Results 1 - 10 of 12

An optimal order error analysis of the one-dimensional quasicontinuum approximation

by Matthew Dobson, Mitchell Luskin - SIAM J. Numer. Anal
"... Abstract. 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 ..."
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Abstract. 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 interactions and is linearized about a uniformly stretched reference lattice. The optimal-order error estimates for the quasi-nonlocal quasicontinuum approximation are given for all strains up to the continuum limit strain for fracture. The analysis is based on an explicit treatment of the coupling error at the atomistic-to-continuum interface, combined with an analysis of the error due to the atomistic and continuum schemes using the stability of the quasicontinuum approximation.
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AN ANALYSIS OF THE EFFECT OF GHOST FORCE OSCILLATION ON QUASICONTINUUM ERROR

by Matthew Dobson, Mitchell Luskin - TO APPEAR IN MATHEMATICAL MODELING AND NUMERICAL ANALYSIS , 2009
"... The atomistic to continuum interface for quasicontinuum energies exhibits nonzero forces under uniform strain that have been called ghost forces. In this paper, we prove for a linearization of a one-dimensional quasicontinuum energy around a uniform strain that the effect of the ghost forces on th ..."
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The atomistic to continuum interface for quasicontinuum energies exhibits nonzero forces under uniform strain that have been called ghost forces. In this paper, we prove for a linearization of a one-dimensional quasicontinuum energy around a uniform strain that the effect of the ghost forces on the displacement nearly cancels and has a small effect on the error away from the interface. We give optimal order error estimates that show that the quasicontinuum displacement converges to the atomistic displacement at the rate O(h) in the discrete ℓ ∞ and w 1,1 norms where h is the interatomic spacing. We also give a proof that the error in the displacement gradient decays away from the interface to O(h) at distance O(h|log h|) in the atomistic region and distance O(h) in the continuum region. E, Ming, and Yang previously gave a counterexample to convergence in the w 1, ∞ norm for a harmonic interatomic potential. Our work gives an explicit and simplified form for the decay of the effect of the atomistic to continuum coupling error in terms of a general underlying interatomic potential and gives the estimates described above in the discrete ℓ ∞ and w 1,p norms.

A quadrature-rule type approximation for the quasicontinuum method. Multiscale Modeling and Simulation

by Yanzhi Zhang , Max Gunzburger
"... Keywords: Atomistic models Quasicontinuum method Quadrature-rule type approximation Long-range interactions Coulomb potential a b s t r a c t A quadrature-rule type method is presented to approximate the quasicontinuum method for atomistic mechanics. For both the short-range and long-range interact ..."
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Keywords: Atomistic models Quasicontinuum method Quadrature-rule type approximation Long-range interactions Coulomb potential a b s t r a c t A quadrature-rule type method is presented to approximate the quasicontinuum method for atomistic mechanics. For both the short-range and long-range interaction cases, the complexity of this method depends on the number of representative particles but not on the total number of particles. Simple analysis and numerical experiments are provided to illustrate the accuracy and performance of the method. It is shown that, for the same accuracy, the quadrature-rule type method is much less costly than the quasicontinuum method.
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Iterative solution of the quasicontinuum equilibrium equations with continuation

by Matthew Dobson, Mitchell Luskin - Journal of Scientific Computing
"... Abstract. 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 precondit ..."
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Abstract. 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 efficient strategy for the parameter step size and number of iterations at each parameter value to achieve a solution to a required tolerance. We present computational results for the deformation of a Lennard-Jones chain under tension to demonstrate the necessity of carefully applying continuation to ensure that the computed solution remains in the domain of convergence of the iterative method as the parameter is increased. These results exhibit fracture before the actual load limit if the parameter step size is too large. 1.
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Contents lists available at ScienceDirect

by Lin Yang
"... jo ur nal homepage: www.elsev ier.de / i j leo LiDAR data reduction assisted by optical image fo ..."
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jo ur nal homepage: www.elsev ier.de / i j leo LiDAR data reduction assisted by optical image fo

Goal-Oriented Adaptive Mesh Refinement for the Quasicontinuum Approximation of a Frenkel-Kontorova Model

by Marcel Arndt A, Mitchell Luskin A
"... The quasicontinuum approximation [1] is a method to reduce the atomistic degrees of freedom of a crystalline solid by piecewise linear interpolation from representative atoms that are nodes for a finite element triangulation. In regions of the crystal with a highly nonuniform deformation such as aro ..."
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The quasicontinuum approximation [1] is a method to reduce the atomistic degrees of freedom of a crystalline solid by piecewise linear interpolation from representative atoms that are nodes for a finite element triangulation. In regions of the crystal with a highly nonuniform deformation such as around defects, every atom must be a representative atom to obtain sufficient accuracy, but the mesh can be coarsened away from such regions to remove atomistic degrees of freedom while retaining sufficient accuracy. We present an error estimator and a related adaptive mesh refinement algorithm for the quasicontinuum approximation of a generalized Frenkel-Kontorova model that enables a quantity of interest to be efficiently computed to a predetermined accuracy. Key words: error estimation, adaptive mesh refinement, atomistic-continuum modeling, quasicontinuum, Frenkel-Kontorova model 2000 MSC: 65Z05, 70C20, 70G75 1

Running Head: Iterative Solution of the QC Equilibrium Equations with Continuation

by Matthew Dobson, Mitchell Luskin , 2008
"... We give an analysis of a continuation algorithm for the numerical solution of the force-based 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. Th ..."
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We give an analysis of a continuation algorithm for the numerical solution of the force-based 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 efficient strategy for the parameter step size and number of iterations at each parameter value to achieve a solution to a required tolerance. We present computational results for the deformation of a Lennard-Jones chain under tension to demonstrate the necessity of carefully
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A POSTERIORI ERROR CONTROL FOR A QUASICONTINUUM APPROXIMATION OF A PERIODIC CHAIN

by Christoph Ortner, Hao Wang
"... Abstract. We consider a 1D periodic atomistic model, for which we formulate and analyze an adaptive variant of a quasicontinuum method. We establish a posteriori error estimates for the energy norm and for the energy, based on a posteriori residual and stability estimates. We formulate adaptive mesh ..."
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Abstract. We consider a 1D periodic atomistic model, for which we formulate and analyze an adaptive variant of a quasicontinuum method. We establish a posteriori error estimates for the energy norm and for the energy, based on a posteriori residual and stability estimates. We formulate adaptive mesh refinement algorithms based on these error estimators. Our numerical experiments indicate optimal convergence rates of these algorithms. 1.
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by unknown authors
"... , 1 U rue ont ted, wh em i.e. to find the optimal configuration of the overlap region between the molecular and continuummodels, in order to deliver approximations of quantities of interest within some preset accuracy. Performance of the adaptive strategy is demonstrated on two-dimensional numerical ..."
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, 1 U rue ont ted, wh em i.e. to find the optimal configuration of the overlap region between the molecular and continuummodels, in order to deliver approximations of quantities of interest within some preset accuracy. Performance of the adaptive strategy is demonstrated on two-dimensional numerical examples. 2009 Elsevier B.V. All rights reserved. impo nginee d seve n be si well-known method for multiscale computations of particle sys-tems is the quasi-continuum method [25] even though it shall be considered as a coarsening method rather than one which explic-itly couples two different models. In this paper, we focus on a vol-ume coupling method based on the Arlequin framework [7–9]. Arlequin coupling was recently extended and analyzed for the cou-such as those dealing with heterogeneous materials [18,26] or wave propagation in elastic materials [22]. A general review can be found in [21]. For the particular case of atomic-to-continuum coupling methods, modeling error estimates and adaptive proce-dures have been derived in [23,1] for the quasi-continuummethod. In this paper, we extend the Goals algorithm to multiscale sim-ulations using an atomic-to-continuum coupling method based on the Arlequin approach. The methods are presented on two-dimen-sional model problems in which either harmonic or Lennard–Jones potentials are considered for the atomic model and plane stress linear elasticity is selected for the continuum model. We
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BLENDING METHODS FOR COUPLING ATOMISTIC AND CONTINUUM MODELS

by P. Bochev, R. Lehoucq, M. Parks, S. Badia, M. Gunzburger
"... ..."
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