Complementarity Models for Rolling and Plasticity

1. Complementarity Models for Rolling and Plasticity Mihai Anitescu Argonne National Laboratory At ICCP 2012, Singapore With: A. Tasora (Parma), D. Negrut (Wisconsin), and S. Negrini (Milan)

2. 1. DVI MODELS, STATE OF RESEARCH

3. Nonsmooth contact dynamics—what is it? Newton Equations Non-Penetration Constraints • Differential problem with variational inequality constraints – DVI • Truly, a Differential Problem with Equilibrium Constraints Generalized Velocities Friction Model

4. Granular materials: abstraction: DVI • Differential variational inequalities Mixture of differential equations and variational inequalities. • Target Methodology (only hope for stability): time-stepping schemes.

5. Smoothing/regularization/DEM • Recall, DVI (for C=R+) • Smoothing • Followed by forward Euler. Easy to implement!! • Compare with the complexity of time-stepping • Which is faster?

6. Simulating the PBR nuclear reactor • Generation IV nuclear reactor with continuously moving fuel. • Previous attempts: DEM methods on supercomputers at Sandia Labs regularization) • 40 seconds of simulation for 440,000 pebbles needs 1 week on 64 processors dedicated cluster (Rycroft et al.) Simulations with DEM. Bazant et al. (MIT and Sandia laboratories).

7. Simulating the PBR nuclear reactor • 160’000 Uranium-Graphite spheres, 600’000 contacts on average • Two millions of primal variables, six millions of dual variables • 1 CPU day on a single processor… • We estimate 3CPU days, compare with 150 CPU days for DEM !!!

8. KT= An iterative method • Convexification opens the path to high performance computing. • How to efficiently solve the Cone Complementarity Problem for large-scale systems? • Our method: use a fixed-point iteration (Gauss-Seidel-Jacobi) • with matrices: • ..and a non-extensive separable projection operator onto feasible set

9. 2. Plasticity Models

10. Need for plasticity models • This is a very important effect in materials. • It is also true for granular materials where it is about phenomena of cohesion, friction, compliance and plasticization. • Think of the effect of water bridges and cohesion in powders such as appear in medication.

11. Type of plasticity models in 1D. • Plasticity elicits a different constitutive law compared to rigid contact • Force-displacement relationship at a contact a) Rigid b) Compliant c) Nonlinear d) cohesion e) Cohesion + Compliance f) Cohesion + Compliance + plastic

12. The yield surface. • Standard rigid contact 1-a can be turned to cohesive one by the transformation: • In addition, rule in 1-d satisfies (at contact): • In this context is called the yield surface. • The displacement is normal at the Yield surface, such constitutive rules are called associative.

13. Example: Friction and Cohesion in 3D by playing around with Yield surfaces • Shift the Coulomb cone downward and make it an associative yield surface.

14. Modeling plasticity • The yield surface give by Coulomb needs to be modified as no infinite reaction is now allowed (crushing)

15. The model • Separate elastic and plastic displacement: • The elastic part of the force allows to compute the force at the contact if the plastic part of the displacement is known. • Except, of course, the plastic part is NOT known. But now we use the associated plasticity hypothesis to constrain the plastic displacement evolution with a variational inequality. • Mathematically, it is the same idea with normal velocity at the contact; EXCEPT that the containing set is no longer a cone, proper (it can be a shifted cone, or just some convex set. So we can time-step and close it.

16. Time-stepping • After some notation • After this is solved, all we have to do is to solve the plasticity displacement • Use the velocity update • Replace in the plasticity velocity equation; to obtain the variational inequality • Over the cartesian product of cones:

17. Damping and VI • We can similarly accommodate damping • And obtain the VI • Where • And • Once we compute new contact impulse, we compute velocity, and update position, and iterate.

18. How do we solve it: • There is no need to change the PGS algorithm as the matrix N is still symmetric positive definite. • Only the projection over more complex sets needs to be implemented, but the problem and the algorithm have the same abstraction; that is. • We are expending the proof to the general case of convex sets, we do not expect problems.

19. Numerical Experiments • Compaction and shear test of a granular media. • Advanced cases of earth-moving machines (bulldozzers, vehicles) need such things to understand soil reaction. • Configuration: soil sample put in 0.1m x 0.1 mx0.2 m, • Top part is pulled by a “drawer” after a mass is dropped on top; shear force is measured and compared with experiment.

20. Results. • Configuration: Getting parameters is hard; Normal and tangentia compliances are 0:910e-7 m/N and damping coefficient is = 0.1. Sphere distribution. • Evolution of the shear force and vertical force. These will be measured from experiments. Note spike in forces – including shear! When load is dropped (probably some lock-in before it relaxes and pushes on the drawe)

21. The advantage of a simulator is that we can have a deeper understanding of the forces. • Evolution of the contact network. Note the “rolling” behavior. • Calibration and experiments are in progress.

22. Nonassociative plasticity • However, not all plasticity is associative. • How do we do things in the nonassociative case? • Extension • H is positive definite and symmetric, but N is only symmetric. • Do we have existence?

23. Existence for non-associative plasticity • Consider the eigendecomposition of H • We then have that the nonassociative plasticity problem is equivalent to: the following problem over a rotated cone:

24. Existence result • We use results from Kong, Tuncel and Xiu to obtain: • In addition, we can solve the problem by continuation: • P_0 guarantees the path exists; R_0 that it accumulates to a solution. • What to do about stationary iterations: unclear

25. To do • Lots and lots of things. • Extend the nonassociative plasticity to general compact convex sets • Extend projected iteration to the general convex sets and nonassociative plasticity • Simulations for cohesion and plasticity. • Do physical experiments to compare; choose the right parameters.

26. 3. Rolling Friction Models

27. Need for rolling friction models. • Rolling friction models important phenomena in computational mechanics • A very important example is the one of tires: rolling friction is critical important (as forward motion in vehicles is due to rolling friction) • The configuration of stacked granular materials is also critically dependent on rolling friction. • Ball bearing performance is affected by rolling friction, • … • In the DVI space, rolling friction was approached before (Leine and Glocker 2003), here we discuss a related approach but also other issues such as convexification.

28. Phenomenon • A constant force T is needed in the center to keep a ball rolling at constant speed. • That is equivalent to a resistive momentum M. • In a DVI framework:

29. Components of the rolling friction • The contact plane has a normal vector and consists of two tangential vectors • Force between bodies at contact, has a normal component and a tangential component • Torque between bodies at contact, has a normal component and a tangential component. • Dual motion quantities: tangential velocity, normal rotation and tangential rotation:

30. Model: rolling, spinning, sliding • Sliding friction, friction coefficient • Rolling coefficient, • Spinning: Spinning coefficient,

31. Restating the model to expose the maximum dissipation principle • This exposes the conic constraints:

32. Time-stepping with convex relaxation. • The reaction cone: • Define total cone (inclusive of bilateral constraints) and its polar: • Using the virtually identical sequence of time stepping, notations and relaxation we obtain the cone complementarity problem.

33. Essentially free choice, we use identity blocks Use w overrelaxationfactor to adjust this Always satisfied inmultibody systems General: The iterative method • ASSUMPTIONS • Under the above assumptions, wecan prove convergence. • The method produces in absence of jamming a bounded sequencewith an unique accumulation point • Method is relatively easy to parallelize, even for GPU!

34. Efficient Computation of the Projection over Intersection of Coupled Quadratic Cones • Structure of the variable • Structure of one cone: • Structure of the projection: • Key observation: For each cone the direction of the projected vector is known except for first component there exists 0 <= t_i <= 1 such that :

35. Efficient Computation of the Projection over Intersection of Coupled Quadratic Cones • In the end, we obtain • We can solve in each t_i easily to obtain: • The function is piecewise quadratic and convex (the graph of each component is a convex parabola union with a flat line with matching derivative. • We can compute the answer efficiently.

36. Validation: linear guideway with recirculating balls • We get close matching to analytical result (we allow a bit of compliance to resolve indeterminacy).

37. Rolling Friction Effects on Granular Materials • Converyor belt scenario: Question can we reproduce rolling friction effects over the repose angle of piles (observed)? • Yes, for same sliding friction the results are different.

38. Angle of repose. • Dependence on rolling friction coefficient – confirms trends in other experiments.

39. Conclusions • We presented extensions of DVI time-stepping schemes to rolling contact and elasto-plastic contact . • These include convex relaxation and algorithmic considerations. • The method is implemented in ChronoEngine. • Plasticity and in particular Non-associative plasticity open the need for more sophisticated complementarity VI models and analysis. (R_0 matrices over convex sets?). • This project is very much in progress.