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Computational Flexible Multibody Dynamics pp 173–215Cite as

Time Integration Methods

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Part of the book series: Differential-Algebraic Equations Forum ((DAEF))

Abstract

In flexible multibody dynamics, the time integration of the semi-discretized equations of motion represents a challenging problem due to the simultaneous presence of constraints and different time scales. This combination leads, as analyzed in the foregoing chapter, to a stiff differential-algebraic system. We investigate here the behavior of numerical methods for such problems, with particular focus on the well-established implicit integrators that are either based on the BDF (Backward Differentiation Formulas) methods or on implicit Runge–Kutta methods of collocation type. The chapter starts, however, with an overview on time integration methods for constrained mechanical systems. Using a model equation with smooth and highly oscillatory solution parts, we then show that stiff methods suffer from order reductions which are directly related to the limiting DAE of index 3 to which the stiff mechanical system converges. At the end of this chapter, we also introduce extensions of the generalized-α method and of the implicit midpoint rule to the differential-algebraic case and investigate their potential for mechanical multibody systems.

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References

  1. M. Arnold, A perturbation analysis for the dynamical simulation of mechanical multibody systems. Appl. Numer. Math. 18, 37–56 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Arnold, O. Brüls, Convergence of the generalized-α scheme for constrained mechanical systems. Multibody Syst. Dyn. 18, 185–202 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Arnold, A. Murua, Non-stiff integrators for differential–algebraic systems of index 2. Numer. Algorithms 19(1–4), 25–41 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Arnold, B. Simeon, Pantograph and catenary dynamics: a benchmark problem and its numerical solution. Appl. Numer. Math. 34, 345–362 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. U. Ascher, H. Chin, L. Petzold, S. Reich, Stabilization of constrained mechanical systems with DAEs and invariant manifolds. J. Mech. Struct. Mach. 23, 135–158 (1995)

    Article  MathSciNet  Google Scholar 

  6. O. Bauchau, Flexible Multibody Dynamics (Springer, Berlin, 2010)

    Google Scholar 

  7. P. Betsch, S. Uhlar, Energy-momentum conserving integration of multibody dynamics. Multibody Syst. Dyn. 17, 243–289 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. C.L. Bottasso, O.A. Bauchau, A. Cardona, Time-step-size independent conditioning and sensitivity to perturbations in the numerical solution of index three differential-algebraic equations. SIAM J. Sci. Comput. 29, 397–414 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. K.E. Brenan, S.L. Campbell, L.R. Petzold, The Numerical Solution of Initial Value Problems in Ordinary Differential-Algebraic Equations (SIAM, Philadelphia, 1996)

    Google Scholar 

  10. J.C. Butcher, Integration processes based on Radau quadrature formulas. Math. Comput. 18, 233–244 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  11. J.C. Butcher, Numerical Methods for Ordinary Differential Equations (Wiley, New York, 2008)

    Book  MATH  Google Scholar 

  12. A. Cardona, M. Géradin, Time integration of the equations of motion in mechanism analysis. Comput. Struct. 33(3), 801–820 (1989)

    Article  MATH  Google Scholar 

  13. J. Chang, G. Hulbert, A time integration algorithm for structural dynamics with improved numerical dissipation. ASME J. Appl. Mech. 60, 371–375 (1993)

    Article  Google Scholar 

  14. G. Dahlquist, A special stability problem for linear multistep methods. BIT Numer. Math. 3, 27–43 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  15. C.W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations (Prentice-Hall, Englewood Cliffs, 1971)

    MATH  Google Scholar 

  16. C.W. Gear, G.K. Gupta, B.J. Leimkuhler, Automatic integration of the Euler-Lagrange equations with constraints. J. Comput. Appl. Math. 12 & 13, 77–90 (1985)

    Article  MathSciNet  Google Scholar 

  17. S. González-Pinto, D. Hernández-Abreu, J.I. Montijano, An efficient family of strongly A-stable Runge–Kutta collocation methods for stiff systems and DAE’s. Part I: stability and order results. J. Comput. Appl. Math. 234, 1105–1116 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Guillou, J.L. Soulé, La résolution numérique des problemes differentiels aux conditions initiales par des méthodes de collocation. RAIRO. Anal. Numér. 3, 17–44 (1969)

    MATH  Google Scholar 

  19. E. Hairer, Symmetric projection methods for differential equations on manifolds. BIT Numer. Math. 40, 726–734 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. E. Hairer, Private communication, 2011

    Google Scholar 

  21. E. Hairer, G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems (Springer, Berlin, 1996)

    Book  MATH  Google Scholar 

  22. E. Hairer, G. Wanner, Stiff differential equations solved by Radau methods. J. Comput. Appl. Math. 111, 93–111 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. E. Hairer, Ch. Lubich, M. Roche, The Numerical Solution of Differential-Algebraic Equations by Runge–Kutta Methods. Lecture Notes in Mathematics, vol. 1409 (Springer, Heidelberg, 1989)

    Google Scholar 

  24. E. Hairer, S.P. Nørsett, G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems (Springer, Berlin, 1993)

    MATH  Google Scholar 

  25. E. Hairer, Ch. Lubich, G. Wanner, Geometric Numerical Integration (Springer, Berlin, 2002)

    Book  MATH  Google Scholar 

  26. H. Hilber, T. Hughes, R. Taylor, Improved numerical dissipation for time integration algorithms in structural dynamics. Earthq. Eng. Struct. Dyn. 5, 283–292 (1977)

    Article  Google Scholar 

  27. A.C. Hindmarsh, P.N. Brown, K.E. Grant, S.L. Lee, R. Serban, D.E. Shumaker, C.S. Woodward, Sundials: suite of nonlinear and differential/algebraic equation solvers. ACM Trans. Math. Softw. 31(3), 363–396 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. T.J. Hughes, The Finite Element Method (Prentice Hall, Englewood Cliffs, 1987)

    MATH  Google Scholar 

  29. L. Jay, Structure preservation for constrained dynamics with super partitioned additive Runge–Kutta methods. SIAM J. Sci. Comput. 20, 416–446 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  30. L. Jay, D. Negrut, Extensions of the HHT-method to differential-algebraic equations in mechanics. Electron. Trans. Numer. Anal. 26, 190–208 (2007)

    MathSciNet  MATH  Google Scholar 

  31. P. Lötstedt, Mechanical systems of rigid bodies subject to unilateral constraints. SIAM J. Appl. Math. 42, 281–296 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  32. P. Lötstedt, L.R. Petzold, Numerical solution of nonlinear differential equations with algebraic constraints I: convergence results for BDF. Math. Comput. 46, 491–516 (1986)

    MATH  Google Scholar 

  33. Ch. Lubich, h 2 extrapolation methods for differential-algebraic equations of index-2. Impact Comput. Sci. Eng. 1, 260–268 (1989)

    Article  MATH  Google Scholar 

  34. Ch. Lubich, Integration of stiff mechanical systems by Runge–Kutta methods. Z. Angew. Math. Phys. 44, 1022–1053 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ch. Lunk, B. Simeon, Solving constrained mechanical systems by the family of Newmark and α-methods. Z. Angew. Math. Mech. 86, 772–784 (2007)

    Article  MathSciNet  Google Scholar 

  36. K.G. Murty, Linear Complementarity, Linear and Nonlinear Programming (Heldermann Verlag, Berlin, 1988)

    MATH  Google Scholar 

  37. N. Newmark, A method of computation for structural dynamics. J. Eng. Mech. Div. 85, 67–94 (1959)

    Google Scholar 

  38. L.R. Petzold, Order results for implicit Runge–Kutta methods applied to differential/algebraic systems. SIAM J. Numer. Anal. 23, 837–852 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  39. A. Popp, M.W. Gee, W.A. Wall, A primal-dual active set strategy for finite deformation dual mortar contact, in Recent Advances in Contact Mechanics, ed. by G.E. Stavroulakis. Lecture Notes in Applied and Computational Mechanics, vol. 56 (Springer, Berlin, 2013), pp. 151–171

    Chapter  Google Scholar 

  40. A. Prothero, A. Robinson, On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations. Math. Comput. 28, 145–162 (1974)

    Article  MathSciNet  Google Scholar 

  41. P. Rentrop, G. Steinebach, Model and numerical techniques for the alarm system of river Rhine. Surv. Math. Ind. 6, 245–265 (1997)

    MathSciNet  MATH  Google Scholar 

  42. M. Schaub, Numerische Integration steifer mechanischer Systeme mit impliziten Runge-Kutta-Verfahren. PhD thesis, TU München, Zentrum Mathematik, 2004

    Google Scholar 

  43. M. Schaub, B. Simeon, Automatic h-scaling for the efficient time integration of stiff mechanical systems. Multibody Syst. Dyn. 8, 329–345 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  44. M. Schaub, B. Simeon, Blended Lobatto methods in multibody dynamics. Z. Angew. Math. Mech. 83(10), 720–728 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  45. S. Scholz, Order barriers for the B-convergence of ROW methods. Computing 41, 219–235 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  46. B. Simeon, Order reduction of stiff solvers at elastic multibody systems. Appl. Numer. Math. 28, 459–475 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  47. J.C. Simo, N. Tarnow, The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics. Z. Angew. Math. Phys. 43, 757–792 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  48. J. Solberg, Finite element methods for frictionless dynamic contact between elastic materials. PhD thesis, UC Berkeley, Dept. of Mechanical Engineering, 2000

    Google Scholar 

  49. J.M. Solberg, P. Papadopoulos, An analysis of dual formulations for the finite element solution of two-body contact problems. Comput. Methods Appl. Mech. Eng. 194, 2734–2780 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  50. J. Stoer, R. Bulirsch, Introduction to Numerical Analysis (Springer, Berlin, 2002)

    MATH  Google Scholar 

  51. K. Strehmel, R. Weiner, Numerik gewöhnlicher Differentialgleichungen (Teubner-Verlag, Stuttgart, 1995)

    MATH  Google Scholar 

  52. Th. Stumpp, Asymptotic expansions and attractive invariant manifolds of strongly damped mechanical systems. Z. Angew. Math. Mech. 88(8), 630–643 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  53. P.J. van der Houwen, B.P. Sommeijer, Explicit Runge–Kutta–Nyström methods with reduced phase errors for computing oscillating solutions. SIAM J. Numer. Anal. 24, 595–617 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  54. R. von Schwerin, Multibody System Simulation (Springer, Berlin, 1999)

    Book  MATH  Google Scholar 

  55. B. Wohlmuth, R. Krause, Monotone methods on non-matching grids for non-linear contact problems. SIAM J. Sci. Comput. 25(1), 324–347 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  56. P. Wriggers, Computational Contact Mechanics (Wiley, Chichester, 2002)

    Google Scholar 

  57. K. Wright, Some relationships between implicit Runge–Kutta collocation and Lanczos methods. BIT Numer. Math. 10, 217–227 (1970)

    Article  MATH  Google Scholar 

  58. J. Yen, L. Petzold, S. Raha, A time integration algorithm for flexible mechanism dynamics: the Dae-alpha method. Comput. Methods Appl. Mech. Eng. 158, 341–355 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Felix-Klein-Zentrum für Mathematik, Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Bernd Simeon

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Simeon, B. (2013). Time Integration Methods. In: Computational Flexible Multibody Dynamics. Differential-Algebraic Equations Forum. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35158-7_7

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