Animation of Fluids

Animating Fluid is Hard…

• Too complex to animate by hand– Surface is changing

very quickly– Lots of small details– In short, a nightmare!

• Need automaticsimulations…

Ad-Hoc Methods

• Some simple algorithmsexist for special cases– Mostly waves…

• What about water glass?

• Too much work to comeup with empirical algorithmsfor each case…

Physically-Based Approach

• Look to Fluid Dynamics– Long history. Back to Newton…– Equations that describe fluid motion

• Use numerical methods to approximate fluid equations, simulating fluid motion– Like mass-spring systems

Current State-of-the-art in CG

• Marker-And-Cell (MAC) Method– fedkiw_fluid\glass00.avi– fedkiw_fluid\splash-640.avi

• Smoothed Particle Hydrodynamics (SPH)– muller_particles\sph.avi– muller_particles\pool.avi

Applications

• Mostly Hollywood– Shrek– Antz– Terminator 3– Many others…

• Games

• Engineering…

Fluid Dynamics

(with as little math as possible)

What do we mean by ‘Fluid’?

• liquids or gasses

• Mathematically:– A vector field u (represents the fluid velocity)– A scalar field p (represents the fluid pressure)– fluid density (d) and fluid viscosity (v)

Vector Fields

• 2D Scalar function:– f(x,y) = z – z is a scalar value

• 2D Vector function:– u(x,y) = v – v is a vector value

• v = (x’, y’)

• The set of valuesu(x,y) = v is called avector field

Fluid Velocity == Vector Field

• Can model a fluid as a vector field u(x,y)– u is the velocity of the fluid at (x,y)– Velocity is different at each point in fluid!

• Need to compute change in vector field

Conceptual Leap

• Particle Simulation:– Track particle positions x = (x,y)– Numerically Integrate: change in position

• Fluid Simulation :– Track fluid velocities u = (u,v) at

all points x in some fluid volume D– Numerically Integrate: change in velocity

dt

dx

dt

du

Equations of Fluid Dynamics

• Navier-Stokes Equation:

– Non-linear Partial Differential Equation– Models fluid transport – Derived from Newton’s second law

• conservation of momentum – all the forces go “somewhere”

• Mass-Conservation condition:– If we have a liter of water at the beginning of the solution, we have

a liter at the end…

Change in Velocity

• Derivative of velocity with respect to time

• Change in velocity, or acceleration– So this equation models acceleration of fluids

Advection Term

• Advection term– Force exerted on a particle

of fluid by the other particlesof fluid surrounding it

– How the fluid “pushes itself around”

Change inVelocity

Particle Advection

Video

Diffusion Term

• Viscosity constant controls velocity diffusion

• Essentially, this term describes how fluid motion is damped

• Highly viscous fluids stick together– Like maple syrup

• Low-viscosity fluids flow freely– Gasses have low viscosity

AdvectionChange inVelocity

Weather: Advection & Diffusion

• “Jet-Stream”

Pressure Term

• Pressure follows a diffusion process– Fluid moves from high-pressure

areas to low-pressure areas

• Moving == velocity– So fluid moves in direction of

largest change in pressure– This direction is the gradient

DiffusionAdvectionChange inVelocity

p = 0.5

p = 1p = 0

Time…

Weather: Pressure

• “Fronts” are the boundaries between regions of air with different pressure…

• “High Pressure Zones” will diffuse into “Low Pressure Zones”

Fluid Example

• Fast moving fluid is “pulled” towards slower-moving fluid

Body Force

• Body force term represents external forces that act on the fluid– Gravity– Wind– Etc…

DiffusionAdvectionChange inVelocity Pressure

Summary

• And 1 liter == 1 liter constraint:

• Need to simulate these equations…

DiffusionAdvectionChange inVelocity Pressure

Intermission

• Smoke– fedkiw_octree\smoke_octree.avi

• Fire– fedkiw_fire\flammable.avi

Implementation Overview

Fluid Representation

• Want to simulate motion of some fluid body– fluid is represented by a vector field

• Two problems:– Need to compute change in

vector field (using Navier-Stokes equation)

– Need to track fluid position

Solution: Discretization

• Create regular grid

Solution: Discretization

• Create regular grid

• Discretize fluid into grid cells

Solution: Discretization

• Create regular grid

• Discretize fluid into grid cells

• Track single velocityvector in eachgrid cell

Simulation Step

• “Solve” Navier-Stokes equation for each grid cell to compute change in velocity:

finite-difference Diffusion and Pressure terms are linear systems of equations

Body Force is just like mass-spring systems

fuuuu

pd

vt

1)()(

non-linear Advection term is difficult. Can finite-difference, but is not robust…

Free Surface Tracking with Marker Particles

• Want higher-resolution surface for rendering

• Add a bunch of particles

• Passively Advect thembased on fluid velocity

So the rest is easy, right?

• No

• Still have to enforce mass-conservation constaint:

• Standard equation does not take boundary conditions into account– Boundary conditions are things like walls, fluid/air

boundaries, rubber duckies, and so on– Have to ‘hack’ the equations…this is hard…

• Numerical Stability is elusive…

Intermission 2

• Melting– carlson_melting\bunnyside.mpg

• Rigid Body + Fluid– carlson_rigidfluid\rigidfluid.avi

Problems with Fluid Simulation

Surface Resolution…

Weird Behavior….

Water or Vegetable Oil?

• fedkiw_fluid\glass00.avi

• Oh, and it’s very, very slow– 7 minutes

per frame forwater glass…

Hard to Control

• Animators want to control fluid behavior

• Fluid simulation has a lot of free variables

• There has been limited success so far…

Numerical Stability

Maya Fluid Effects

What does fluid effects support?

• Naver-Stokes-based Fluids:– Smoke– Clouds– Explosions– Fire– “Goo” type-stuff

• Ad-Hoc / Mass-Spring fluids:– Oceans– Ponds

Fluid Effects Algorithms

• Unconditionally Stable Navier-Stokes simulation– Means they never explode, even with large timesteps – Jos Stam, “Stable Fluids”, SIGGRAPH 99

• Do not preserve volume very well– Ok for smoke– Problematic for water glass…– Gets worse w/ larger timestep

Maya Smoke

• Smoke demo

Maya Fire

• Looks like smoke…

More Fluid Effects Demos

Maya Ocean

• 2D height field – no crashing waves

• Can attach ‘bouyant’ objects

Maya Pond

• 2D height field– No splashing

• Mass-spring system

• Bouys, Boats, Wakes

• Can run in real-time

Fin

(questions?)