Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation
maximize
 

to

SPH

Title Smoothed Particle Hydrodynamics with applications in astrophysics 
Participant  Attila Caglar , Michael Griebel 
Keywords  smoothed-particle-hydrodynamics, meshless methods, particle methods, multilevel, parallelization 
Description  Smoothed Particle Hydrodynamics (SPH) is a meshless, Lagrangian particle method for the numerical solution of partial differential equations. Among the meshless is SPH one of the oldest methods (Lucy 1977, Monaghan 1982). SPH offers several additional benefits. Unlike conventional Lagrangian techniques, SPH avoids mesh tangling and is therefore much more robust in its treatment of problems with large material distortions.   The following Figure sketches the principles of SPH:

 


Derivation of the SPH-Method:

  • A scalar function f(x) satisfies:


  • One obtain the continuous SPH approximation by replacing the Dirac delta distribution by a kernel with compact support
    (radial Basisfunction) The particles become Lagrangian
    fluid elements

The kernel has the following properties:

  • (symmetry of the kernel)
  • is normed, i.e.
  • for , h is the smoothing length

Continous SPH formula


  • Let be the density (number of particles, mass, charge, etc.), approximated by N particles in the phase space V.

    : set of N particles with weights . Approximate by discrete measures:

Basis of the SPH method:





Applications to astrophysics


Aim of the project:

  • Constructing a SPH-Method of higher consistency-order and better stability
  • Efficient parallelization based on a multilevel approach



Bibliography 
  • Bader M., Simulation einer rotierenden Gasscheibe durch numerisches Lösen der 2D-Navier-Stokes-Gleichungen, Diplomarbeit TU München 1996
  • Flebbe O., Münzel S., Herold H., Riffert H., Ruder H., Smoothed Particle Hydrodynamics:Physical Viscosity and the Simulatgion of Accretion Disks, 1994, Ap. J., 431,754.
  • Hernquist L., Katz N., TREESPH: A Unification of SPH with the Hierarchical Tree Method, IAS Princeton 1989
  • Ott F., Smoothed Particle Hydrodynamics, Diplomarbeit Universität Tübingen 1995
  • Ben Moussa B., Vila J.P., Convergence of SPH Method for Scalar Nonlinear Conservation Laws, preprint INSA Toulouse 1997
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    Related projects  Partition of unity methods for instationary convection-diffusion equations  
    In cooperation with  SFB 256