1. Matthew Zhu

3. At each time step calculate each particle’s acceleration and use Verlet numerical integration to update its position, velocity, and grid cell Particles — Sequential Particle Interactions Particle objects have data fields such as:  Density, pressure,  position, velocity, acceleration, and force.  For each grid cell:  Retrieve all particles in cell  For each particle:  Find interacting particles within distance H  Calculate inter-particle viscosity forces, pressure forces, surface tension forces, external forces, properties, and acceleration  Account for each neighbor particle’s contribution to density  Calculate pressure from density

4. Trivially Parallelizable SPH Simulation  Each thread accesses a particle,  finds the particle’s neighbors,  and computes their force contributions.

5. Symmetrizing Particle Interactions

6. Accounting for Symmetric Contributions A data race occurs if two particles try to access each other simultaneously  Each thread accesses a particle,  finds the particle’s neighbors,  and applies forces to both the particle and its neighbors.

7. Dividing Workload Particles keep track of grid cells and z-index Particle array sorted by z-indices Apply cell-fitting procedure

8. Two-dimensional Coloring Algorithm Interaction Pattern On cell (0, 0) Chromatic Number: 4  Interactions of a single color may be processed in parallel without data races  Four passes are required to process all interactions

9. Generalization to Three Dimensions (0,0,0) (1,0,0) (1,0,1) (0,0,1) (1,1,0) (0,1,0) (0,1,1) (1,1,1) 3D Interaction Pattern  To determine the color of a cell: Chromatic Number: 8  (x mod 2, y mod 2, z mod 2) yields one of eight possible colors  For example,the grid cell at (9, 9, 8) is assigned the color red

10. Simulation Algorithm — Particle Interactions (9,9,8) (10,9,8) (10,9,9) (9,9,9) (9,10,8) (9,10,9) Thread 1 Local Interaction ListColor Interaction List[red]  For each local cell:  For each pair of cells in the interaction pattern: (pattern includes interactions within a cell represented by dot) (9,9,8)  If cells are local to thread add pair to local interaction list  Otherwise add pair to interaction list of the cell’s color (9,9,8)(9,9,9) (9,9,8)(10,9,9) (9,9,8)(10,10,9) (9,9,9)(9,10,8) (9,9,8)(10,9,8) (9,9,8)(9,10,9)(9,9,8)(9,10,8) (9,9,9)(10,10,8) (9,9,8)(10,10,8) (9,9,9)(10,9,8) (9,10,8)(10,9,9) (9,10,9)(10,9,8) (9,10,8)(10,9,8)

11. Simulation Algorithm — Physics Computation (9,9,8) (9,9,9) (9,9,8)(10,9,9) (9,9,8)(10,10,9) (9,9,9)(9,10,8) (9,9,8)(10,9,8) (9,9,8)(9,10,9) (9,9,8)(9,10,8) (9,9,9)(10,10,8) (9,9,8)(10,10,8) (9,9,9)(10,9,8) (9,10,8)(10,9,9) (9,10,9)(10,9,8) (9,10,8)(10,9,8)

12. References [1] P. Goswami et al., “Interactive SPH Simulation and Rendering on the GPU,” in ACM SIGGRAPH Symposium on Computer Animation., 2010, pp. 1-10. [2] Y. R. López and D. Roose, “A Parallel SPH Implementation on Shared Memory Systems,” in 9th International SPHERIC Workshop., Paris, France, 2014, pp. 1-8. [3] M. Müller et al., “Particle-Based Fluid Simulation for Interactive Applications,” in SIGGRAPH Symposium on Computer Animation., 2003, pp. 1-7.