Advertisement

The European Physical Journal Special Topics

, Volume 227, Issue 14, pp 1501–1514 | Cite as

Fully implicit time integration in truly incompressible SPH

  • Manuel Hopp-HirschlerEmail author
  • Ulrich Nieken
Regular Article
  • 17 Downloads
Part of the following topical collections:
  1. Particle Methods in Natural Science and Engineering

Abstract

In the last years, the Smoothed Particle Hydrodynamics (SPH) method was introduced in new fields of engineering applications where highly viscous flow like polymer flow or very small geometries, like in micro-flow, are involved. In most of these applications it was not possible to use realistic fluid parameters, larger simulation domains or higher resolution because of restriction of time step due to the viscous time step criterion of the explicit time integration scheme. The computational effort was too high even for highly scalable SPH codes. In this article, we present a first-order implicit time stepping scheme for truly incompressible SPH (ISPH) to eliminate the viscous time step criterion. We propose a consistent time stepping approach where both velocity and particle position are solved implicitly and compare the results to traditional semi-implicit ISPH and implicit schemes where only velocity is integrated implicitly. We study Poiseuille flow, Taylor–Green flow and Rayleigh–Taylor instability and compare results of the implicit schemes to ISPH. Energy conservation is investigated using Taylor–Green vortex to highlight differences between the approaches. We find that the fully implicit time integration scheme is numerically stable. Since no error estimation of accuracy is used, the larger time step size leads to deviations of the trajectories. In future work, higher-order time integration schemes as well as error estimators should be investigated.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R.A. Gingold, J.J. Monaghan, Mon. Not. R. Astron. Soc. 181, 375 (1977)ADSCrossRefGoogle Scholar
  2. 2.
    L.B. Lucy, Astron. J. 82, 1013 (1977)ADSCrossRefGoogle Scholar
  3. 3.
    M. Shadloo, G. Oger, G. Le Touzé, Comput. Fluids 136, 11 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    M. Hopp-Hirschler, U. Nieken, J. Membr. Sci. 564, 820 (2018)CrossRefGoogle Scholar
  5. 5.
    M. Hirschler, P. Kunz, M. Huber, F. Hahn, U. Nieken, J. Comput. Phys. 307, 614 (2016)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    M. Hopp-Hirschler, M.S. Shadloo, U. Nieken, Comput. Fluids 176, 1 (2018)MathSciNetCrossRefGoogle Scholar
  7. 7.
    D. Barcarolo, D.L. Touz, G. Oger, F. de Vuyst, J. Comput. Phys. 273, 640 (2014)ADSCrossRefGoogle Scholar
  8. 8.
    R. Vacondio, B. Rogers, P. Stansby, P. Mignosa, J. Feldman, Comput. Methods Appl. Mech. Eng. 256, 132 (2013)ADSCrossRefGoogle Scholar
  9. 9.
    J. Feldman, J. Bonet, Int. J. Numer. Methods Eng. 72, 295 (2010)CrossRefGoogle Scholar
  10. 10.
    F. Spreng, D. Schnabel, A. Mueller, P. Eberhard, Comput. Part. Mech. 1, 131 (2014)CrossRefGoogle Scholar
  11. 11.
    X. Xu, P. Yu, J. Non-Newtonian Fluid Mech. 229, 27 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    S.J. Cummins, M. Rudman, J. Comput. Phys. 152, 584 (1999)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    J.J. Monaghan, J. Comput. Phys. 138, 801 (1997)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    S. Litvinov, M. Ellero, X. Hu, N. Adams, J. Comput. Phys. 229, 5457 (2010)ADSCrossRefGoogle Scholar
  15. 15.
    P. Espanol, M. Revenga, Phys. Rev. E 67, 026705 (2003)ADSCrossRefGoogle Scholar
  16. 16.
    X.-J. Fan, R. Tanner, R. Zheng, J. Non-Newtonian Fluid Mech. 165, 219 (2010)CrossRefGoogle Scholar
  17. 17.
    Y. Han, H. Qiang, Q. Huang, J. Zhao, Sci. China Technol. Sci. 56, 2480 (2013)CrossRefGoogle Scholar
  18. 18.
    Y.-W. Han, H.-F. Qiang, H. Liu, W.-R. Gao, Acta Mech. Sin. 30, 37 (2014)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    T. Takahashi, Y. Dobashi, I. Fujishiro, T. Nishita, M.C. Lin, Eurographics 34, 493 (2015)Google Scholar
  20. 20.
    M. Weiler, D. Koschier, M. Brand, J. Bender, Comput. Graph. Forum 37, 145 (2018)CrossRefGoogle Scholar
  21. 21.
    M. Ihmsen, J. Cornelis, B. Solenthaler, C. Horvath, M. Teschner, IEEE Trans. Visual. Comput. Graphics 20, 26 (2014)CrossRefGoogle Scholar
  22. 22.
    J. Cornelis, M. Ihmsen, A. Peer, M. Teschner, Comput. Graph. Forum 33, 255 (2014)CrossRefGoogle Scholar
  23. 23.
    P. Goswami, A. Eliasson, Implicit incompressible SPH on the GPU, in Workshop on Virtual Reality Interaction and Physical Simulation VRIPHYS(2015)Google Scholar
  24. 24.
    R. Rook, M. Yildiz, S. Dost, Numer. Heat Transfer, Part B: Fundam.: Int. J. Comput. Method. 51, 1 (2007)ADSCrossRefGoogle Scholar
  25. 25.
    P.V. Liedekerke, B. Smeets, T. Odenthal, E. Tijskens, H. Ramon, Comput. Phys. Commun. 184, 1686 (2013)ADSCrossRefGoogle Scholar
  26. 26.
    W. Pan, K. Kim, M. Perego, A.M. Tartakovsky, M.L. Parks, J. Comput. Phys. 334, 125 (2017)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    J.J. Monaghan, Annu. Rev. Astron. Astrophys. 30, 543 (1992)ADSCrossRefGoogle Scholar
  28. 28.
    M.B. Liu, G.R. Liu, Arch. Comput. Method. Eng. 17, 25 (2010)CrossRefGoogle Scholar
  29. 29.
    J. Bonet, T.-S.L. Lok, Comput. Method. Appl. Mech. Eng. 180, 97 (1999)ADSCrossRefGoogle Scholar
  30. 30.
    J.P. Morris, Int. J. Numer. Methods. Fluids 33, 333 (2000)ADSCrossRefGoogle Scholar
  31. 31.
    S. Shao, E.Y.M. Lo, Adv. Water Resour. 26, 787 (2003)ADSCrossRefGoogle Scholar
  32. 32.
    R. Falgout, A. Cleary, J. Jones, E. Chow, V. Henson, C. Baldwin, P. Brown, P. Vassilevski, U.M. Yang Hypre Web page (2016), http://acts.nersc.gov/hypre
  33. 33.
    S. Balay, S. Abhyankar, M.F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, V. Eijkhout, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, K. Rupp, B.F. Smith, S. Zampini, H. Zhang, H. Zhang, PETSc Web page (2016), http://www.mcs.anl.gov/petsc, version 3.3
  34. 34.
    R. Xu, P. Stansby, D. Laurence, J. Comp. Phys. 228, 6703 (2009)ADSCrossRefGoogle Scholar
  35. 35.
    S.J. Lind, R. Xu, P.K. Stansby, B.D. Rogers, J. Comp. Phys. 231, 1499 (2012)ADSCrossRefGoogle Scholar
  36. 36.
    P. Kunz, M. Hirschler, M. Huber, U. Nieken, J. Comput. Phys. 326, 171 (2016)ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    M. Huber, F. Keller, W. Sckel, M. Hirschler, P. Kunz, S. Hassanizadeh, U. Nieken, J. Comput. Phys. 310, 459 (2016)ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    A.M. Tartakovsky, P. Meakin, J. Comput. Phys. 207, 610 (2005)ADSCrossRefGoogle Scholar
  39. 39.
    X.Y. Hu, N.A. Adams, J. Comput. Phys. 227, 264 (2007)ADSCrossRefGoogle Scholar
  40. 40.
    N. Grenier, M. Antuono, A. Colagrossi, D. Le Touzé, B. Alessandrini, J. Comput. Phys. 228, 8380 (2009)ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    M.S. Shadloo, A. Zainali, M. Yildiz, Comput. Mech. 51, 699 (2012)CrossRefGoogle Scholar
  42. 42.
    G. Oger, S. Marrone, D. Le Touzé, M. de Leffe, J. Comput. Phys. 313, 76 (2016)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Chemical Process Engineering, University of StuttgartStuttgartGermany

Personalised recommendations