Surface effects in nanoscale structures investigated by a fully-nonlocal energy-based quasicontinuum method

Amelang, Jeffrey S.; Kochmann, Dennis M.

doi:10.1016/j.mechmat.2015.04.004

Cited by 12 publications

(9 citation statements)

References 83 publications

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“…There is clearly room for improvement here but even this rather ad hoc choice performs orders of magnitude better than the other summation rules. A more detailed analysis of optimal values for the effective interaction radii in particular in the presence of free surfaces can be found in Amelang and Kochmann (2015).…”

Section: Local Position and Energy Errorsmentioning

confidence: 99%

“…In contrast, the new second-order rule assigns sampling atoms to the faces of tetrahedral elements, which automatically takes care of free surfaces by representing all surface-nearest lattice sites by those sampling atoms on the surface. We will report details of how surfaces and interfaces can be modeled by the new scheme in Amelang and Kochmann (2015); here, we will show that the second-order scheme can successfully master the combined challenges of free surfaces and microstructure formation.…”

Section: Potentialmentioning

confidence: 99%

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Summation rules for a fully nonlocal energy-based quasicontinuum method

Amelang

Venturini

Kochmann

2015

Journal of the Mechanics and Physics of Solids

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a b s t r a c tThe quasicontinuum (QC) method coarse-grains crystalline atomic ensembles in order to bridge the scales from individual atoms to the micro-and mesoscales. A crucial cornerstone of all QC techniques, summation or quadrature rules efficiently approximate the thermodynamic quantities of interest. Here, we investigate summation rules for a fully nonlocal, energy-based QC method to approximate the total Hamiltonian of a crystalline atomic ensemble by a weighted sum over a small subset of all atoms in the crystal lattice. Our formulation does not conceptually differentiate between atomistic and coarsegrained regions and thus allows for seamless bridging without domain-coupling interfaces. We review traditional summation rules and discuss their strengths and weaknesses with a focus on energy approximation errors and spurious force artifacts. Moreover, we introduce summation rules which produce no residual or spurious force artifacts in centrosymmetric crystals in the large-element limit under arbitrary affine deformations in two dimensions (and marginal force artifacts in three dimensions), while allowing us to seamlessly bridge to full atomistics. Through a comprehensive suite of examples with spatially non-uniform QC discretizations in two and three dimensions, we compare the accuracy of the new scheme to various previous ones. Our results confirm that the new summation rules exhibit significantly smaller force artifacts and energy approximation errors. Our numerical benchmark examples include the calculation of elastic constants from completely random QC meshes and the inhomogeneous deformation of aggressively coarse-grained crystals containing nano-voids. In the elastic regime, we directly compare QC results to those of full atomistics to assess global and local errors in complex QC simulations. Going beyond elasticity, we illustrate the performance of the energy-based QC method with the new second-order summation rule by the help of nanoindentation examples with automatic mesh adaptation. Overall, our findings provide guidelines for the selection of summation rules for the fully nonlocal energy-based QC method.

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Section: Local Position and Energy Errorsmentioning

confidence: 99%

Section: Potentialmentioning

confidence: 99%

Summation rules for a fully nonlocal energy-based quasicontinuum method

Amelang

Venturini

Kochmann

2015

Journal of the Mechanics and Physics of Solids

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“…Here, we briefly summarize the key theoretical concepts at the level required for subsequent discussions. The interested reader is referred to [15,16,32] for further details. As in atomistics, an ensemble of N atoms is described by their positions q = {q 1 , .…”

Section: The Fully-nonlocal Quasicontinuum Methodsmentioning

confidence: 99%

Automatic adaptivity in the fully nonlocal quasicontinuum method for coarse‐grained atomistic simulations

Tembhekar

Amelang

Munk

et al. 2016

Numerical Meth Engineering

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The quasicontinuum (QC) method is a concurrent scale-bridging technique that extends atomistic accuracy to significantly larger length scales by reducing the full atomic ensemble to a small set of representative atoms and using interpolation to recover the motion of all lattice sites where full atomistic resolution is not necessary. While traditional QC methods thereby create interfaces between fully-resolved and coarsegrained regions, the recently introduced fully-nonlocal QC framework does not fundamentally differentiate between atomistic and coarsened domains. Adding adaptive refinement enables us to tie atomistic resolution to evolving regions of interest such as moving defects. However, model adaptivity is challenging because large particle motion is described based on a reference mesh (even in the atomistic regions). Unlike in the context of, e.g., finite element meshes, adaptivity here requires that (i) all vertices lie on a discrete point set (the atomic lattice), (ii) model refinement is performed locally and provides sufficient mesh quality, and (iii) Verlet neighborhood updates in the atomistic domain are performed against a Lagrangian mesh. With the suite of adaptivity tools outlined here, the nonlocal QC method is shown to bridge across scales from atomistics to the continuum in a truly seamless fashion, as illustrated for nanoindentation and void growth.

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“…A wide range of existing approaches have been proposed to solve this issue. In general, existing models may be classified into three groups: (I) enhanced classical continuum mechanics [9,[24][25][26][27][28][29][30][31][32][33][34][35][36][37][38], (II) atomic-based continuum analysis [39][40][41][42][43] and (III) coarse-grained atomistic approaches [44][45][46][47][48].…”

Section: Introductionmentioning

confidence: 99%

“…This aforementioned drawback calls for a powerful multiscale model that may not depend on the continuum framework such as the constitutive stress-strain relationship but pure atomistic description, which motivated the development of coarse-grained atomistic models [45][46][47][48][53][54][55][56][57][58][59] in group III. The nonlocal quasicontinuum (QC) method [45-47, 53-56, 60] is a typical representative method and has been recently applied to study surface effect in nanoscale structures based on the newly proposed summation rules in [45]. Since the proposed summation rules employed in the quasicontinuum framework are specifically designed for linear interpolation shape functions, the consistent extension to high order interpolation shape functions is not clear, as has been clearly observed in [61,62].…”

Section: Introductionmentioning

confidence: 99%

Multiresolution molecular mechanics: Surface effects in nanoscale materials

Yang

2017

Journal of Computational Physics

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Surface effects have been observed to contribute significantly to the mechanical response of nanoscale structures. The newly proposed energy-based coarse-grained atomistic method Multiresolution Molecular Mechanics (MMM) [Q. Yang, A.C. To, Comput. Methods in Appl. Mech. Eng. 283 (2015) 384-418] is applied to capture surface effect for nanosized structures by designing a surface summation rule SR S within the framework of MMM. Combined with previously proposed bulk summation rule SR B , the MMM summation rule SR MMM is completed. SR S and SR B are consistently formed within SR MMM for general finite element shape functions. Analogous to quadrature rules in finite element method (FEM), the key idea to the good performance of SR MMM lies in that the order or distribution of energy for coarse-grained atomistic model is mathematically derived such that the number, position and weight of quadrature-type (sampling) atoms can be determined. Mathematically, the derived energy distribution of surface area is different from that of bulk region. Physically, the difference is due to the fact that surface atoms lack neighboring bonding. As such, SR S and SR B are employed for surface and bulk domains, respectively. Two-and three-dimensional numerical examples using the respective 4-node bilinear quadrilateral, 8-node quadratic quadrilateral and 8-node hexahedral meshes are employed to verify and validate the proposed approach. It is shown that MMM with SR MMM accurately captures corner, edge and surface effects with less 0.3% degrees of freedom of the original atomistic system, compared against full atomistic simulation. The effectiveness of SR MMM with respect to high order element is also demonstrated by employing the 8-node quadratic quadrilateral to solve a beam bending problem considering surface effect. In addition, the introduced sampling error with SR MMM that is analogous to numerical integration error with quadrature rule in FEM is very small.

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Surface effects in nanoscale structures investigated by a fully-nonlocal energy-based quasicontinuum method

Cited by 12 publications

References 83 publications

Summation rules for a fully nonlocal energy-based quasicontinuum method

Summation rules for a fully nonlocal energy-based quasicontinuum method

Automatic adaptivity in the fully nonlocal quasicontinuum method for coarse‐grained atomistic simulations

Multiresolution molecular mechanics: Surface effects in nanoscale materials

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