Enhancing SPH using moving least-squares and radial basis functions

TL;DR

This paper introduces enhancements to the SPH method by incorporating moving least-squares and radial basis functions to improve accuracy and eliminate the need for smoothing-length parameter tuning.

Contribution

It develops new shape functions for SPH using MLS and RBFs, ensuring polynomial consistency and removing the smoothing-length parameter, while maintaining conservation and meshfree properties.

Findings

Improved accuracy of SPH with MLS and RBF-based shape functions.

Elimination of smoothing-length parameter in SPH.

Successful demonstration on Sod shock tube problem.

Abstract

In this paper we consider two sources of enhancement for the meshfree Lagrangian particle method smoothed particle hydrodynamics (SPH) by improving the accuracy of the particle approximation. Namely, we will consider shape functions constructed using: moving least-squares approximation (MLS); radial basis functions (RBF). Using MLS approximation is appealing because polynomial consistency of the particle approximation can be enforced. RBFs further appeal as they allow one to dispense with the smoothing-length -- the parameter in the SPH method which governs the number of particles within the support of the shape function. Currently, only ad hoc methods for choosing the smoothing-length exist. We ensure that any enhancement retains the conservative and meshfree nature of SPH. In doing so, we derive a new set of variationally-consistent hydrodynamic equations. Finally, we demonstrate the performance of the new equations on the Sod shock tube problem.