Computer Animation Algorithms and Techniques

1. Computer AnimationAlgorithms and Techniques Fluids Rick Parent Computer Animation

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 as a single flex object Clouds as implicit surfaces Computational Fluid Dynamics Cloth weave Physical properties of a strand of hair Computational Fluid Dynamics Rick Parent Computer Animation

3. Superficial Models for Water Water is complex Changes shape Changes topology Governed by fluid dynamics Model specific features Still waters Small amplitude waves Ocean waves Running downhill Rick Parent Computer Animation

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

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

6. Simple Wave Rick Parent Computer Animation

7. Simple Wave Rick Parent Computer Animation

8. Sum of Sinusoidals Rick Parent Computer Animation

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

10. Ocean Waves Darwyn R. Peachey, “Modeling waves and surf”, SIGGRAPH '86. s – distance from source t – time A – maximum amplitude C – propogation speed L – wavelength T – period of wave Rick Parent Computer Animation

11. Movement of a particle In idealized wave, no transport of water L – wavelength A - amplitude of wave H – twice the amplitude C – propagation speed T – period of wave S – steepness of the wave Q – average orbital speed Rick Parent Computer Animation

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

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 Rick Parent Computer Animation

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

15. Wave in shallow water propogation speed, C, and wavelength, L, are reduced period of wave, T, remains the same amplitude, A (H), remains the same or increase orbital speed, Q, remains the same Waves break Rick Parent Computer Animation

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

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

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

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

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

21. Clouds Backdrop - static or slow moving, 2D Distant - 2 1/2 D, relative movement, opaque Close - 3D, translucent, amorphous Immersion - 3D, transparent, ethereal, misty, foggy Sets mood - ominous to playful Rick Parent Computer Animation

22. Cloud Formation air at certain temperature can hold certain amount of moisture formed when moisture content approaches moisture limit increase moisture content decrease temperature Example forces of formation Air mass is lifted and cooled as result (front, mountains) Ground water evaporates Air mass travels over something cold (e.g. water) Rick Parent Computer Animation

23. Basic Cloud Terms Altitude Cirrus/cirro - high (alone, means ‘fibrous’) Altus/alto - middle level Shape Cumulus/cumulo - puffy Stratus/strato - layers, sheet Moisture Nimbus/nimbo - water bearing Rick Parent Computer Animation

24. Cloud Formation Transpiration, evaporation Condensation Precipitation and loop ... Rick Parent Computer Animation

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

26. Approaches to Clouds Particle systems - but massive amount of particles needed for any significant cloud mass Volumetric representation - possible, but computationally expensive and must pre-determine extent of cloud to descretize space Implicit functions - partitions clouds into semi-transparent pieces; can animate independently Rick Parent Computer Animation

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

28. Dave Ebert Rick Parent Computer Animation

29. Models for Fire Procedural 2D Particle system Ad hoc approaches Rick Parent Computer Animation

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

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

32. Particle System Fire Rick Parent Computer Animation

33. Flames for film production Arnauld Lamorlette, Nick Foster, Structural modeling of flames for a production environment, Siggraph 02 (PDI/DreamWorks ) Rick Parent Computer Animation

34. 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. Rick Parent Computer Animation

35. CFD - terms Compressible – changeable density Steady state flow – motion attributes are constant at a point Viscosity – resistance to flow Newtonian Fluid has linear stresss-strain rate Vortices – circular swirls Rick Parent Computer Animation

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

37. CFD equations mass is conserved momentum is conserved energy is conserved Usually not modeled in computer animation To solve: discretize cells discretize equations solve iteratively by numerical methods Rick Parent Computer Animation

38. CFD Rick Parent Computer Animation

39. Conservation of mass (2D) vx A vy Small control volume: Δx by Δy A = Δx * Δy Mass inside: Time rate of mass change in volume Rick Parent Computer Animation

40. Conservation of mass (2D) vx A A = Δx * Δy vy Amount of mass entering from left: Time rate of mass change in volume = difference in rate of mass entering and rate of mass exiting Difference between left and right: Rick Parent Computer Animation

41. A Conservation of mass (2D) vx vy Divergence operator: If incompressible Rick Parent Computer Animation

42. 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 Rick Parent Computer Animation

43. Conservation of momentum in 2D Rate of change of vx-momentum vx-Momentum entering: Difference of x-momentum in x: Difference of x-momentum in y: Pressure difference in x : Rick Parent Computer Animation

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

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

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

47. 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 Rick Parent Computer Animation

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

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

50. Conservation of momentum with viscosity Rick Parent Computer Animation