A Computational and Theoretical Investigation of the Accuracy of Quasicontinuum Methods

Koten, Brian Van; Li, Xingjie Helen; Luskin, Mitchell; Ortner, Christoph

doi:10.1007/978-3-642-22061-6_3

Cited by 20 publications

(22 citation statements)

References 40 publications

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“…We are continuing the development of the BQCE energy by modifying the code developed to study the accuracy of quasicontinuum methods for two benchmark problems -the stability of a Lomer dislocation pair under shear and the stability of a lattice to plastic slip under tensile loading [40]. We note that the potential to significantly improve the accuracy of existing quasicontinuum codes by easily implemented modifications is a very desirable feature of the BQCE approach.…”

Section: Resultsmentioning

confidence: 99%

“…We are investigating the extension of our one-dimensional analysis to the multi-dimensional setting, but we expect that any multi-dimensional analysis would likely be restricted to perturbations from the ground state which are far from lattice instability. We will thus need to rely on our one-dimensional analysis to attempt to understand the computational results from our multi-dimensional benchmark studies [40].…”

Section: Resultsmentioning

confidence: 99%

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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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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Analysis of Energy-Based Blended Quasi-Continuum Approximations

Koten¹,

Luskin²

2011

SIAM J. Numer. Anal.

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“…Finally, it is important to mention that we ignore here the core energy, that is, the energy contribution proportional to | log ε| in (1.6), which comes from the small cores that were removed to obtain J ε . We refer to [27,33,35] for a more detailed discussion of the core energy.…”

Section: Introductionmentioning

confidence: 99%

Dynamics for Systems of Screw Dislocations

Blass¹,

Fonseca²,

Leoni³

et al. 2015

SIAM J. Appl. Math.

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The goal of this paper is the analytical validation of a model of Cermelli and Gurtin [12] for an evolution law for systems of screw dislocations under the assumption of antiplane shear. The motion of the dislocations is restricted to a discrete set of glide directions, which are properties of the material. The evolution law is given by a "maximal dissipation criterion", leading to a system of differential inclusions. Short time existence, uniqueness, cross-slip, and fine cross-slip of solutions are proved.

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“…[25,9,26,2,24,32,21]), and the corresponding coupled methods for crystalline materials (cf. [39,7,23,10,8,4,3,5,6,15,14,36,17,16,27,30,18,19,20,35,28,37,38,43,44,1,22,34,35,42,41]). …”

Section: Introductionmentioning

confidence: 99%

Finite Element Analysis of Cauchy–Born Approximations to Atomistic Models

Makridakis

Süli

2012

Arch Rational Mech Anal

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ABSTRACT. This paper is devoted to a new finite element consistency analysis of Cauchy-Born approximations to atomistic models of crystalline materials in two and three space dimensions. Through this approach new "atomistic Cauchy-Born" models are introduced and analyzed. These intermediate models can be seen as first level atomistic/quasicontinuum approximations in the sense that they involve only short-range interactions. The analysis and the models developed herein are expected to be useful in the design of coupled atomistic/continuum methods in more than one dimension. Taking full advantage of the symmetries of the atomistic lattice we show that the consistency error of the models considered both in energies and in dual W 1,p type norms is O(ε 2 ), where ε denotes the interatomic distance in the lattice.

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A Computational and Theoretical Investigation of the Accuracy of Quasicontinuum Methods

Cited by 20 publications

References 40 publications

Analysis of Energy-Based Blended Quasi-Continuum Approximations

Analysis of Energy-Based Blended Quasi-Continuum Approximations

Dynamics for Systems of Screw Dislocations

Finite Element Analysis of Cauchy–Born Approximations to Atomistic Models

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