A coupling generalized multiscale finite element method for coupled thermomechanical problems

It is crucial to build multiscale modeling for the coupling effects between microstructure and the physical mechanisms in multiphysics problems. In the paper, we develop a coupling formulation of the generalized multiscale finite element method (GMsFEM) to solve coupled thermomechanical problems, and it is referred as the coupling generalized multiscale finite element method (CGMsFEM). The approach consists in defining the coupling multiscale basis functions through local coupling spectral problems in each coarse-grid block, which can be solved by a novel design of two relaxation parameters. Compared to the standard GMsFEM, the proposed strategy can not only accurately capture the multiscale coupling correlation effects of multiphysics problems, but also greatly improve the 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 approach shows better robustness and efficiency than uncoupled GMsFEM.