1. Fundamentals of Computer Animation Natural Phenomena: Water Surfaces

2. Natural Phenomena (Chap. 5) Plants Water Gaseous Phenomena + Fire

3. Water Surfaces A tightly stretched elastic membrane in which gravity can be ignored As infinitesimal sections are displaced, their external neighbors exert linear “spring” forces (surface tension) to minimize the space between them Since horizontal forces are equalized, particles move in only the z-direction

4. The vertical position with respect to time and space can be described with the partial differential equation: Speed at which waves travel across the surface Boundary conditions  homogeneous (i.e., edges don’t move up and down) Initial velocity of surface  zero General solution for a square L X L section of water ?

5. Approximation of water surface: an L X L height field with N points along each side (evenly spaced grid of z-values)

6. Using central differences to approximate the partial derivatives: Height at time t(+1) = t(0) + delta t Height at time t(-1) = t(0) – delta t Height of the (i,j)th grid position at time t(0)

8. The motion z(i,j) is influenced only by its nearest neighbors

9. Why? Islands and Shorelines!

10. d(i,j) = 1  free movement of the height value without any energy loss d(i,j) = 0  restricts all movement of water at that location Wave motion d (i,j) If coefficients d (i,j) are distributed and scaled according to the terrain features, wave react to the shoreline more naturally. Examples: If the bank is steep, d(i,j) should make a quick transition from 1 (water) to 0 (land) If the bank is gently sloped, the d(i,j) should make a gradual transition from 1 to 0.

11. If not met: Integration methods becomes unstable Successive z-values grow exponentially Instability We are using an explicit integration method (it uses only previous and current values of z (i, j) to evaluate z n+1 An implicit solution  solving sets of simultaneous equations for z n+1

12. Stability Explicit Euler schemeImplicit Euler scheme vs.

13. Interacting with the Surface : Splashes Created by instantaneously displacing one or several z-values at a particular location As the solution progresses, waves radiate from this location

14. Interacting with the Surface : Buoyancy Objects float because their overall density is less than of the surrounding water Force of buoyancy of an object is equal to the weight of the water displaced by that object This force is the direction of the pressure gradient, but in most cases, the direction normal to the water surface is appropriate

15. Interacting with the Surface : Buoyant Objects If the shape of the hull is approximated as a set of discrete of points, normals, and area patches Buoyancy Force  volume integral over the submerged part Bilinearly interpolated water height at p k Local area patches

16. Torque at p k  Buoyancy Force at p k  Vector from the center of mass to p k Total Force and Torque  summing the contributions from each hull vertex (submerged portions only)

17. Interacting with the Surface : Buoyant Objects Water Surface Approximating the first derivatives with central differences: Normal at the (i,j) th grid location:

18. To keep the object from sliding over the surface (like a surfboard), a drag force can also be calculated by summing contributions from each vertex: Velocity of the hull relative to the water at r k (vector from the center of mass to p k )

19. Rendering You can use alpha blending to give the appearance of transparency Draw triangles farthest from the viewer first (without the help of the Z- buffer)

20. Water also reflects light from its surroundings Use environment mapping Reflecting and refracting rays of light from the eye and intersecting them with an environment map surface to calculate a texture coordinate Rendering

21. Lasse Jensen Deep-Water Animation and Rendering Gamasutra September 26, 2001 Color of water (refraction cube-map)

22. Rendering Bump-Mapping to Reduce Geometry

23. Rendering Caustics Light sinuous shifting patterns due to sunlight transmitted from the specular water surface.

24. Rendering Godrays As the rays pass the water matter, they scatter from small particles floating in the water (plankton, dirt), making them visible and causing streaks of light known as Godrays Slices of the volume in front of camera Used for rendering Godrays Resulting image for the volume slices

25. Rendering Foam and Bubbles Texture layers

26. Rendering Foam and Bubbles Texture layers

27. Rendering Spray Particle Systems

28. Rendering: Spray  Particle Systems When water collides against obstacles we generate spray of water using a particle system with simple dynamics Each particle is given an initial velocity taken directly from the water-surface’s velocity, at the spawning position, with added turbulence. It’s then updated according to gravity, wind and other global forces thereafter. Rendering of the particles are done with a mixture of alpha-transparency and additive-alpha sprites (image objects).

29. Rendering: Spray  Particle Systems The particle system is also used for drawing bobbles from objects dropped into the water. For this effect we simply move the bobbles on a sinus path around the buoyancy vector up to the surface were they are killed.

30. The Perfect Storm (Industrial Light and Magic) ILM employed custom and off-the-shelf software for fluid dynamics simulations and animation on Silicon Graphics® O2® workstations, then rendered them on SGI® Origin® family systems to create the storm sequences

33. A shaded version of the Andrea Gail

34. The colored and textured version of the Andrea Gail

35. Close-up showing the CG actors and buoys

36. The simulated ocean. Note that the boat's movement in this ocean must be completely matched to the wave motion to be realistic.

37. Run-off from a wave crashing over the boat in an earlier frame

38. A wave of particles created by the boat's movement in the water

39. The splash created by the boat crashing into a wave

40. A shaded version of the Andrea Gail

41. One frame of the final shot with all of the elements integrated together