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BIT Numerical Mathematics

, Volume 56, Issue 2, pp 777–804 | Cite as

Structural-algebraic regularization for coupled systems of DAEs

  • Lena ScholzEmail author
  • Andreas Steinbrecher
Article

Abstract

In the automated modeling of multi-physics dynamical systems, frequently different subsystems are coupled together via interface or coupling conditions. This approach often results in large-scale high-index differential-algebraic equations (DAEs). Since the direct numerical simulation of these kinds of systems leads to instabilities and possibly non-convergence of numerical methods, a regularization or remodeling of such systems is required. In many simulation environments, a structural method that analyzes the system based on its sparsity pattern is used to determine the index and an index-reduced system model. However, this approach is not reliable for certain problem classes, and in particular not suited for coupled systems of DAEs. We present a new approach for the regularization of coupled dynamical systems that combines the Signature method (\({\varSigma }\)-method) for the structural analysis with algebraic regularization techniques. This allows to handle structurally singular systems and also enables a proper treatment of redundancies or inconsistencies in the system.

Keywords

Differential-algebraic equation Coupled system Regularization Structural analysis \({\varSigma }\)-method Structurally singular 

Mathematics Subject Classification

65L80 34A09 37M05 

Notes

Acknowledgments

We would like to thank two anonymous referees for their careful reading and for thoughtful suggestions for the improvement of the paper. We also thank Jakob Schneck for performing the computations and comparisons of the different simulation environments.

References

  1. 1.
    Altmeyer, R., Steinbrecher, A.: Regularization and Numerical Simulation of Dynamical Systems Modeled with Modelica. Institut für Mathematik, TU Berlin (2013) (preprint 29-2013)Google Scholar
  2. 2.
    Barton, P.I., Martinson, W.S., Reißig, G.: Differential-algebraic equations of index 1 may have an arbitrarily high structural index. SIAM J. Sci. Comput. 21, 1987–1990 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brenan, K., Campbell, S., Petzold, L.: Numerical Solution of Initial-Value Problems in Differential Algebraic Equations, Classics in Applied Mathematics, vol. 14. SIAM, Philadelphia (1996)zbMATHGoogle Scholar
  4. 4.
    Bunse-Gerstner, A., Byers, R., Mehrmann, V., Nichols, N.K.: Numerical computation of an analytic singular value decomposition of a matrix valued function. Numer. Math. 60(1), 1–39 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Campbell, S., Gear, C.: The index of general nonlinear DAEs. Numer. Math. 72(2), 173–196 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dieci, L., Eirola, T.: On smooth decompositions of matrices. SIAM J. Matrix Anal. Appl. 20(3), 800–819 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Giesbrecht, M., Pham, N.: A symbolic approach to compute a null-space basis in the projection method. In: Computer Mathematics, pp. 243–259. Springer, New York (2014)Google Scholar
  8. 8.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II—Stiff and Differential-Algebraic Problems, 2nd edn. Springer, Berlin (1996)zbMATHGoogle Scholar
  9. 9.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kunkel, P., Mehrmann, V.: Index reduction for differential-algebraic equations by minimal extension. Z. Angew. Math. Mech. 84(9), 579–597 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations—Analysis and Numerical Solution. EMS Publishing House, Zürich (2006)CrossRefzbMATHGoogle Scholar
  12. 12.
    Nedialkov, N., Pryce, J.: Solving differential-algebraic equations by Taylor series (I): computing Taylor coefficients. BIT Numer. Math. 45(3), 561–591 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Nedialkov, N., Pryce, J., Tan, G.: DAESA—a Matlab tool for structural analysis of DAEs: software. In: Technical Report CAS-12-01-NN, Department of Computing and Software, McMaster University, Hamilton (2012)Google Scholar
  14. 14.
    Nilsson, H.: Type-based structural analysis for modular systems of equations. In: Proceedings of the 2nd International Workshop on Equation-Based Object-Oriented Languages and Tools, vol. 029, pp. 71–81 (2008)Google Scholar
  15. 15.
    Pantelides, C.: The consistent initialization of differential-algebraic systems. SIAM J. Sci. Stat. Comput. 9, 213–231 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Pryce, J.: A simple structural analysis method for DAEs. BIT Numer. Math. 41, 364–394 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Scholz, L., Steinbrecher, A.: Efficient numerical integration of dynamical systems based on structural-algebraic regularization avoiding state selection. In: Proceedings of the 10th International Modelica Conference, Lund (2014)Google Scholar
  18. 18.
    Scholz, L., Steinbrecher, A.: Regularization of DAEs based on the signature method. BIT Numer. Math. (2014) (submitted)Google Scholar
  19. 19.
    Steinbrecher, A.: Numerical Solution of Quasi-Linear Differential-Algebraic Equations and Industrial Simulation of Multibody Systems. PhD thesis, Technische Universität Berlin (2006)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Institut für Mathematik, Ma 4-5TU BerlinBerlinFederal Republic of Germany

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