Smoothed Particle Hydrodynamics

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Smoothed Particle Hydrodynamics
Carlos Eduardo Aguiar Instituto de Física - UFRJ

• Outline
• Non-relativistic hydrodynamics.
• SPH equations.
• Applications.
• Relativistic hydrodynamics.
• Relativistic SPH.
• High energy nuclear physics.

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Fluid Dynamics
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Hydrodynamic Equations
Continuity equation
Ideal fluid
Eulers equation
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Entropy Equation

• no viscosity
• no thermal conduction

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Energy Equation
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Ideal Gas
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Conservation Laws
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SPH

• Developed to study gas dynamics in
  astrophysical systems.
• Lagrangian method.
• No grids.
• Arbitrary geometries.
• Equally applicable in 1, 2 and 3 space
• dimensions.

- L.Lucy, Astron.J. 82, 1013 (1977) - R.Gingold,
J.Monaghan, MNRAS 181, 378 (1977)
Reviews - J. Monaghan, Annu. Rev. Astron.
Astrophys. 30, 543 (1992) - L. Hernquist, N.
Katz, Ap. J. Suppl. 70, 419 (1989)
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Smoothing
h
x
0
Error
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Particles
"Monte-Carlo" sampling
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Different ways of writing SP estimates (we omit
the SP subscript from now on)
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Derivatives
No need for finite differences and grids
D
D
i1
i-1
i
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More than one way of calculating derivatives
Exact Galilean invariance
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Moving the Particles
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Euler's Equation
Exact momentum conservation
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Entropy
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Energy
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Alternatively
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SPH Equations
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Smoothing Kernels
Gaussian
Spline
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Shock Waves
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Artificial Viscosity

• Galilean invariant.
• Vanishes for rotations.
• Conserves linear and angular momentum.

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Local Resolution Length
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SPH Simulation of the Hubble Volume
Mass density in a thin slice (100x100x20Mpc/h) at
the present epoch. This is a view one would
observe if the speed of light were infinite.
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SPH Simulation of Galaxy Formation
Gas
Dark Matter
Density of gas and dark matter in a group of
galaxies.
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SPH Simulation of Supernova Explosion
75 ms after bounce
z (km)
x (km)
Herant et al., Ap.J. 435, 339 (1994)
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SPH Simulation of Supernova Explosion
Herant et al., Ap.J. 435, 339 (1994)
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SPH Simulation of Stellar Collision
Disruption of a main sequence star by a close
encounter with a high velocity neutron star.
Colors represent log (density).
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SPH Simulation of Colliding Asteroids
An 8 m radius rock strikes the 1.6 km long
asteroid Castalia at 5 km/s. Red is totally
fractured rock, blue is intermediate fractured
rock, and white particles represent the impactor.
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SPH Simulation of Projectile Impact on Sand
Projectile Aluminum cylinder 30x10 cm (2d).
Initial velocity 3 km/sec
Particles
Temperature
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Relativistic Hydrodynamics
Energy-momentum conservation
Baryon-number conservation
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Continuity equation
Entropy equation
s entropy density (rest frame)
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Relativistic Euler equation
w enthalpy per baryon
Momentum equation
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Energy equation
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Relativistic SPH
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Momentum (per baryon number)
Energy (per baryon number)
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Particle Velocity
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RSPH Equations
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Baryon-Free System
entropy density
(Rest frame)
(Lab frame)
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Baryon-Free RSPH
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Ultrarelativistic Pion Gas
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Ultrarelativistic Pions
Rarefaction Wave
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SHASTA et al.
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Ultrarelativistic Pions
Landau Solution
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Artificial Viscosity

• E.Chow and J.Monaghan, Journal of
• Computational Physics 134, 296 (1997)
• See also
• S.Siegler and H.Riffert, astro-ph/9904070

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Shock Tube
Ideal nucleon gas
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SHASTA et al.
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Shock Tube
Ideal nucleon gas
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SHASTA et al.