Multicontinuum homogenization in perforated domains
arxiv(2024)
摘要
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.
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