An analysis of the boundary layer in the 1D surface Cauchy–Born model∗

Kavinda Jayawardana; Christelle Mordacq; Christoph Ortner; Harold S. Park

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 47, Issue: 1, page 109-123
  • ISSN: 0764-583X

Abstract

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The surface Cauchy–Born (SCB) method is a computational multi-scale method for the simulation of surface-dominated crystalline materials. We present an error analysis of the SCB method, focused on the role of surface relaxation. In a linearized 1D model we show that the error committed by the SCB method is 𝒪(1) in the mesh size; however, we are able to identify an alternative “approximation parameter” – the stiffness of the interaction potential – with respect to which the relative error in the mean strain is exponentially small. Our analysis naturally suggests an improvement of the SCB model by enforcing atomistic mesh spacing in the normal direction at the free boundary. In this case we even obtain pointwise error estimates for the strain.

How to cite

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Jayawardana, Kavinda, et al. "An analysis of the boundary layer in the 1D surface Cauchy–Born model∗." ESAIM: Mathematical Modelling and Numerical Analysis 47.1 (2012): 109-123. <http://eudml.org/doc/222172>.

@article{Jayawardana2012,
abstract = {The surface Cauchy–Born (SCB) method is a computational multi-scale method for the simulation of surface-dominated crystalline materials. We present an error analysis of the SCB method, focused on the role of surface relaxation. In a linearized 1D model we show that the error committed by the SCB method is 𝒪(1) in the mesh size; however, we are able to identify an alternative “approximation parameter” – the stiffness of the interaction potential – with respect to which the relative error in the mean strain is exponentially small. Our analysis naturally suggests an improvement of the SCB model by enforcing atomistic mesh spacing in the normal direction at the free boundary. In this case we even obtain pointwise error estimates for the strain.},
author = {Jayawardana, Kavinda, Mordacq, Christelle, Ortner, Christoph, Park, Harold S.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Surface-dominated materials; surface Cauchy–Born rule; coarse-graining; surface-dominated materials; surface Cauchy-Born rule},
language = {eng},
month = {7},
number = {1},
pages = {109-123},
publisher = {EDP Sciences},
title = {An analysis of the boundary layer in the 1D surface Cauchy–Born model∗},
url = {http://eudml.org/doc/222172},
volume = {47},
year = {2012},
}

TY - JOUR
AU - Jayawardana, Kavinda
AU - Mordacq, Christelle
AU - Ortner, Christoph
AU - Park, Harold S.
TI - An analysis of the boundary layer in the 1D surface Cauchy–Born model∗
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2012/7//
PB - EDP Sciences
VL - 47
IS - 1
SP - 109
EP - 123
AB - The surface Cauchy–Born (SCB) method is a computational multi-scale method for the simulation of surface-dominated crystalline materials. We present an error analysis of the SCB method, focused on the role of surface relaxation. In a linearized 1D model we show that the error committed by the SCB method is 𝒪(1) in the mesh size; however, we are able to identify an alternative “approximation parameter” – the stiffness of the interaction potential – with respect to which the relative error in the mean strain is exponentially small. Our analysis naturally suggests an improvement of the SCB model by enforcing atomistic mesh spacing in the normal direction at the free boundary. In this case we even obtain pointwise error estimates for the strain.
LA - eng
KW - Surface-dominated materials; surface Cauchy–Born rule; coarse-graining; surface-dominated materials; surface Cauchy-Born rule
UR - http://eudml.org/doc/222172
ER -

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