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An analysis of node-based cluster summation rules in the quasicontinuum method. (English) Zbl 1196.82122

Summary: We investigate two examples of node-based cluster summation rules that have been proposed for the quasicontinuum (QC) method: a force-based approach and an energy-based approach which is a generalization of the nonlocal QC method. We show that, even for the case of nearest-neighbor interaction in a one-dimensional periodic chain, both of these approaches create large errors that cannot be removed by increasing the cluster size when used with graded and, more generally, nonsmooth meshes. We offer some suggestions for how the accuracy of (cluster) summation rules may be improved.

MSC:

82D25 Statistical mechanics of crystals
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
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