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<p><b>Abstract</b>—We consider optimal placements of two-dimensional flexible (elastic, deformable) objects. The objects are discs of equal size placed within a rigid boundary. The paper is divided into two parts. In the first part, analytical results for three types of regular, periodic arrangements—the hexagonal, square, and triangular placements—are presented. The regular arrangements are analyzed for rectangular boundaries and radii of discs that are small compared to the area of the placement region, because, in this case, the influence of boundary conditions can be neglected. This situation is called the unbounded case. We show that, for 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 from the largest possible number of generally shaped units in arbitrary arrangements. These analytical results are used as guidances for testing stochastic algorithms optimizing placements of flexible objects. In the second part of the paper, mainly two problems are considered: The underlying physical model and a simulated annealing algorithm maximizing the number of flexible discs in equilibrium placements. Along with the physical model, an approximate formula is derived, reflecting the deformation/force relationship for a large range of deformations. This formula is obtained from numerical experiments which were performed for various sizes of discs and several elastic materials. The potential applications of the presented approach are in the design of new amorphous polymeric and related materials as well as in the design of package cushioning systems.</p>
Combinatorial optimization, placement problems, stochastic algorithms, simulated annealing, composite materials.

C. Wong, K. Leung, K. Hui, S. Cheung and A. Albrecht, "Optimal Placements of Flexible Objects: Part I: Analytical Results for the Unbounded Case," in IEEE Transactions on Computers, vol. 46, no. , pp. 890-904, 1997.
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