A Quasicontinuum Method With Optimized Local Maximum-Entropy Interpolation and Heaviside Enrichment for Heterogeneous Lattices
B. Werner, O. Rokos and J. Zeman
International Journal for Numerical Methods in Engineering, 127, e70390 (2026).

ABSTRACT
Lattice systems are indispensable for modeling and analyzing physical phenomena
in materials with discrete or heterogeneous micro- or meso-structures. However,
the computational requirements for practical engineering applications of lattice
systems remain high. The quasicontinuum (QC) method addresses this by reducing
the system of equations using a finite element mesh interpolation, rather than
considering all nodes of a fully resolved lattice. Nevertheless, interfaces
between separate phases in heterogeneous materials, such as concrete, require
fine meshes throughout the domain, diminishing the effectiveness of QC.
Enrichment strategies originally introduced in the extended finite element
method (XFEM) can also account for material interfaces in discrete systems using
nonconforming meshes, thereby resolving this issue. In a previous study, we
applied Heaviside enrichment to investigate concrete mesostructures with the
extended QC method, achieving a tenfold reduction in the number of unknowns
while maintaining similar accuracy compared to discretizations with fully
resolved interfaces. In the present study, we employ the meshless local maximum
entropy (LME) interpolation, which transitions seamlessly from widespread
meshfree to linear basis functions. Additionally, we combine LME interpolation
with Heaviside enrichment and systematically investigate the role of the
locality parameter and its optimization in heterogeneous lattices. This
combination of optimized LME basis functions with Heaviside enrichment leads to
an order-of-magnitude improvement in displacement accuracy while using the same
number of degrees of freedom (DOF) compared to QC with linear interpolation.
Moreover, we identify optimized distributions of the LME locality parameter and
propose simple, non-optimized rules that deliver comparable accuracy at a
fraction of the computational cost. Results from three numerical examples show
that the optimal locality-parameter fields are nonuniform near interfaces and
can be approximated by simple pattern-based rules that retain much of the
benefit of full optimization.