1. INTEGRATOR CLASSIFICATION & CHARACTERISTICS
SECTION 7
INTEGRATOR CLASSIFICATION & CHARACTERISTICS

3. COURSE OBJECTIVES What’s in this section:
Implicit vs Explicit Integrators
Integrator Classification Terminology
Physical vs Numerical Stiffness
Integrator Accuracy & Stability
Integrator Index (I3 vs SI2)

4. IMPLICIT VS EXPLICIT INTEGRATORS
Forward (Explicit) Euler
Easier to implement (explicit)
Very small ratio
Less stable – is limited to take small step-sizes
Backward (Implicit) Euler
More difficult to implement (implicit)
Large ratio
More stable – can take very large step sizes
where h = step size

5. CLASSIFICATION OF INTEGRATORS
Most numerical integrators are either Taylor series based or polynomial based *. Other classifications included
Major:
Explicit vs. implicit
Stiff vs. non-stiff
Minor:
Single step vs. multi-step
Fixed step size vs. variable step size
Variable order vs. fixed order
Each has advantages/disadvantages for different classes of problems!
* With the notable exception of the RKF integrator, most presented herein are polynomial based. The simple forward Euler integrator results from either approach.

6. PHYSICAL STIFFNESS F(t) Physical Stiffness
High frequencies (Im(l) is large when mass small compared to stiffness)
High frequency oscillations; takes time to die out because little damping (c2)
Mass 2
F(t)
k1=10
c1=0.1
Mass 1
k2=10000
c2=0.1

7. NUMERICAL STIFFNESS F(t) Numerical Stiffness
Damped high frequencies (Re(l) is large and negative)
High frequency vibration dies out very quickly
Only lower frequency modes are active in system
System is classified as stiff when
The ratio for non-stiff systems is less than 20
Mass 2
F(t)
k1=10
c1=0.1
Mass 1
k2=10000
c2=1000

8. NUMERICAL VS PHYSICAL STIFFNESS
All methods have problems with physically stiff systems
Stability issues (for explicit integrators)
Convergence issue (for implicit integrators)
Numerically stiff systems handled well by stiff integrators (e.g., Backward Euler, BDF integrators)

9. INTEGRATOR ACCURACY Does numerical solution converge as ? Case Study:
True solution:
Numerical solution (Forward Euler): As
Control over accuracy is by means of step-size control ( )

10. INTEGRATOR STABILITIY
Does the error in the solution stay bounded?
What is the effect of step size on error growth?
Case Study:
Forward Euler:
Backward Euler:

11. INTEGRATOR STABILITY (CONT.)
Constraints on the value of step-size:
Forward Euler is stable provided
The Backward Euler is stable provided
What happens when l is large and negative?
F.Euler
The step-size should be such that is in the stability region
For numerically stiff systems, staying in the region of stability is:
Easy for Backward Euler
Hard for Forward Euler
B.Euler

12. INTEGRATOR INDEX
The index of a set of differential algebraic equations (DAE) is the number of times the equations must be differentiated to get a system of ordinary differential equations
The DAE derived for mechanical systems have index 3, which is considered high
This is because you would have to differentiate the constraint equations twice to get a set of acceleration constraint equations. You would also have to define a derivative of the Lagrange multipliers.
The general rule is that the higher the index, the more challenging the numerical solution becomes
The default solver setting is Formulation=I3 (i.e., index-3)

13. INTEGRATOR INDEX: INDEX-3 ISSUES
When solving the original system index-3 DAE:
the Jacobian matrix becomes ill-conditioned as the step size decreases (because h appears in the denominator of several terms)
Another problem is that the integration error cannot be monitored on either velocities, u, or reaction forces, l
Consequently, the corrector may fail when the Jacobian matrix becomes ill-conditioned or the predictions are not good

14. INTEGRATOR INDEX: MOTIVATION FOR THE SI2 FORMULATION
For some integrators there is an option that reduces the original index-3 problem to an analytically equivalent index-2 problem. While slower, it is typically more robust and accurate. This:
Improves robustness of solutions
Decreases corrector failures due to ill-conditioning of Jacobian matrix with small step sizes
Reduces spikes in accelerations
Improves accuracy of solution
More accurate velocities
More accurate accelerations

15. INTEGRATOR INDEX: MOTIVATION FOR THE SI2 FORMULATION (CONT.)
However if you were to simply replace F = 0 with F = 0, you would satisfy the latter but not the former. Thus eventually one or more joint constraints would drift.
To alleviate this problem, another set of Lagrange multipliers m associated with the velocities can be created to satisfy both the F = 0 and F = 0 equations. Thus the system becomes: • •

16. INTEGRATOR INDEX: MOTIVATION FOR THE SI2 FORMULATION (CONT.)
The numerical solution to these DAE are guaranteed to satisfy both F = 0 and F = 0, so the above equations are called a stabilized index-2 (SI2) representation
The Jacobian matrix does not become ill-conditioned with small h because all differential equations are multiplied by h (no terms of the form 1/h in the Jacobian)
The integrators can monitor error on both position and velocity (unlike I3, velocities are guaranteed to satisfy ERROR with SI2 formulation)
Can generally loosen ERROR by /10 or /100 when switching from I3 to SI2 •

17. INTEGRATOR INDEX: MOTIVATION FOR THE SI2 FORMULATION (CONT.)
The constraint derivative equations require velocities to be tangential to the constraint surface which removes noise seen in Index-3
A slightly different corrector is used that solves for hl and hm during corrector calculations

18. INTEGRATOR INDEX: THE SI2 FORMULATION
Full equations:
translational motion equation
angular momentum definition
rotational motion equation
translational velocity definition
rotational velocity definition
constraint equation
constraint derivative

19. INTEGRATOR INDEX: SI2 VERSUS INDEX-3 FOR PENDULUM PROBLEM
Stabilized Index 2:

20. INTEGRATOR INDEX: SI2 JACOBIAN FOR PENDULUM PROBLEM
Note that there is no 1/h terms in the Jacobian matrix for an SI2 formulation. This adds robustness.

21. Useful References – Books on Numerical Methods
Uri M. Ascher, Linda R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, ISBN – good introductory level book
K.E. Brenan, S.L. Campbell, and L.R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, Classics in Applied Mathematics Series, SIAM Vol.14 - good introductory level book on the subject of DAE
Ernst Hairer, Syvert Paul Nørsett, and Gerhard Wanner, Numerical Methods for the solution of ODE and DAE, Springer Series in Comput. Mathematics (Second revised edition 1993) – tough book, excellent reference though
Ernst Hairer, Gerhard Wanner, Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer Series in Comput. Mathematics, Vol. 14, Springer-Verlag (Second revised edition 1996) – tough book, excellent reference though
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