Efficient Algorithms for the Dense Packing of Congruent Circles Inside a Square

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