Fluid Simulation

1. Fluid Simulation

2. Contents • Introduction • Diffusion • Semi-Lagrangian advection • Poisson Equation • Implementation

3. Motivation Fedkiw et al. 2001

4. Fluids in Computer Graphics • Fast • Looks good • Easy to code

5. Fluid Mechanics • Natural framework for fluid modeling • Full Navier-Stokes Equations • Has a long history • Reuse code/algorithms • Equations are hard to solve • Non-linear

6. Application • Use velocity to move densities

7. Equations • Density field • Velocity field

8. Finite Difference Scheme

9. Equations • Evolution of density (assume velocity known) Over a time step...

10. Equations • Evolution of density (assume velocity known) Density changes in the direction of the flow

11. Equations • Evolution of density (assume velocity known) Density diffuses over time

12. Equations • Evolution of density (assume velocity known) Increases due to sources from the UI

13. Algorithm Subdivide space into voxels Velocity + density defined in the center of each voxel

14. Algorithm add source diffuse move

15. Diffusing Densities dt Dn Dn+1

16. Diffusing Densities Exchange of density between neighbors

17. Diffusing Densities Exchange of density between neighbors

18. Diffusing Densities i,j+1 i-1,j i,j i+1,j i,j-1 Dn+1i,j = Dn i,j + k dt (Dn i-1,j + Dn i+1,j + Dn i,j-1 + Dn i,j+1 - 4Dn i,j)/h2

19. Algorithm add source diffuse move

20. Moving Densities dt Velocity known

21. Moving Densities • Finite differences • Transfer only between neighbors • Unstable with large time steps

22. Semi-Lagrangian Advection

23. Semi-Lagrangian Advection

24. Semi-Lagrangian Advection

25. Semi-Lagrangian Advection Interpolate the density at new location

26. Semi-Lagrangian Advection Set interpolated density at grid location Requires two grids

27. Computing Velocities Use same algorithms as for density Velocity is moved by itself

28. Moving Velocity Trace particle backwards in time

29. Moving Velocity Interpolate the velocity at new location

30. Moving Velocity Set interpolated velocity at grid location Requires two grids

31. Conservation of Mass Inflow = Outflow Ui+1,j - Ui-1,j + Vi,j+1 - Vi,j-1 = 0

32. Poisson Equation • Poisson Equation • Step 1 • Step 2

33. /* Running */ while(simulating) { velocityfield.advect(dt); velocityfield += gravityfield*(densityfield*(dt*10.0f)); fvelocityfield = velocityfield; fvelocityfield.removeDivergence(); velocityfield = fvelocityfield; densityfield.advect(velocityfield, dt); } Implementation

34. Implicit Diffusion

35. Implicit Diffusion • Explicit diffusion • Dn+1i,j = Dn i,j + k dt (Dn i-1,j + Dn i+1,j + Dn i,j-1 + Dn i,j+1 - 4Dn i,j)/h2 • Implicit diffusion • Dn+1i,j – k dt (Dn+1i-1,j+Dn+1i+1,j+Dn+1i,j-1+Dn+1 i,j+1-4Dn+1 i,j)/h2 = Dni,j A x = b

36. Diffusing Densities Linear solvers: Name Cost Comments Gaussian elimination N3 Use only for very small N (test code) Jacobi/SOR relaxation N2 Easy to code but slow FFT/cyclical reduction N logN Use when no internal boundaries Conjugate gradient N1.5 Use when internal boundaries Multi-grid N Slower than FFT in practice. Hard to code when internal boundaries present

37. Mass Conservation- Numerical Example-

38. Given Velocity Field 10.0f

39. Poisson Equation

40. Solution

41. Superposition + 10.0f 5.83

42. Divergence-Free Velocity Field