Elastic Collisions in Complementarity-based Time

Elastic Collisions in Complementarity-based
Time-stepping Methods
T. Preclik? , U. Rüde?
? Friedrich-Alexander University of Erlangen-Nürnberg
Chair of Computer Science 10 (System Simulation), Cauerstr. 6, 91058 Erlangen, Germany
e-mails: [email protected],
[email protected]
web page: http://www10.informatik.uni-erlangen.de/
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Introduction
Rigid multibody systems with non-smooth contact dynamics.
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Introduction
Rigid multibody systems with non-smooth contact dynamics.
2.5
position
velocity
2
1.5
1
0.5
0
-0.5
-1
-1.5
-1
-0.5
0
0.5
1
time
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Introduction
Rigid multibody systems with non-smooth contact dynamics.
Switching points separate smooth system evolutions (e.g. stick-slip
transitions and impacts).
2.5
position
velocity
2
1.5
1
0.5
0
-0.5
-1
-1.5
-1
-0.5
0
0.5
1
time
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Introduction
Rigid multibody systems with non-smooth contact dynamics.
Switching points separate smooth system evolutions (e.g. stick-slip
transitions and impacts).
Time-stepping methods do not detect switching points in contrast to
event-driven schemes.
T. Preclik? , U. Rüde?
(? LSS)
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Introduction
Rigid multibody systems with non-smooth contact dynamics.
Switching points separate smooth system evolutions (e.g. stick-slip
transitions and impacts).
Time-stepping methods do not detect switching points in contrast to
event-driven schemes.
Talk is in the context of complementarity-based time-stepping
schemes by Anitescu and Potra [1997].
[1997] M. Anitescu and F.A. Potra. Formulating Dynamic Multi-Rigid-Body
Contact Problems with Friction as Solvable Linear Complementarity Problems.
ASME Nonlinear Dynamics, 14(3):231–247, 1997.
T. Preclik? , U. Rüde?
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Elastic Collisions in Time-Stepping Methods
Restitution hypotheses are only meaningful for non-simultaneous
collisions.
Application of restitution hypotheses can cause constraint violation
for simultaneous collisions.
Figure: Newton’s cradle integrated with Moreau’s midpoint rule at t.
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Elastic Collisions in Time-Stepping Methods
Restitution hypotheses are only meaningful for non-simultaneous
collisions.
Application of restitution hypotheses can cause constraint violation
for simultaneous collisions.
Figure: Newton’s cradle integrated with Moreau’s midpoint rule at (t + 12 δt)− .
T. Preclik? , U. Rüde?
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Elastic Collisions in Time-Stepping Methods
Restitution hypotheses are only meaningful for non-simultaneous
collisions.
Application of restitution hypotheses can cause constraint violation
for simultaneous collisions.
Figure: Newton’s cradle integrated with Moreau’s midpoint rule at (t + 21 δt)+ .
T. Preclik? , U. Rüde?
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Elastic Collisions in Time-Stepping Methods
Restitution hypotheses are only meaningful for non-simultaneous
collisions.
Application of restitution hypotheses can cause constraint violation
for simultaneous collisions.
Figure: Newton’s cradle integrated with Moreau’s midpoint rule at t + δt.
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Solution Set of a Simultaneous Collision
v0− = v , v1− = v2− = 0
E − = 21 mv 2
perfectly elastic collisions
λ1
λ2
v0+
v1+
v2+
E+
2
3 mv
1
3 mv
1
3v
1
3v
1
3v
Newton’s collision hypothesis
Collision resolution
mv
0
0
v
0
1 −
3E
E−
4
3 mv
2
3 mv
− 31 v
2
3v
2
3v
E−
Poisson’s collision hypothesis
mv
mv
0
0
v
E−
“like Newton’s cradle”
inelastic collision
Table: Collision resolution options for three spheres in a row without gaps.
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Solution Set of a Simultaneous Collision (cont.)
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Non-dispersive Solutions
An ideal Newton’s Cradle is free of dispersion as noted e.g. in
Herrmann and Seitz [1982] and thus collisions can be treated as
occurring successively.
Recipe for computing a non-dispersive solution:
Let fi,j : R6nb → R6nb calculate the two-body collision between bodies i, j
and f : R6nb → R6nb be the composition of all two-body collision operators
then the fixed point iteration
xn+1 = f (xn ),
where x0 = ġ(q(t − ), ϕ(t − )) (possibly) converges to the non-dispersive
solution and we can assume ġ(q(t + ), ϕ(t + )) = limn→∞ xn .
Does not necessarily converge for perfectly elastic collisions (billiard
frame, sphere trapped between walls).
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Consequences for Time-Stepping Methods
Option 1: Resolve time of impact for elastic collisions.
Integrate non-smooth system until elastic impact occurs, resolve
elastic impact and continue integration.
Would be step backwards towards event-driven schemes.
Fails on accumulative switching points.
Option 2: Integrate impacts inelastically and correct velocities after time
step according to restitution hypothesis.
No constraint violation.
Elastic response shifted in time by up to δt.
Limits collision frequency to
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δt .
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Summary
Restitution hypothesis for Newton’s cradle.
Flexible restitution hypothesis insertion.
Suggestion for treatment of elastic collisions in time-stepping
methods.
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Questions? Comments?
Suggestions? . . .
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References
M. Anitescu and F.A. Potra. Formulating Dynamic Multi-Rigid-Body
Contact Problems with Friction as Solvable Linear Complementarity
Problems. ASME Nonlinear Dynamics, 14(3):231–247, 1997.
F. Herrmann and M. Seitz. How does the ball-chain work? American
Journal of Physics, 50(11):977–981, 1982.
T. Preclik. Elastic Collisions in Complementarity-based Time-stepping
Methods. Technical report, Friedrich-Alexander University
Erlangen-Nuremberg, December 2010.
C. Studer. Augmented time-stepping integration of non-smooth dynamical
systems. PhD thesis, ETH Zürich, 2008.
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