Embed to rigorously and accurately homogenise quasi-periodic multi-scale heterogeneous PDEs, with computer algebra

TL;DR

This paper develops a novel, rigorous methodology for homogenizing quasi-periodic heterogeneous PDEs at multiple scales without requiring scale separation, using dynamical systems theory and computer algebra.

Contribution

It introduces a new systematic approach for asymptotic homogenization of quasi-periodic PDEs that works without scale separation and can be constructed to any order.

Findings

Homogenization valid down to lengths twice the microscale.

Method provides correct initial and boundary conditions.

Homogenization applicable to uncertain and forced systems.

Abstract

For microscale heterogeneous PDEs, this article further develops novel theory and methodology for their macroscale mathematical/asymptotic homogenization. This article specifically encompasses the case of quasi-periodic heterogeneity with finite scale separation: no scale separation limit is required. Dynamical systems theory frames the homogenization as a slow manifold of the ensemble of all phase-shifts of the heterogeneity. Depending upon any perceived scale separation within the quasi-periodic heterogeneity, the homogenization may be done in either one step, or two sequential steps: the results are equivalent. The theory not only assures us of the existence and emergence of the homogenization, it also provides a practical systematic method to construct the homogenization to any specified order. For a class of heterogeneities, we show that the macroscale homogenization is potentially valid down to lengths which are just twice that of the microscale heterogeneity! This methodology provides a new rigorous and flexible approach to homogenization that potentially also provides correct initial and boundary conditions, treatment of forcing and control, and analysis of uncertainty.