1. Computer Animation Rick Parent Computer Animation Algorithms and Techniques Fluids

2. Computer Animation Rick Parent Superficial models v. Deep models Water waves Wrinkles in skin and cloth Hair Clouds OR Computational Fluid Dynamics Cloth weave Physical properties of a strand of hair Computational Fluid Dynamics Directly model visible properties Model underlying processes that produce the visible properties (comes up throughout graphics, but particularly relevant here)

3. Computer Animation Rick Parent Superficial Models for Water Still waters Small amplitude waves The anatomy of waves ocean waves running downhill Main problem with water Changes shape Changes topology

4. Computer Animation Rick Parent Simple Wave Model - Sinusoidal Distance-amplitude

5. Computer Animation Rick Parent Simple Wave Time-amplitude at a location

6. Computer Animation Rick Parent Simple Wave

7. Computer Animation Rick Parent Simple Wave

8. Computer Animation Rick Parent Sum of Sinusoidals

9. Computer Animation Rick Parent Sum of Sinusoidals Normal vector perturbation Height field

10. Computer Animation Rick Parent Ocean Waves s – distance from source t – time A – maximum amplitude C – propogation speed L – wavelength T – period of wave

11. Computer Animation Rick Parent Movement of a particle Q – average orbital speed S – steepness of the wave T – time to complete orbit H – twice the amplitude In idealized wave, no transport of water

12. Computer Animation Rick Parent Breaking waves If Q exceeds C => breaking wave If non-breaking wave, steepness is limited Observed steepness between 0.5 and 1.0

13. Computer Animation Rick Parent Airy model of waves Relates depth of water, propagation speed and wavelength g - gravity As depth decreases, C approaches As depth increases, C approaches

14. Computer Animation Rick Parent Implication of depth on waves approaching beach at an angle Wave tends to straighten out – closer sections slow down

15. Computer Animation Rick Parent Wave in shallow water C an L are reduced T remains the same A (H) remain the same or increase Q remains the same Waves break

16. Computer Animation Rick Parent See book for details of modeling ocean waves from article by Peachey

17. Computer Animation Rick Parent Model for Transport ofWater h – water surface b – ground v – water velocity

18. Computer Animation Rick Parent Model for Transport ofWater Relates Acceleration Difference in adjacent velocities Acceleration due to gravity

19. Computer Animation Rick Parent Model for Transport ofWater Relates Temporal change in the height Spatial change in amount of water d = h(x) – b(x)

20. Computer Animation Rick Parent Model for Transport ofWater Small fluid velocity Slowly varying depth Use finite differences to model - see book

21. Computer Animation Rick Parent Models for Clouds Basic cloud types Physics of clouds Visual characteristics, rendering issues Early approaches Volumetric cloud modeling

22. Computer Animation Rick Parent Basic Cloud Types Example forces of formation Convection Convergence lifting along frontal boundaries Lifting due to mountains Height – water v. ice composition cumulus stratus cirrus nimbus heap layer curl of hair rain cirr – high alto – mid-level

23. Computer Animation Rick Parent Visual characteristics 3D Amorphous Turbulent Complex shading Semi-transparent Self-shadowing Reflective (albedo)

24. Computer Animation Rick Parent Early Approach - Gardner Early flight simulator research Static model for the most part Sum of overlaping semi-transparent hollow ellipsoids Taper transparency from edges to center See his paper from SIGGRAPH 1985

25. Computer Animation Rick Parent Other approaches Particle systems implicit functions Volumetric representations

26. Computer Animation Rick Parent Dave Ebert

27. Computer Animation Rick Parent Models for Fire Procedural 2D Particle system Other approaches

28. Computer Animation Rick Parent Models for Fire - 2D

29. Computer Animation Rick Parent Models for Fire - particle system Derived from Reeves’ paper on particle systems

30. Computer Animation Rick Parent Particle System Fire

31. Computer Animation Rick Parent Combustion examples Show Fedkiw’s work

32. Computer Animation Rick Parent Computational Fluid Dynamics (CFD) Fluid - a substance, as a liquid or gas, that is capable of flowing and that changes its shape at a steady rate when acted upon by a force. Compressible – changeable density Steady state flow – motion attributes are constant at a point Viscous – resists flow; Newtonian fluid has linear stresss-strain rate Vortices – circular swirls

33. Computer Animation Rick Parent Grid-based Particle-based method Hybrid method General Approaches

34. Computer Animation Rick Parent CFD equations To solve: discretize cells discrete equations numerically solve mass is conserved momentum is conserved energy is conserved Usually not modeled in computer animation

35. Computer Animation Rick Parent CFD

36. Computer Animation Rick Parent Conservation of mass (2D) u A r v Time rate of mass change in volume = rate of mass entering Small control volume:  x by  y by 1 A =  x *  y Mass inside CV: Time rate of mass change in volume

37. Computer Animation Rick Parent Conservation of mass (2D) u A r v Time rate of mass change in volume = rate of mass entering Small control volume:  x by  y by 1 A =  x *  y Amount of mass entering from left: Difference between left and right:

38. Computer Animation Rick Parent Conservation of mass (2D) u A r v Time rate of mass change in volume = rate of mass entering If incompressible Divergence operator

39. Computer Animation Rick Parent Conservation of momentum Momentum in CV changes as the result of: Mass flowing in and out Collisions of adjacent fluid (pressure) Random interchange of fluid at boundary

40. Computer Animation Rick Parent Conservation of momentum in 2D Rate of change of u-momentum within CV u-Momentum entering: Difference in x: Difference in y: Pressure difference in x :

41. Computer Animation Rick Parent Conservation of momentum in 2D

42. Computer Animation Rick Parent Conservation of momentum (3D) x direction y direction z direction Material derivative

43. Computer Animation Rick Parent Navier-Stokes (for graphics)

44. Computer Animation Rick Parent Viscosity, etc. h V Hooke’s law: in solid, stress is proportional to strain Fluid continuously deforms under an applied shear stress Newtonian fluid: stress is linearly proportional to time rate of strain Water, air are Newtonian; blood in non-Newtonian

45. Computer Animation Rick Parent Stokes Relations Extended Newtonian idea to multi-dimensional flows

46. Computer Animation Rick Parent Stokes Hypothesis Choose so that normal stresses sum to zero

47. Computer Animation Rick Parent Conservation of momentum with viscosity

48. Computer Animation Rick Parent Incompressible, Steady 2-D flow Kinematic viscosity

49. Computer Animation Rick Parent 2D Euler Equations – no viscosity If incompressible

50. Computer Animation Rick Parent 2D Equations review