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

Published online by Cambridge University Press:  31 July 2012

Kavinda Jayawardana ,
Christelle Mordacq ,
Christoph Ortner  and
Harold S. Park
Kavinda Jayawardana
Affiliation:
Department of Mathematics, University College London, Gower Street, WC1E 6BT London, UK. k.guruge@ucl.ac.uk
Christelle Mordacq
Affiliation:
École Normale Supérieure de Cachan, Antenne de Bretagne, Avenue Robert Schuman, 35170 Bruz, France; Christelle.Mordacq@gmail.com
Christoph Ortner
Affiliation:
Mathematics Institute, Zeeman Building, University of Warwick, CV4 7AL Coventry, UK; c.ortner@warwick.ac.uk
Harold S. Park
Affiliation:
Boston University, Department of Mechanical Engineering, 730 Commonwealth Avenue, ENA 212, Boston, 02215 MA, USA; parkhs@acs.bu.edu
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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.

Keywords

Surface-dominated materialssurface Cauchy–Born rulecoarse-graining
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 1 , January 2013 , pp. 109 - 123
Copyright
© EDP Sciences, SMAI, 2012

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