Regularized coupling multiscale method for thermomechanical coupled problems

TL;DR

This paper introduces a regularized multiscale finite element method for coupled thermomechanical problems, improving accuracy and efficiency by capturing multiscale coupling effects and providing robust convergence analysis.

Contribution

The paper develops a novel coupling generalized multiscale finite element method (CGMsFEM) with regularized spectral problems and relaxation parameters, enhancing multiscale simulation accuracy and computational efficiency.

Findings

Accurately captures multiscale coupling effects in thermomechanical problems.

Achieves better computational efficiency with fewer basis functions.

Demonstrates robustness and efficiency over uncoupled GMsFEM.

Abstract

The coupling effects in multiphysics processes are often neglected in designing multiscale methods. The coupling may be described by a non-positive definite operator, which in turn brings significant challenges in multiscale simulations. In the paper, we develop a regularized coupling multiscale method based on the generalized multiscale finite element method (GMsFEM) to solve coupled thermomechanical problems, and it is referred to as the coupling generalized multiscale finite element method (CGMsFEM). The method consists of defining the coupling multiscale basis functions through local regularized coupling spectral problems in each coarse-grid block, which can be implemented by a novel design of two relaxation parameters. Compared to the standard GMsFEM, the proposed method can not only accurately capture the multiscale coupling correlation effects of multiphysics problems but also greatly improve computational efficiency with fewer multiscale basis functions. In addition, the convergence analysis is also established, and the optimal error estimates are derived, where the upper bound of errors is independent of the magnitude of the relaxation coefficient. Several numerical examples for periodic, random microstructure, and random material coefficients are presented to validate the theoretical analysis. The numerical results show that the CGMsFEM shows better robustness and efficiency than uncoupled GMsFEM.