Multibody System Dynamics

, Volume 39, Issue 1–2, pp 3–20 | Cite as

N-body gravitational and contact dynamics for asteroid aggregation

  • Fabio FerrariEmail author
  • Alessandro Tasora
  • Pierangelo Masarati
  • Michèle Lavagna


The development of dedicated numerical codes has recently pushed forward the study of N-body gravitational dynamics, leading to a better and wider understanding of processes involving the formation of natural bodies in the Solar System. A major branch includes the study of asteroid formation: evidence from recent studies and observations support the idea that small and medium size asteroids between 100 m and 100 km may be gravitational aggregates with no cohesive force other than gravity. This evidence implies that asteroid formation depends on gravitational interactions between different boulders and that asteroid aggregation processes can be naturally modeled with N-body numerical codes implementing gravitational interactions. This work presents a new implementation of an N-body numerical solver. The code is based on Chrono::Engine (2006). It handles the contact and collision of large numbers of complex-shaped objects, while simultaneously evaluating the effect of N to N gravitational interactions. A special case of study is considered, investigating the relative dynamics between the N bodies and highlighting favorable conditions for the formation of a stable gravitationally bound aggregate from a cloud of N boulders. The code is successfully validated for the case of study by comparing relevant results obtained for typical known dynamical scenarios. The outcome of the numerical simulations shows good agreement with theory and observation, and suggests the ability of the developed code to predict natural aggregation phenomena.


N-body problem Asteroid aggregation Contact dynamics Rigid body Non-spherical shape 


  1. 1.
    Tasora, A., Negrut, D., Serban, R., Mazhar, H., Heyn, T., Pazouki, A., Melanz, D.: Chrono::engine web pages at (2006)
  2. 2.
    Biele, J., Ulamec, S.: Capabilities of Philae, the Rosetta lander. Space Sci. Rev. 138, 275–289 (2008) CrossRefGoogle Scholar
  3. 3.
    Heggy, E., Palmer, E.M., Kofman, W., Clifford, S.M., Righter, K., Hérique, A.: Radar properties of comets: parametric dielectric modeling of comet 67p/Churyumov–Gerasimenko. Icarus 221, 925–939 (2012) CrossRefGoogle Scholar
  4. 4.
    Kaula, W.M.: Theory of Satellite Geodesy. Blaisdell, Waltham (1966) zbMATHGoogle Scholar
  5. 5.
    Scheeres, D.J.: Dynamics about uniformly rotating triaxial ellipsoids: applications to asteroids. Icarus 110, 225–238 (1994) CrossRefGoogle Scholar
  6. 6.
    Werner, R.A., Scheeres, D.J.: Exterior gravitation of a polyhedron derived and compared with harmonic and mascon gravitation representations of asteroid 4769 Castalia. Celest. Mech. Dyn. Astron. 65, 313–344 (1997) CrossRefzbMATHGoogle Scholar
  7. 7.
    Scheeres, D.J., Ostro, S.J., Hudson, R.S., DeJong, E.M., Suzuki, S.: Dynamics of orbits close to asteroid 4179 Toutatis. Icarus 132, 53–79 (1998) CrossRefGoogle Scholar
  8. 8.
    Chapman, C.R.: Asteroid collisions, craters, regolith and lifetimes. In: Asteroids: an Exploration Assessment. NASA Conf. Publ., vol. 2053, pp. 145–160 (1978) Google Scholar
  9. 9.
    Richardson, D.C., Leinhardt, Z.M., Melosh, H.J., Bottke, W.F. Jr., Asphaug, E.: Gravitational aggregates: evidence and evolution. In: Asteroids III, pp. 501–515. University of Arizona Press, Tucson (2002) Google Scholar
  10. 10.
    Morbidelli, A.: Modern integrations of solar system dynamics. Annu. Rev. Earth Planet. Sci. 30 (2002) Google Scholar
  11. 11.
    Stadel, J.: Cosmological \(N\)-body simulations and their analysis. PhD thesis, University of Washington, Seattle, WA, USA (2001) Google Scholar
  12. 12.
    Richardson, D.C., Quinn, T., Stadel, J., Lake, G.: Direct large-scale \(n\)-body simulations of planetesimal dynamics. Icarus 143, 45–59 (2000) CrossRefGoogle Scholar
  13. 13.
    Richardson, D.C., Michel, P., Walsh, K.J., Flynn, K.W.: Numerical simulations of asteroids modelled as gravitational aggregates. Planet. Space Sci. 57, 183–192 (2009) CrossRefGoogle Scholar
  14. 14.
    Aarseth, S.J.: Nbody2: a direct \(n\)-body integration code. New Astron. 6, 277–291 (2001) CrossRefGoogle Scholar
  15. 15.
    Pruett, C.D., Rudmin, J.W., Lacy, J.M.: An adaptive \(n\)-body algorithm of optimal order. J. Comput. Phys. 187, 298–317 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dorband, E.N., Hemsendorf, M., Merritt, D.: Systolic and hyper-systolic algorithms for the gravitational \(n\)-body problem, with an application to Brownian motion. J. Comput. Phys. 185, 484–511 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Wisdom, J., Holman, M.: Symplectic maps for the \(n\)-body problem. Astron. J. 102, 1528–1538 (1991) CrossRefGoogle Scholar
  18. 18.
    Duncan, M.J., Levison, H.F., Lee, M.H.: A multiple time step symplectic algorithm for integrating close encounters. Astron. J. 116, 2067–2077 (1998) CrossRefGoogle Scholar
  19. 19.
    Chambers, J.E.: A hybrid symplectic integrator that permits close encounters between massive bodies. Mon. Not. R. Astron. Soc. 304, 793–799 (1999) CrossRefGoogle Scholar
  20. 20.
    Michel, P., Tanga, P., Benz, W., Richardson, D.C.: Formation of asteroid families by catastrophic disruption: simulations with fragmentation and gravitational reaccumulation. Icarus 160, 10–23 (2002) CrossRefGoogle Scholar
  21. 21.
    Mazhar, H., Heyn, T., Pazouki, A., Melanz, D., Seidl, A., Barthlomew, A., Tasora, A., Negrut, D.: Chrono: a parallel multi-physics library for rigid-body, flexible-body and fluid dynamics. Mech. Sci. (2013) Google Scholar
  22. 22.
    Anitescu, M., Tasora, A.: An iterative approach for cone complementarity problems for nonsmooth dynamics. Comput. Optim. Appl. 47(2), 207–235 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Moreau, J.J.: Numerical aspects of the sweeping process. Comput. Methods Appl. Mech. Eng. 177(3–4), 329–349 (1999) MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Stewart, D.E., Trinkle, J.C.: An implicit time-stepping scheme for rigid body dynamics with inelastic collisions and Coulomb friction. Int. J. Numer. Methods Eng. 39(15), 281–287 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Potra, F.A., Anitescu, M., Gavrea, B., Trinkle, J.: A linearly implicit trapezoidal method for integrating stiff multibody dynamics with contact and friction. Int. J. Numer. Methods Eng. 66(7), 1079–1124 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Renouf, M., Alart, P.: Conjugate gradient type algorithms for frictional multi-contact problems: applications to granular materials. Comput. Methods Appl. Mech. Eng. 194(18–20), 2019–2041 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Leine, R., Glocker, C.: A set-valued force law for spatial Coulomb–Contensou friction. Eur. J. Mech. 22(2), 193–216 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Acary, V., Brogliato, B.: Numerical methods for nonsmooth dynamical systems: applications in mechanics and electronics. In: Lect. N. App. Comput. Mech., vol. 35. Springer, Berlin (2008) Google Scholar
  29. 29.
    Tasora, A., Anitescu, M.: A complementarity-based rolling friction model for rigid contacts. Meccanica 48(7), 1643–1659 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Pang, J.S., Stewart, D.E.: Differential variational inequalities. Math. Program. 113, 1–80 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    De Saxcé, G., Feng, Z.-Q.: Recent advances in contact mechanics the bipotential method: a constructive approach to design the complete contact law with friction and improved numerical algorithms. Math. Comput. Model. 28(4), 225–245 (1998) CrossRefzbMATHGoogle Scholar
  32. 32.
    Pfeiffer, F., Glocker, C.: Multibody Dynamics with Unilateral Contacts. Wiley, New York (1996) CrossRefzbMATHGoogle Scholar
  33. 33.
    Heyn, T., Anitescu, M., Tasora, A., Negrut, D.: Using Krylov subspace and spectral methods for solving complementarity problems in many-body contact dynamics simulation. Int. J. Numer. Methods Eng. 95(7), 541–561 (2013) MathSciNetCrossRefGoogle Scholar
  34. 34.
    Mazhar, H., Heyn, T., Negrut, D., Tasora, A.: Using Nesterov’s method to accelerate multibody dynamics with friction and contact. ACM Trans. Graph. 34(3), 32:1–32:14 (2015) CrossRefzbMATHGoogle Scholar
  35. 35.
    Munthe-Kaas, H.: High order Runge–Kutta methods on manifolds. Appl. Numer. Math. 29(1), 115–127 (1999). Proceedings of the NSF/CBMS Regional Conference on Numerical Analysis of Hamiltonian Differential Equations MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Terze, Z., Müller, A., Zlatar, D.: Singularity-free time integration of rotational quaternions using non-redundant ordinary differential equations. Multibody Syst. Dyn., 1–25 (2016) Google Scholar
  37. 37.
    Tasora, A., Anitescu, M.: A matrix-free cone complementarity approach for solving large-scale, nonsmooth, rigid body dynamics. Comput. Methods Appl. Mech. Eng. 200, 439–453 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Tasora, A., Negrut, D., Anitescu, M.: Large-scale parallel multi-body dynamics with frictional contact on the graphical processing unit. J. Multi-Body Dyn. 222, 315–326 (2008) Google Scholar
  39. 39.
    Tasora, A., Anitescu, M.: A convex complementarity approach for simulating large granular flows. J. Comput. Nonlinear Dyn. 5, 1–10 (2010) zbMATHGoogle Scholar
  40. 40.
    Bradford Barber, C., Dobkin, D.P., Huhdanpaa, H.: The quickhull algorithm for convex hulls. ACM Trans. Math. Softw. 22(4), 469–483 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Keerthi, S.S., Gilbert, E.G., Johnson, D.W.: A fast procedure for computing the distance between complex objects in three-dimensional space. Robot. Autom. 4(2), 193–203 (1988) CrossRefGoogle Scholar
  42. 42.
    Johnston, R.: Johnston’s archive web pages at (2016)
  43. 43.
    NASA, J.P.L.: Jpl small-body database web pages at (2016)
  44. 44.
    Edelsbrunner, H., Mücke, E.P.: Three-dimensional alpha shapes. Trans. Graph. 13(1), 43–72 (1994) CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Dipartimento di Scienze e Tecnologie AerospazialiPolitecnico di MilanoMilanoItaly
  2. 2.Dipartimento di Ingegneria IndustrialeUniversità degli Studi di ParmaParmaItaly

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