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Christopher Batty and Robert Bridson University of British Columbia

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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…


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