# Analysis of a force-based quasicontinuum approximation

Matthew Dobson; Mitchell Luskin

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

- Volume: 42, Issue: 1, page 113-139
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topDobson, Matthew, and Luskin, Mitchell. "Analysis of a force-based quasicontinuum approximation." ESAIM: Mathematical Modelling and Numerical Analysis 42.1 (2008): 113-139. <http://eudml.org/doc/250331>.

@article{Dobson2008,

abstract = {
We analyze a force-based quasicontinuum approximation to a
one-dimensional system of atoms that interact by a classical
atomistic potential. This force-based quasicontinuum approximation
can be derived as the modification of an energy-based
quasicontinuum approximation by the addition of nonconservative
forces to correct nonphysical “ghost” forces that occur in the
atomistic to continuum interface during constant strain. The algorithmic
simplicity and consistency with the purely atomistic model
at constant strain has made the force-based
quasicontinuum approximation popular for large-scale
quasicontinuum computations.
We prove that the force-based quasicontinuum equations have
a unique solution when the magnitude of the external forces satisfy
explicit bounds. For Lennard-Jones next-nearest-neighbor
interactions, we show that unique solutions exist
for external forces that extend the system nearly to its tensile limit.
We give an analysis of the convergence of the ghost force iteration method
to 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 its
convergence rate.
},

author = {Dobson, Matthew, Luskin, Mitchell},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Quasicontinuum; ghost force; atomistic to continuum.; Lennard-Jones potential; convergence; ghost force iteration method},

language = {eng},

month = {1},

number = {1},

pages = {113-139},

publisher = {EDP Sciences},

title = {Analysis of a force-based quasicontinuum approximation},

url = {http://eudml.org/doc/250331},

volume = {42},

year = {2008},

}

TY - JOUR

AU - Dobson, Matthew

AU - Luskin, Mitchell

TI - Analysis of a force-based quasicontinuum approximation

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2008/1//

PB - EDP Sciences

VL - 42

IS - 1

SP - 113

EP - 139

AB -
We analyze a force-based quasicontinuum approximation to a
one-dimensional system of atoms that interact by a classical
atomistic potential. This force-based quasicontinuum approximation
can be derived as the modification of an energy-based
quasicontinuum approximation by the addition of nonconservative
forces to correct nonphysical “ghost” forces that occur in the
atomistic to continuum interface during constant strain. The algorithmic
simplicity and consistency with the purely atomistic model
at constant strain has made the force-based
quasicontinuum approximation popular for large-scale
quasicontinuum computations.
We prove that the force-based quasicontinuum equations have
a unique solution when the magnitude of the external forces satisfy
explicit bounds. For Lennard-Jones next-nearest-neighbor
interactions, we show that unique solutions exist
for external forces that extend the system nearly to its tensile limit.
We give an analysis of the convergence of the ghost force iteration method
to 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 its
convergence rate.

LA - eng

KW - Quasicontinuum; ghost force; atomistic to continuum.; Lennard-Jones potential; convergence; ghost force iteration method

UR - http://eudml.org/doc/250331

ER -

## References

top- S. Antman, Nonlinear problems of elasticity, Applied Mathematical Sciences107. Springer, New York, second edition (2005). Zbl1098.74001
- X. Blanc, C. Le Bris and F. Legoll, Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics. ESAIM: M2AN39 (2005) 797–826. Zbl1330.74066
- X. Blanc, C. Le Bris and P.-L. Lions, Atomistic to continuum limits for computational materials science. ESAIM: M2AN41 (2007) 391–426. Zbl1144.82018
- R.F. Brown, A Topological Introduction to Nonlinear Analysis. Birkhäuser (2004). Zbl1061.47001
- W. E and P. Ming, Analysis of multiscale methods. J. Comput. Math.22 (2004) 210–219. Zbl1046.65108
- 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.
- W. E and P. Ming, Cauchy-born rule and the stabilitiy of crystalline solids: Static problems. Arch. Ration. Mech. Anal.183 (2007) 241–297. Zbl1106.74019
- W. E, J. Lu and J. Yang, Uniform accuracy of the quasicontinuum method. Phys. Rev. B74 (2006) 214115.
- W. Fleming, Functions of Several Variables. Springer-Verlag (1977). Zbl0348.26002
- J. Knap and M. Ortiz, An analysis of the quasicontinuum method. J. Mech. Phys. Solids49 (2001) 1899–1923. Zbl1002.74008
- P. Lin, Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model. Math. Comp.72 (2003) 657–675 (electronic). Zbl1010.74003
- P. Lin, Convergence analysis of a quasi-continuum approximation for a two-dimensional material. SIAM J. Numer. Anal.45 (2007) 313–332. Zbl1220.74010
- M. Marder, Condensed Matter Physics. John Wiley & Sons (2000).
- R. Miller and E. Tadmor, The quasicontinuum method: Overview, applications and current directions. J. Comput. Aided Mater. Des.9 (2002) 203–239.
- R. Miller, L. Shilkrot and W. Curtin, A coupled atomistic and discrete dislocation plasticity simulation of nano-indentation into single crystal thin films. Acta Mater.52 (2003) 271–284.
- J.T. Oden, S. Prudhomme, A. Romkes and P. Bauman, Multi-scale modeling of physical phenomena: Adaptive control of models. SIAM J. Sci. Comput.28 (2006) 2359–2389. Zbl1126.74006
- 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). Zbl1139.74004
- S. Prudhomme, P.T. Bauman and J.T. Oden, Error control for molecular statics problems. Int. J. Multiscale Comput. Eng.4 (2006) 647–662.
- D. Rodney and R. Phillips, Structure and strength of dislocation junctions: An atomic level analysis. Phys. Rev. Lett.82 (1999) 1704–1707.
- D. Serre, Matrices: Theory and applications, Graduate Texts in Mathematics216. Springer-Verlag, New York (2002). Translated from the 2001 French original. Zbl1008.15002
- V. Shenoy, R. Miller, E. Tadmor, D. Rodney, R. Phillips and M. Ortiz, An adaptive finite element approach to atomic-scale mechanics — the quasicontinuum method. J. Mech. Phys. Solids47 (1999) 611–642. Zbl0982.74071
- T. Shimokawa, J. Mortensen, J. Schiotz and K. Jacobsen, Matching conditions in the quasicontinuum method: Removal of the error introduced at the interface between the coarse-grained and fully atomistic regions. Phys. Rev. B69 (2004) 214104.
- E. Tadmor, M. Ortiz and R. Phillips, Quasicontinuum analysis of defects in solids. Phil. Mag. A73 (1996) 1529–1563.

## Citations in EuDML Documents

top- Christoph Ortner, Endre Süli, Analysis of a quasicontinuum method in one dimension
- Christoph Ortner, The role of the patch test in 2D atomistic-to-continuum coupling methods
- Christoph Ortner, The role of the patch test in 2D atomistic-to-continuum coupling methods
- Matthew Dobson, Mitchell Luskin, An analysis of the effect of ghost force oscillation on quasicontinuum error

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.