Computer Animation Algorithms and Techniques

1. Computer AnimationAlgorithms and Techniques Fluids

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

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

4. Simple Wave Model - Sinusoidal Distance-amplitude

5. Simple Wave Time-amplitude at a location

6. Simple Wave

7. Simple Wave

8. Sum of Sinusoidals

9. Sum of Sinusoidals Height field Normal vector perturbation

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

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

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

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

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

15. 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. See book for details of modeling ocean waves from article by Peachey

17. Model for Transport ofWater h – water surface b – ground v – water velocity

18. Model for Transport ofWater Relates Acceleration Difference in adjacent velocities Acceleration due to gravity

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

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

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

22. 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. Visual characteristics • 3D • Amorphous • Turbulent • Complex shading • Semi-transparent • Self-shadowing • Reflective (albedo)

24. 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. Other approaches Particle systems implicit functions Volumetric representations

26. Dave Ebert

27. Models for Fire Procedural 2D Particle system Other approaches

28. Models for Fire - 2D

29. Models for Fire - particle system Derived from Reeves’ paper on particle systems

30. Particle System Fire

31. Combustion examples Show Fedkiw’s work

32. 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. General Approaches Grid-based Particle-based method Hybrid method

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

35. CFD

36. Conservation of mass (2D) u r A Small control volume: Dx by Dy by 1 A = Dx * Dy v Time rate of mass change in volume = rate of mass entering Mass inside CV: Time rate of mass change in volume

37. Conservation of mass (2D) u r A Small control volume: Dx by Dy by 1 A = Dx * Dy v Time rate of mass change in volume = rate of mass entering Amount of mass entering from left: Difference between left and right:

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

39. 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. 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. Conservation of momentum in 2D

42. Conservation of momentum (3D) x direction y direction z direction Material derivative

43. Navier-Stokes (for graphics)

44. V h Viscosity, etc. 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. Stokes Relations Extended Newtonian idea to multi-dimensional flows

46. Stokes Hypothesis Choose l so that normal stresses sum to zero

47. Conservation of momentum with viscosity

48. Incompressible, Steady 2-D flow Kinematic viscosity

49. 2D Euler Equations – no viscosity If incompressible

50. 2D Equations review