Simulation of bubble dynamics in three-dimensional potential flows on heterogeneous computing systems using the fast multipole and boundary element methods

Authors

  • Yu.A. Itkulova
  • O.A. Abramova
  • N.A. Gumerov
  • I.Sh. Akhatov

Keywords:

bubble dynamics
potential flow
boundary element method
fast multipole method
parallel computing
graphics processors

Abstract

The bubble dynamics in potential flows of incompressible liquid is studied. The proposed approach is based on the boundary element method for the Laplace equation, which is especially efficient for the 3D bubble dynamics. In order to increase the problem size and to accelerate computations, an efficient numerical algorithm is developed and implemented. Depending on the problem size, for the acceleration of computations we used the direct matrix-vector multiplication on graphics processors (GPU) or the fast multipole method (FMM) implemented on heterogeneous computing systems (multicore CPUs and graphics processors). For the simulation of bubble surfaces, a new method based on the filtration of spherical harmonics is proposed. The proposed approach allows one to directly calculate the 3D dynamics of a single bubble, two and three interacting bubbles as well as a bubble cluster with a high degree of surface discretization. The developed method can be used to solve a wide range of problems related to the potential flow of bubble liquids.

New computational technologies

Downloads

Published

2014-04-18

Issue

Vol. 15 (2014): Issue 2.

Section

Section 1. Numerical methods and applications

Author Biographies

Yu.A. Itkulova

Bashkir State University
• Research Intern

O.A. Abramova

Bashkir State University
• Junior Researcher

N.A. Gumerov

University of Maryland, Baltimore,
620 W. Lexington St., Baltimore, MD 2120, USA
• Professor

I.Sh. Akhatov

University of North Dakota,
Faculty of Mechanical Engineering
• Professor

References

  1. Knapp R.T., Daily J.W., Hammitt F.G. Cavitation. New York: McGraw-Hill, 1970.
  2. Brennen C.E. Cavitation and bubble dynamics. Oxford: Oxford Univ. Press, 2013.
  3. Xi X., Cegla F., Mettin R., Holsteyns F., Lippert A. Collective bubble dynamics near a surface in a weak acoustic standing wave field // J. Acoust. Soc. Am. 2012. 132, N 1. 37-47.
  4. Gumerov N.A., Akhatov I.Sh. Numerical simulation of 3D self-organization of bubbles in acoustic fields // Proc. of the 8th International Symposium on Cavitation. Singapore: National Univ. of Singapore, 2012. Article No. 189.
  5. Gumerov N.A., Ohl C.-D., Akhatov I.S., Sametov S., Khasimullin M. Waves of acoustically induced transparency in bubbly liquids: theory and experiment // J. Acoust. Soc. Am. 2013. 133, N 5. 3277-3286.
  6. Lauterborn W., Kurz T., Akhatov I. Nonlinear acoustics in fluids // Springer Handbook of Acoustics. New York: Springer, 2007. 257-297.
  7. Nasibullaeva E.S., Akhatov I.S. Bubble cluster dynamics in an acoustic field // J. Acoust. Soc. Am. 2013. 133, N 6. 3727-3738.
  8. Konovalova S.I., Akhatov I.S. Structure formation in acoustic cavitation // Multiphase Science and Technology. 2005. 17, N 4. 343-371.
  9. Parlitz U., Mettin R., Luther S., Akhatov I., Voss M., Lauterborn W. Spatio-temporal dynamics of acoustic cavitation bubble clouds // Phil. Trans. R. Soc. Lond. A. 1999. 357. 313-334.
  10. Akhatov I., Parlitz U., Lauterborn W. Towards a theory of self-organization phenomena in bubble-liquid mixtures // Phys. Rev. E. 1996. 54. 4990-5003.
  11. Plesset M.S., Prosperetti A. Bubble dynamics and cavitation // Ann. Rev. Fluid Mech. 1977. N 9. 145-185.
  12. Akhatov I., Gumerov N., Ohl C.-D., Parlitz U., Lauterborn W. The role of surface tension in stable single-bubble sonoluminescence // Phys. Rev. Lett. 1997. 78, N 2. 227-230.
  13. Pozrikidis C. Expansion of a compressible gas bubble in Stokes flow // J. Fluid Mech. 2001. 442. 171-189.
  14. Pozrikidis C. Computation of the pressure inside bubbles and pores in Stokes flow // J. Fluid Mech. 2003. 474. 319-337.
  15. Blake J.R., Gibson D.C. Cavitation bubbles near boundaries // Ann. Rev. Fluid Mech. 1987. 19. 99-123.
  16. Best J.P., Kucera A. A numerical investigation of non-spherical rebounding bubbles // J. Fluid Mech. 1992. 245. 137-154.
  17. Sangani A.S., Didwania A.K. Dynamic simulations of flows of bubbly liquids at large Reynolds numbers // J. Fluid Mech. 1993. 250. 307-337.
  18. Lucca G., Prosperetti A. A numerical method for the dynamics of non-spherical cavitation bubbles // Proc. 2nd Int. Colloq. on Drops and Bubbles. Monterey: Jet Propulsion Lab., 1982. 175-181.
  19. Chahine G.L., Duraiswami R. Dynamical interactions in a multi-bubble cloud // J. Fluids Eng. 1992. 114, N 4. 680-686.
  20. Chahine G.L. Strong interactions bubble/bubble and bubble/flow // Bubble Dynamics and Interface Phenomena. Dordrecht: Kluwer, 1994. 195-206.
  21. Oguz H.N., Prosperetti A. Dynamics of bubble growth and detachment from a needle // J. Fluid Mech. 1993. 257. 111-145.
  22. Zhang Y.L., Yeo K.S., Khoo B.C., Wang C. 3D jet impact and toroidal bubbles // J. Comput. Phys. 2001. 166, N 2. 336-360.
  23. Воинов О.В., Петров А.Г. Движение пузырей в жидкости // Гидромеханика. 10. М.: ВИНИТИ, 1976. 86-147.
  24. Bui T.T., Ong E.T., Khoo B.C., Klaseboer E., Hung K.C. A fast algorithm for modeling multiple bubbles dynamics // J. Comp. Physics. 2006. 216, N 2. 430-453.
  25. Prosperetti A., Tryggvason G. Computational methods for multiphase flow // New York: Cambridge Univ. Press, 2007.
  26. Magnaudet J., Eames I. The motion of high-Reynolds-number bubbles in inhomogeneous flows // Ann. Rev. Fluid Mech. 2000. 32. 659-708.
  27. Zhang S., Duncan J.H., Chahine G.L. The final stage of the collapse of cavitation bubble near a rigid wall // J. Fluid Mech. 1993. 257. 147-181.
  28. Brebbia C.A., Telles J.C. F., Wrobel L.C. Boundary element techniques: theory and applications in engineering. Berlin: Springer, 1984.
  29. Itkulova Yu.A., Abramova O.A., Gumerov N.A. Boundary element simulations of compressible bubble dynamics in Stokes flow // Proc. of ASME 2013 International Mechanical Engineering Congress and Exposition. San Diego, 2013. Article No. 63284.
  30. Greengard L., Rokhlin V. A fast algorithm for particle simulations // J. Comput. Phys. 1987. 73, N 2. 325-348.
  31. Gumerov N.A., Duraiswami R. Fast multipole methods for the Helmholtz equation in three dimensions. Oxford: Elsevier, 2005.
  32. Gumerov N.A., Duraiswami R. Comparison of the efficiency of translation operators used in the fast multipole method for the 3D Laplace equation. Technical Report CS-TR-4701. College Park: Univ. of Maryland, 2005.
  33. Gumerov N.A., Duraiswami R. FMM accelerated BEM for 3D Laplace &; Helmholtz equations // Proc. of the Int. Conf. on Boundary Element Techniques VII. Paris, 2006. 79-84.
  34. Gumerov N.A., Duraiswami R. Fast multipole methods on graphics processors // J. Comput. Phys. 2008. 227, N 18. 8290-8313.
  35. Hu Q., Gumerov N.A., Duraiswami R. Scalable fast multipole methods on distributed heterogeneous architectures // Proc. 2011 Int. Conf. for High Performance Computing, Networking, Storage and Analysis. Article N 36. New York: ACM Press, 2011.
  36. Hu Q., Gumerov N.A., Duraiswami R. Scalable distributed fast multipole methods // Proc. 14th International Conference on High Performance Computing and Communications. New York: IEEE Press, 2012. 270-279.
  37. Марьин Д.Ф, Малышев В.Л., Моисеева Е.Ф., Гумеров Н.А., Ахатов И.Ш., Михайленко К.И. Ускорение молекулярно-динамических расчетов с помощью быстрого метода мультиполей и графических процессоров // Вычислительные методы и программирование. 2013. 14. 483-495.
  38. Солнышкина О.А., Иткулова Ю.А., Гумеров Н.А. Ускорение расчетов на графических процессорах при исследовании течения Стокса методом граничных элементов // Вестник Уфимского гос. авиационного техн. ун-та. 2013. 17, № 2. 92-100.
  39. Itkulova Yu.A., Solnyshkina O.A., Gumerov N.A. Toward large scale simulations of emulsion flows in microchannels using fast multipole and graphics processor accelerated boundary element method // Proc. of ASME 2012 International Mechanical Engineering Congress and Exposition. Vol. 7. Houston, 2012. Article No. 86238, pp. 873-881.
  40. Абрамова О.А, Иткулова Ю.А., Гумеров Н.А., Ахатов И.Ш. Трехмерное моделирование динамики деформируемых капель эмульсии методом граничных элементов и быстрым методом мультиполей на гетерогенных вычислительных системах // Вычислительные методы и программирование. 2013. 14. 438-450.
  41. Abramova O.A., Itkulova Yu.A., Gumerov N.A. FMM/GPU accelerated BEM simulations of emulsion flow in microchannels // Proc. of ASME 2013 International Mechanical Engineering Congress and Exposition. San Diego, 2013. Article No. 63193. 8 pp.
  42. Zinchenko A.Z., Rother M.A., Davis R.H. A novel boundary-integral algorithm for viscous interaction of deformable drops // Phys. Fluids. 1997. 9, N 6. 1493-1511.
  43. Абрамова О.А., Иткулова Ю.А., Гумеров Н.А. Моделирование трехмерного движения деформируемых капель в стоксовом режиме методом граничных элементов // Вычислительная механика сплошных сред. 2013. 6, № 2. 214-223.
  44. Saad Y. Iterative methods for sparse linear systems. Philadelphia: SIAM, 2000.
  45. Гумеров Н.А. Быстрый метод мультиполей // Вестник АН Республики Башкортостан. 2013. 18, № 4. 11-24.
  46. Gumerov N.A., Duraiswami R., Borovikov E.A. Data structures, optimal choice of parameters, and complexity results for generalized multilevel fast multipole method in d dimensions. Technical Report CS-TR-4458. College Park: Univ. of Maryland, 2003.
  47. Mettin R., Akhatov I., Parlitz U., Ohl C.-D., Lauterborn W. Bjerknes forces between small cavitation bubbles in a strong acoustic field // Phys. Rev. E. 1997. 56, N 3. 2924-2931.