This Article 
 Bibliographic References 
 Add to: 
Optimal Placements of Flexible Objects: Part I: Analytical Results for the Unbounded Case
August 1997 (vol. 46 no. 8)
pp. 890-904

Abstract—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.

[1] D.C. Allen, "Package Cushioning Systems," The Packaging Media, F.A. Paine, ed., pp. 5.44-5.64. Blackie&Son Ltd, 1977.
[2] ASM Int'l, ASM Engineered Materials Reference Book. ASM Int'l Reference Publications, 1989.
[3] Simulated Annealing: Parallelization Techniques, R. Azencott, ed. John Wiley&Sons, 1992.
[4] A. Bos, "Sphere-Packings in Euclidean Space," Packing and Covering in Combinatorics, A. Schrijver, ed., pp. 161-177,Amsterdam, 1979. Mathematical Centre Tracts 106.
[5] E.G. Coffman Jr. and G.S. Lueker, Probabilistic Analysis of Packing and Partitioning Algorithms.New York: Wiley&Sons, 1991.
[6] G. Galambos and A. van Vliet, "Lower Bounds for 1-, 2- and 3-Dimensional On-Line Bin-Packing Algorithms," Computing, vol. 52, pp. 281-297, 1994.
[7] A. Jagota and G.W. Scherer, "Viscosities and Sintering Rates of a Two-Dimensional Granular Composite," J. Am. Ceramic Soc., vol. 76, pp. 3,123-3,135, 1993.
[8] V.B. Kashirin and E.V. Kozlov, "New Approach to the Dense Random Packing of Soft Spheres," J. Non-Crystalline Solids, vol. 163, pp. 24-28, 1993.
[9] S. Kirkpatrick, C.D. Gelatt Jr., and M.P. Vecchi, "Optimization by Simulated Annealing," Science, vol. 220, pp. 671-680, May 1983.
[10] A. Engberts, W. Kozaczynski, and J. Ning, "Concept Recognition-Based Program Transformation," Proc. IEEE Conf. Software Maintenance, 1991.
[11] F. Romeo and A. Sangiovanni-Vincentelli, "A Theoretical Framework for Simulated Annealing," Algorithmica, vol. 6, no. 3, pp. 302-345, 1991.
[12] C. Sechen and A. Sangiovanni-Vincentelli, "The TimberWolf Placement and Routing Package," IEEE J. Solid-State Circuits, vol. 20, no. 3, pp. 510-522, Apr. 1985.
[13] W. Swartz and C. Sechen, "Time Driven Placement for Large Standard Cell Circuits," Proc. 32nd Design Automation Conf., pp. 211-215, 1995.
[14] I.M. Ward, Mechanical Properties of Solid Polymers. John Wiley&Sons, 1985.
[15] G. Woeginger, "Improved Space for Bounded-Space, One-Line Bin-Packing," SIAM J. Discrete Math., vol. 6, pp. 575-581, 1993.
[16] D.F. Wong and C.L. Liu, "Floorplan Design of VLSI Circuits," Algorithmica, vol. 4, pp. 263-291, 1989.
[17] A.Z. Zinchenko, "Algorithms for Random Close Packing of Spheres with Periodic Boundary Conditions," J. Computational Physics, vol. 114, pp. 298-307, 1994.

Index Terms:
Combinatorial optimization, placement problems, stochastic algorithms, simulated annealing, composite materials.
A. Albrecht, S.k. Cheung, K.c. Hui, K.s. Leung, C.k. Wong, "Optimal Placements of Flexible Objects: Part I: Analytical Results for the Unbounded Case," IEEE Transactions on Computers, vol. 46, no. 8, pp. 890-904, Aug. 1997, doi:10.1109/12.609278
Usage of this product signifies your acceptance of the Terms of Use.