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Animation of Fluids

Learn about the challenges of animating fluid, the physics-based approach, and the current state-of-the-art methods in computer graphics. Explore applications in movies, games, and engineering.

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Animation of Fluids

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Presentation Transcript

  1. Animation of Fluids

  2. 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…

  3. 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…

  4. 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

  5. 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

  6. Applications • Mostly Hollywood • Shrek • Antz • Terminator 3 • Many others… • Games • Engineering…

  7. Fluid Dynamics (with as little math as possible)

  8. 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)

  9. 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

  10. 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

  11. Conceptual Leap • Particle Simulation: • Track particle positionsx = (x,y) • Numerically Integrate: change in position • Fluid Simulation : • Track fluid velocitiesu = (u,v) atall pointsx in some fluid volume D • Numerically Integrate: change in velocity

  12. 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…

  13. Change in Velocity • Derivative of velocity with respect to time • Change in velocity, or acceleration • So this equation models acceleration of fluids

  14. Change inVelocity Advection Term • Advection term • Force exerted on a particleof fluid by the other particlesof fluid surrounding it • How the fluid “pushes itself around”

  15. Particle AdvectionVideo

  16. Change inVelocity Advection 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

  17. Weather: Advection & Diffusion • “Jet-Stream”

  18. Change inVelocity Advection Diffusion p = 0 p = 1 p = 0.5 Pressure Term • Pressure follows a diffusion process • Fluid moves from high-pressureareas to low-pressure areas • Moving == velocity • So fluid moves in direction oflargest change in pressure • This direction is the gradient Time…

  19. Weather: Pressure • “Fronts” are the boundaries between regions of air with different pressure… • “High Pressure Zones” will diffuse into “Low Pressure Zones”

  20. Fluid Example • Fast moving fluid is “pulled” towards slower-moving fluid

  21. Change inVelocity Advection Diffusion Pressure Body Force • Body force term represents external forces that act on the fluid • Gravity • Wind • Etc…

  22. Change inVelocity Advection Diffusion Pressure Summary • And 1 liter == 1 liter constraint: • Need to simulate these equations…

  23. Intermission • Smoke • fedkiw_octree\smoke_octree.avi • Fire • fedkiw_fire\flammable.avi

  24. Implementation Overview

  25. Fluid Representation • Want to simulate motion of some fluid body • fluid is represented by a vector field • Two problems: • Need to compute change invector field (using Navier-Stokes equation) • Need to track fluid position

  26. Solution: Discretization • Create regular grid

  27. Solution: Discretization • Create regular grid • Discretize fluid into grid cells

  28. Solution: Discretization • Create regular grid • Discretize fluid into grid cells • Track single velocityvector in eachgrid cell

  29. u 1 = - × Ñ + Ñ × Ñ - Ñ + ( u ) u ( v u ) p f ¶ t d Simulation Step • “Solve” Navier-Stokes equation for each grid cell to compute change in velocity: Body Force is just like mass-spring systems non-linear Advection term is difficult. Can finite-difference, but is not robust… finite-difference Diffusion and Pressure terms are linear systems of equations

  30. Free Surface Tracking with Marker Particles • Want higher-resolution surface for rendering • Add a bunch of particles • Passively Advect thembased on fluid velocity

  31. 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…

  32. Intermission 2 • Melting • carlson_melting\bunnyside.mpg • Rigid Body + Fluid • carlson_rigidfluid\rigidfluid.avi

  33. Problems with Fluid Simulation

  34. Surface Resolution…

  35. Weird Behavior….

  36. Water or Vegetable Oil? • fedkiw_fluid\glass00.avi • Oh, and it’s very, very slow • 7 minutes per frame forwater glass…

  37. Hard to Control • Animators want to control fluid behavior • Fluid simulation has a lot of free variables • There has been limited success so far…

  38. Numerical Stability

  39. Maya Fluid Effects

  40. What does fluid effects support? • Naver-Stokes-based Fluids: • Smoke • Clouds • Explosions • Fire • “Goo” type-stuff • Ad-Hoc / Mass-Spring fluids: • Oceans • Ponds

  41. 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

  42. Maya Smoke • Smoke demo

  43. Maya Fire • Looks like smoke…

  44. More Fluid Effects Demos

  45. Maya Ocean • 2D height field – no crashing waves • Can attach ‘bouyant’ objects

  46. Maya Pond • 2D height field • No splashing • Mass-spring system • Bouys, Boats, Wakes • Can run in real-time

  47. Fin (questions?)

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