An analysis of the effect of ghost force oscillation on quasicontinuum error

Dobson, Matthew; Luskin, Mitchell

doi:10.1051/m2an/2009007

Cited by 46 publications

(56 citation statements)

References 24 publications

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“…Since the equilibrium bond length is ǫ, it follows from the above corollary that the width of the interface is O(ǫ|ln ǫ|) (see [22]). Essentially the same result was presented firstly in print by Dobson and Luskin in their recent manuscript [5].…”

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confidence: 72%

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Analysis of a One-Dimensional Nonlocal Quasi-Continuum Method

Ming¹,

Yang²

2009

Multiscale Model. Simul.

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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

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confidence: 72%

“…In this special case, a simpler expression (see Lemma 3.8 below) can be found for the error of the QC method, as was firstly noted by Dobson and Luskin [5] in a slightly different set-up, although we derived this result independently. In the absence of the external force, the atomistic system is at the equilibrium state, i.e., y ǫ = x.…”

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Analysis of a One-Dimensional Nonlocal Quasi-Continuum Method

Ming¹,

Yang²

2009

Multiscale Model. Simul.

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“…Assumption (4.7) is justified since y ′ ξ ≤ r * 2 only under extreme compression, and in that case the second nearest neighbor pair interaction model (2.1) itself can be expected to be invalid. The authors of [31], [13], and [12] all consider energies similar to (2.1), and they all make assumptions similar to (4.7) (see Section 2.3 of [31] for further discussion of this point). …”

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confidence: 99%

Analysis of Energy-Based Blended Quasi-Continuum Approximations

Koten¹,

Luskin²

2011

SIAM J. Numer. Anal.

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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.

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“…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].…”

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Positive Definiteness of the Blended Force-Based Quasicontinuum Method

Li¹,

Luskin²,

Ortner³

2012

Multiscale Model. Simul.

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Abstract. The development of consistent and stable quasicontinuum models for multidimensional crystalline solids remains a challenge. For example, proving the stability of the force-based quasicontinuum (QCF) model [M. Dobson and M. Luskin, M 2AN Math. Model. Numer. Anal., 42 (2008), pp. 113-139] remains an open problem. In one and two dimensions, we show that by blending atomistic and Cauchy-Born continuum forces (instead of a sharp transition as in the QCF method) one obtains positive-definite blended force-based quasicontinuum (B-QCF) models. We establish sharp conditions on the required blending width.Key words. quasicontinuum, atomistic-to-continuum, blending, stability AMS subject classifications. 65Z05, 70C20 DOI. 10.1137/1108592701. 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]. 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].An alternative approach to a/c coupling is the force-based quasicontinuum (QCF) approximation [7,12,13,23,24], but the nonconservative and indefinite equilibrium equations make the iterative solution and the determination of lattice stability more challenging [13,14,15]. Indeed, it is an open problem whether the (sharp-interface) QCF method is stable in dimensions greater than one.Many blended a/c coupling methods have been proposed in the literature; see, e.g., [1,2,3,4,17,22,33,34,40]. In the present work, we formulate a blended force-based quasicontinuum (B-QCF) method, similar to the method proposed in [23], which smoothly blends the forces of the atomistic and continuum model instead of the sharp transition in the QCF method. In one and two dimensions, we establish sharp conditions under which a linearized B-QCF operator is positive definite.Our results have three advantages over the stability result proven in [23]. First, we establish H 1 -stability (instead of H 2 -stability), which opens up the possibility of including defects in the analysis, along the lines of [15,29]. Second, our conditions for

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An analysis of the effect of ghost force oscillation on quasicontinuum error

Cited by 46 publications

References 24 publications

Analysis of a One-Dimensional Nonlocal Quasi-Continuum Method

Analysis of a One-Dimensional Nonlocal Quasi-Continuum Method

Analysis of Energy-Based Blended Quasi-Continuum Approximations

Positive Definiteness of the Blended Force-Based Quasicontinuum Method

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