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

Viscous Liquids Many common liquids exhibit viscosity…

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

Viscous Buckling

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

Eulerian Fluid Simulation Advection External Forces Viscosity Pressure Projection

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)

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

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

The Usual Simplification (Full form)

The Usual Simplification (Full form) (Constant viscosity)

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

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

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

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?

Free Surface Condition Air applies zero force on the liquid surface

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

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

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.)

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.

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

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

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

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

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

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

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

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

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

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

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