Computational Mechanics

, Volume 55, Issue 4, pp 617–642 | Cite as

A goal-oriented adaptive procedure for the quasi-continuum method with cluster approximation

  • Arash MemarnahavandiEmail author
  • Fredrik Larsson
  • Kenneth Runesson
Original Paper


We present a strategy for adaptive error control for the quasi-continuum (QC) method applied to molecular statics problems. The QC-method is introduced in two steps: Firstly, introducing QC-interpolation while accounting for the exact summation of all the bond-energies, we compute goal-oriented error estimators in a straight-forward fashion based on the pertinent adjoint (dual) problem. Secondly, for large QC-elements the bond energy and its derivatives are typically computed using an appropriate discrete quadrature using cluster approximations, which introduces a model error. The combined error is estimated approximately based on the same dual problem in conjunction with a hierarchical strategy for approximating the residual. As a model problem, we carry out atomistic-to-continuum homogenization of a graphene monolayer, where the Carbon–Carbon energy bonds are modeled via the Tersoff–Brenner potential, which involves next-nearest neighbor couplings. In particular, we are interested in computing the representative response for an imperfect lattice. Within the goal-oriented framework it becomes natural to choose the macro-scale (continuum) stress as the “quantity of interest”. Two different formulations are adopted: The Basic formulation and the Global formulation. The presented numerical investigation shows the accuracy and robustness of the proposed error estimator and the pertinent adaptive algorithm.


Goal-oriented adaptivity Quasi-continuum Cluster approximation Graphene 



The support from Swedish Research Council is gratefully acknowledged. We thank Prof. Ragnar Larsson for helpful discussions.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Arash Memarnahavandi
    • 1
    • 2
    Email author
  • Fredrik Larsson
    • 1
  • Kenneth Runesson
    • 1
  1. 1.Department of Applied MechanicsChalmers University of TechnologyGöteborgSweden
  2. 2.University of WaterlooWaterlooCanada

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