Packing problem of non-oriented ellipses

Н.І. Gil', A.V. Pankratov, T.E. Romanova, I.A. Subbota
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The paper considers a problem of packing non-oriented ellipses within rectangular area (container) of minimal sizes. In order to model non-overlapping of rotating ellipses and containment of ellipses into a container phi-functions and quasi-phifunctions are derived. Mathematical model of the packing problem is constructed in the form of nonlinear programming problem. Efficient algorithm is proposed. The algorithm employs an original approach to construct starting points, the multistart method and IPOPT to search for a local minimum. Numerical experiments are given.
Keywords: Mathematical modeling, packing, continuous rotation, ellipses, phi-function, quasi-phi-function, optimization