arXiv subjects

Mitchell Luskin

Publications and source records attributed to Mitchell Luskin.

85 records · Page 5Linked to original sources

arXiv2009

Sharp Stability Estimates for the Force-based Quasicontinuum Method

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 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 energy-based quasi-nonlocal quasicontinuum operator (QNL), which we know from our previous analyses to be positive below the critical load. However, the QCF operator is non-normal 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. Force-based multi-physics coupling methods are popular techniques to circumvent the difficulties faced in formulating consistent energy-based coupling pproaches. Even though the QCF method is possibly the simplest coupling method of this kind, we anticipate that many of our observations apply more generally.

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arXiv2009

Stability, Instability, and Error of the Force-based Quasicontinuum Approximation

Due to their algorithmic simplicity and high accuracy, force-based model coupling techniques are an exciting development in computational physics. For example, the force-based quasicontinuum approximation is the only known pointwise consistent quasicontinuum approximation for coupling a general atomistic model with a finite element continuum model. In this paper, we analyze the stability of the force-based quasicontinuum approximation. We then use our stability result to obtain an optimal order error analysis of this coupling method that provides theoretical justification for the high accuracy of the force-based quasicontinuum approximation -- the computational efficiency of continuum modeling can be utilized without the loss of significant 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 quasicontinuum operator is typically not positive definite. Moreover, we prove that no uniform inf-sup stability condition holds for discrete versions of the $W^{1,p}$-$W^{1,q}$ "duality pairing" with $1/p+1/q=1$, if $1 \leq p < \infty$. We must therefore derive an inf-sup stability condition for a discrete version of the $W^{1,\infty}$-$W^{1,1}$ "duality pairing" which then leads to optimal order error estimates in a discrete $W^{1,\infty}$-norm.

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arXiv2009

An Optimal Order Error Analysis of the One-Dimensional Quasicontinuum Approximation

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. 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 atomistic and continuum schemes using the stability of the quasicontinuum approximation.

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arXiv2008

Iterative Solution of the Quasicontinuum Equilibrium Equations with Continuation

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 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.

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arXiv2008

Non-ergodicity of Nose-Hoover dynamics

The numerical integration of the Nose-Hoover dynamics gives a deterministic method that is used to sample the canonical Gibbs measure. The Nose-Hoover dynamics extends the physical Hamiltonian dynamics by the addition of a "thermostat" variable, that is coupled nonlinearly with the physical variables. The accuracy of the method depends on the dynamics being ergodic. Numerical experiments have been published earlier that are consistent with non-ergodicity of the dynamics for some model problems. The authors recently proved the non-ergodicity of the Nose-Hoover dynamics for the one-dimensional harmonic oscillator. In this paper, this result is extended to non-harmonic one-dimensional systems. It is also shown for some multidimensional systems that the averaged dynamics for the limit of infinite thermostat "mass" have many invariants, thus giving theoretical support for either non-ergodicity or slow ergodization. Numerical experiments for a two-dimensional central force problem and the one-dimensional pendulum problem give evidence for non-ergodicity.

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arXiv2008

An Analysis of the Effect of Ghost Force Oscillation on Quasicontinuum Error

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 optimal rate O($h$) in the discrete $\ell^\infty$ norm and O($h^{1/p}$) in the $w^{1,p}$ norm for $1 \leq p < \infty.$ 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 $w^{1,\infty}$ 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 $\ell^\infty$ and $w^{1,p}$ norms.

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arXiv2008

An Analysis of Node-Based Cluster Summation Rules in the Quasicontinuum Method

We investigate two examples of node-based cluster summation rules that have been proposed for the quasicontinuum method: a force-based approach (Knap & Ortiz, J. Mech. Phys. Solids 49, 2001), and an energy-based approach which is a generalization of the non-local quasicontinuum method (Eidel & Stukowski, J. Mech. Phys. Solids, to appear). We show that, even for the case of nearest neighbour interaction in a one-dimensional periodic chain, both of these approaches create large errors when used with graded and, more generally, non-smooth meshes. These errors cannot be removed by increasing the cluster size. We offer some suggestions how the accuracy of (cluster) summation rules may be improved.

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arXiv2007

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

The quasicontinuum approximation 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.

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arXiv2007

Goal-oriented Atomistic-Continuum Adaptivity for the Quasicontinuum Approximation

We give a goal-oriented a posteriori error estimator for the atomistic-continuum modeling error in the quasicontinuum method, and we use this estimator to design an adaptive algorithm to compute a quantity of interest to a given tolerance by using a nearly minimal number of atomistic degrees of freedom. We present computational results that demonstrate the effectiveness of our algorithm for a periodic array of dislocations described by a Frenkel-Kontorova type model.

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arXiv2007

Error Estimation and Atomistic-Continuum Adaptivity for the Quasicontinuum Approximation of a Frenkel-Kontorova Model

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 computation of the structure of a crystallographic defect described by a Frenkel-Kontorova 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 atomistic-continuum modeling strategy.

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arXiv2006

Analysis of a Force-Based Quasicontinuum Approximation

We analyze a force-based quasicontinuum approximation to a one-dimensional system of atoms that interact by a classical atomistic potential. This force-based quasicontinuum approximation is derived as the modification of an energy-based 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 force-based quasicontinuum equations have a unique solution under suitable restrictions on the loads. For Lennard-Jones next-nearest-neighbor 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 force-based quasicontinuum approximation. We show that the ghost force iteration is a contraction and give an analysis for its convergence rate.

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arXiv2006

Non-ergodicity of the Nose-Hoover Thermostatted Harmonic Oscillator

The Nose-Hoover thermostat is a deterministic dynamical system designed for computing phase space integrals for the canonical Gibbs distribution. Newton's equations are modified by coupling an additional reservoir variable to the physical variables. The correct sampling of the phase space according to the Gibbs measure is dependent on the Nose-Hoover dynamics being ergodic. Hoover presented numerical experiments that show the Nose-Hoover dynamics to be non-ergodic when applied to the harmonic oscillator. In this article, we prove that the Nose-Hoover thermostat does not give an ergodic dynamics for the one-dimensional harmonic oscillator when the ``mass'' of the reservoir is large. Our proof of non-ergodicity uses KAM theory to demonstrate the existence of invariant tori for the Nose-Hoover dynamical system that separate phase space into invariant regions. We present numerical experiments motivated by our analysis that seem to show that the dynamics is not ergodic even for a moderate thermostat mass. We also give numerical experiments of the Nose-Hoover chain with two thermostats applied to the one-dimensional harmonic oscillator. These experiments seem to support the non-ergodicity of the dynamics if the masses of the reservoirs are large enough and are consistent with ergodicity for more moderate masses.

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arXiv2003

Computational modeling of microstructure

Many materials such as martensitic or ferromagnetic crystals are observed to be in metastable states exhibiting a fine-scale, structured spatial oscillation called microstructure; and hysteresis is observed as the temperature, boundary forces, or external magnetic field changes. We have developed a numerical analysis of microstructure and used this theory to construct numerical methods that have been used to compute approximations to the deformation of crystals with microstructure.

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