Smoothed Particle Hydrodynamics (SPH)
The fully Lagrangian Smoothed Particle Hydrodynamics (SPH) method was originally invented to deal with non axisymmetric problems in astrophysics (Lucy 1977, Gingold & Monaghan 1977). Since then the use of SPH has expanded in many areas of solid and fluid dynamics (involving large deformations, impacts, free-surface and multiphase flows). For example, collision of rubber cylinders (Swegle et al., 1995) in solid mechanics, dam breaking and free-surface waves (Monaghan, 1994) and two-phase flows such as Rayleigh-Benard instability (Violeau, 1999) in fluid mechanics. A major advantage of SPH over Eulerian methods is that the method does not need a grid to calculate spatial derivatives. Instead, they are found by summation of analytical differentiated interpolation formulae (Monaghan, 1992). The momentum and energy equations become sets of ordinary differential equations which are easy to understand in mechanical and thermodynamical terms. For example, the pressure gradient becomes a force between pairs of particles. While Eulerian methods have difficulties to construct a mesh for the simulation domain when it has very complex interfaces, SPH is able to do it without any special front tracking treatment. Nevertheless, despite good agreements in general, some limitations are found in the SPH method such as very small time step, which lead to very expensive CPU cost, and pressure fluctuation.
