Computational Mechanics

, Volume 62, Issue 1, pp 81–96 | Cite as

The maximum dissipation principle in rigid-body dynamics with inelastic impacts

  • Tobias PreclikEmail author
  • Sebastian Eibl
  • Ulrich Rüde
Original Paper


Formulating a consistent theory for rigid-body dynamics with impacts is an intricate problem. Twenty years ago Stewart published the first consistent theory with purely inelastic impacts and an impulsive friction model analogous to Coulomb friction. In this paper we demonstrate that the consistent impact model can exhibit multiple solutions with a varying degree of dissipation even in the single-contact case. Replacing the impulsive friction model based on Coulomb friction by a model based on the maximum dissipation principle resolves the non-uniqueness in the single-contact impact problem. The paper constructs the alternative impact model and presents integral equations describing rigid-body dynamics with a non-impulsive and non-compliant contact model and an associated purely inelastic impact model maximizing dissipation. An analytic solution is derived for the single-contact impact problem. The models are then embedded into a time-stepping scheme. The macroscopic behaviour is compared to Coulomb friction in a large-scale granular flow problem.


Impulse (physics) Coulomb friction Collisions (physics) Rigid body dynamics Contact dynamics Impact dynamics Measure differential inclusions Complementarity problems 



The authors would like to acknowledge the support through the Cluster of Excellence Engineering of Advanced Materials (EAM).


  1. 1.
    Anitescu M, Potra F (1997) Formulating dynamic multi-rigid-body contact problems with friction as solvable linear complementarity problems. Nonlinear Dyn 14(3):231–247MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bannerman MN, Sargant R, Lue L (2011) DynamO: a free O(N) general event-driven molecular dynamics simulator. J Comput Chem 32(15):3329–3338CrossRefGoogle Scholar
  3. 3.
    Bonnefon O, Daviet G (2011) Quartic formulation of coulomb 3D frictional contact. Technical report RT-0400. INRIAGoogle Scholar
  4. 4.
    Diebel J (2006) representing attitude: Euler angles, unit quaternions, and rotation vectors. Matrix 58:1–35Google Scholar
  5. 5.
    Erleben K (2004) Stable, robust, and versatile multibody dynamics animation. PhD thesis. University of CopenhagenGoogle Scholar
  6. 6.
    Gavrea BI, Anitescu M, Potra FA (2008) Convergence of a class of semi-implicit timestepping schemes for nonsmooth rigid multibody dynamics. SIAM J Optim 19(2):969–1001MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hassanpour A et al (2011) Analysis of particle motion in a paddle mixer using discrete element method (DEM). Powder Technol 206(1):189–194CrossRefGoogle Scholar
  8. 8.
    Jayasundara C et al (2011) CFD-DEM modelling of particle flow in IsaMills–Comparison between simulations and PEPT measurements. Miner Eng 24(3):181–187CrossRefGoogle Scholar
  9. 9.
    Jean M (1999) The non-smooth contact dynamics method. Comput Methods Appl Mech Eng 177(3):235–257MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jia Y-B (2013) Three-dimensional impact: energybased modeling of tangential compliance. Int J Robot Res 32(1):56–83CrossRefGoogle Scholar
  11. 11.
    Jia Y-B, Wang F (2016) Analysis and computation of two body impact in three dimensions. J Comput Nonlinear Dyn 12:041012CrossRefGoogle Scholar
  12. 12.
    Lubliner J (1984) A maximum-dissipation principle in generalized plasticity. Acta Mech 52(3):225–237MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mirtich B (1996) Impulse-based dynamic simulation of rigid body systems. PhD thesis. University of CaliforniaGoogle Scholar
  14. 14.
    Mirtich B, Canny J (1995) Impulse-based simulation of rigid bodies. In: Proceedings of the 1995 symposium on interactive 3D graphics. ACM, pp 181–ffGoogle Scholar
  15. 15.
    Mishra B, Rajamani R (1992) The discrete element method for the simulation of ball mills. Appl Math Model 16(11):598–604CrossRefzbMATHGoogle Scholar
  16. 16.
    Mitarai N, Nakanishi H (2012) Granular flow: dry and wet. Eur Phys J Spec Topic 204(1):5–17CrossRefGoogle Scholar
  17. 17.
    Moreau JJ (1988) Unilateral contact and dry friction in finite freedom dynamics. In: Moreau JJ, Panagiotopoulos PD (eds) Nonsmooth mechanics and applications. Springer, New York, pp 1–82Google Scholar
  18. 18.
    Nuseirat AA-F, Stavroulakis G (2000) A complementarity problem formulation of the frictional grasping problem. Comput Methods Appl Mech Eng 190:941–952CrossRefzbMATHGoogle Scholar
  19. 19.
    Painlevé P (1895) Sur les lois du frottement de glissement. C R Acad Sci Paris 121:112–115zbMATHGoogle Scholar
  20. 20.
    Popa C, Preclik T, Rüde U (2015) Regularized solution of LCP problems with application to rigid body dynamics. Numer Algorithms 69(1):145–156MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Preclik T (2014) Models and algorithms for ultrascale simulations of non-smooth granular dynamics. PhD thesis. Friedrich-Alexander-Universität Erlangen-NürnbergGoogle Scholar
  22. 22.
    Preclik T, Rüde U (2015) Ultrascale simulations of non-smooth granular dynamics. Comput Part Mech 2(2):173–196CrossRefGoogle Scholar
  23. 23.
    Sauer J, Schömer E (1998) A constraint-based approach to rigid body dynamics for virtual reality applications. In: Proceedings of the ACM symposium on virtual reality software and technology, pp 153–162Google Scholar
  24. 24.
    Shen Y, Stronge W (2011) Painlevé paradox during oblique impact with friction. Eur J Mech A/Solids 30(4):457–467MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Stewart DE (1998) Convergence of a time-stepping scheme for rigid-body dynamics and resolution of Painlevé’s problem. Arch Ration Mech Anal 145(3):215–260MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Stewart DE (2011) Dynamics with inequalities: impacts and hard constraints. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  27. 27.
    Stewart D (2000) Rigid-body dynamics with friction and impact. SIAM Rev 42(1):3–39MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Stewart D, Trinkle J (1996) An implicit timestepping scheme for rigid body dynamics with inelastic collisions and coulomb friction. Int J Numer Methods Eng 39(15):2673–2691CrossRefzbMATHGoogle Scholar
  29. 29.
    Stronge WJ (1990 ) Rigid body collisions with friction. In: Proceedings of the royal society of London A: mathematical, physical and engineering sciences, vol. 431, no. 1881. The Royal Society, pp 169–181Google Scholar
  30. 30.
    Stronge WJ (2004) Impact mechanics. Cambridge University Press, CambridgezbMATHGoogle Scholar
  31. 31.
    Tasora A, Anitescu M (2010) A convex complementarity approach for simulating large granular flows. J Comput Nonlinear Dyn 5(3):1–10CrossRefzbMATHGoogle Scholar
  32. 32.
    Tasora A, Anitescu M (2011) A matrix-free cone complementarity approach for solving large-scale, nonsmooth, rigid body dynamics. Comput Methods Appl Mech Eng 200(5):439–453MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Lehrstuhl für Informatik 10 (Systemsimulation)Friedrich-Alexander Universität Erlangen-NürnbergErlangenGermany
  2. 2.CERFACSToulouse Cedex 01France

Personalised recommendations