Development of an Optimization-Based Atomistic-to-Continuum Coupling Method

Olson, Derek; Bochev, Pavel B.; Luskin, Mitchell; Shapeev, Alexander V.

doi:10.1007/978-3-662-43880-0_3

Cited by 12 publications

(10 citation statements)

References 22 publications

(32 reference statements)

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“…The coupling error can be minimized but never altogether removed [13,15]. Our analytical results have been numerically confirmed in [23] in one dimension; however, our analysis in the present is restricted to two and three dimensions. This paper is organized as follows.…”

Section: Introductionsupporting

confidence: 54%

“…The purpose of this paper is to analyze an optimization-based AtC, introduced in [23,24], which couches the coupling of the two models into a constrained minimization problem. Specifically, a suitable measure of the mismatch between the atomistic and continuum states, the "mismatch energy," is minimized over a common overlap region, subject to the atomistic and continuum force balance equations holding in atomistic and continuum subdomains.…”

Section: Introductionmentioning

confidence: 99%

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Analysis of an optimization-based atomistic-to-continuum coupling method for point defects

et al. 2015

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Abstract. We formulate and analyze an optimization-based Atomistic-to-Continuum (AtC) coupling method for problems with point defects. Near the defect core the method employs a potential-based atomistic model, which enables accurate simulation of the defect. Away from the core, where site energies become nearly independent of the lattice position, the method switches to a more efficient continuum model. The two models are merged by minimizing the mismatch of their states on an overlap region, subject to the atomistic and continuum force balance equations acting independently in their domains. We prove that the optimization problem is well-posed and establish error estimates. IntroductionAtomistic-to-continuum (AtC) coupling methods combine the accuracy of potential-based atomistic models of solids with the efficiency of coarse-grained continuum elasticity models by using the former only in small regions where the deformation of the material is highly variable such as near a crack tip or dislocation. The past two decades have seen an abundance of interest in AtC methods both in the engineering community to enable predictive simulations of crystalline materials and in the mathematical community to understand the errors introduced by AtC approximations. Of prime importance is the use of AtC methods to model material defects such dislocations and interacting point defects, which play roles in determining the elastic and plastic response of a material [35].A prototypical AtC method is an instance of heterogeneous domain decomposition in which different parts of the domain are treated by different mathematical models. In particular, AtC divides the domain into an atomistic and continuum parts and uses a discrete system involving non-local interactions between atoms on the former and a continuum model, such as hyperelastic continuum mechanics, on the latter.Depending on how these two models are coupled, AtC methods can be broadly classified as as either force or energy-based [22]. Energy-based couplings define a hybrid energy functional as a combination of atomistic and continuum energy functionals, and this hybrid energy functional is then minimized over a class of admissible deformations. Force-based couplings instead derive atomistic and continuum forces from the separate energies and then equilibrate them. We refer to [20,22] for a review of many existing AtC methods. The primary challenge in developing energy-based methods has been the existence of "ghost forces" [20,25] near the interface between the atomistic and continuum regions. These ghost forces may lead to uncontrollable errors in predicted strains, and to date, no method has been implemented that completely eliminates these errors for general many-body potentials and general interface geometry in two and three dimensions. Many force-based methods do not suffer from the perils of ghost forces; however, for two and three dimensions, establishing the stability of these methods in the absence of an energy functional remains a difficult task.Owing to the prac...

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Section: Introductionsupporting

confidence: 54%

Section: Introductionmentioning

confidence: 99%

Analysis of an optimization-based atomistic-to-continuum coupling method for point defects

et al. 2015

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“…In order to preserve mesh regularity (3.8), one would need to impose that h(x) |x|. Note that this does not violate any of our foregoing assumptions for suitable choices of α; see [29] for further discussion.…”

Section: 22mentioning

confidence: 99%

“…The result immediately follows from (4.26), which is proven in § 6.4.1, and from Lemma 4.6. 29) and denote…”

Section: Theorem 47 (Consistency Of B-qcf)mentioning

confidence: 99%

Analysis of blended atomistic/continuum hybrid methods

et al. 2015

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Abstract. We present a comprehensive error analysis of two prototypical atomistic-tocontinuum coupling methods of blending type: the energy-based and the force-based quasicontinuum methods.Our results are valid in two and three dimensions, for finite range many-body interactions (e.g., EAM type), and in the presence of lattice defects (we consider point defects and dislocations). The two key ingredients in the analysis are (i) new force and energy consistency error estimates; and (ii) a new technique for proving energy norm stability of a/c couplings that requires only the assumption that the exact atomistic solution is a stable equilibrium.

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“…In order to balance the sources of error, one should take R c = R 2/d+1 a . Finally, by simply writing the number of degrees of freedom as the sum of those in the atomistic and continuum regions, it is possible to derive the result that #DoF ≈ R d a ; further details can be found in [32,33,28,25]. After making the estimation γ tr ≤ (log DoF) 1/2 [25] for d = 2, the main error estimate, (3.5), currently written in terms of solution regularity, may now be replaced by an estimate of (3.6) in terms of computational cost since #DoF ≈ R d a :…”

Section: 3mentioning

confidence: 99%

Force-based atomistic/continuum blending for multilattices

Olson

Ortner

et al. 2018

Numer. Math.

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We formulate the blended force-based quasicontinuum (BQCF) method for multilattices and develop rigorous error estimates in terms of the approximation parameters: atomistic region, blending region and continuum finite element mesh. Balancing the approximation parameters yields a convergent atomistic/continuum multiscale method for multilattices with point defects, including a rigorous convergence rate in terms of the computational cost. The analysis is illustrated with numerical results for a Stone-Wales defect in graphene.

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Development of an Optimization-Based Atomistic-to-Continuum Coupling Method

Cited by 12 publications

References 22 publications

Analysis of an optimization-based atomistic-to-continuum coupling method for point defects

Analysis of an optimization-based atomistic-to-continuum coupling method for point defects

Analysis of blended atomistic/continuum hybrid methods

Force-based atomistic/continuum blending for multilattices

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