CS 231

Fluid simulation

Why Simulate Fluids?

Feature film special effects

Computer games

Medicine (e.g. blood flow in heart)

Because it’s fun

Fluid Simulation

Called Computational Fluid Dynamics (CFD)

Many approaches from math and

engineering

Graphics favors finite difference

Introduced fast and stable methods to

graphics [Stam 1999]

Vector Fields (Fluid Velocity)

uij = (ux,uy)

Navier-Stokes Equations

u=0

ut = k2u –(u)u – p + f

Incompressibility

Change in Velocity

Advection

Diffusion

Body Forces

Pressure

Navier-Stokes Equations

u=0

ut = k2u –(u)u – p + f

Incompressibility

Change in Velocity

Advection

Diffusion Pressure

Body Forces

Fluid Grids

All values live on regular grids

Need scalar and vector fields

Scalar fields: amount of smoke or dye

Vector fields: fluid velocity

Subtract adjacent quantities to approximate

derivatives

Scalar Field (Smoke, Dye)

cij

1.2 3.7 5.1 …

Diffusion

cij

Diffusion

1

1

1

1

-4

cijnew = cij + k Dt (ci-1j + ci+1j + cij-1 + cij+1 - 4cij)

ct = k2c

value relative

to neighbors

Diffusion = Blurring

Original Some Diffusion More Diffusion

Vector Field Diffusion

ut = k2u

… blur the x-velocity and the y-velocity

uxt = k2ux

uyt = k2uy

Two separate diffusions:

viscosity

Effect of Viscosity

• Each one is ten times higher

viscosity than the last

Low Medium High Very High

Variable Viscosity

Viscosity can vary based on position

Viscosity field k can change with temperature

Navier-Stokes Equations

u=0

ut = k2u –(u)u – p + f

Incompressibility

Change in Velocity

Advection

Diffusion Pressure

Body Forces

Advection = Pushing Stuff

Scalar Field Advection

ct = –(u)c

change in value advection current values

Vector Field Advection

ut = –(u)u

uxt = –(u)ux

uyt = –(u)uy

Two separate advections:

… push around x-velocity and y-velocity

Advection

Easy to code

Method stable even at large time steps

Problem: numerical inaccuracy diffuses flow

Diffusion/dissipation

in first order advection

Original Image After 360 degree rotationusing first order advection

Solution: Error Correction

1) Perform forward advection

2) Do backward advection from new position

3) Compute error and take correction step

4) Do forward advection from corrected

position

error

Compensate

Forward

BackwardForward

Solution: Error Correction

Navier-Stokes Equations

u=0

ut = k2u –(u)u – p + f

Incompressibility

Change in Velocity

Advection

Diffusion Pressure

Body Forces

Divergence

High divergence Low divergence

Zero divergence

Enforcing Incompressibility

First do velocity diffusion and advection

Find “closest” vector field that is divergence-

free

Need to calculate divergence

Need to find and use pressure

Measuring Divergence

u=?

uij=(uxi+1j - ux

i-1j) + (uyij+1-u

yij-1)

?

-uyij-1

uyij+1

-uxi-1j uxi+1j

Pressure Term

unew = u – p

Take divergence of both sides…

unew = u – p

u = 2p

zero

Pressure Term

u = 2p

pnewij = pij + ( uij - (pi-1j + pi+1j + pij-1 + pij+1 - 4pij))

known unknown

?

pij-1

pij+1

pi-1j pi+1j

Pressure Term

unew = u – p

…and velocity is now divergence-free

Fluid Simulator

1) Diffuse velocity

2) Advect velocity

3) Add body forces (e.g. gravity)

4) Pressure projection

5) Diffuse dye/smoke

6) Advect dye/smoke

Where is the Water?

Water is often modeled as a fluid with a

free surface

The only thing we’re missing so far is how to

track where the water is:

Voxelized: which cells are fluid vs. empty?

Better: where does air/water interface cut

through grid lines?

Marker Particles

Simplest approach is to use marker particles

Sample fluid volume with particles

At each time step, mark grid cells containing any

marker particles as Fluid, rest as Empty or Solid

Move particles in incompressible grid velocity field

(interpolating as needed)

Rendering Marker Particles

Need to wrap a surface around the particles

E.g. blobbies, or Zhu & Bridson ‘05

Problem: rendered surface has bumpy detail

that the fluid solver doesn’t have access to

The simulation can’t animate that detail properly

if it can’t “see” it.

Result: looks fine for splashy, noisy surfaces,

but fails on smooth, glassy surfaces.

Improvement

Sample (implicit) surface function on simulation grid

Even just union of spheres is reasonable if we smooth

and render from the grid

Issues, must make sure that each particle always shows up

on grid or droplets can flicker in and out of existence…

Or: you must delete stray particles

But still not perfect for flat fluid surfaces.

Level Sets

Idea: drop the particles all together, and just sample the

implicit surface function on the grid

In particular, best behaved type of implicit surface function is

signed distance

Computing Signed Distance

Many algorithms exist

Simplest and most robust (and accurate enough for

animation) are geometric in nature

Our example algorithm is based on a Fast Sweeping method

[Tsai’02]

Level Sets

Surface is exactly where =0

We need to evolve on the grid

The surface (=0) moves at the velocity of the fluid (any fluid

particle with =0 should continue with =0)

See e.g. the book [Osher&Fedkiw’02]

Reinitializing

One problem is that advecting this way

doesn’t preserve signed distance property

Eventually gets badly distorted

Causes problems particularly for extrapolation,

also for accuracy in general

Reinitialize: recompute distance to surface

Ideally shouldn’t change surface at all, just

values in the volume

Small-scale liquid-solid

Interactions

Lake ( >1 meter) Water drops (millimeters)

What makes large water and small water behave

differently?

Surface Tension

Viscosity

Surface Tension

Normal (always pointing outward)

Surface Tension Force

surfF N

0 0

Virtual Surface Method

Solid

Virtual

Surface

Virtual Liquid

Liquid

Air

60s 90s 120s

hydrophillic hydrophobic

Virtual Surface Method