A method of generation of starting arrangements in a problem of structure modelling of systems of densely packed objects
Keywords:
mathematical modelling, quasi -function, non-oriented convex polytopes, objects dense packingAbstract
In this paper a mathematical model of a dense packing problem of non-oriented convex polytopes into a cuboid of minimum height is constructed by using the quasi Ф-function.
An application of quasi Ф-functions allows to formulate mutual non-intersections conditions for a pair of objects as a set of inequalities systems left sides of which are infinitely differentiable functions. Owing to this fact a mathematical model of the problem is presented as a classical non-linear programming problem.
For construction of different starting points a special method is proposed. The method includes three stages. On the first and second stages helper problems are solved. The first helper problem allows us to find a covering of polytopes by spheres of minimal radius. The second one allows us to find a dense packing of spheres in an arrangement region. At the third stage parameters of separating planes between the dense packing spheres are calculated.
In order to find local extrema of the helper problems the IPOPT library is used.References
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