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.
Highlights
In the Lagrangian meshless/particle methods, for example, the smoothed particle hydrodynamics (SPH, e.g. Monaghan 1994; Shao and Lo 2003; Shao et al 2006; Khayyer et al 2008; Lind et al 2012), moving particle semi-implicit method (MPS, e.g. Koshizuka and Oka 1996; Gotoh and Khayyer 2016; Khayyer and Gotoh 2010), the meshless local PetrovGalerkin method (MLPG, e.g., Ma 2005a, b; Zhou and Ma 2010), interpolating element-free Galerkin method (Abbaszadeh and Dehghan 2019; Dehghan and Abbaszadeh 2018, 2019), the computational domain is represented by particles and the governing equations with associated boundary conditions are discretised to form a linear algebraic equation system, which leads to the approximation of physical quantities at particle locations
This paper develops a new scheme called quadric semi-analytical finite-difference interpolation (QSFDI), which adopts the same principle of SFDI, to discretise the Laplacian operator for Lagrangian meshless methods, in which the particles move during the numerical simulation and exhibit a disordered/random distribution
The accuracy and consistency of the QSFDI are similar to the LSMPS and LP-MPS but higher than the CSPM and CSPH for randomly distributed particles
Summary
In the Lagrangian meshless/particle methods, for example, the smoothed particle hydrodynamics (SPH, e.g. Monaghan 1994; Shao and Lo 2003; Shao et al 2006; Khayyer et al 2008; Lind et al 2012), moving particle semi-implicit method (MPS, e.g. Koshizuka and Oka 1996; Gotoh and Khayyer 2016; Khayyer and Gotoh 2010), the meshless local PetrovGalerkin method (MLPG, e.g., Ma 2005a, b; Zhou and Ma 2010), interpolating element-free Galerkin method (Abbaszadeh and Dehghan 2019; Dehghan and Abbaszadeh 2018, 2019), the computational domain is represented by particles and the governing equations with associated boundary conditions are discretised to form a linear algebraic equation system, which leads to the approximation of physical quantities at particle locations. Adams (2007); Gotoh et al (2014); and Khayyer and Gotoh (2010, 2012), converge at a rate less than first order for estimating the Laplacian of a given function, they do not need inversions of any matrices and have relatively low computational costs Their performances may be improved by reducing the randomness of the particle distribution, e.g., using the particle shifting scheme proposed by Lind et al (2012), or by introducing error correction and compensating terms, e.g. Patch tests by Schwaiger (2008), Zheng et al (2014) and Tamai et al (2017) have shown that the CSPM, LP-MPS and quadric LSMPS have a higher convergent rate, compared with the type 1 schemes Formulating these schemes requires a significant computational cost on inversing matrices at all particle locations and at every time step of the transient simulations. The performance of the present scheme will be assessed by systematic patch tests considering both directly estimating the Laplacian of specific functions and solving Poisson’s equation
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
