Semi-decoupled first-order correction for smoothed particle hydrodynamics

Abstract

The Finite Particle Method (FPM) and the Decoupled Finite Particle Method (DFPM) are variants of the Smoothed Particle Hydrodynamics (SPH) in which the estimation of a field and its gradient for an arbitrary distribution of particles is performed by imposing first order consistency equations (C1). A modification of the kernel is introduced that involves the inversion of a correction matrix for each interpolation point in the system. In the FPM, the inversion is rigorously performed and therefore the method has first-order consistency (C1). However, in DPFM the vanishingly small non-diagonal terms in the correction matrix are neglected to obtain a more straightforward method, at expense of consistency. Following the idea of DFPM, the Semi-Decoupled Finite Particle Method (SDFPM) and the Corrected Semi Decoupled Finite Particle Method (CSDFPM) are introduced. In the SDFPM, the kernel is normalized by a Shepard factor and the first order consistency equations are solved by neglecting non-diagonal terms in the correction matrix as in the DFPM method. In the corrected version of this method (CSDFPM), the SDFPM estimation is used as the initial guess to get a second corrected estimation in the C1 equations. The precision of FPM, DFPM, SDFPM, and CSDFPM is tested by evaluating the gradient components of a field around a cylindrical obstacle and around a cylindrical hole by using (i) trial field functions with an ordered array of particles and (ii) the pressure field with a distribution of particles taken from a flow simulation. Drag coefficients for flow around a cylinder are obtained by the different methods and compared in a wide range of Reynolds numbers, including the laminar and turbulent regimes. The gradient components and drag coefficients calculated with the proposed methods show a precision comparable to the FPM at a lower computational cost.