volume | PIER Journals

Abstract

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.

1. Leonard, A., "Vortex methods for flow simulations," J. Comput. Phys., Vol. 37, 289-335, 1980.
doi:10.1016/0021-9991(80)90040-6 Google Scholar

2. Barnes, J. E. and P. Hut, "A hierarchical O(N logN) force calculation algorithm," Nature, Vol. 324, 446-449, 1986.
doi:10.1038/324446a0 Google Scholar

3. Greengard, L. and V. Rokhlin, "A fast algorithm for particle simulations," J. Comput. Phys., Vol. 73, 325-348, 1987.
doi:10.1016/0021-9991(87)90140-9 Google Scholar

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. Google Scholar

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.
doi:10.1038/345033a0 Google Scholar

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.
doi:10.1016/j.compfluid.2007.01.006 Google Scholar

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

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. Google Scholar

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

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

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.
doi:10.1006/jcph.1993.1216 Google Scholar

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. Google Scholar

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.
doi:10.1109/TAP.2005.851859 Google Scholar

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.
doi:10.1006/jcph.1999.6355 Google Scholar

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.
doi:10.1002/mop.24963 Google Scholar

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.
doi:10.1002/nme.2467 Google Scholar

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.
doi:10.2528/PIER08013003 Google Scholar

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. Google Scholar

22. Anderson, C. R., "An implementation of the fast multipole method without multipoles," SIAM J. Sci. Stat. Comput., Vol. 13, 923-947, 1992.
doi:10.1137/0913055 Google Scholar

23. Makino, J., "Yet another fast multipole method without multipoles-pseudo-particle multipole method," J. Comput. Phys., Vol. 151, 910-920, 1999.
doi:10.1006/jcph.1999.6226 Google Scholar

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.
doi:10.1299/jfst.2.248 Google Scholar

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.
doi:10.1006/jcph.2001.6963 Google Scholar

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. Google Scholar

Dr. Tarun Kumar Sheel