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XProtoSphere: an eXtended multi-sized sphere packing algorithm driven by particle size distribution

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Abstract

The sphere packing problem, which involves filling an arbitrarily shaped geometry with the maximum number of non-overlap** spheres, is a critical research challenge. ProtoSphere is a prototype-oriented algorithm designed for solving sphere packing problems. Due to its easily parallelizable design, it exhibits high versatility and has wide-ranging applications. However, the controllable regulation of particle size distribution (PSD) produced by ProtoSphere is often neglected, which limits its application on algorithm. This paper proposes a novel PSD-driven technique that extends the ProtoSphere algorithm to achieve multi-sized sphere packing with distribution-specific characteristics, as dictated by a pre-defined cumulative distribution function. The proposed approach improves the controllability and flexibility of the packing process, and enables users to generate packing configurations that meet their specific requirements. In addition, by combining the relaxation method with the ProtoSphere algorithm, we can further improve the packing density and ensure the average overlap below 1%. Our method generates multi-sized particles that can be used to simulate the behavior of various granular materials, including sand-like and clay-like soils.

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References

  1. Adams, B., Pauly, M., Keiser, R., Guibas, L.J.: Adaptively sampled particle fluids. ACM Trans. Graph. (TOG) 26(3), 48 (2007)

    Article  Google Scholar 

  2. Aurenhammer, F.: Power diagrams: properties, algorithms and applications. SIAM J. Comput. 16(1), 78–96 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baran, I., Popović, J.: Automatic rigging and animation of 3D characters. ACM Trans. Graph. (TOG) 26(3), 72 (2007)

    Article  Google Scholar 

  4. Barber, C.B., Dobkin, D.P., Huhdanpaa, H.: The quickhull algorithm for convex hulls. ACM Trans. Math. Softw. (TOMS) 22(4), 469–483 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bell, N., Yu, Y., Mucha, P.J.: Particle-based simulation of granular materials. In: Proceedings of the 2005 ACM SIGGRAPH/Eurographics symposium on Computer animation, pp. 77–86 (2005)

  6. Bonneau, F., Scholtes, L., Rambure, H.: An algorithm for generating mechanically sound sphere packings in geological models. Comput. Particle Mech. 8(2), 201–214 (2021)

  7. Borkovec, M., De Paris, W., Peikert, R.: The fractal dimension of the apollonian sphere packing. Fractals 2(04), 521–526 (1994)

  8. Bridson, R.: Fast Poisson disk sampling in arbitrary dimensions. In: ACM SIGGRAPH 2007 Sketches, pp. 22 (2007)

  9. Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups, vol. 290. Springer Science & Business Media, Cham (2013)

    Google Scholar 

  10. Cui, L., O’Sullivan, C.: Analysis of a triangulation based approach for specimen generation for discrete element simulations. Granular Matter 5, 135–145 (2003)

    Article  MATH  Google Scholar 

  11. Cundall, P.A., Strack, O.D.: A discrete numerical model for granular assemblies. Geotechnique 29(1), 47–65 (1979)

    Article  Google Scholar 

  12. Desbrun, M., Cani, M.P.: Space-time adaptive simulation of highly deformable substances. Ph.D. thesis, INRIA (1999)

  13. Devroye, L.: Sample-based non-uniform random variate generation. In: Proceedings of the 18th conference on Winter simulation, pp. 260–265 (1986)

  14. George, P.L., Borouchaki, H.: Delaunay triangulation and meshing: application to finite elements. HermHermÃs (1998)

  15. Hales, T.C.: A proof of the kepler conjecture. Ann. Math. pp. 1065–1185 (2005)

  16. Hentz, S., Donzé, F.V., Daudeville, L.: Discrete element modelling of concrete submitted to dynamic loading at high strain rates. Comput. Struct. 82(29–30), 2509–2524 (2004)

    Article  Google Scholar 

  17. Hifi, M., M’Hallah, R.: A literature review on circle and sphere packing problems: models and methodologies. Adv. Oper. Res. 2009, 150624 (2009)

    MATH  Google Scholar 

  18. Jerier, J.F., Imbault, D., Donze, F.V., Doremus, P.: A geometric algorithm based on tetrahedral meshes to generate a dense polydisperse sphere packing. Granular Matter 11, 43–52 (2009)

    Article  MATH  Google Scholar 

  19. Jerier, J.F., Richefeu, V., Imbault, D., Donzé, F.V.: Packing spherical discrete elements for large scale simulations. Comput. Methods Appl. Mech. Eng. 199(25–28), 1668–1676 (2010)

    Article  MATH  Google Scholar 

  20. Jiang, M., Southern, R., Zhang, J.J.: Energy-based dissolution simulation using SPH sampling. Comput. Anim. Virtual Worlds 29(2), 1–20 (2018)

    Article  Google Scholar 

  21. Jiang, M., Zhou, Y., Wang, R., Southern, R., Zhang, J.J.: Blue noise sampling using an SPH-based method. ACM Trans. Graph. (TOG) 34(6), 1–11 (2015)

    Google Scholar 

  22. Kita, N., Miyata, K.: Multi-class anisotropic blue noise sampling for discrete element pattern generation. Vis. Comput. 32, 1035–1044 (2016)

    Article  Google Scholar 

  23. Lopes, L.G., Cintra, D.T., Lira, W.W.: A geometric separation method for non-uniform disk packing with prescribed filling ratio and size distribution. Comput. Particle Mech. 8, 169–182 (2021)

    Article  Google Scholar 

  24. Lopes, L.G., Cintra, D.T., Lira, W.W.: A particle packing parallel geometric method using GPU. Comput. Particle Mech. 8, 931–942 (2021)

    Article  Google Scholar 

  25. Schechter, H., Bridson, R.: Ghost SPH for animating water. ACM Trans. Graph. (TOG) 31(4), 1–8 (2012)

    Article  Google Scholar 

  26. Schwarz, M., Seidel, H.P.: Fast parallel surface and solid voxelization on GPUs. ACM Trans. Graph. (TOG) 29(6), 1–10 (2010)

    Article  Google Scholar 

  27. Šmilauer, V., Chareyre, B.: Yade dem formulation. http://yade-dem.org/doc/formulation.html

  28. Teuber, J., Weller, R., Zachmann, G., Guthe, S.: Fast sphere packings with adaptive grids on the gpu. In: GI AR/VRWorkshop, 12 pages (2013)

  29. Torquato, S., Haslach, H., Jr.: Random heterogeneous materials: microstructure and macroscopic properties. Appl. Mech. Rev. 55(4), B62–B63 (2002

  30. Tsuzuki, S., Aoki, T.: Large-scale granular simulations using dynamic load balance on a gpu supercomputer. In: Poster at the 26th IEEE/ACM International Conference on High Performance Computing, Networking, Storage and Analysis (2014)

  31. Wang, X., Fujisawa, M., Mikawa, M.: Visual simulation of soil-structure destruction with seepage flows. Proc. ACM Comput. Graph. Interact. Tech. 4(3), 1–18 (2021)

    Article  Google Scholar 

  32. Wang, X., Fujisawa, M., Mikawa, M.: Multi-sized particle sampling method based on porosity optimization in 2D space. IIEEJ Trans. Image Electron. Vis. Comput. 10(2), 150–161 (2022)

    Google Scholar 

  33. Weller, R., Zachmann, G.: A unified approach for physically-based simulations and haptic rendering. In: Proceedings of the 2009 ACM SIGGRAPH Symposium on Video Games, pp. 151–159 (2009)

  34. Weller, R., Zachmann, G.: Protosphere: A gpu-assisted prototype guided sphere packing algorithm for arbitrary objects. In: ACM SIGGRAPH ASIA 2010 Sketches, pp. 8:1–2 (2010)

  35. Winchenbach, R., Hochstetter, H., Kolb, A.: Infinite continuous adaptivity for incompressible SPH. ACM Trans. Graph. (TOG) 36(4), 1–10 (2017)

    Article  Google Scholar 

  36. Winchenbach, R., Kolb, A.: Optimized refinement for spatially adaptive SPH. ACM Trans. Graph. 40(1), 1–15 (2021)

    Article  Google Scholar 

  37. Wong, K.M., Wong, T.T.: Blue noise sampling using an n-body simulation-based method. Vis. Comput. 33, 823–832 (2017)

    Article  Google Scholar 

  38. Yan, D.M., Guo, J.W., Wang, B., Zhang, X.P., Wonka, P.: A survey of blue-noise sampling and its applications. J. Comput. Sci. Technol. 30(3), 439–452 (2015)

    Article  MathSciNet  Google Scholar 

  39. Zhang, K., Liu, F., Zhao, G., **a, K.: Fast and efficient particle packing algorithms based on triangular mesh. Powder Technol. 366, 448–459 (2020)

    Article  Google Scholar 

  40. Zheng, X., Si, J., Dai, S.: Blue noise sampling with a PBF-based method. In: Proceedings of the 33rd Computer Graphics International, pp. 77–80 (2016)

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Authors and Affiliations

  1. University of Tsukuba, Ibaraki, Japan

    Xu Wang, Makoto Fujisawa & Masahiko Mikawa

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  1. Xu Wang
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  2. Makoto Fujisawa
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Wang, X., Fujisawa, M. & Mikawa, M. XProtoSphere: an eXtended multi-sized sphere packing algorithm driven by particle size distribution. Vis Comput 39, 3333–3346 (2023). https://doi.org/10.1007/s00371-023-02977-w

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