Iterative Methods for the Force-based Quasicontinuum Approximation

TL;DR

This paper investigates iterative solution methods for the force-based quasicontinuum approximation, demonstrating that preconditioned GMRES effectively handles its stability challenges and provides a reliable computational approach.

Contribution

The paper introduces an analysis of GMRES for linearized force-based quasicontinuum equations, showing its effectiveness with proper preconditioning.

Findings

GMRES accurately captures the stability properties of the QCF approximation.

Preconditioning makes GMRES a reliable and efficient solver.

The approach addresses the indefiniteness of the linearized QCF equations.

Abstract

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.