Multicontinuum homogenization in perforated domains

TL;DR

This paper introduces a multicontinuum homogenization framework for perforated domains, enabling accurate flow predictions across different perforation sizes without requiring scale separation.

Contribution

It develops a novel multicontinuum homogenization method based on cell problems tailored for various perforation sizes, improving accuracy over existing approaches.

Findings

Accurate flow predictions in perforated domains.

Effective handling of multiple perforation scales.

Validation through numerical experiments.

Abstract

In this paper, we develop a general framework for multicontinuum homogenization in perforated domains. The simulations of problems in perforated domains are expensive and, in many applications, coarse-grid macroscopic models are developed. Many previous approaches include homogenization, multiscale finite element methods, and so on. In our paper, we design multicontinuum homogenization based on our recently proposed framework. In this setting, we distinguish different spatial regions in perforations based on their sizes. For example, very thin perforations are considered as one continua, while larger perforations are considered as another continua. By differentiating perforations in this way, we are able to predict flows in each of them more accurately. We present a framework by formulating cell problems for each continuum using appropriate constraints for the solution averages and their gradients. These cell problem solutions are used in a multiscale expansion and in deriving novel macroscopic systems for multicontinuum homogenization. Our proposed approaches are designed for problems without scale separation. We present numerical results for two continuum problems and demonstrate the accuracy of the proposed methods.