figure 11. experiments with a bed of nails highlight the method’s ability to deal with sharp boundaries, isolated points of contact, sliver triangles, and localized points of high pressure between two nearly incident surfaces.

of separating plane events, respectively, 5.2% and 23.0% for processing and rescheduling of k-DOP events, respectively. The incident figure demonstrates how per frame runtime increases as the stress on the ribbons elevates.

6. 5. Parameters

We list parameters for the various examples. Bending and stretching stiffness refers to the Discrete Shells10 and common edge spring models. COR refers to coefficient of restitution, or the “bounciness” of the collisions.

7. DiScuSSion
7. 1. Parameters and the triad of safety, correctness,
and progress

One of our driving goals is to investigate methods that ensure safety, correctness, and progress regardless of the choice of parameters. The method proposed here does expose some parameters to the user, such as the proximity h. These parameters affect performance, not the triad of guarantees. Our experience in running the problem

verticles
Simulation
seconds
event
processing
(hours)

KDS event
rescheduling
(hours)

total (hours)

10,642 2.00 1. 5 16. 7 18. 5 3,995 5.00 3.0 141.1 144.5

examples reef knot Bowline knot trash compactor two sheets draped two sheets pulled

714

3.08

0.5

53.0

53. 6

15,982

3. 95

4. 5

260.8

265.5

15,982

3. 83

13. 6

310.5

325.6

scenarios, therefore, was qualitatively different than when using other methods, in that we did not need to search for parameters to ensure a successful modeling of contact. On the other hand, our method does not address the spatial discretization of elasticity (stretching and bending models), which can also require user tuning.

Although in theory the nested penalty barrier has infinitely many penalty layers at its disposal, it is impractical to

0 500 1000 Wall clock time (in seconds)

Total Event Processing Event Rescheduling

2 10

3

Simulation time (in seconds)

activate penalty layers whose stable time steps are too small, e.g., below the floating point epsilon. Simulations with thick-nesses h ( 1) too small, or velocities or masses too high, can thus fail to make progress (but remain safe). This limitation can be worked around by choosing a slow-shrinking layer distribution function, which is why we recommend h(l) = h(l )l−1/4.

example Density coR r( 1) h( 1)

Stretching
stiffness
Stretching
damping
Bending
stiffness

reefknot 0.1 0.0 1000.00.1 750.0 0.1 0.01 Bowline knot

0.01 0.0 1000.0 0.1 100.0 0.1 0.01

Bunny compactor

0.01 0.01 10000.0 0.05 1000.0 0.0 1000.0

trash compactor

0.001 0.01 1000.0 0.05 1000.0 15.0 10.0

two sheets draped

0.001 0.0 1000.0 0.1 1000.0 1.0 0.1

reef knot untied

0.1 0.0 1000.0 0.1 1000.0 0.1 0.01

two sheets pulled

0.001 0.0 1000.0 0.1 1000.0 1.0 0.1

Balls on nails

0.016 0.3 10000.0 0.1 50000.0 1.0 100000.0

 

2d sludge –

0.0 1000.0 0.1

–
–
–

Multistepping methods such as AVIs are known to have resonance instabilities, 8, 14 particularly if the simulation contains adjacent mesh elements of very different size. However, we have not observed any such instabilities or artifacts that we can attribute to such instabilities in our use of the method.

7. 2. Broader exploration

In this paper we were concerned with building the most robust contact implementation we could; therefore, we tied the knots as tight as possible, until each triangle was packed as tightly as possible into its neighbors. In the tightest configurations the spatial discretization becomes evident. It would therefore be interesting to introduce spatial adaptation, refining the mesh where curvature is high. Another alternative would be to improve the smoothness at render time, using for example the collision-aware subdivision of Bridson et al. 4

Dissipation and friction are critical for expressing the widest possible range of scenarios in physical simulation. We have omitted their discussion in this extended abstract, but refer the reader to the original publication for simple models that fit this criteria. Nevertheless, future work might explore efficient algorithms to handle stacking and static friction while still fitting the multisymplectic treatment.

7. 3. immediate and future impact

In considering this method for immediate industrial use, we anticipate two important hurdles. From the standpoint of incorporation into animation systems the first hurdle is the method’s insistence on safety even at the cost of artistic freedom. This effectively disallows all pinching, 2, 21 as well as commencing from invalid configurations. We believe that the method can be extended to permit shallow (“skimming”) pinching, but handling extremely unphysical boundary conditions within this framework seems at least initially at odds