Does it Flow? SPH, a Lagrangian Meshless Method
By: Ricardo Canelas
From: IST, UL
At: Faculdade de Ciências, Ed. C8, 8.2.06
[2015-05-21]
Smooth Particle Hydrodynamics stands, uncomfortably, between numerical and mathematical bastions. Traditional discretizations for PDE systems span from innocent Finite Difference schemes to Monte Carlo simulations, covering several approaches in one stroke: Eulerian to Lagrangian and deterministic to stochastic simulations. SPH follows a particulate approach, but the particles have a diffuse nature, as the system’s equations are derived using kernel estimates. By applying this technique to interpolation, estimates of any quantity at any point are known, using the value of such quantities at the particles. Differentiation of those estimates is exact if the kernel is differentiable, allowing the system to be written in terms of particle properties.Â
Applying these concepts to fluid flow, the particles move, carrying their mass. This implies that our interpolation nodes change position at each integration step, generating our discomfort: in a more regular lattice traditional techniques for stability analysis and convergence studies are well known; in a purely random sampling, as a Monte Carlo method, these qualities are also simple to compute, at least approximately. In SPH however, any lattice deforms, not randomly, but according to the conservation equations being solved, introducing an unexpected difficulty in analyzing the method under normal conditions.Â
Nonetheless, it flows. SPH has been used extensively in hydrodynamics, from astrophysical simulations to viscous fluid flow, with countless variations. A solver for a form of the Navier-Stokes system is presented, coupled with a rigid body solver and Discrete Element Methods to account for solid-solid interactions. The result is a high-performance implementation, capable of providing novel results concerning free-surface flows with arbitrarily shaped boundaries and emerged rigid bodies with no kinematic restrictions. Canonical and non-conventional solutions are explored, in an effort to estimate the qualities of the model.
