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

Kavinda Jayawardana, Christelle Mordacq, Christoph Ortner, Harold S. Park (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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...

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

Kavinda Jayawardana, Christelle Mordacq, Christoph Ortner, Harold S. Park (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

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...

An analysis of the effect of ghost force oscillation on quasicontinuum error

Matthew Dobson, Mitchell Luskin (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

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...

Analysis of a force-based quasicontinuum approximation

Matthew Dobson, Mitchell Luskin (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

We analyze a force-based quasicontinuum approximation to a one-dimensional system of atoms that interact by a classical atomistic potential. This force-based quasicontinuum approximation can be derived as the modification of an energy-based quasicontinuum approximation by the addition of nonconservative forces to correct nonphysical “ghost” forces that occur in the atomistic to continuum interface during constant strain. The algorithmic simplicity and consistency with the purely atomistic model at...

Analysis of a quasicontinuum method in one dimension

Christoph Ortner, Endre Süli (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

The quasicontinuum method is a coarse-graining technique for reducing the complexity of atomistic simulations in a static and quasistatic setting. In this paper we aim to give a detailed a priori and a posteriori error analysis for a quasicontinuum method in one dimension. We consider atomistic models with Lennard–Jones type long-range interactions and a QC formulation which incorporates several important aspects of practical QC methods. First, we prove the existence, the local uniqueness...

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