A Quasicontinuum Method with Optimized Local Maximum-Entropy Interpolation and Heaviside Enrichment for Heterogeneous Lattices

Abstract: Lattice systems are effective for modeling heterogeneous materials, but their computational cost is often prohibitive. The QuasiContinuum (QC) method reduces this cost by interpolating the lattice response over a coarse finite-element mesh, yet material interfaces in heterogeneous systems still require fine discretizations. Enrichment strategies from the eXtended Finite Element Method (XFEM) address this by representing interfaces on nonconforming meshes. In this work, we combine Heaviside enrichment with meshless Local Maximum Entropy (LME) interpolation in the QC framework for heterogeneous lattice systems. We systematically investigate the role of the LME locality parameter and its optimization. The results show that optimized LME interpolation improves displacement accuracy by about one order of magnitude over QC with linear interpolation at the same number of degrees of freedom. In addition, the optimal locality-parameter fields are nonuniform near interfaces and exhibit systematic spatial structure. Based on these observations, we derive simple pattern-based rules that retain much of the benefit of full optimization at a fraction of the computational cost. The approach is demonstrated on three numerical examples.