ALE or Arbitrary Lagrangian Eulerian formulation is used to model the interaction between fluids and solids; in particular,
the fluid loading on structures. It can also be used to model fluid-like behavior, as seen in plastic deformation of materials.

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.

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.

Let a set of particles $i$=1, n at positions ${x}_{i}$ ($i$=1,n) with mass ${m}_{i}$ and density ${\rho}_{i}$. The smoothed approximation of the function $f$ is (summation over neighboring particles and the
particle $i$ itself):(4)

This kernel has compact support, so that for each particle $i$, only the closest particles contribute to
approximations at $i$ (this feature is computationally efficient). The
accuracy of approximating Equation 1 by Equation 4 depends on
the order of the particles.