Mitchell Luskin

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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)
We give error estimates for the approximation of a laminated microstructure which minimizes the energy R (rv(x))dx for a rotationally invariant, double well energy density (A). We present error estimates for the convergence of the deformation in L; the convergence of directional derivatives of the deformation in the \twin planes," the weak convergence of(More)
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 tetragonal (triple well) transformation. We rst establish a series of error(More)
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)
We consider the enhancement of accuracy, by means of a simple post-processing technique, for finite element approximations to transient hyperbolic equations. The post-processing is a convolution with a kernel whose support has measure of order one in the case of arbitrary unstructured meshes; if the mesh is locally translation invariant, the support of the(More)
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(More)
The quasicontinuum (QC) method is applied to materials possessing a multilattice 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 analysis(More)
We give an analysis of a continuation algorithm for the numerical solution of the forcebased quasicontinuum equations. The approximate solution of the force-based quasicontinuum equations is computed by an iterative method using an energy-based quasicontinuum approximation as the preconditioner. The analysis presented in this paper is used to determine an(More)