Packing stretched convex polygons in an optimized rectangle

A nonstandard optimized packing convex polygons in a rectangular container is considered. The shapes of the polygons are not fixed: the polygons can be compressed/stretched in certain limits along the principal axes, but their areas remain constant. The polygons must be placed completely in the container without overlapping under free translations and rotations. The objective is to define the shapes of the polygons and their locations to minimize the height of the container. Non-overlapping and containment conditions for the stretched polygons are presented using the phi-function technique. The overall packing problem is formulated in the form of a nonconvex nonlinear programming problem. A solution approach is proposed based on the multistart strategy. To illustrate the main steps of the solution technique a numerical example is provided. Directions for future research are discussed. Our interest in this class of packing problems is motivated by studying properties of porous media under external force. Elements of porous media can be deformed under pressure, but the mass of each particle is conserved. In two-dimensional case this corresponds to the area conservation.