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An iterative approach for cone complementarity problems for nonsmooth dynamics

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Abstract

Aiming at a fast and robust simulation of large multibody systems with contacts and friction, this work presents a novel method for solving large cone complementarity problems by means of a fixed-point iteration. The method is an extension of the Gauss-Seidel and Gauss-Jacobi method with overrelaxation for symmetric convex linear complementarity problems. The method is proved to be convergent under fairly standard assumptions and is shown by our tests to scale well up to 500,000 contact points and more than two millions of unknowns.

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References

  1. Anitescu, M.: Optimization-based simulation of nonsmooth rigid multibody dynamics. Math. Program. 105(1), 113–143 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Anitescu, M., Hart, G.D.: A constraint-stabilized time-stepping approach for rigid multibody dynamics with joints, contact and friction. Int. J. Numer. Methods Eng. 60(14), 2335–2371 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Anitescu, M., Hart, G.D.: A fixed-point iteration approach for multibody dynamics with contact and friction. Math. Program. Ser. B 101(1), 3–32 (2004)

    MATH  MathSciNet  Google Scholar 

  4. Anitescu, M., Potra, F.A.: Formulating dynamic multi-rigid-body contact problems with friction as solvable linear complementarity problems. Nonlinear Dyn. 14, 231–247 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  5. Anitescu, M., Potra, F.A.: Time-stepping schemes for stiff multi-rigid-body dynamics with contact and friction. Int. J. Numer. Methods Eng. 55(7), 753–784 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Anitescu, M., Cremer, J.F., Potra, F.A.: Formulating 3d contact dynamics problems. Mech. Struct. Mach. 24(4), 405–437 (1996)

    Article  MathSciNet  Google Scholar 

  7. Anitescu, M., Potra, F.A., Stewart, D.: Time-stepping for three-dimensional rigid-body dynamics. Comput. Methods Appl. Mech. Eng. 177, 183–197 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  8. Baraff, D.: Issues in computing contact forces for non-penetrating rigid bodies. Algorithmica 10, 292–352 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  9. Baraff, D.: Fast contact force computation for nonpenetrating rigid bodies. In: Computer Graphics (Proceedings of SIGGRAPH), pp. 23–34 (1994)

  10. Cottle, R., Dantzig, G.: Complementary pivot theory of mathematical programming. Linear Algebra Appl. 1, 103–125 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem. Academic, Boston (1992)

    MATH  Google Scholar 

  12. Donald, B.R., Pai, D.K.: On the motion of compliantly connected rigid bodies in contact: a system for analyzing designs for assembly. In: Proceedings of the Conf. on Robotics and Automation, pp. 1756–1762. IEEE, New York (1990)

    Chapter  Google Scholar 

  13. Gougar, H.D.: Advanced core design and fuel management for pebble-bed reactors. Ph.D. thesis, Department of Nuclear Engineering, Penn State University (2004)

  14. Haug, E.J.: Computer Aided Kinematics and Dynamics of Mechanical Systems. Allyn and Bacon, Boston (1989)

    Google Scholar 

  15. Haug, E.J., Wu, S., Yang, S.: Dynamic mechanical systems with coulomb friction, stiction, impact and constraint addition-deletion. Mech. Mach. Theory 21(5), 407–416 (1986)

    Article  Google Scholar 

  16. Hiriart-Urruty, J.-B., Lemarechal, C.: Convex Analysis and Minimization Algorithms. Springer, Berlin (1993)

    Google Scholar 

  17. Jourdan, F., Alart, P., Jean, M.: A Gauss Seidel like algorithm to solve frictional contract problems. Comput. Methods Appl. Mech. Eng. 155, 31–47 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kim, Y.J., Lin, M.C., Manocha, D.: Deep: Dual-space expansion for estimating penetration depth between convex polytopes. In: Proceedings of the 2002 International Conference on Robotics and Automation, vol. 1, pp. 921–926. Institute for Electrical and Electronics Engineering, New York (2002)

    Google Scholar 

  19. Lotstedt, P.: Mechanical systems of rigid bodies subject to unilateral constraints. SIAM J. Appl. Math. 42(2), 281–296 (1982)

    Article  MathSciNet  Google Scholar 

  20. Marques, M.D.P.: Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction. Progress in Nonlinear Differential Equations and Their Applications, vol. 9. Birkhäuser, Basel (1993)

    MATH  Google Scholar 

  21. Moreau, J.J.: Standard inelastic shocks and the dynamics of unilateral constraints. In: Piero, G.D., Macieri, F. (eds.) Unilateral Problems in Structural Analysis, pp. 173–221. New York (1983). CISM Courses and Lectures no. 288

  22. Moreau, J.J., Jean, M.: Numerical treatment of contact and friction: The contact dynamics method. In: Proceedings of the Third Biennial Joint Conference on Engineering Systems and Analysis, pp. 201–208. Montpellier, France, July 1996

  23. Murray, R.M., Li, Z., Sastry, S.S.: A Mathematical Introduction to Robotic Manipulation. CRC Press, Boca Raton (1993)

    Google Scholar 

  24. Murty, K.G.: Linear Complementarity, Linear and Nonlinear Programming. Helderman, Berlin (1988)

    MATH  Google Scholar 

  25. Pang, J.-S., Stewart, D.: A unified approach to frictional contact problems. Int. J. Eng. Sci. 37(13), 1747–1768 (1999)

    Article  MathSciNet  Google Scholar 

  26. Pang, J.-S., Stewart, D.: Differential variational inequalities. Math. Program. 113(2), 345–424 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  27. Pang, J.-S., Stewart, D.: Solution dependence on initial conditions in differential variational inequalities. Math. Program. 116(1), 429–460 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  28. Pang, J.-S., Trinkle, J.C.: Complementarity formulations and existence of solutions of dynamic multi-rigid-body contact problems with coulomb friction. Math. Program. 73(2), 199–226 (1996)

    Article  MathSciNet  Google Scholar 

  29. Pang, J.-S., Kumar, V., Song, P.: Convergence of time-stepping method for initial and boundary-value frictional compliant contact problems. SIAM J. Numer. Anal. 43(5), 2200–2226 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  30. Pang, J.-S., Kumar, V., Trinkle, J.: On a continuous-time quasistatic frictional contact model with local compliance. Int. J. Numer. Methods Eng. (2007, submitted)

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

    Book  MATH  Google Scholar 

  32. 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  MATH  MathSciNet  Google Scholar 

  33. Rycroft, C., Grest, G., Landry, J., Bazant, M.: Analysis of granular flow in a pebble-bed nuclear reactor. Phys. Rev. E 74, 021306 (2006)

    Article  Google Scholar 

  34. Song, P., Kraus, P., Kumar, V., Dupont, P.: Analysis of rigid-body dynamic models for simulation of systems with frictional contacts. J. Appl. Mech. 68(1), 118–128 (2001)

    Article  MATH  Google Scholar 

  35. Song, P., Pang, J.-S., Kumar, V.: A semi-implicit time-stepping model for frictional compliant contact problems. Int. J. Numer. Methods Eng. 60(13), 267–279 (2004)

    Article  MathSciNet  Google Scholar 

  36. Stewart, D.E.: Convergence of a time-stepping scheme for rigid body dynamics and resolution of Painleve’s problems. Arch. Ration. Mech. Anal. 145(3), 215–260 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  37. Stewart, D.E.: Rigid-body dynamics with friction and impact. SIAM Rev. 42(1), 3–39 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  38. 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, 2673–2691 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  39. Tasora, A.: Chrono::engine project, web page. www.deltaknowledge.com/chronoengine (2006)

  40. Tasora, A., Manconi, E., Silvestri, M.: Un nuovo metodo del simplesso per il problema di complementarità lineare mista in sistemi multibody con vincoli unilateri. In: Proceedings of AIMETA 05, Firenze, Italy (2005)

  41. Trinkle, J., Pang, J.-S., Sudarsky, S., Lo, G.: On dynamic multi-rigid-body contact problems with coulomb friction. Z. Angew. Math. Mech. 77, 267–279 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  42. Tseng, P., Yun, S.: A coordinate gradient descent method for nonsmooth separable minimization. Math. Program. 117, 387–423 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL, 60439, USA

    Mihai Anitescu

  2. Dipartimento di Ingegneria Industriale, Università degli Studi di Parma, 43100, Parma, Italy

    Alessandro Tasora

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  1. Mihai Anitescu
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  2. Alessandro Tasora
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Correspondence to Mihai Anitescu.

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Anitescu, M., Tasora, A. An iterative approach for cone complementarity problems for nonsmooth dynamics. Comput Optim Appl 47, 207–235 (2010). https://doi.org/10.1007/s10589-008-9223-4

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  • DOI: https://doi.org/10.1007/s10589-008-9223-4

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