Quadric SFDI for Laplacian Discretisation in Lagrangian Meshless Methods
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RESEARCH ARTICLE
Quadric SFDI for Laplacian Discretisation in Lagrangian Meshless Methods Shiqiang Yan 1 & Q. W. Ma 1 & Jinghua Wang 1 Received: 7 September 2019 / Accepted: 12 May 2020 # The Author(s) 2020
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. Keywords Laplacian discretisation . Lagrangian meshless methods . QSFDI . Random/disordered particle distribution . Poisson’s equation . Patch tests
1 Introduction Article Highlights • It develops a new scheme, QSFDI, for numerical interpolation, gradient calculation and Laplacian estimations using disordered particles, which is more efficient than others at the same accuracy level. • The QSFDI can achieve the same degree of the convergent rate as the best scheme but requires inversing significant lower-order matrices, i.e. 3 × 3 for 3D cases, compared with 6 × 6 or 10 × 10 in the latter, and, consequently, requires much less CPU time. • It presents a systematic patch test comparing different schemes in terms of accuracy, convergence and robustness. * Q. W. Ma [email protected] 1
School of Mathematics, Computer Science and Engineering, University of London, EC1V 0HB, London, UK
In the Lagrangian meshless/particle methods, for example, the smoothed particle hydrodynamics (SPH
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