We propose and analyze a goal-oriented a posteriori error estimator for the atomisticcontinuum modeling error in the quasicontinuum method. Based on this error estimator, we develop an algorithm which adaptively determines the atomistic and continuum regions to compute a quantity of interest to within a given tolerance. We apply the algorithm to the computation of the structure of a crystallogr...

Force-based atomistic-continuum hybrid methods are the only known pointwise consistent methods for coupling a general atomistic model to a finite element continuum model. For this reason, and due to their algorithmic simplicity, force-based coupling methods have become a popular approach for atomistic-continuum hybrid methods as well as other types of multiphysics model coupling. However, the r...

The force-based quasicontinuum (QCF) approximation is a non-conservative atomistic/continuum hybrid model for the simulation of defects in crystals. We present an a priori error analysis of the QCF method, applied to a one-dimensional periodic chain, that is valid for an arbitrary interaction range, large deformations, and takes coarse-graining into account. Our main tool in this analysis is a ...

We propose a new upscaling scheme for the passage from atomistic to continuum mechanical models for crystalline solids. It is based on a Taylor expansion of the deformation function and allows us to capture the microscopic properties and the discreteness effects of the underlying atomistic system up to an arbitrary order. The resulting continuum mechanical model involves higher order terms and ...

In this paper we present two atomistic models for the energy of a one-dimensional elastic crystal. We assume that the macroscopic displacement equals the microscopic one. The energy of the first model is given by a two-body interaction potential, and we assume that the atoms follow a continuous and piecewise smooth macroscopic (continuum) deformation. We calculate the first terms of the Taylor ...

Atomistic modelling of magnetic materials provides unprecedented detail about the underlying physical processes that govern their macroscopic properties, and allows the simulation of complex effects such as surface anisotropy, ultrafast laser-induced spin dynamics, exchange bias, and microstructural effects. Here we present the key methods used in atomistic spin models which are then applied to...

We present an adaptive mesh and algorithmic refinement (AMAR) scheme for modeling multi–scale hydrodynamics. The AMAR approach extends standard conservative adaptive mesh refinement (AMR) algorithms by providing a robust flux–based method for coupling an atomistic fluid representation to a continuum model. The atomistic model is applied locally in regions where the continuum description is inva...