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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...
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...
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...
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...
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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