Fluid Simulation for Computer Animation

1. Fluid Simulation forComputer Animation Greg Turk College of Computing and GVU Center Georgia Institute of Technology

2. Why Simulate Fluids? • Feature film special effects • Computer games • Medicine (e.g. blood flow in heart) • Because it’s fun

3. Fluid Simulation • Called Computational Fluid Dynamics (CFD) • Many approaches from math and engineering • Graphics favors finite differences • Jos Stam introduced fast and stable methods to graphics [Stam 1999]

4. Diffusion Pressure Advection Body Forces Change in Velocity Navier-Stokes Equations u=0 ut= k2u –(u)u – p + f Incompressibility

5. Navier-Stokes Equations u=0 ut= k2u–(u)u – p + f Incompressibility Diffusion Pressure Advection Body Forces Change in Velocity

6. Finite Differences Grids • All values live on regular grids • Need scalar and vector fields • Scalar fields: amount of smoke or dye • Vector fields: fluid velocity • Subtract adjacent quantities to approximate derivatives

7. Scalar Field (Smoke, Dye) … 5.1 1.2 3.7 cij

8. Diffusion cij

9. 1 1 -4 1 1 change in value value relative to neighbors Diffusion ct = k2c cijnew = cij + k Dt (ci-1j + ci+1j + cij-1 + cij+1 - 4cij)

10. More Diffusion Some Diffusion Original Diffusion = Blurring

11. Vector Fields (Fluid Velocity) uij = (ux,uy)

12. viscosity Two separate diffusions: uxt= k2ux uyt= k2uy Vector Field Diffusion ut= k2u … blur the x-velocity and the y-velocity

13. Effect of Viscosity Low Medium High Very High • Each one is ten times higher viscosity than the last “Melting and Flowing” Mark Carlson, Peter J. Mucha, Greg Turk Symposium on Computer Animation 2002

14. Low Viscosity

15. High Viscosity

16. Variable Viscosity • Viscosity can vary based on position • Viscosity field k can change with temperature • Need implicit solver for high viscosity

17. Wax

20. Navier-Stokes Equations u=0 ut= k2u –(u)u – p + f Incompressibility Diffusion Pressure Advection Body Forces Change in Velocity

21. Advection = Pushing Stuff

22. Advection

23. Advection 0.3 2.7 0.3

24. change in value advection current values Scalar Field Advection ct=–(u)c

25. Two separate advections: uxt=–(u)ux uyt=–(u)uy Vector Field Advection ut=–(u)u … push around x-velocity and y-velocity

26. Advection • Easy to code • Method stable even at large time steps • Problem: numerical inaccuracy diffuses flow

27. Diffusion/dissipation in first order advection After 360 degree rotation using first order advection Original Image

28. Solution: BFECC 1) Perform forward advection 2) Do backward advection from new position 3) Compute error and take correction step 4) Do forward advection from corrected position “Flowfixer: Using BFECC for Fluid Simulation” ByungMoon Kim, Yingjie Liu, Ignacio Llamas, Jarek Rossignac Eurographics Workshop on Natural Phenomena 2005

29. Forward Compensate Forward Backward error Intuition to BFECC

31. Navier-Stokes Equations u=0 ut= k2u –(u)u– p + f Incompressibility Diffusion Pressure Advection Body Forces Change in Velocity

32. Low divergence Zero divergence Divergence High divergence

33. Enforcing Incompressibility • First do velocity diffusion and advection • Find “closest” vector field that is divergence-free • Need to calculate divergence • Need to find and use pressure

34. uyij+1 ? -uxi-1j uxi+1j -uyij-1 Measuring Divergence u=? uij=(uxi+1j - uxi-1j) + (uyij+1-uyij-1)

35. zero Pressure Term unew= u – p Take divergence of both sides…  unew=  u – p  u = 2p

36. 1 known unknown 1 1 -4 1 Pressure Term  u = 2p pnew = p + e( u - 2p) Let dij =  uij pnewij = pij + e(dij - (pi-1j + pi+1j + pij-1 + pij+1 - 4pij))

37. Pressure Term unew= u – p …and velocity is now divergence-free Found “nearest” divergence-free vector field to original.

38. Fluid Simulator Diffuse velocity Advect velocity Add body forces (e.g. gravity) Pressure projection Diffuse dye/smoke Advect dye/smoke

39. “Real-Time Fluid Dynamics for Games” Jos Stam, March 2003 (CDROM link is to source code) www.dgp.toronto.edu/people/stam/reality/Research/pubs.html

40. Rigid Objects • Want rigid objects in fluid • Use approach similar to pressure projection “Rigid Fluid: Animating the Interplay Between Rigid Bodies and Fluid” Mark Carlson, Peter J. Mucha and Greg Turk Siggraph 2004

41. Rigid Fluid Method 1) Solve Navier-Stokes on entire grid, treating solids exactly as if they were fluid 2) Calculate forces from collisions and relative density 3) Enforce rigid motion for cells inside rigid bodies

43. Small-scale liquid-solid Interactions What makes large water and small water behave differently? Surface Tension (water: 72 dynes/cm at 25º C) Viscosity (water: 1.002 x 10-3 N·s/m2 at 20º C) Lake ( >1 meter) Water drops (millimeters)

44. Surface Tension Normal (always pointing outward) Surface Tension Force

45. Water/Surface Contact hydrophillic hydrophobic

46. Contact Front

47. Virtual Liquid Virtual Surface Virtual Surface Method Air Liquid Solid

48. Air Liquid Virtual Liquid Solid Virtual Surface Advancing to right:qc>qs Virtual Surface Method 1) Creating a virtual surface 2) Estimate curvature from new fluid surface 3) Kink will “push” the fluid towards stable contact angle

49. Air Virtual Liquid Solid Virtual Surface Receding to left:qc<qs Virtual Surface Method 1) Creating a virtual surface 2) Estimate curvature from new fluid surface 3) Kink will “push” the fluid towards stable contact angle