Skip to main content
SpringerLink
Log in

Numerical Methods for Constructing Suboptimal Packings of Nonconvex Domains with Curved Boundary

Journal of Applied and Industrial Mathematics Aims and scope Submit manuscript

Abstract

We study the problem of constructing some optimal packings of simply-connected nonconvex plane domains with a union of congruent circles. We consider the minimization of the radius of circles if the number of the circles is fixed. Using subdifferential calculus, we develop theoretical methods for solution of the problem and propose an approach for constructing some suboptimal packings close to optimal. In the numerical algorithms, we use the iterative procedures and take into account mainly the location of the current center of a packing element, the centers of the nearest neighboring elements, and the points of the boundary of the domain. The algorithms use the same supergradient ascent scheme whose parameters can be adapted to the number of packing elements and the geometry of the domain. We present a new software package whose efficiency is demonstrated by several examples of numerical construction of some suboptimal packings of the nonconvex domains bounded by the Cassini oval, a hypotrochoid, and a cardioid.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price includes VAT (Canada)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.

REFERENCES

  1. N. N. Krasovskii and A. I. Subbotin, Positional Differential Games (Nauka, Moscow, 1974) [in Russian].

    MATH  Google Scholar 

  2. V. N. Ushakov, A. A. Uspenskii, and A. G. Malev, “An Estimate of the Stability Defect for a Positional Absorption Set Subjected to Discriminant Transformations,” Proc. Steklov Inst. Math. 279 (Suppl. 1), 113–129 (2012).

    Article  Google Scholar 

  3. V. N. Ushakov, P. D. Lebedev, and N. G. Lavrov, “Algorithms for Optimal Packings in Ellipses,” Vestnik Yuzhno-Ural. Gos. Univ. Ser. Mat. Model. Program. 10 (3), 67–79 (2017).

    MATH  Google Scholar 

  4. P. D. Lebedev and A. L. Kazakov, “Iterative Methods for Constructing Planar Packings of Circles of Different Sizes,” Trudy Inst. Mat. Mekh. 24 (2), 141–151 (2018).

    MathSciNet  Google Scholar 

  5. J. Machchhar and G. Elber, “Dense Packing of Congruent Circles in Free-Form Nonconvex Containers,” Comput. Aided Geom. Des. 52–53, 13–27 (2017).

    Article  Google Scholar 

  6. L. Meng, C. Wang, and X. Yao, “Nonconvex Shape Effects on the Dense Random Packing Properties of Assembled Rods,” Physica A, 490, 212–221 (2017).

    Article  Google Scholar 

  7. M. Locatelli and U. Raber, “Packing Equal Circles in a Square: A Deterministic Global Optimization Approach,” Discrete Appl. Math. 122, 139–166 (2017).

    Article  MathSciNet  Google Scholar 

  8. Y. Li and H. Akeb, Basic Heuristics for Packing a Large Number of Equal Circles (Univ. Picardie Jules Verne, Amiens, 2005). [Available at www.researchgate.net/publication/250761942_Basic_Heuristics_ for_Packing_a_Great_Number_of_Equal_Circles (accessed August 13, 2020)].

  9. I. Litvinchev and L. Ozuna, “Approximate Packing Circles in a Rectangular Container: Valid Inequalities and Nesting„” J. Appl. Res. Technol. 12 (4), 716–723 (2014).

    Article  Google Scholar 

  10. Yu. G. Stoyan and G. Yas’kov, “A Mathematical Model and a Solution Method for the Problem of Placing Various-Sized Circles into a Strip,” European J. Oper. Res. 156, 590–600 (2004).

    Article  MathSciNet  Google Scholar 

  11. Yu. G. Stoyan and G. Yas’kov, “Packing Identical Spheres into a Rectangular Parallelepiped,” in Intelligent Decision Support: Current Challenges and Approaches (Gabler, Wiesbaden, 2008), pp. 46–67.

  12. M. Hifi and R. M’Hallah, “Approximate Algorithms for Constrained Circular Cutting Problems,” Comput. Oper. Res. 31, 675–694 (2004).

    Article  Google Scholar 

  13. A. M. Chugai, “A Solution to the Disk Packing Problem in a Convex Polygon with the Use of a Modified Method of Tapering Neighborhoods,” Radioélektron. Inform. No. 1, 58–63 (2005).

  14. P. D. Lebedev and A. A. Uspenskii, “Construction of the Optimal Resultant Function and Dispersing Lines in Time-Optimal Problems with a Nonconvex Target Set,” Trudy Inst. Mat. Mekh. 22 (2), 188–198 (2016).

    MathSciNet  Google Scholar 

  15. A. L. Kazakov, A. A. Lempert, and T. T. Ta, “The Sphere Packing Problem into Bounded Containers in Three-Dimension Non-Euclidean Space,” IFAC-PapersOnLine 51 (32), 782–787 (2018).

    Article  Google Scholar 

  16. A. L. Kazakov, A. A. Lempert, and H. L. Nguyen, “An Algorithm of Packing Congruent Circles in a Multiply Connected Set with Non-Euclidean Metrics,” Vychisl. Metody Program. 17 (2), 177–188 (2016).

    Google Scholar 

  17. A. L. Kazakov, A. A. Lempert, and H. L. Nguyen, “The Problem of the Optimal Packing of the Equal Circles for Special Non-Euclidean Metric,” Comm. Comput. Inform. Sci. 661, 58–68 (2017).

    Article  Google Scholar 

  18. P. D. Lebedev and A. A. Uspenskii, “Algorithms of Optimal Packing Construction in a \(3\)-Dimensional Euclidian Space,” in Modern Problems in Mathematics and Its Applications. (Proceedings of 47th International Youth School-Conference MPMA-2016, Yekaterinburg, Russia, January 31–February 6, 2016) (RWTH Aachen Univ., Aachen, 2016), pp. 84–93

  19. A. I. Subbotin, Generalized Solutions of First-Order PDEs: The Dynamical Optimization Perspective (Birkhäuser, Boston, 1995; Inst. Comput. Technol., Moscow, 2003).

    Book  Google Scholar 

  20. V. F. Demyanov and L. V. Vasilyev, Nondifferentiable Optimization (Nauka, Moscow, 1981) [in Russian].

    Google Scholar 

  21. I. V. Bychkov, A. L. Kazakov, A. A. Lempert, D. S. Bukharov, and A. B. Stolbov, “An Intelligent Management System for the Development of a Regional Transport Logistics Infrastructure,” Automat. Remote Control 77 (2), 332–343 (2016).

    Article  MathSciNet  Google Scholar 

  22. V. L. Rvachev and Yu. G. Stoyan, “The Problem of Optimal Placement of Circular Ingots,” Kibernetika No. 3, 77–83 (1965).

  23. I. Castillo, F. J. Kampas, and J. D. Pinter, “Solving Circle Packing Problems by Global Optimization: Numerical Results and Industrial Applications,” European J. Oper. Res. 191 (3), 786–802 (2008).

    Article  MathSciNet  Google Scholar 

  24. F. Harary, W. Randolph, and P. G. Mezey, “A Study of Maximum Unit-Circle Caterpillars—Tools for the Study of the Shape of Adsorption Patterns,” Discrete Appl. Math. 67 (1–3), 127–135 (1996).

    Article  MathSciNet  Google Scholar 

  25. A. V. Eremeev, L. A. Zaozerskaya, and A. A. Kolokolov, “The Set Covering Problem: Complexity, Algorithms, and Experimental Studies,” Diskret. Anal. Issled. Oper. Ser. 2, 7 (2), 22–46 (2000).

    MATH  Google Scholar 

  26. R. Rockafellar, Convex Analysis (Princeton Univ., Princeton, 1970; Mir, Moscow, 1973).

    Book  Google Scholar 

  27. P. D. Lebedev and A. V. Ushakov, “Approximating Sets on a Plane with Optimal Sets of Circles,” Automat. Remote Control 73 (3), 485–493 (2012).

    Article  MathSciNet  Google Scholar 

  28. A. G. Sukharev, A. V. Timokhov, and V. V. Fedorov, A Course in Optimization Methods (Nauka, Moscow, 1986) [in Russian].

    MATH  Google Scholar 

  29. E A. Nurminskii and D. Tien, “Method of Conjugate Subgradients with Constrained Memory,” Automat. Remote Control 75 (4), 646–656 (2012).

    Article  MathSciNet  Google Scholar 

  30. E. A. Vorontsova, “Linear Tolerance Problem for Input-Output Model with Interval Data,” Vychisl. Tekhnol. 22 (2), 67–84 (2017).

    MATH  Google Scholar 

  31. A. V. Gasnikov, P. E. Dvurechenskii, D. I. Kamzolov, Yu. E. Nesterov, V. G. Spokoiny, P. I. Stetsyuk, A. L. Suvorikova, and A. V. Chernov, “Searching for Equilibriums in Multistage Transport Models,” Trudy Moskov. Fiz.-Tekh. Inst. 7 (4), 143–155 (2015).

    Google Scholar 

  32. P. D. Lebedev, “A Program for Calculating the Optimal Coverage of a Hemisphere with a Set of Spherical Segments,” Certificate of State Registration No. 2015661543 from 29.10.2015 [in Russian].

  33. L. F. Tóth, Lagerungen in der Ebene, auf der Kugel und im Raum (Springer, Heidelberg, 1953; Fizmatgiz, Moscow, 1958).

    Book  Google Scholar 

  34. L. M. Mestetskii, Continuous Morphology of Binary Images: Figures, Skeletons, and Circulars (Fizmatlit, Moscow, 2009) [in Russian].

    Google Scholar 

  35. P. D. Lebedev and A. A. Uspenskii, “Construction of a Nonsmooth Solution to the Time-Optimal Problem with a Low Order of Smoothness of the Target Set Boundary,” Trudy Inst. Mat. Mekh. 25 (1), 108–119 (2019).

    MathSciNet  Google Scholar 

  36. A. A. Savyolov, Flat Curves: Systematics, Properties, and Applications (Librokom, Moscow, 2010) [in Russian].

    Google Scholar 

  37. E. Specht, Packomania (2018). Available at www.packomania.com (accessed August 13, 2020).

Download references

Funding

P. D. Lebedev and V. N. Ushakov were supported by the Russian Science Foundation (project no. 19–11–00105).

Author information

Authors and Affiliations

  1. Krasovskii Institute of Mathematics and Mechanics, Yekaterinburg, 620990, Russia

    P. D. Lebedev, V. N. Ushakov & A. A. Uspenskii

Authors
  1. P. D. Lebedev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. N. Ushakov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. A. Uspenskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to P. D. Lebedev, V. N. Ushakov or A. A. Uspenskii.

Additional information

Translated by G.A. Chumakov

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lebedev, P.D., Ushakov, V.N. & Uspenskii, A.A. Numerical Methods for Constructing Suboptimal Packings of Nonconvex Domains with Curved Boundary. J. Appl. Ind. Math. 14, 681–692 (2020). https://doi.org/10.1134/S1990478920040079

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1990478920040079

Keywords

search

Navigation