Differential-algebraic equations (DAEs) arise naturally in many technical and industrial applications. By incorporating the special structure of the DAE systems arising in certain physical domains, the general approach for the regularization of DAEs can be efficiently adapted to the system structure. We will present the analysis and regularization approaches for DAEs arising in mechanical multibody systems, electrical circuit equations, and flow problems. In each of these cases the DAEs exhibit a certain structure that can be used for an efficient analysis and regularization. Moreover, we discuss the numerical treatment of hybrid DAE systems, that also occur frequently in industrial applications. For such systems, the framework of DAEs provides essential information for a robust numerical treatment.


Hybrid System Multibody System Nonholonomic Constraint Minimal Extension Switching Surface 
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.



This work is dedicated to Prof. Dr. Volker Mehrmann. We would like to express our gratitude to him for initiating our work on DAEs and their applications many years ago. We would like to thank him for his support, his valuable criticism and discussions, but also for the freedom he gave us in our research.


  1. 1.
    Altmann, R., Heiland, J.: Finite element decomposition and minimal extension for flow equations. Preprint 2013-11, Institute of Mathematics, TU Berlin (2013)Google Scholar
  2. 2.
    Arnold, M.: Half-explicit Runge-Kutta methods with explicit stages for differential-algebraic systems of index 2. BIT 38(3), 415–438 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Auslander, L., MacKenzie, R.E.: Introduction to Differentiable Manifolds. Dover, New York (1977). Corrected reprintingGoogle Scholar
  4. 4.
    Bächle, S.: Numerical solution of differential-algebraic systems arising in circuit simulation. Ph.D. thesis, TU Berlin (2007)Google Scholar
  5. 5.
    Bächle, S., Ebert, F.: Index reduction by element-replacement for electrical circuits. In: G. Ciprina and D. Ioan (eds): Scientific Computing in Electrical Engineering. Mathematics in Industry, vol. 11, pp. 191–198. Springer, Berlin/Heidelberg (2007)Google Scholar
  6. 6.
    Baumgarte, J.: Asymptotische Stabilisierung von Integralen bei gewöhnlichen Differentialgleichungen 1. Ordnung. ZAMM 53, 701–704 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Bremer, H.: Dynamik und Regelung mechanischer Systeme. Teubner Studienbücherei, Stuttgart (1988)CrossRefzbMATHGoogle Scholar
  8. 8.
    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
  9. 9.
    Eich, E., Führer, C., Leimkuhler, B., Reich, S.: Stabilization and projection methods for multibody dynamics. Technical Report A281, Helsinki University of Technology (1990)Google Scholar
  10. 10.
    Eich-Soellner, E., Führer, C.: Numerical Methods in Multibody Dynamics. Teubner, B.G., Stuttgart (1998)CrossRefzbMATHGoogle Scholar
  11. 11.
    Eichberger, A.: Simulation von Mehrkörpersystemen auf parallelen Rechnerarchitekturen. Number 332 in Fortschritt-Berichte VDI, Reihe 8: Meß-, Steuerungs- und Regelungstechnik. VDI-Verlag Düsseldorf (1993)Google Scholar
  12. 12.
    Estévez Schwarz, D., Tischendorf, C.: Structural analysis of electric circuits and consequences for MNA. Int. J. Circuit Theory Appl. 28(2), 131–162 (2000)CrossRefzbMATHGoogle Scholar
  13. 13.
    Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Kluwer Academic, Dordrecht (1988)CrossRefGoogle Scholar
  14. 14.
    Führer, C.: Differential-algebraische Gleichungssysteme in mechanischen Mehrkörpersystemen - Theorie, numerische Ansätze und Anwendungen. Ph.D. thesis, TU München (1988)Google Scholar
  15. 15.
    Führer, C., Leimkuhler, B.J.: Numerical solution of differential-algebraic equations for constrained mechanical motion. Numer. Math. 59, 55–69 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Gear, C.W.: Differential-algebraic equation index transformations. SIAM J. Sci. Stat. Comput. 9, 39–47 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    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)Google Scholar
  18. 18.
    Gresho, P.M., Sani, R.L.: Incompressible Flow and the Finite Element Method. Isothermal Laminar Flow, vol. 2. Wiley, Chichester (2000)Google Scholar
  19. 19.
    Griepentrog, E., März, R.: Differential-Algebraic Equations and Their Numerical Treatment. Teubner-Texte zur Mathematik, vol. 88. BSB B.G.Teubner Verlagsgesellschaft, Leipzig (1986)Google Scholar
  20. 20.
    Günther, M., Feldmann, U.: CAD-based electric-circuit modeling in industry, I. Mathematical structure and index of network equations. Surv. Math. Ind. 8, 97–129 (1999)zbMATHGoogle Scholar
  21. 21.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd edn. Springer, Berlin (1996)CrossRefzbMATHGoogle Scholar
  22. 22.
    Hamann, P., Mehrmann, V.: Numerical solution of hybrid systems of differential-algebraic equations. Comput. Methods Appl. Mech. Eng. 197(6–8), 693–705 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Hamel, G.: Theoretische Mechanik. Springer, Berlin (1967)zbMATHGoogle Scholar
  24. 24.
    Haug, E.J.: Computer Aided Kinematics and Dynamics of Mechanical Systems. Basic Methods, vol. 1. Allyn & Bacon, Boston (1989)Google Scholar
  25. 25.
    Heuser, H.: Gewöhnliche Differentialgleichungen. B.G. Teubner, Stuttgart (1991)zbMATHGoogle Scholar
  26. 26.
    Kunkel, P., Mehrmann, V.: Analysis of over- and underdetermined nonlinear differential-algebraic systems with application to nonlinear control problems. Math. Control Signals Syst. 14, 233–256 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Kunkel, P., Mehrmann, V.: Index reduction for differential-algebraic equations by minimal extension. Zeitschrift für Angewandte Mathematik und Mechanik 84, 579–597 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations—Analysis and Numerical Solution. EMS Publishing House, Zürich (2006)CrossRefzbMATHGoogle Scholar
  29. 29.
    Kunkel, P., Mehrmann, V., Rath, W., Weickert, J.: GELDA: A software package for the solution of general linear differential algebraic equations. SIAM J. Sci. Comput. 18, 115–138 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Kunkel, P., Mehrmann, V., Seufer, I.: GENDA: A software package for the numerical solution of general nonlinear differential-algebraic equations. Report 730-02, Institute of Mathematics, TU Berlin (2002)Google Scholar
  31. 31.
    Kunkel, P., Mehrmann, V., Seidel, S.: A Matlab package for the numerical solution of general nonlinear differential-algebraic equations. Preprint, Institute of Mathematics, TU Berlin(2005)Google Scholar
  32. 32.
    Lagrange, J.L.: Méchanique analytique. Desaint, Paris (1788)Google Scholar
  33. 33.
    Lötstedt, P.: Mechanical systems of rigid bodies subject to unilateral constraints. SIAM J. Appl. Math. 42, 281–296 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Mehrmann, V., Wunderlich, L.: Hybrid systems of differential-algebraic equations – Analysis and numerical solution. J. Process Control 19, 1218–1228 (2009)Google Scholar
  35. 35.
    Mosterman, P.J.: An overview of hybrid simulation phenomena and their support by simulation packages. In: F.W. Vaandrager and J.H. van Schuppen (eds): Hybrid Systems: Computation and Control. Lecture Notes in Computer Science, vol. 1569, pp. 165–177. Springer, Berlin(1999)Google Scholar
  36. 36.
    Newton, S.I.: Philosophiae naturalis principia mathematica. (1687)Google Scholar
  37. 37.
    Petzold, L.R.: Differential/algebraic equations are not ODEs. SIAM J. Sci. Stat. Comput. 3, 367–384 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Reis, T.: Mathematical modeling and analysis of nonlinear time-invariant RLC circuits. In: P. Benner, R. Findeisen, D. Flockerzi, U. Reichl and K. Sundmacher (eds): Large-Scale Networks in Engineering and Life Sciences, pp. 125–198. Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, Basel, (2014)Google Scholar
  39. 39.
    Rentrop, P., Strehmel, K., Weiner, R.: Ein Überblick über Einschrittverfahren zur numerischen Integration in der technischen Simulation. GAMM-Mitteilungen 1, 9–43 (1996)MathSciNetGoogle Scholar
  40. 40.
    Roberson, R.E., Schwertassek, R.: Dynamics of Multibody Systems. Springer, Berlin (1988)CrossRefzbMATHGoogle Scholar
  41. 41.
    Rulka, W.: Effiziente Simulation der Dynamik mechatronischer Systeme für industrielle Anwendungen. Technical Report IB 532–01–06, DLR German Aerospace Center, Institute of Aeroelasticity, Vehicle System Dynamics Group (2001)Google Scholar
  42. 42.
    Schiehlen, W.O.: Multibody System Handbook. Springer, Berlin (1990)CrossRefGoogle Scholar
  43. 43.
    Schiehlen, W.O.: Advanced Multibody System Dynamics. Kluwer Academic, Dordrecht (1993)CrossRefzbMATHGoogle Scholar
  44. 44.
    Schiehlen, W.O.: Multibody system dynamics: Roots and perspectives. Multibody Syst. Dyn. 1, 149–188 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  45. 45.
    Scholz, L., Steinbrecher, A.: Regularization by minimal extension for multibody systems. Preprint, Institute of Mathematics, TU Berlin (in preparation)Google Scholar
  46. 46.
    De Schutter, B., Heemels, W.P.M.H., Lunze, J., Prieur, C.: Survey of modeling, analysis, and control of hybrid systems. In: J. Lunze, and F. Lamnabhi-Lagarrigue, (eds): Handbook of Hybrid Systems Control – Theory, Tools, Applications. Cambridge University Press, Cambridge, UK, 31–55 (2009). doi:10.1017/CBO9780511807930.003
  47. 47.
    Simeon, B.: MBSPACK – Numerical integration software for constrained mechanical motion. Surv. Math. Ind. 5(3), 169–202 (1995)Google Scholar
  48. 48.
    Steinbrecher, A.: Regularization of nonlinear equations of motion of multibody systems by index reduction with preserving the solution manifold. Preprint 742-02, Institute of Mathematics, TU Berlin (2002)Google Scholar
  49. 49.
    Steinbrecher, A.: Numerical solution of quasi-linear differential-algebraic equations and industrial simulation of multibody systems. Ph.D. thesis, TU Berlin (2006)Google Scholar
  50. 50.
    Steinbrecher, A.: GEOMS: A software package for the numerical integration of general model equations of multibody systems. Preprint 400, Matheon – DFG Research Center “Mathematics for key techonolgies”, Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin (2007)Google Scholar
  51. 51.
    Steinbrecher, A.: GEOMS: A new software package for the numerical simulation of multibody systems. In: 10th International Conference on Computer Modeling and Simulation (EUROSIM/UKSIM 2008), Cambridge, pp. 643–648. IEEE (2008)Google Scholar
  52. 52.
    Steinbrecher, A.: Strangeness-index concept for multibody systems. Preprint, Institute of Mathematics, TU Berlin (in preparation)Google Scholar
  53. 53.
    Temam, R.: Navier-Stokes Equations. Theory and Numerical Analysis. North-Holland, Amsterdam (1977)zbMATHGoogle Scholar
  54. 54.
    Utkin, V.I., Zhao, F.: Adaptive simulation and control of variable-structure control systems in sliding regimes. Automatica 32(7), 1037–1042 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  55. 55.
    Vlach, J., Singhal, K.: Computer Methods for Circuit Analysis and Design. Van Nostrand Reinhold, New York (1994)Google Scholar
  56. 56.
    Weickert, J.: Applications of the theory of differential-algebraic equations to partial differential equations of fluid dynamics. Ph.D. thesis, TU Chemnitz, Fakultät Mathematik, Chemnitz (1997)Google Scholar
  57. 57.
    Wunderlich, L.: Analysis and numerical solution of structured and switched differential-algebraic systems. Ph.D. thesis, TU Berlin (2008)Google Scholar
  58. 58.
    Wunderlich, L.: GESDA: A software package for the numerical solution of general switched differential-algebraic equations. Preprint 576-2009, Matheon – DFG Research Center “Mathematics for key techonolgies”, TU Berlin (2009)Google Scholar
  59. 59.
    Yu, M., Wang, L., Chu, T., Xie, G.: Stabilization of networked control systems with data packet dropout and network delays via switching system approach. In: IEEE Conference on Decision and Control, Nassau, pp. 3539–3544 (2004)Google Scholar
  60. 60.
    Zeidler, E.: Nonlinear Functional Analysis and its Applications II/A: Linear Monotone Operators. Springer, New York (1990)CrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Institut für Mathematik, Technische Universität BerlinBerlinGermany

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