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

¹ Department of Mathematics, University College London, Gower Street, WC1E 6 BT London, UK
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² École Normale Supérieure de Cachan, Antenne de Bretagne, Avenue Robert Schuman, 35170 Bruz, France
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³ Mathematics Institute, Zeeman Building, University of Warwick, CV4 7AL Coventry, UK
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⁴ Boston University, Department of Mechanical Engineering, 730 Commonwealth Avenue, ENA 212, Boston, 02215 MA, USA
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Received: 5 December 2011
Revised: 6 April 2012

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.