# 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

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topDobson, 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 -

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## Citations in EuDML Documents

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