OPTIMAL PLACEMENTS OF FLEXIBLE OBJECTS .1. ANALYTICAL RESULTS FOR THEUNBOUNDED CASE

We consider optimal placements of two-dimensional flexible (elastic, d eformable) objects. The objects are discs of equal size placed within a rigid boundary. The paper is divided into two parts. In the first pa rt, analytical results for three types of regular, periodic arrangemen ts-the hexagonal, square, and triangular placements-are presented. The regular arrangements are analyzed for rectangular boundaries and radi i of discs that are small compared to the area of the placement region , because, in this case, the influence of boundary conditions can be n eglected. This situation is called the unbounded case. We show that, f or the unbounded case among the three regular placements, the type of hexagonal arrangements provides the largest number of placed units for the same deformation depth. Furthermore, it can be proved that these regular placements are not too far from the truly optimal arrangements . For example, hexagonal placements differ at most by the factor 1.1 f rom the largest possible number of generally shaped units in arbitrary arrangements. These analytical results are used as guidances for test ing stochastic algorithms optimizing placements of flexible objects. I n the second part of the paper, mainly two problems are considered: Th e underlying physical model and a simulated annealing algorithm maximi zing the number of flexible discs in equilibrium placements. Along wit h the physical model, an approximate formula is derived, reflecting th e deformation/force relationship for a large range of deformations. Th is formula is obtained from numerical experiments which were performed for various sizes of discs and several elastic materials. The potenti al applications of the presented approach are in the design of new amo rphous polymeric and related materials as well as in the design of pac kage cushioning systems.