1. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Introduction to Modeling Fluid Dynamics 1

2. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 2 Different Kind of Problem Can be particles, but lots of them Solve instead on a uniform grid

3. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 3 No Particles => New State Particle Mass Velocity Position Fluid Density Velocity Field Pressure Viscosity

4. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 4 No Particles => New Equations Navier-Stokes equations for viscous, incompressible liquids.

5. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 5 What goes in must come out Gradient of the velocity field= 0 Conservation of Mass

6. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 6 Time derivative Time derivative of velocity field Think acceleration

7. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 7 Advection term Field is advected through itself Velocity goes with the flow

8. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 8 Diffusion term Kinematic Viscosity times Laplacian of u Differences in Velocity damp out

9. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 9 Pressure term Fluid moves from high pressure to low pressure Inversely proportional to fluid density, ρ

10. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 10 External Force Term Can be or represent anythying Used for gravity or to let animator “stir”

11. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 11 Navier-Stokes How do we solve these equations?

12. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 12 Discretizing in space and time We have differential equations We need to put them in a form we can compute Discetization – Finite Difference Method

13. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 13 Discretize in Space X Velocity Y Velocity Pressure Staggered Grid vs Regular

14. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 14 Discretize the operators Just look them up or derive them with multidimensional Taylor Expansion Be careful if you used a staggered grid

15. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 15 Example 2D Discetizations 01 1 1-41 1 1 Divergence Operator Laplacian Operator

16. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 16 Make a linear system It all boils down to Ax=b.

17. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 17 Simple Linear System Exact solution takes O(n 3 ) time where n is number of cells In 3D k 3 cells where k is discretization on each axis Way too slow O(n 9 )

18. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 18 Need faster solver Our matrix is symmetric and positive definite….This means we can use ♦ Conjugate Gradient Multigrid also an option – better asymptotic, but slower in practice.

19. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 19 Time Integration Solver gives us time derivative Use it to update the system state U(t+Δt) UtUt U(t)

20. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 20 Discetize in Time Use some system such as forward Euler. RK methods are bad because derivatives are expensive Be careful of timestep

21. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 21 Time/Space relation? Courant-Friedrichs- Lewy (CFL) condition Comes from the advection term

22. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 22 Now we have a CFD simulator We can simulate fluid using only the aforementioned parts so far This would be like Foster & Metaxas first full 3D simulator What if we want it real-time?

23. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 23 Time for Graphics Hacks Unconditionally stable advection ♦ Kills the CFL condition Split the operators ♦ Lets us run simpler solvers Impose divergence free field ♦ Do as post process

24. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 24 Semi-lagrangian Advection CFL Condition limits speed of information travel forward in time Like backward Euler, what if instead we trace back in time? p(x,t) back-trace

25. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 25 Divergence Free Field Helmholtz-Hodge Decomposition ♦ Every field can be written as w is any vector field u is a divergence free field q is a scalar field

26. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 26 Helmholtz-Hodge STAM 2003

27. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 27 Divergence Free Field We have w and we want u Projection step solves this equation

28. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 28 Ensures Mass Conservation Applied to field before advection Applied at the end of a step Takes the place of first equation in Navier-Stokes

29. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 29 Operator Splitting We can’t use semi-lagrangian advection with a Poisson solver We have to solve the problem in phases Introduces another source of error, first order approximation

30. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 30 Operator Splitting

31. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 31 Operator Splitting 1.Add External Forces 2.Semi-lagrangian advection 3.Diffusion solve 4.Project field

32. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 32 Operator Splitting W0W0 W1W1 W2W2 W3W3 W4W4 u(x,t) u(x,t+Δt)

33. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 33 Various Extensions Free surface tracking Inviscid Navier-Stokes Solid Fluid interaction

34. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 34 Free Surfaces Level sets ♦ Loses volume ♦ Poor surface detail Particle-level sets ♦ Still loses volume ♦ Osher, Stanley, & Fedkiw, 2002 MAC grid ♦ Harlow, F.H. and Welch, J.E., "Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with a Free Surface", The Physics of Fluids 8, 2182-2189 (1965).

35. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 35 Free Surfaces + - ++ ++ + + + + ++ + + + + + + + + + ++ + + + + + + + + + + + + + - - - - - + 0 0 - - - - - - - - -- - - - - - - - - - - - -- - - - - - - - - -- -- - MAC GridLevel Set

36. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 36 Inviscid Navier-Stokes Can be run faster Only 1 Poisson Solve needed Useful to model smoke and fire ♦ Fedkiw, Stam, Jensen 2001

37. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 37 Solid Fluid Interaction Long history in CFD Graphics has many papers on 1 way coupling ♦ Way back to Foster & Metaxas, 1996 Two way coupling is a new area in past 3-4 years ♦ Carlson 2004

38. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 38 Where to get more info Simplest way to working fluid simulator (Even has code) ♦ STAM 2003 Best way to learn enough to be dangerous ♦ CARLSON 2004

39. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 39 References CARLSON, M., “Rigid, Melting, and Flowing Fluid,” PhD Thesis, Georgia Institute of Technology, Jul. 2004. FEDKIW, R., STAM, J., and JENSEN, H. W., “Visual simulation of smoke,” in Proceedings of ACM SIGGRAPH 2001, Computer Graphics Proceedings, Annual Conference Series, pp. 15–22, Aug. 2001. FOSTER, N. and METAXAS, D., “Realistic animation of liquids,” Graphical Models and Image Processing, vol. 58, no. 5, pp. 471–483, 1996. HARLOW, F.H. and WELCH, J.E., "Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with a Free Surface", The Physics of Fluids 8, 2182-2189 (1965). LOSASSO, F., GIBOU, F., and FEDKIW, R., “Simulating water and smoke with an octree data structure,” ACM Transactions on Graphics, vol. 23, pp. 457–462, Aug. 2004. OSHER, STANLEY J. & FEDKIW, R. (2002). Level Set Methods and Dynamic Implicit Surfaces. Springer- Verlag. STAM, J., “Real-time fluid dynamics for games,” in Proceedings of the Game Developer Conference, Mar. 2003.