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- Mitchell Luskin
- 1996

Microstructure is a feature of crystals with multiple symmetry-related energy-minimizing states. Continuum models have been developed explaining mi-crostructure as the mixture of these symmetry-related states on a fine scale to minimize energy. This article is a review of numerical methods and the numerical analysis for the computation of crystalline… (More)

- Matthew Dobson, Mitchell Luskin
- SIAM J. Numerical Analysis
- 2009

We derive a model problem for quasicontinuum approximations that allows a simple , yet insightful, analysis of the optimal-order convergence rate in the continuum limit for both the energy-based quasicontinuum approximation and the quasi-nonlocal quasicontinuum approximation. For simplicity, the analysis is restricted to the case of second-neighbor… (More)

- Mitchell Luskin
- 1996

We give error estimates for the approximation of a laminated mi-crostructure which minimizes the energy R rvx dx for a rotationally invariant, double well energy density A. We present error estimates for the convergence of the deformation in L 2 ; the convergence of directional derivatives of the deformation in the twin planes," the weak convergence of the… (More)

A sharp stability analysis of atomistic-to-continuum coupling methods is essential for evaluating their capabilities for predicting the formation and motion of lattice defects. We formulate a simple one-dimensional model problem and give a detailed analysis of the stability of the force-based quasicontinuum (QCF) method. The focus of the analysis is the… (More)

The formation and motion of lattice defects such as cracks, dislocations, or grain boundaries, occurs when the lattice configuration loses stability, that is, when an eigenvalue of the Hessian of the lattice energy functional becomes negative. When the atomistic energy is approximated by a hybrid energy that couples atomistic and continuum models, the… (More)

- Bo Li, Mitchell Luskin
- Math. Comput.
- 1998

We consider a class of nonconforming nite element approximations of a simply laminated microstructure which minimizes the nonconvex variational problem for the deformation of martensitic crystals which can undergo either an orthorhombic to monoclinic (double well) or a cubic to tetrag-onal (triple well) transformation. We rst establish a series of error… (More)

- Marcel Arndt, Mitchell Luskin
- Multiscale Modeling & Simulation
- 2008

We propose and analyze a goal-oriented a posteriori error estimator for the atomistic-continuum 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… (More)

The quasicontinuum (QC) method is applied to materials possessing a mul-tilattice crystal structure. Cauchy-Born (CB) kinematics, which accounts for the shifts of the crystal basis, is used in continuum regions to relate atomic motions to continuum deformation gradients. To avoid failures of the CB kinematics, QC is augmented with a phonon stability… (More)

- Matthew Dobson, Mitchell Luskin, Christoph Ortner
- Multiscale Modeling & Simulation
- 2010

The accuracy of atomistic-to-continuum hybrid methods can be guaranteed only for deformations where the lattice configuration is stable for both the atomistic energy and the hybrid energy. For this reason, a sharp stability analysis of atomistic-to-continuum coupling methods is essential for evaluating their capabilities for predicting the formation of… (More)

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… (More)