Robust Algorithm to Generate a Diverse Class of Dense Disordered and Ordered Sphere Packings via Linear Programming

TL;DR

This paper introduces a linear programming-based algorithm to generate a wide variety of dense and disordered sphere packings across multiple dimensions efficiently and with high robustness, surpassing previous methods in speed and accuracy.

Contribution

The authors develop a sequential linear programming approach for the Adaptive Shrinking Cell formulation, enabling the generation of diverse jammed sphere packings in multiple dimensions with improved efficiency and robustness.

Findings

Successfully generated jammed packings in dimensions 2 to 6.

Produced packings with densities ranging from low to near maximum.

Demonstrated higher robustness and speed compared to previous LS procedures.

Abstract

We have formulated the problem of generating periodic dense paritcle packings as an optimization problem called the Adaptive Shrinking Cell (ASC) formulation [S. Torquato and Y. Jiao, Phys. Rev. E {\bf 80}, 041104 (2009)]. Because the objective function and impenetrability constraints can be exactly linearized for sphere packings with a size distribution in $d$-dimensional Euclidean space $\mathbb{R}^d$, it is most suitable and natural to solve the corresponding ASC optimization problem using sequential linear programming (SLP) techniques. We implement an SLP solution to produce robustly a wide spectrum of jammed sphere packings in $\mathbb{R}^d$ for $d=2,3,4,5$ and $6$ with a diversity of disorder and densities up to the maximally densities. This deterministic algorithm can produce a broad range of inherent structures besides the usual disordered ones with very small computational cost by tuning the radius of the {\it influence sphere}. In three dimensions, we show that it can produce with high probability a variety of strictly jammed packings with a packing density anywhere in the wide range $[0.6, 0.7408...]$. We also apply the algorithm to generate various disordered packings as well as the maximally dense packings for $d=2,3, 4,5$ and 6. Compared to the LS procedure, our SLP protocol is able to ensure that the final packings are truly jammed, produces disordered jammed packings with anomalously low densities, and is appreciably more robust and computationally faster at generating maximally dense packings, especially as the space dimension increases.