1 / 19

Stable Fluids

Stable Fluids. A paper by Jos Stam. Contributions. Real-Time unconditionally stable solver for Navier-Stokes fluid dynamics equations Implicit methods allow for large timesteps Excessive damping – damps out swirling vortices Easy to implement Controllable (?). Some Math(s).

zeal
Download Presentation

Stable Fluids

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Stable Fluids A paper by Jos Stam

  2. Contributions • Real-Time unconditionally stable solver for Navier-Stokes fluid dynamics equations • Implicit methods allow for large timesteps • Excessive damping – damps out swirling vortices • Easy to implement • Controllable (?)

  3. Some Math(s) • Nabla Operator: • Laplacian Operator: • Gradient:

  4. More Math(s) • Vector Gradient: • Divergence: • Directional Derivative:

  5. Navier-Stokes Fluid Dynamics • Velocity field u, Pressure field p • Viscosity v, density d (constants) • External force f • Navier-Stokes Equation: • Mass Conservation Condition:

  6. Navier-Stokes Equation • Derived from momentum conservation condition • 4 Components: • Advection/Convection • Diffusion (damping) • Pressure • External force (gravity, etc)

  7. Mass Conservation Condition • Velocity field u has zero divergence • Net mass change of any sub-region is 0 • Flow in == flow out • Incompressible fluid • Comes from continuum assumption

  8. Enforcing Zero Divergence • Pressure and Velocity fields related • Say we have velocity field w with non-zero divergence • Can decompose into • Helmholtz-Hodge Decomposition • u has zero divergence • Define operator P that takes w to u: • Apply P to Navier-Stokes Equation: • (Used facts that and )

  9. Operator P • Need to find • Implicit definition: • Poisson equation for scalar field p • Neumann boundary condition • Sparse linear system when discretized

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

  11. Step 1 – Add Force • Assume change in force is small during timestep • Just do a basic forward-Euler step • Note: f is actually an acceleration?

  12. Step 2 - Advection

  13. Method of Characteristics • p is called the characteristic • Partial streamline of velocity field u • Can show u does not vary along streamline • Determine p by tracing backwards • Unconditionally stable • Maximum value of w2 is never greater than maximum value of w1

  14. Step 3 – Diffusion • Standard diffusion equation • Use implicit method: • Sparse linear system

  15. Step 4 - Projection • Enforces mass-conservation condition • Poisson Problem: • Discretize q using central differences • Sparse linear system • Maybe banded diagonal… • Relaxation methods too inaccurate • Method of characteristics more precise for divergence-free field

  16. Complexity Analysis • Have to solve 2 sparse linear systems • Theoretically O(N) with multigrid methods • Advection solver is also O(N) • However, have to take lots of steps in particle tracer, or vortices are damped out very quickly • So solver is theoretically O(N) • I think the constant is going to be pretty high…

  17. Periodic Boundaries • Allows transformation into Fourier domain • In Fourier domain, nabla operator is equivalent to ik • New Algorithm: • Compute force and advection • Transform to Fourier domain • Compute diffusion and projection steps • Trivial because nabla is just a multiply • Transform back to time domain

  18. Diffusing Substances • Diffuse scalar quantity a (smoke, dust, texture coordinate) • Advected by velocity field while diffusing • kais diffusion constant, da is dissipation rate, Sa is source term • Similar to Navier-Stokes • Can use same methods to solve equations, Except dissipation term

  19. The End

More Related