Adaptive multiscale model reduction for linear elasticity equation in perforated domains

Abstract

In this paper, we develop a Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) for solving linear elasticity problems in heterogeneous perforated domains. The presence of numerous perforations introduces multiple scales into the computational domain, making direct fine-grid simulations computationally expensive. The proposed method follows the standard offline--online decomposition of CEM-GMsFEM. In the offline stage, local spectral problems are solved on coarse elements to construct auxiliary spaces, and localized energy-minimizing basis functions are then computed on oversampled regions to capture fine-scale geometric information induced by the perforations. In the online stage, residual-driven basis functions are constructed in enlarged coarse neighborhoods to incorporate source-term information and improve the accuracy of the multiscale approximation adaptively. We establish convergence results for both the offline and online stages. In particular, we derive error estimates for the localized multiscale approximation and prove the convergence of the adaptive online enrichment algorithm. Moreover, we show that the oversampling regions used in the online stage can be determined locally, leading to a reduction in computational cost while maintaining convergence properties. Numerical experiments on perforated media with different geometric configurations demonstrate the accuracy and efficiency of the proposed method.