Presentation is loading. Please wait.

Presentation is loading. Please wait.

Stable Fluids A paper by Jos Stam.

Similar presentations

Presentation on theme: "Stable Fluids A paper by Jos Stam."— 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 ka is 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

Download ppt "Stable Fluids A paper by Jos Stam."

Similar presentations

Ads by Google