A conservative and consistent Lagrangian gradient smoothing method for simulating free surface flows in hydrodynamics

  • Zirui MaoEmail author
  • G. R. Liu
  • Xiangwei Dong
  • Tao Lin


A novel particle method, Lagrangian gradient smoothing method (L-GSM), has been proposed in our earlier work to avoid the tensile instability problem inherently existed in SPH, through replacing the SPH gradient operator with a robust GSM gradient operator. However, the nominal area of each L-GSM particle determined by the relative location of particles is always inconsistent with the real representative area of it in simulation, especially in large-deformation problems. This is why the earlier L-GSM model has to be limited to the solid-like flow simulations where the deformation is not very serious. In this work, a conservative and consistent Lagrangian gradient smoothing method (CCL-GSM) is developed for handling large-deformation problems in hydrodynamics with an arbitrarily changing free surface profile. This is achieved by deriving a conservative and consistent form for the discretized Navier–Stokes governing equations in L-GSM, which even holds in the neighbor-updating or ‘remeshing’ process. Special techniques are also devised for free surface treatment, which is important to restore the conservation and consistency manner of CCL-GSM simulation on free surface boundary. The effectiveness of the proposed CCL-GSM framework is evaluated with a number of benchmarking examples, including dam break, wall impacts of breaking dam, water discharge and water splash. It shows that the CCL-GSM model can handle the incompressible flows with complicated free surfaces effectively and easily. The results comparison with experiments and SPH solutions demonstrates that the CCL-GSM can give a desirable result for all these examples.


Conservative and consistent Lagrangian gradient smoothing method L-GSM SPH Free surface flows Incompressible flows Hydrodynamics 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon Not R Astron Soc 181(3):375–389zbMATHGoogle Scholar
  2. 2.
    Lucy LB (1977) A numerical approach to the testing of the fission hypothesis. Astron J 82:1013–1024Google Scholar
  3. 3.
    Liu GR, Liu MB (2003) Smoothed particle hydrodynamics: a meshfree particle method. World Scientific, SingaporezbMATHGoogle Scholar
  4. 4.
    Liu GR, Quek SS (2014) The finite element method: a practical course, 2nd edn. Butterworth-Heinemann, OxfordzbMATHGoogle Scholar
  5. 5.
    Müller M, Schirm S, Teschner M (2004) Interactive blood simulation for virtual surgery based on smoothed particle hydrodynamics. Technol Health Care 12(1):25–31Google Scholar
  6. 6.
    Hieber SE, Walther JH, Koumoutsakos P (2004) Remeshed smoothed particle hydrodynamics simulation of the mechanical behavior of human organs. Technol Health Care 12(4):305–314Google Scholar
  7. 7.
    Monaghan JJ, Kocharyan A (1995) SPH simulation of multi-phase flow. Comput Phys Commun 87(1–2):225–235zbMATHGoogle Scholar
  8. 8.
    Bin Wang Z, Chen R, Wang H, Liao Q, Zhu X, Li SZ (2016) An overview of smoothed particle hydrodynamics for simulating multiphase flow. Appl Math Model 40(23–24):1339–1351MathSciNetGoogle Scholar
  9. 9.
    Yan X, Jiang Y-T, Li C-F, Martin RR, Hu S-M (2016) Multiphase SPH simulation for interactive fluids and solids. ACM Trans Graph 35(4):1–11Google Scholar
  10. 10.
    Swegle JW, Attaway SW (1995) On the feasibility of using Smoothed Particle Hydrodynamics for underwater explosion calculations. Comput Mech 17(3):151–168zbMATHGoogle Scholar
  11. 11.
    Liu MB, Liu GR, Lam KY, Zong Z (2003) Smoothed particle hydrodynamics for numerical simulation of underwater explosion. Comput Mech 30(2):106–118zbMATHGoogle Scholar
  12. 12.
    Liu MB, Feng DL, Guo ZM (2013) Recent developments of SPH in modelling explosion and impact problems. In: International conference on particle-based methods—fundamentals and applications, pp 1–8Google Scholar
  13. 13.
    Rabczuk T, Eibl J (2003) Simulation of high velocity concrete fragmentation using SPH/MLSPH. Int J Numer Methods Eng 56(10):1421–1444zbMATHGoogle Scholar
  14. 14.
    Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H (2010) A simple and robust three-dimensional cracking-particle method without enrichment. Comput Methods Appl Mech Eng 199(37–40):2437–2455zbMATHGoogle Scholar
  15. 15.
    Fan H, Li S (2017) Parallel peridynamics–SPH simulation of explosion induced soil fragmentation by using OpenMP. Comput Part Mech 4(2):199–211Google Scholar
  16. 16.
    Seo S, Min O, Lee J (2008) Application of an improved contact algorithm for penetration analysis in SPH. Int J Impact Eng 35(6):578–588Google Scholar
  17. 17.
    Kulak RF (2011) Modeling of cone penetration test using SPH and MM-ALE approaches. In: 8th European LS-DYNA users conference, pp 1–10Google Scholar
  18. 18.
    Johnson GR, Stryk RA, Beissel SR (1996) SPH for high velocity impact computations. Comput Methods Appl Mech Eng 139(1–4):347–373zbMATHGoogle Scholar
  19. 19.
    Mehra V, Chaturvedi S (2006) High velocity impact of metal sphere on thin metallic plates: a comparative smooth particle hydrodynamics study. J Comput Phys 212(1):318–337zbMATHGoogle Scholar
  20. 20.
    Alhussan KA, Babenko VA, Kozlov IM, Smetannikov AS (2012) Development of modified SPH approach for modeling of high-velocity impact. Int J Heat Mass Transf 55(23–24):6340–6348Google Scholar
  21. 21.
    Bui HH, Fukagawa R, Sako K, Ohno S (2008) Lagrangian meshfree particles method (SPH) for large deformation and failure flows of geomaterial using elastic-plastic soil constitutive model. Int J Numer Anal Methods Geomech 32(12):1537–1570zbMATHGoogle Scholar
  22. 22.
    Dai Z, Huang Y, Cheng H, Xu Q (2014) 3D numerical modeling using smoothed particle hydrodynamics of flow-like landslide propagation triggered by the 2008 Wenchuan earthquake. Eng Geol 180:21–33Google Scholar
  23. 23.
    Hu M, Liu MB, Xie MW, Liu GR (2015) Three-dimensional run-out analysis and prediction of flow-like landslides using smoothed particle hydrodynamics. Environ Earth Sci 73(4):1629–1640Google Scholar
  24. 24.
    Nguyen CT, Nguyen CT, Bui HH, Nguyen GD, Fukagawa R (2017) A new SPH-based approach to simulation of granular flows using viscous damping and stress regularisation. Landslides 14(1):69–81Google Scholar
  25. 25.
    Mao Z, Liu GR (2018) A smoothed particle hydrodynamics model for electrostatic transport of charged lunar dust on the moon surface. Comput Part Mech 5:539–551Google Scholar
  26. 26.
    Zhang C, Hu XY, Adams NA (2017) A generalized transport-velocity formulation for smoothed particle hydrodynamics. J Comput Phys 337:216–232MathSciNetzbMATHGoogle Scholar
  27. 27.
    Mehra V, Cd S, Mishra V, Chaturvedi S (2018) Tensile instability and artificial stresses in impact problems in SPH. J Phys Conf Ser 377:012102Google Scholar
  28. 28.
    Xu X, Ouyang J, Yang B, Liu Z (2013) SPH simulations of three-dimensional non-Newtonian free surface flows. Comput Methods Appl Mech Eng 256:101–116MathSciNetzbMATHGoogle Scholar
  29. 29.
    Mao Z, Liu GR, Dong X (2017) A comprehensive study on the parameters setting in smoothed particle hydrodynamics (SPH) method applied to hydrodynamics problems. Comput Geotech 92:77–95Google Scholar
  30. 30.
    Fang HS, Bao K, Wei JA, Zhang H, Wu EH, Zheng LL (2009) Simulations of droplet spreading and solidification using an improved SPH model. Numer Heat Transf Part A Appl 55(2):124–143Google Scholar
  31. 31.
    Sirotkin FV, Yoh JJ (2012) A new particle method for simulating breakup of liquid jets. J Comput Phys 231(4):1650–1674MathSciNetzbMATHGoogle Scholar
  32. 32.
    Yang X, Liu M, Peng S (2014) Smoothed particle hydrodynamics modeling of viscous liquid drop without tensile instability. Comput Fluids 92:199–208MathSciNetzbMATHGoogle Scholar
  33. 33.
    Naceur H, Lin J, Coutellier D, Laksimi A (2015) Efficient smoothed particle hydrodynamics method for the analysis of planar structures undergoing geometric nonlinearities. J Mech Sci Technol 29(5):2147–2155Google Scholar
  34. 34.
    Swegle JW, Hicks DL, Attaway SW (1995) Smoothed particle hydrodynamics stability analysis. J Comput Phys 116(1):123–134MathSciNetzbMATHGoogle Scholar
  35. 35.
    Liu GR, Xu GX (2008) A gradient smoothing method (GSM) for fluid dynamics problems. Int J Numer Methods Fluids 58(10):1101–1133zbMATHGoogle Scholar
  36. 36.
    Chen J-S, Wu C-T, Yoon S, You Y (2001) A stabilized conforming nodal integration for Galerkin mesh-free methods. Int J Numer Methods Eng 50(2):435–466zbMATHGoogle Scholar
  37. 37.
    Chen J-S, Yoon S, Wu C-T (2002) Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methods. Int J Numer Methods Eng 53(12):2587–2615zbMATHGoogle Scholar
  38. 38.
    Li E, Tan V, Xu GX, Liu GR, He ZC (2011) A novel linearly-weighted gradient smoothing method (LWGSM) in the simulation of fluid dynamics problem. Comput Fluids 50(1):104–119MathSciNetzbMATHGoogle Scholar
  39. 39.
    Li E, Tan V, Xu GX, Liu GR, He ZC (2012) A novel alpha gradient smoothing method (αGSM) for fluid problems. Numer Heat Transf Part B Fundam 61(3):204–228Google Scholar
  40. 40.
    Li E, Liu GR, Xu GX, Vincent T, He ZC (2012) Numerical modeling and simulation of pulsatile blood flow in rigid vessel using gradient smoothing method. Eng Anal Bound Elem 36(3):322–334MathSciNetzbMATHGoogle Scholar
  41. 41.
    Yao J, Liu GR, Qian D, Chen CL, Xu G (2013) A moving-mesh gradient smoothing method for compressible CFD problems. Math Model Methods Appl Sci 23(02):273–305MathSciNetzbMATHGoogle Scholar
  42. 42.
    Wang S, Khoo BC, Liu GR, Xu GX (2013) An arbitrary Lagrangian–Eulerian gradient smoothing method (GSM/ALE) for interaction of fluid and a moving rigid body. Comput Fluids 71:327–347MathSciNetzbMATHGoogle Scholar
  43. 43.
    Liu GR, Zhang J, Lam KY, Li H, Xu G, Zhong ZH, Li GY, Han X (2008) A gradient smoothing method (GSM) with directional correction for solid mechanics problems. Comput Mech 41(3):457–472zbMATHGoogle Scholar
  44. 44.
    Zhang J, Liu GR, Lam KY, Li H, Xu G (2008) A gradient smoothing method (GSM) based on strong form governing equation for adaptive analysis of solid mechanics problems. Finite Elem Anal Des 44(15):889–909MathSciNetGoogle Scholar
  45. 45.
    Liu GR (2008) A generalized gradient smoothing technique and the smoothed bilinear form for galerkin formulation of a wide class of computational methods. Int J Comput Methods 05(02):199–236MathSciNetzbMATHGoogle Scholar
  46. 46.
    Mao Z, Liu GR (2018) A Lagrangian gradient smoothing method for solid-flow problems using simplicial mesh. Int J Numer Methods Eng 113(5):858–890MathSciNetGoogle Scholar
  47. 47.
    Mao Z, Liu G, Huang Y, Bao Y (2019) A conservative and consistent Lagrangian gradient smoothing method for earthquake-induced landslide simulation. Eng Geol. Google Scholar
  48. 48.
    Mao Z, Liu G, Huang Y (2019) A local Lagrangian gradient smoothing method for fluids and fluid-like solids: a novel particle-like method. Eng Anal Boun Elements. Google Scholar
  49. 49.
    Monaghan JJ (1994) Simulating Free Surface Flows with SPH. J Comput Phys 110(2):399–406MathSciNetzbMATHGoogle Scholar
  50. 50.
    Monaghan JJ (1985) Artificial viscosity for particle methods J.J. Monaghan and H. Pongracic. Appl Numer Math 1:187–194Google Scholar
  51. 51.
    Lee DT, Schachter BJ (1980) Two algorithms for constructing a Delaunay triangulation. Int J Comput Inf Sci 9(3):219–242MathSciNetzbMATHGoogle Scholar
  52. 52.
    Sloan S (1987) A fast algorithm for constructing Delaunay triangulations in the plane. Adv Eng Softw 9(1):34–55MathSciNetzbMATHGoogle Scholar

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© OWZ 2019

Authors and Affiliations

  1. 1.Department of Aerospace Engineering and Engineering MechanicsUniversity of CincinnatiCincinnatiUSA
  2. 2.Taiyuan University of TechnologyTaiyuanChina
  3. 3.College of Mechanical and Electronic EngineeringChina University of PetroleumQingdaoChina

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