Derek%20C.%20Richardson%20(U%20Maryland)

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PKDGRAV A Parallel k-D Tree Gravity Solver for
N-Body Problems

• Derek C. Richardson (U Maryland)

FMM 2004
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Overview

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

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Sample Movie
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Spatial Binary Tree

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

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Spatial Binary Tree
k-D Tree
k-D with Squeeze
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Tree Walking

• Construct particle-particle and particle-cell
  interaction lists from top down.
• 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.

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

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

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Parallel Implementation

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

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Domain Decomposition
PST
Binary tree balanced by work factors. Nodes
construct local trees.
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Scaling at Fixed Accuracy
Clustered cosmology simulation (N 3106) (?
0.8)
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Some Science Planetesimals

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

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Planetesimal Evolution

• Evolutionary models based on laboratory
  experiments may be inadequate.
• Outcomes determined by gravitational
  reaccumulation.

Leinhardt et al. 2000
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Incorporating Collision Models

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

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Near-Earth Asteroid Binaries

• Reaccumulation following tidal disruption could
  form NEA binaries.
• Need reaccumulated (or shattered/fragile) body to
  start with

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New Code Rigid Aggregates

• 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.
• Technique solve Euler rigid body equations of
  motion with torque and allow for non-central
  impacts.

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Rigid Aggs Movie 1
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Rigid Aggs Movie 2
Bouncing Allowed
No Bouncing
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Rigid Aggs Movie 3
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Rigid Aggs Movie 4
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Future Work

• PKDGRAV continues to be improved.
• Algorithms currently being considered
• Local expansions.
• Tree repair.
• Improved collision detection.
• Other possibilities include combining PKDGRAV
  with a Grape board!