1. Christopher Batty and Robert Bridson University of British Columbia
Accurate Viscous Free Surfaces for Buckling, Coiling, and Rotating Liquids
Christopher Batty and Robert Bridson
University of British Columbia

2. Viscous Liquids
Many common liquids exhibit viscosity…

3. Viscous Buckling and Coiling
Characteristic of highly viscous liquids
Dependent on correct forces at the surface
Video: buckling_n_coiling.mov

4. Viscous Buckling

5. Goals Accurate free surface behavior
Fully implicit, for large stable time steps
Handle variable viscosity
Easy implementation & efficient solution

6. Eulerian Fluid Simulation
Advection
External Forces
Viscosity
Pressure Projection

7. Related Work Carlson et al. 2002, Roble et al. 2003
Viscous liquids with a simplified implicit solve
Rasmussen et al. 2004
Variable viscosity liquids with IMEX integration
Goktekin et al. 2004, Zhu & Bridson 2005
Non-Newtonian liquids (viscoelastic, granular)

8. Fundamentals Viscosity is analogous to a fluid friction
Nearby elements of fluid exchange velocity, affecting their flow
Shear stress tensor, , is:
a measure of the resulting force per unit area
dependent on the gradient of velocity

9. Complete Form Shear stress is expressed as:
To apply the resulting forces to the fluid:
This is the full PDE form for viscosity

10. The Usual Simplification
(Full form)

11. The Usual Simplification
(Full form)
(Constant viscosity)

12. The Usual Simplification
(Full form)
(Constant viscosity)
(Expand)

13. The Usual Simplification
(Full form)
(Constant viscosity)
(Expand)
(Calculus identity)

14. The Usual Simplification
(Full form)
(Constant viscosity)
(Expand)
(Calculus identity)
(Incompressibility, )

15. The Simplified Form Looks like diffusion/smoothing of velocity
Velocity components are decoupled
3 implicit Poisson-like systems, solved with PCG
Eg. [Carlson et al, 2002]
What about the free surface?

16. Free Surface Condition
Air applies zero force on the liquid surface

17. Free Surface Condition
Air applies zero force on the liquid surface
The term is needed to enforce the constraint - it can’t simplify!
Free surfaces require the full stress expression even for constant viscosity

18. Incorrect Free Surfaces
What are the side effects?
Neumann BC:
Adds erroneous “ghost” forces
halts rotation
Dirichlet BC:
prevents viscosity from acting at the surface
liquid seems less viscous
Buckling fails to arise in either case.
Video: bending.mov

19. Correct Free Surfaces …are very difficult to discretize directly.
GENSMAC method (Tomé, McKee, et al.) is the only other MAC-based approach
Velocity gradients aren’t naturally co-located
The constraint should be applied only at the surface
Difficult to avoid special cases
Can it be solved implicitly?
How is the linear system affected? (symmetry, definiteness, etc.)

20. Key Idea
The free surface is actually a natural boundary condition in this setting
Using the proper variational form, it will fall out automatically
Idea: Replace the viscosity solve with minimization of a variational principle.

21. Characterizing Viscous Flow
Minimum Dissipation Theorem
The solution to a Stokes problem minimizes viscous dissipation [Helmholtz, 1868]

22. Characterizing Viscous Flow
Minimum Dissipation Theorem
The solution to a Stokes problem minimizes viscous dissipation [Helmholtz, 1868]
Viscous dissipation:
Kinetic energy dissipated by viscosity

23. Variational Form
Minimize dissipation while perturbing velocity as little as possible
This is equivalent to the full PDE form

24. Variational Form Benefits: Caveat…
No need to enforce the free surface discretely
Just estimate integrals and minimize
Fully implicit, SPD system
Take large timesteps, solve with CG
Supports variable viscosity
Exhibits the correct behaviour
Caveat…
Velocity components are no longer decoupled
Get a single 3x larger linear system

25. Discretization Use the classic (MAC) staggered grid
Velocities at cell faces
Stress at cell centres and edges
See [Goktekin et al, 2004]
syncs up naturally with positions of velocity gradients

26. Discretization Compute terms at each sample point
Faces for 1st integral, edges/centres for 2nd integral
Use centred differencing for velocity gradients
Scale by the liquid fraction
in the surrounding cube

27. Linear System
Identical to a MAC-based discretization of the full viscosity PDE…
but with new volume weights added!
Before:
After:

28. Results Artifact-free rotation and bending
Viscous buckling and coiling
Efficient, stable, highly variable viscosity
Video: results.mov

29. Future Work The linear system is no longer an M-matrix
Incomplete Cholesky may be less effective
Can we find better preconditioners?
Full free surface condition involves pressure, viscosity & surface tension
Can we solve all three simultaneously?
Should we? (speed vs. accuracy tradeoff)
Accuracy
Further analytical and ground truth comparisons

30. Conclusions Don’t solve the PDE – minimize the variational principle!
For viscosity, this approach…
drastically simplifies complex boundary conditions
yields efficient, straightforward, robust code
produces convincing simulations of purely viscous liquids

31. Thanks!
I’ll be happy to take any questions…