2. Motivation  Movie  Game  Engineering

3. Introduction  Ideally  Looks good  Fast simulation  Looks good?  Look plausible  Doesn’t need to be exactly correct  doesn’t suffer from excessive numerical dissipation

4. Overview  Simulation  Compressible/Imcompressible Equation  Vortex particle method  Using octree data structure  Rendering  Thick smoke: plain particles  Thin smoke: adaptive particles  Using Compensated Ray Marching

5. Compressible Fluids  Yngve, G. D., O'Brien, J. F., Hodgins, J. K., 2000, Animating Explosions. The proceedings of ACM SIGGRAPH 2000, New Orleans, July 23-28, pp. 29-36.Animating Explosions.

6. Operator Splitting  Stam used the technique in his Stable Fluids paper, 1999.Stable Fluids  Allows us to solve the Navier-Stokes equation in Parts.  As long as each part is stable the technique will be stable.

7. Semi-Lagrangian Convection  Trace each point p in the field backwards in time. The new velocity at x is therefore the velocity that the particle had a time step ago at the old location.

8. Smoke Viscous Diffusion  In the simulation of smoke, it is reasonable to consider the fluid inviscid. Therefore this term is zero and need not be solved for.  In fact, the implicit solver that is used will take energy away from the system anyway, so there is a numerical dampening that can look like a non-zero viscosity.

9. Body Forces  Gravity, user forces, and object interaction.  The visible smoke is from a density field.  Areas of high density fall in the direction of gravity.  Heat moves against gravity (hot air rises).

10. Incompressibility  At each time step we start with a velocity field that is divergence free. We need this to be true at the end of the time step because the fluid is incompressible.  After solving for convection, and body forces the velocity field is not divergence free.

11. Incompressibility  Fedkiw, R., Stam, J. and Jensen, H.W., "Visual Simulation of Smoke", SIGGRAPH 2001, 23-30 (2001).Visual Simulation of Smoke

12. Incompressible Euler Equations self-advection Forces incompressible (Navier-Stokes without viscosity)

13. Additional Equations smoke’s density temperature We now know all the finite differences we need to add the forces for heat and density…

14. Vorticity Confinement  The numerical dissipation, and the coarse grid size, cause the fine scale detail of turbulent swirling smoke to vanish.  Identify where the curl is highest, and add back in a rotational force there.  “Vorticity Confinement” force preserves  swirling nature of fluids.

15. What is vorticity?  A measure of the local rotation in a fluid flow.

16. Pressure Boundary Conditions  A Neumann boundary condition is a restriction on the derivative of a function.

17. Velocity Boundary Condition  A Dirichlet boundary condition is a restriction on the value of a function.

18. Vorticity Confinement

21. Solving the System  Need to calculate:  Start with initial state  Calculate new velocity fields  New state:

22. Smoke Simulation  While (simulating)  Get external forces (if any) from UI  Get density/heat sources (from UI or init grid cells)  Update velocity  Update density  Update temperature  Display density

23. Smoke Simulation  While (simulating)  Get external forces (if any) from UI  Get density/heat sources (from UI or init grid cells)  Update velocity  Update density  Update temperature  Display density

24. Update Velocity  Equation:  First term:  Advection  Move the fluid through its velocity field (Du/Dt=0)  Second term:  external forces  Final term:  pressure update

25. Update Velocity  Add external forces (fbuoy + forces from UI + fconf)  Advection (semi-lagrangian step – trace particles back)  Pressure update (solve linear system)

26. Smoke Simulation  While (simulating)  Get external forces (if any) from UI  Get density/heat sources (from UI or init grid cells)  Update velocity  Update density  Update temperature  Display density

27. Update density  Equation  First term:  Advection  Move the temperature through velocity field  Second term:  Diffusion  Can skip this term  Second term:  external sources

28. Update density  Add Sources (pick grid cells or from UI)  Advection (semi-lagrangian step – trace particles back)  Diffusion (solve linear system – can skip this step)

29. Smoke Simulation  While (simulating)  Get external forces (if any) from UI  Get density/heat sources (from UI or init grid cells)  Update velocity  Update density  Update temperature  Display density

30. Update temperature  Equation  First term:  Advection  Move the temperature through velocity field  Second term:  Diffusion  Can skip this term

31. Update Temperature  Add Sources (grid cells or objects or UI)  Advection (semi-lagrangian step – trace particles back)  Diffusion (solve linear system – can skip this step)

32. Smoke Simulation  While (simulating)  Get external forces (if any) from UI  Get density/heat sources (from UI or init grid cells)  Update velocity  Update density  Update temperature  Display density

33. Display density  Use any approach you want  Visualize the density field:  just opengl render a bunch of cubes (corresponding to grid cells) that have alpha values based on how dense the smoke is.

34. Numerical Dissipation  ‘Stable Fluids’ method dampens the flow  Typical with semi- Lagrangian methods  Improve using  “Vorticity Confinement” force

35. Total Forces  User supplied fields  Buoyancy force  New confinement force

36. Results

38. Other Methods  Vortex particle method  Using octree data structure

39. Vortex particle method -Lagrangian primitives  Curves carry the vorticity  Each local vortex induces a weighted rotation

40. Vortex particle method -Lagrangian primitives  Curves carry the vorticity  Each local vortex induces a weighted rotation

41. Method of simulation  Vortex particles (for motion) organized as curves. = tangent  Smoke particles (for visualisation)  Curves carry vortices  Vortices induce a velocity field  velocity field deforms curves & smoke At every step:  Advect the curves  Stretch  Advect the smoke

42. Method of simulation  Vortex particles (for motion) organized as curves. = tangent  Smoke particles (for visualisation)  Curves carry vortices  Vortices induce a velocity field  velocity field deforms curves & smoke At every step:  Advect the curves  Stretch  Advect the smoke

43. Method of simulation  Vortex particles (for motion) organized as curves. = tangent  Smoke particles (for visualisation)  Curves carry vortices  Vortices induce a velocity field  velocity field deforms curves & smoke At every step:  Advect the curves  Stretch  Advect the smoke

44. Video

45. Smoke Rendering  Thick smoke: plain particles  Thin smoke: adaptive particles [AN05]  accumulate stretching

46. Smoke Rendering  Thin smoke behaves like a surface [ William Brennan ] e n l

47. Smoke Rendering  Using Compensated Ray Marching

48. References