Linear stationary iterative methods for the force-based quasicontinuum approximation

Mitchell Luskin; Christoph Ortner

doi:10.1007/978-3-642-21943-6_14

Linear stationary iterative methods for the force-based quasicontinuum approximation

Mitchell Luskin, Christoph Ortner

Mathematics

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

2 Scopus citations

Abstract

Force-based multiphysics coupling methods have become popular since they provide a simple and efficient coupling mechanism, avoiding the difficulties in formulating and implementing a consistent coupling energy. They are also the only known pointwise consistent methods for coupling a general atomistic model to a finite element continuum model. However, the development of efficient and reliable iterative solution methods for the force-based approximation presents a challenge due to the non-symmetric and indefinite structure of the linearized force-based quasicontinuum approximation, as well as to its unusual stability properties. In this paper, we present rigorous numerical analysis and computational experiments to systematically study the stability and convergence rate for a variety of linear stationary iterative methods.

Original language	English (US)
Title of host publication	Numerical Analysis of Multiscale Computations - Proceedings of a Winter Workshop at the Banff International Research Station 2009
Editors	Olof Runborg, Bjorn Engquist, Yen-Hsi R. Tsai
Publisher	Springer Verlag
Pages	331-368
Number of pages	38
ISBN (Print)	9783642219429
DOIs	https://doi.org/10.1007/978-3-642-21943-6_14
State	Published - 2012
Event	Workshop on Numerical Analysis and Multiscale Computations, 2009 - Banff, Canada Duration: Dec 6 2009 → Dec 11 2009

Publication series

Name	Lecture Notes in Computational Science and Engineering
Volume	82
ISSN (Print)	1439-7358

Other

Other	Workshop on Numerical Analysis and Multiscale Computations, 2009
Country	Canada
City	Banff
Period	12/6/09 → 12/11/09

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.” Fig. 4 Graphs of the operator norm kGqcekUk;p, .k;p/ 2 f.1;1/;.2;1/g, for f ixed N D 256;K D 15, F00 D 1, plotted against AF. For the case U1;1 only estimates are available and upper and lower bounds are shown instead (cf. Appendix 7). The graphs confirm the result of Corollary 2 that kGqcekUk;p ! 1 as AF C K 2F00 ! 0. Moreover, they clearly indicate that kGqcekUk;p > 1 already for strains F in the region AF 0:5, which are much lower than the critical strain at which LFqce becomes singular

Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2012.

Access

10.1007/978-3-642-21943-6_14

Cite this

Luskin, M., & Ortner, C. (2012). Linear stationary iterative methods for the force-based quasicontinuum approximation. In O. Runborg, B. Engquist, & Y-H. R. Tsai (Eds.), Numerical Analysis of Multiscale Computations - Proceedings of a Winter Workshop at the Banff International Research Station 2009 (pp. 331-368). (Lecture Notes in Computational Science and Engineering; Vol. 82). Springer Verlag. https://doi.org/10.1007/978-3-642-21943-6_14

Linear stationary iterative methods for the force-based quasicontinuum approximation. / Luskin, Mitchell; Ortner, Christoph.

Numerical Analysis of Multiscale Computations - Proceedings of a Winter Workshop at the Banff International Research Station 2009. ed. / Olof Runborg; Bjorn Engquist; Yen-Hsi R. Tsai. Springer Verlag, 2012. p. 331-368 (Lecture Notes in Computational Science and Engineering; Vol. 82).

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

Luskin, M & Ortner, C 2012, Linear stationary iterative methods for the force-based quasicontinuum approximation. in O Runborg, B Engquist & Y-HR Tsai (eds), Numerical Analysis of Multiscale Computations - Proceedings of a Winter Workshop at the Banff International Research Station 2009. Lecture Notes in Computational Science and Engineering, vol. 82, Springer Verlag, pp. 331-368, Workshop on Numerical Analysis and Multiscale Computations, 2009, Banff, Canada, 12/6/09. https://doi.org/10.1007/978-3-642-21943-6_14

Luskin M, Ortner C. Linear stationary iterative methods for the force-based quasicontinuum approximation. In Runborg O, Engquist B, Tsai Y-HR, editors, Numerical Analysis of Multiscale Computations - Proceedings of a Winter Workshop at the Banff International Research Station 2009. Springer Verlag. 2012. p. 331-368. (Lecture Notes in Computational Science and Engineering). https://doi.org/10.1007/978-3-642-21943-6_14

Luskin, Mitchell ; Ortner, Christoph. / Linear stationary iterative methods for the force-based quasicontinuum approximation. Numerical Analysis of Multiscale Computations - Proceedings of a Winter Workshop at the Banff International Research Station 2009. editor / Olof Runborg ; Bjorn Engquist ; Yen-Hsi R. Tsai. Springer Verlag, 2012. pp. 331-368 (Lecture Notes in Computational Science and Engineering).

@inproceedings{1945ed99096a4eadbda9a7654177e7d2,

title = "Linear stationary iterative methods for the force-based quasicontinuum approximation",

abstract = "Force-based multiphysics coupling methods have become popular since they provide a simple and efficient coupling mechanism, avoiding the difficulties in formulating and implementing a consistent coupling energy. They are also the only known pointwise consistent methods for coupling a general atomistic model to a finite element continuum model. However, the development of efficient and reliable iterative solution methods for the force-based approximation presents a challenge due to the non-symmetric and indefinite structure of the linearized force-based quasicontinuum approximation, as well as to its unusual stability properties. In this paper, we present rigorous numerical analysis and computational experiments to systematically study the stability and convergence rate for a variety of linear stationary iterative methods.",

author = "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.” Fig. 4 Graphs of the operator norm kGqcekUk;p, .k;p/ 2 f.1;1/;.2;1/g, for f ixed N D 256;K D 15, F00 D 1, plotted against AF. For the case U1;1 only estimates are available and upper and lower bounds are shown instead (cf. Appendix 7). The graphs confirm the result of Corollary 2 that kGqcekUk;p ! 1 as AF C K 2F00 ! 0. Moreover, they clearly indicate that kGqcekUk;p > 1 already for strains F in the region AF 0:5, which are much lower than the critical strain at which LFqce becomes singular Publisher Copyright: {\textcopyright} Springer-Verlag Berlin Heidelberg 2012.; Workshop on Numerical Analysis and Multiscale Computations, 2009 ; Conference date: 06-12-2009 Through 11-12-2009",

year = "2012",

doi = "10.1007/978-3-642-21943-6_14",

language = "English (US)",

isbn = "9783642219429",

series = "Lecture Notes in Computational Science and Engineering",

publisher = "Springer Verlag",

pages = "331--368",

editor = "Olof Runborg and Bjorn Engquist and Tsai, {Yen-Hsi R.}",

booktitle = "Numerical Analysis of Multiscale Computations - Proceedings of a Winter Workshop at the Banff International Research Station 2009",

}

TY - GEN

T1 - Linear stationary iterative methods for the force-based quasicontinuum approximation

AU - Luskin, Mitchell

AU - Ortner, Christoph

N1 - 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.” Fig. 4 Graphs of the operator norm kGqcekUk;p, .k;p/ 2 f.1;1/;.2;1/g, for f ixed N D 256;K D 15, F00 D 1, plotted against AF. For the case U1;1 only estimates are available and upper and lower bounds are shown instead (cf. Appendix 7). The graphs confirm the result of Corollary 2 that kGqcekUk;p ! 1 as AF C K 2F00 ! 0. Moreover, they clearly indicate that kGqcekUk;p > 1 already for strains F in the region AF 0:5, which are much lower than the critical strain at which LFqce becomes singular Publisher Copyright: © Springer-Verlag Berlin Heidelberg 2012.

PY - 2012

Y1 - 2012

N2 - Force-based multiphysics coupling methods have become popular since they provide a simple and efficient coupling mechanism, avoiding the difficulties in formulating and implementing a consistent coupling energy. They are also the only known pointwise consistent methods for coupling a general atomistic model to a finite element continuum model. However, the development of efficient and reliable iterative solution methods for the force-based approximation presents a challenge due to the non-symmetric and indefinite structure of the linearized force-based quasicontinuum approximation, as well as to its unusual stability properties. In this paper, we present rigorous numerical analysis and computational experiments to systematically study the stability and convergence rate for a variety of linear stationary iterative methods.

AB - Force-based multiphysics coupling methods have become popular since they provide a simple and efficient coupling mechanism, avoiding the difficulties in formulating and implementing a consistent coupling energy. They are also the only known pointwise consistent methods for coupling a general atomistic model to a finite element continuum model. However, the development of efficient and reliable iterative solution methods for the force-based approximation presents a challenge due to the non-symmetric and indefinite structure of the linearized force-based quasicontinuum approximation, as well as to its unusual stability properties. In this paper, we present rigorous numerical analysis and computational experiments to systematically study the stability and convergence rate for a variety of linear stationary iterative methods.

UR - http://www.scopus.com/inward/record.url?scp=81555197399&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=81555197399&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-21943-6_14

DO - 10.1007/978-3-642-21943-6_14

M3 - Conference contribution

AN - SCOPUS:81555197399

SN - 9783642219429

T3 - Lecture Notes in Computational Science and Engineering

SP - 331

EP - 368

BT - Numerical Analysis of Multiscale Computations - Proceedings of a Winter Workshop at the Banff International Research Station 2009

A2 - Runborg, Olof

A2 - Engquist, Bjorn

A2 - Tsai, Yen-Hsi R.

PB - Springer Verlag

T2 - Workshop on Numerical Analysis and Multiscale Computations, 2009

Y2 - 6 December 2009 through 11 December 2009

ER -

Linear stationary iterative methods for the force-based quasicontinuum approximation

Abstract

Publication series

Other

Bibliographical note

Access

Other files and links

Fingerprint

Cite this