Binh Nguyen, Jeff Trinkle ([nguyeb2,trink]@cs.rpi) Rensselaer Polytechnic Institute

1. A Comparison of prox and Complementarity Formulations Thorsten Schindler (thorsten.schindler@mytum.de)INRIA Grenoble – Rhône-Alpes Binh Nguyen, Jeff Trinkle ([nguyeb2,trink]@cs.rpi.edu)Rensselaer Polytechnic Institute Euromech Colloquium ‘Nonsmooth Contact and Impact Laws in Mechanics’Grenoble, 08.07.2011

2. Multibody Dynamics Equations of bodies with bilateral and unilateral contacts • Examplaryvideos: grasping, double track, pushbeltcvt (wmv) • Essential: effective parallelcollissiondetection, constraintrepresentation and solution by highlyparallelsupercomputers • Here: comparison of twoconstraintrepresentationsconcerninganalytical and numerical issues

3. Complementarity Formulation How can looklike? Bilateralconstraint: e.g. idealizedknee joint Unilateralconstraint: e.g. imperfect joints, Newton’scradle • Wesee the contact laws on position level; increasing the number of dots over g givesvelocity and accelerationlevel.

4. Complementarity Formulation How can looklike? Coulomb friction: e.g. idealizedclutches • The contact lawisnaturally on velocitylevel; increasing the number of dots over g gives the accelerationlevel.

5. Complementarity Formulation Mathematical model:nonlinear differential complementarity problem (DCP) discretization • Numerical model:nonlinearcomplementarityproblem (NCP) solution • Numericalalgorithm:pivotingscheme(whatdoes PATH do in the linear case?)exponentialworstcomplexity / polynomial averagecomplexity

6. Formulation with prox Function How can looklike? Wecanfigureit out! bilateral unilateral Coulomb

7. 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

8. Formulation with prox Function Unilateral constraint • separateproximality yields If andso If and

9. Formulation with prox Function Bilateral constraint • Coulomb friction • Increasing the number of dots over g changes kinematiclevels. • Numerical model:nonsmooth, nonlinearequations • Numericalalgorithm: fixed-point iterationor Newton method

10. Formulation with prox Function How shood we choose ? Figure out prox functions! bilateral unilateral Coulomb

11. Point Mass on Frictional Plane Equations of motion • Unilateralconstraint and Coulomb friction

12. Point Mass on Frictional Plane: prox Fixed-point equation for normal contact force • Assumption: contact withzero normal velocity and pushingexternal force • Slope of proxfunction varies with: • Convergence: • One iterationstep (horizontal line):

13. Point Mass on Frictional Plane: prox Fixed-point equation for tangential contact force • Assumption: particlestays on the plane withpushingexternal force, sticking • Slope of proxfunction varies with: • Convergence: • One iterationstep (horizontal line):

14. Painlevé’s Paradox Solution of dynamics not unique: • Assumption:objectistranslatingtoward the left:objectisleaningtoward the right • Unilateralconstraint and Coulomb friction on accelerationlevel:

15. Painlevé’s Paradox Complementarity formulation with and • Formulation withproxfunction • From the point of view of solution existence, the prox formulation completelyagreeswithcomplementaritytheory (we have shown). • Adifferenceappears for attempting to find a solution via fixed-point iteration.

16. Painlevé’s Paradox Comparison of complementarity and prox formulation Globally convergent unique solutions • everythingworksquitewell

17. Painlevé’s Paradox Comparison of complementarity and prox formulation No or several solutions no solution: fixed-point schemediverges two solutions: fixed-point schememaydiverge

18. 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.