Effect of the integration scheme on the rotation of non-spherical particles with the discrete element method

Abstract

The discrete element method (DEM) is an emerging tool for the calculation of the behaviour of bulk materials. One of the key features of this method is the explicit integration of the motion equations. Explicit methods are rapid, at the cost of a limited time step to achieve numerical stability. First- or second-order integration schemes based on a Taylor series are frequently used in this framework and shown to be accurate for the translational and rotational motion of spherical particles. However, they may lead to relevant inaccuracies when non-spherical particles are used since the orientation implies a modification in the second-order inertia tensor in the inertial reference frame. Specific integration schemes for non-spherical particles have been proposed in the literature, such as the fourth-order Runge–Kutta scheme presented by Munjiza et al. and the predictor–corrector scheme developed by Zhao and van Wachem which applies the direct multiplication algorithm for integrating the orientation. In this work, both methods are adapted to be used together with a velocity Verlet scheme for the translational integration. The performance of the resulting schemes, as well as that of the direct integration method, is assessed, both in benchmark tests with analytical solution and in real-scale problems. The results suggest that the fourth-order Runge–Kutta and the Zhao and van Wachem schemes are clearly more accurate than the direct integration method without increasing the computational time.

This is a preview of subscription content, access via your institution.

Fig. 1

Adapted from [32]

Fig. 2
Fig. 3

Adapted from Irazábal [22]

Fig. 4
Fig. 5
Fig. 6
Fig. 7

Taken from Zhao and van Wachem [43]

Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Adapted from Chen et al. [5]

Fig. 14
Fig. 15
Fig. 16

References

  1. 1.

    Alonso-Marroquín F (2008) Spheropolygons: a new method to simulate conservative and dissipative interactions between 2d complex-shaped rigid bodies. EPL 83(1):14001

    Article  Google Scholar 

  2. 2.

    Alonso-Marroquín F, Wang Y (2009) An efficient algorithm for granular dynamics simulations with complex-shaped objects. Granul Matter 11(5):317–329

    Article  MATH  Google Scholar 

  3. 3.

    Belytschko T, Liu WK, Moran B, Elkhodary K (2013) Nonlinear finite elements for continua and structures. Wiley, Hoboken

    MATH  Google Scholar 

  4. 4.

    Chakraborty N, Peng J, Akella S, Mitchell JE (2008) Proximity queries between convex objects: an interior point approach for implicit surfaces. IEEE Trans Robot 24(1):211–220

    Article  Google Scholar 

  5. 5.

    Chen C, McDowell GR, Thom NH (2014) Investigating geogrid-reinforced ballast: experimental pull-out tests and discrete element modelling. Soils Found 54(1):1–11

    Article  Google Scholar 

  6. 6.

    Chen W, Qiu T (2012) Numerical simulations for large deformation of granular materials using smoothed particle hydrodynamics method. Int J Geomech 12(2):127–135

    Article  Google Scholar 

  7. 7.

    Cundall PA (1988) Formulation of a three-dimensional distinct element model—Part I. A scheme to detect and represent contacts in a system composed of many polyhedral blocks. Int J Rock Mech Min Sci 25(3):107–116

    Article  Google Scholar 

  8. 8.

    Cundall PA, Strack ODL (1979) A discrete numerical model for granular assemblies. Géotechnique 29(1):47–65

    Article  Google Scholar 

  9. 9.

    Dafalias Y, Manzari M (1997) A critical state two-surface plasticity model for sands. Géotechnique 47(2):255–272

    Article  Google Scholar 

  10. 10.

    Deresiewicz H, Mindlin RD (1952) Elastic spheres in contact under varying oblique forces. Columbia University, Department of Civil Engineering, New York

    MATH  Google Scholar 

  11. 11.

    Duncan J (1996) State of the art: limit equilibrium and finite-element analysis of slopes. J Geotech Eng ASCE 122(7):577–596

    Article  Google Scholar 

  12. 12.

    Eliáš J (2014) Simulation of railway ballast using crushable polyhedral particles. Powder Technol 264:458–465

    Article  Google Scholar 

  13. 13.

    Galindo-Torres SA, Pedroso DM (2010) Molecular dynamics simulations of complex-shaped particles using Voronoi-based spheropolyhedra. Phys Rev E 81(6):061303

    Article  Google Scholar 

  14. 14.

    Hart R, Cundall PA, Lemos J (1988) Formulation of a three-dimensional distinct element model—Part II. Mechanical calculations for motion and interaction of a system composed of many polyhedral blocks. Int J Rock Mech Min Sci 25(3):117–125

    Article  Google Scholar 

  15. 15.

    Hemingway E, O’Reilly OM (2018) Perspectives on Euler angle singularities, gimbal lock, and the orthogonality of applied forces and applied moments. Multibody Syst Dyn 44(1):31–56

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Henderson DM (1977) Euler angles, quaternions, and transformation matrices for space shuttle analysis. Tech. rep., NASA

  17. 17.

    Huang H, Tutumluer E (2014) Image-aided element shape generation method in discrete-element modeling for railroad ballast. J Mater Civ Eng 26(3):527–535

    Article  Google Scholar 

  18. 18.

    Höhner D, Wirtz S, Kruggel-Emden H, Scherer V (2011) Comparison of the multi-sphere and polyhedral approach to simulate non-spherical particles within the discrete element method: Influence on temporal force evolution for multiple contacts. Powder Technol 208(3):643–656

    Article  Google Scholar 

  19. 19.

    Iaconeta I, Larese A, Rossi R, Oñate E (2017) An implicit material point method applied to granular flows. Procedia Eng 175:226–232

    Article  Google Scholar 

  20. 20.

    Indraratna B, Ngo NT, Rujikiatkamjorn C (2011) Behavior of geogrid-reinforced ballast under various levels of fouling. Geotext Geomembr 29(3):313–322

    Article  Google Scholar 

  21. 21.

    Indraratna B, Ngo NT, Rujikiatkamjorn C, Vinod JS (2014) Behavior of fresh and fouled railway ballast subjected to direct shear testing: discrete element simulation. Int J Geomech 14(1):34–44

    Article  Google Scholar 

  22. 22.

    Irazábal J (2017) Numerical analysis of railway ballast behaviour using the discrete element method. PhD thesis. https://www.scipedia.com/public/Iraz

  23. 23.

    Irazábal J, Salazar F, Oñate E (2017) Numerical modelling of granular materials with spherical discrete particles and the bounded rolling friction model. Application to railway ballast. Comput Geotech 85:220–229

    Article  Google Scholar 

  24. 24.

    Kuipers JB (2002) Quaternions and rotation sequences: a primer with applications to orbits, aerospace, and virtual reality. Princeton University Press, Princeton

    MATH  Google Scholar 

  25. 25.

    Liu Z (2011) Measuring the angle of repose of granular systems using hollow cylinders. Techreport, University of Pittsburgh

  26. 26.

    Lopes DS, Silva MT, Ambrósio JA, Flores P (2010) A mathematical framework for rigid contact detection between quadric and superquadric surfaces. Multibody Syst Dyn 24(3):255–280

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Luding S (2005) Molecular dynamics simulations of granular materials. In: The physics of granular media, Wiley, pp 297–324

  28. 28.

    Munjiza A, Latham JP, John NWM (2003) 3d dynamics of discrete element systems comprising irregular discrete elements-integration solution for finite rotations in 3d. Int J Numer Methods Eng 56(1):35–55

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Nezami EG, Hashash YMA, Zhao D, Ghaboussi J (2006) Shortest link method for contact detection in discrete element method. Int J Numer Anal Methods Geomech 30(8):783–801

    Article  MATH  Google Scholar 

  30. 30.

    Oñate E, Idelsohn S, Del Pin F, Aubry R (2004) The particle finite element method. An overview. Int J Comput Methods 1(2):267–307. https://doi.org/10.1142/S0219876204000204

    Article  MATH  Google Scholar 

  31. 31.

    Oñate E, Celigueta MA, Idelsohn SR, Salazar F, Suárez B (2011) Possibilities of the particle finite element method for fluid-soil-structure interaction problems. Comput Mech 48(3):307

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Oñate E, Zárate F, Miquel J, Santasusana M, Celigueta MA, Arrufat F, Gandikota R, Valiullin K, Ring L (2015) A local constitutive model for the discrete element method. Application to geomaterials and concrete. Comput Particle Mech 2(2):139–160

    Article  Google Scholar 

  33. 33.

    O’Sullivan C, Bray JD (2004) Selecting a suitable time step for discrete element simulations that use the central difference time integration scheme. Eng Comput 21(2/3/4):278–303

    Article  MATH  Google Scholar 

  34. 34.

    Podlozhnyuk A, Pirker S, Kloss C (2017) Efficient implementation of superquadric particles in discrete element method within an open-source framework. Comput Part Mech 4(1):101–118

    Article  Google Scholar 

  35. 35.

    Richefeu V, Mollon G, Daudon D, Villard P (2012) Dissipative contacts and realistic block shapes for modeling rock avalanches. Eng Geol 149–150:78–92

    Article  Google Scholar 

  36. 36.

    Salazar F, Irazábal J, Larese A, Oñate E (2015) Numerical modelling of landslide-generated waves with the particle finite element method (PFEM) and a non-Newtonian flow model. Int J Numer Anal Methods Geomech 40(6):809–826

    Article  Google Scholar 

  37. 37.

    Santasusana M (2016) Numerical techniques for non-linear analysis of strutures combining discrete element and finite element methods. PhD thesis

  38. 38.

    Santasusana M, Irazábal J, Oñate E, Carbonell JM (2016) The double hierarchy method. A parallel 3D contact method for the interaction of spherical particles with rigid FE boundaries using the DEM. Comput Part Mech 3:407–428

    Article  Google Scholar 

  39. 39.

    Taylor DW (1948) Fundamentals of soil mechanics. Wiley, Hoboken

    Book  Google Scholar 

  40. 40.

    Tian T, Su J, Zhan J, Geng S, Xu G, Liu X (2018) Discrete and continuum modeling of granular flow in silo discharge. Particuology 36:127–138

    Article  Google Scholar 

  41. 41.

    Whitmore SA (2000) U.S. Patent No. 6,061,611. U.S. Patent and Trademark Office, Washington, DC

  42. 42.

    Zhang Y, Jia F, Zeng Y, Han Y, Xiao Y (2018) DEM study in the critical height of flow mechanism transition in a conical silo. Powder Technol 331:98–106

    Article  Google Scholar 

  43. 43.

    Zhao F, van Wachem BGM (2013) A novel Quaternion integration approach for describing the behaviour of non-spherical particles. Acta Mech 224(12):3091–3109

    MathSciNet  Article  MATH  Google Scholar 

  44. 44.

    Zhou T, Hu B, Sun J, Liu Z (2013) Discrete element method simulation of railway ballast compactness during tamping process. Open Electr Electron Eng J 7(1):103–109

    Article  Google Scholar 

  45. 45.

    Zhou W, Yang L, Ma G, Xu K, Lai Z, Chang X (2017) DEM modeling of shear bands in crushable and irregularly shaped granular materials. Granul Matter 19(2):25

    Article  Google Scholar 

  46. 46.

    Zhu HP, Zhou ZY, Yang RY, Yu AB (2008) Discrete particle simulation of particulate systems: a review of major applications and findings. Chem Eng Sci 63(23):5728–5770

    Article  Google Scholar 

Download references

Acknowledgements

This work was carried out with the financial support Spanish MINECO within the MONICAB (BIA2015-67263-R) project.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Joaquín Irazábal.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Irazábal, J., Salazar, F., Santasusana, M. et al. Effect of the integration scheme on the rotation of non-spherical particles with the discrete element method. Comp. Part. Mech. 6, 545–559 (2019). https://doi.org/10.1007/s40571-019-00232-5

Download citation

Keywords

  • Discrete element method
  • Granular material
  • Non-spherical particles
  • Clusters of spheres
  • Explicit rotational integration