Conservative, density-based smoothed particle hydrodynamics with improved partition of the unity and better estimation of gradients

TL;DR

This paper introduces a new volume element approach in smoothed particle hydrodynamics that improves gradient estimation accuracy and partition of unity, enhancing the method's robustness and conservation properties in complex flow simulations.

Contribution

It proposes a novel volume element variant that improves partition of unity and gradient accuracy, compatible with Lagrangian SPH, and introduces a flexible scheme for handling sharp density contrasts.

Findings

Improved partition of unity enhances gradient estimation accuracy.

The new scheme performs well in standard tests with contact discontinuities.

Conservation of energy, momentum, and entropy remains robust with the new approach.

Abstract

The correct evaluation of gradients is at the cornerstone of the smoothed particle hydrodynamics (SPH) technique. Using an integral approach to estimate gradients has proven to enhance accuracy substantially. Such approach retains the Lagrangian structure of SPH equations and is fully conservative. But, in practice, it is difficult to make the Lagrangian formulation totally consistent to an exact partition of the unity. In this paper we study, among other things, the connection between the choice of the volume elements (VEs), which enters in the SPH summations, and the accuracy in the gradient estimation within the integral approach scheme (ISPH). A new variant of VEs are proposed which improve the partition of the unity and is fully compatible with the Lagrangian formulation of SPH, including the grad-h corrections. Using analytic considerations, simple static toy models in 1D, and a few full 3D test cases, we show that any improvement in the partition of the unity also leads to a better calculation of gradients when the integral approach is used jointly. Additionally, we propose an easy-to-implement modification of the ISPH scheme, which makes it more flexible and better suited to handle sharp density contrasts. The ISPH code built with the proposed scheme has been validated with a good number of standard tests, some of them involving contact discontinuities. The performance of the code was excellent in all of them, showing that an improvement in the partition of the unity is not detrimental of the good conservation of energy, momentum, and entropy typical of Lagrangian schemes.