Efficient Simulation of Physical System Models Using Inlined Implicit Runge-Kutta Algorithms

1. Efficient Simulation of Physical System Models Using Inlined Implicit Runge-Kutta Algorithms Vicha Treeaporn Department of Electrical & Computer Engineering The University of Arizona Tucson, Arizona 85721 U.S.A

2. Topics • Introduction • Techniques for Simulation • Results • An Application

3. Introduction • Stiffness • Widely varying eigenvalues • Explicit algorithms • Straightforward to implement • Step size limited by numerical stability • Implicit algorithms • More difficult to implement • Additional computational load • Needed to simulate stiff systems • May use larger step sizes

4. Inline-Integration • Merges the integration algorithm with the model • Eliminates differential equations • Results in difference equations (∆Es) • Easily implement implicit algorithms • Circuit example inlining Rad3

5. Simple Circuit

6. Circuit Equations

7. Inlined with Rad3 Evaluate at Rad3 time instants Eliminate derivatives Integrator equations

8. Sorting

9. Sorting

10. Sorting

11. Sorting • 10 equations immediately causalized • Need to perform tearing • Make assumptions about variables being ‘known’

12. Tearing Tearing variable Residual Eq.

13. Tearing Residual Eq. #2 Tearing variable #2

14. Tearing • Completely causalized equations • 2 iteration variables, vc and i1 • Could use this set of equations for simulation • Want step-size control

15. Step-Size Control • Want larger step sizes • Reduce the overall computational cost • Maintain desired accuracy • Compute error estimate • Embedding method • Shares computations with original method

16. Step-Size Control • Explicit RKs • Embedding methods have been found • Implicit RKs • Difficult problem • Algorithms are compact • Can find embedding methods using two steps • Linear polynomial approximation

17. HW-SDIRK Embedding • 3rd-order accurate • Behaves like an explicit method • May unnecessarily restrict step size for stiff systems • Search for an alternate embedding method

18. Alt. HW-SDIRK Embedding • 3rd-order accurate • Implicit method

19. Alt. HW-SDIRK Embedding Stability Domain Damping Plots

20. Lobatto IIIC(6) • No embedding method exists • Expensive to perform step size control • Can search for an embedding method

21. Lobatto IIIC(6) Embedding Method • 5th-order accurate • A-Stable • Large asymptotic region

22. Lobatto IIIC(6) Embedding Method Stability Domain Damping Plots

23. Numerical Experiments

24. Numerical Experiments • Tested various algorithms with selected benchmark ODEs • Implemented in Dymola/Modelica

25. ODE Set B Inlined with HWSDIRK and alternate error method ode15s

26. ODE Set B Error estimate stays near 10-3 Step size grows and shrinks appropriately

27. ODE Set D Inlined with Lobatto IIIC(6) ode15s

28. ODE Set D

29. An Application

30. An Application • Real-Time, Limited Resources • Embedded control systems • Model Predictive • Add additional system dynamics • Simulate missile dynamics in flight for trajectory shaping • First solution is faster computer • Model may still be too complex • Try inlining

31. Questions?