Toward local effective parameter theories using multiresolution decomposition

Abstract

Using the recently developed theory of multiresolution decomposition, a formulation that governs the response of a linear, dynamical system with nonstationary, microscale heterogeneity is reduced to two coupled formulations: one governing the response smoothed on an arbitrary chosen reference scale with the response fine details as forcing, and one governing the response detail with the smoothed response as a forcing. By substituting the solution of the latter in the former, a new framework specifically tuned to macroscale variations of the response, in which the effects of the nonstationary, microscale heterogeneity are described via a macroscale-effective material operator, is obtained. Localization of across-scale couplings and the dependence of the smoothed response and the effective material operator on the microscale and macroscale geometries are investigated asymptotically and numerically. The numerical results are the response of a fluid-loaded elastic plate with nonstationary, microscale mass heterogeneity.