Packing ovals in optimized regular polygons

Abstract

We present a model development framework and numerical solution approach to the general problem-class of packing convex objects into optimized convex containers. Specifically, here we discuss the problem of packing ovals (egg-shaped objects, defined here as generalized ellipses) into optimized regular polygons in ℝ 2 . Our solution strategy is based on the use of embedded Lagrange multipliers, followed by nonlinear (global-local) optimization. The numerical results are attained using randomized starting solutions refined by a single call to a local optimization solver. We obtain credible, tight packings for packing 4 to 10 ovals into regular polygons with 3 to 10 sides in all (224) test problems presented here, and for other similarly difficult packing problems.

A New Class of Irregular Packing Problems Reducible to Sphere Packing in Arbitrary Norms

“…Applications range from planning radio-surgical treatment of tumors [3,4] to studying structure of nanomaterials [5][6][7]; from spherical packing in coding theory [8,9] to modeling power bed fusion in additive manufacturing [10,11]. See also [12][13][14][15][16][17] and the references therein.…”

Applications of the Symmetrical Structures of Cassini Ovals

“…Cassini ovals can be used in various industrial engineering applications. The packing arrangement of solid objects for storage or transportation was discussed in [13]. An application of Cassini ovals for the formation of the 24-h light attention of the runway is provided in [14].…”

Packing spheres with quasi-containment conditions

“…Numerical techniques for sphere packing problems rely on different approaches. Exact solution methods based on principles like branch-and-bound are of particular importance for small problem instances [15][16][17]35]. For larger instances, combinations of heuristic and deterministic techniques are applied frequently [2,29].…”