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...
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...
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...
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,...
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...
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...
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