Acta Mechanica Sinica

, Volume 34, Issue 4, pp 601–613 | Cite as

Application of particle splitting method for both hydrostatic and hydrodynamic cases in SPH

  • W. T. Liu
  • P. N. SunEmail author
  • F. R. Ming
  • A. M. Zhang
Research Paper


Smoothed particle hydrodynamics (SPH) method with numerical diffusive terms shows satisfactory stability and accuracy in some violent fluid–solid interaction problems. However, in most simulations, uniform particle distributions are used and the multi-resolution, which can obviously improve the local accuracy and the overall computational efficiency, has seldom been applied. In this paper, a dynamic particle splitting method is applied and it allows for the simulation of both hydrostatic and hydrodynamic problems. The splitting algorithm is that, when a coarse (mother) particle enters the splitting region, it will be split into four daughter particles, which inherit the physical parameters of the mother particle. In the particle splitting process, conservations of mass, momentum and energy are ensured. Based on the error analysis, the splitting technique is designed to allow the optimal accuracy at the interface between the coarse and refined particles and this is particularly important in the simulation of hydrostatic cases. Finally, the scheme is validated by five basic cases, which demonstrate that the present SPH model with a particle splitting technique is of high accuracy and efficiency and is capable for the simulation of a wide range of hydrodynamic problems.


Smoothed particle hydrodynamics Particle splitting Particle refinement Solid wall boundary Fluid–structure interaction 



The project was supported by the National Natural Science Foundation of China (Grant 51609049), the Science Foundation of Heilongjiang Province (Grant QC2016061), and the Fundamental Research Funds for the Central Universities (Grants HEUGIP201701, HEUCFJ170109).


  1. 1.
    Marrone, S., Bouscasse, B., Colagrossi, A., et al.: Study of ship wave breaking patterns using 3D parallel SPH simulations. Comput. Fluids 69, 54–66 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Sun, P., Ming, F., Zhang, A.: Numerical simulation of interactions between free surface and rigid body using a robust SPH method. Ocean Eng. 98, 32–49 (2015)CrossRefGoogle Scholar
  3. 3.
    Zhang, A.-M., Sun, P.-N., Ming, F.-R., et al.: Smoothed particle hydrodynamics and its applications in fluid–structure interactions. J. Hydrodyn. Ser. B 29, 187–216 (2017)CrossRefGoogle Scholar
  4. 4.
    Shao, J.R., Li, H.Q., Liu, G.R., et al.: An improved SPH method for modeling liquid sloshing dynamics. Comput. Struct. 100–101, 18–26 (2012)CrossRefGoogle Scholar
  5. 5.
    Bouscasse, B., Colagrossi, A., Marrone, S., et al.: Nonlinear water wave interaction with floating bodies in SPH. J. Fluids Struct. 42, 112–129 (2013)CrossRefGoogle Scholar
  6. 6.
    Jiang, T., Lu, L.G., Lu, W.G.: The numerical investigation of spreading process of two viscoelastic droplets impact problem by using an improved SPH scheme. Comput. Mech. 53, 977–999 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Jiang, T., Ren, J.L., Lu, W.G., et al.: A corrected particle method with high-order Taylor expansion for solving the viscoelastic fluid flow. Acta Mech. Sin. 33, 1–20 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ming, F.R., Sun, P.N., Zhang, A.M.: Numerical investigation of rising bubbles bursting at a free surface through a multiphase SPH model. Meccanica 52, 2665–2684 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Zhang, A., Sun, P., Ming, F.: An SPH modeling of bubble rising and coalescing in three dimensions. Comput. Methods Appl. Mech. Eng. 294, 189–209 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ming, F.R., Zhang, A.M., Xue, Y.Z., et al.: Damage characteristics of ship structures subjected to shockwaves of underwater contact explosions. Ocean Eng. 117, 359–382 (2016)CrossRefGoogle Scholar
  11. 11.
    Zhang, A.M., Yang, W.S., Huang, C., et al.: Numerical simulation of column charge underwater explosion based on SPH and BEM combination. Comput. Fluids 71, 169–178 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Zhang, Z., Wang, L., Silberschmidt, V.V., et al.: SPH–FEM simulation of shaped-charge jet penetration into double hull: a comparison study for steel and SPS. Compos. Struct. 155, 135–144 (2016)CrossRefGoogle Scholar
  13. 13.
    Liu, M.B., Liu, G.R.: Smoothed particle hydrodynamics (SPH): an overview and recent developments. Arch. Comput. Methods Eng. 17, 25–76 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ren, J., Jiang, T., Lu, W., et al.: An improved parallel SPH approach to solve 3D transient generalized Newtonian free surface flows. Comput. Phys. Commun. 205, 87–105 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gan, B.S., Nguyen, D.K., Han, A.L., et al.: Proposal for fast calculation of particle interactions in SPH simulations. Comput. Fluids 104, 20–29 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Sun, P., Colagrossi, A., Marrone, S., et al.: The \(\delta \)plus-SPH model: simple procedures for a further improvement of the SPH scheme. Comput. Methods Appl. Mech. Eng. 315, 25–49 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Antuono, M., Colagrossi, A., Marrone, S.: Numerical diffusive terms in weakly-compressible SPH schemes. Comput. Phys. Commun. 183, 2570–2580 (2012)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Barcarolo, D.A. Improvement of the precision and the efficiency of the SPH method: theoretical and numerical study. [Ph.D. Thesis], Centrale Nantes, France (2013)Google Scholar
  19. 19.
    Liu, G.R., Liu, M.B.: Smoothed Particle Hydrodynamics: A Meshfree Particle Method. World Scientific, Singapore (2003)CrossRefzbMATHGoogle Scholar
  20. 20.
    Marsh, A., Oger, G., Touzé, D.l., et al.: Validation of a conservative variable-resolution SPH scheme including \(\nabla \)h terms. In: The 6th International SPHERIC Workshop, Hambourg, Germany (2011)Google Scholar
  21. 21.
    Koukouvinis, P.K., Anagnostopoulos, J.S., Papantonis, D.E.: Simulation of 2D wedge impacts on water using the SPH–ALE method. Acta Mech. Sin. 224, 2559–2575 (2013)CrossRefzbMATHGoogle Scholar
  22. 22.
    Feldman, J., Bonet, J.: Dynamic refinement and boundary contact forces in SPH with applications in fluid flow problems. Int. J. Numer. Methods Eng. 72, 295–324 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Omidvar, P., Stansby, P.K., Rogers, B.D.: Wave body interaction in 2D using smoothed particle hydrodynamics (SPH) with variable particle mass. Int. J. Numer. Methods Fluids 68, 686–705 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    López, Y.R., Roose, D., Morfa, C.R.: Dynamic particle refinement in SPH: application to free surface flow and non-cohesive soil simulations. Comput. Mech. 51, 731–741 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Antuono, M., Bouscasse, B., Colagrossi, A., et al.: A measure of spatial disorder in particle methods. Comput. Phys. Commun. 185, 2609–2621 (2014)CrossRefGoogle Scholar
  26. 26.
    Vacondio, R., Rogers, B.D., Stansby, P.K., et al.: Variable resolution for SPH: a dynamic particle coalescing and splitting scheme. Comput. Methods Appl. Mech. Eng. 256, 132–148 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Omidvar, P., Stansby, P.K., Rogers, B.D.: SPH for 3D floating bodies using variable mass particle distribution. Int. J. Numer. Methods Fluids 72, 427–452 (2013)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Barcarolo, D.A., Oger, G., Vuyst, F.D.: Adaptive particle refinement and derefinement applied to the smoothed particle hydrodynamics method. J. Comput. Phys. 273, 640–657 (2014)CrossRefzbMATHGoogle Scholar
  29. 29.
    Monaghan, J.J.: Simulating free surface flows with SPH. J. Comput. Phys. 110, 399–406 (1994)CrossRefzbMATHGoogle Scholar
  30. 30.
    Colagrossi, A., Landrini, M.: Numerical simulation of interfacial flows by smoothed particle hydrodynamics. J. Comput. Phys. 191, 448–475 (2003)CrossRefzbMATHGoogle Scholar
  31. 31.
    Marrone, S., Antuono, M., Colagrossi, A., et al.: \(\delta \)-SPH model for simulating violent impact flows. Comput. Methods Appl. Mech. Eng. 200, 1526–1542 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Dehnen, W., Aly, H.: Improving convergence in smoothed particle hydrodynamics simulations without pairing instability. Mon. Not. R. Astron. Soc. 425, 1068–1082 (2012)CrossRefGoogle Scholar
  33. 33.
    Adami, S., Hu, X.Y., Adams, N.A.: A generalized wall boundary condition for smoothed particle hydrodynamics. J. Comput. Phys. 231, 7057–7075 (2012)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Zhang, A.M., Cao, X.Y., Ming, F.R., et al.: Investigation on a damaged ship model sinking into water based on three dimensional SPH method. Appl. Ocean Res. 42, 24–31 (2013)CrossRefGoogle Scholar
  35. 35.
    Colagrossi, A., Soutoiglesias, A., Antuono, M., et al.: Smoothed particle hydrodynamics modeling of dissipation mechanisms in gravity waves. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 87, 023302 (2013)CrossRefGoogle Scholar
  36. 36.
    Marrone, S., Colagrossi, A., Antuono, M., et al.: An accurate SPH modeling of viscous flows around bodies at low and moderate Reynolds numbers. J. Comput. Phys. 245, 456–475 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Guo, K., Sun, P.-N., Cao, X.-Y., et al.: A 3-D SPH model for simulating water flooding of a damaged floating structure. J. Hydrodyn. Ser. B 29, 831–844 (2017)CrossRefGoogle Scholar
  38. 38.
    Monaghan, J.J., Gingold, R.A.: Shock simulation by the particle method SPH. J. Comput. Phys. 52, 374–389 (1983)CrossRefzbMATHGoogle Scholar
  39. 39.
    Colagrossi, A., Bouscasse, B., Antuono, M., et al.: Particle packing algorithm for SPH schemes. Comput. Phys. Commun. 183, 1641–1653 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Greco, M.: A two-dimensional study of green-water loading. [Ph.D. Thesis], Norwegian University of Science and Technology, Norway (2001)Google Scholar
  41. 41.
    Buchner, B. Green water on ship-type offshore structures. [Ph.D. Thesis], Technische Universiteit Delft, Netherlands (2002)Google Scholar
  42. 42.
    Yettou, E.M., Desrochers, A., Champoux, Y.: Experimental study on the water impact of a symmetrical wedge. Fluid Dyn. Res. 38, 47–66 (2006)CrossRefzbMATHGoogle Scholar
  43. 43.
    Federico, I., Marrone, S., Colagrossi, A., et al.: Simulating 2D open-channel flows through an SPH model. Eur. J. Mech. 34, 35–46 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Sun, P., Colagrossi, A., Marrone, S., et al.: Detection of Lagrangian coherent structures in the SPH framework. Comput. Methods Appl. Mech. Eng. 305, 849–868 (2016)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Lind, S., Xu, R., Stansby, P., et al.: 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, 1499–1523 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Huang, C., Zhang, D.H., Shi, Y.X., et al.: Coupled finite particle method with a modified particle shifting technology. Int. J. Numeri. Methods Eng. 113, 197–207 (2018).
  47. 47.
    Pinelli, A., Naqavi, I.Z., Piomelli, U., et al.: Immersed-boundary methods for general finite-difference and finite-volume Navier–Stokes solvers. J. Comput. Phys. 229, 9073–9091 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Vanella, M., Balaras, E.: A moving-least-squares reconstruction for embedded-boundary formulations. J. Comput. Phys. 228, 6617–6628 (2009)CrossRefzbMATHGoogle Scholar
  49. 49.
    Cai, S.G., Ouahsine, A., Favier, J., et al.: Improved implicit immersed boundary method via operator splitting. In: Ibrahimbegovic, A. (ed.) Computational Methods for Solids and Fluids. Computational Methods in Applied Sciences, vol. 41. Springer, Cham (2016)Google Scholar
  50. 50.
    Constant, E., Favier, J., Meldi, M., et al.: An immersed boundary method in OpenFOAM: verification and validation. Comput. Fluids 157, 55–72 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • W. T. Liu
    • 1
  • P. N. Sun
    • 1
    Email author
  • F. R. Ming
    • 1
  • A. M. Zhang
    • 1
  1. 1.College of Shipbuilding EngineeringHarbin Engineering UniversityHarbinChina

Personalised recommendations