters [50] and steady-state heat conduction in densely packed composites [51]. The approximations have also proven to be useful in modelling flows involving fluids interacting with many small obstacles within narrow spaces [52], important for understanding CO2-sequestration processes.…”