Transcript Fluids and Cellular Automata

Fluid Animation
CSE 3541
By: Matt Boggus
• Procedural approximations
• Mathematical background
• Computational models for representing fluids
• Using the computational models
– Forces
– “Stable fluids” by Stam
• Goals:
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Cheap to compute
Low memory consumption
Stability
Plausibility
Interactivity
• Simulate the effect,
not the cause
Real-time fluids
• Procedural water
– Unbounded surfaces, oceans
• Particle systems
– Splashing, spray, puddles,
smoke, bubbles, rain
• Heightfield fluids
– Ponds, lakes, rivers
Procedural water – cos wave
• Visualization of parameter changes using
graphing calculator
Superimposed linear waves
Normal vector displacement
Height displacement
Heightfield fluid
• Function u(x,y) gives height at (x,y)
• Store height values in an array u[x,y]
Can this wave be represented using
a heightfield?
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
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
Example videos
• Real-Time Eulerian Water Simulation
– Grid based
– https://www.youtube.com/watch?v=Jl54WZtm0Q
E
• SPH based real-time liquid simulation
– Particle based
– https://www.youtube.com/watch?v=6CP5QvfuD_
w
Fluid Models
Grid-based (Eulerian)
d is density
Particle-based (Lagrangian)
Hybrid
Animate the particles
“Collect” particles to compute density
d
d d
d
d
d d
d
d
d d
d
d
d d
d
• Momentum equation
ut = k2u –(u)u – p + f
Change in Diffusion/
velocity
Viscosity
• Incompressibility
u=0
Advection Pressure Body
Forces
u: the velocity field
k: kinematic viscosity
Diffusion/Viscosity Force
• Limit shear movement of particles in the liquid
• The momentum between the neighbour particles
are exchanged
Pexch 
v  k (d pn )( Pp  Ppn )t
2n 1 k (d i )
n
v : viscosity coefficient
k () : A weighting function (Gaussian)
d i : distance from the processed particle
Pp , Ppn : momentum of p and pn
Pexch : momentum exchanced
Pp  Pp  Pexch
Ppn  Ppn  Pexch
Pp
Ppn
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
Advection Force
Velocity grid
Pressure force
Pressure force
Friction Force
• Dampen movement of particles in contact with objects
in the environment
• Scale down the velocity by a constant value
vˆt  fvt
Rendering particles
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)
Case Study: A 2D Fluid Simulator
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Incompressible, viscous fluid
Assuming the gravity is the only external force
No inflow or outflow
Constant viscosity, constant density
everywhere in the fluid
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: http://www.dgp.utoronto.ca/~stam/reality/Talks/FluidsTalk/FluidsTalkNotes.pdf
Source code: http://www.autodeskresearch.com/publications/games
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)
ADDITIONAL SLIDES
Heightfield mesh creation and cell
iteration
• Covered earlier with terrains
Heightfield or Heightmap terrain data
2D greyscale image
Surface in 3D space
Images from http://en.wikipedia.org/wiki/Heightmap
Heightfield mesh creation and cell
iteration
u[x,y] ; dimensions n by n
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]
Heightfield mesh creation and cell
iteration
Inner loop iterates over i, the x coordinate
Last quad: i=5
(i = n-1)
Heightfield mesh creation and cell
iteration
Outer loop iterates over j, the y coordinate
Last row: j=5
(j = n-1)
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