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A Survey on Numerical Methods for the Simulation of Initial Value Problems with sDAEs

  • Michael Burger
  • Matthias Gerdts
Chapter
Part of the Differential-Algebraic Equations Forum book series (DAEF)

Abstract

This paper provides an overview on numerical aspects in the simulation of differential-algebraic equations (DAEs). Amongst others we discuss the basic construction principles of frequently used discretization schemes, such as BDF methods, Runge–Kutta methods, and ROW methods, as well as their adaption to DAEs. Moreover, topics like consistent initialization, stabilization, parametric sensitivity analysis, co-simulation techniques, aspects of real-time simulation, and contact problems are covered. Finally, some illustrative numerical examples are presented.

Keywords

BDF methods Consistent initialization Contact problems Co-simulation Differential-algebraic equations Real-time simulation ROW methods Runge–Kutta methods Sensitivity analysis Stabilization 

Subject Classifications:

65L80 65L05 65L06 

References

  1. 1.
    Amodio, P., Mazzia, F.: Numerical solution of differential algebraic equations and computation of consistent initial/boundary conditions. J. Comput. Appl. Math. 87, 135–146 (1997)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Anitescu, M.: Optimization-based simulation of nonsmooth rigid multibody dynamics. Math. Program. 105 (1(A)), 113–143 (2006)Google Scholar
  3. 3.
    Anitescu, M., Potra, F.A.: Formulating dynamic multi-rigid-body contact problems with friction as solvable linear complementarity problems. Nonlinear Dyn. 14 (3), 231–247 (1997)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Anitescu, M., Potra, F.A.: A time-stepping method for stiff multibody dynamics with contact and friction. Int. J. Numer. Methods Eng. 55 (7), 753–784 (2002)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Anitescu, M., Tasora, A.: An iterative approach for cone complementarity problems for nonsmooth dynamics. Comput. Optim. Appl. 47 (2), 207–235 (2010)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Anitescu, M., Potra, F.A., Stewart, D.E.: Time-stepping for three-dimensional rigid body dynamics. Comput. Methods Appl. Mech. Eng. 177 (3–4), 183–197 (1999)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Arnold, M.: Half-explicit Runge-Kutta methods with explicit stages for differential-algebraic systems of index 2. BIT 38 (3), 415–438 (1998)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Arnold, M.: Multi-rate time integration for large scale multibody system models. In: IUTAM Symposium on Multiscale Problems in Multibody System Contacts: Proceedings of the IUTAM Symposium held in Stuttgart, Germany, February 20–23, 2006, pp. 1–10. Springer, Dordrecht (2007)Google Scholar
  9. 9.
    Arnold, M.: Stability of sequential modular time integration methods for coupled multibody system models. J. Comput. Nonlinear Dyn. 5, 031003 (2010)CrossRefGoogle Scholar
  10. 10.
    Arnold, M.: Modular time integration of block-structured coupled systems without algebraic loops. In: Schöps, S., Bartel, A., Günther, M., ter Maten, E.J.W., Müller, P.C. (eds.) Progress in Differential-Algebraic Equations. Differential-Algebraic Equations Forum, pp. 97–106. Springer, Berlin/Heidelberg (2014)Google Scholar
  11. 11.
    Arnold, M., Günther, M.: Preconditioned dynamic iteration for coupled differential-algebraic systems. BIT Numer. Math. 41 (1), 001–025 (2001)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Arnold, M., Murua, A.: Non-stiff integrators for differential-algebraic systems of index 2. Numer. Algorithm. 19 (1–4), 25–41 (1998)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Arnold, M., Strehmel, K., Weiner, R.: Half-explicit Runge–Kutta methods for semi-explicit differential-algebraic equations of index 1. Numer. Math. 64 (1), 409–431 (1993)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Arnold, M., Burgermeister, B., Eichberger, A.: Linearly implicit time integration methods in real-time applications: DAEs and stiff ODEs. Multibody Syst. Dyn. 17 (2–3), 99–117 (2007)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Arnold, M., Burgermeister, B., Führer, C., Hippmann, G., Rill, G.: Numerical methods in vehicle system dynamics: state of the art and current developments. Veh. Syst. Dyn. 49 (7), 1159–1207 (2011)CrossRefGoogle Scholar
  16. 16.
    Arnold, M., Clauß, C., Schierz, T.: Error analysis and error estimates for co-simulation in FMI for model exchange and co-simulation v2.0. Arch. Mech. Eng. LX, 75–94 (2013)Google Scholar
  17. 17.
    Arnold, M., Hante, S., Köbis, M.A.: Error analysis for co-simulation with force-displacement coupling. Proc. Appl. Math. Mech. 14 (1), 43–44 (2014)CrossRefGoogle Scholar
  18. 18.
    Ascher, U.M., Petzold, L.R.: Projected implicit Runge-Kutta methods for differential-algebraic equations. SIAM J. Numer. Anal. 28 (4), 1097–1120 (1991)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Balzer, M., Burger, M., Däuwel, T., Ekevid, T., Steidel, S., Weber, D.: Coupling DEM particles to MBS wheel loader via co-simulation. In: Proceedings of the 4th Commercial Vehicle Technology Symposium (CVT 2016), pp. 479–488 (2016)Google Scholar
  20. 20.
    Bartel, A., Brunk, M., Günther, M., Schöps, S.: Dynamic iteration for coupled problems of electric circuits and distributed devices. SIAM J. Sci. Comput. 35 (2), B315–B335 (2013)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Bartel, A., Brunk, M., Schöps, S.: On the convergence rate of dynamic iteration for coupled problems with multiple subsystems. J. Comput. Appl. Math. 262, 14–24 (2014). Selected Papers from NUMDIFF-13Google Scholar
  22. 22.
    Baumgarte, J.: Stabilization of constraints and integrals of motion in dynamical systems. Comput. Methods Appl. Mech. Eng. 1, 1–16 (1972)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Becker, U.: Efficient time integration and nonlinear model reduction for incompressible hyperelastic materials. Ph.D. thesis, TU Kaiserslautern (2012)Google Scholar
  24. 24.
    Becker, U., Simeon, B., Burger, M.: On rosenbrock methods for the time integration of nearly incompressible materials and their usage for nonlinear model reduction. J. Comput. Appl. Math. 262, 333–345 (2014). Selected Papers from NUMDIFF-13Google Scholar
  25. 25.
    Bock, H.G.: Randwertproblemmethoden zur Parameteridentifizierung in Systemen nichtlinearer Differentialgleichungen, vol. 183. Bonner Mathematische Schriften, Bonn (1987)MATHGoogle Scholar
  26. 26.
    Brasey, V., Hairer, E.: Half-explicit RungeKutta methods for differential-algebraic systems of index 2. SIAM J. Numer. Anal. 30 (2), 538–552 (1993)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Brenan, K.E., Engquist, B.E.: Backward differentiation approximations of nonlinear differential/algebraic systems. Math. Comput. 51 (184), 659–676 (1988)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Brenan, K.E., Petzold, L.R.: The numerical solution of higher index differential/algebraic equations by implicit methods. SIAM J. Numer. Anal. 26 (4), 976–996 (1989)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. Classics in Applied Mathematics, vol. 14. SIAM, Philadelphia (1996)Google Scholar
  30. 30.
    Brown, P.N., Hindmarsh, A.C., Petzold, L.R.: Consistent initial condition calculation for differential-algebraic systems. SIAM J. Sci. Comput. 19 (5), 1495–1512 (1998)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Burgermeister, B., Arnold, M., Esterl, B.: DAE time integration for real-time applications in multi-body dynamics. Z. Angew. Math. Mech. 86 (10), 759–771 (2006)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Burgermeister, B., Arnold, M., Eichberger, A.: Smooth velocity approximation for constrained systems in real-time simulation. Multibody Syst. Dyn. 26 (1), 1–14 (2011)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Büskens, C., Gerdts, M.: Differentiability of consistency functions for DAE systems. J. Optim. Theory Appl. 125 (1), 37–61 (2005)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Campbell, S.L., Gear, C.W.: The index of general nonlinear DAEs. Numer. Math. 72, 173–196 (1995)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Campbell, S.L., Kelley, C.T., Yeomans, K.D.: Consistent initial conditions for unstructured higher index DAEs: a computational study. In: Computational Engineering in Systems Applications, France, pp. 416–421 (1996)Google Scholar
  36. 36.
    Cao, Y., Li, S., Petzold, L.R., Serban, R.: Adjoint sensitivity analysis for differential-algebraic equations: the adjoint DAE system and its numerical solution. SIAM J. Sci. Comput. 24 (3), 1076–1089 (2003)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Caracotsios, M., Stewart, W.E.: Sensitivity analysis of initial-boundary-value problems with mixed PDEs and algebraic equations. Comput. Chem. Eng. 19 (9), 1019–1030 (1985)CrossRefGoogle Scholar
  38. 38.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)MATHGoogle Scholar
  39. 39.
    Cuadrado, J., Cardenal, J., Bayo, E.: Modeling and solution methods for efficient real-time simulation of multibody dynamics. Multibody Syst. Dyn. 1 (3), 259–280 (1997)MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Curtiss, C.F., Hirschfelder, J.O.: Integration of stiff equations. Proc. Nat. Acad. Sci. U.S.A. 38, 235–243 (1952)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Deuflhard, P., Hairer, E., Zugck, J.: One-step and extrapolation methods for differential-algebraic systems. Numer. Math. 51 (5), 501–516 (1987)MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Diehl, M., Bock, H.G., Schlöder, J.P., Findeisen, R., Nagy, Z., Allgöwer, F.: Real-time optimization and nonlinear model predictive control of processes governed by differential-algebraic equations. J. Process Control 12 (4), 577–585 (2002)CrossRefGoogle Scholar
  43. 43.
    Diehl, M., Bock, H.G., Schlöder, J.P.: A real-time iteration scheme for nonlinear optimization in optimal feedback control. SIAM J. Control Optim. 43 (5), 1714–1736 (2005)MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Dopico, D., Lugris, U., Gonzalez, M., Cuadrado, J.: Two implementations of IRK integrators for real-time multibody dynamics. Int. J. Numer. Methods Eng. 65 (12), 2091–2111 (2006)MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Duff, I.S., Gear, C.W.: Computing the structural index. SIAM J. Algebr. Discrete Methods 7 (4), 594–603 (1986)MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Ebrahimi, S., Eberhard, P.: A linear complementarity formulation on position level for frictionless impact of planar deformable bodies. Z. Angew. Math. Mech. 86 (10), 807–817 (2006)MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Eich, E.: Convergence results for a coordinate projection method applied to mechanical systems with algebraic constraints. SIAM J. Numer. Anal. 30 (5), 1467–1482 (1993)MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    Eichberger, A., Rulka, W.: Process save reduction by macro joint approach: the key to real time and efficient vehicle simulation. Veh. Syst. Dyn. 41 (5), 401–413 (2004)CrossRefGoogle Scholar
  49. 49.
    Engelhardt, L., Burger, M., Bitsch, G.: Real-time simulation of multibody systems for on-board applications. In: Proceedings of the First Joint International Conference on Multibody System Dynamics (IMSD2010) (2010)Google Scholar
  50. 50.
    Esterl, B., Butz, T., Simeon, B., Burgermeister, B.: Real-time capable vehicletrailer coupling by algorithms for differential-algebraic equations. Veh. Syst. Dyn. 45 (9), 819–834 (2007)CrossRefGoogle Scholar
  51. 51.
    Estévez Schwarz, D.: Consistent initialization for index-2 differential-algebraic equations and its application to circuit simulation. Ph.D. thesis, Mathematisch-Naturwissenschaftlichen Fakultät II, Humboldt-Universität Berlin (2000)Google Scholar
  52. 52.
    Feehery, W.F., Tolsma, J.E., Barton, P.I.: Efficient sensitivity analysis of large-scale differential-algebraic systems. Appl. Numer. Math. 25, 41–54 (1997)MathSciNetMATHCrossRefGoogle Scholar
  53. 53.
    Feng, A., Holland, C.D., Gallun, S.E.: Development and comparison of a generalized semi-implicit Runge–Kutta method with Gear’s method for systems of coupled differential and algebraic equations. Comput. Chem. Eng. 8 (1), 51–59 (1984)CrossRefGoogle Scholar
  54. 54.
    Fiacco, A.V.: Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. Mathematics in Science and Engineering, vol. 165. Academic Press, New York (1983)Google Scholar
  55. 55.
    Fischer, A.: A special Newton-type optimization method. Optimization 24, 269–284 (1992)MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    Führer, C.: Differential-algebraische Gleichungssysteme in mechanischen Mehrkörpersystemen: Theorie, numerische Ansätze und Anwendungen. Ph.D. thesis, Fakultät für Mathematik und Informatik, Technische Universität München (1988)Google Scholar
  57. 57.
    Führer, C., Leimkuhler, B.J.: Numerical solution of differential-algebraic equations for constraint mechanical motion. Numer. Math. 59, 55–69 (1991)MathSciNetMATHCrossRefGoogle Scholar
  58. 58.
    Gallrein, A., Baecker, M., Burger, M., Gizatullin, A.: An advanced flexible realtime tire model and its integration into Fraunhofer’s driving simulator. SAE Technical Paper 2014-01-0861 (2014)Google Scholar
  59. 59.
    Garavello, M., Piccoli, B.: Hybrid necessary principle. SIAM J. Control Optim. 43 (5), 1867–1887 (2005)MathSciNetMATHCrossRefGoogle Scholar
  60. 60.
    Gavrea, B.I., Anitescu, M., Potra, F.A.: Convergence of a class of semi-implicit time-stepping schemes for nonsmooth rigid multibody dynamics. SIAM J. Optim. 19 (2), 969–1001 (2008)MathSciNetMATHCrossRefGoogle Scholar
  61. 61.
    Gear, C.W.: Simultaneous numerical solution of differential-algebraic equations. IEEE Trans. Circuit Theory 18 (1), 89–95 (1971)CrossRefGoogle Scholar
  62. 62.
    Gear, C.W.: Differential-algebraic equation index transformations. SIAM J. Sci. Stat. Comput. 9, 39–47 (1988)MathSciNetMATHCrossRefGoogle Scholar
  63. 63.
    Gear, C.W., Petzold, L.R.: ODE methods for the solution of differential/algebraic systems. SIAM J. Numer. Anal. 21 (4), 716–728 (1984)MathSciNetMATHCrossRefGoogle Scholar
  64. 64.
    Gear, C.W., Leimkuhler, B., Gupta, G.K.: Automatic integration of Euler-Lagrange equations with constraints. J. Comput. Appl. Math. 12 (13), 77–90 (1985)MathSciNetMATHCrossRefGoogle Scholar
  65. 65.
    Geier, T., Foerg, M., Zander, R., Ulbrich, H., Pfeiffer, F., Brandsma, A., van der Velde, A.: Simulation of a push belt CVT considering uni- and bilateral constraints. Z. Angew. Math. Mech. 86 (10), 795–806 (2006)MathSciNetMATHCrossRefGoogle Scholar
  66. 66.
    Gerdts, M.: Optimal control and real-time optimization of mechanical multi-body systems. Z. Angew. Math. Mech. 83 (10), 705–719 (2003)MathSciNetMATHCrossRefGoogle Scholar
  67. 67.
    Gerdts, M.: Parameter optimization in mechanical multibody systems and linearized runge-kutta methods. In: Buikis, A., Ciegis, R., Flitt, A.D. (eds.) Progress in Industrial Mathematics at ECMI 2002. Mathematics in Industry, vol. 5, pp. 121–126. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  68. 68.
    Gerdts, M.: Optimal Control of ODEs and DAEs. Walter de Gruyter, Berlin/Boston (2012)MATHCrossRefGoogle Scholar
  69. 69.
    Gerdts, M., Büskens, C.: Consistent initialization of sensitivity matrices for a class of parametric DAE systems. BIT Numer. Math. 42 (4), 796–813 (2002)MathSciNetMATHCrossRefGoogle Scholar
  70. 70.
    Gerdts, M., Kunkel, M.: A nonsmooth Newton’s method for discretized optimal control problems with state and control constraints. J. Ind. Manag. Optim. 4 (2), 247–270 (2008)MathSciNetMATHCrossRefGoogle Scholar
  71. 71.
    Gopal, V., Biegler, L.T.: A successive linear programming approach for initialization and reinitialization after discontinuities of differential-algebraic equations. SIAM J. Sci. Comput. 20 (2), 447–467 (1998)MathSciNetMATHCrossRefGoogle Scholar
  72. 72.
    Griewank, A., Walther, A.: Evaluating Derivatives. Principles and Techniques of Algorithmic Differentiation, 2nd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2008)Google Scholar
  73. 73.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer Series in Computational Mathematics, vol. 14, 2nd edn. Springer, Berlin/Heidelberg/New York (1996)Google Scholar
  74. 74.
    Hairer, E., Lubich, C., Roche, M.: Error of Rosenbrock methods for stiff problems studied via differential algebraic equations. BIT 29 (1), 77–90 (1989)MathSciNetMATHCrossRefGoogle Scholar
  75. 75.
    Hairer, E., Lubich, C., Roche, M.: The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods. Lecture Notes in Mathematics, vol. 1409. Springer, Berlin/Heidelberg/New York (1989)Google Scholar
  76. 76.
    Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer Series in Computational Mathematics, vol. 8, 2nd edn. Springer, Berlin/Heidelberg/New York (1993)Google Scholar
  77. 77.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations. Reprint of the Second 2006 edition. Springer, Berlin (2010)Google Scholar
  78. 78.
    Hansen, B.: Computing consistent initial values for nonlinear index-2 differential-algebraic equations. Seminarberichte Humboldt-Universität Berlin, 92-1, 142–157 (1992)Google Scholar
  79. 79.
    Heim, A.: Parameteridentifizierung in differential-algebraischen Gleichungssystemen. Master’s thesis, Mathematisches Institut, Technische Universität München (1992)Google Scholar
  80. 80.
    Hindmarsh, A.C., Brown, P.N., Grant, K.E., Lee, S.L., Serban, R., Shumaker, D.E., Woodward, C.S.: Sundials: suite of nonlinear and differential/algebraic equation solvers. ACM Trans. Math. Softw. 31 (3), 363–396 (2005)MathSciNetMATHCrossRefGoogle Scholar
  81. 81.
    INTEC GmbH. SIMPACK – Analysis and Design of General Mechanical Systems. WeßlingGoogle Scholar
  82. 82.
    Jackiewicz, Z., Kwapisz, M.L Convergence of waveform relaxation methods for differential-algebraic systems. SIAM J. Numer. Anal. 33 (6), 2303–2317 (1996)Google Scholar
  83. 83.
    Jackson, K.R.: A survey of parallel numerical methods for initial value problems for ordinary differential equations. IEEE Trans. Magn. 27 (5), 3792–3797 (1991)CrossRefGoogle Scholar
  84. 84.
    Jay, L.: Collocation methods for differential-algebraic equations of index 3. Numer. Math. 65, 407–421 (1993)MathSciNetMATHCrossRefGoogle Scholar
  85. 85.
    Jay, L.: Convergence of Runge-Kutta methods for differential-algebraic systems of index 3. Appl. Numer. Math. 17, 97–118 (1995)MathSciNetMATHCrossRefGoogle Scholar
  86. 86.
    Jiang, H.: Global convergence analysis of the generalized Newton and Gauss-Newton methods of the Fischer-Burmeister equation for the complementarity problem. Math. Oper. Res. 24 (3), 529–543 (1999)MathSciNetMATHCrossRefGoogle Scholar
  87. 87.
    Kiehl, M.: Sensitivity analysis of ODEs and DAEs - theory and implementation guide. Optim. Methods Softw. 10 (6), 803–821 (1999)MathSciNetMATHCrossRefGoogle Scholar
  88. 88.
    Kleinert, J., Simeon, B., Dreßler, K.: Nonsmooth contact dynamics for the large-scale simulation of granular material. Technical report, Fraunhofer ITWM, Kaiserslautern, Germany. J. Comput. Appl. Math. (2015, in press). http://dx.doi.org/10.1016/j.cam.2016.09.037
  89. 89.
    Kübler, R., Schiehlen, W.: Two methods of simulator coupling. Math. Comput. Model. Dyn. Syst. 6 (2), 93–113 (2000)MATHCrossRefGoogle Scholar
  90. 90.
    Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations. Analysis and Numerical Solution, vol. viii, 377 p. European Mathematical Society Publishing House, Zürich (2006)Google Scholar
  91. 91.
    Küsters, F., Ruppert, M.G.-M., Trenn, S.: Controllability of switched differential-algebraic equations. Syst. Control Lett. 78, 32–39 (2015)MathSciNetMATHCrossRefGoogle Scholar
  92. 92.
    Lamour, R., März, R., Tischendorf, C.: Differential-Algebraic Equations: A Projector Based Analysis. Differential-Algebraic Equations Forum. Springer, Berlin (2013)MATHCrossRefGoogle Scholar
  93. 93.
    Leimkuhler, B., Petzold, L.R., Gear, C.W.: Approximation methods for the consistent initialization of differential-algebraic equations. SIAM J. Numer. Anal. 28 (1), 205–226 (1991)MathSciNetMATHCrossRefGoogle Scholar
  94. 94.
    Lelarasmee, E., Ruehli, A.E., Sangiovanni-Vincentelli, A.L.: The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 1 (3), 131–145 (1982)CrossRefGoogle Scholar
  95. 95.
    Lemke, C.E.: The dual method of solving the linear programming problem. Naval Res. Log. Q. 1, 36–47 (1954)MathSciNetMATHCrossRefGoogle Scholar
  96. 96.
    Leyendecker, S., Ober-Blöbaum, S.: A variational approach to multirate integration for constrained systems. In: Multibody Dynamics. Computational Methods and Applications. Selected Papers Based on the Presentations at the ECCOMAS Thematic Conference, Brussels, Belgium, 4–7 July 2011, pp. 97–121. Springer, Dordrecht (2013)Google Scholar
  97. 97.
    Liberzon, D., Trenn, S.: Switched nonlinear differential algebraic equations: solution theory, Lyapunov functions, and stability. Automatica 48 (5), 954–963 (2012)MathSciNetMATHCrossRefGoogle Scholar
  98. 98.
    Linn, J., Stephan, T., Carlson, J.S., Bohlin, R.: Fast simulation of quasistatic rod deformations for VR applications. In: Bonilla, L.L., Moscoso, M., Platero, G., Vega, J.M. (eds.) Progress in Industrial Mathematics at ECMI 2006. Springer, New York (2007)Google Scholar
  99. 99.
    Lötstedt, P., Petzold, L.R.: Numerical solution of nonlinear differential equations with algebraic constraints I: convergence results for backward differentiation formulas. Math. Comput. 46, 491–516 (1986)MathSciNetMATHCrossRefGoogle Scholar
  100. 100.
    Lubich, C., Engstler, C., Nowak, U., Pöhle, U.: Numerical integration of constrained mechanical systems using MEXX*. Mech. Struct. Mach. 23 (4), 473–495 (1995)CrossRefGoogle Scholar
  101. 101.
    Maly, T., Petzold, L.R.: Numerical methods and software for sensitivity analysis of differential-algebraic systems. Appl. Numer. Math. 20 (1), 57–79 (1996)MathSciNetMATHCrossRefGoogle Scholar
  102. 102.
    Michael, J., Gerdts, M.: A method to model impulsive multi-body-dynamics using Riemann-Stieltjes- Integrals. In: 8th Vienna International Conference on Mathematical Modelling, International Federation of Automatic Control, pp. 629–634 (2015)Google Scholar
  103. 103.
    Michael, J., Chudej, K., Gerdts, M., Pannek, J.: Optimal rendezvous path planning to an uncontrolled tumbling target. In: IFAC Proceedings Volumes (IFAC-PapersOnline), 19th IFAC Symposium on Automatic Control in Aerospace, ACA 2013, Wurzburg, Germany, 2–6 September 2013, vol. 19, pp. 347–352 (2013)Google Scholar
  104. 104.
    Miekkala, U., Nevanlinna, O.: Convergence of dynamic iteration methods for initial value problems. SIAM J. Sci. Stat. Comput. 8 (4), 459–482 (1987)MathSciNetMATHCrossRefGoogle Scholar
  105. 105.
    Murua, A.: Partitioned half-explicit Runge–Kutta methods for differential-algebraic systems of index 2. Computing 59 (1), 43–61 (1997)MathSciNetMATHCrossRefGoogle Scholar
  106. 106.
    Negrut, D., Sandu, A., Haug, E.J., Potra, F.A., Sandu, C.: A Rosenbrock-Nystrom state space implicit approach for the dynamic analysis of mechanical systems: II –the method and numerical examples. J. Multi-body Dyn. 217 (4), 273–281 (2003)Google Scholar
  107. 107.
    Ostermann, A.: A class of half-explicit Runge–Kutta methods for differential-algebraic systems of index 3. Appl. Numer. Math. 13 (1), 165–179 (1993)MathSciNetMATHCrossRefGoogle Scholar
  108. 108.
    Pantelides, C.C.: The consistent initialization of differential-algebraic systems. SIAM J. Sci. Stat. Comput. 9 (2), 213–231 (1988)MathSciNetMATHCrossRefGoogle Scholar
  109. 109.
    Petzold, L.R.: A description of DASSL: a differential/algebraic system solver. Rep. Sand 82-8637, Sandia National Laboratory, Livermore (1982)Google Scholar
  110. 110.
    Petzold, L.R.: Differential/algebraic equations are not ODE’s. SIAM J. Sci. Stat. Comput. 3 (3), 367–384 (1982)MathSciNetMATHCrossRefGoogle Scholar
  111. 111.
    Petzold, L.R.: Recent developments in the numerical solution of differential/algebraic systems. Comput. Methods Appl. Mech. Eng. 75, 77–89 (1989)MathSciNetMATHCrossRefGoogle Scholar
  112. 112.
    Pfeiffer, A.: Numerische Sensitivitätsanalyse unstetiger multidisziplinärer Modelle mit Anwendungen in der gradientenbasierten Optimierung. Fortschritt-Berichte VDI Reihe 20, Nr. 417. VDI–Verlag, Düsseldorf (2008)Google Scholar
  113. 113.
    Potra, F.A., Anitescu, M., Gavrea, B., Trinkle, J.: A linearly implicit trapezoidal method for integrating stiff multibody dynamics with contact, joints, and friction. Int. J. Numer. Methods Eng. 66 (7), 1079–1124 (2006)MathSciNetMATHCrossRefGoogle Scholar
  114. 114.
    Pytlak, R., Suski, D.: On solving hybrid optimal control problems with higher index DAEs. Institute of Automatic Control and Robotics, Warsaw University of Technology, Warsaw, Poland (2015, Preprint)Google Scholar
  115. 115.
    Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18 (1), 227–244 (1993)MathSciNetMATHCrossRefGoogle Scholar
  116. 116.
    Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58 (3), 353–367 (1993)MathSciNetMATHCrossRefGoogle Scholar
  117. 117.
    Rentrop, P., Roche, M., Steinebach, G.: The application of Rosenbrock-Wanner type methods with stepsize control in differential-algebraic equations. Numer. Math. 55 (5), 545–563 (1989)MathSciNetMATHCrossRefGoogle Scholar
  118. 118.
    Rill, G.: A modified implicit Euler algorithm for solving vehicle dynamic equations. Multibody Syst. Dyn. 15 (1), 1–24 (2006)MATHCrossRefGoogle Scholar
  119. 119.
    Rill, G., Chucholowski, C.: Real time simulation of large vehicle systems. In: Proceedings of Multibody Dynamics 2007 (ECCOMAS Thematic Conference) (2007)Google Scholar
  120. 120.
    Roche, M.: Rosenbrock methods for differential algebraic equations. Numer. Math. 52 (1), 45–63 (1988)MathSciNetMATHCrossRefGoogle Scholar
  121. 121.
    Rosenbrock, H.H.: Some general implicit processes for the numerical solution of differential equations. Comput. J. 5 (4), 329–330 (1963)MathSciNetMATHCrossRefGoogle Scholar
  122. 122.
    Rulka, W., Pankiewicz, E.: MBS approach to generate equations of motions for hil-simulations in vehicle dynamics. Multibody Syst. Dyn. 14 (3), 367–386 (2005)MATHCrossRefGoogle Scholar
  123. 123.
    Sandu, A., Negrut, D., Haug, E.J., Potra, F.A., Sandu, C.: A Rosenbrock-Nystrom state space implicit approach for the dynamic analysis of mechanical systems: I—theoretical formulation. J. Multi-body Dyn. 217 (4), 263–271 (2003)Google Scholar
  124. 124.
    Schaub, M., Simeon, B.: Blended Lobatto methods in multibody dynamics. Z. Angew. Math. Mech. 83 (10), 720–728 (2003)MathSciNetMATHCrossRefGoogle Scholar
  125. 125.
    Schierz, T., Arnold, M.: Stabilized overlapping modular time integration of coupled differential-algebraic equations. Appl. Numer. Math. 62 (10), 1491–1502 (2012). Selected Papers from NUMDIFF-12Google Scholar
  126. 126.
    Schneider, F., Burger, M., Arnold, M., Simeon, B.: A new approach for force-displacement co-simulation using kinematic coupling constraints. Submitted to Z. Angew. Math. Mech. (2016)Google Scholar
  127. 127.
    Schulz, V.H., Bock, H.G., Steinbach, M.C.: Exploiting invariants in the numerical solution of multipoint boundary value problems for DAE. SIAM J. Sci. Comput. 19 (2), 440–467 (1998)MathSciNetMATHCrossRefGoogle Scholar
  128. 128.
    Schwartz, W., Frik, S., Leister, G.: Simulation of the IAVSD Road Vehicle Benchmark Bombardier Iltis with FASIM, MEDYNA, NEWEUL and SIMPACK. Technical Report IB 515/92-20, Robotik und Systemdynamik, Deutsche Forschungsanstalt für Luft- und Raumfahrt (1992)Google Scholar
  129. 129.
    Schweizer, B., Lu, D.: Stabilized index-2 co-simulation approach for solver coupling with algebraic constraints. Multibody Syst. Dyn. 34 (2), 129–161 (2014)MathSciNetMATHCrossRefGoogle Scholar
  130. 130.
    Schweizer, B., Li, P., Lu, D.: Implicit co-simulation methods: stability and convergence analysis for solver coupling approaches with algebraic constraints. Z. Angew. Math. Mech. 96 (8), 986–1012 (2016)MathSciNetCrossRefGoogle Scholar
  131. 131.
    Stetter, H.J.: Analysis of Discretization Methods for Ordinary Differential Equations. Springer Tracts in Natural Philosophy, vol. 23. Springer, Berlin/Heidelberg/New York (1973)Google Scholar
  132. 132.
    Stewart, D.E.: Rigid-body dynamics with friction and impact. SIAM Rev. 42 (1), 3–39 (2000)MathSciNetMATHCrossRefGoogle Scholar
  133. 133.
    Stewart, D.E., Anitescu, M.: Optimal control of systems with discontinuous differential equations. Numer. Math. 114 (4), 653–695 (2010)MathSciNetMATHCrossRefGoogle Scholar
  134. 134.
    Strehmel, K., Weiner, R.: Numerik gewöhnlicher Differentialgleichungen. Teubner, Stuttgart (1995)MATHGoogle Scholar
  135. 135.
    Strehmel, K., Weiner, R., Dannehl, I.: On error behaviour of partitioned linearly implicit Runge–Kutta methods for stiff and differential algebraic systems. BIT 30 (2), 358–375 (1990)MathSciNetMATHCrossRefGoogle Scholar
  136. 136.
    Sussmann, H.J.: A nonsmooth hybrid maximum principle. In: Stability and Stabilization of Nonlinear Systems. Proceedings of the 1st Workshop on Nonlinear Control Network, Held in Gent, Belgium, 15–16 March 1999, pp. 325–354. Springer, London (1999)Google Scholar
  137. 137.
    Tasora, A., Anitescu, M.: A fast NCP solver for large rigid-body problems with contacts, friction, and joints. In: Multibody Dynamics. Computational Methods and Applications. Revised, extended and selected papers of the ECCOMAS Thematic Conference on Multibody Dynamics 2007, Milano, Italy, 25–28 June 2007, pp. 45–55. Springer, Dordrecht (2009)Google Scholar
  138. 138.
    Tasora, A., Anitescu, M.: A matrix-free cone complementarity approach for solving large-scale, nonsmooth, rigid body dynamics. Comput. Methods Appl. Mech. Eng. 200 (5–8), 439–453 (2011)MathSciNetMATHCrossRefGoogle Scholar
  139. 139.
    Tasora, A., Anitescu, M.: A complementarity-based rolling friction model for rigid contacts. Meccanica 48 (7), 1643–1659 (2013)MathSciNetMATHCrossRefGoogle Scholar
  140. 140.
    Tasora, A., Negrut, D., Anitescu, M.: GPU-based parallel computing for the simulation of complex multibody systems with unilateral and bilateral constraints: an overview. In: Multibody Dynamics. Computational Methods and Applications. Selected papers based on the presentations at the ECCOMAS Conference on Multibody Dynamics, Warsaw, Poland, June 29–July 2, 2009, pp. 283–307. Springer, New York, NY (2011)Google Scholar
  141. 141.
    Trenn, S.: Solution concepts for linear DAEs: a survey. In: Ilchmann, A., Reis, T. (eds.) Surveys in Differential-Algebraic Equations I. Differential-Algebraic Equations Forum, pp. 137–172. Springer, Berlin (2013)CrossRefGoogle Scholar
  142. 142.
    van der Schaft, A., Schumacher, H.: An Introduction to Hybrid Dynamical Systems. Springer, London (1989)MATHGoogle Scholar
  143. 143.
    Veitl, A., Gordon, T., van de Sand, A., Howell, M., Valasek, M., Vaculin, O., Steinbauer, P.: Methodologies for coupling simulation models and codes in mechatronic system analysis and design. In: Proceedings of the 16th IAVSD Symposium on Dynamics of Vehicles on Roads and Tracks. Pretoria. Supplement to Vehicle System Dynamics, vol. 33, pp. 231–243. Swets & Zeitlinger (1999)Google Scholar
  144. 144.
    von Schwerin, R.: Multibody System Simulation: Numerical Methods, Algorithms, and Software. Lecture Notes in Computational Science and Engineering, vol. 7. Springer, Berlin/Heidelberg/New York (1999)Google Scholar
  145. 145.
    Wensch, J.: An eight stage fourth order partitioned Rosenbrock method for multibody systems in index-3 formulation. Appl. Numer. Math. 27 (2), 171–183 (1998)MathSciNetMATHCrossRefGoogle Scholar
  146. 146.
    Wensch, J., Strehmel, K., Weiner, R.: A class of linearly-implicit Runge–Kutta methods for multibody systems. Appl. Numer. Math. 22 (13), 381–398 (1996). Special Issue Celebrating the Centenary of Runge–Kutta MethodsGoogle Scholar
  147. 147.
    Wolfbrandt, A., Steihaug, T.: An attempt to avoid exact Jacobian and nonlinear equations in the numerical solution of stiff differential equations. Math. Comput. 33 (146), 521–534 (1979)MathSciNetMATHCrossRefGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department Mathematical Methods in Dynamics and Durability MDFFraunhofer Institute for Industrial Mathematics ITWMKaiserslauternGermany
  2. 2.Department of Aerospace Engineering, Institute of Mathematics and Applied ComputingUniversität der Bundeswehr MünchenNeubibergGermany

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