Higher order asymptotic homogenization for dynamical problems

Abstract

In general, asymptotic homogenization methods (AHMs) are based on the hypothesis of perfect scale separation. In practice, this is not always the case. The problem arises of improving the solution in such a way that it becomes applicable if the inhomogeneity parameter is not small. Our study focuses on the higher order asymptotic homogenization for dynamical problems. Systems with continuous and piecewise continuous parameters, discrete systems, and also continuous systems with discrete elements are considered. Both low-frequency and high-frequency vibrations are analyzed. For low-frequency vibrations, several approximations of the AHM are constructed. The influence of the boundary conditions and the system parameters is investigated. The accuracy of the resulting approximation is assessed in numerical simulations.