# Smooth Particle Hydrodynamics

Smooth Particle Hydrodynamics (SPH) is a meshless numerical method based on interpolation theory. It allows any function to be expressed in terms of its values at a set of disordered point's so-called particles.

SPH is not based on the particle physics theory. The conservation laws of continuum
dynamics, in the form of partial differential equations, are transformed into integral
equations through the use of kernel approximation. A comprehensive state-of-the-art of
the method. ^{1}
^{2}
^{3} These techniques were initially developed in astrophysics. ^{4}
^{5} During the 1991-1995 periods, SPH has become widely recognized
and has been used extensively for fluid and solid mechanics type of applications. SPH
method is implemented in Radioss in Lagrangian approach
whereby the motion of a discrete number of particles is followed in time.

SPH is a complementary approach with respect to ALE method. When the ALE mesh is too distorted to handle good results (for example in the case of vortex creation), SPH method allows getting a sufficiently accurate solution.

^{1}Bonet J., TSL Lok, “Variational and Momentum Preservation Aspects of Smooth Particle Hydrodynamic Formulations”, Computer Methods in Applied Mechanics and Engineering, Vol. 180, pp. 97-115 (1999).

^{2}Randles P.W. and Libersky L.D., “Smoothed Particle Hydrodynamics: Some recent improvements and applications”, Computer Methods Appl. Mech. Engrg. Vol. 139, pp. 375-408, 1996.

^{3}Balsara D.S., “Von Neumann Stability Analysis of Smoothed Particle Hydrodynamics Suggestions for Optimal Algorithms”, Journal of Computational Physics, Vol. 121, pp. 357-372, 1995.

^{4}Lucy L.B., “A numerical approach to the testing of the fission hypothesis”, Astro. J., Vol. 82, 1013, 1977.

^{5}Gingold R.A. and Monaghan J.J., “SPH: Theory and application to non-spherical stars”, Mon. Not. R. Astron. Soc., Vol. 181, 375, 1977.