Journal of Scientific Computing

, Volume 64, Issue 3, pp 858–897 | Cite as

Evaluation of Accuracy and Stability of the Classical SPH Method Under Uniaxial Compression

  • R. Das
  • P. W. Cleary


The accuracy and stability of the classical formulation of the smoothed particle hydrodynamics (SPH) method for modelling compression of elastic solids is studied to assess its suitability for predicting solid deformation. SPH has natural advantages for simulating problems involving compression of deformable solids arising from its ability to handle large deformation without re-meshing, complex free surface behaviour and tracking of multiple material interfaces. The ‘classical SPH method’, as originally proposed by Monaghan (in Ann Rev Astron 30:543–574, 1992, Rep Prog Phys 68:1703–1759, 2005), has become broadly established as a robust method in different areas, especially involving fluid flows. However, limited attention has been paid to understanding of its numerical performance for elastic deformation problems. To address this, we evaluate the classical SPH method to explore its stability, accuracy and convergence and the effect of numerical parameters on elastic solutions using a generic uniaxial stress test. Short term transient and long term uniform state SPH solutions agree well with those from the finite element method (FEM). The SPH elastic deformation solution showed good convergence with increasing particle resolution. The tensile instability stabilisation method was found to have little impact on the solution, except for higher values of the correction factor which then produce small amplitude benign artificial banded stress patterns. The use of artificial viscosity is able to eliminate the instability and improve the accuracy of the solutions. Overall, the classical SPH method appears to be robust and suitable for accurate modelling of elastic solids under compression.


Smoothed particle hydrodynamics Elasticity Uniaxial compression  Stress waves Convergence Stability 


  1. 1.
    Monaghan, J.J.: Smoothed particle hydrodynamics. Ann. Rev. Astron. Astrophys. 30, 543–574 (1992)CrossRefGoogle Scholar
  2. 2.
    Monaghan, J.J.: Smoothed particle hydrodynamics. Rep. Prog. Phys. 68, 1703–1759 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Gingold, R.A., Monaghan, J.J.: Smoothed particle hydrodynamics—theory and application to non-spherical stars. MNRAS 181, 375–389 (1977)CrossRefMATHGoogle Scholar
  4. 4.
    Lucy, L.B.: A numerical approach to the testing of the fission hypothesis. Astron. J. 82, 1013–1024 (1977)CrossRefGoogle Scholar
  5. 5.
    Monaghan, J.J., Price, D.J.: Variational principles for relativistic smoothed particle hydrodynamics. Mon. Not. R. Astron. Soc. 328(2), 381–392 (2001)CrossRefGoogle Scholar
  6. 6.
    Cleary, P.W.: Modelling confined multi-material heat and mass flows using SPH. Appl. Math. Model. 22(12), 981–993 (1998)CrossRefGoogle Scholar
  7. 7.
    Cleary, P.W., Ha, J., Prakash, M., Nguyen, T.: Simulation of casting complex shaped objects using SPH. In: In San Francisco, CA, United States. pp. 317–326. Minerals, Metals and Materials Society, Warrendale, PA 15086, United States (2005)Google Scholar
  8. 8.
    Cummins, S.J., Rudman, M.J.: Truly incompressible SPH. In: In Washington, DC, USA. p. 8. ASME, Fairfield, NJ, USA (1998)Google Scholar
  9. 9.
    Cleary, P.W., Monaghan, J.J.: Conduction modelling using smoothed particle hydrodynamics. J. Comput. Phys. 148(1), 227–264 (1999)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Bonet, J., Kulasegaram, S.: A simplified approach to enhance the performance of smooth particle hydrodynamics methods. Appl. Math. Comput. (N. Y.) 126(2–3), 133–155 (2002)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Cleary, P., Ha, J., Alguine, V., Nguyen, T.: Flow modelling in casting processes. Appl. Math. Model. 26(2), 171–190 (2002)CrossRefMATHGoogle Scholar
  12. 12.
    Cleary, P.W., Prakash, M., Ha, J., Stokes, N., Scott, C.: Smooth particle hydrodynamics: status and future potential. Prog. Comput. Fluid Dyn. 7(2–4), 70–90 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Libersky, L.D., Petschek, A.G.: Smooth particle hydrodynamics with strength of materials. In: Trease, H., Crowley, W.P. (eds.) Advances in the Free-Lagrange Method. Springer, Berlin (1990)Google Scholar
  14. 14.
    Wingate, C.A., Fisher, H.N.: Strength Modeling in SPHC. Los Alamos National Laboratory, Report No. LA-UR-93-3942 (1993)Google Scholar
  15. 15.
    Gray, J.P., Monaghan, J.J., Swift, R.P.: SPH elastic dynamics. Comput. Methods Appl. Mech. Eng. 190(49–50), 6641–6662 (2001)CrossRefMATHGoogle Scholar
  16. 16.
    Cleary, P.W., Prakash, M., Ha, J.: Novel applications of smoothed particle hydrodynamics (SPH) in metal forming. J. Mater. Process. Technol. 177(1–3), 41–48 (2006)CrossRefGoogle Scholar
  17. 17.
    Das, R., Cleary, P.W.: The potential for SPH modelling of solid deformation and fracture. In: Reddy, D. (ed.) IUTAM Proceedings Book Series Volume on “Theoretical, Computational and Modelling Aspects of Inelastic Media”, pp. 287–296. Springer, Capetown (2008)Google Scholar
  18. 18.
    Karekal, S., Das, R., Mosse, L., Cleary, P.W.: Application of a mesh-free continuum method for simulation of rock caving processes. Int. J. Rock Mech. Min. Sci. 48(5), 703–711 (2011)CrossRefGoogle Scholar
  19. 19.
    Cleary, P.W., Prakash, M., Das, R., Ha, J.: Modelling of metal forging using SPH. Appl. Math. Model. 36(8), 3836–3855 (2012)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Das, R., Cleary, P.W.: A mesh-free approach for fracture modelling of gravity dams under earthquake. Int. J. Fract. 179(1–2), 9–33 (2013)CrossRefGoogle Scholar
  21. 21.
    Das, R., Cleary, P.W.: Effect of rock shapes on brittle fracture using Smoothed Particle Hydrodynamics. Theor. Appl. Fract. Mech. 53(1), 47–60 (2010)CrossRefGoogle Scholar
  22. 22.
    Fagan, T., Das, R., Lemiale, V., Estrin, Y.: Modelling of equal channel angular pressing using a mesh-free method. J. Mater. Sci. 47 (11), 4514–4519 (2012)Google Scholar
  23. 23.
    Islam, S., Ibrahim, R., Das, R., Fagan, T.: Novel approach for modelling of nanomachining using a mesh-less method. Appl. Math. Model. 36 (11), 5589–5602 (2012)Google Scholar
  24. 24.
    Bradley, G.L., Chang, P.C., McKenna, G.B.: Rubber modeling using uniaxial test data. J. Appl. Polym. Sci. 81(4), 837–848 (2001)CrossRefGoogle Scholar
  25. 25.
    Liu, W.K., Jun, S., Li, S., Adee, J., Belytschko, T.: Reproducing kernel particle methods for structural dynamics. Int. J. Numer. Methods Eng. 38(10), 1655–1679 (1995)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Chen, J.K., Beraun, J.E., Jih, C.J.: Improvement for tensile instability in smoothed particle hydrodynamics. Comput. Mech. 23(4), 279–287 (1999)CrossRefMATHGoogle Scholar
  27. 27.
    Liu, M.B., Liu, G.R.: Restoring particle consistency in smoothed particle hydrodynamics. Appl. Numer. Math. 56(1), 19–36 (2006)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Bonet, J., Kulasegaram, S.: Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations. Int. J. Numer. Methods Eng. 47(6), 1189–1214 (2000)CrossRefMATHGoogle Scholar
  29. 29.
    Bonet, J., Kulasegaram, S.: Remarks on tension instability of Eulerian and Lagrangian corrected smooth particle hydrodynamics (CSPH) methods. Int. J. Numer. Methods Eng. 52(11), 1203–1220 (2001)CrossRefMATHGoogle Scholar
  30. 30.
    Vidal, Y., Bonet, J., Huerta, A.: Stabilized updated Lagrangian corrected SPH for explicit dynamic problems. Int. J. Numer. Methods Eng. 69(13), 2687–2710 (2007)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Dyka, C.T., Ingel, R.P.: An approach for tension instability in smoothed particle hydrodynamics. Comput. Struct. 57, 573–580 (1995)CrossRefMATHGoogle Scholar
  32. 32.
    Dyka, C.T., Randles, P.W., Ingel, R.P.: Stress points for tension instability in SPH. Int. J. Numer. Methods Eng. 40(13), 2325–2341 (1997)CrossRefMATHGoogle Scholar
  33. 33.
    Randles, P.W., Libersky, L.D.: Normalized SPH with stress points. Int. J. Numer. Methods Eng. 48(10), 1445–1462 (2000)CrossRefMATHGoogle Scholar
  34. 34.
    Vignjevic, R., Campbell, J., Libersky, L.: A treatment of zero-energy modes in the smoothed particle hydrodynamics method. Comput. Methods Appl. Mech. Eng. 184(1), 67–85 (2000)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Belytschko, T., Xiao, S.: Stability analysis of particle methods with corrected derivatives. Comput. Math. Appl. 43(3–5), 329–350 (2002)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Xiao, S.R., Belytschko, T.: Material stability analysis of particle methods. Adv. Comput. Math. 23(1–2), 171–190 (2005)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Shaw, A., Roy, D.: Stabilized SPH-based simulations of impact dynamics using acceleration-corrected artificial viscosity. Int. J. Impact Eng 48, 98–106 (2012)CrossRefGoogle Scholar
  38. 38.
    Shaw, A., Roy, D., Reid, S.R.: Optimised form of acceleration correction algorithm within SPH-based simulations of impact mechanics. Int. J. Solids Struct. 48(25–26), 3484–3498 (2011)CrossRefGoogle Scholar
  39. 39.
    Shaw, A., Reid, S.R.: Applications of SPH with the acceleration correction algorithm in structural impact computations. Curr. Sci. 97(8), 1177–1186 (2009)MathSciNetGoogle Scholar
  40. 40.
    Shaw, A., Reid, S.R.: Heuristic acceleration correction algorithm for use in SPH computations in impact mechanics. Comput. Methods Appl. Mech. Eng. 198(49–52), 3962–3974 (2009)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Wang, S.L.N.: A large-deformation Galerkin SPH method for fracture. J. Eng. Math. 71(3), 305–318 (2011)CrossRefMATHGoogle Scholar
  42. 42.
    Wong, S., Shie, Y.: Large deformation analysis with Galerkin based smoothed particle hydrodynamics. CMES 36(2), 97–118 (2008)MathSciNetMATHGoogle Scholar
  43. 43.
    Gray, J.P., Monaghan, J.J.: Numerical modelling of stress fields and fracture around magma chambers. J. Volcanol. Geotherm. Res. 135, 259–283 (2004)CrossRefGoogle Scholar
  44. 44.
    Swegle, J.W., Hicks, D.L., Attaway, S.W.: Smoothed particle hydrodynamics stability analysis. J. Comput. Phys. 116(1), 123–134 (1995)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Morris, J.P.: A study of the stability properties of smooth particle hydrodynamics. Publ. Astron. Soc. Aust. 13(1), 97–102 (1996)Google Scholar
  46. 46.
    Monaghan, J.J.: SPH without a tensile instability. J. Comput. Phys. 159(2), 290–311 (2000)CrossRefMATHGoogle Scholar
  47. 47.
    Melean, Y., Sigalotti, L.D.G., Hasmy, A.: On the SPH tensile instability in forming viscous liquid drops. Comput. Phys. Commun. 157(3), 191–200 (2004)CrossRefGoogle Scholar
  48. 48.
    Liu, Z.S., Swaddiwudhipong, S., Koh, C.G.: High velocity impact dynamic response of structures using SPH method. Int. J. Comput. Eng. Sci. 5(2), 315–326 (2004)CrossRefGoogle Scholar
  49. 49.
    Monaghan, J.J.: Simulating free surface flows with SPH. J. Comput. Phys. 110, 399–406 (1994)CrossRefMATHGoogle Scholar
  50. 50.
    Kulasegaram, S., Bonet, J., Lewis, R.W., Profit, M.: High pressure die casting simulation using a Lagrangian particle method. Commun. Numer. Methods Eng. 19(9), 679–687 (2003)CrossRefMATHGoogle Scholar
  51. 51.
    Cedric, T., Janssen, L.P.B.M., Pep, E.: Smoothed particle hydrodynamics model for phase separating fluid mixtures. I. General equations. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72, 016713 (2005)CrossRefGoogle Scholar
  52. 52.
    Cleary, P.W., Ha, J., Prakash, M., Nguyen, T.: 3D SPH flow predictions and validation for high pressure die casting of automotive components. Appl. Math. Model. 30(11), 1406–1427 (2006)CrossRefGoogle Scholar
  53. 53.
    Fang, J., Owens, R.G., Tacher, L., Parriaux, A.: A numerical study of the SPH method for simulating transient viscoelastic free surface flows. J. Nonnewton. Fluid Mech. 139(1–2), 68–84 (2006)CrossRefMATHGoogle Scholar
  54. 54.
    Imaeda, Y., Inutsuka, S.-I.: Shear flows in smoothed particle hydrodynamics. Astrophys. J. 569(1), 501–518 (2002)CrossRefGoogle Scholar
  55. 55.
    Monaghan, J.J., Gingold, R.A.: Shock simulation by the particle method SPH. J. Comput. Phys. 52(2), 374–389 (1983)CrossRefMATHGoogle Scholar
  56. 56.
    Monaco, A.D., Manenti, S., Gallati, M., Sibilla, S., Agate, G., Guandalini, R.: SPH modeling of solid boundaries through a semi-analytic approach. Eng. Appl. Comput. Fluid Mech. 5(1), 1–15 (2011)Google Scholar
  57. 57.
    Libersky, L.D., Randles, P.W., Carney, T.C., Dickinson, D.L.: Recent improvements in SPH modeling of hypervelocity impact. Int. J. Impact Eng 20(6–10 pt 2), 525–532 (1997)CrossRefGoogle Scholar
  58. 58.
    Libersky, L.D., Petscheck, A.G., Carney, T.C., Hipp, J.R., Allahdadi, F.A.: High strain Lagrangian hydrodynamics—a three dimensional SPH code for dynamic material response. J. Comput. Phys. 109(1), 67–75 (1993)CrossRefMATHGoogle Scholar
  59. 59.
    Randles, P.W., Libersky, L.D.: Smoothed particle hydrodynamics: some recent improvements and applications. Comput. Methods Appl. Mech. Eng. 139(1–4), 375–408 (1996)MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    Mayrhofer, A., Rogers, B.D., Violeau, D., Ferrand, M.: Investigation of wall bounded flows using SPH and the unified semi-analytical wall boundary conditions. Comput. Phys. Commun. 184(11), 2515–2527 (2013)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Wu, B., Tan, C.P.: Sand production prediction of gas field: methodology and laboratory verification. In: SPE Asia Pacific Oil & Gas Conference and Exhibition, Melbourne, Australia (2002)Google Scholar
  62. 62.
    Timoshenko, S., Goodier, J.N.: Theory of Elasticity, 3rd edn. McGraw-Hill, New York (1984)Google Scholar
  63. 63.
    Bathe, K.J.: Finite Element Procedures. Prentice-Hall, Englewood Cliffs (1995)Google Scholar
  64. 64.
    Babuska, I., Suri, M.: On locking and robustness in the finite element method. SIAM J. Numer. Anal. 29(5), 1261–1293 (1992)MathSciNetCrossRefMATHGoogle Scholar
  65. 65.
    Suri, M., Babuska, I., Schwab, C.: Locking effects in the finite element approximation of plate models. Math. Comput. 64(210), 461 (1995)MathSciNetCrossRefMATHGoogle Scholar
  66. 66.
    Ozkul, T.A., Ture, U.: The transition from thin plates to moderately thick plates by using finite element analysis and the shear locking problem. Thin-Walled Struct. 42(10), 1405–1430 (2004)CrossRefGoogle Scholar
  67. 67.
    Suri, M.: Analytical and computational assessment of locking in the hp finite element method. Comput. Methods Appl. Mech. Eng. 133(3–4), 347–371 (1996)MathSciNetCrossRefMATHGoogle Scholar
  68. 68.
    Hansbo, P.: New approach to quadrature for finite elements incorporating hourglass control as a special case. Comput. Methods Appl. Mech. Eng. 158(3–4), 301–309 (1998)MathSciNetCrossRefMATHGoogle Scholar
  69. 69.
    Reese, S., Wriggers, P.: Stabilization technique to avoid hourglassing in finite elasticity. Int. J. Numer. Methods Eng. 48(1), 79–109 (2000)CrossRefMATHGoogle Scholar
  70. 70.
    Fernandez-Mendez, S., Bonet, J., Huerta, A.: Continuous blending of SPH with finite elements. Comput. Struct. 83, 1448–1458 (2005)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Mechanical Engineering, Centre for Advanced Composite MaterialsUniversity of AucklandAucklandNew Zealand
  2. 2.CSIRO Mathematics, Informatics and StatisticsClaytonAustralia

Personalised recommendations