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N-body gravitational and contact dynamics for asteroid aggregation

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Abstract

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.

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Notes

  1. As opposed to classical Ordinary Differential Equation (ODE) or Differential Algebraic Equations (DAE) formulations for smooth multibody problems that just imply one or more linear problems per time step.

  2. Concave shapes can be defined as well, by using convex decompositions. However, we assume that our initial bodies have only convex shapes.

References

  1. Tasora, A., Negrut, D., Serban, R., Mazhar, H., Heyn, T., Pazouki, A., Melanz, D.: Chrono::engine web pages at projectchrono.org (2006)

  2. Biele, J., Ulamec, S.: Capabilities of Philae, the Rosetta lander. Space Sci. Rev. 138, 275–289 (2008)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  4. Kaula, W.M.: Theory of Satellite Geodesy. Blaisdell, Waltham (1966)

    MATH  Google Scholar 

  5. Scheeres, D.J.: Dynamics about uniformly rotating triaxial ellipsoids: applications to asteroids. Icarus 110, 225–238 (1994)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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. Morbidelli, A.: Modern integrations of solar system dynamics. Annu. Rev. Earth Planet. Sci. 30 (2002)

  11. Stadel, J.: Cosmological \(N\)-body simulations and their analysis. PhD thesis, University of Washington, Seattle, WA, USA (2001)

  12. Richardson, D.C., Quinn, T., Stadel, J., Lake, G.: Direct large-scale \(n\)-body simulations of planetesimal dynamics. Icarus 143, 45–59 (2000)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  14. Aarseth, S.J.: Nbody2: a direct \(n\)-body integration code. New Astron. 6, 277–291 (2001)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wisdom, J., Holman, M.: Symplectic maps for the \(n\)-body problem. Astron. J. 102, 1528–1538 (1991)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  19. Chambers, J.E.: A hybrid symplectic integrator that permits close encounters between massive bodies. Mon. Not. R. Astron. Soc. 304, 793–799 (1999)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  22. Anitescu, M., Tasora, A.: An iterative approach for cone complementarity problems for nonsmooth dynamics. Comput. Optim. Appl. 47(2), 207–235 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Moreau, J.J.: Numerical aspects of the sweeping process. Comput. Methods Appl. Mech. Eng. 177(3–4), 329–349 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  27. Leine, R., Glocker, C.: A set-valued force law for spatial Coulomb–Contensou friction. Eur. J. Mech. 22(2), 193–216 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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. Tasora, A., Anitescu, M.: A complementarity-based rolling friction model for rigid contacts. Meccanica 48(7), 1643–1659 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Pang, J.S., Stewart, D.E.: Differential variational inequalities. Math. Program. 113, 1–80 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  32. Pfeiffer, F., Glocker, C.: Multibody Dynamics with Unilateral Contacts. Wiley, New York (1996)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Tasora, A., Anitescu, M.: A convex complementarity approach for simulating large granular flows. J. Comput. Nonlinear Dyn. 5, 1–10 (2010)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  42. Johnston, R.: Johnston’s archive web pages at johnstonsarchive.net (2016)

  43. NASA, J.P.L.: Jpl small-body database web pages at ssd.jpl.nasa.gov (2016)

  44. Edelsbrunner, H., Mücke, E.P.: Three-dimensional alpha shapes. Trans. Graph. 13(1), 43–72 (1994)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, Via La Masa 34, 20156, Milano, Italy

    Fabio Ferrari, Pierangelo Masarati & Michèle Lavagna

  2. Dipartimento di Ingegneria Industriale, Università degli Studi di Parma, Viale delle Scienze 181/A, 43100, Parma, Italy

    Alessandro Tasora

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  1. Fabio Ferrari
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  2. Alessandro Tasora
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  3. Pierangelo Masarati
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  4. Michèle Lavagna
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Correspondence to Fabio Ferrari.

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Ferrari, F., Tasora, A., Masarati, P. et al. N-body gravitational and contact dynamics for asteroid aggregation. Multibody Syst Dyn 39, 3–20 (2017). https://doi.org/10.1007/s11044-016-9547-2

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