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Mathematical modeling of the interaction of non-oriented convex polytopes

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Abstract

An Φ-function for two non-oriented convex polytopes is set up. The Φ-function can be used to construct a mathematical model of packing optimization problem for non-oriented polytopes. An example of an Φ-function for two non-oriented parallelepipeds is given.

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References

  1. Yu. G. Stoyan and S. V. Yakovlev, Mathematical Models and Optimization Methods of Geometric Design [in Russian], Naukova Dumka, Kyiv (1986).

    Google Scholar 

  2. N. Mayer, E. Fogel, and D. Halperin, “Fast and robust retrieval of Minkowski sums of rotating convex polyhedra in 3-space,” in: Proc. 14th ACM Symp. on Solid and Physical Modeling, Haifa, Israel (2010), pp. 1–10.

  3. O. Takayuki, Approaches to 3D Free-Form Cutting and Packing Problems and their Applications: A Survey, IBM Research Laboratory, Tokyo (1998).

    Google Scholar 

  4. J. Egeblad, B. K. Nielsen, and M. Brazil, “Translational packing of arbitrary polytopes,” Computational Geometry: Theory and Applications, 42, No. 4, 269–288 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  5. S. Torquato and F. H. Stillinger, “Jammed hard-particle packings: From Kepler to Bernal and beyond,” Reviews of Modern Physics, 82, No. 3, 2633–2672 (2010).

    Article  MathSciNet  Google Scholar 

  6. M. Gan, N. Gopinathan, X. Jia, and R. A. Williams, “Predicting packing characteristics of particles of arbitrary shapes,” KONA, No. 22, 82–93 (2004).

    Google Scholar 

  7. V. S. Doroshenko, “3D-technologies in foundry,” Vynakhidnyk i Ratsionalizator, No. 2, 12–15 (2009).

  8. F. Eisenbrand, S. Funke, A. Karrenbauer, J. Reichel, and E. Schömer, “Packing a trunk — now with a twist!,” in: Proc. SPM 2005 ACM Symp. on Solid and Physical Modeling, New York, USA (2005), pp. 197–206.

  9. U. Betke and M. Henk, “Densest lattice packings of 3-polytopes,” Computational Geometry, 16, No. 3, 157–186 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Bennell and J. Oliveira, “The geometry of nesting problems: A tutorial,” Europ. J. of Operational Research, No. 184, 397–415 (2008).

  11. Yu. G. Stoyan, “Φ-function and its basic properties,” Dokl. NAN Ukr., Ser. A, No. 8, 112–117 (2001).

  12. N. Chernov, Y. Stoyan, and T. Romanova, “Mathematical model and efficient algorithms for object packing problem,” Computational Geometry: Theory and Applications, 43, No. 5, 535–553 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  13. G. Scheithauer, Yu. G. Stoyan, and T. Ye. Romanova, “Mathematical modeling of interactions of primary geometric 3D objects,” Cybern. Syst. Analysis, 41, No. 3, 332–342 (2005).

    Article  MATH  Google Scholar 

  14. Y. Stoyan, J. Terno, G. Scheithauer, M. Gil, and T. Romanova, “Construction of a Phi-function for two convex polytopes,” Applicationes Mathematicae, 29, No. 2, 199–218 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  15. Yu. Stoyan, A. Pankratov, and G. Sheithauer, “Translational packing non-convex polytopes into a parallelepiped,” J. of Mechanical Engineering, 12, No. 3, 67–76 (2009).

    Google Scholar 

  16. K. Kuratowski, Topology, Vol. 1, Academic Press, New York (1966).

    Google Scholar 

  17. Y. Stoyan, G. Scheithauer, M. Gil, and T. Romanova, “Φ-function for complex 2D objects,” 4OR, Quarterly Journal of the Belgian, French and Italian Operations Research Societies, 2, No. 1, 69–84 (2004).

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. A. N. Podgorny Institute for Problems of Mechanical Engineering, National Academy of Sciences of Ukraine,, Kharkov, Ukraine

    Yu. G. Stoyan & A. M. Chugay

Authors
  1. Yu. G. Stoyan
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  2. A. M. Chugay
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Correspondence to Yu. G. Stoyan.

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Translated from Kibernetika i Sistemnyi Analiz, No. 6, November–December, 2012, pp. 36–44.

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Stoyan, Y.G., Chugay, A.M. Mathematical modeling of the interaction of non-oriented convex polytopes. Cybern Syst Anal 48, 837–845 (2012). https://doi.org/10.1007/s10559-012-9463-2

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  • DOI: https://doi.org/10.1007/s10559-012-9463-2

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