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

Mitchell Luskin, Christoph Ortner
2009 SIAM Journal on Numerical Analysis  
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
more » ... oth meshes. We offer some suggestions for how the accuracy of (cluster) summation rules may be improved. Introduction. The quasicontinuum (QC) method [17, 14, 19, 20 ] is a prototypical coarse-graining technique for the static and quasistatic simulation of crystalline solids. One of its key features is that, instead of coupling an atomistic model to a continuum model, it uses the atomistic model also in the continuum region where degrees of freedom are removed from the model by means of piecewise linear interpolation. However, the nonlocal nature of the atomistic interactions makes further approximation necessary to enable the computation of energies or forces with complexity proportional to the number of coarse degrees of freedom. Two families of approximations have been developed to achieve this goal. One family of approximations localizes the interactions by a strain energy density (based on the Cauchy-Born rule) which provides sufficient accuracy in regions away from defects where the strain gradient varies slowly. Classical finite element methodology can then be utilized in those regions modeled by the strain energy density. This class of QC approximations has been the subject of many recent mathematical analyses [5, 12, 13, 6, 2, 3, 4, 16, 18] . The purpose of the present paper is to investigate the second family of approximations that has been developed to reduce the computational complexity of the QC method. These methods, which have received far less attention in the mathematics literature, use summation rules (discrete variants of quadrature rules) to approximate the sums that define the QC energy or forces. To the best of our knowledge, the forcebased cluster summation rule of Knap and Ortiz [10], the nonlocal QC method based on a simple trapezoidal rule [14, sect. 3.3], and the latter's extension to energy-based cluster summation rules [7] have not been analyzed to date. These cluster summation rules approximate the sum over atom-based quantities by uniformly averaging over *
doi:10.1137/080743391 fatcat:rixxmbxlmrbhdoow2ita36m3ka