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. 1. Parameters and the triad of safety, correctness,
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
10,642 2.00 1. 5 16. 7 18. 5
3,995 5.00 3.0 141.1 144.5
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)
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)
reefknot 0.1 0.0 1000.00.1 750.0 0.1 0.01
0.01 0.0 1000.0 0.1 100.0 0.1 0.01
0.01 0.01 10000.0 0.05 1000.0 0.0 1000.0
0.001 0.01 1000.0 0.05 1000.0 15.0 10.0
0.001 0.0 1000.0 0.1 1000.0 1.0 0.1
0.1 0.0 1000.0 0.1 1000.0 0.1 0.01
0.001 0.0 1000.0 0.1 1000.0 1.0 0.1
0.016 0.3 10000.0 0.1 50000.0 1.0 100000.0
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