A goal-oriented adaptive procedure for the quasi-continuum method with cluster approximation
A. Memarnahavandi, F. Larsson and K. Runesson
Computational Mechanics, 55, 617-642 (2015).

ABSTRACT
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.