1. VEHICLE DYNAMICS SIMULATIONS USING NUMERICAL METHODS VIYAT JHAVERI

2. Motivation Model Limitations Numerical Methods Implicit Runge-Kutta Numerical Integration Methods Topologically-Based Sparse Matrix Linear Equation Solvers Dual-Rate Numerical Integration Methods INTRODUCTION

3. Utilized by engineers to design vehicles to perform vehicle simulations Understand behavior of vehicle Inform design changes to achieve goal Self-Driving Cars MOTIVATION

4. 14 Body System 4 Independent Suspension incl. 1 Wheel & 2 Control Arms 1 Car Chassis 1 Steering Rack Complex System 10 Degrees of Freedom 98 Coordinates 88 Non Linear Equations MODEL

5. The displacement of all the nodes are put into a matrix: This can be differentiated twice to generate the acceleration term, and combined with F=ma to generate: MODEL

6. The previous equation is partitioned into dependent and independent u & v variables The general coordinate system is partitioned and the system of DAE is partitioned into a state-space ODE The system in generated as state space ODE in terms of v MODEL

7. 1.Real-time simulation models incompatible with CAE models Models had to be simplified with complex formulation 2.DAE (Differential Algebraic Equations) solutions only with Explicit Predictor-Corrector Unsuitable for stiff Vehicle Dynamics models Required numerical solution to differentials for implicit 3.High frequency response characteristics were not real-time Components such as tires were grossly simplified LIMITATIONS

8. Implicit Runge-Kutta Numerical Integration Methods Topologically-Based Sparse Matrix Linear Equation Solvers Dual-Rate Numerical Integration Methods APPROACH

9. IMPLICIT RUNGE-KUTTA BASIC METHOD

10. The y term is treated as an approximate solution and is recombined into the RK Method. This leads to the following: The constraints aij = 0 for j>I and sum of bi = 1. Many different methods can be generated using this method IMPLICIT RUNGE-KUTTA MODIFYING FOR 2 ND ORDER DAE

11. This method can be applied to the underlying state space equations These equations can be substituted in the system model to create discretized equations of motion involving independent and dependent accelerations and LaGrange multiplier. These can be solved using the RK algorithm that was developed IMPLICIT RUNGE-KUTTA MODIFYING FOR 2 ND ORDER DAE

12. 2 Methods were developed 4 th Order, 5 Stage, A-Stable, SDIRK formula Trapezoidal Formula Used to generate the time evolved model of the car in bump Comp Time Trap MethodComp Time SDIRK Method IMPLICIT RUNGE-KUTTA EXAMPLE

13. Compared to Explicit Solver Explicit Solver required 100 times longer to complete the simulation to the same error tolerances and simulation time Small time steps have to be takes otherwise model becomes unstable IMPLICIT RUNGE-KUTTA EXAMPLE

14. TOPOLOGICALLY-BASED SPARSE MATRIX LINEAR EQUATION SOLVER

15. Steps to Use: 1.Generate the reduced matrix 2.Solve the linear system of equations 3.Recover the accelerations based on the following equations: TOPOLOGICALLY-BASED SPARSE MATRIX LINEAR EQUATION SOLVER

16. Benefit gained from generating the reduced matrix This reduction takes advantage of the sparsity induced by the topological graph of the mechanical system Traditional Method: LU Decomposition (Algorithm 1 & 2) New Method (Algorithm 3 & 4) TOPOLOGICALLY-BASED SPARSE MATRIX LINEAR EQUATION SOLVER

17. Useful when different frequencies present in the system Different time steps used to speed up computations 1-2Hz for Chassis, 10 to 15 Hz for Unsprung Mass, Much higher for wheel spin and tire force cals Following Equations describe the system The last equation is a set of ODE used to model subsystem interaction DUAL RATE NUMERICAL INTEGRATION SOLVER

18. Split up the ODE into fast and slow variables Solve slow variables first, extrapolate fast variables if implicit method is used Solve fast variables using interpolated values of slow variables DUAL RATE NUMERICAL INTEGRATION SOLVER

19. Step size for slow variables can be 6 times larger than fast variables This method is convergent under certain conditions & can be derived from absolute stability of the linear multistep integration formula Using the cost of method, we can determined ratio of step size Useful if function evaluation cost is larger than interpolation cost mr = ratio, ns = # of slow vars Cinteg = cost of interpolation Cs = cost of diff slow time DUAL RATE NUMERICAL INTEGRATION SOLVER

20. 1.Using Implicit RK Method to solve stiff equations speeds up solution by 2 orders of magnitude. 2.Topologically-Based Sparse Matrix Linear Equation Solvers reduce the size of the system 3.Dual-Rate Simulation can be used to utilize more resources on high frequency variables CONCLUSION

21. 1. Hairer, E., Nørsett S. P., Wanner, G., 1993, Solving Ordinary Differential Equations I. Nonstiff Problems, Springer-Verlag, Berlin. 2. Haug, E. J., Iancu, M., Negrut, D., 1997, “Implicit Integration of the Equations of Multibody Dynamics in Descriptor Form,” Advances in Design Automation, 1997 ASME Design Automation Conference. 3. Haug, E.J., Negrut, D., Serban, R. and Solis, D., 1999. Numerical methods for high-speed vehicle dynamic simulation. Journal of Structural Mechanics,27(4), pp.507-533. 4. Wells, D. R., 1982, “Multirate Linear Multistep Methods for the Solution of Systems of Ordinary Differential Equations,” Report UIUCDCS-R-82-1093 Department of Computer Science, University of Illinois at Urbana-Champaign REFERENCES