|Table of Contents|

Citation:
 hiqiang Yan,Q. W. Ma,Jinghua Wang.Quadric SFDI for Laplacian Discretisation in Lagrangian Meshless Methods[J].Journal of Marine Science and Application,2020,(3):362-380.[doi:10.1007/s11804-020-00159-x]
Click and Copy

Quadric SFDI for Laplacian Discretisation in Lagrangian Meshless Methods

Info

Title:
Quadric SFDI for Laplacian Discretisation in Lagrangian Meshless Methods
Author(s):
hiqiang Yan Q. W. Ma Jinghua Wang
Affilations:
Author(s):
hiqiang Yan Q. W. Ma Jinghua Wang
School of Mathematics, Computer Science and Engineering, University of London, EC1V 0HB, London, UK
Keywords:
LaplaciandiscretisationLagrangianmeshlessmethodsQSFDIRandom/disorderedparticledistributionPoisson’s equationPatch tests
分类号:
-
DOI:
10.1007/s11804-020-00159-x
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.

References:

Abbaszadeh M, Dehghan M (2019) The interpolating element-free Galerkin method for solving Korteweg-de Vries-Rosenau-regularized long-wave equation with error analysis. Nonlinear Dyn 96(2):1345-1365. https://doi.org/10.1007/S11071-019-04858-1
Brookshaw L (1985) A method of calculating radiative heat diffusion in particle simulations. Publ Astron Soc Aust 6(2):207-210. https://doi.org/10.1017/S1323358000018117
Chen JK, Beraun JE, Carney TC (1999) A corrective smoothed particle method for boundary value problems in heat conduction. Int J Numer Methods Eng 46(2):231-252. https://doi.org/10.1002/(SICI)1097-020719990920)46:2<231::AID-NME672>3.0.CO;2-K
Cummins SJ, Rudman M (1999) An SPH projection method. J Comput Phys 152(2):584-607. https://doi.org/10.1006/jcph.1999.6246
Dehghan M, Abbaszadeh M (2018) Variational multiscale element-free Galerkin method combined with the moving Kriging interpolation for solving some partial differential equations with discontinuous solutions. Comput Appl Math 37(3):3869-3905. https://doi.org/10.1007/s40314-017-0546-6
Dehghan M, Abbaszadeh M (2019) Error analysis and numerical simulation of magnetohydrodynamics (MHD) equation based on the interpolating element free Galerkin (IEFG) method. Appl Numer Math 137:252-273. https://doi.org/10.1016/j.apnum.2018.10.004
Fatehi R, Manzari MT (2011) Error estimation in smoothed particle hydrodynamics and a new scheme for second derivatives. Comput Math Appl 61(2):482-498. https://doi.org/10.1016/j.camwa.2010.11.028
Gotoh H, Khayyer A (2016) Current achievements and future perspectives for projection-based particle methods with applications in ocean engineering. J Ocean Eng Marine Energy 2(3):251-278. https://doi.org/10.1007/s40722-016-0049-3
Gotoh H, Khayyer A, Ikari H, Arikawa T, Shimosako K (2014) On enhancement of incompressible SPH method for simulation of violent sloshing flows. Appl Ocean Res 46:104-115. https://doi.org/10.1016/j.apor.2014.02.005
Hu XY, Adams NA (2007) An incompressible multi-phase SPH method. J Comput Phys 227(1):264-278. https://doi.org/10.1016/j.jcp.2007.07.013
Ikari H, Khayyer A, Gotoh H (2015) Corrected higher order Laplacian for enhancement of pressure calculation by projection-based particle methods with applications in ocean engineering. J Ocean Eng Marine Energy 1(4):361-376. https://doi.org/10.1007/s40722-015-0026-2
Khayyer A, Gotoh H (2010) A higher order Laplacian model for enhancement and stabilization of pressure calculation by the MPS method. Appl Ocean Res 32(1):124-131. https://doi.org/10.1016/j.apor.2010.01.001
Khayyer A, Gotoh H (2012) A 3D higher order Laplacian model for enhancement and stabilization of pressure calculation in 3D MPS-based simulations. Appl Ocean Res 37:120-126. https://doi.org/10.1016/j.apor.2012.05.003
Khayyer A, Gotoh H, Shao SD (2008) Corrected incompressible SPH method for accurate water-surface tracking in breaking waves. Coast Eng 55(3):236-250. https://doi.org/10.1016/j.coastaleng.2007.10.001
Koshizuka S, Oka Y (1996) Moving-particle semi-implicit method for fragmentation of incompressible fluid. Nucl Sci Eng 123(3):421-434. https://doi.org/10.13182/NSE96-A24205
Lee ES, Moulinec C, Xu R, Violeau D, Laurence D, Stansby PK (2008) Comparisons of weakly compressible and truly incompressible algorithms for the SPH mesh free particle method. J Comput Phys 227(18):8417-8436. https://doi.org/10.1016/j.jcp.2008.06.005
Lind SJ, Stansby PK (2016) High-order Eulerian incompressible smoothed particle hydrodynamics with transition to Lagrangian free surface motion. J Comput Phys 326:290-311. https://doi.org/10.1016/j.jcp.2016.08.047
Lind SJ, Xu R, Stansby PK, Rogers BD (2012) Incompressible smoothed particle hydrodynamics for free-surface flows:a generalised diffusion-based algorithm for stability and validations for impulsive flows and propagating waves. J Comput Phys 231(4):1499-1523. https://doi.org/10.1016/j.jcp.2011.10.027
Lo EY, Shao S (2002) Simulation of near-shore solitary wave mechanics by an incompressible SPH method. Appl Ocean Res 24(5):275-286. https://doi.org/10.1016/S0141-1187(03)00002-6
Ma QW (2005a) MLPG method based on Rankine source solution for simulating nonlinear water waves. Comput Model Eng Sci 9(2):193-210. https://doi.org/10.3970/cmes.2005.009.193
Ma QW (2005b) Meshless local Petrov-Galerkin method for two-dimensional nonlinear water wave problems. J Comput Phys 205(2):611-625. https://doi.org/10.1016/j.jcp.2004.11.010
Ma QW (2008) A new meshless interpolation scheme for MLPG_R method. Comput Model Eng Sci 23(2):75-90. https://doi.org/10.3970/cmes.2008.023.075
Ma QW, Zhou Y, Yan S (2016) A review on approaches to solving Poisson’s equation in projection-based meshless methods for modelling strongly nonlinear water waves. J Ocean Eng Marine Energy 2(3):279-299. https://doi.org/10.1007/s40722-016-0063-5
Monaghan JJ (1994) Simulating free surface flows with SPH. J Comput Phys 110(2):399-406. https://doi.org/10.1006/jcph.1994.1034
Oger G, Doring M, Alessandrini B, Ferrant P (2007) An improved SPH method:towards higher order convergence. J Comput Phys 225(2):1472-1492. https://doi.org/10.1016/j.jcp.2007.01.039
Quinlan NJ, Basa M, Lastiwka M (2006) Truncation error in mesh-free particle methods. Int J Numer Methods Eng 66(13):2064-2085. https://doi.org/10.1002/nme.1617
Schwaiger HF (2008) An implicit corrected SPH formulation for thermal diffusion with linear free surface boundary conditions. Int J Numer Methods Eng 75(6):647-671. https://doi.org/10.1002/nme.2266
Shao S, Ji C, Graham DI, Reeve DE, James PW, Chadwick AJ (2006) Simulation of wave overtopping by an incompressible SPH model. Coast Eng 53(9):723-735. https://doi.org/10.1016/j.coastaleng.2006.02.005
Shao S, Lo EYM (2003) Incompressible SPH method for simulating Newtonian and non-Newtonian flows with free surface. Adv Water Resour 26(7):787-800. https://doi.org/10.1016/S0309-1708(03)00030-7
Tamai T, Koshizuka S (2014) Least squares moving particle semi-implicit method. Comput Part Mech 1(3):277-305. https://doi.org/10.1007/s40571-014-0029-0
Tamai T, Murotani K, Koshizuka S (2017) On the consistency and convergence of particle-based meshfree discretization schemes for the Laplace operator. Comput Fluids 142:79-85. https://doi.org/10.1016/j.compfluid.2016.02.012
Xu R, Stansby P, Laurence D (2009) Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach. J Comput Phys 228(18):6703-6025. https://doi.org/10.1016/j.jcp.2009.05.032
Zheng X, Ma QW, Duan WY (2014) Incompressible SPH method based on Rankine source solution for violent water wave simulation. J Comput Phys 276:291-314. https://doi.org/10.1016/j.jcp.2014.07.036
Zheng X, Ma QW, Shao S (2018) Study on SPH Viscosity Term Formulations. Appl Sci 8(2):249. https://doi.org/10.3390/app8020249
Zhou JT, Ma QW (2010) MLPG method based on Rankine source solution for modelling 3D breaking waves. Comput Model Eng Sci 56(2):179. https://doi.org/10.3970/cmes.2010.056.179

Memo

Memo:
Received date:2019-09-07;Accepted date:2020-05-12。
Corresponding author:Q. W. Ma,q.ma@city.ac.uk
Last Update: 2020-11-21