Multicontinuum homogenization in perforated domains

TL;DR

This work tackles efficient simulation of diffusion in perforated domains by introducing a multicontinuum homogenization framework that separates perforations into continua based on size. It builds a macroscopic model by solving constraint cell problems on oversampled regions and performing a multiscale expansion to derive a coupled system for macroscopic variables $U_i$, without assuming scale separation. The method integrates ideas from MsFEM, GMsFEM, and CEM-GMsFEM to construct constrained basis information and produce macroscopic equations in the form of a convection-diffusion-reaction system, with coefficients computed from local problems. Numerical results on Laplace problems with two continua demonstrate accurate prediction of averaged fine-grid solutions and show the practicality of the approach for complex perforated geometries.

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.

Figures (15)

• Figure 1: Illustration of two continua (red and blue).
• Figure 2: Left: Periodic perforated domain $\Omega^{\epsilon}$. Right: Non-periodic perforated domain.
• Figure 3: Illustration.
• Figure 4: Illustration of the computational process for a cell problem.
• Figure 5: Left: single structure 1. Right: single structure 2.