A temperature‐related homogenization technique and its implementation in the meshfree particle method for nanoscale simulations

Xiao, Shaoping; Yang, Weixuan

doi:10.1002/nme.1841

Cited by 34 publications

(19 citation statements)

References 40 publications

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“…Gauss-Lobatto quadrature (Gunzburger and Zhang, 2010;Yang et al, 2013). Quadrature summation rules sample over all repatoms plus quadrature-type lattice sites comparable to stress-point integration in particle methods (Xiao and Yang, 2007). In addition, one can formulate mixed summation rules having clusters around repatoms or quadrature-type sites, or around both (Gunzburger and Zhang, 2010).…”

Section: General Framework Of Summation Rulesmentioning

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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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: General Framework Of Summation Rulesmentioning

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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“…A major issue in the latter class of problems is the coupling of domains where the modelling is done at different scales, see for instance [1][2][3][4], or the book by Liu et al [5]. Broadly speaking, two classes of coupling methods have been put forward: methods where the coupling is achieved at a discrete interface, and the methods where a zone of a finite size is employed, often called overlap or zonal coupling.…”

Section: Introductionmentioning

confidence: 99%

Energy conservation of atomistic/continuum coupling

Aubertin

Réthoré

Borst

2009

Numerical Meth Engineering

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“…Arroyo and Belytschko [41] developed an atomistic-based continuum theory for nanoscale membranes. Xiao and Yang [42,43] developed a temperature-related Cauchy-Born rule for crystalline solids. Chung and Narnburu [44] proposed a framework involving atoms and continuums.…”

Section: Introductionmentioning

confidence: 99%

Atomistic-based continuum constitutive relation for microtubules: elastic modulus prediction

Jiang

Posner

et al. 2008

Comput Mech

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Mechanical behavior of cells is primarily governed by the cytoskeleton (CSK), a remarkable system of filaments consisting of microtubules, actin filaments and intermediate filaments. This system defines the shape and bulk mechanical properties of the cell. In order to understand how the CSK network influences the mechanical behavior of living cells from a theoretical perspective, the mechanical properties of an individual CSK filament must first be properly described. Existing atomistic simulation methods have computational size limitations; conversely, conventional continuum mechanics lack fundamental nanoscale information. Here a new simulation method is developed that bridges the gap between these two simulation regimes using an atomistic-based continuum constitutive relation for microtubules based on the interatomic potential for proteins and specific atomic structures. This theory is used to predict the elastic modulus of microtubules, which agrees with the range of experimentally measured values without any parameter fitting. The proposed method is applicable to other biopolymers if the subunits are bonded through noncovalent bonds.

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A temperature‐related homogenization technique and its implementation in the meshfree particle method for nanoscale simulations

Cited by 34 publications

References 40 publications

Summation rules for a fully nonlocal energy-based quasicontinuum method

Summation rules for a fully nonlocal energy-based quasicontinuum method

Energy conservation of atomistic/continuum coupling

Atomistic-based continuum constitutive relation for microtubules: elastic modulus prediction

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