Homogenization and Modeling of Processes in Сomposite Materials with a Layered Structure

Abstract

Mathematical models and asymptotic methods for wave and thermal processes in composite materials consisting of layers with different characteristics will be considered. It is taken into account that the relevant initial-boundary value problems depend on small parameters characterizing the microscale and density of such composite materials. Using the methods of homogenization theory, such problems are approximately reduced to modeling wave and thermal processes in a homogeneous material. Asymptotic expansions and homogenized problems are defined, the solutions of which determine the approximate asymptotics of the solutions of problems for models of wave and thermal processes. Formulas for calculating the characteristics of a homogeneous material and estimating the accuracy of such an approximation are derived. Estimates of the accuracy of such expansions also are presented, which are of great value in numerical calculations and computer simulation for such models with guaranteed accuracy. In addition, such approximations can be useful for understanding wave and thermal processes in devices similar to transformers and capacitors. This approach can be considered as a basis for the study of more general electromagnetic processes modeled by Maxwell's equations with taking into account thermal effects.