Binh Nguyen, Jeff Trinkle ([nguyeb2,trink]@cs.rpi) Rensselaer Polytechnic Institute
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Transcript of Binh Nguyen, Jeff Trinkle ([nguyeb2,trink]@cs.rpi) Rensselaer Polytechnic Institute
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A Comparison of prox and Complementarity Formulations
Euromech Colloquium ‘Nonsmooth Contact and Impact Laws in Mechanics’Grenoble, 08.07.2011
Thorsten Schindler ([email protected])INRIA Grenoble – Rhône-Alpes
Binh Nguyen, Jeff Trinkle ([nguyeb2,trink]@cs.rpi.edu)Rensselaer Polytechnic Institute
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Multibody Dynamics
• Equations of bodies with bilateral and unilateral contacts
• Examplary videos: grasping, double track, pushbelt cvt (wmv) Essential: effective parallel collission detection, constraint
representation and solution by highly parallel supercomputers Here: comparison of two constraint representations concerning
analytical and numerical issues
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Complementarity Formulation
• How can
look like?
1. Bilateral constraint: e.g. idealized knee joint
• We see the contact laws on position level; increasing the number of dots over g gives velocity and acceleration level.
2. Unilateral constraint: e.g. imperfect joints, Newton’s cradle
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Complementarity Formulation
• How can
look like?
3. Coulomb friction: e.g. idealized clutches
• The contact law is naturally on velocity level; increasing the number of dots over g gives the acceleration level.
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Complementarity Formulation
• Mathematical model:nonlinear differential complementarity problem (DCP)
• Numerical model:nonlinear complementarity problem (NCP)
• Numerical algorithm:pivoting scheme (what does PATH do in the linear case?)exponential worst complexity / polynomial average complexity
discretization
solution
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Formulation with prox Function
• How can
look like? We can figure it out!
bilateral unilateral Coulomb
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Formulation with prox Function
• Unilateral constraint
force and gap are always positive and elements of
common description: distinguish the branches of the corner law
If we assume
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Formulation with prox Function
• Unilateral constraint
separate proximality
Ifyieldsand so
and If
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Formulation with prox Function
• Bilateral constraint
• Coulomb friction
Increasing the number of dots over g changes kinematic levels. Numerical model:
nonsmooth, nonlinear equations Numerical algorithm: fixed-point iteration or Newton method
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Formulation with prox Function
• How shood we choose ? Figure out prox functions!
bilateral unilateral Coulomb
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Point Mass on Frictional Plane
• Equations of motion
• Unilateral constraint and Coulomb friction
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Point Mass on Frictional Plane: prox
• Fixed-point equation for normal contact force
• Assumption: contact with zero normal velocity and pushing external force
Slope of prox function varies with:
Convergence: One iteration step
(horizontal line):
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• Assumption: particle stays on the plane with pushing external force, sticking
Slope of prox function varies with:
Convergence: One iteration step
(horizontal line):
Point Mass on Frictional Plane: prox
• Fixed-point equation for tangential contact force
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• Assumption:object is translating toward the left:object is leaning toward the right
Unilateral constraint and Coulomb friction on acceleration level:
Painlevé’s Paradox
• Solution of dynamics not unique:
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Painlevé’s Paradox
• Complementarity formulation
• Formulation with prox function
andwith
• From the point of view of solution existence, the prox formulation completely agrees with complementarity theory (we have shown).
• A difference appears for attempting to find a solution via fixed-point iteration.
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Painlevé’s Paradox• Comparison of complementarity and prox formulation
1. Globally convergent unique solutions
everything works quite well
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Painlevé’s Paradox
• Comparison of complementarity and prox formulation2. No or several solutions
no solution: fixed-point scheme diverges
two solutions: fixed-point scheme may diverge
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Conclusion
• Prox and complementarity formulations are equivalent from the point of view of solution existence.
• Prox formulations can be solved via fixed-point or Newton schemes.
• Complementarity formulations can be solved via pivoting schemes.
• Fixed-point schemes can diverge when a solution exists; one does not recognize the case of solution non-existence.
• Fixed-point schemes are worth pursuing to explore the exploitation of fine-grained parallelism in the solution process.