A nonsmooth program for jamming hard spheres

TL;DR

This paper introduces a nonsmooth optimization approach to identify jammed sphere packings in high-dimensional spaces, providing a finite set of solutions and an algorithm for maximizing packing radii.

Contribution

It presents a novel nonsmooth function whose extrema correspond to jammed packings and an algorithm to find these configurations, applicable to various container shapes.

Findings

Finite number of jammed packing radii for fixed sphere count

Algorithm successfully finds jammed packings in 2D and 3D cases

Method generalizes to different container geometries

Abstract

We study packings of $n$ hard spheres of equal radius in the $d$-dimensional unit cube. We present a nonsmooth function whose local extrema are the radii of jammed packings (where no subset of spheres can be moved keeping all others fixed) and show that for a fixed number of spheres there are only finitely many radii of such jammed configurations. We propose an algorithm for the maximization of this maximal radius function and present examples for 5 - 8 disks in the unit square and 4 - 6 spheres in the unit cube. The method allows straightforward generalization to packings of spheres in other compact containers.