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The role of the patch test in 2D atomistic-to-continuum coupling methods

Christoph Ortner — 2012

ESAIM: Mathematical Modelling and Numerical Analysis

For a general class of atomistic-to-continuum coupling methods, coupling multi-body interatomic potentials with a P-finite element discretisation of Cauchy–Born nonlinear elasticity, this paper adresses the question whether (or, absence of ) implies a first-order error estimate. In two dimensions it is shown that this is indeed true under the following additional technical assumptions: (i) an energy consistency condition, (ii) locality of the interface correction, (iii) volumetric scaling of the...

The role of the patch test in 2D atomistic-to-continuum coupling methods

Christoph Ortner — 2012

ESAIM: Mathematical Modelling and Numerical Analysis

For a general class of atomistic-to-continuum coupling methods, coupling multi-body interatomic potentials with a P-finite element discretisation of Cauchy–Born nonlinear elasticity, this paper adresses the question whether (or, absence of ) implies a first-order error estimate. In two dimensions it is shown that this is indeed true under the following additional technical assumptions: (i) an energy consistency condition, (ii) locality of the interface correction, (iii) volumetric scaling of the...

On the stability of Bravais lattices and their Cauchy–Born approximations

Thomas HudsonChristoph Ortner — 2012

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

We investigate the stability of Bravais lattices and their Cauchy–Born approximations under periodic perturbations. We formulate a general interaction law and derive its Cauchy–Born continuum limit. We then analyze the atomistic and Cauchy–Born stability regions, that is, the sets of all matrices that describe a stable Bravais lattice in the atomistic and Cauchy–Born models respectively. Motivated by recent results in one dimension on the stability of atomistic/continuum coupling methods, we analyze...

Analysis of a quasicontinuum method in one dimension

Christoph OrtnerEndre 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 and 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 and the stability...

On the stability of Bravais lattices and their Cauchy–Born approximations

Thomas HudsonChristoph Ortner — 2011

ESAIM: Mathematical Modelling and Numerical Analysis

We investigate the stability of Bravais lattices and their Cauchy–Born approximations under periodic perturbations. We formulate a general interaction law and derive its Cauchy–Born continuum limit. We then analyze the atomistic and Cauchy–Born stability regions, that is, the sets of all matrices that describe a stable Bravais lattice in the atomistic and Cauchy–Born models respectively. Motivated by recent results in one dimension on the stability of atomistic/continuum coupling methods,...

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

Kavinda JayawardanaChristelle MordacqChristoph OrtnerHarold 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 JayawardanaChristelle MordacqChristoph OrtnerHarold 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...

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