An analysis of the effect of ghost force oscillation on quasicontinuum error

Matthew Dobson; Mitchell Luskin

ESAIM: Mathematical Modelling and Numerical Analysis (2009)

  • Volume: 43, Issue: 3, page 591-604
  • ISSN: 0764-583X

Abstract

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The atomistic to continuum interface for quasicontinuum energies exhibits nonzero forces under uniform strain that have been called ghost forces. In this paper, we prove for a linearization of a one-dimensional quasicontinuum energy around a uniform strain that the effect of the ghost forces on the displacement nearly cancels and has a small effect on the error away from the interface. We give optimal order error estimates that show that the quasicontinuum displacement converges to the atomistic displacement at the rate O(h) in the discrete and w1,1 norms where h is the interatomic spacing. We also give a proof that the error in the displacement gradient decays away from the interface to O(h) at distance O(h|logh|) in the atomistic region and distance O(h) in the continuum region. Our work gives an explicit and simplified form for the decay of the effect of the atomistic to continuum coupling error in terms of a general underlying interatomic potential and gives the estimates described above in the discrete and w1,p norms.

How to cite

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Dobson, Matthew, and Luskin, Mitchell. "An analysis of the effect of ghost force oscillation on quasicontinuum error." ESAIM: Mathematical Modelling and Numerical Analysis 43.3 (2009): 591-604. <http://eudml.org/doc/250624>.

@article{Dobson2009,
abstract = { The atomistic to continuum interface for quasicontinuum energies exhibits nonzero forces under uniform strain that have been called ghost forces. In this paper, we prove for a linearization of a one-dimensional quasicontinuum energy around a uniform strain that the effect of the ghost forces on the displacement nearly cancels and has a small effect on the error away from the interface. We give optimal order error estimates that show that the quasicontinuum displacement converges to the atomistic displacement at the rate O(h) in the discrete $\ell^\infty$ and w1,1 norms where h is the interatomic spacing. We also give a proof that the error in the displacement gradient decays away from the interface to O(h) at distance O(h|logh|) in the atomistic region and distance O(h) in the continuum region. Our work gives an explicit and simplified form for the decay of the effect of the atomistic to continuum coupling error in terms of a general underlying interatomic potential and gives the estimates described above in the discrete $\ell^\infty$ and w1,p norms. },
author = {Dobson, Matthew, Luskin, Mitchell},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Quasicontinuum; atomistic to continuum; ghost force.; quasicontinuum; ghost force},
language = {eng},
month = {4},
number = {3},
pages = {591-604},
publisher = {EDP Sciences},
title = {An analysis of the effect of ghost force oscillation on quasicontinuum error},
url = {http://eudml.org/doc/250624},
volume = {43},
year = {2009},
}

TY - JOUR
AU - Dobson, Matthew
AU - Luskin, Mitchell
TI - An analysis of the effect of ghost force oscillation on quasicontinuum error
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2009/4//
PB - EDP Sciences
VL - 43
IS - 3
SP - 591
EP - 604
AB - The atomistic to continuum interface for quasicontinuum energies exhibits nonzero forces under uniform strain that have been called ghost forces. In this paper, we prove for a linearization of a one-dimensional quasicontinuum energy around a uniform strain that the effect of the ghost forces on the displacement nearly cancels and has a small effect on the error away from the interface. We give optimal order error estimates that show that the quasicontinuum displacement converges to the atomistic displacement at the rate O(h) in the discrete $\ell^\infty$ and w1,1 norms where h is the interatomic spacing. We also give a proof that the error in the displacement gradient decays away from the interface to O(h) at distance O(h|logh|) in the atomistic region and distance O(h) in the continuum region. Our work gives an explicit and simplified form for the decay of the effect of the atomistic to continuum coupling error in terms of a general underlying interatomic potential and gives the estimates described above in the discrete $\ell^\infty$ and w1,p norms.
LA - eng
KW - Quasicontinuum; atomistic to continuum; ghost force.; quasicontinuum; ghost force
UR - http://eudml.org/doc/250624
ER -

References

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  1. M. Arndt and M. Luskin, Goal-oriented atomistic-continuum adaptivity for the quasicontinuum approximation. Int. J. Mult. Comp. Eng.5 (2007) 407–415.  
  2. M. Arndt and M. Luskin, Error estimation and atomistic-continuum adaptivity for the quasicontinuum approximation of a Frenkel-Kontorova model. Multiscale Model. Simul.7 (2008) 147–170.  
  3. M. Arndt and M. Luskin, Goal-oriented adaptive mesh refinement for the quasicontinuum approximation of a Frenkel-Kontorova model. Comp. Meth. App. Mech. Eng.197 (2008) 4298–4306.  
  4. S. Badia, M.L. Parks, P.B. Bochev, M. Gunzburger and R.B. Lehoucq, On atomistic-to-continuum (AtC) coupling by blending. Multiscale Model. Simul.7 (2008) 381–406.  
  5. X. Blanc, C. Le Bris and F. Legoll, Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics. ESAIM: M2AN39 (2005) 797–826.  
  6. W. Curtin and R. Miller, Atomistic/continuum coupling in computational materials science. Model. Simul. Mater. Sc.11 (2003) R33–R68.  
  7. M. Dobson and M. Luskin, Analysis of a force-based quasicontinuum method. ESAIM: M2AN42 (2008) 113–139.  
  8. 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 (2005) 18–32.  
  9. W. E., J. Lu and J. Yang, Uniform accuracy of the quasicontinuum method. Phys. Rev. B74 (2006) 214115.  
  10. J. Knap and M. Ortiz, An analysis of the quasicontinuum method. J. Mech. Phys. Solids49 (2001) 1899–1923.  
  11. P. Lin, Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model. Math. Comp.72 (2003) 657–675 (electronic).  
  12. P. Lin, Convergence analysis of a quasi-continuum approximation for a two-dimensional material. SIAM J. Numer. Anal.45 (2007) 313–332.  
  13. R. Miller and E. Tadmor, The quasicontinuum method: Overview, applications and current directions. J. Comput. Aided Mater. Des.9 (2002) 203–239.  
  14. 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.  
  15. P. Ming and J.Z. Yang, Analysis of a one-dimensional nonlocal quasicontinuum method. Preprint.  
  16. 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.  
  17. C. Ortner and E. Süli, A-posteriori analysis and adaptive algorithms for the quasicontinuum method in one dimension. Research Report NA-06/13, Oxford University Computing Laboratory (2006).  
  18. C. Ortner and E. Süli, Analysis of a quasicontinuum method in one dimension. ESAIM: M2AN42 (2008) 57–91.  
  19. M.L. Parks, P.B. Bochev and R.B. Lehoucq, Connecting atomistic-to-continuum coupling and domain decomposition. Multiscale Model. Simul.7 (2008) 362–380.  
  20. S. Prudhomme, P.T. Bauman and J.T. Oden, Error control for molecular statics problems. Int. J. Mult. Comp. Eng.4 (2006) 647–662.  
  21. D. Rodney and R. Phillips, Structure and strength of dislocation junctions: An atomic level analysis. Phys. Rev. Lett.82 (1999) 1704–1707.  
  22. 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.  
  23. 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.  
  24. G. Strang and G. Fix, Analysis of the Finite Elements Method. Prentice Hall (1973).  
  25. E. Tadmor, M. Ortiz and R. Phillips, Quasicontinuum analysis of defects in solids. Phil. Mag. A73 (1996) 1529–1563.  

Citations in EuDML Documents

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  1. Christoph Ortner, The role of the patch test in 2D atomistic-to-continuum coupling methods
  2. Kavinda Jayawardana, Christelle Mordacq, Christoph Ortner, Harold S. Park, An analysis of the boundary layer in the 1D surface Cauchy–Born model
  3. Kavinda Jayawardana, Christelle Mordacq, Christoph Ortner, Harold S. Park, An analysis of the boundary layer in the 1D surface Cauchy–Born model
  4. Christoph Ortner, The role of the patch test in 2D atomistic-to-continuum coupling methods

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