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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 (thorsten.schindler@mytum.de)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.