Elena Shmoylova
Maplesoft, Canada
Jürgen Gerhard
Maplesoft, Canada
Erik Postma
Maplesoft, Canada
Austin Roche
Maplesoft, Canada
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Ingår i: Proceedings of the 5th International Workshop on Equation-Based Object-Oriented Modeling Languages and Tools; April 19; University of Nottingham; Nottingham; UK
Linköping Electronic Conference Proceedings 84:10, s. 87-96
Publicerad: 2013-03-27
ISBN: 978-91-7519-621-3 (print)
ISSN: 1650-3686 (tryckt), 1650-3740 (online)
Reduction of a differential algebraic equation (DAE) system to an ordinary differential equation system (ODE) is an important step in solving the DAE numerically. When the ODE is obtained; an ODE solution technique can be used to obtain the final solution. In this paper we consider combining index reduction with projection onto the constraint manifold.We show that the reduction benefits from the projection for DAEs of certain form. We demonstrate that one of the applications where DAEs of this form appear is optimization under constraints. We emphasize the importance of optimization problems in physical systems and provide an example application of the projection method to an electric circuit formulated as an optimization problem where Kirchhoff’s laws are acting as constraints.
[1] K. Arczewski and W. Blajer; A unified approach to the modeling of holonomic and nonholonomic mechanical systems; Math. Modeling of Systems 2 (1996); 157–174.
[2] U. M. Ascher; L. R. Petzold; Computer Methods for Ordinary Differential Equations and Differential Algebraic Equations; SIAM; 1998.
[3] J. Bakus; L. Bernardin; K. Kowalska; M. Léger; A.Wittkopf; High-Level Physical Modeling Description and Symbolic Computing; IFAC Proceedings of the 17th World Congress;n 2008; 1054–1055.
[4] J.-L. Basdevant; Variational principles in physics; Springer; New York; 2007.
[5] L. D. Berkovitz; Variational methods in problems of control and programming; J. Math. Analysis and Applications 3 (1961); 145–169.
[6] W. Blajer; A projection method approach to constrained dynamic analysis; J. Appl. Mech. 59 (1992); 643–649.
[7] W. Blajer; Projective formulation of Maggi’s method for nonholonomic system analysis; J. Guidance Control Dyn. 15 (1992); 522–525.
[8] W. Blajer; Elimination of constraint violation and accuracy aspects in numerical simulation of multibody systems; Multibody System Dynamics; 7 (2002); 265–284.
[9] P. Blanchard and E. Brüning; Variational Methods in Mathematical Physics: A Unified Approach; Springer- Verlag; Berlin; Heidelberg; New York 1992.
[10] K. E. Brenan; S. L. Campbell; L. R. Petzold; Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations; SIAM; 1996.
[11] A. N. Kolmagorov; S. V. Fomin; Elements of the Theory of Functions and Functional Analysis; Courier Dover Publications; 1999.
[12] S. G. Krantz; H. R. Parks; The Implicit Function Theorem: History; Theory; and Applications; Birkhäuser; 2002.
[13] R. K. Nesbet; Variational Principles and Methods in Theoretical Physics and Chemistry; Cambridge University Press; Cambridge; 2003.
[14] A. Ohata; H. Ito; S. Gopalswamy; K. Furuta; Plant Modeling Environment Based on Conservation Laws and Projection Method for Automotive Control Systems; SICE Journal of Control; Measurement and System Integration 1 (2008); 227–234.
[15] C. M. Roithmayr; Relating Constrained Motion to Force Through Newton’s Second Law; Ph.D. thesis; Georgia Institute of Technology; 2007.
[16] D. Scott; Can a projection method of obtaining equations of motion compete with Lagrange’s equations? Am. J. Phys 56 (1988); 451–456.
