An improvement in MPS method using Voronoi diagram and a new kernel function

  • Gholamreza Shobeyri
  • HamidReza Madadi
Technical Paper


In this study, accuracy of the mesh-less moving particles semi-implicit (MPS) method in solving 2-D elliptic partial differential equations over a unit square domain with known analytical solutions is improved using Voronoi diagram drawing technique to approximate nodal volumes and a new kernel function. Voronoi diagram is employed as an alternative method for estimation of nodal volumes instead of kernel approximations. In addition to this, a kernel function with flat shape is introduced to enhance accuracy of MPS method. The numerical results obtained over highly irregular computational node distributions for the proposed kernel function show higher accuracy in comparison with two other employed kernel functions having steeper shapes.


MPS Voronoi diagram Mesh-less methods Elliptic partial differential equations Kernel function 


  1. 1.
    Afshar MH, Shobeyri G (2010) Efficient simulation of free surface flows with discrete least-squares meshless method using a priori error estimator. Int J Comput Fluid Dyn 24(9):349–367MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ataie-Ashtiani B, Farhadi L (2006) A stable moving-particle semi-implicit method for free surface flows. Fluid Dyn Res 38(4):241–256MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Barcarolo DA, Touze´ DL, Oger G, De Vuyst F (2014) Voronoi-SPH: on the analysis of a hybrid finite volumes—smoothed particle hydrodynamics method. In: 9th International SPHERIC workshop ParisGoogle Scholar
  4. 4.
    Chiaki G, Yoshida N (2015) Particle splitting in smoothed particle hydrodynamics based on Voronoi diagram. Mon Not R Astron Soc 451(4):3955–3963CrossRefGoogle Scholar
  5. 5.
    Ghaffari MA, Xiao S (2016) Smoothed particle hydrodynamics with stress points and centroid Voronoi tessellation (CVT) topology optimization. Int J Comput Methods 13(6):1650031MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics: theory and application to non spherical stars. Mon Not R Astron Soc 181:375–389CrossRefMATHGoogle Scholar
  7. 7.
    Gomez-Gesteria M, Dalrymple R (2004) Using a three-dimensional smoothed particle hydrodynamics method for wave impact on a tall structure. J Waterw Port Coast Eng 130(2):63–69CrossRefGoogle Scholar
  8. 8.
    Heb S, Springel V (2010) Particle hydrodynamics with tessellation techniques. Mon Not R Astron Soc 406(4):2289–2311CrossRefGoogle Scholar
  9. 9.
    Koshizuka S, Nobe A, Oka Y (1998) Numerical analysis of breaking waves using the moving particle semi-implicit method. Int J Numer Methods Fluids 26:751–769CrossRefMATHGoogle Scholar
  10. 10.
    Koshizuka S, Oka Y (1996) Moving-particle semi-implicit method for fragmentation of incompressible fluid. Nucl Sci Eng 123:421–434CrossRefGoogle Scholar
  11. 11.
    Koshizuka S, Tamako H, Oka Y (1995) A particle method for incompressible viscous flow with fluid fragmentation. Comput Fluid Dyn 4(1):29–46Google Scholar
  12. 12.
    Lee BH, Park JC, Kim MH, Hwang SC (2011) Step-by-step improvement of MPS method in simulating violent free-surface motions and impact-loads. Comput Methods Appl Mech Eng 200:1113–1125CrossRefMATHGoogle Scholar
  13. 13.
    Liu GR (2002) Mesh free methods: moving beyond the finite element method. CRC Press, Boca Raton, p 1420040588CrossRefGoogle Scholar
  14. 14.
    Sanchez-Mondragon J (2016) On the stabilization of unphysical pressure oscillations in MPS method simulations. Int J Numer Methods Fluids 82(8):471–492MathSciNetCrossRefGoogle Scholar
  15. 15.
    Shao SD, Lo EYM (2003) Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface. Adv Water Resour 26(7):787–800CrossRefGoogle Scholar
  16. 16.
    Shobeyri G, Afshar MH (2012) Adaptive simulation of free surface flows with discrete least squares meshless (DLSM) method using a posteriori error estimator. Eng Comput 29(8):794–813CrossRefGoogle Scholar
  17. 17.
    Shobeyri G (2017) Improving efficiency of SPH method for simulation of free surface flows using a new treatment of Neumann boundary conditions. J Br Soc Mech Sci Eng. Google Scholar
  18. 18.
    Shobeyri G, Rasti R (2017) Improving accuracy of SPH method using Voronoi diagram. Iran J Sci Technol Trans Civ Eng 41:345–350CrossRefGoogle Scholar
  19. 19.
    Tang Z, Wan D, Chen G, Xiao Q (2016) Numerical simulation of 3D violent free-surface flows by multi-resolution MPS method. J. Ocean Eng Mar Energy 2:355–364CrossRefGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2018

Authors and Affiliations

  1. 1.Department of Civil EngineeringAllaodoleh Semnani Institute of Higher Education (ASIHE)GarmsarIran
  2. 2.Department of Civil EngineeringEast Tehran Branch, Islamic Azad UniversityTehranIran

Personalised recommendations