Analysis of Energy-Based Blended Quasi-Continuum Approximations

Abstract

Abstract. The development of patch test consistent quasicontinuum energies for multi-dimensional crystalline solids modeled by many-body potentials remains a challenge. The original quasicontinuum energy (QCE) [28] has been implemented for many-body potentials in two and three space dimensions, but it is not patch test consistent. We propose that by blending the atomistic and corresponding Cauchy-Born continuum models of QCE in an interfacial region with thickness of a small number k of blended atoms, a general quasicontinuum energy (BQCE) can be developed with the potential to significantly improve the accuracy of QCE near lattice instabilities such as dislocation formation and motion.In this paper, we give an error analysis of the blended quasicontinuum energy (BQCE) for a periodic one-dimensional chain of atoms with next-nearest neighbor interactions. Our analysis includes the optimization of the blending function for an improved convergence rate. We show that the ℓ 2 strain error for the non-blended QCE energy (QCE), which has low order O(ε 1/2 ) where ε is the atomistic length scale [13,29], can be reduced by a factor of k 3/2 for an optimized blending function where k is the number of atoms in the blending region. The QCE energy has been further shown to suffer from a O(1) error in the critical strain at which the lattice loses stability [16]. We prove that the error in the critical strain of BQCE can be reduced by a factor of k 2 for an optimized blending function, thus demonstrating that the BQCE energy for an optimized blending function has the potential to give an accurate approximation of the deformation near lattice instabilities such as crack growth.

Positive Definiteness of the Blended Force-Based Quasicontinuum Method

“…The estimate (2.10) is stated for a single connected interface region; however, an analogous result holds if the interface has connected components with comparable width. A similar result can also be found in [19].…”

“…Appropriate extensions of the stability results in this paper can likely be obtained for more general smooth deformations by utilizing the more technical formalism developed, for example, in [19,28,29]. Here and throughout we use the notation φ F := φ (F ) and φ 2F := φ (2F ), where φ is the potential in (2.1).…”

“…Introduction. Atomistic-to-continuum (a/c) coupling methods have been proposed to increase the computational efficiency of atomistic computations involving the interaction between local crystal defects with long-range elastic fields [6,7,16,19,21,24,25,37]. Energy-based methods in this class, such as the quasicontinuum model (denoted QCE [38]), exhibit spurious interfacial forces ("ghost forces") even under uniform strain [8,36].…”

Analysis of blended atomistic/continuum hybrid methods

“…The B-QCE method, as we formulated it, was introduced for one-dimensional lattices in [38], and was later extended to two and three-dimensions in [26] in a formulation which differs only marginally from the one given in (3.1): in [26] the operator Q h defined a trapezoidal rule instead of a midpoint rule. As a matter of fact, all of our results can be adapted to this case.…”

“…Precisely, we will consider (i) the B-QCE scheme formulated in [38,26], which is closely related to methods proposed in [41,2,1]; and (ii) the B-QCF scheme formulated in [24,20,21], which is closely related to methods proposed in [1,2,3,14,22,35,37,41]. While our results are not be immediately applicable to these related schemes [41,2,1,3,14,22,35,37], we expect that many of the techniques we develop can be employed to develop such extensions.…”