Short note: Smoothed particle hydrodynamics

Smoothed particle hydrodynamics (SPH) is a mesh-free, Lagrangian particle method originally developed for astrophysics (Lucy, and Gingold and Monaghan, 1977) and now used for fluids and solids with large deformation. The medium is represented by particles carrying mass, density, velocity and pressure; there is no grid.

Principle: the value of a field $f$ at a point is the weighted sum of the values at neighbouring particles within a smoothing length $h$:

$$f(x_i)\approx\sum_j \frac{m_j}{\rho_j}\,f_j\,W(x_i-x_j,h)$$

where $W$ is the smoothing (kernel) function. Gradients are obtained by differentiating $W$, so the governing equations (continuity and momentum) become ordinary differential equations for each particle, which are integrated in time.

Advantages: no mesh distortion; free surfaces and large deformation are handled easily; mass is conserved automatically (each particle carries a fixed mass).

Disadvantages: costly neighbour search; boundary treatment and tensile instability are difficult; lower accuracy than FEM for small deformation.

Applications: dam break, wave impact, flood flow, landslide and debris flow, soil flow.