Skip to main content
SpringerLink
Log in

Iterative Solution of the Quasicontinuum Equilibrium Equations with Continuation

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We give an analysis of a continuation algorithm for the numerical solution of the force-based quasicontinuum equations. The approximate solution of the force-based quasicontinuum equations is computed by an iterative method using an energy-based quasicontinuum approximation as the preconditioner.

The analysis presented in this paper is used to determine an efficient strategy for the parameter step size and number of iterations at each parameter value to achieve a solution to a required tolerance. We present computational results for the deformation of a Lennard-Jones chain under tension to demonstrate the necessity of carefully applying continuation to ensure that the computed solution remains in the domain of convergence of the iterative method as the parameter is increased. These results exhibit fracture before the actual load limit if the parameter step size is too large.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arndt, M., Luskin, M.: Goal-oriented atomistic-continuum adaptivity for the quasicontinuum approximation. Int. J. Multiscale Comput. Eng. 5, 407–415 (2007)

    Article  Google Scholar 

  2. Arndt, M., Luskin, M.: Error estimation and atomistic-continuum adaptivity for the quasicontinuum approximation of a Frenkel-Kontorova model. SIAM J. Multiscale Model. Simul. 7, 147–170 (2008)

    Article  MathSciNet  Google Scholar 

  3. Arndt, M., Luskin, M.: Goal-oriented adaptive mesh refinement for the quasicontinuum approximation of a Frenkel-Kontorova model. Comput. Methods Appl. Mech. Eng. (2008, to appear)

  4. Bank, R.E., Rose, D.J.: Analysis of a multilevel iterative method for nonlinear finite element equations. Math. Comput. 39(160), 453–465 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  5. Blanc, X., Le Bris, C., Legoll, F.: Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics. M2AN Math. Model. Numer. Anal. 39(4), 797–826 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Curtin, W., Miller, R.: Atomistic/continuum coupling in computational materials science. Model. Simul. Mater. Sci. Eng. 11(3), R33–R68 (2003)

    Article  Google Scholar 

  7. Dobson, M., Luskin, M.: Analysis of a force-based quasicontinuum method. M2AN Math. Model. Numer. Anal. 42, 113–139 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Doedel, E.: Lecture notes on numerical analysis of bifucation problems, March 1997. Electronic Source: http://cmvl.cs.concordia.ca/publications.html

  9. E, W., Lu, J., Yang, J.: Uniform accuracy of the quasicontinuum method. Phys. Rev. B 74, 214115 (2006)

    Article  Google Scholar 

  10. E, W., Ming, P.: Analysis of the local quasicontinuum method. In: Frontiers and Prospects of Contemporary Applied Mathematics, pp. 18–32. World Scientific, Singapore (2005)

    Google Scholar 

  11. E, W., Ming, P.: Cauchy-born rule and the stabilitiy of crystalline solids: Static problems. Arch. Ration. Mech. Anal. 183, 241–297 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Keller, H.B.: Numerical solution of bifurcation and nonlinear eigenvalue problems. In: Applications of Bifurcation Theory. Proc. Advanced Sem., Univ. Wisconsin, Madison, WI, 1976. Publ. Math. Res. Center, vol. 38, pp. 359–384. Academic Press, New York (1977)

    Google Scholar 

  13. Knap, J., Ortiz, M.: An analysis of the quasicontinuum method. J. Mech. Phys. Solids 49, 1899–1923 (2001)

    Article  MATH  Google Scholar 

  14. Lin, P.: Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model. Math. Comput. 72(242), 657–675 (2003) (electronic)

    MATH  Google Scholar 

  15. Lin, P.: Convergence analysis of a quasi-continuum approximation for a two-dimensional material. SIAM J. Numer. Anal. 45(1), 313–332 (2007)

    MATH  MathSciNet  Google Scholar 

  16. Miller, R., Shilkrot, L., Curtin, W.: A coupled atomistic and discrete dislocation plasticity simulation of nano-indentation into single crystal thin films. Acta Mater. 52(2), 271–284 (2003)

    Article  Google Scholar 

  17. Miller, R., Tadmor, E.: The QC code. http://www.qcmethod.com/

  18. Miller, R., Tadmor, E.: The quasicontinuum method: Overview, applications and current directions. J. Comput. Aided Mater. Des. 9(3), 203–239 (2002)

    Article  Google Scholar 

  19. Oden, J.T., Prudhomme, S., Romkes, A., Bauman, P.: Multi-scale modeling of physical phenomena: Adaptive control of models. SIAM J. Sci. Comput. 28(6), 2359–2389 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Ortner, C., Süli, E.: A-posteriori analysis and adaptive algorithms for the quasicontinuum method in one dimension. Research Report NA-06/13, Oxford University Computing Laboratory (2006)

  21. Ortner, C., Süli, E.: Analysis of a quasicontinuum method in one dimension. M2AN 42, 57–91 (2008)

    Article  MATH  Google Scholar 

  22. Prudhomme, S., Bauman, P.T., Oden, J.T.: Error control for molecular statics problems. Int. J. Multiscale Comput. Eng. 4(5–6), 647–662 (2006)

    Article  Google Scholar 

  23. Rodney, D., Phillips, R.: Structure and strength of dislocation junctions: An atomic level analysis. Phys. Rev. Lett. 82(8), 1704–1707 (1999)

    Article  Google Scholar 

  24. Shimokawa, T., Mortensen, J., Schiotz, J., Jacobsen, K.: Matching conditions in the quasicontinuum method: Removal of the error introduced at the interface between the coarse-grained and fully atomistic regions. Phys. Rev. B 69(21), 214104 (2004)

    Article  Google Scholar 

  25. Tadmor, E., Ortiz, M., Phillips, R.: Quasicontinuum analysis of defects in solids. Philos. Mag. A 73(6), 1529–1563 (1996)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Matthew Dobson & Mitchell Luskin

Authors
  1. Matthew Dobson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mitchell Luskin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mitchell Luskin.

Additional information

This work was supported in part by DMS-0304326, by the University of Minnesota Supercomputing Institute and by the Department of Energy under Award Number DE-FG02-05ER25706.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dobson, M., Luskin, M. Iterative Solution of the Quasicontinuum Equilibrium Equations with Continuation. J Sci Comput 37, 19–41 (2008). https://doi.org/10.1007/s10915-008-9208-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-008-9208-6

Keywords

Search

Navigation