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: tobias.preclik@informatik.uni-erlangen.de,
ulrich.ruede@informatik.uni-erlangen.de
web page: http://www10.informatik.uni-erlangen.de/
T. Preclik? , U. Rüde?
(? LSS)
1 / 10
Introduction
Rigid multibody systems with non-smooth contact dynamics.
T. Preclik? , U. Rüde?
(? LSS)
2 / 10
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
T. Preclik? , U. Rüde?
(? LSS)
2 / 10
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
T. Preclik? , U. Rüde?
(? LSS)
2 / 10
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)
2 / 10
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?
(? LSS)
2 / 10
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.
T. Preclik? , U. Rüde?
(? LSS)
3 / 10
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?
(? LSS)
3 / 10
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?
(? LSS)
3 / 10
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.
T. Preclik? , U. Rüde?
(? LSS)
3 / 10
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.
T. Preclik? , U. Rüde?
(? LSS)
4 / 10
Solution Set of a Simultaneous Collision (cont.)
T. Preclik? , U. Rüde?
(? LSS)
5 / 10
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).
T. Preclik? , U. Rüde?
(? LSS)
6 / 10
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
T. Preclik? , U. Rüde?
(? LSS)
1
δt .
7 / 10
Summary
Restitution hypothesis for Newton’s cradle.
Flexible restitution hypothesis insertion.
Suggestion for treatment of elastic collisions in time-stepping
methods.
T. Preclik? , U. Rüde?
(? LSS)
8 / 10
Questions? Comments?
Suggestions? . . .
T. Preclik? , U. Rüde?
(? LSS)
9 / 10
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.
T. Preclik? , U. Rüde?
(? LSS)
10 / 10

Download Report