Granular Matter

, 18:70 | Cite as

Sequential sphere packing by trilateration equations

  • Huu Duc To
  • Sergio Andres Galindo-Torres
  • Alexander Scheuermann
Original Paper


Specimen generation by packing of particles is the initial step in most numerical simulations for discrete media. In many cases, especially for virtual soil compaction, filtration, or penetration tests, this preparation work is essential for the success of the tests because the discrete particles are not able to be redistributed during the simulation. This paper presents a novel sphere sequential packing method for specimen generation using the trilateration method and its relevant equations. The method is developed on an assumption that spherical particles must be in contact with at least three neighbour particles to be kept in balance. This semi-analytical method has three advantages: quick generation speed, adjustable porosity, and good control over the spatial distribution of particles at local scale. This paper is a study on two typical spatial distributions: (1) layer-wise, where particles with similar sizes have priority of being positioned next to each other; and (2) discrete, where small particles are located preferentially in between large particles.


Sphere Sequential packing Trilateration equations Semi-analytical Spatial distribution Porosity Discrete element General shape 



The first author was granted a scholarship from the Vietnamese Ministry of Education and Training (MOET). The support from Australian Research Council (ARC) through the Discovery Grant (DP120102188) Hydraulic erosion of granular structures: experiments and computational simulations is gratefully acknowledged. The simulations were based on Mechsys, an open source library and carried out using the Macondo Cluster from the School of Civil Engineering at The University of Queensland.


  1. 1.
    O’Sullivan, C.: Particulate Discrete Element Modelling: A Geomechanics Perspective. Applied Geotechnics. Spon Press/Taylor, Francis (2011)Google Scholar
  2. 2.
    Matuttis, H., Luding, S., Herrmann, H.: Discrete element simulations of dense packings and heaps made of spherical and non-spherical particles. Powder Technol. 109(1), 278 (2000)CrossRefGoogle Scholar
  3. 3.
    Reboul, N., Vincens, E., Cambou, B.: A statistical analysis of void size distribution in a simulated narrowly graded packing of spheres. Granul. Matter. 10(6), 457 (2008)CrossRefzbMATHGoogle Scholar
  4. 4.
    Shire, T., O’Sullivan, C.: Micromechanical assessment of an internal stability criterion. Acta Geotech. 8(1), 81 (2013)CrossRefGoogle Scholar
  5. 5.
    Winkler, P., Sadaghiani, M.S., Jentsch, H., Witt, K.: Scour and Erosion. In: Proceedings of the 7th International Conference on Scour and Erosion, Perth, Australia, 2–4 December 2014. CRC Press, p. 345 (2014)Google Scholar
  6. 6.
    Ogarko, V.A.: Microstructure and Macroscopic Properties of Polydisperse Systems of Hard Spheres. Universiteit Twente, Enschede (2014)CrossRefGoogle Scholar
  7. 7.
    Bagi, K.: An algorithm to generate random dense arrangements for discrete element simulations of granular assemblies. Granul. Matter 7(1), 31 (2005)CrossRefzbMATHGoogle Scholar
  8. 8.
    Bezrukov, A., Bargieł, M., Stoyan, D.: Statistical analysis of simulated random packings of spheres. Part. Part. Syst. Charact. 19(2), 111 (2002)CrossRefGoogle Scholar
  9. 9.
    Rodriguez, J., Allibert, C., Chaix, J.: A computer method for random packing of spheres of unequal size. Powder Technol. 47(1), 25 (1986)CrossRefGoogle Scholar
  10. 10.
    Belheine, N., Plassiard, J.P., Donzé, F.V., Darve, F., Seridi, A.: Numerical simulation of drained triaxial test using 3D discrete element modeling. Comput. Geotech. 36(1), 320 (2009)CrossRefGoogle Scholar
  11. 11.
    Sherwood, J.: Packing of spheroids in three-dimensional space by random sequential addition. J. Phys. A Math. Gen. 30(24), L839 (1997)ADSCrossRefGoogle Scholar
  12. 12.
    Buechler, S., Johnson, S.: Efficient generation of densely packed convex polyhedra for 3D discrete and finite-discrete element methods. Int. J. Numer. Meth. Eng. 94(1), 1 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Oñate, E., Idelsohn, S., Celigueta, M., Rossi, R., Marti, J., Carbonell, J., Ryzhakov, P., Suárez, B.: Particle-Based Methods, pp. 1–49. Springer, Berlin (2011)Google Scholar
  14. 14.
    Galindo-Torres, S., Pedroso, D., Williams, D., Li, L.: Breaking processes in three-dimensional bonded granular materials with general shapes. Comput. Phys. Commun. 183(2), 266 (2012)ADSCrossRefGoogle Scholar
  15. 15.
    To, H.D., Torres, S.A.G., Scheuermann, A.: Primary fabric fraction analysis of granular soils. Acta Geotech. 10(3), 375–387 (2015)CrossRefGoogle Scholar
  16. 16.
    To, H.D., Galindo-Torres S.A.: Dem gasket website. (2014)
  17. 17.
  18. 18.
    Fang, B.T.: Trilateration and extension to global positioning system navigation. J. Guid. Control Dyn. 9(6), 715 (1986)ADSCrossRefGoogle Scholar
  19. 19.
    Silveira, A., de Lorena Peixoto, T., Nogueira, J.: In: Proceedings of the 5th Pan-American Conference on Soil Mechanics and Foundation Engineering, Buenos Aires , pp. 161–176 (1975)Google Scholar
  20. 20.
    To, H.D., Scheuermann, A., Williams, D.J.: In: 6th International Conference on Scour and Erosion (ICSE-6) (Société Hydrotechnique de France (SHF) ), pp. 295–303 (2012)Google Scholar
  21. 21.
    Langlet, G.: A new fast direct solution to the problem of the sphere tangent to four spheres. Acta Crystallogr. Sect. A Cryst. Phys. Diffr. Theor. Gen. Crystallogr. 35(5), 836 (1979)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Sickafus, E., Mackie, N.: Interstitial space in hardsphere clusters. Acta Crystallogr. Sect. A Cryst. Phys. Diffr. Theor. Gen. Crystallogr. 30(6), 850 (1974)ADSCrossRefGoogle Scholar
  23. 23.
    Lagarias, J.C., Mallows, C.L., Wilks, A.R.: Beyond the Descartes Circle Theorem. Am. Math. Mon. 109(4), 338–361 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Galindo-Torres, S., Muñoz, J., Alonso-Marroquin, F.: Minkowski-Voronoi diagrams as a method to generate random packings of spheropolygons for the simulation of soils. Phys. Rev. E 82(5), 056713 (2010)Google Scholar
  25. 25.
    D’Addetta, G.A.: Discrete models for cohesive frictional materials. Dissertation for Dr. Ing (PhD in Engineering), Stuttgart University, Stuttgart, Germany. (2004)
  26. 26.
    O’Sullivan, C., Bray, J.D.: Selecting a suitable time step for discrete element simulations that use the central difference time integration scheme. Eng. Comput. 21(2/3/4), 278 (2004)CrossRefzbMATHGoogle Scholar
  27. 27.
    Galindo-Torres, S., Scheuermann, A., Mühlhaus, H., Williams, D.: A micro-mechanical approach for the study of contact erosion. Acta Geotech. 10(3), 357–368 (2015)CrossRefGoogle Scholar
  28. 28.
    Cundall, P.A., Strack, O.D.: A discrete numerical model for granular assemblies. Geotechnique 29(1), 47 (1979)CrossRefGoogle Scholar
  29. 29.
    Green, S.: Particle simulation using cuda, NVIDIA Whitepaper (2010)Google Scholar
  30. 30.
    Xu, J., Qi, H., Fang, X., Lu, L., Ge, W., Wang, X., Xu, M., Chen, F., He, X., Li, J.: Quasi-real-time simulation of rotating drum using discrete element method with parallel GPU computing. Particuology 9(4), 446 (2011)CrossRefGoogle Scholar
  31. 31.
    Thornton, A., Weinhart, T., Luding, S., Bokhove, O.: Modeling of particle size segregation: calibration using the discrete particle method. Int. J. Mod. Phys. C 23(08), 1240014 (2012)ADSCrossRefGoogle Scholar
  32. 32.
    Saucke, U., Brauns, J., Schuler, U.: Zur Entmischungsneigung körniger Schüttstoffe. Geotechnik Zeitschrift 22, 259 (1999)Google Scholar
  33. 33.
    Chang, C.S., Misra, A.: Packing structure and mechanical properties of granulates. J. Eng. Mech. 116(5), 1077 (1990)CrossRefGoogle Scholar
  34. 34.
    Oda, M., Nemat-Nasser, S., Mehrabadi, M.: A statistical study of fabric in a random assembly of spherical granules. Int. J. Numer. Anal. Meth. Geomech. 6(1), 77 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Powell, M.: Computer-simulated random packing of spheres. Powder Technol. 25(1), 45 (1980)CrossRefGoogle Scholar
  36. 36.
    Kumar, N., Imole, O.I., Magnanimo, V., Luding, S.: Advanced Materials Research, vol. 508, pp. 160–165. Trans Tech Publ, Clausthal-Zellerfeld (2012)Google Scholar
  37. 37.
    Göncü, F., Luding, S.: Effect of particle friction and polydispersity on the macroscopic stress–strain relations of granular materials. Acta Geotech. 8(6), 629 (2013)CrossRefGoogle Scholar
  38. 38.
    An, X., Yang, R., Zou, R., Yu, A.: Effect of vibration condition and inter-particle frictions on the packing of uniform spheres. Powder Technol. 188(2), 102 (2008)CrossRefGoogle Scholar
  39. 39.
    Budhu, M.: Soil Mechanics and Foundations (With CD). Wiley, New York (2011)Google Scholar
  40. 40.
    Cho, G.C., Dodds, J., Santamarina, J.C.: Particle shape effects on packing density, stiffness, and strength: natural and crushed sands. J. Geotech. Geoenvironmental Eng. 132(5), 591 (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Huu Duc To
    • 1
  • Sergio Andres Galindo-Torres
    • 2
  • Alexander Scheuermann
    • 2
  1. 1.College of Science and Engineering, James Cook UniversityTownsvilleAustralia
  2. 2.Research Group on Complex Processes in Geo-Systems, Geotechnical Engineering Centre, School of Civil EngineeringThe University of QueenslandBrisbaneAustralia

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