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.

Figures (17)

• Figure 1: Schematic illustration of the kinematic variables of an X-braced lattice.
• Figure 2: Local Maximum Entropy interpolation functions with $\gamma^{\mathrm{LME}} = 0.9$ for (a) the center repatom, (b) a boundary repatom, and (c) a corner repatom of the two dimensional domain. The repatoms are indicated by black dots
• Figure 3: Local Maximum Entropy interpolation functions with $\gamma^{\mathrm{LME}} = 4.0$ for (a) the center repatom, (b) a boundary repatom, and (c) a corner repatom of the two dimensional domain. The repatoms are indicated by black dots.
• Figure 4: (a) Square inclusion example with the interface shown in red, the underlying X-braced lattice in grey, and repatoms in black; (b) the corresponding signed distance function; (c) the Heaviside function derived from the signed distance function.
• Figure 5: Enriched LME interpolation functions using the shifted Heaviside sign function with $\gamma = 0.9$ for (a) the center repatom, (b) a boundary repatom, and (c) a corner repatom of the two dimensional domain. The repatoms are indicated by black dots