1. Particle-based fluid simulation for interactive applications
Matthias Müller
David Charypar
Markus Gross
陳岳澤

2. Outline Introduction Navier-Stokes Equation
SPH (Smoothed Particle Hydrodynamics )
Smooth Kernel
Marching Cubes
Result

3. Introduction
Navier-Stokes Equation describe the motion of fluid substances such as liquids and gases
Use Smoothed Particle Hydrodynamics (SPH) to simulate fluids with free surfaces.
Interactive simulation (about 5 fps).

4. Navier-Stokes Equation -1
Conservation of momentum equation
Three components:
Pressure term
External force term
Viscosity term
v: velocity, : density, p: pressure,
g: external force, : viscosity coefficient

5. Navier-Stokes Equation -2
The acceleration ai of particle i is
(fi is body force)
Using ai , we can get velocity and position of particle i

6. SPH -1 Originally developed for astrophysical problems (1977).
Interpolation method for particles.
Properties that are defined at discrete particles can be evaluated anywhere in space.
Uses smoothing kernels to distribute quantities.

7. SPH -2 Smoothing of attribute A mj: mass rj : density
Aj: quantity to be interpolated
W: smoothing kernel h

8. Particle density
Smoothing of attribute A
Particle density

9. Pressure Term
Navier-Stokes Equation
Pressure Term

10. Viscosity term
Navier-Stokes Equation
Viscosity Term

11. External force term
Other external forces are directly applied to the particles.
Collisions: In case of collision the normal component of the velocity is flipped.

12. Smoothing Kernel -1
Has an impact on the stability and speed of the simulation.
ex: Avoid square-roots for distance computation.
Sample smoothing kernel:

13. Smoothing Kernel -2
all points inside a radius of ‘h’ are considered for “smoothing”.
Thick line: the kernel
Thin line: the gradient of kernel
Dashed line: the laplacian of kernel

14. Smoothing Kernel -3 For n particles n2 potential interactions!
To reduce to linear complexity O(n2) define interaction cutoff distance h

15. Smoothing Kernel -4 Fill particles into grid with spacing h
Only search potential neighbors in adjacent cells

16. Marching Cubes -1
To visualize the free surface

17. Marching Cubes -2

18. Result Interactive Simulation (5fps) Marching Cubs 2200 particle
Point Splatting