Consistent Energy-Based Atomistic/Continuum Coupling For Two-Body Potentials In One And Two Dimensions

This paper addresses the problem of consistent energy-based coupling of atomistic and continuum models of materials, limited to zero-temperature statics of simple crystals. It has been widely recognized that the most practical coupled methods exhibit large errors on the atomistic/continuum interface (which are often attributed to spurious forces called "ghost forces"). There are only few existing works that propose a coupling which is sufficiently accurate near the interface under certain limitations. In this paper a novel coupling that is free from "ghost forces" is proposed for a two-body interaction potential under the assumptions of either (i) one spatial dimension, or (ii) two spatial dimensions and piecewise affine finite elements for describing the continuum deformation. The performance of the proposed coupling is demonstrated with numerical experiments. The coupling strategy is based on judiciously defining the contributions of the atomistic bonds to the discrete and the continuum potential energy. The same method in one dimension has been independently developed and analyzed in Li and Luskin [IMA J. Numer. Anal, to appear].