## 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 mathematicalÂ bastions. Traditional discretizations for PDE systems span from innocent Finite DifferenceÂ schemes to Monte Carlo simulations, covering several approaches in one stroke: Eulerian toÂ Lagrangian and deterministic to stochastic simulations. SPH follows a particulate approach, butÂ the particles have a diffuse nature, as the systemâ€™s equations are derived using kernelÂ estimates. By applying this technique to interpolation, estimates of any quantity at any pointÂ are known, using the value of such quantities at the particles. Differentiation of thoseÂ estimates is exact if the kernel is differentiable, allowing the system to be written in terms ofÂ particle properties.Â

Applying these concepts to fluid flow, the particles move, carrying their mass. This implies thatÂ our interpolation nodes change position at each integration step, generating our discomfort: inÂ a more regular lattice traditional techniques for stability analysis and convergence studies areÂ well known; in a purely random sampling, as a Monte Carlo method, these qualities are alsoÂ simple to compute, at least approximately. In SPH however, any lattice deforms, not randomly,Â but according to the conservation equations being solved, introducing an unexpected difficultyÂ in analyzing the method under normal conditions.Â

Nonetheless, it flows. SPH has been used extensively in hydrodynamics, from astrophysicalÂ 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 toÂ account for solid-solid interactions. The result is a high-performance implementation, capableÂ of providing novel results concerning free-surface flows with arbitrarily shaped boundaries andÂ emerged rigid bodies with no kinematic restrictions. Canonical and non-conventional solutionsÂ are explored, in an effort to estimate the qualities of the model.

**Project**UIDB/00618/2020