A computational and theoretical investigation of the accuracy of quasicontinuum methods

Brian Van Koten; Xingjie Helen Li; Mitchell Luskin; Christoph Ortner

doi:10.1007/978-3-642-22061-6_3

A computational and theoretical investigation of the accuracy of quasicontinuum methods

Brian Van Koten, Xingjie Helen Li, Mitchell Luskin, Christoph Ortner

School of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

15 Scopus citations

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 onedimensional 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.

Original language	English (US)
Title of host publication	Numerical Analysis of Multiscale Problems
Editors	Thomas Y. Hou, Omar Lakkis, Ivan G. Graham, Robert Scheichl
Publisher	Springer Verlag
Pages	67-96
Number of pages	30
ISBN (Print)	9783642220609
DOIs	https://doi.org/10.1007/978-3-642-22061-6_3
State	Published - 2012
Event	91st LMS Durham Symposium on Numerical Analysis of Multiscale Problems, 2010 - Durham, United Kingdom Duration: Jul 5 2010 → Jul 15 2010

Publication series

Name	Lecture Notes in Computational Science and Engineering
Volume	83
ISSN (Print)	1439-7358

Other

Other	91st LMS Durham Symposium on Numerical Analysis of Multiscale Problems, 2010
Country/Territory	United Kingdom
City	Durham
Period	7/5/10 → 7/15/10

Bibliographical note

Funding Information:
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.

Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2012.

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10.1007/978-3-642-22061-6_3

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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 T. Y. Hou, O. Lakkis, I. G. Graham, & R. Scheichl (Eds.), Numerical Analysis of Multiscale Problems (pp. 67-96). (Lecture Notes in Computational Science and Engineering; Vol. 83). Springer Verlag. https://doi.org/10.1007/978-3-642-22061-6_3

A computational and theoretical investigation of the accuracy of quasicontinuum methods. / Van Koten, Brian; Li, Xingjie Helen; Luskin, Mitchell et al.

Numerical Analysis of Multiscale Problems. ed. / Thomas Y. Hou; Omar Lakkis; Ivan G. Graham; Robert Scheichl. Springer Verlag, 2012. p. 67-96 (Lecture Notes in Computational Science and Engineering; Vol. 83).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Van Koten, B, Li, XH, Luskin, M & Ortner, C 2012, A computational and theoretical investigation of the accuracy of quasicontinuum methods. in TY Hou, O Lakkis, IG Graham & R Scheichl (eds), Numerical Analysis of Multiscale Problems. Lecture Notes in Computational Science and Engineering, vol. 83, Springer Verlag, pp. 67-96, 91st LMS Durham Symposium on Numerical Analysis of Multiscale Problems, 2010, Durham, United Kingdom, 7/5/10. https://doi.org/10.1007/978-3-642-22061-6_3

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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 onedimensional 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.",

author = "{Van Koten}, Brian and Li, {Xingjie Helen} and Mitchell Luskin and Christoph Ortner",

note = "Funding Information: 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. Publisher Copyright: {\textcopyright} Springer-Verlag Berlin Heidelberg 2012.; 91st LMS Durham Symposium on Numerical Analysis of Multiscale Problems, 2010 ; Conference date: 05-07-2010 Through 15-07-2010",

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N2 - 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 onedimensional 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.

AB - 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 onedimensional 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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A computational and theoretical investigation of the accuracy of quasicontinuum methods

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