Advertisement

A Variational Approach to Multirate Integration for Constrained Systems

  • Sigrid Leyendecker
  • Sina Ober-Blöbaum
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 28)

Abstract

The simulation of systems with dynamics on strongly varying time scales is quite challenging and demanding with regard to possible numerical methods. A rather naive approach is to use the smallest necessary time step to guarantee a stable integration of the fast frequencies. However, this typically leads to unacceptable computational loads. Alternatively, multirate methods integrate the slow part of the system with a relatively large step size while the fast part is integrated with a small time step. In this work, a multirate integrator for constrained dynamical systems is derived in closed form via a discrete variational principle on a time grid consisting of macro and micro time nodes. Being based on a discrete version of Hamilton’s principle, the resulting variational multirate integrator is a symplectic and momentum preserving integration scheme and also exhibits good energy behaviour. Depending on the discrete approximations for the Lagrangian function, one obtains different integrators, e.g. purely implicit or purely explicit schemes, or methods that treat the fast and slow parts in different ways. The performance of the multirate integrator is demonstrated by means of several examples.

Keywords

Time Grid Time Step Scheme Multirate System Discrete Trajectory Lagrangian Flow 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This chapter was partly developed and published in the course of the Collaborative Research Centre 614 “Self-Optimizing Concepts and Structures in Mechanical Engineering” funded by the German Research Foundation (DFG) under grant number SFB 614.

References

  1. 1.
    Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis, and Applications. Springer, New York (1988) zbMATHCrossRefGoogle Scholar
  2. 2.
    Arnold, M.: Multi-rate time integration for large scale multibody system models. In: Proceedings of the IUTAM Symposium on Multiscale Problems in Multibody System Contacts, Stuttgart, Germany (2006) Google Scholar
  3. 3.
    Barth, E., Schlick, T.: Extrapolation versus impulse in multiple-timestepping schemes. II. Linear analysis and applications to Newtonian and Langevin dynamics. J. Chem. Phys. 109, 1633–1642 (1998) CrossRefGoogle Scholar
  4. 4.
    Betsch, P., Leyendecker, S.: The discrete null space method for the energy consistent integration of constrained mechanical systems. Part II: Multibody dynamics. Int. J. Numer. Methods Eng. 67(4), 499–552 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bou-Rabee, N., Owhadi, H.: Stochastic variational integrators. IMA J. Numer. Anal. 29, 421–443 (2008) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cohen, D., Jahnke, T., Lorenz, K., Lubich, C.: Numerical integrators for highly oscillatory Hamiltonian systems: a review. In: Analysis, Modeling and Simulation of Multiscale Problems, pp. 553–576 (2006) CrossRefGoogle Scholar
  7. 7.
    Fetecau, R.C., Marsden, J.E., Ortiz, M., West, M.: Nonsmooth Lagrangian mechanics and variational collision integrators. SIAM J. Appl. Dyn. Syst. 2(3), 381–416 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Fong, W., Darve, E., Lew, A.: Stability of asynchronous variational integrators. J. Comput. Phys. 227, 8367–8394 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Ge, Z., Marsden, J.E.: Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators. Phys. Lett. A 133(3), 134–139 (1988) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gear, C.W., Wells, R.R.: Multirate linear multistep methods. BIT 24, 484–502 (1984) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Günther, M., Kærnø, A., Rentrop, P.: Multirate partitioned Runge-Kutta methods. BIT Numer. Math. 41(3), 504–514 (2001) zbMATHCrossRefGoogle Scholar
  12. 12.
    Hairer, E., Wanner, G., Lubich, C.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer, New York (2004) Google Scholar
  13. 13.
    Kane, C., Marsden, J.E., Ortiz, M., West, M.: Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems. Int. J. Numer. Methods Eng. 49(10), 1295–1325 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Kobilarov, M., Marsden, J.E., Sukhatme, G.S.: Geometric discretization of nonholonomic systems with symmetries. Discrete Contin. Dyn. Syst., Ser. S 1(1), 61–84 (2010) MathSciNetGoogle Scholar
  15. 15.
    Leimkuhler, B., Patrick, G.: A sympletic integrator for Riemannian manifolds. J. Nonlinear Sci. 6, 367–384 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Leimkuhler, B., Reich, S.: Symplectic integration of constrained Hamiltonian systems. Math. Comput. 63, 589–605 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Leimkuhler, B., Reich, S.: Simulating Hamiltonian Dynamics. Cambridge University Press, Cambridge (2004) zbMATHGoogle Scholar
  18. 18.
    Lew, A., Marsden, J.E., Ortiz, M., West, M.: An overview of variational integrators. In: Finite Element Methods: 1970’s and Beyond, pp. 85–146. CIMNE, Barcelona (2003) Google Scholar
  19. 19.
    Lew, A., Marsden, J.E., Ortiz, M., West, M.: An overview of variational integrators. In: Franca, L.P., Tezduyar, T.E., Masud, A. (eds.) Finite Element Methods: 1970’s and Beyond, pp. 98–115. CIMNE, Barcelona (2004) Google Scholar
  20. 20.
    Lew, A., Marsden, J.E., Ortiz, M., West, M.: Variational time integrators. Int. J. Numer. Methods Eng. 60(1), 153–212 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Leyendecker, S., Marsden, J.E., Ortiz, M.: Variational integrators for constrained dynamical systems. Z. Angew. Math. Mech. 88, 677–708 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Leyendecker, S., Ober-Blöbaum, S., Marsden, J.E., Ortiz, M.: Discrete mechanics and optimal control for constrained systems. Optim. Control Appl. Methods 31(6), 505–528 (2010) zbMATHCrossRefGoogle Scholar
  23. 23.
    Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry. Springer, New York (1994) zbMATHCrossRefGoogle Scholar
  24. 24.
    Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numer. 10, 357–514 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    McLachlan, R., Quispel, G.: Geometric integrators for ODEs. J. Phys. A 39(19), 5251–5286 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Ober-Blöbaum, S., Junge, O., Marsden, J.E.: Discrete mechanics and optimal control: an analysis. ESAIM Control Optim. Calc. Var. 17(2), 322–352 (2010) CrossRefGoogle Scholar
  27. 27.
    Reich, S.: Momentum conserving symplectic integrations. Physica D 76(4), 375–383 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Stern, A., Grinspun, E.: Implicit-explicit integration of highly oscillatory problems. Multiscale Model. Simul. 7, 1779–1794 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Striebel, M., Bartel, A., Günther, M.: A multirate ROW-scheme for index-1 network equations. Appl. Numer. Math. 59, 800–814 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Tao, M., Owhadi, H., Marsden, J.E.: Nonintrusive and structure preserving multiscale integration of stiff ODEs, SDEs, and Hamiltonian systems with hidden slow dynamics via flow averaging. Multiscale Model. Simul. 8(4), 1269–1324 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Verhoeven, A., Tasić, B., Beelen, T.G.J., ter Maten, E.J.W, Mattheij, R.M.M.: BDF compound-fast multirate transient analysis with adaptive stepsize control. J. Numer. Anal. Ind. Appl. Math. 3(3–4), 275–297 (2008) zbMATHGoogle Scholar
  32. 32.
    Weinan, E., Engquist, B., Li, X., Ren, W., Vanden-Eijnden, E.: Heterogeneous multiscale methods: a review. Commun. Comput. Phys. 2(3), 367–450 (2007) MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Chair of Applied DynamicsUniversity of Erlangen-NurembergErlangenGermany
  2. 2.Computational Dynamics and Optimal ControlUniversity of PaderbornPaderbornGermany

Personalised recommendations