Analysis of a One-Dimensional Nonlocal Quasi-Continuum Method

Abstract

Abstract. The accuracy of the quasicontinuum method is analyzed using a series of models with increasing complexity. It is demonstrated that the existence of the ghost force may lead to large errors. It is also shown that the ghost force removal strategy proposed by E, Lu and Yang leads to a version of the quasicontinuum method with uniform accuracy.Key words. Quasicontinuum method, ghost force, geometrically consistent scheme AMS subject classifications. 65N12, 65N06, 74G20, 74G151. Introduction. The quasicontinuum (QC) method [33] is among the most successful multiscale methods for modeling the mechanical deformation of crystalline solids. It is designed to deal with situations when the crystal is undergoing mostly elastic deformation except at isolated regions with defects. The QC method is usually formulated as an adaptive finite element method. But instead of relying on a continuum model, the QC method is based on an atomistic model. Its main ingredients are: adaptive selection of representative atoms (rep-atoms), with fewer atoms selected in regions with smooth deformation; division of the whole sample into local and nonlocal regions, with the defects covered by the nonlocal regions; and the application of the Cauchy-Born (CB) approximation in the local region as a device for reducing the complexity involved in computing the total energy of the system.The quasicontinuum method has several distinct advantages. First of all, it has a reasonably simple formulation. In fact, it can be considered as a natural extension of adaptive finite element methods in which one simply uses the atomistic model where the mesh is refined to the atomic scale. Secondly, in the QC method, the treatment in different regions is based on the same model, the atomistic model, with the additional Cauchy-Born approximation used in the local region. For this reason, it is also considered to be more seamless than methods that are based on an explicit coupling between continuum and atomistic models. We refer to the review articles [6,21] for a discussion of methods that are based on explicitly coupling atomistic and continuum models.However, this does not mean that the QC method is free of the problems that one encounters when formulating coupled atomistic-continuum methodologies. In some sense, one may also regard the QC method as an example of such a strategy, with the local region playing the role of the continuum region, and the Cauchy-Born nonlinear elasticity model playing the role of the continuum model. In particular, the issue of consistency between the continuum and atomistic models across the coupling interface is very much manifested in the accuracy at the local-nonlocal interface for the QC method. This is the issue we will focus on in this paper. In fact, even though the atomistic models are used in both the local and the nonlocal regions, the CauchyBorn approximation made in the local regions means that the effective model in this

Analysis of Energy-Based Blended Quasi-Continuum Approximations

“…Our error estimate in Theorem 5.1 shows that the ℓ 2 strain error for the BQCE method can be reduced by a factor of k 3/2 where k is the number of atoms in the blending region. This result is suggested by the results of [13,29] on the decay of the coupling error for QCE. It has been shown that the QCE method suffers from a O(1) error in the critical strain at which the lattice loses stability (which models fracture or the formation of a defect) [16].…”

“…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].…”

Positive Definiteness of the Blended Force-Based Quasicontinuum Method

“…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]. The effect of the ghost force on the error in computing the deformation and the lattice stability by the QCE approximation has been analyzed in [8,9,11,26]. The development of more accurate energy-based a/c methods is an ongoing process [5,16,20,31,35,37].…”

A priori error analysis of two force-based atomistic/continuum models of a periodic chain

“…The error of the QCL method is of course dominated by the "defect". It is also very interesting to note that, in this particular problem, the error in the QCE method due to the ghost force [5,23,31] is even larger than the error committed by the QCL method due to describing the "defect" with the continuum model. Repeating the test with κ = 3, which is shown in Figure 5, gives rise to an interesting observation.…”

“…By contrast, force-based coupling mechanisms [4,16,15,31] are comparatively simple in their formulation and, more importantly, they do not suffer from the interfacial consistency errors exhibited by most energybased methods [5,23]. The force-based quasicontinuum (QCF) method [4] is a prototypical example of a force-based atomistic/continuum hybrid method and is the focus of the present paper.…”