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

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 &#x1d4aa;(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∗

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...

### Analysis of a force-based quasicontinuum approximation

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

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...

### Kinetic and hydrodynamic equations for granular media

Journées équations aux dérivées partielles

In this lecture i present some open mathematical problems concerning some PDE arising in the study of one-dimensional models for granular media.

### Models for phase separation and their mathematics.

Electronic Journal of Differential Equations (EJDE) [electronic only]

### Molecular Simulation in the Canonical Ensemble and Beyond

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we discuss advanced thermostatting techniques for sampling molecular systems in the canonical ensemble. We first survey work on dynamical thermostatting methods, including the Nosé-Poincaré method, and generalized bath methods which introduce a more complicated extended model to obtain better ergodicity. We describe a general controlled temperature model, projective thermostatting molecular dynamics (PTMD) and demonstrate that it flexibly accommodates existing alternative thermostatting...

### Multiscale Materials Modelling: Case Studies at the Atomistic and Electronic Structure Levels

ESAIM: Mathematical Modelling and Numerical Analysis

Although the intellectual merits of computational modelling across various length and time scales are generally well accepted, good illustrative examples are often lacking. One way to begin appreciating the benefits of the multiscale approach is to first gain experience in probing complex physical phenomena at one scale at a time. Here we discuss materials modelling at two characteristic scales separately, the atomistic level where interactions are specified through classical potentials and the...

### Qualitative properties of a continuum theory for thin films

Annales de l'I.H.P. Analyse non linéaire

### Sufficient conditions for the validity of the Cauchy-Born rule close to $\mathrm{SO}\left(n\right)$

Journal of the European Mathematical Society

The Cauchy–Born rule provides a crucial link between continuum theories of elasticity and the atomistic nature of matter. In its strongest form it says that application of affine displacement boundary conditions to a monatomic crystal will lead to an affine deformation of the whole crystal lattice. We give a general condition in arbitrary dimensions which ensures the validity of the Cauchy–Born rule for boundary deformations which are close to rigid motions. This generalizes results of Friesecke...

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