An Analysis of Node-Based Cluster Summation Rules in the Quasicontinuum Method

TL;DR

This paper critically examines node-based cluster summation rules in the quasicontinuum method, revealing significant errors with non-smooth meshes and discussing potential improvements for accuracy.

Contribution

It provides a detailed analysis of two cluster summation approaches, highlighting their limitations and proposing ways to enhance their accuracy in non-smooth mesh scenarios.

Findings

Both approaches produce large errors with graded meshes.

Errors persist despite increasing cluster size.

Suggestions for improving summation rule accuracy are discussed.

Abstract

We investigate two examples of node-based cluster summation rules that have been proposed for the quasicontinuum method: a force-based approach (Knap & Ortiz, J. Mech. Phys. Solids 49, 2001), and an energy-based approach which is a generalization of the non-local quasicontinuum method (Eidel & Stukowski, J. Mech. Phys. Solids, to appear). We show that, even for the case of nearest neighbour interaction in a one-dimensional periodic chain, both of these approaches create large errors when used with graded and, more generally, non-smooth meshes. These errors cannot be removed by increasing the cluster size. We offer some suggestions how the accuracy of (cluster) summation rules may be improved.