Meshfree GMsFEM for Elasticity Problems in Multicontinuum Media

This paper presents a meshfree generalized multiscale finite element method (MGMsFEM) for solving elasticity problems in multicontinuum heterogeneous media. The proposed approach constructs multiscale basis functions that capture both the directional coupling within each continuum and the cross-continuum coupling between different physical phases. We investigate two distinct approaches for basis function construction: a split approach, which independently constructs basis functions for each continuum, and a coupled approach, which directly incorporates the inter-continuum coupling. The performance of these methods is evaluated through numerical experiments on two model problems with varying coupling strengths and point cloud distributions. The results demonstrate that the coupled basis functions yield good accuracy, particularly for problems with strong coupling, and that unstructured point clouds adapted to the material properties further enhance the method’s efficiency.