Improving Accuracy of Laplacian Model of Incompressible SPH Method Using Higher-Order Interpolation

  • Gholamreza ShobeyriEmail author
Research paper


In this study, a new model of Laplacian operator is formulated as a hybrid of an incompressible SPH (I-SPH) method with Taylor expansion and moving least-squares method. Accuracy of the proposed Laplacian model in solving 2-D elliptic partial differential equations for a unit square computational domain is compared with the conventional I-SPH Laplacian operator. The results show significant improvement in accuracy for the proposed model on regular, highly irregular and multi-resolution irregular node distributions employed for computational domain discretization. The proposed Laplacian model because of notable accuracy can be applied for more efficient simulation of free surface flows.


Incompressible SPH Laplacian model MLS Taylor expansion Elliptic partial differential equations Mesh less 


  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):349–367MathSciNetzbMATHCrossRefGoogle Scholar
  2. Ataie-Ashtiani B, Shobeyri G (2008) Numerical simulation of landslide impulsive waves by incompressible smoothed particle hydrodynamics. Int J Numer Methods Fluids 56(2):209–232MathSciNetzbMATHCrossRefGoogle Scholar
  3. Ataie-Ashtiani B, Shobeyri G, Farhadi L (2008) Modified incompressible SPH method for simulating free surface problems. Fluid Dyn Res 40(9):637–661zbMATHCrossRefGoogle Scholar
  4. Becker M, Teschner M (2007) Weakly compressible SPH for free surface flows. In: SCA '07 proceedings of the 2007 ACM SIGGRAPH/Eurographics symposium on computer animation, pp 209–217Google Scholar
  5. Chow AD, Rogers BD, Lind SJ, Stansby PK (2018) Incompressible SPH (ISPH) with fast Poisson solver on a GPU. Comput Phys Commun 226:81–103CrossRefGoogle Scholar
  6. Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics: theory and application to non spherical stars. Mon Not R Astron Soc 181:375–389zbMATHCrossRefGoogle Scholar
  7. Gotoh H, Khayyer A (2018) On the state-of-the-art of particle methods for coastal and ocean engineering. Coast Eng J. CrossRefGoogle Scholar
  8. 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 46:104–115CrossRefGoogle Scholar
  9. Gui Q, Dong P, Shao SD (2015) Numerical study of PPE source term errors in the incompressible SPH models. Int J Numer Methods Fluids 77(6):358–379MathSciNetCrossRefGoogle Scholar
  10. Guo X, Rogers DB, Lind S, Stansby PK (2018) New massively parallel scheme for incompressible smoothed particle hydrodynamics (ISPH) for highly nonlinear and distorted flow. Comput Phys Commun 233:16–28MathSciNetCrossRefGoogle Scholar
  11. Hosseini SM, Feng JJ (2011) Pressure boundary conditions for computing incompressible flows with SPH. J Comput Phys 230(19):7473–7487MathSciNetzbMATHCrossRefGoogle Scholar
  12. Huang C, Lei JM, Liu MB, Peng XY (2016) An improved KGF-SPH with a novel discrete scheme of Laplacian operator for viscous incompressible fluid flows. Int J Numer Methods Fluids 81:377–396MathSciNetCrossRefGoogle Scholar
  13. 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 Mar Energy 1:361–376CrossRefGoogle Scholar
  14. 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):124–131CrossRefGoogle Scholar
  15. Khayyer A, Gotoh H, Shimizu Y, Gotoh K (2017a) On enhancement of energy conservation properties of projection-based particle methods. Eur J Mech B/Fluids 66:20–37MathSciNetzbMATHCrossRefGoogle Scholar
  16. Khayyer A, Gotoh H, Shimizu Y (2017b) Comparative study on accuracy and conservation properties of two particle regularization schemes and proposal of an optimized particle shifting scheme in ISPH context. J Comput Phys 332:236–256MathSciNetzbMATHCrossRefGoogle Scholar
  17. Khayyer A, Gotoh H, Falahaty H, Shimizu Y (2018a) An enhanced ISPH-SPH coupled method for simulation of incompressible fluid-elastic structure interactions. Comput Phys Commun. MathSciNetCrossRefGoogle Scholar
  18. Khayyer A, Gotoh H, Shimizu Y, Gotoh K, Falahaty H, Shao SD (2018b) Development of a projection-based SPH method for numerical wave flume with porous media of variable porosity. Coast Eng 140:1–22CrossRefGoogle Scholar
  19. 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
  20. Koshizuka S, Nobe A, Oka Y (1998) Numerical analysis of breaking waves using the moving particle semi-implicit method. Int J Numer Methods Fluids 26:751–769zbMATHCrossRefGoogle Scholar
  21. Lee ES, Moulinec C, Xu R, Violeau D, Laurence D, Stansby P (2008) Comparisons of weakly compressible and truly incompressible algorithms for the SPH free particle method. J Comput Phys 227:8417–8436MathSciNetzbMATHCrossRefGoogle Scholar
  22. Liu GR (2002) Mesh free methods: moving beyond the finite element method. CRC Press, Boca Raton, p 1420040588CrossRefGoogle Scholar
  23. 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 Mar Energy 2:279–299CrossRefGoogle Scholar
  24. Monaghan JJ (1996) Gravity currents and solitary waves. Phys D 98:523–533zbMATHCrossRefGoogle Scholar
  25. Monaghan JJ (2000) SPH without a tensile instability. J Comput Phys 159(2):290–311MathSciNetzbMATHCrossRefGoogle Scholar
  26. Omidvar P, Stansby PK, Rogers BD (2012) Wave body interaction in 2D using smoothed particle hydrodynamics (SPH) with variable particle mass. Int J Numer Methods Fluids 68:686–705MathSciNetzbMATHCrossRefGoogle Scholar
  27. Pu JH, Huang Y, Shao SD, Hussain K (2016) Three-Gorges Dam fine sediment pollutant transport: turbulence SPH model simulation of multi-fluid flows. J Appl Fluid Mech 9(1):1–10CrossRefGoogle Scholar
  28. Rezavand M, Taeibi-Rahni M, Rauch W (2018) An ISPH scheme for numerical simulation of multiphase flows with complex interfaces and high density ratios. Comput Math Appl 75(8):2658–2677MathSciNetCrossRefGoogle Scholar
  29. Schwaiger HF (2008) An implicit corrected SPH formulation for thermal diffusion with linear free surface boundary conditions. Int J Numer Methods Eng 75(6):647–671MathSciNetzbMATHCrossRefGoogle Scholar
  30. Shao SD (2010) Incompressible SPH flow model for wave interactions with porous media. Coast Eng 57:304–316CrossRefGoogle Scholar
  31. Shao SD, Gotoh H (2005) Turbulence particle models for tracking free surfaces. J Hydraul Res 43(3):276–289CrossRefGoogle Scholar
  32. 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
  33. Shobeyri G (2017) Improving efficiency of SPH method for simulation of free surface flows using a new treatment of Neumann boundary conditions. J Braz Soc Mech Sci Eng 39:5001–5014CrossRefGoogle Scholar
  34. Shobeyri G (2018) A simplified SPH method for simulation of free surface flows. Iran J Sci Technol Trans Civ Eng. CrossRefGoogle Scholar
  35. 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
  36. Shobeyri G, Ardakani RR (2017) Improving accuracy of SPH method using Voronoi Diagram. Iran J Sci Technol Trans Civ Eng 41:345–350CrossRefGoogle Scholar
  37. Shobeyri G, Yourdkhani M (2017) A new meshless approach in simulating free surface flows using continuous MLS shape functions and Voronoi Diagram. Eng Comput 34(8):2565–2581CrossRefGoogle Scholar
  38. Tamai T, Koshizuka S (2014) Least squares moving particle semi implicit method. Comput Part Mech 1(3):277–305CrossRefGoogle Scholar
  39. Tamai T, Murotani K, Koshizuka S (2017) On the consistency and convergence of particle-based meshfree discretization schemes for the Laplace operator. Comput Fluids 142:79–85MathSciNetzbMATHCrossRefGoogle Scholar
  40. Violeau D, Leroy A, Joly A, Hérault A (2018) Spectral properties of the SPH Laplacian operator. Comput Math Appl 75(10):3649–3662MathSciNetCrossRefGoogle Scholar
  41. Young DL, Chen KH, Lee CW (2005) Novel meshless method for solving the potential problems with arbitrary domain. J Comput Phys 209:290–321zbMATHCrossRefGoogle Scholar
  42. Zheng X, Duan WY, Ma QW (2010) Comparison of improved meshless interpolation schemes for SPH method and accuracy analysis. Int J Mar Sci Appl 9:223–230CrossRefGoogle Scholar
  43. Zheng X, Ma QW, Duan WY (2014) Incompressible SPH method based on Rankine source solution for violent water wave simulation. J Comput Phys 276:291–314MathSciNetzbMATHCrossRefGoogle Scholar
  44. Zheng X, Ma Q, Shao SD, Khayyer A (2017) Modelling of violent water wave propagation and impact by incompressible SPH with first-order consistent kernel interpolation scheme. Water 9:400CrossRefGoogle Scholar

Copyright information

© Shiraz University 2019

Authors and Affiliations

  1. 1.Department of Civil EngineeringAllaodoleh Semnani Institute of Higher Education (ASIHE)GarmsarIran

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