PIER
 
Progress In Electromagnetics Research
ISSN: 1070-4698, E-ISSN: 1559-8985
Home | Search | Notification | Authors | Submission | PIERS Home | EM Academy
Home > Vol. 89 > pp. 275-289

WEAK FORM NONUNIFORM FAST FOURIER TRANSFORM METHOD FOR SOLVING VOLUME INTEGRAL EQUATIONS

By Z. Fan, R.-S. Chen, H. Chen, and D.-Z. Ding

Full Article PDF (324 KB)

Abstract:
Electromagnetic scattering problems involving inhomogeneous objects can be numerically solved by applying a method of moment's discretization to the hypersingular volume integral equation in which a grad-div operator acts on a vector potential. The vector potential is a spatial convolution of the free space Green's function and the contrast source over the domain of interest. For electrically large problems, the direct solution of the resulting linear system is expensive, both computationally and in memory use. Conventionally, the fast Fourier transform method (FFT) combined Krylov subspace iterative approaches are adopted. However, the uniform discretization required by FFT is not ideal for those problems involving inhomogeneous scatterers and sharp discontinuities. In this paper, a nonuniform FFT method combined weak form integral equation technique is presented. The method performs better in terms of speed and memory use than FFT on the configuration involving both the electrically large and fine structures. This is illustrated by a representative numerical test case.

Citation:
Z. Fan, R.-S. Chen, H. Chen, and D.-Z. Ding, " weak form nonuniform fast fourier transform method for solving volume integral equations ," Progress In Electromagnetics Research, Vol. 89, 275-289, 2009.
doi:10.2528/PIER08121308
http://www.jpier.org/PIER/pier.php?paper=08121308

References:
1. Livesay, D. E. and K. M. Chen, "Electromagnetic fields induced inside arbitrarily shaped biological bodies," IEEE Trans. on MTT , Vol. 22, 1273-1280, 1974.
doi:10.1109/TMTT.1974.1128475

2. Schaubert, D., D. Wilton, and A. Glisson, "A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies," IEEE Transactions on Antennas and Propagation , Vol. 32, No. 1, 77-85, 1984.
doi:10.1109/TAP.1984.1143193

3. Davidson, D. B., Computational Electromagnetics for RF and Microwave Engineering, Cambridge University Press, Apr. 2005.

4. Nie, X.-C., N. Yuan, L.-W. Li, T. S. Yeo, and Y.-B. Gan, "Fast analysis of electromagnetic transmission through arbitrarily shaped airborne radomes using precorrected-FFT method," Progress In Electromagnetics Research, Vol. 54, 37-59, 2005.
doi:10.2528/PIER04100601

5. Li, L.-W., Y.-J. Wang, and E.-P. Li, "MPI-based parallelized precorrected FFT algorithm for analyzing scattering by arbitrarily shaped three-dimensional objects," Progress In Electromagnetics Research, Vol. 42, 247-259, 2003.
doi:10.2528/PIER03030701

6. Volakis, J. L. and K. Barkeshli, "Applications of the conjugate gradient FFT method to radiation and scattering," Progress In Electromagnetics Research, Vol. 05, 159-239, 1991.

7. Peterson, A. F., S. L. Ray, C. H. Chen, and R. Mittra, "Numerical implementations of the conjugate gradient method and the CGFFT for electromagnetic scattering," Progress In Electromagnetics Research, Vol. 05, 241-300, 1991.

8. Gago, E. and M. F. Catedra, "Analysis of finite sized conducting patches in multilayer media using the CG-FFT method and discretizing Green's function in the spectral domain," Progress In Electromagnetics Research, Vol. 05, 301-327, 1991.

9. Gan, H. and W. C. Chew, "A discrete BCG-FFT algorithm for solving 3D inhomogeneous scatterer problems," Journal of Electromagnetic Waves and Applications, Vol. 9, 1339-1357, 1995.

10. Xin, Y. F. and P.-L. Rui, "Adaptively accelerated GMRES fast fourier transform method for electromagnetic scattering," Progress In Electromagnetics Research, Vol. 81, 303-314, 2008.
doi:10.2528/PIER08011603

11. Rui, P.-L. and R.-S. Chen, "Implicitly restarted GMRES fast Fourier transform method for electromagnetic scattering," Journal of Electromagnetic Waves and Applications , Vol. 21, No. 7, 973-986, 2007.
doi:10.1163/156939307780748968

12. Fan, Z. H., D. X. Wang, R. S. Chen, and E. K. N. Yung, "The application of iterative solvers in discrete dipole approximation method for computing electromagnetic scattering," Microwave and Optical Technology Letters, Vol. 48, No. 9, 1741-1746, Sep. 2006.
doi:10.1002/mop.21760

13. Rui, P. L., R. S. Chen, Z. H. Fan, E. K. N. Yung, C. H. Chan, Z. Nie, and J. Hu, "Fast analysis of electromagnetic scattering of 3-D dielectric bodies with augmented GMRES-FFT method ," IEEE Transactions on Antennas and Propagation, Vol. 53, No. 11, 3848-3852, Nov. 2005.
doi:10.1109/TAP.2005.858833

14. Zhao, L., T.-J. Cui, and W.-D. Li, "An efficient algorithm for EM scattering by electrically large dielectric objects using MR-QEB iterative scheme and CG-FFT method ," Progress In Electromagnetics Research, Vol. 67, 341-355, 2007.
doi:10.2528/PIER06121902

15. Saad, Y., Iterative Methods for Sparse Linear Systems, PWS, Publishing Company, 1996.

16. Xu, X. M. and Q. H. Liu, "Conjugate-gradient nonuniform fast Fourier transform (CG-NUFFT) method for one- and twodimensional media," Microwave and Optical Technology Letters, Vol. 24, No. 6, 385-389, 2000.
doi:10.1002/(SICI)1098-2760(20000320)24:6<385::AID-MOP8>3.0.CO;2-W

17. Liu, Q. H. and X. Y. Tang, "Iterative algorithm for nonuniform inverse fast Fourier transform (NU-IFFT)," Electronics Letters, Vol. 34, No. 20, 1913-1914, Oct. 1, 1998.
doi:10.1049/el:19981372

18. Liu, Q. H. and N. Nguyen, "Accurate algorithm for nonuniform fast Fourier transforms (NUFFT’s)," IEEE Microwave and Guided Wave Letters, Vol. 8, No. 1, 18-20, Jan. 1998.
doi:10.1109/75.650975

19. Liu, Q. H., X. M. Xu, B. Tian, and Z. Q. Zhang, "Applications of nonuniform fast transform algorithms in numerical solutions of differential and integral equations," IEEE Transactions on Geoscience and Remote Sensing, Vol. 38, No. 4, Part I, 1551-1560, July 2000.
doi:10.1109/36.851955

20. Liu, Q. H., X. M. Xu, and Z. Q. Zhang, "Applications of nonuniform fast transform algorithms in numerical solutions of integral equations," Annual Review of Progress in Applied Computational Electromagnetics, Vol. 2, 897-904, 2000.

21. Zwamborn, P. and P. M. van den Berg, "The three-dimensional weak form of the conjugate gradient FFT method for solving scattering problems," IEEE Transactions on Microwave Theory and Techniques, Vol. 40, No. 9, 1757-1766, 1992.
doi:10.1109/22.156602

22. Zhang, Z. Q. and Q. H. Liu, "Three-dimensional weak-form conjugate- and biconjugate-gradient FFT methods for volume integral equations," Microwave and Optical Technology Letters, Vol. 29, No. 5, 350-356, 2001.
doi:10.1002/mop.1176

23. Potts, D. and G. Steidl, "Fast summation at nonequispaced knots by NFFTs," SIAM J. on Sci. Comput., Vol. 24, 2013-2037, 2003.
doi:10.1137/S1064827502400984

24. Dutt, A. and V. Rokhlin, "Fast Fourier transforms for nonequispaced data," SIAM J. Sci. Stat. Comput., Vol. 14, 1368-1393, 1993.
doi:10.1137/0914081

25. Kunis, S. and D. Potts, "Time and memory requirements of the Nonequispaced FFT," Sampling Theory in Signal and Image Processing, Vol. 7, 77-100, 2008.

26. Fessler, J. A. and B. P. Sutton, "Nonuniform fast Fourier transforms using min-max interpolation," IEEE Transactions on Signal Processing, Vol. 51, No. 2, 560-574, Feb.2003.

27. Morgan, R. B., "GMRES with deflated restarting," SIAM J. Sci. Comput., Vol. 24, 20-37, 2002.

28. Jin, J. M., The Finite Element Method in Electromagnetics, 2nd Ed., John Wiley & Sons, New York, 2002.

29. Chen, R. S., Z. H. Fan, and E. K. N. Yung, "Analysis of electromagnetic scattering of three-dimensional dielectric bodies using Krylov subspace FFT iterative methods," Source: Microwave and Optical Technology Letters, Vol. 39, No. 4, 261-267, Nov. 20, 2003.

30. Ding, D.-Z., R.-S. Chen, and Z. Fan, "An efficient SAI preconditioning technique for higher order hierarchical MLFMM implementation," Progress In Electromagnetics Research, Vol. 88, 255-273, 2008.


© Copyright 2014 EMW Publishing. All Rights Reserved