Conference article

Robust Initialization of Differential-Algebraic Equations Using Homotopy

Michael Sielemann
DLR Institute of Robotics and Mechatronics, Germany

Francesco Casella
Dipartimento di Elettronica e Informazione, Politecnico di Milano, Italy

Martin Otter
DLR Institute of Robotics and Mechatronics, Germany

Christop Clauß
Fraunhofer EAS, Dresden, Germany

Jonas Eborn
Modelon AB, Lund, Sweden

Sven Erik Matsson
Dassault Systèmes AB, Lund Sweden

Hans Olsson
Dassault Systèmes AB, Lund Sweden

Download article

Published in: Proceedings of the 8th International Modelica Conference; March 20th-22nd; Technical Univeristy; Dresden; Germany

Linköping Electronic Conference Proceedings 63:10, p. 75-85

Show more +

Published: 2011-06-30

ISBN: 978-91-7393-096-3

ISSN: 1650-3686 (print), 1650-3740 (online)


The new operator homotopy(..) was introduced in Modelica 3.2 to improve the solution of difficult ini-tialization problems. The background and motivation for this approach is discussed and it is demonstrated how to apply it for mechanical; electrical and fluid systems. Furthermore; it is shown at hand of several examples how an inappropriate formulation might lead to ill-posed problems.


Initialization; DAE; homotopy; nonlinear equations


Allgower E.L.; Georg K. (2003): Introduction to numerical continuation methods. SIAM Classics in Applied Mathematics. doi: 10.1137/1.9780898719154.

Casella F.; Sielemann M.; Savoldelli L. (2011): Steadystate initialization of object-oriented thermo-fluid models by homotopy methods. Modelica’2011 Conference; Dresden; March 20-22.

Choi S.H.; Book N.L. (1991): Unreachable roots for global homotopy continuation methods. AIChE Journal 37; pp. 1093-1095. doi: 10.1002/aic.690370713.

Chow S.N.; Mallet-Paret J.; Yorke J.A. (1978): Finding Zeroes of Maps: Homotopy Methods That are Constructive With Probability One. Mathematics of Computation 32; pp. 887-899. doi: 10.1090/S0025-5718-1978-0492046-9.

Clavel R. (1990): Device for the Movement and Positioning of an Element in Space. US Patent No. 4;976;582; December 11; 1990. Download:

Elib (2010): . Accessed November 2010.

Dennis J.E.; Schnabel R.B. (1996): Numerical methods for unconstrained optimization and nonlinear equations. SIAM Classics in Applied Mathematics. doi: 10.1137/1.9781611971200.

Deuflhard P.; Fiedler B.; Kunkel P. (1987): Efficient numerical path following beyond critical points. SIAM Journal on Numerical Analysis; Society for Industrial and Applied Mathematics; 24; 912-927.

Deuflhard P. (2004): Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms. Springer.

Dymola (2010): Dymola 7.4.

Heroux M.A.; Bartlett R.A.; Howle V.E.; Hoekstra R.J.; Hu J.J.; Kolda T.G.; Lehoucq R.B.; Long K.R.; Pawlowski R.P.; Phipps E.T.; Salinger A.G.; Thornquist H.K.; Tuminaro R.S.; Willenbring J.M.; Williams A.; Stanley K.S. (2005): An overview of the Trilinos project. ACM Transactions on Mathematical Software; 31; 397-423. doi: 10.1145/1089014.1089021.

Hompack (2010): . Accessed November 2010.

Horowitz P.; Hill W. (1989): The Art of Electronics. Cambridge University Press; page 189.

Keller H. (1978): Global homotopies and Newton methods. C. de Boor and G. Golub; eds.; Academic Press; New York; pp. 73-94.

Kelley C.T. (2003): Solving nonlinear equations with Newton’s method. SIAM. doi: 10.1137/1.9780898718898.

Mattsson S.E.; Elmqvist H.; Otter M.; Olsson H. (2002): Initialization of Hybrid Differential-Algebraic Equations in Modelica 2.0. Proceedings of the Second International Modelica Conference; Munich; Germany; pp. 9-15. Download:

Modelica (2010): Modelica – A Unified Object-Oriented Language for Physical Systems Modeling. Language Specification; Version 3.2. March 24. Download:

Tietze U.; Schenk C. (2002): Halbleiterschaltungstechnik. Springer; 12th edition; page 1150.

Watson L.T.; Billups S.C.; Morgan A. P. (1987): Algorithm 652: HOMPACK; A suite of codes for globally convergent homotopy algorithms. ACM Transactions on Mathematical Software; 13; 281-310. doi: 10.1145/29380.214343.

Wayburn T.; Seader J. (1987): Homotopy continuation methods for computer-aided process design. Computers & Chemical Engineering 11; pp. 7-25. doi: 10.1016/0098-1354(87)80002-9.

Citations in Crossref