Periodic cubic six-way cylinder packing structures

Packing problems are an important aspect of crystallography. In particular, sphere packings have played an important role in improving our understanding of crystal structures. Cylinder packings are also important for the same reason and have been investigated in the fields of both science and engineering. In the field of science, the complex structure of garnet has been explained on the basis of cylinder packing to be a periodic structure with a cubic <111> four-way cylinder packing [1a]. In the field of engineering, cylinder packings are important for determining the fiber packings of composite materials. Some regular fiber packing structures have been designed. Motivated by structures of composite materials, periodic cubic <110> six-way cylinder packing structures have also been investigated [1b]. The known <110> six-way cylinder packings can be classified into three categories on the basis of packing density: (√2)π/9 ≍ 0.494 (Type-I), (√2)π/18 ≍ 0.247 (Type-II), and (351√2 + 108√6)π/1936 ≍ 0.376 (Type-III). Recently, Teshima and Matsumoto studied the space group of the Type-III structure [2]. And Moore reported another type of periodic cubic <110> six-way cylinder packing structure (packing density ≍ 0.133) [3a,b]. In this study, authors consider a general description of periodic cubic <110> six-way cylinder packing structures.

Cylinder packing by simulated annealing

This paper is motivated by the problem of loading identical items of circular base (tubes, rolls, ...) into a rectangular base (the pallet). For practical reasons, all the loaded items are considered to have the same height. The resolution of this problem consists in determining the positioning pattern of the circular bases of the items on the rectangular pallet, while maximizing the number of items. This pattern will be repeated for each layer stacked on the pallet. Two algorithms based on the meta-heuristic Simulated Annealing have been developed and implemented. The tuning of these algorithms parameters implied running intensive tests in order to improve its efficiency. The algorithms developed were easily extended to the case of non-identical circles.