Polynomial-basis model reduction

Abstract

A preceding companion paper [1] provides an abstract setting for the process of mathematically reducing a complicated mathematical model of a physical system to a more manageable model while preserving, in the reduced model, the appropriate fundamental laws of the original model. The polynomial-basis method of model reduction is developed in this paper using the formalism of the preceding paper as the framework for its construction. The method utilizes projection operators whose range is the span of ordinary polynomials, which serve as basis functions. As a demonstration, the method is applied to the problem of steady heat conduction in an isotropic, heterogeneous material. The method does not make explicit use of either Floquet theory or the assumption of a periodic material microstructure. Upon specialization to periodic media, however, (along with the usually accompanying assumptions) the results reduce to that known for the multiple scales perturbation method, at least to lowest (bulk property) order. In this sense, the method generalizes the multiple scales method of homogenization, since periodicity and Floquet theory are part of the underlying assumptions and implementation of multiple scales. As a special case, an explicit polynomial-basis-method solution for the alternating-layer laminate case is also obtained.