Clustering analysis accounting for boundary conditions

Multiscale methods with efficient and accurate homogenization provide reliable estimates for structural analysis of composites. However, existing homogenization methods relying on periodic or immersed boundary conditions fail to treat complex applications, such as cracked or non-periodic structures. In this work, we introduce a numerical homogenization method, called clustering analysis accounting for boundary conditions. By an efficient algorithm proposed to evaluate the fundamental solutions, we derive discrete governing equations relating strain to polarization stress for an FEM-discretized linear system under displacement or mixed boundary conditions. By clustering, the governing equations are significantly condensed, reducing computational costs. The incorporation of boundary conditions ensures accuracy, which is validated by examples of cracked, non-periodic random fiber-reinforced, and particle-reinforced structures.