1. Fluid Animation
CSE 3541
By: Matt Boggus

2. Overview Procedural approximations Mathematical background
Computational models for representing fluids
Using the computational models
Forces
“Stable fluids” by Stam

3. Real-time fluids Goals: Simulate the effect, not the cause
Cheap to compute
Low memory consumption
Stability
Plausibility
Interactivity
Simulate the effect,
not the cause

4. Real-time fluids Procedural water Particle systems Heightfield fluids
Unbounded surfaces, oceans
Particle systems
Splashing, spray, puddles, smoke, bubbles, rain
Heightfield fluids
Ponds, lakes, rivers

5. Procedural water – cos wave
y = A cos (2 * pi * x / T)
Visualization of parameter changes using graphing calculator

6. Superimposed linear waves
Normal vector displacement
Height displacement

7. Heightfield fluid Function u(x,y) gives height at (x,y)
Store height values in an array u[x,y]

8. Can this wave be represented using a heightfield?

9. Setting and updating heightfield fluids
Methods
Pressure or height differences between cells
Collision detection and displacement
Additional considerations
When updating, a second heightfield may be used to preserve the values from the previous frame
Handle boundary cases differently

10. Pressure/Height differences
Initialize u[i,j] with some interesting function Initialze v[i,j]=0 loop v[i,j] += (u[i-1,j] + u[i+1,j] + u[i,j-1] + u[i,j+1])/4 – u[i,j] v[i,j] *= 0.99 u[i,j] += v[i,j] endloop Clamp at boundary
Algorithm from Fast Water Simulation for Games
Using Height Fields
Linked at end of slides

11. Example videos Real-Time Eulerian Water Simulation
Grid based
SPH based real-time liquid simulation
Particle based

12. Fluid Models Grid-based (Eulerian) d is density
Particle-based (Lagrangian)
Hybrid
Animate the particles
“Collect” particles to compute density

13. Navier-Stokes Equation
Momentum equation
Incompressibility
ut = k2u –(u)u – p + f
Change in velocity
Diffusion/ Viscosity
Advection
Pressure
Body Forces
u: the velocity field
u=0
k: kinematic viscosity

14. Diffusion/Viscosity Force
Limit shear movement of particles in the liquid
The momentum between the neighbour particles are exchanged Pp
Ppn

15. Adhesion Force
Attract particles to each other and to other objects (similar to gravitational force)
The adhesion force between (left) honey-honey, (middle) honey-ceramic and (right) non-mixing liquid

16. Advection Force
Velocity grid

17. Pressure force
Pressure figure from
Huamin Wang’s 3541 slides

18. Pressure force

19. Friction Force
Dampen movement of particles in contact with objects in the environment
Scale down the velocity by a constant value

20. Rendering particles
Figure from

21. Heightfield mesh particle collection
Step 1. Zero out all u[i,j]
Step 2. For each u[i,j], determine which particles are closet to it
(bounding box collision test)
Alternative Step 2. For each particle, determine which (i,j) it is closest to
(translate position half a cell, then floor it)

22. Case Study: A 2D Fluid Simulator
Incompressible, viscous fluid
Assuming the gravity is the only external force
No inflow or outflow
Constant viscosity, constant density everywhere in the fluid

23. Stable Fluids – overview of data
Velocity grid
Density grid
Move densities around using velocity grid and dissipate densities
Move velocities around using velocity grid and dissipate velocities
Walkthrough:
Source code:

24. Additional readings
Fluid Simulation for Computer Animation (SIGGRAPH course)
The original stable fluids paper
An update to the stable fluids work
Fast Water Simulation for Games Using Height Fields (GDC talk)

25. Additional Slides

26. Heightfield mesh creation and cell iteration
Covered earlier with terrains

27. Heightfield or Heightmap terrain data
2D greyscale image
Surface in 3D space
Images from

28. Heightfield mesh creation and cell iteration
u[x,y] ; dimensions n by n

29. Heightfield mesh creation and cell iteration
Quad[i,j] such that i = 0, j = 0; Vertices are U[0,0], U[1,0], U[1,1], U[0,1]
U[i,j], U[i+1,j], U[i+1,j+1], U[i,j+1]

30. Heightfield mesh creation and cell iteration
Inner loop iterates over i, the x coordinate
Last quad: i=5 (i = n-1)

31. Heightfield mesh creation and cell iteration
Outer loop iterates over j, the y coordinate
Last row: j=5 (j = n-1)

32. Smoothing
For every grid cell u[i,j], set it to average of itself and neighbors
Implementation concerns:
A. looping order
B. boundary cases
C. both
D. none