Numerical Methods for Multilattices

TL;DR

This paper reviews existing numerical methods for multilattices, introduces a new concurrent macro-to-micro homogenization approach, and discusses extensions to dynamic problems and random materials.

Contribution

It proposes a novel concurrent macro-to-micro homogenization method for multilattices and demonstrates its equivalence to existing approaches.

Findings

Unified mathematical formulation of methods

Equivalence between new and existing methods

Extensions to time-dependent and random materials

Abstract

Among the efficient numerical methods based on atomistic models, the quasicontinuum (QC) method has attracted growing interest in recent years. The QC method was first developed for crystalline materials with Bravais lattice and was later extended to multilattices (Tadmor et al, 1999). Another existing numerical approach to modeling multilattices is homogenization. In the present paper we review the existing numerical methods for multilattices and propose another concurrent macro-to-micro method in the numerical homogenization framework. We give a unified mathematical formulation of the new and the existing methods and show their equivalence. We then consider extensions of the proposed method to time-dependent problems and to random materials.