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.

翻译：本文针对穿孔域中的多连续介质均质化问题，构建了一个通用框架。穿孔域问题的数值模拟成本高昂，在许多应用中需开发粗网格宏观模型。已有方法包括均质化、多尺度有限元法等。本文基于我们近期提出的框架，设计了面向穿孔域的多连续介质均质化方法。在该框架中，我们根据穿孔尺寸区分不同空间区域：极细穿孔视为一种连续介质，较大穿孔则视为另一种连续介质。通过这种差异化处理，可更精确地预测各穿孔内的流动。我们通过为各连续介质构建具有适当解平均值及梯度约束的单元问题，提出了相应框架。这些单元问题的解被用于多尺度展开，并推导出新颖的多连续介质均质化宏观系统。所提方法适用于无尺度分离的问题。文中展示了双连续介质问题的数值结果，验证了所提方法的准确性。