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A Computational and Theoretical Investigation of the Accuracy of Quasicontinuum Methods

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Numerical Analysis of Multiscale Problems

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 83))

Abstract

We give computational results to study the accuracy of several quasicontinuum methods for two benchmark problems – the stability of a Lomer dislocation pair under shear and the stability of a lattice to plastic slip under tensile loading. We find that our theoretical analysis of the accuracy near instabilities for one-dimensional model problems can successfully explain most of the computational results for these multi-dimensional benchmark problems. However, we also observe some clear discrepancies, which suggest the need for additional theoretical analysis and benchmark problems to more thoroughly understand the accuracy of quasicontinuum methods.

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Acknowledgements

This work was supported in part by the National Science Foundation under DMS-0757355, DMS-0811039, the PIRE Grant OISE-0967140, the Institute for Mathematics and Its Applications, and the University of Minnesota Supercomputing Institute. This work was also supported by the Department of Energy under Award Number DE-SC0002085. CO was supported by the EPSRC grant EP/H003096/1 “Analysis of Atomistic-to-Continuum Coupling Methods.”

We wish to thank Ellad Tadmor for helpful discussions.

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Authors and Affiliations

  1. School of Mathematics, University of Minnesota, 206 Church St. SE, Minneapolis, MN, 55455, USA

    Brian Van Koten, Xingjie Helen Li & Mitchell Luskin

  2. Mathematical Institute, St. Giles’ 24–29, Oxford, OX1 3LB, UK

    Christoph Ortner

Authors
  1. Brian Van Koten
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  2. Xingjie Helen Li
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  3. Mitchell Luskin
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  4. Christoph Ortner
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Corresponding author

Correspondence to Mitchell Luskin .

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Editors and Affiliations

  1. , Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom

    Ivan G. Graham

  2. , Applied & Computational Mathematics, California Institute of Technology, Pasadena, 91125, California, USA

    Thomas Y. Hou

  3. , Department of Mathematics, University of Sussex, Brighton, BN1 9QH, United Kingdom

    Omar Lakkis

  4. , Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom

    Robert Scheichl

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Van Koten, B., Li, X.H., Luskin, M., Ortner, C. (2012). A Computational and Theoretical Investigation of the Accuracy of Quasicontinuum Methods. In: Graham, I., Hou, T., Lakkis, O., Scheichl, R. (eds) Numerical Analysis of Multiscale Problems. Lecture Notes in Computational Science and Engineering, vol 83. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22061-6_3

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