Matthew Dobson

Matthew Dobson, Mitchell Luskin

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.

Matthew Dobson, Mitchell Luskin

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.

Matthew Dobson, Mitchell Luskin

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.

Matthew Dobson, Christoph Ortner, Alexander V. Shapeev

We show under general conditions that the linearized force-based quasicontinuum (QCF) operator has a positive spectrum, which is identical to the spectrum of the quasinonlocal quasicontinuum (QNL) operator in the case of second-neighbour interactions. Moreover, we establish a bound on the condition number of a matrix of eigenvectors that is uniform in the number of atoms and the size of the atomistic region. These results establish the validity of and improve upon recent conjectures ([arXiv:0907.3861, Conjecture 2] and [arXiv:0910.2013, Conjecture 8]) which were based on numerical experiments. As immediate consequences of our results we obtain rigorous estimates for convergence rates of (preconditioned) GMRES algorithms, as well as a new stability estimate for the QCF method.

Matthew Dobson, Claude Le Bris, Frederic Legoll

We follow up on our previous works which presented a possible approach for deriving symplectic schemes for a certain class of highly oscillatory Hamiltonian systems. The approach considers the Hamilton-Jacobi form of the equations of motion, formally homogenizes it and infers an appropriate symplectic integrator for the original system. In our previous work, the case of a system exhibiting a single constant fast frequency was considered. The present work successfully extends the approach to systems that have either one varying fast frequency or several constant frequencies. Some related issues are also examined.

Matthew Dobson

This work presents a generalization of the Kraynik-Reinelt (KR) boundary conditions for nonequilibrium molecular dynamics simulations. In the simulation of steady, homogeneous flows with periodic boundary conditions, the simulation box moves with the flow, and it is possible for particle replicas to become arbitrarily close, causing a breakdown in the simulation. The KR boundary conditions avoid this problem for planar elongational flow and general planar mixed flow [J. Chem. Phys 133, 14116 (2010)] through careful choice of the initial simulation box and by periodically remapping the simulation box in a way that conserves replica locations. In this work, the ideas are extended to a large class of three dimensional flows by using multiple remappings for the simulation box. The simulation box geometry is no longer time-periodic (which was shown to be impossible for uniaxial and biaxial stretching flows in the original work by Kraynik and Reinelt [Int. J. Multiphase Flow 18, 1045 (1992)]). The presented algorithm applies to all flows with nondefective flow matrices, and in particular, to uniaxial and biaxial flows.

Matthew Dobson, Manh Hong Duong, Christoph Ortner

We develop a rigorous error analysis for coarse-graining of defect-formation free energy. For a one-dimensional constrained atomistic system, we establish the thermodynamic limit of the defect-formation free energy and obtain explicitly the rate of convergence. We then construct a sequence of coarse-grained energies with the same rate but significantly reduced computational cost. We illustrate our analytical results through explicit computations for the case of harmonic potentials and through numerical simulations.

Matthew Dobson

Much work has gone into the construction of quasicontinuum energies that reduce the coupling error along the interface between atomistic and continuum regions. The largest consistency errors are typically pointwise $O(\frac{1}{\eps})$ errors, and in some cases this has been reduced to pointwise O(1) errors. In this paper we show that one cannot create a coupling method using a finite-range coupling interface that has o(1)-consistency in the interface, and we use this to give an upper bound on the order of convergence in discrete $w^{1,p}$-norms in 1D.

Matthew Dobson, Abdel Kader Geraldo

Several numerical schemes are proposed for the solution of Nonequilibrium Langevin Dynamics (NELD), and the rate of convergence is analyzed. Due to the special deforming boundary conditions used, care must be taken when using standard stochastic integration schemes, and we demonstrate a loss of convergence for a naive implementation. We then present several first and second order schemes, in the sense of strong convergence.

Matthew Dobson, Ian Fox, Alexandra Saracino

We present two modifications of the standard cell list algorithm for nonequilibrium molecular dynamics simulations of homogeneous, linear flows. When such a flow is modeled with periodic boundary conditions, the simulation box deforms with the flow, and recent progress has been made developing boundary conditions suitable for general 3D flows of this type. For the typical case of short-ranged, pairwise interactions, the cell list algorithm reduces computational complexity of the force computation from O($N^2$) to O($N$), where $N$ is the total number of particles in the simulation box. The new versions of the cell list algorithm handle the dynamic, deforming simulation geometry. We include a comparison of the complexity and efficiency of the two proposed modifications of the standard algorithm.

Matthew Dobson, Mitchell Luskin, Christoph Ortner

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 atomistic-to-continuum 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 {\em up to the onset of instability of the atomistic energy}. We formulate a one-dimensional model problem with nearest and next-nearest 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 quasi-nonlocal QC approximation correctly reproduces the stability of the atomistic system, whereas the inconsistent energy-based 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.

Matthew Dobson, Mitchell Luskin, Christoph Ortner

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 class of atomistic-continuum hybrid models as well as other types of multiphysics models. However, the recently discovered unusual stability properties of the linearized force-based quasicontinuum (QCF) approximation, especially its indefiniteness, present a challenge to the development of efficient and reliable iterative methods. We present analytic and computational results for the generalized minimal residual (GMRES) solution of the linearized QCF equilibrium equations. We show that the GMRES method accurately reproduces the stability of the force-based approximation and conclude that an appropriately preconditioned GMRES method results in a reliable and efficient solution method.

Matthew Dobson, Mitchell Luskin, Christoph Ortner

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.

Matthew Dobson, Mitchell Luskin, Christoph Ortner

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.

Matthew Dobson, Mitchell Luskin

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.