Efficient Simulation of Large Bodies of Water by Coupling Two and Three Dimensional Techniques Geoffrey Irving Stanford University Pixar Animation Studios Eran Guendelman Stanford University Frank Losasso Ron Fedkiw Stanford University Industrial Light + Magic

Motivation Large scale water phenomena important – Rivers, lakes, oceans, floods Fast option: height field methods – Nice wave propagation – Can’t handle overturning Accurate option: 3D Navier Stokes – Captures three dimensional behavior – Slow at high resolutions: O(N 4 )

Solution: use both Uniform 3D Navier-Stokes near interface Coarsen elsewhere using tall cells

Solution: use both

Related work: 2D Deep water – Fournier and Reeves 1986, Peachy 1986 – Recent: Thon et al. 2000, Hinsinger 2002 Shallow Water – Kass and Miller 1990, O’Brien and Hodgins 1995 Rivers and streams – Chen and Lobo 1994, Thon and Ghazanfarpour 2001

Related work: 3D Uniform Navier-Stokes water – Foster and Metaxas 1997, Foster and Fedkiw 2001 – Enright et. al 2002: Particle level set method Large bodies of water – Takahashi et al. 2003: spray and foam – Mihalef et al. 2004: breaking waves Adaptive simulation – Losasso et al. 2004: Octree grids – Houston et al. 2006: Run-Length Encoded (RLE) grids

Outline Grid structure Uniform solver Advection on tall cells Pressure solver on tall cells Parallel implementation Discussion and Results

Outline Grid structure Uniform solver Advection on tall cells Pressure solver on tall cells Parallel implementation Discussion and Results

Why height fields work Water likes to stay flat Only water-air interface is visible Vertical structure simpler than horizontal

Mixing height fields and 3D Specify “optical depth” where we expect turbulent motion Use uniform 3D cells within optical depth Use height field model elsewhere optical depth

Grid structure Start with uniform MAC grid Keep cells within optical depth of the interface Outside optical depth, merge vertical sequences of cells into single tall cells

Grid structure: storing values Start with MAC grid storage – Level set values in cell centers near interface – Pressure values in cell centers – Velocity components on corresponding faces

Grid structure: pressure Two pressure samples per tall cell Linear interpolation between Allows

Grid structure: velocity Velocity corresponds to pressure gradients

Refinement and coarsening Grid is rebuilt whenever fluid moves based on current level set Linear time (Houston et al. 2006) Velocity must be transferred to new grid – optionally transfer pressure as initial guess

Transferring velocity InterpolateLeast squares

Transferring velocity (cont) Interpolate: Least squares:

Outline Grid structure Uniform solver Advection on tall cells Pressure solver on tall cells Parallel implementation Discussion and Results

Uniform solver Navier-Stokes equations plus level set Level set exists only in uniform cells Velocity updated using projection method

Uniform solver (cont) Advect velocity and add gravity – use semi-Lagrangian for uniform cells Solve Laplace equation for pressure Apply pressure correction to velocity

Outline Grid structure Uniform solver Advection on tall cells Pressure solver on tall cells Parallel implementation Discussion and Results

Tall cell advection Can’t use semi-Lagrangian for Use conservative method for plausible motion Simplest option: first order upwinding

First order upwinding Conservation form of advection equation Compute flux on each control volume face Add fluxes to current velocity

First order upwinding (tall cells) Pretend to do the following – Refine to uniform grid – Advect – Coarsen back to original grid Simulate this by applying least squares directly to uniform discretization Same answer but faster

First order upwinding (tall cells) u along uu along vv along uv along v

Advection issues Occasional instabilities near steep terrain – fix by clamping to affine combination Lots of numerical dissipation – good enough for bulk motion Future work: generalize higher order schemes used in shallow water

Outline Grid structure Uniform solver Advection on tall cells Pressure solver on tall cells Parallel implementation Discussion and Results

Pressure solve on tall cells Pressure projection is Need gradient and divergence operators

Pressure solve: gradient Gradient is easy

Pressure solve: divergence Two divergence samples per cell Interpolate velocity to uniform face Divide flux between samples with same weights used for interpolation

Pressure solve: Laplacian Compose divergence and gradient to get linear system Symmetric and positive definite since we used the same weights in both Solve using preconditioned conjugate gradients

Outline Grid structure Uniform solver Advection on tall cells Pressure solver on tall cells Parallel implementation Discussion and Results

Parallel implementation Parallelize only along horizontal dimensions – No harder than parallelizing a uniform code – Vertical dimension already cheap Exchange data with neighbors every step

Parallel implementation Pressure solved globally on all processor together Block diagonal preconditioner built out of incomplete Choleski on each processor AB CD

Results: splash (300 x 200) Optical depth equal to water depth

Results: splash (300 x 200) Optical depth 1/4 th water depth

Results: splash (300 x 200) Optical depth 1/16 th water depth

Results: deep splash Water depth doubled

Results: boat (1500 x 300)

Results: river (2000 x 200)

Matching bottom topography Tall cells match ground for free Octrees would require extra refinement Less important for deep problems

Comparison with octrees Advantages over octrees: – Easy to parallelize – Reduces to MAC discretization with refinement – Matches bottom topography for free Main disadvantage: relies on vertical simplicity for efficiency Not applicable for all flows – rising bubbles, colliding droplets, etc.

Conclusion Want high resolution near interface – Uniform resolution at interface sufficient Plausible bulk motion enough elsewhere Many flows have simple vertical structure Use this to create hybrid 2D/3D method

Future work Improved advection scheme – Match ENO/WENO schemes for shallow water Better parallelism – Remove global linear system solve Find optimal adaptive structure – Hybrid RLE / octree grid?

The End Questions?