Goal-oriented adaptive mesh refinement for the quasicontinuum approximation of a Frenkel-Kontorova model

Marcel Arndt, Mitchell Luskin

Research output: Contribution to journalArticlepeer-review

20 Scopus citations


The quasicontinuum approximation [E.B. Tadmor, M. Ortiz, R. Phillips, Quasicontinuum analysis of defects in solids, Philos. Mag. A 73(6) (1996) 1529-1563] is a method to reduce the atomistic degrees of freedom of a crystalline solid by piecewise linear interpolation from representative atoms that are nodes for a finite element triangulation. In regions of the crystal with a highly nonuniform deformation such as around defects, every atom must be a representative atom to obtain sufficient accuracy, but the mesh can be coarsened away from such regions to remove atomistic degrees of freedom while retaining sufficient accuracy. We present an error estimator and a related adaptive mesh refinement algorithm for the quasicontinuum approximation of a generalized Frenkel-Kontorova model that enables a quantity of interest to be efficiently computed to a predetermined accuracy.

Original languageEnglish (US)
Pages (from-to)4298-4306
Number of pages9
JournalComputer Methods in Applied Mechanics and Engineering
Issue number49-50
StatePublished - Sep 15 2008

Bibliographical note

Funding Information:
This work was supported in part by National Science Foundation Grant DMS-0304326, the Department of Energy under Award Number DE-FG02-05ER25706, and by the Minnesota Supercomputing Institute.


  • Adaptive mesh refinement
  • Atomistic-continuum modeling
  • Error estimation
  • Frenkel-Kontorova model
  • Quasicontinuum


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