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Computational Mechanics
Erschienen in: Computational Mechanics 4/2015

01.04.2015 | Original Paper

A goal-oriented adaptive procedure for the quasi-continuum method with cluster approximation

verfasst von: Arash Memarnahavandi, Fredrik Larsson, Kenneth Runesson

Erschienen in: Computational Mechanics | Ausgabe 4/2015

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Abstract

We present a strategy for adaptive error control for the quasi-continuum (QC) method applied to molecular statics problems. The QC-method is introduced in two steps: Firstly, introducing QC-interpolation while accounting for the exact summation of all the bond-energies, we compute goal-oriented error estimators in a straight-forward fashion based on the pertinent adjoint (dual) problem. Secondly, for large QC-elements the bond energy and its derivatives are typically computed using an appropriate discrete quadrature using cluster approximations, which introduces a model error. The combined error is estimated approximately based on the same dual problem in conjunction with a hierarchical strategy for approximating the residual. As a model problem, we carry out atomistic-to-continuum homogenization of a graphene monolayer, where the Carbon–Carbon energy bonds are modeled via the Tersoff–Brenner potential, which involves next-nearest neighbor couplings. In particular, we are interested in computing the representative response for an imperfect lattice. Within the goal-oriented framework it becomes natural to choose the macro-scale (continuum) stress as the “quantity of interest”. Two different formulations are adopted: The Basic formulation and the Global formulation. The presented numerical investigation shows the accuracy and robustness of the proposed error estimator and the pertinent adaptive algorithm.
Nächster Artikel Multiscale modeling of polycrystalline materials with Jacobian-free multiscale method (JFMM)

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Fußnoten
1
Note that \(\bar{B}_{ij}\) is a function of atom positions, although no argument list is presented.
 
2
The notion of a representative unit lattice (RUL) is considered representative in terms of the stress-strain response on the macro-level. In this context, an RUL with defects pertain to the situation with defects distributed with the same intensity over a large region periodically. Although we study a discrete structure, we need to relate it to a volume, or area in this case, in order to define effective continuum properties.
 
3
This choice is suitable as long as the solution on the lower scale deviates moderately from its equilibrium, and the state at any moment in time is well defined by \(\bar{\varvec{F}}\).
 
4
The dyadic open product is introduced, defined such that \((\varvec{a}\otimes \varvec{b})_{ij}=a_i b_j\).
 
5
Thus assuming, \(\underline{\varvec{S}}^{\textit{CI}} \mathop {=}\limits ^{!} \underline{\varvec{0}}\).
 
6
For the equation to hold exactly, \(\underline{\varvec{K}}_{\textit{II}}\) would have to be stated as the secant between the approximate and exact solutions, whereof the latter is unknown.
 
7
We note that the dual problem, however, does not involve a non-linear solution.
 
8
This holds for 2D. In 3D, we would consider tetrahedrons.
 
9
Here we note that \(\tilde{\underline{\varvec{S}}}^\mathrm{F}\ne {\underline{\varvec{S}}}^\mathrm{F}\) is possible, depending on how \(\tilde{\underline{\varvec{S}}}\) is enriched.
 
10
Here we introduce the pertinent consistent Jacobian \(\tilde{\underline{\varvec{K}}}_{r^+}\mathop {=}\limits ^\mathrm{def}\mathrm{d} \tilde{\underline{\varvec{R}}}_{r^+}/\mathrm{d}\tilde{\underline{\varvec{x}}}\) computed using the enriched clusters.
 
11
Here, we make the distinction that the adaptive change \(\delta r\) might be different from the hierarchical increase \(\Delta r\) used to compute the approximate error estimator.
 
12
Note that for brevity, these are provided as a guide and the rest of the expressions can be derived accordingly.
 
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Metadaten
Titel
A goal-oriented adaptive procedure for the quasi-continuum method with cluster approximation
verfasst von
Arash Memarnahavandi
Fredrik Larsson
Kenneth Runesson
Publikationsdatum
01.04.2015
Verlag
Springer Berlin Heidelberg
Erschienen in
Computational Mechanics / Ausgabe 4/2015
Print ISSN: 0178-7675
Elektronische ISSN: 1432-0924
DOI
https://doi.org/10.1007/s00466-015-1127-4

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