Computer Animation Algorithms and Techniques

1. Computer AnimationAlgorithms and Techniques Computational Fluid Dynamics

2. Fluids conservation of mass momentum energy types viscous v. non-viscous compressible v. incompressible properties characteristics velocity laminar flow density turbulent pressure steady state flow heat gravity

3. d d d d d d d d d d d d d d d d Models grid-based (Eulerian) particle-based (Lagrangian) Hybrid

4. Notation Gradient Divergence

5. Divergence Theorem The integral of the flow field’s divergence over the volume of the cell is the same as the intgral of the flow field over the cell’s boundary

6. Conservation of Mass dx

7. Conservation of Mass If incompressible:

8. Conservation of Momentum in x dy dx

9. Conservation of Momentum

10. Velocity: u,v Coordinates: x,y Density: Pressure: p Euler Equations - 2D

11. Euler Equations - incompressible form

12. Stable Fluids Velocity grid Density grid Move densities around using velocity grid and dissipate densities Move velocities around using velocity grid and dissipate velocities

13. Stable Fluids Source Force Dissipation Advection