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SECTION 7
INTEGRATOR CLASSIFICATION &
CHARACTERISTICS
ADM703b, Section 7, February 2013
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ADM703b, Section 7, February 2013
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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)
ADM703b, Section 7, February 2013
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IMPLICIT VS EXPLICIT INTEGRATORS
• Forward (Explicit) Euler
yn1  yn  hyn
• Easier to implement (explicit)
• Very small
CPU Time
ratio
Integration-Step
• Less stable – is limited to take small step-sizes
• Backward (Implicit) Euler

yn1  yn  hyn+1
• More difficult to implement (implicit)
• Large
CPU Time
Integration-Step
ratio
• More stable – can take very large step sizes
ADM703b, Section 7, February 2013
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where h = step size
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.
ADM703b, Section 7, February 2013
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PHYSICAL STIFFNESS
• 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)
F(t)
Mass 1
k1=10
c1=0.1
ADM703b, Section 7, February 2013
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Mass 2
k2=10000
c2=0.1
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NUMERICAL STIFFNESS
• 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
highest overdamped eigenvalue
 200
highest underdampe d eigenvalue
– The ratio for non-stiff systems is less than 20
F(t)
Mass 1
k1=10
c1=0.1
ADM703b, Section 7, February 2013
Copyright© 2013 MSC.Software Corporation
Mass 2
k2=10000
c2=1000
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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)
ADM703b, Section 7, February 2013
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INTEGRATOR ACCURACY
• Does numerical solution converge as h  0
?
• Case Study: y=l y, y(0) = y0
y0 elt
– True solution:
– Numerical solution (Forward Euler):
yn+1=yn +hyn =(1+hλ)yn =(1+hλ)2 yn-1=...=(1+hλ)n+1 y0
• As h  0
n+1
yn+1=(1+hλ)
1/hl
y0 = (1+hλ)

(n+1)hl
y0  y0 el ( n1) h  y0 elt
• Control over accuracy is by means of step-size control ( h )
ADM703b, Section 7, February 2013
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INTEGRATOR STABILITIY
• Does the error in the solution stay bounded?
• What is the effect of step size on error growth?
 y =l y
• Case Study: 
 y(0) = y0

– Forward Euler: yn 1  yn  hyn
yn+1=(1+hλ)n+1 y0
– Backward Euler:

yn+1=yn+hyn+1
 =yn +hλyn+1  (1-hl )1 yn  (1-hl )2 yn-1  (1-hl )( n1) y0
yn+1=yn +hyn+1
ADM703b, Section 7, February 2013
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INTEGRATOR STABILITY (CONT.)
• Constraints on the value of step-size: h
F.Euler
– Forward Euler is stable provided 1+hλ  1
– The Backward Euler is stable provided 1-hλ  1
• What happens when l is large and negative?
• The step-size h should be such that hl is
in the stability region
• For numerically stiff systems, staying in the
region of stability is:
– Easy for Backward Euler
– Hard for Forward Euler
ADM703b, Section 7, February 2013
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B.Euler
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)
ADM703b, Section 7, February 2013
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INTEGRATOR INDEX: INDEX-3 ISSUES
•
When solving the original system index-3 DAE:
Mu  Φ Tq λ  B T F  0
u  q  0
Φ0
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
ADM703b, Section 7, February 2013
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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
ADM703b, Section 7, February 2013
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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:
Mu  ΦTq λ  BT F  0
u  q  ΦTq μ  0
 (q, μ, t )  0
Φ
Φ(q, t )  0
ADM703b, Section 7, February 2013
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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
ADM703b, Section 7, February 2013
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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
ADM703b, Section 7, February 2013
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INTEGRATOR INDEX: THE SI2 FORMULATION
•
Full equations:
mu  ΦTp λ p  Q p  0
translational motion equation
T
 K 
Γ
 0

ζ


angular momentum definition
 K 
T
Γ  
  Φε λ ε  Qε  0
 ε 
rotational motion equation
p  u  ΦTp μ p  0
translational velocity definition
ε  ζ  Φ Tε μ ε  0
rotational velocity definition
Φ0
constraint equation
 0
Φ
constraint derivative
T
ADM703b, Section 7, February 2013
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INTEGRATOR INDEX: SI2 VERSUS INDEX-3 FOR PENDULUM
PROBLEM
Index 3:
Stabilized Index 2:
 ux 
u 
 y 
 u 
  


 x 


x y 
  


l
 1 
 l2 


h
m
 1
hm 
2
mu x  l1  0
mu y  l2  mg  0
  l1 L sin   l2 L cos   0
  Ju  0
u x  x  m1  0
u y  y  m2  0
u    m1 L sin   m2 L cos   0
x  L cos   0
mu x  l1  0
mu y  l2  mg  0
  l1 L sin   l2 L cos   0
  Ju  0
u x  x  0
u y  y  0
u    0
x  L cos   0
y  L sin   0
y  L sin   0
u x  u L sin   0
u y  u L cos   0
ADM703b, Section 7, February 2013
Copyright© 2013 MSC.Software Corporation
u x 
u 
 y
u 

 
xx
 
y
 
 
 l1 
l2 
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INTEGRATOR INDEX: SI2 JACOBIAN FOR PENDULUM
PROBLEM
M
0

0
0



Gx   0
0

0
0

1
0

0
0
0
0
0
0
1
0
0
M
0
0
0
0
0
0
1
0
0
0
1
0
0
C1
L sin 
 L cos
0
0
J
1
0
0
0
0
0
0
0
0
0 1
0
0
0
0
1

0
0
1
0
0
0
0
0

0
0
1
0
0
L sin 
0
0
0
1
0
L sin 
0
0
0
0
0
0
0
1
 L cos
0
0
0
0
L sin 
0
0
0
u L cos
0
0
0
1
 L cos
0
0
0
u L sin 
0
0
0
T  l1 L cos  l2 L sin 
0
M  mh 0


0

0


0


0

1

 L cos 

0


0

0


0

0
  h 0
Note that there is no 1/h terms in the Jacobian matrix for an SI2 formulation. This adds robustness.
ADM703b, Section 7, February 2013
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Useful References – Books on Numerical Methods

Uri M. Ascher, Linda R. Petzold, Computer Methods for Ordinary Differential
Equations and Differential-Algebraic Equations, ISBN 0-89871-412-5 – 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
ADM703b, Section 7, February 2013
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ADM703b, Section 7, February 2013
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