热力耦合问题的正则化耦合多尺度方法
Regularized coupling multiscale method for thermomechanical coupled problems
AI总结：
本文提出正则化耦合多尺度方法（CGMsFEM）求解热力耦合问题，通过局部正则化耦合谱问题和松弛参数设计准确捕捉多尺度耦合效应，提高计算效率，并给出收敛性分析与数值验证。
摘要：
在设计多尺度方法时，多物理过程中的耦合效应常被忽略。这种耦合可能由非正定算子描述，进而给多尺度模拟带来重大挑战。本文基于广义多尺度有限元方法（GMsFEM）开发了一种正则化耦合多尺度方法来求解热力耦合问题，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求解局部正则化耦合谱问题来定义耦合多尺度基函数，并可通过两个松弛参数的新颖设计来实现。与标准GMsFEM相比，该方法不仅能准确捕捉多物理问题的多尺度耦合相关效应，还能以更少的多尺度基函数大幅提高计算效率。此外，本文还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。文中给出了周期性、随机微结构和随机材料系数的几个数值算例以验证理论分析。数值结果表明，CGMsFEM比非耦合GMsFEM表现出更好的鲁棒性和效率。
英文摘要：
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.
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