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.

翻译：多物理过程中的耦合效应在多尺度方法设计中常被忽略。耦合作用可能由非正定算子描述，这给多尺度模拟带来了重大挑战。本文基于广义多尺度有限元方法发展了一种正则化耦合多尺度方法，称为耦合广义多尺度有限元方法，用于求解耦合热力问题。该方法通过在每个粗网格块中定义局部正则化耦合谱问题来构建耦合多尺度基函数，这一过程可通过两个弛豫参数的新型设计实现。与标准广义多尺度有限元方法相比，本方法不仅能精确捕捉多物理问题的多尺度耦合相关效应，还能通过更少的基函数大幅提升计算效率。此外，本文建立了收敛性分析并推导了最优误差估计，其中误差上界与弛豫系数的大小无关。通过周期型、随机微结构及随机材料系数等数值算例验证了理论分析结果。数值结果表明，耦合广义多尺度有限元方法比非耦合广义多尺度有限元方法具有更好的鲁棒性和效率。