Convergence analysis of a heuristic collective sphere packing algorithm

Abstract

Computer simulation of random sphere packing is important for the study of densely packed particulate systems. In previous work, quasi dynamics method (QDM), a heuristic collective random sphere packing algorithm was developed to effectively handle large numbers of densely packed spheres in complex geometries. In this work, a theoretical analysis of the convergence of QDM is performed and the impact of algorithm step size on the convergence is discussed. System potential functions that measure the overall system overlaps are introduced and defined. By using different system potentials, the convergence/stability of QDM for a sphere packing domain with and without active boundary conditions is investigated. QDM is proved to be strictly convergent with small step size when no active boundary constraint exists. When active boundary constraint is imposed, the upper limit of step size for convergence and the criteria for step size selection are theoretically analysed and obtained. Our analyses focus on systems packed with mono-dispersed spheres. The mathematical approaches for the analysis, however, can be easily modified for poly-dispersed sphere systems and extended to analyse other collective packing algorithms.