Locally periodic unfolding method and two-scale convergence on surfaces of locally periodic microstructures

TL;DR

This paper extends the periodic unfolding and two-scale convergence methods to locally periodic microstructures, enabling analysis of non-periodic perforated domains and deriving macroscopic equations for complex microstructures.

Contribution

It introduces locally periodic unfolding and two-scale convergence on surfaces, broadening the applicability of these methods to non-periodic microstructures.

Findings

Generalized unfolding method for non-periodic microstructures

Derived macroscopic equations for perforated domains

Analyzed differential equations on non-periodic surfaces

Abstract

In this paper we generalize the periodic unfolding method and the notion of two-scale convergence on surfaces of periodic microstructures to locally periodic situations. The methods that we introduce allow us to consider a wide range of non-periodic microstructures, especially to derive macroscopic equations for problems posed in domains with perforations distributed non-periodically. Using the methods of locally periodic two-scale convergence (l-t-s) on oscillating surfaces and the locally periodic (l-p) boundary unfolding operator, we are able to analyze differential equations defined on boundaries of non-periodic microstructures and consider non-homogeneous Neumann conditions on the boundaries of perforations, distributed non-periodically.