Publicerad: 2019-02-21
ISBN: 978-91-7685-266-8
ISSN: 1650-3686 (tryckt), 1650-3740 (online)
In the current FMI standard the dynamical behavior of a model can only be defined as a system of Ordinary Differential Equations (ODE). The dynamics of many physical systems, such as the equations of motion of constrained mechanical multibody systems, are expressed by high-index Differential Algebraic Equations (DAE) so they cannot be simulated directly using standard ODE or DAE solvers. These systems can be converted through index-reduction into ODE or index 1 DAE systems. However FMUs based solely on these latter systems suffer from drift in hidden constraints on the states. As a consequence, the simulation may results in physically meaningless solutions. In this paper, we propose an extension of the FMI standard to handle DAE Systems of index 1 or higher and ODE with constraints. This FMI extension requires only few additions to the FMI specification, all of which can be omitted for FMUs that represent ODE systems or FMUs that do not support DAE handling. The extension has been implemented in solid-Thinking ActivateTM and two examples that illustrate the ease of implementation and the effectiveness of the method will be discussed.
Simcenter System Synthesis, Simcenter Amesim, FMI, Architecture-driven simulation, heterogeneous simulation
U. Ascher and L. Petzold. Computer methods for ordinary differential equations and differential-algebraic equations. SIAM, 1988.
U. M. Ascher and L. R. Petzold. Projected implicit rungekutta methods for differential-algebraic equations. SIAM, Numerical Analysis, 28(4), 1097-1120., 1992.
U. M. Ascher, H. Chin, and S. Reich. Stabilization of daes and invariant manifolds. Numer. Math. 67: 131, 1994.
J. Baumgarte. Stabilization of constraints and integrals of motion in dynamical systems. Computer Methods in Applied Mechanics and Engineering Volume 1, Issue 1, Pages 1-16, 1972.
E. Eich. Convergence results for a coordinate projection method applied to mechanical systems with algebraic constraints. SIAM J. on Numerical Analysis 30(5):1467-1482, 1993.
E. Eich-Soellner and C. Fuhrer. Numerical Methods in Multibody Dynamics. European Consortium for Mathematics in Industry, B.G. Teubner, 1998.
C.W. Gear. Maintaining solution invariants in the numerical solution of odes. Journal on Scientific and Statistical Computing, Vol. 7, No. 3, 1986.
E. Hairer and G. Wanner. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer, 1996.
S.E. Mattsson and G. Soderlind. Index reduction in differential-algebraic equations using dummy derivatives. SIAM Journal of Scientific Computing. 14(3), pp. 677-692, 1993.
R. Nikoukhah. A simulation environment for efficiently mixing signal blocks and modelica components. 12’th International Modelica conference, 2017.
M. Otter and H. Elmqvist. Transformation of differential algebraic array equations to index one form. Proceedings of the 12th International Modelica Conference, Prag, Czech Republic, 2017.
C. C. Pantelides. The consistent initialization of differentialalgebraic systems. SIAM Journal on Scientific and Statistical Computing, 9(2):213–231, 1988. doi:10.1137/0909014.
H. Olsson S.E. Mattsson and H. Elmqvist. Dynamic selection of states in dymola. Modelica Workshop 2000, Lund, Sweden, pp. 61-67, 2000.
L. Shampine. Conservation laws and the numerical solution of odes,. Comput. Math. Appls, Part B., 12, pp. 1287-1296, 1986.
