Packing Soft Convex Polygons in an Optimized Convex Container

Packing soft convex polygonal objects in an optimized convex container is considered. The shape of the soft object can be changed in certain limits, but the object remains convex and maintains its area unchanged under all shape transformations. Non-overlapping, containment, as well as convexity and area conservation constraints are presented. A nonlinear programming model is formulated to find the optimal container for fixed values of elasticity parameters. An inverse problem is considered to find the minimal value of the elasticity parameter resulting in the same optimal container. Numerical experiments for packing soft triangles and pentagons in optimized circular and square containers are provided.