Efficient algorithms for the dense packing of congruent circles inside a square

TL;DR

This paper introduces two algorithms to efficiently find dense arrangements of congruent circles inside a square, significantly improving packing density through iterative optimization.

Contribution

The paper presents novel algorithms for dense circle packing inside a square, combining initial configuration generation and subsequent enhancement.

Findings

Algorithms effectively increase packing density for large numbers of circles.

Numerical tests demonstrate high performance and improved configurations.

Methods are versatile, working sequentially or independently.

Abstract

We study dense packings of a large number of congruent non-overlapping circles inside a square by looking for configurations which maximize the packing density, defined as the ratio between the area occupied by the disks and the area of the square container. The search for these configurations is carried out with the help of two algorithms that we have devised: a first algorithm is in charge of obtaining sufficiently dense configurations starting from a random guess, while a second algorithm improves the configurations obtained in the first stage. The algorithms can be used sequentially or independently. The performance of these algorithms is assessed by carrying out numerical tests for configurations with a large number of circles.