Fast Two-Scale Analysis via Clustering

Both man-made and natural materials exhibit heterogeneous properties at smaller observation scales. The multiscale analysis allows the inclusion of fine-scale information in coarse-scale simulations. One of the commonly used methods is homogenization, replacing the detailed fine-scale structures with their locally homogeneous effective material properties. When fine-scale material structures are stationary, representative volume elements (RVE) are often identified for their effective material properties to be applied over the entire structure. However, in non-stationary material structures, it is inappropriate to assume a single representative material. In this case, homogenization is often required for every individual cell, resulting in significant increases in computational cost. We propose a stiffness-based clustering algorithm that reduces the total number of homogenization computations needed for multiscale analysis. Cells with similar effective stiffness tensors are clustered together such that only a single homogenization is required for each cluster. Specifically, the clustering algorithm is based on the novel concept of Eigenstiffness, which represents the relative directional stiffness of a given material structure. The rotation invariant property of Eigenstiffness allows material structure with similar intrinsic stiffness but different orientations to be clustered together, further decreasing the number of clusters required for the multiscale analysis. Without a priori knowledge of the accurate homogenized material properties, approximated elasticity tensors and Eigenstiffness estimated through FFT-based homogenization methods are used for rapid clustering. The effectiveness of the method is verified by numerical simulations on various multiscale structures, including Voronoi foams and fiber-reinforced composites.