Iterative Solution of the Quasicontinuum Equilibrium Equations with Continuation

TL;DR

This paper analyzes a continuation algorithm for solving force-based quasicontinuum equations, using an iterative method with an energy-based approximation as a preconditioner, and demonstrates its effectiveness through computational experiments on a Lennard-Jones chain.

Contribution

It introduces an efficient continuation strategy for quasicontinuum equations and analyzes parameter step size and iteration count for convergence.

Findings

Careful continuation prevents fracture prediction errors.

Optimal parameter step size ensures convergence.

Demonstrates the importance of continuation in fracture simulations.

Abstract

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.