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A Γ-Convergence Analysis of the Quasicontinuum Method

Español, Malena I. and Kochmann, Dennis M. and Conti, Sergio and Ortiz, Michael (2013) A Γ-Convergence Analysis of the Quasicontinuum Method. Multiscale Modeling and Simulation, 11 (3). pp. 766-794. ISSN 1540-3459. http://resolver.caltech.edu/CaltechAUTHORS:20131104-155146267

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Abstract

We present a Γ-convergence analysis of the quasicontinuum method focused on the behavior of the approximate energy functionals in the continuum limit of a harmonic and defect-free crystal. The analysis shows that, under general conditions of stability and boundedness of the energy, the continuum limit is attained provided that the continuum---e.g., finite-element---approximation spaces are strongly dense in an appropriate topology and provided that the lattice size converges to zero more rapidly than the mesh size. The equicoercivity of the quasicontinuum energy functionals is likewise established with broad generality, which, in conjunction with Γ-convergence, ensures the convergence of the minimizers. We also show under rather general conditions that, for interatomic energies having a clusterwise additive structure, summation or quadrature rules that suitably approximate the local element energies do not affect the continuum limit. Finally, we propose a discrete patch test that provides a practical means of assessing the convergence of quasicontinuum approximations. We demonstrate the utility of the discrete patch test by means of selected examples of application.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1137/120895354DOIArticle
http://epubs.siam.org/doi/abs/10.1137/120895354PublisherArticle
ORCID:
AuthorORCID
Kochmann, Dennis M.0000-0002-9112-6615
Additional Information:© 2013 Society for Industrial and Applied Mathematics. Received by the editors October 16, 2012; accepted for publication (in revised form) May 13, 2013; published electronically August 1, 2013. The first and fourth authors gratefully acknowledge the support of the U.S. National Science Foundation through the Partnership for International Research and Education (PIRE) on Science at the Triple Point Between Mathematics, Mechanics and Materials Science, Award 0967140. The second author gratefully acknowledges the support provided by Germany’s Alexander von Humboldt Stiftung through a Feodor Lynen Postdoctoral Fellowship. This author gratefully acknowledges support provided by the Hausdorff Trimester Program “Mathematical challenges of materials science and condensed matter physics: From quantum mechanics through statistical mechanics to nonlinear PDEs”, Hausdorff Research Institute for Mathematics (HIM), University of Bonn, Bonn, Germany.
Group:GALCIT
Funders:
Funding AgencyGrant Number
NSFOISE-0967140
Alexander von Humboldt FoundationUNSPECIFIED
Hausdorff Research Institute for MathematicsUNSPECIFIED
Subject Keywords:quasicontinuum method, atomistic-to-continuum models, Γ-convergence
Classification Code:AMS subject classifications: 70C20, 74G15, 74G65
Record Number:CaltechAUTHORS:20131104-155146267
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20131104-155146267
Official Citation:A Γ-Convergence Analysis of the Quasicontinuum Method Espan͂ol, M., Kochmann, D., Conti, S., and Ortiz, M. Multiscale Modeling & Simulation 2013 11:3, 766-794
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:42235
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:05 Nov 2013 15:41
Last Modified:14 Sep 2016 00:01

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