Application of hourglass control to Eulerian smoothed particle hydrodynamics

Computational Particle Mechanics volume 8, pages51–67(2021)Cite this article

Abstract

Being a truly meshless method, smoothed particle hydrodynamics (SPH) raises expectations to naturally handle solid mechanics problems of large deformations. However, in a simple formulation it severely suffers from two instabilities, namely tensile instability and zero-energy modes, which hinders SPH from being an popular numerical tool in that area. Although Lagrangian SPH completely removes tensile instability, it is not yet able to prevent zero-energy modes. Furthermore, kernel updates are required to properly handle very large deformations which again triggers tensile instability. Additionally, Lagrangian SPH cannot naturally deal with contact problems. Pursuing an alternative route, this paper aims at stabilizing Eulerian SPH in order to accurately deal with large deformations while preserving the fundamental properties of SPH to easily handle contact problems as well as fluid–structure interaction in a straightforward monolithic manner. For this purpose, an hourglass control scheme already employed to prevent zero-energy modes in Lagrangian SPH framework is used. The advantage of the present scheme is that the stabilization method can be easily implemented in any Eulerian SPH code by making only few changes to the code. The proposed scheme is employed to simulate several cases of elasticity, plasticity, fracture and fluid–structure interaction in order to assess its accuracy and effectiveness. The obtained results are compared with analytical solutions and finite element results where very good agreement is found.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

US$ 39.95

Tax calculation will be finalised during checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

References

  1. 1.

    Adami S, Hu XY, Adams NA (2012) A generalized wall boundary condition for smoothed particle hydrodynamics. J Comput Phys 231(21):7057–7075. https://doi.org/10.1016/j.jcp.2012.05.005

    MathSciNet  Article  Google Scholar 

  2. 2.

    Antoci C, Gallati M, Sibilla S (2007) Numerical simulation of fluid-structure interaction by SPH. Comput Struct 85(11–14):879–890. https://doi.org/10.1016/j.compstruc.2007.01.002

    Article  Google Scholar 

  3. 3.

    Antuono M, Colagrossi A, Marrone S (2012) Numerical diffusive terms in weakly-compressible SPH schemes. Comput Phys Commun 183(12):2570–2580. https://doi.org/10.1016/j.cpc.2012.07.006

    MathSciNet  Article  Google Scholar 

  4. 4.

    Ba K, Gakwaya A (2018) Thermomechanical total Lagrangian SPH formulation for solid mechanics in large deformation problems. Comput Methods Appl Mech Eng 342:458–473. https://doi.org/10.1016/j.cma.2018.07.038

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Beissel S, Belytschko T (1996) Nodal integration of the element-free Galerkin method. Comput Methods Appl Mech Eng 139(1–4):49–74. https://doi.org/10.1016/S0045-7825(96)01079-1

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Belytschko T, Guo Y, Kam Liu W, Ping Xiao S (2000) A unified stability analysis of meshless particle methods. Int J Numer Methods Eng 48(9):1359–1400. https://doi.org/10.1002/1097-0207(20000730)48:9\(<\)1359::AID-NME829\(>\)3.0.CO;2-U

  7. 7.

    Bonet J, Kulasegaram S (2000) Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations. Int J Numer Methods Eng 47(6):1189–1214. https://doi.org/10.1002/(SICI)1097-0207(20000228)47:6%3C1189::AID-NME830%3E3.0.CO;2-I

  8. 8.

    Bonet J, Kulasegaram S (2001) Remarks on tension instability of eulerian and lagrangian corrected smooth particle hydrodynamics (CSPH) methods. Int J Numer Methods Eng 52(11):1203–1220. https://doi.org/10.1002/nme.242

    Article  MATH  Google Scholar 

  9. 9.

    Chiron L, Marrone S, Di Mascio A, Le Touzé D (2018) Coupled SPH-FV method with net vorticity and mass transfer. J Comput Phys 364:111–136. https://doi.org/10.1016/j.jcp.2018.02.052

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Cleary PW, Prakash M, Das R, Ha J (2012) Modelling of metal forging using SPH. Appl Math Model 36(8):3836–3855. https://doi.org/10.1016/j.apm.2011.11.019

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Dehnen W, Aly H (2012) Improving convergence in smoothed particle hydrodynamics simulations without pairing instability. Mon Not R Astron Soc 425(2):1068–1082. https://doi.org/10.1111/j.1365-2966.2012.21439.x

    Article  Google Scholar 

  12. 12.

    Dong X, Li Z, Feng L, Sun Z, Fan C (2017) Modeling, simulation, and analysis of the impact(s) of single angular-type particles on ductile surfaces using smoothed particle hydrodynamics. Powder Technol 318:363–382. https://doi.org/10.1016/j.powtec.2017.06.011

    Article  Google Scholar 

  13. 13.

    Dyka CT, Randles PW, Ingel RP (1997) Stress points for tension instability in SPH. Int J Numer Methods Eng 40(13):2325–2341. https://doi.org/10.1002/(SICI)1097-0207(19970715)40:13\(<\)2325::AID-NME161\(>\)3.0.CO;2-8

  14. 14.

    Flanagan DP, Belytschko T (1981) A uniform strain hexahedron and quadrilateral with orthogonal hourglass control. Int J Numer Methods Eng 17(5):679–706. https://doi.org/10.1002/nme.1620170504

    Article  MATH  Google Scholar 

  15. 15.

    Ganzenmüller GC (2015) An hourglass control algorithm for lagrangian smooth particle hydrodynamics. Comput Methods Appl Mech Eng 286:87–106. https://doi.org/10.1016/j.cma.2014.12.005

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Ganzenmüller GC, Sauer M, May M, Hiermaier S (2016) Hourglass control for smooth particle hydrodynamics removes tensile and rank-deficiency instabilities. Eur Phys J Spec Top 225(2):385–395. https://doi.org/10.1140/epjst/e2016-02631-x

    Article  Google Scholar 

  17. 17.

    Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon Not R Astron Soc 181(3):375–389. https://doi.org/10.1093/mnras/181.3.375

    Article  MATH  Google Scholar 

  18. 18.

    Gray JP, Monaghan JJ, Swift RP (2001) SPH elastic dynamics. Comput Methods Appl Mech Eng 190(49–50):6641–6662. https://doi.org/10.1016/S0045-7825(01)00254-7

    Article  MATH  Google Scholar 

  19. 19.

    Hirschler M, Oger G, Nieken U, Le Touzé D (2017) Modeling of droplet collisions using smoothed particle hydrodynamics. Int J Multiph Flow 95:175–187. https://doi.org/10.1016/j.ijmultiphaseflow.2017.06.002

    MathSciNet  Article  Google Scholar 

  20. 20.

    House JW, Lewis JC, Gillis PP, Wilson LL (1995) Estimation of flow stress under high rate plastic deformation. Int J Impact Eng 16(2):189–200. https://doi.org/10.1016/0734-743X(94)00042-U

    Article  Google Scholar 

  21. 21.

    Johnson GR, Cook WH (1985) Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Eng Fract Mech 21(1):31–48. https://doi.org/10.1016/0013-7944(85)90052-9

    Article  Google Scholar 

  22. 22.

    Krongauz Y, Belytschko T (1997) Consistent pseudo-derivatives in meshless methods. Comput Methods Appl Mech Eng 146(3–4):371–386. https://doi.org/10.1016/S0045-7825(96)01234-0

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Libersky LD, Randles PW, Carney TC, Dickinson DL (1997) Recent improvements in sph modeling of hypervelocity impact. Int J Impact Eng 20(6–10):525–532. https://doi.org/10.1016/S0734-743X(97)87441-6

    Article  Google Scholar 

  24. 24.

    Lind SJ, Xu R, Stansby PK, Rogers BD (2012) 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(4):1499–1523. https://doi.org/10.1016/j.jcp.2011.10.027

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Liu MB, Zhang ZL, Feng DL (2017) A density-adaptive SPH method with kernel gradient correction for modeling explosive welding. Comput Mech 60(3):513–529. https://doi.org/10.1007/s00466-017-1420-5

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Lucy LB (1977) A numerical approach to the testing of the fission hypothesis. Astron J 82:1013. https://doi.org/10.1086/112164

    Article  Google Scholar 

  27. 27.

    Maurel B, Potapov S, Fabis J, Combescure A (2009) Full SPH fluid-shell interaction for leakage simulation in explicit dynamics. Int J Numer Methods Eng 80(2):210–234. https://doi.org/10.1002/nme.2629

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Monaghan JJ (1992) Smoothed particle hydrodynamics. Ann Rev Astron Astrophys 30(1):543–574. https://doi.org/10.1146/annurev.aa.30.090192.002551

    Article  Google Scholar 

  29. 29.

    Monaghan JJ, Mériaux CA (2018) An SPH study of driven turbulence near a free surface in a tank under gravity. Eur J Mech B Fluids 68:201–210. https://doi.org/10.1016/j.euromechflu.2017.12.008

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Neto EAdS, Perić D, Owens D (2008) Computational methods for plasticity: theory and applications. Wiley, Oxford

    Google Scholar 

  31. 31.

    Polfer P, Kraft T, Bierwisch C (2016) Suspension modeling using smoothed particle hydrodynamics: accuracy of the viscosity formulation and the suspended body dynamics. Appl Math Model 40(4):2606–2618. https://doi.org/10.1016/j.apm.2015.10.013

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Rabczuk T, Belytschko T, Xiao SP (2004) Stable particle methods based on lagrangian kernels. Comput Methods Appl Mech Eng 193(12–14):1035–1063. https://doi.org/10.1016/j.cma.2003.12.005

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Rafiee A, Thiagarajan KP (2009) An SPH projection method for simulating fluid-hypoelastic structure interaction. Comput Methods Appl Mech Eng 198(33–36):2785–2795. https://doi.org/10.1016/j.cma.2009.04.001

    Article  MATH  Google Scholar 

  34. 34.

    Rafiee A, Manzari MT, Hosseini M (2007) An incompressible SPH method for simulation of unsteady viscoelastic free-surface flows. Int J Non-Linear Mech 42(10):1210–1223. https://doi.org/10.1016/j.ijnonlinmec.2007.09.006

    Article  MATH  Google Scholar 

  35. 35.

    Randles PW, Libersky LD (2000) Normalized SPH with stress points. Int J Numer Methods Eng 48(10):1445–1462. https://doi.org/10.1002/1097-0207(20000810)48:10%3C1445::AID-NME831%3E3.0.CO;2-9

  36. 36.

    Sun PN, Colagrossi A, Marrone S, Zhang AM (2017) The \(\delta \)plus-sph model: Simple procedures for a further improvement of the SPH scheme. Comput Methods Appl Mech Eng 315:25–49. https://doi.org/10.1016/j.cma.2016.10.028

    MathSciNet  Article  MATH  Google Scholar 

  37. 37.

    Swegle JW, Hicks DL, Attaway SW (1995) Smoothed particle hydrodynamics stability analysis. J Comput Phys 116(1):123–134. https://doi.org/10.1006/jcph.1995.1010

    MathSciNet  Article  MATH  Google Scholar 

  38. 38.

    Takaffoli M, Papini M (2012) Material deformation and removal due to single particle impacts on ductile materials using smoothed particle hydrodynamics. Wear 274–275:50–59. https://doi.org/10.1016/j.wear.2011.08.012

    Article  Google Scholar 

  39. 39.

    Vidal Y, Bonet J, Huerta A (2007) Stabilized updated Lagrangian corrected SPH for explicit dynamic problems. Int J Numer Methods Eng 69(13):2687–2710. https://doi.org/10.1002/nme.1859

    MathSciNet  Article  MATH  Google Scholar 

  40. 40.

    Vignjevic R, Campbell J, Libersky L (2000) A treatment of zero-energy modes in the smoothed particle hydrodynamics method. Comput Methods Appl Mech Eng 184(1):67–85. https://doi.org/10.1016/S0045-7825(99)00441-7

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Lars Pastewka and Matthias Teschner for helpful discussions. The financial support by the Deutsche Forschungsgemeinschaft (DFG) under Grant No. BI 1859/1-1 within the M-ERA.NET framework is greatly acknowledged. All SPH simulations are carried out using the SimPARTIX code developed by Fraunhofer IWM.

Author information

Affiliations

  1. Fraunhofer Institute for Mechanics of Materials IWM, Freiburg, Germany

    Shoya Mohseni-Mofidi & Claas Bierwisch

Authors
  1. Shoya Mohseni-Mofidi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Claas Bierwisch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shoya Mohseni-Mofidi.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mohseni-Mofidi, S., Bierwisch, C. Application of hourglass control to Eulerian smoothed particle hydrodynamics. Comp. Part. Mech. 8, 51–67 (2021). https://doi.org/10.1007/s40571-019-00312-6

Download citation

Keywords

  • Meshless methods
  • Smoothed particle hydrodynamics
  • Stabilization
  • Hourglass control