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.

翻译：构建多物理场问题中微结构与其物理机制之间耦合效应的多尺度建模至关重要。本文提出了一种耦合公式化的广义多尺度有限元方法（GMsFEM）用于求解热力耦合问题，并将其称为耦合广义多尺度有限元方法（CGMsFEM）。该方法通过在每个粗网格块中构造局部耦合谱特征值问题来定义耦合多尺度基函数，并通过创新性地设计两个松弛参数求解该特征值问题。相较于标准GMsFEM，本文所提策略不仅能准确捕捉多物理场问题的多尺度耦合关联效应，还能通过更少的多尺度基函数显著提升计算效率。此外，本文建立了收敛性分析并推导了最优误差估计，其中误差上界与松弛系数的量级无关。通过周期性微结构、随机微结构及随机材料系数等数值算例验证了理论分析的正确性。数值结果表明，CGMsFEM方法比非耦合GMsFEM方法具有更强的鲁棒性和更高的效率。