On Atomistic-to-Continuum Couplings without Ghost Forces in Three Dimensions

Abstract

ABSTRACT. In this paper we construct energy based numerical methods free of ghost forces in three dimensional lattices arising in crystalline materials. The analysis hinges on establishing a connection of the coupled system to conforming finite elements. Key ingredients are: (i) a new representation of discrete derivatives related to long range interactions of atoms as volume integrals of gradients of piecewise linear functions over bond volumes, and (ii) the construction of an underlying globally continuous function representing the coupled modeling method.

Analysis of Boundary Conditions for Crystal Defect Atomistic Simulations

“…Many variants of a/c couplings exist that fit within the above framework [25]. In order to demonstrate how to apply our framework to this setting, we shall restrict ourselves to QNL type schemes [14,35,41], but our discussion applies essentially verbatim to other force-consistent energy-based schemes such as [28,38,39]. For other types of a/c couplings the general framework is still applicable; see in particular [24] for a complete analysis of blending-type a/c methods.…”

“…As a starting point of our present analysis we assume a result that is proven in various forms in the literature, for example, in [28,33,35]: we assume that there exist η > 0 and c 1 > 0 such that there exists a strongly stable solutionū ac R to (24) satisfying…”

“…Briefly, let BZ 3 = Z 3 ∪ (Z 3 + (1/2, 1/2, 1/2) T ) denote a BCC crystal, then both the dislocation core and Burgers vector point in the (1, 1, 1) T direction. Upon rotating and possibly dilating, the projection AZ 2 of the BCC crystal is a triangular lattice, hence we again assume (28). The linear elasticity predictor is now given by u lin (x) = 1 2π arg(x −x), where we assumed that the Burgers vector is b = (0, 0, 1) T andx is the centre of the dislocation core.…”

Approximation of Crystalline Defects at Finite Temperature

“…There has been considerable progress recently in developing a numerical analysis theory of defect computation at zero temperature (we refer to recent a review [13] and more recent papers [5,11,14,17]), as well as new numerical methods whose accuracy could be rigorously quantified [15,16,[19][20][21][22]24]. Efforts towards rigorous analysis of the accuracy of computation of defects at finite temperature, however, are just beginning [10].…”

Force-based atomistic/continuum blending for multilattices

“…Analyses in two and three dimensional Bravais lattices also exist for the AtC method of [26], but this has not yet been extended to allow for defects. Meanwhile, the methods [30,46,45] have been shown to be consistent (or free of ghost forces) for pair potential interactions only.…”