This document discusses testing and comparing different numerical integration methods for simulating Cosserat rod and rigid contact mechanics (RCM) models. It finds that the θ-method is most stable for the Cosserat rod model, while RCM simulations are less stable. The stiff second-order differential equations mean implicit methods like RADAU5 and θ are required. Performance profiling shows the θ-method has the best performance for the Cosserat rod model, while initialization is most expensive for both models.

1. Testing for Cosserat Rods
Zhan Wang
30 November 2012
Contents
1 Characterization of stiffness[wiki stiffEquation] 2
2 Some Notes 2
3 Oscillating Ring 2
4 Pearl Chain CVT Setting 3
4.1 Results of θ method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
4.2 Results of SSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4.3 Comparison of RCM and Cosserat . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.3.1 Different Parameters Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.3.2 Integrator for the RCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.3.3 Elastic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.3.5 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
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2. 1 Characterization of stiffness[wiki stiffEquation]
A linear constant coefficient system is stiff if all of its eigenvalues have negative real
part and the stiffness ratio is large. Stiffness occurs when stability requirements, rather
than those of accuracy, constrain the step length. Stiffness occurs when some
components of the solution decay much more rapidly than others.[2]
As the stiffness ratio is determined by the property of the system, which includes the deformation
flexible rod and movement of rigid body. The stiffness cannot eliminate by improving the formula.
The only way is to use A-stable Integrator.
2 Some Notes
• LSODE: adaptive timestep size, to look for angular and position
• h can not larger than 1e-6, otherwise collapse for all solver? theta method?? see the new
simulation result.
• trapezoidal method (θ = 0.5) does not have perfect stability: it does damp all decaying
components, but rapidly decaying components are damped only very mildly.
• There are no explicit A-stable and linear multistep methods. The implicit ones have order of
convergence at most 2.
• The trapezoidal rule has the smallest error constant amongst the A-stable linear multistep
methods of order 2.
3 Oscillating Ring
The following simulation results are based on nine layer model.
• In the following simulation, ∆R = 0.1R, actually ∆R ≈ 1e-3 ∼ 1e-4 R in general.
• For the SSC: setMaxOrder(4,1), 4 means choose the order 4, which is the maximum, 1 means
method 1 which is embedded method (compare maxOrder maxOrder+1); proceed with
maxOrder (recommended).
• TimeSteppingIntegrator: if the number of elements is small, and time step size is extremely
small, it will not crash during certain simulation time.
Table 1: Crash Time
Timestep elements number=20 elements number=8
1e-6 0.00001s 0.00002s
1e-7 0.00001s -
1e-8 0.001s 0.01229s
1e-9 0.0089 -
• LSODE: does not improve so much,
– crash time(elements number=20): 0.001s
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3. – crash time(elements number=10): 0.002s
– crash time(elements number=8): 0.19818s (the step size is during 1e-8 ∼ 1e-9)
– use the option set.stiff does not helps so much, crash at 0.009s(N=20) 0.1469s(N=8)
• SSC: when N=20, it does not improve so much. It crash as soon as it reach the minimum
time step size, which is 5.96046e-08s. So during simulation, as soon as we see it reach teh
minimu time setp size, we should stop.
The simulation result in the table is based on ∆R = 0.01R .
Table 2: Stability Test
Model TimeStepping LSODE SSC RADAU5 Theta
one layer Y(1e-7) Y(1e-7) Y Y Y
nine layer N(1e-9) N N Y Y
If using one layer model to calculate the area inertial moment I, all these method converge. If
using nine layer model, RADAU5 and integrator work.
4 Pearl Chain CVT Setting
I1 also have to be divided by pow(Nr, 2) as it is also consist of two separate part.
Table 3: Stability Test
I TimeStepping LSODE SSC RADAU5 Theta
I0 = I1 + I2 N N lack of Memory - Y
I0 = 0.32 ∗ (2wr) ∗ hr3
N N lack of Memory - Y
For the area inertial moment I2, one layer model and nine layer model are corresponding to two
extreme cases. The system will have the biggest stiffness ratio if the nine layer model is chosen.
If the nine layer cannot be solved by the θ method, then we should improve this value by
• choose some value between the the one layer model and nine layer model, use bisection to
find the critical point
• experiment data
• use ANSYS to simulate to get the value if the friction coefficient is known.
• analysis how much does the inertial moment effects the simulation result which we need.
In the testing, for nine layer model, θ method is even faster than the SSC method, as the SSC
method is reaching the minimum time step size,which is 5.96046e-08s or 2.98023e-08(with new
setting).
4.1 Results of θ method
It seems that the area inertial moment I2 calculated by the nine layer model is too small so that
the ring seems to be too elastic. The correct value should between the values calculated by the
nine layer model and the one layer model. (I2one/I2nine = 81) we try to use bisection to find the
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4. critical point, testing different value of I2 which is between I2one and I2nine:
I2 = K ∗ I2nine, K = 1, 21, 41, 61, 81.
After testing, we decide to choose the I2 = 41 ∗ I2nine temporarily.
Table 4: Stability Test of θ Method: I0 = I1 + I2, θ = 0.5
I 1e-5 5e-5 1e-4 5e-4 1e-3
I2 = 1 ∗ I2nine
I2 = 21 ∗ I2nine
I2 = 41 ∗ I2nine Y(0.19805) 0.0752 0.0767 0.075
I2 = 61 ∗ I2nine
I2 = 81 ∗ I2nine
Table 5: Stability Test of θ Method: I0 = I1 + I2, θ = 0.878
I 1e-5 5e-5 1e-4 5e-4 1e-3
I2 = 1 ∗ I2nine Y(may stop around 0.19s) -
I2 = 21 ∗ I2nine Y(may stop around 0.19s) 0.0753s
I2 = 41 ∗ I2nine Y(0.1997s) 0.0745s 0.0756s 0.082s
I2 = 61 ∗ I2nine Y(may stop around 0.19s) 0.0753s
I2 = 81 ∗ I2nine Y(may stop around 0.19s) 0.0737s 0.0745s 0.0835s
Table 6: Stability Test of θ Method: I0 = I1 + I2, θ = 1
I 1e-5 5e-5 1e-4 5e-4 1e-3())
I2 = 1 ∗ I2nine
I2 = 21 ∗ I2nine
I2 = 41 ∗ I2nine Y(0.19325s) 0.0737s 0.0658s
I2 = 61 ∗ I2nine 0.092s
I2 = 81 ∗ I2nine
Comparing the results of simulation by setting θ = 1,θ = 0.5, it shows a slightly difference after
about 0.095s. But the overall tends of them are the same. When θ = 0.5, the solution is more
oscillating locally than that of when θ = 1. But when θ = 0.5, the global error is controlled better
than that when θ = 1. So it diverges later. It indicates that a more smaller stepsize is needed for
more accurate solution and longer simulation.
4.2 Results of SSC
The SSC is very slow for solving Pearl Chain CVT Setting as it always reach the minimum stepsize
which is too small. And the memory consumption is quite high, causing the simulation stops at
about 0.05s. So the SSC is infeasible for this simulation.
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5. Table 7: Stability Test of SSC Method: I0 = I1 + I2, SSC, SSCNewSetting
I SSC(setMaxOrder(2,1)) SSC(setMaxOrder(1,0), setFlagErrorTest(2,false),
SetGapControl(true, 1))
I2 = 1 ∗ I2nine N N(0.01383s, lack of Memory)
I2 = 21 ∗ I2nine
I2 = 41 ∗ I2nine Y N(0.058s, lack of Memory)
I2 = 61 ∗ I2nine
I2 = 81 ∗ I2nine Y
Table 8: Stability Test of SSC Method: I0 = 0.32 ∗ (2wr) ∗ hr3
I SSC(setMaxOrder(2,1)) SSC(setMaxOrder(1,0), setFlagErrorTest(2,false),
SetGapControl(true, 1))
I2 = 1 ∗ I2nine
I2 = 21 ∗ I2nine
I2 = 41 ∗ I2nine
I2 = 61 ∗ I2nine
I2 = 81 ∗ I2nine
4.3 Comparison of RCM and Cosserat
4.3.1 Different Parameters Setting
4.3.2 Integrator for the RCM
TimeSteppingIntegrator: 1e-6(Converge) 5e-5(No convergence) 1e-5(No convergence)
4.3.3 Elastic
When I2 = 1 ∗ I2nine, the results of both methods show a high elastic property.
4.3.4 Results
The overall tend of results of these two methods are the same. The results of RCM model is
oscillating locally. A new simulation with time step size equals 1e − 9 is executed to check whether
the oscillating is generated as the timestep size is large(1e − 6). The simulation is still running.
4.3.5 Performance Analysis
1. Preparation: compile the code with debugging info (the -g option) and with optimization
(-O3) turned on.
2. To start a profile run for a program, execute: ”valgrind –tool=callgrind –cache-sim=yes
–branch-sim=yes ./main”, an output file named callgrind.out.pid will be generated.
3. Visualization the output file: use the tool caller Kcachegrind. Usage: ”kcachegrind
cachegrind.out.pid”
The meaning of some abbreviation in Kcachegrind:
Ir : Instruction Read
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6. Table 9: Different Parameters Setting of RCM and Cosserat
I RCM Cosserat
I1 t-n t-b
I2 t-b t-n
I0
setCurlRadius R1: t-n R2: t-b R1: t-b R2: t-n
Incl. : Inclusive Cost. Cost attributes for functions regarding some event type, including all called
functions.
Self : Exclusive Cost. Cost attributes for functions regarding some event type, only of the
function itself.
Table 10: Performance of Cosserat
initialize updateStateDependentVariables updateJacobians updateG
TimeStepping 17.27 - 19.74 34.2
SSC 5.92 5.06 15.67 54.37
theta - 29.52 45.66 -
Table 11: Performance of RCM
initialize updateJacobians updateG updateg updateh
TimeStepping 35.8 9.03 22.43 11.91 9.63
SSC 19.33 6.67 33.04 10.08 7.01
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