Analysis of the incompressibility constraint in the Smoothed Particle Hydrodynamics method

TL;DR

This paper compares different incompressibility treatments in Smoothed Particle Hydrodynamics, revealing issues with density errors in current methods and proposing a correction algorithm to improve accuracy.

Contribution

The study provides a comprehensive comparison of incompressibility methods in SPH and introduces a novel density correction algorithm to enhance simulation accuracy.

Findings

Truly incompressible methods suffer from density accumulation errors.

A new correction algorithm improves density consistency in SPH simulations.

Boundary conditions were standardized across methods for fair comparison.

Abstract

Smoothed particle hydrodynamics is a particle-based, fully Lagrangian, method for fluid-flow simulations. In this work, fundamental concepts of the method are first briefly recalled. Then, we present a thorough comparison of three different incompressibility treatments in SPH: the weakly compressible approach, where a suitably-chosen equation of state is used; and two truly incompressible methods, where the velocity field projection onto a divergence-free space is performed. A noteworthy aspect of the study is that, in each incompressibility treatment, the same boundary conditions are used (and further developed) which allows a direct comparison to be made. Problems associated with implementation are also discussed and an optimal choice of the computational parameters has been proposed and verified. Numerical results show that the present state-of-the-art truly incompressible method (based on a velocity correction) suffer from density accumulation errors. To address this issue, an algorithm, based on a correction for both particle velocities and positions, is presented. The usefulness of this density correction is examined and demonstrated in the last part of the paper.