Embed in Ensemble to Rigorously and Accurately Homogenize Quasi-Periodic Multi-Scale Heterogeneous Material
Files
(Published version)
Date
2024
Authors
ROBERTS, A.
Editors
Advisors
Journal Title
Journal ISSN
Volume Title
Type:
Journal article
Citation
Australia and New Zealand Industrial and Applied Mathematics (ANZIAM) Journal, 2024; 66(1):1-34
Statement of Responsibility
Anthony Roberts
Conference Name
Abstract
For microscale heterogeneous partial differential equations (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. A key innovation herein is to analyse the ensemble of all phase-shifts of the heterogeneity. Dynamical systems theory then frames the homogenization as a slow manifold of the ensemble. 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 an exact homogenization at finite scale separation, 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 complements existing well-established results by providing a new rigorous and flexible approach to homogenization that potentially also provides correct macroscale initial and boundary conditions, treatment of forcing and control, and analysis of uncertainty.
School/Discipline
Dissertation Note
Provenance
Description
Access Status
Rights
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.