Analysis of Boundary Conditions for Crystal Defect Atomistic Simulations

Ehrlacher, Virginie; Ortner, Christoph; Shapeev, Alexander V.

doi:10.1007/s00205-016-1019-6

Cited by 81 publications

(332 citation statements)

References 45 publications

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“…After extending results from [20] and employing regularity estimates for the elastic fields generated by crystal defects [12], we are able to also conclude stability of the B-QCF scheme; see § 4.6. Aside from technical conditions, our stability results only require the assumption that the atomistic equilibrium we are aiming to approximate is itself stable, but no assumptions on the magnitude or smoothness of the solution as in [32] or [24] are required.…”

Section: Introductionmentioning

confidence: 67%

“…The atomistic model. We review an atomistic model from [12] for a defect in a homogeneous crystalline environment, which will form the "exact problem" that we will subsequently aim to approximate using atomistic/continuum blending schemes.…”

Section: 2mentioning

confidence: 99%

“…Other types of point defects, such as vacancies, interstitials and impurities, can be modeled (to some extent) by adding an external defect potential P ∈ C 4 (Y ) to the total energy (see [13]). We shall assume throughout that (A.P1) P is localised: there exists R P > 0 so that P depends only on (y(ξ); |ξ| ≤ R P ).…”

Section: 33mentioning

confidence: 99%

“…Conceptually, our analysis is easy to extend to such cases, but we keep our simplifying assumptions for the sake of a convenient notation. We refer to [13] for details of the ideas required to carry out this extension.…”

Section: 33mentioning

confidence: 99%

“…Our starting assumption is that the error analysis ought to be performed in the energynorm as this provides, to the best of our knowledge, the only route at present to include crystal defects in the analysis following [32,12,25].…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 67%

Section: 2mentioning

confidence: 99%

Section: 33mentioning

confidence: 99%

Section: 33mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Analysis of blended atomistic/continuum hybrid methods

et al. 2015

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Fast tensor method for summation of long‐range potentials on 3D lattices with defects

Khoromskaia

Khoromskij

2015

Numerical Linear Algebra App

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In this paper, we present a method for fast summation of long-range potentials on 3D lattices with multiple defects and having non-rectangular geometries, based on rank-structured tensor representations. This is a significant generalization of our recent technique for the grid-based summation of electrostatic potentials on the rectangular L L L lattices by using the canonical tensor decompositions and yielding the O.L/ computational complexity instead of O.L 3 / by traditional approaches. The resulting lattice sum is calculated as a Tucker or canonical representation whose directional vectors are assembled by the 1D summation of the generating vectors for the shifted reference tensor, once precomputed on large N N N representation grid in a 3D bounding box. The tensor numerical treatment of defects is performed in an algebraic way by simple summation of tensors in the canonical or Tucker formats. To diminish the considerable increase in the tensor rank of the resulting potential sum, the "-rank reduction procedure is applied based on the generalized reduced higher-order SVD scheme. For the reduced higher-order SVD approximation to a sum of canonical/Tucker tensors, we prove the stable error bounds in the relative norm in terms of discarded singular values of the side matrices. The required storage scales linearly in the 1D grid-size, O.N /, while the numerical cost is estimated by O.NL/. The approach applies to a general class of kernel functions including those for the Newton, Slater, Yukawa, Lennard-Jones, and dipole-dipole interactions. Numerical tests confirm the efficiency of the presented tensor summation method; we demonstrate that a sum of millions of Newton kernels on a 3D lattice with defects/impurities can be computed in seconds in Matlab implementation. The tensor approach is advantageous in further functional calculus with the lattice potential sums represented on a 3D grid, like integration or differentiation, using tensor arithmetics of 1D complexity. POTENTIAL SUMS ON 3D LATTICES WITH DEFECTS Á denotes the Hadamard (entrywise) product of rank-1 tensors. The expression (3.7) can be calculated in terms of 1D Hadamard and scalar products with linear complexity O.n/.Similar to the case of Galerkin projection onto the well separable basis set, many other

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

Olson

Bochev

Luskin

et al. 2014

Large-Scale Scientific Computing

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Abstract. Atomistic-to-Continuum (AtC) coupling methods are a novel means of computing the properties of a discrete crystal structure, such as those containing defects, that combine the accuracy of an atomistic (fully discrete) model with the efficiency of a continuum model. In this note we extend the optimization-based AtC, formulated in [17] for linear, one-dimensional problems to multi-dimensional settings and arbitrary interatomic potentials. We conjecture optimal error estimates for the multidimensional AtC, outline an implementation procedure, and provide numerical results to corroborate the conjecture for a 1D Lennard-Jones system with next-nearest neighbor interactions.

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Analysis of Boundary Conditions for Crystal Defect Atomistic Simulations

Cited by 81 publications

References 45 publications

Analysis of blended atomistic/continuum hybrid methods

Analysis of blended atomistic/continuum hybrid methods

Fast tensor method for summation of long‐range potentials on 3D lattices with defects

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

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