Matthew Dobson

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. 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.

Accuracy of quasicontinuum approximations near instabilities

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 up to the onset of instability of the atomistic energy. We formulate a one-dimensional model problem with nearest and next-nearest neighbour 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.

Sharp Stability Estimates for the Force-Based Quasicontinuum Approximation of Homogeneous Tensile Deformation

The accuracy of atomistic-to-continuum hybrid methods can be guaranteed only for deformations where the lattice configuration is stable for both the atomistic energy and the hybrid energy. For this reason, a sharp stability analysis of atomistic-to-continuum coupling methods is essential for evaluating their capabilities for predicting the formation of lattice defects. We formulate a simple one-dimensional model problem and give a detailed analysis of the linear stability of the force-based quasicontinuum (QCF) method at homogeneous deformations. The focus of the analysis is the question of 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 (QNL) operator, 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.

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.

The Spectrum of the Force-Based Quasicontinuum Operator for a Homogeneous Periodic Chain

We show under general conditions that the linearized force-based quasicontinuum (QCF) operator has a real, positive spectrum. The spectrum is identical to that of the quasinonlocal quasicontinuum (QNL) operator in the case of second-neighbor interactions. We construct an eigenbasis for the linearized QCF operator whose condition number is uniform in the number of atoms and the size of the atomistic region. These results establish the validity of and improve upon recent numerical observations [M. Dobson, M. Luskin, and C. Ortner, Comput. Methods Appl. Mech. Engrg., 200 (2011), pp. 2697--2709, Multiscale Model. Simul., 8 (2010), pp. 782--802]. 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.

Periodic boundary conditions for long-time nonequilibrium molecular dynamics simulations of incompressible flows

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 deforms with the flow, and it is possible for image particles 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 [T. A. Hunt, S. Bernardi, and B. D. Todd, J. Chem. Phys. 133, 154116 (2010)] through careful choice of the initial simulation box and by periodically remapping the simulation box in a way that conserves image 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.

Cell list algorithms for nonequilibrium molecular dynamics

We present two modifications of the standard cell list algorithm that handle molecular dynamics simulations with deforming periodic geometry. Such geometry naturally arises in the simulation of homogeneous, linear nonequilibrium flow modeled with periodic boundary conditions, and recent progress has been made developing boundary conditions suitable for general 3D flows of this type. Previous works focused on the planar flows handled by Lees-Edwards or Kraynik-Reinelt boundary conditions, while the new versions of the cell list algorithm presented here are formulated to handle the general 3D deforming simulation geometry. As in the case of equilibrium, for short-ranged pairwise interactions, the cell list algorithm reduces the 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. We include a comparison of the complexity and efficiency of the two proposed modifications of the standard algorithm.