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Acceleration of Vortex Methods Calculation Using Fmm and Mdgrape-3

By Tarun Kumar Sheel
Progress In Electromagnetics Research B, Vol. 27, 327-348, 2011


The present study discusses some numerical techniques on the simultaneous use of the Fast Multipole Method (FMM) and specialpurpose computer (MDGRAPE-3) to make the impractically expensive calculation feasible without the loss of numerical accuracy. In the present calculations, the impingement of two identical inclined vortex rings has been studied, and the computation time has been reduced by a factor of 1000 at N=1.18 × 106 where N is the number of vortex elements. The direct and MDGRAPE-3 calculations both have a scaling of O(N2), and the use of the FMM brings them both down to O(N). The global kinetic energy, enstrophy and energy spectra have been investigated to address the numerical accuracy and have good agreement with other similar works.


Tarun Kumar Sheel, "Acceleration of Vortex Methods Calculation Using Fmm and Mdgrape-3," Progress In Electromagnetics Research B, Vol. 27, 327-348, 2011.


    1. Leonard, A., "Vortex methods for flow simulations," J. Comput. Phys., Vol. 37, 289-335, 1980.

    2. Barnes, J. E. and P. Hut, "A hierarchical O(N logN) force calculation algorithm," Nature, Vol. 324, 446-449, 1986.

    3. Greengard, L. and V. Rokhlin, "A fast algorithm for particle simulations," J. Comput. Phys., Vol. 73, 325-348, 1987.

    4. Narumi, T., Y. Ohno, N. Okimoto, T. Koishi, A. Suenaga, N. Futatsugi, R. Yanai, R. Himeno, S. Fujikawa, M. Ikei, and M. Taiji, "A 55 TFLOPS simulation of amyloid-forming peptides from yeast prion sup35 with the specialpurpose computer system MDGRAPE-3," Proceedings of the SC06 (High Performance Computing, Networking, Storage and Analysis), CDROM, Tampa, USA, 2006.

    5. Sugimoto, D., Y. Chikada, J. Makino, T. Ito, T. Ebisuzaki, and M. Umemura, "A special-purpose computer for gravitational many-body problems," Nature, Vol. 345, 33-35, 1990.

    6. Sheel, T. K., K. Yasuoka, and S. Obi, "Fast vortex method calculation using a special-purpose computer," Computers and Fluids, Vol. 36, 1319-26, 2007.

    7. Makino, J., "Treecode with a special-purpose processor," Pub. of the Astronomical Society of Japan, Vol. 43, 621-638, 1991.

    8. Chau, N. H., A. Kawai, and T. Ebisuzaki, "Implementation of fast multipole algorithm on special-purpose computer MDGRAPE-2," Proc. of the 6th World Multiconference on Systematics, Cybernetics and Informatics SCI 2002', Vol. XVI(2002), 477-481, USA, 2002.

    9. Shankar, S., "A new mesh-free vortex method,", Ph.D. Thesis, The Florida State University, 1996.

    10. Chatelain, P., "Contributions to the three-dimensional vortex element method and spinning bluff body flows,", Ph.D. Thesis, California Institute of Technology, 2005.

    11. Winckelmans, G. S. and A. Leonard, "Contributions to vortex particle methods for the computation of three-dimensional incompressible unsteady flows," J. Comput. Phys., Vol. 109, 247-273, 1993.

    12. Sheel, T. K., R. Yokota, K. Yasuoka, and S. Obi, "The study of colliding vortex rings using a special-purpose computer and FMM," Transactions of the Japan Society for Computational Engineering and Science, Vol. 2008, 20080003, 2008.

    13. Greengard, L. and V. Rokhlin, Rapid Evaluation of Potential Fields in Three Dimensions, in Vortex Methods, Edited by C. Anderson and C. Greengard, Number 1360 in Lecture Notes in Mathematics, 121-141, Springer-Verlag, Berlin, 1988.

    14. Sanjay, V. and W. C. Chew, "Analysis and performance of a distributed memory multilevel fast multipole algorithm," IEEE Trans. Antennas Propag., Vol. 53, 2719-2727, 2005.

    15. Chew, W. C., J. M. Jin, and M. Eric, Fast and Efficient Algorithms in Computational Electromagnetics, Artech House Publishers, 2001.

    16. Cheng, H., L. Greengard, and V. Rokhlin, "A fast adaptive multipole algorithm in three dimensions," J. Comp. Phys, Vol. 155, 468-498, 1999.

    17. Gumerov, N. A. and R. Duraiswami, Fast Multipole Methods for the Helmholtz Equation in Three Dimensions, Elsevier, 2004.

    18. Xu, K., D. Z. Ding, Z. H. Fan, and R. S. Chen, "Multilevel fast multipole algorithm enhanced by GPU parallel technique for electromagnetic scattering problems," Microwave and Optical Technology Letters, Vol. 53, 502-507, 2010.

    19. Ravnik, J., S. Leopold, and Z. Zoran, "Fast single domainsubdomain BEM algorithm for 3D incompressible fluid flow and heat transfer," IJNME, Vol. 77, 1627-1645, 2009.

    20. Rui, P.-L., R.-S. Chen, Z.-W. Liu, and Y.-N. Gan, "Schwarz-Krylov subspace method for MLFMM analysis of electromagnetic wave scattering problems," Progress In Electromagnetics Research, Vol. 82, 51-63, 2008.

    21. Taiji, M., T. Narumi, Y. Ohno, N. Futatsugi, A. Suenaga, N. Takada, and A. Konagaya, "Protein explorer: A petaflops special-purpose computer system for molecular dynamics simulations," Proc. Supercomputing, in CD-ROM, USA, 2003.

    22. Anderson, C. R., "An implementation of the fast multipole method without multipoles," SIAM J. Sci. Stat. Comput., Vol. 13, 923-947, 1992.

    23. Makino, J., "Yet another fast multipole method without multipoles-pseudo-particle multipole method," J. Comput. Phys., Vol. 151, 910-920, 1999.

    24. Totsuka, Y. and S. Obi, "A validation of viscous dissipation models for fast vortex methods in simulations of decaying turbulence," Journal of Fluid Science and Technology, Vol. 2, No. 1, 248-257, 2007.

    25. Cottet, G.-H., B. Michaux, S. Ossia, and G. VanderLinden, "A comparison of spectral and vortex methods in three-dimensional incompressible flows," J. Comp. Phys., Vol. 175, 702-712, 2002.

    26. Fukuda, K. and K. Kamemoto, "Application of a redistribution model incorporated in a vortex method to turbulent flow analysis," The 3rd International Conference on Vortex Flows and Vortex Methods, 131-136, Japan, 2005.