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 ℓ ∞ 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 ℓ ∞ 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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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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