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GENGA: Gravitational Encounters in N-bodysimulations with GPU Acceleration
Institute for Computational ScienceUniversity of Zürich
Perspectives of GPU Computing in Physics and AstrophysicsRoma 2014
Simon GrimmJoachim Stadel
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Outline
● Physical problem
● Theory and integration scheme
● Some details about the kernels
● Applications
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Overview
Goals:Study the formation process of terrestrial planetary systems due to planetesimal dynamics.Analyze the long term evolution of (exo-)planetary systems.
Requirements: Good energy conservation over billions of time steps.Resolve close encounters accurately.Collisions between bodies are possible.
Limits:The number of massive bodies is limited to 2048 due to the close encounter handling of the algorithm.
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GENGA
GENGA supports 3 Computational Modes: ● Up to 2048 massive bodies
● Up to 1 million test particles
● Up to 100000 parallel small simulations
We need to study relatively small simulations, but we need a lot of them to cover a large parameter space.
All of the computation is done on the GPU to avoid memory transfer with the CPU.
Simulations run on only one node, could use multiple GPUs.
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Theory: The hybrid symplectic integrator
Using democratic coordinates:(Heliocentric positions and barycentric velocities)
Drift Keplerian arcsanalytical with FG
Use direct n-body integrator if K < 1
KickO(N2)
Sun Kick
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Close encounter pairs form groups
Rcrit = max(3 * RHill, 0.4 *dt * v)
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Close encounter pairs form groups
Limit of N = 2048
Rcrit = max(3 * RHill, 1.5 *dt * v)
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The integration scheme
No close encounter candidates With close encounter candidates
Prechecker findsCE candidates forthe next time step
Polynomialinterpolation
Parallel groupfinding
algorithm
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The Kick kernel
● The number of bodies is too small to usea standard N2 kernel as described in e.g. Gems 3
● We need more work to be done in paralleland perform the summations in shared memory withinone thread block.
● We need different versions of the kernel, depending onthe number of bodies.
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All kernels are optimized for a certain number of bodies
switch(N){case 16: D.step_16(t);break;case 32: D.step_32(t);break;case 64: D.step_64(t);break;case 128: D.step_128(t);break;case 256: D.step_256(t);break;case 512: D.step_512(t);break;case 1024: D.step_1024(t);break;case 2048: D.step_2048(t);break;
}
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Find independent close encounter groups in parallel
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Timing of the main kernels
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Comparison between different GPUs
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Timing of the main kernelsfor the multi simulation mode
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Comparison between different GPUsfor the multi simulation mode
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Comparison with pkdgrav and Mercury
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Comparison of the energy conservation between GENGA, pkdgrav and Mercury, for a set of 40
simulations with 32 planetesimals.
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Tools in GENGA:
● Analytic Gas disc modelAdapted from Morishima et al. 2010
● Poincaré surface of section● Fraction of time spent in a-e limits
Count how much time a body spend in a specified semi-major-axis and eccentricity regime.
● a-e gridPlot how long regions in semi-major-axis and eccentricity are populated from bodies.
● Switch for exact reproduction of random rounding errors.For studying chaotic systems the order of arithmetic operations can be fixed to reproduce exact the same results
● openGL visualization
http://www.youtube.com/watch?v=74O-8P49iJk
http://www.youtube.com/watch?v=a_4cjXVDEAw
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Planet formation with a gas disc
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Stability analysis of addition super Earths in exoplanetary systems
Elser et Al. 2013
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Use test particles to find stable orbits withina planetary system
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Conclusions
● GENGA runs up to 30 times faster than a CPU code (Mercury)
● We want to run many relatively small simulations
● A new algorithm is needed for higher N
● The best hardware setup is one node with many GPUs.
● GENGA is available at:
https://bitbucket.org/sigrimm/genga
