Quadric SFDI for Laplacian Discretisation in Lagrangian Meshless Methods

Abstract

In the Lagrangian meshless (particle) methods, such as the smoothed particle hydrodynamics (SPH), moving particle semi-implicit (MPS) method and meshless local Petrov-Galerkin method based on Rankine source solution (MLPG_R), the Laplacian discretisation is often required in order to solve the governing equations and/or estimate physical quantities (such as the viscous stresses). In some meshless applications, the Laplacians are also needed as stabilisation operators to enhance the pressure calculation. The particles in the Lagrangian methods move following the material velocity, yielding a disordered (random) particle distribution even though they may be distributed uniformly in the initial state. Different schemes have been developed for a direct estimation of second derivatives using finite difference, kernel integrations and weighted/moving least square method. Some of the schemes suffer from a poor convergent rate. Some have a better convergent rate but require inversions of high order matrices, yielding high computational costs. This paper presents a quadric semi-analytical finite-difference interpolation (QSFDI) scheme, which can achieve the same degree of the convergent rate as the best schemes available to date but requires the inversion of significant lower-order matrices, i.e. 3 × 3 for 3D cases, compared with 6 × 6 or 10 × 10 in the schemes with the best convergent rate. Systematic patch tests have been carried out for either estimating the Laplacian of given functions or solving Poisson’s equations. The convergence, accuracy and robustness of the present schemes are compared with the existing schemes. It will show that the present scheme requires considerably less computational time to achieve the same accuracy as the best schemes available in literatures, particularly for estimating the Laplacian of given functions.

A QSFDI based Laplacian discretisation for modelling wave-structure interaction using ISPH

“…A detailed error analysis of these two schemes has been given in Yan et al (2020), which shows the leading truncation errors of the CSPM (Eq. ( 11)) and CSPH2Γ (Eq.…”

“…This limits the applications of high-order finite difference schemes which require regular and even particle distribution. Detailed review on the such schemes can be found in Ma et al (2016) and Yan et al (2020) and will not be repeated here.…”

“…However, the CSPH2Γ scheme downgrades the accuracy because of ignoring the cross-derivative terms of the 2nd derivatives. By adopting the principle of the linear semi-analytical finite difference interpolation (SFDI), we developed a quadric version, which is referred to as the QSFDI, for Laplacian discretization (Yan et al, 2020). The QSFDI can achieve the same degree of the convergent rate as the best schemes available to date, e.g., the CSPM, LP-MPS and quadric LSMPS, but requires inversion of significant lower order matrices, i.e., 3×3 for 3D cases, compared with 6×6 or 10×10 in the schemes with the best convergent rate.…”

“…The QSFDI can achieve the same degree of the convergent rate as the best schemes available to date, e.g., the CSPM, LP-MPS and quadric LSMPS, but requires inversion of significant lower order matrices, i.e., 3×3 for 3D cases, compared with 6×6 or 10×10 in the schemes with the best convergent rate. Systematic patch tests have been carried out for either estimating the Laplacian of given functions or solving Poisson's equations using disordered particles (Yan et al, 2020).…”

“…For the purpose of comparison, other schemes developed for SPH applications, including the classic ISPH scheme, the CSPM and CSPH2Γ, have been considered in the present comparative study. It shall be noted that the QSFDI (Yan et al, 2020) includes consistent schemes for numerical interpolation and gradient (spatial derivative) estimations, which can be implemented in the ISPH procedures, e.g., the treatment of the boundary condition, the calculation of pressure gradient. However, these are not implemented in this paper in order to focus our evaluation on discretizing the Laplacian operator.…”