Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-29T05:30:41.209Z Has data issue: false hasContentIssue false
  • English
  • Français

Analysis of a force-based quasicontinuum approximation

Published online by Cambridge University Press:  12 January 2008

Matthew Dobson  and
Mitchell Luskin
Matthew Dobson
Affiliation:
School of Mathematics, University of Minnesota, 206 Church Street SE, Minneapolis, MN 55455, USA. dobson@math.umn.edu; luskin@umn.edu
Mitchell Luskin
Affiliation:
School of Mathematics, University of Minnesota, 206 Church Street SE, Minneapolis, MN 55455, USA. dobson@math.umn.edu; luskin@umn.edu
Get access

Abstract

We analyze a force-based quasicontinuum approximation to aone-dimensional system of atoms that interact by a classicalatomistic potential. This force-based quasicontinuum approximationcan be derived as the modification of an energy-basedquasicontinuum approximation by the addition of nonconservativeforces to correct nonphysical “ghost” forces that occur in theatomistic to continuum interface during constant strain. The algorithmicsimplicity and consistency with the purely atomistic modelat constant strain has made the force-basedquasicontinuum approximation popular for large-scalequasicontinuum computations.We prove that the force-based quasicontinuum equations havea unique solution when the magnitude of the external forces satisfyexplicit bounds. For Lennard-Jones next-nearest-neighborinteractions, we show that unique solutions existfor external forces that extend the system nearly to its tensile limit.We give an analysis of the convergence of the ghost force iteration methodto solve the equilibrium equations for the force-based quasicontinuum approximation.We show that the ghost force iteration is a contraction and give an analysis for itsconvergence rate.

Keywords

Quasicontinuumghost forceatomistic to continuum.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 1 , January 2008 , pp. 113 - 139
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

S. Antman, Nonlinear problems of elasticity, Applied Mathematical Sciences 107. Springer, New York, second edition (2005).
X. Blanc, C. Le Bris and F. Legoll, Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics. ESAIM: M2AN 39 (2005) 797–826.
X. Blanc, C. Le Bris and P.-L. Lions, Atomistic to continuum limits for computational materials science. ESAIM: M2AN 41 (2007) 391–426.
R.F. Brown, A Topological Introduction to Nonlinear Analysis. Birkhäuser (2004).
E, W. and Ming, P., Analysis of multiscale methods. J. Comput. Math. 22 (2004) 210–219.
W. E and P. Ming, Analysis of the local quasicontinuum method, in Frontiers and Prospects of Contemporary Applied Mathematics, T. Li and P. Zhang Eds., Higher Education Press, World Scientific, Singapore (2005) 18–32.
E, W. and Ming, P., Cauchy-born rule and the stabilitiy of crystalline solids: Static problems. Arch. Ration. Mech. Anal. 183 (2007) 241–297. CrossRef
E, W., Lu, J. and Yang, J., Uniform accuracy of the quasicontinuum method. Phys. Rev. B 74 (2006) 214115. CrossRef
W. Fleming, Functions of Several Variables. Springer-Verlag (1977).
Knap, J. and Ortiz, M., An analysis of the quasicontinuum method. J. Mech. Phys. Solids 49 (2001) 1899–1923. CrossRef
Lin, P., Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model. Math. Comp. 72 (2003) 657–675 (electronic). CrossRef
Lin, P., Convergence analysis of a quasi-continuum approximation for a two-dimensional material. SIAM J. Numer. Anal. 45 (2007) 313–332. CrossRef
M. Marder, Condensed Matter Physics. John Wiley & Sons (2000).
Miller, R. and Tadmor, E., The quasicontinuum method: Overview, applications and current directions. J. Comput. Aided Mater. Des. 9 (2002) 203–239. CrossRef
Miller, R., Shilkrot, L. and Curtin, W., A coupled atomistic and discrete dislocation plasticity simulation of nano-indentation into single crystal thin films. Acta Mater. 52 (2003) 271–284. CrossRef
Oden, J.T., Prudhomme, S., Romkes, A. and Bauman, P., Multi-scale modeling of physical phenomena: Adaptive control of models. SIAM J. Sci. Comput. 28 (2006) 2359–2389. CrossRef
C. Ortner and E. Süli, A posteriori analysis and adaptive algorithms for the quasicontinuum method in one dimension. Technical report, Oxford Numerical Analysis Group (2006).
C. Ortner and E. Süli, A priori analysis of the quasicontinuum method in one dimension. Technical report, Oxford Numerical Analysis Group (2006).
Prudhomme, S., Bauman, P.T. and Error, J.T. Oden control for molecular statics problems. Int. J. Multiscale Comput. Eng. 4 (2006) 647–662. CrossRef
Rodney, D. and Phillips, R., Structure and strength of dislocation junctions: An atomic level analysis. Phys. Rev. Lett. 82 (1999) 1704–1707. CrossRef
D. Serre, Matrices: Theory and applications, Graduate Texts in Mathematics 216. Springer-Verlag, New York (2002). Translated from the 2001 French original.
Shenoy, V., Miller, R., Tadmor, E., Rodney, D., Phillips, R. and Ortiz, M., An adaptive finite element approach to atomic-scale mechanics — the quasicontinuum method. J. Mech. Phys. Solids 47 (1999) 611–642. CrossRef
Shimokawa, T., Mortensen, J., Schiotz, J. and 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 (2004) 214104. CrossRef
Tadmor, E., Ortiz, M. and Phillips, R., Quasicontinuum analysis of defects in solids. Phil. Mag. A 73 (1996) 1529–1563. CrossRef