Fluid Animation CSE 3541 By: Matt Boggus

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

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

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

Procedural water – cos wave y = A cos (2 * pi * x / T) https://www.desmos.com/calculator 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 Algorithm from Fast Water Simulation for Games Using Height Fields Linked at end of slides

Example videos Real-Time Eulerian Water Simulation Grid based https://www.youtube.com/watch?v=Jl54WZtm0QE 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

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

Diffusion/Viscosity Force Limit shear movement of particles in the liquid The momentum between the neighbour particles are exchanged 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 figure from Huamin Wang’s 3541 slides

Pressure force

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

Rendering particles Figure from https://mdzahidh.github.io/cs148/assets/slides/13-Physically%20Based%20Animation.pdf

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 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