Comparison of improved meshless interpolation schemes for SPH method and accuracy analysis

Abstract

In the smoothed particle hydrodynamics (SPH) method, a meshless interpolation scheme is needed for the unknown function in order to discretize the governing equation. A particle approximation method has so far been used for this purpose. Traditional particle interpolation (TPI) is simple and easy to do, but its low accuracy has become an obstacle to its wider application. This can be seen in the cases of particle disorder arrangements and derivative calculations. There are many different methods to improve accuracy, with the moving least square (MLS) method one of the most important meshless interpolation methods. Unfortunately, it requires complex matrix computing and so is quite time-consuming. The authors developed a simpler scheme, called higher-order particle interpolation (HPI). This scheme can get more accurate derivatives than the MLS method, and its function value and derivatives can be obtained simultaneously. Although this scheme was developed for the SPH method, it has been found useful for other meshless methods.

Keywords

higher order particle interpolation (HPI) / SPH / meshless method / moving least square (MLS)

References

[1] Atluri S.N., Shen S. The meshless local Petrove-Galerkin (MLPG) method: a simple & less-costly alternative to the finite element and boundary element methods. Computer Modeling in Engineering & Sciences, 2002, 3(1): 11-52

[2] Belytschko T., krongauz Y., Organ D., Fleming M., Krysl P. Meshless method: an overview and recent developments. Computer Methods in Applied Mechanics and Engineering, 1996, 169: 3-47

[3] Colagrossi A., Landrini M. Numerical simulation of interfacial flow by smoothed particle hydrodynamics. Journal of Computational Physics, 2003, 191: 448-475

[4] Gingold R.A., Monaghan J.J. Smoothed particle hydrodynamics: theory and application to non-spherical stars. Monthly Notices of Royal Astronomical Society, 1997, 181: 375-389

[5] Lo E.Y.M., Shao S. Simulation of near-shore solitary wave mechanics by an incompressible SPH method. Applied Ocean Research, 2002, 24: 275-286

[6] Lucy L.B. A numerical approach to the testing of the fission hypothesis. The Astronomical Journal, 1997, 82(12): 1013-1024

[7] Ma Q. Meshless local Petrov-Galerkin method for two-dimensional nonlinear water wave problems. Journal of Computational Physics, 2005, 205(2): 611-625

[8] Ma Q. MLPG Method Based on Rankine source solution for simulating nonlinear water waves. Computer Modeling in Engineering & Sciences, 2005, 9(2): 193-210

[9] Ma Q. A new meshless interpolation scheme for MLPG_R method. Computer Modeling in Engineering & Sciences, 2008, 23(2): 75-89

[10] Monaghan J.J. Simulating free surface flows with SPH. Journal of Computational Physics, 1994, 110: 399-406

[11] Souto I.A., Delorme L., Perez R.L., Abril-Perez S. Liquid moment amplitude assessment in sloshing type problems with smooth particle hydrodynamics. Ocean Engineering, 2006, 33: 1462-1484

[12] Souto I.A., Perez R.L., Zamora R.R. Simulation of anti-roll tanks and sloshing type problems with smoothed particle hydrodynamics. Ocean Engineering, 2004, 31: 1169-1192

[13] Wu G., Hu Z. A Taylor series based finite volume method for the Navier-Stokes equations. Journal for Numerical Methods in Fluids, 2008, 58(12): 1299-1405

[14] Zheng X., Duan W. Study on the precision of second order algorithm for smoothed particle hydrodynamics. Advances in Water Science, 2008, 19(6): 86-92