Iterative methods for the force-based quasicontinuum approximation: Analysis of a 1D model problem

M. Dobson, M. Luskin, C. Ortner

Research output: Contribution to journalArticlepeer-review

15 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)2697-2709
Number of pages13
JournalComputer Methods in Applied Mechanics and Engineering
Volume200
Issue number37-40
DOIs
StatePublished - Sep 1 2011

Bibliographical note

Funding Information:
This work was supported in part by DMS-0757355, DMS-0811039, the Department of Energy under Award Numbers DE-FG02-05ER25706 and DE-SC0002085, the University of Minnesota Supercomputing Institute, the University of Minnesota Doctoral Dissertation Fellowship, the NSF Mathematical Sciences Postdoctoral Research Fellowship, and the EPSRC critical mass programme “New Frontier in the Mathematics of Solids”.

Keywords

  • Atomistic-to-continuum coupling
  • Iterative methods
  • Quasicontinuum method
  • Stability

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