Results 1  10
of
50
Analysis of a forcebased quasicontinuum approximation
 M2AN Math. Model. Numer. Anal
"... Abstract. We analyze a forcebased quasicontinuum approximation to a onedimensional system of atoms that interact by a classical atomistic potential. This forcebased quasicontinuum approximation is derived as the modification of an energybased quasicontinuum approximation by the addition of nonco ..."
Abstract

Cited by 56 (25 self)
 Add to MetaCart
(Show Context)
Abstract. We analyze a forcebased quasicontinuum approximation to a onedimensional system of atoms that interact by a classical atomistic potential. This forcebased quasicontinuum approximation is derived as the modification of an energybased quasicontinuum approximation by the addition of nonconservative forces to correct nonphysical “ghost ” forces that occur in the atomistic to continuum interface. We prove that the forcebased quasicontinuum equations have a unique solution under suitable restrictions on the loads. For LennardJones nextnearestneighbor interactions, we show that unique solutions exist for loads in a symmetric region extending nearly to the tensile limit. We give an analysis of the convergence of the ghost force iteration method to solve the equilibrium equations for the forcebased quasicontinuum approximation. We show that the ghost force iteration is a contraction and give an analysis for its convergence rate. 1.
Analysis of a onedimensional nonlocal quasicontinuum method
 SIMULATION 7(4), 1838–1875 (2009). DOI 10.1137/080725842. URL HTTP://LINK.AIP.ORG/LINK/?MMS/7/1838/1
"... The accuracy of the quasicontinuum method is analyzed using a series of models with increasing complexity. It is demonstrated that the existence of the ghost force may lead to large errors. It is also shown that the ghost force removal strategy proposed by E, Lu and Yang leads to a version of the qu ..."
Abstract

Cited by 55 (4 self)
 Add to MetaCart
The accuracy of the quasicontinuum method is analyzed using a series of models with increasing complexity. It is demonstrated that the existence of the ghost force may lead to large errors. It is also shown that the ghost force removal strategy proposed by E, Lu and Yang leads to a version of the quasicontinuum method with uniform accuracy.
An optimal order error analysis of the onedimensional quasicontinuum approximation
 SIAM J. Numer. Anal
"... Abstract. We derive a model problem for quasicontinuum approximations that allows a simple, yet insightful, analysis of the optimalorder convergence rate in the continuum limit for both the energybased quasicontinuum approximation and the quasinonlocal quasicontinuum approximation. For simplicity ..."
Abstract

Cited by 51 (18 self)
 Add to MetaCart
(Show Context)
Abstract. We derive a model problem for quasicontinuum approximations that allows a simple, yet insightful, analysis of the optimalorder convergence rate in the continuum limit for both the energybased quasicontinuum approximation and the quasinonlocal quasicontinuum approximation. For simplicity, the analysis is restricted to the case of secondneighbor interactions and is linearized about a uniformly stretched reference lattice. The optimalorder error estimates for the quasinonlocal 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 atomistictocontinuum interface, combined with an analysis of the error due to the atomistic and continuum schemes using the stability of the quasicontinuum approximation.
Stability, Instability, and Error of the Forcebased Quasicontinuum Approximation
, 2009
"... Due to their algorithmic simplicity and high accuracy, forcebased model coupling techniques are popular tools in computational physics. For example, the forcebased quasicontinuum approximation is the only known pointwise consistent quasicontinuum (QC) approximation for coupling a general atomistic ..."
Abstract

Cited by 51 (32 self)
 Add to MetaCart
(Show Context)
Due to their algorithmic simplicity and high accuracy, forcebased model coupling techniques are popular tools in computational physics. For example, the forcebased quasicontinuum approximation is the only known pointwise consistent quasicontinuum (QC) approximation for coupling a general atomistic model with a finite element continuum model. In this paper, we present a detailed stability and error analysis of this method. Our optimal order error estimates provide a theoretical justification for the high accuracy of the forcebased QC approximation: they clearly demonstrate that the computational efficiency of continuum modeling can be utilized without a significant loss of accuracy if defects are captured in the atomistic region. The main challenge we need to overcome is the fact (which we prove) that the linearized QC operator is typically not positive definite. Moreover, we prove that no uniform infsup stability condition holds for discrete versions of the W 1,pW 1,q “duality pairing ” with 1/p +1/q = 1, if 1 ≤ p < ∞. We therefore derive an infsup stability condition for a discrete version of the W 1, ∞W 1,1 “duality pairing ” which then leads to optimal order error estimates in a discrete W 1, ∞norm.
Accuracy of quasicontinuum approximations near instabilities
 J. Mech. Phys. Solids
"... Abstract. 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 ..."
Abstract

Cited by 45 (34 self)
 Add to MetaCart
(Show Context)
Abstract. 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 accuracy of the approximation can only be guaranteed near deformations where both the atomistic energy as well as the hybrid energy are stable. We propose, therefore, that it is essential for the evaluation of the predictive capability of atomistictocontinuum coupling methods near instabilities that a theoretical analysis be performed, at least for some representative model problems, that determines whether the hybrid energies remain stable up to the onset of instability of the atomistic energy. We formulate a onedimensional model problem with nearest and nextnearest neighbor interactions and use rigorous analysis, asymptotic methods, and numerical experiments to obtain such sharp stability estimates for the basic conservative quasicontinuum (QC) approximations. Our results show that the consistent quasinonlocal QC approximation correctly reproduces the stability of the atomistic system, whereas the inconsistent energybased QC approximation incorrectly predicts instability at a significantly reduced applied load that we describe by an analytic criterion in terms of the derivatives of the atomistic potential. 1.
AN ANALYSIS OF THE EFFECT OF GHOST FORCE OSCILLATION ON QUASICONTINUUM ERROR
 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 onedimensional quasicontinuum energy around a uniform strain that the effect of the ghost forces on th ..."
Abstract

Cited by 44 (14 self)
 Add to MetaCart
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 onedimensional 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(hlog 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.
Sharp stability estimates for the forcebased quasicontinuum approximation of homogeneous tensile deformation. Multiscale Model
 Simul
"... Abstract. A sharp stability analysis of atomistictocontinuum coupling methods is essential for evaluating their capabilities for predicting the formation and motion of lattice defects. We formulate a simple onedimensional model problem and give a detailed analysis of the stability of the forceba ..."
Abstract

Cited by 37 (26 self)
 Add to MetaCart
(Show Context)
Abstract. A sharp stability analysis of atomistictocontinuum coupling methods is essential for evaluating their capabilities for predicting the formation and motion of lattice defects. We formulate a simple onedimensional model problem and give a detailed analysis of the stability of the forcebased quasicontinuum (QCF) method. The focus of the analysis is the question whether the QCF method is able to predict a critical load at which fracture occurs. Numerical experiments show that the spectrum of a linearized QCF operator is identical to the spectrum of a linearized energybased quasinonlocal quasicontinuum operator (QNL), which we know from our previous analyses to be positive below the critical load. However, the QCF operator is nonnormal and it turns out that it is not generally positive definite, even when all of its eigenvalues are positive. Using a combination of rigorous analysis and numerical experiments, we investigate in detail for which choices of “function spaces ” the QCF operator is stable, uniformly in the size of the atomistic system. Forcebased multiphysics coupling methods are popular techniques to circumvent the difficulties faced in formulating consistent energybased coupling approaches. Even though the QCF method is possibly the simplest coupling method of this kind, we anticipate that many of our observations apply more generally. 1.
The role of the patch test in 2d atomistictocontinuum coupling methods
, 2011
"... Copyright and reuse: The Warwick Research Archive Portal (WRAP) makes this work of researchers of the University of Warwick available open access under the following conditions. Copyright © and all moral rights to the version of the paper presented here belong to the individual author(s) and/or othe ..."
Abstract

Cited by 25 (14 self)
 Add to MetaCart
(Show Context)
Copyright and reuse: The Warwick Research Archive Portal (WRAP) makes this work of researchers of the University of Warwick available open access under the following conditions. Copyright © and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable the material made available in WRAP has been checked for eligibility before being made available. Copies of full items can be used for personal research or study, educational, or notforprofit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way. Publisher’s statement:
M.: Error estimation and atomisticcontinuum adaptivity for the quasicontinuum approximation of a FrenkelKontorova model. Multiscale Model
 Simul
, 2008
"... Abstract. We propose and analyze a goaloriented 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 intere ..."
Abstract

Cited by 23 (8 self)
 Add to MetaCart
(Show Context)
Abstract. We propose and analyze a goaloriented 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 crystallographic defect described by a FrenkelKontorova model and present the results of numerical experiments. The numerical results show that our method gives an efficient estimate of the error and a nearly optimal atomisticcontinuum modeling strategy. 1.