期刊
COMPUTATIONAL MECHANICS
卷 55, 期 4, 页码 617-642出版社
SPRINGER
DOI: 10.1007/s00466-015-1127-4
关键词
Goal-oriented adaptivity; Quasi-continuum; Cluster approximation; Graphene
资金
- Swedish Research Council
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.
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