Simplified Smoothed Particle Hydrodynamics for Interactive Applications Zakiya Tamimi Richard McDaniel Based on work done at Siemens Corporate Research, Princeton NJ, Jun 2010-Jun 2011

2 1. Review (Numerical Methods) In a discrete world: approximate derivative as a difference. Given an initial value and rate of change, calculate next value.

3 1. Example (Bouncing Ball) x0x0 x1x1 xixi x2x2 F G m

4 1. Simple Animation Example Need to detect collision with the ground at each time step). Collision happens if distance between x(t+1) and ground less than ball radius. Render animation after calculating position of center of ball at each time step.

5 2. Computer Graphics

6 Outline 1.Review 2.Definitions (CFD) 3.Background (SPH) 4.Problem 5.Solution 6.Algorithm 7.Results 8.Conclusion

7 2. Definitions (CFD) CFD = Computational Fluid Dynamics Fluids {liquids, gas} U {granular materials, fire} advection = integration of velocity and positions (moving fluids) Internal forces {pressure force, viscosity force} External forces {gravity, surface tension, buoyancy forces, other} Models {grid based, particle based} Applications {animation, games, scientific computation, weather, aviation}

8 3. Background (SPH) SPH = Smoothed Particle Hydrodynamics Smoothing = approximating function using discrete values. Based on Monte Carlo sampling (statistics) Particles are samples that move with flow Kernel function: a particle is affected only by particles within its neighborhood 8

9 3. SPH Basics |r|=h |r|=0|r|=h W(r,h)

10 3. Kernel-Based Calculations

11 3. Traditional SPH Algorithm i.for all particles calculate density (interpolation) i.for all particles calculate internal forces (pressure + viscosity) i.for all particles integrate velocity and position i.for all particles detect and resolve collision (render frame)

12 3. SPH Calculations

13 4. Problem More particles  better results  computationally expensive  slow. Order of computation is O(n 2 ) Rendering is done offline  not interactive Ultimate goal is real time What is real-time? What is interactive? Industrial applications required real-time.

14 4. Neighborhood Search

15 4. Improving Neighborhood Search

16 5. Solution Introduce two levels of granularity {particle-level, cell-level} Large particles (centroid) properties are based on properties of particles within one cell Small particles (fluid) calculations between cells are based on large particles.

17 6. Density Calculations

18 6. Algorithm Initialize centroid particle {position, radius, mass} Calculate centroid density using fluid particles within same cell. Calculate fluid particle density using fluid particles within same cell and centroid density in neighboring cells. Centroid particles don’t move Discarded at the end of each cycle

19 6. Comparison Position Radius Smoothing Radius Density

20 6. Simplified SPH Algorithm i.for all non-empty cells initialize centroid particles calculate centroid particle density ii.for all particles calculate density (modified) iii.for all particles calculate internal forces (pressure + viscosity) iv.for all particles integrate velocity and position v.for all particles detect and resolve collision

21 6. Complexity n is number of particles Without grid O(n 2 ) m is average number of particles per cell With grid O(mn)  O(27mn) Our method O(mn)  O(mn+26) Although our algorithm has same complexity, in practice it runs faster. Our force calculations are still same as traditional SPH. Thus overall computation is bounded by O(mn)

22 7. Results Traditional SPH Simplified SPH

23 Traditional SPH Simplified SPH

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26 8. Conclusion Presented an approximate SPH based fluid simulation that is faster than traditional SPH. The presented algorithm allows interactive simulation speed. Future work: employ centroid particles in force calculations.