1. Visual Simulation of Smoke SIGGRAPH’01 Ronald Fedkiw, Jos Stam and Henrik Wann Jensen Stanford University & Alias|wavefront

2. Outline Introduction Previous Work The Equations of Fluid Flow Self-Advection by Semi-Lagrangian Numerical Dissipation Results

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

4. Previous Work Kajiya and Von Herzen (SIGGRAPH 84) N. Foster and D. Metaxas (SIGGRAPH 97) Improve Stam’99 (Stable Fluids)  Handle moving boundaries  Reduce numerical dissipation  Add high quality volume rendering Method still fast but looks more “smoke-like”

5. The Equations of Fluid Flow Review: Navier-Stokes Equations

6. Incompressible Euler Equations self-advectionforces incompressible (Navier-Stokes without viscosity) Fluid velocity: u=

7. Gradient Implement

8. Additional Equations Density Temperature Buoyancy force

9. Algorithm t=0t = t + dt add forces self-advection project Helmholtz-Hodge Decomposition

10. Self-Advection tt+dt Semi-Lagrangian solver (Courant, Issacson & Rees 1952)

11. Semi- Lagrangian The semi-Lagrangian integrator itself is broken down into three main parts  1. Vector field extraction  2. Particle tracing  3. Interpolation

12. Self-Advection tt+dt Semi-Lagrangian solver (Courant, Isaacson & Rees 1952)

13. Self-Advection tt+dt For each u-component…

14. Self-Advection tt+dt For each u-component…

15. Self-Advection tt+dt Set interpolated value in new grid

16. Self-Advection Repeat for all u-nodes

17. Self-Advection Repeat for all v-nodes

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

19. Vorticity Confinement Basic idea:  Add energy lost as an external force Use “Vorticity Confinement” force  invented by John Steinhoff 1994

20. Vorticity Confinement  : vorticity

21. Vorticity Confinement Vorticity location vector :  Normalized vorticity Location vector : N

22. Vorticity Confinement Spatial discretization: h

23. Total Forces User supplied fields Buoyancy force New confinement force

24. Results

26. Any Problem ?