An Online Generalized Multiscale finite element method for heat and mass transfer problem with artificial ground freezing

Abstract

The Online Generalized Multiscale Finite Element Method (Online GMsFEM) is presented in this study for heat and mass transfer problem in heterogeneous media with artificial ground freezing process. The mathematical model is based on the classical Stefan model, which depicts heat transfer with a phase change and includes filtration in a porous media. The model is described by a set of temperature and pressure equations. We employ a finite element method with the fictitious domain method to solve the problem on a fine grid. We apply a model reduction approach based on Online GMsFEM to derive a solution on the coarse grid. We can use the online version of GMsFEM to take less offline multiscale basis functions. We use decoupled offline basis functions built with snapshot space and based on spectral problems in our method. This is the standard approach of basis construction. We calculate additional basis functions in the offline stage to account for artificial ground freezing pipes. We use online multiscale basis functions to get a more precise approximation of phase change. We create an online basis that reduces error using local residual values. The accuracy of standard GMsFEM is greatly improved by using an online approach. Numerical results in a two-dimensional domain with layered heterogeneity are presented. To test the method’s accuracy, we show results from a variety of offline and online basis functions. The results suggest that Online GMsFEM can deliver high-accuracy solutions with minimal processing resources.