1. Derek C. Richardson (U Maryland) PKDGRAV : A Parallel k-D Tree Gravity Solver for N-Body Problems FMM 2004

2. Overview High-performance cosmology code. High-performance cosmology code. Written at U Washington N-body shop by Joachim Stadel (U Zurich) & Tom Quinn. Written at U Washington N-body shop by Joachim Stadel (U Zurich) & Tom Quinn. Now adapted for a variety of applications, including planet formation. Now adapted for a variety of applications, including planet formation. Lastest version has implementation of SPH (smoothed particle hydrodynamics). Lastest version has implementation of SPH (smoothed particle hydrodynamics).

3. Sample Movie

4. Spatial Binary Tree Ironically, PKDGRAV no longer uses a k-D tree for gravity… Ironically, PKDGRAV no longer uses a k-D tree for gravity… The problem: elongated cells can give large multipole moments and other pathologies. The problem: elongated cells can give large multipole moments and other pathologies. The solution: spatially bisect cells and squeeze the cell boundaries. The solution: spatially bisect cells and squeeze the cell boundaries. Also good for neighbour searching. Also good for neighbour searching.

5. Spatial Binary Tree k-D Treek-D with Squeeze

6. Tree Walking Construct particle-particle and particle-cell interaction lists from top down. Construct particle-particle and particle-cell interaction lists from top down. Define opening ball (based on  ) to test for ball-bucket intersection. Define opening ball (based on  ) to test for ball-bucket intersection. – If bucket outside ball, apply multipole (c-list). – Otherwise open cell and test its children, etc., until leaves reached (which go on p-list). Nearby cells have similar lists: amortize. Nearby cells have similar lists: amortize.

7. Tree Walking Note multipole Q acceptable to all particles in cell d.

8. Other Issues Multipole expansion order. Multipole expansion order. – Use hexadecapole (best bang for buck). Force softening. Force softening. – Use spline-softened gravity kernel. Periodic boundary conditions. Periodic boundary conditions. – Use Ewald summation technique. Time steps. Time steps. – Use hierarchical steps (adaptive leapfrog).

9. Parallel Implementation 1.Master layer (serial). –Controls overall flow of program. 2.Processor Set Tree (PST) layer (parallel). –Assigns tasks to processors. 3.Parallel k-D (PKD) layer (serial). –MIMD execution of tasks. 4.Machine-dependent Layer (MDL, separate). –Interface to parallel primitives.

10. Domain Decomposition Binary tree balanced by work factors. Nodes construct local trees. PST

11. Scaling at Fixed Accuracy Clustered cosmology simulation (N = 3·10 6 ) (  = 0.8)

12. Some Science: Planetesimals One of the recent additions to PKDGRAV is the ability to treat particle collisions. One of the recent additions to PKDGRAV is the ability to treat particle collisions. Applications: Applications: – Planet formation (planetesimal growth). – Asteroid disruption. – Planetary rings. Technique: predict collision during leapfrog drift step using fast neighbour search. Technique: predict collision during leapfrog drift step using fast neighbour search.

13. Planetesimal Evolution Evolutionary models based on laboratory experiments may be inadequate. Evolutionary models based on laboratory experiments may be inadequate. – Outcomes determined by gravitational reaccumulation. Leinhardt et al. 2000

14. Incorporating Collision Models Based on individual test cases, collision outcomes can be incorporated into large-scale sims of planet formation. Based on individual test cases, collision outcomes can be incorporated into large-scale sims of planet formation.

15. Near-Earth Asteroid Binaries Reaccumulation following tidal disruption could form NEA binaries. Reaccumulation following tidal disruption could form NEA binaries. Need reaccumulated (or shattered/fragile) body to start with… Need reaccumulated (or shattered/fragile) body to start with…

16. New Code: Rigid Aggregates Recently the ability to treat collections of particles as rigid aggregates has been added. Recently the ability to treat collections of particles as rigid aggregates has been added. Applications include realistic asteroid shape models and more complex granular shapes. Applications include realistic asteroid shape models and more complex granular shapes. Technique: solve Euler rigid body equations of motion with torque and allow for non- central impacts. Technique: solve Euler rigid body equations of motion with torque and allow for non- central impacts.

17. Rigid Aggs: Movie 1

18. Rigid Aggs: Movie 2 Bouncing Allowed No Bouncing

19. Rigid Aggs: Movie 3

20. Rigid Aggs: Movie 4

21. Future Work PKDGRAV continues to be improved. PKDGRAV continues to be improved. Algorithms currently being considered: Algorithms currently being considered: – Local expansions. – Tree repair. – Improved collision detection. Other possibilities include combining PKDGRAV with a Grape board!… Other possibilities include combining PKDGRAV with a Grape board!…