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2018-03-10
Fast Low-Frequency Surface Integral Equation Solver Based on Hierarchical Matrix Algorithm
By
Progress In Electromagnetics Research, Vol. 161, 19-33, 2018
Abstract
A fast low-frequency surface integral equation solver based on hierarchical matrix algorithm is proposed. First, the augmented electric field integral equation (A-EFIE) formulation is introduced to eliminate the low-frequency breakdown of traditional EFIE. To deal with large-scale problems, the low-frequency multilevel fast multipole algorithm (LF-MLFMA) is employed to construct a hierarchical (H-) matrix representation of the A-EFIE system matrix. Moreover, a recompression method is developed to further compress the H-matrix generated by LF-MLFMA. The H-matrix-based triangular factorization algorithm can be performed with almost linear computational complexity and memory requirement, which produces a fast direct solver for multiple right-hand-side (RHS) problems, and a good preconditioner to accelerate the convergence rate of an iterative solver. Numerical examples demonstrate the effectiveness of the proposed method for the analysis of various low-frequency problems.
Citation
Ting Wan, Qi Dai, and Weng Cho Chew, "Fast Low-Frequency Surface Integral Equation Solver Based on Hierarchical Matrix Algorithm," Progress In Electromagnetics Research, Vol. 161, 19-33, 2018.
doi:10.2528/PIER17111701
References

1. Morita, N., N. Kumagai, and J. R. Mautz, Integral Equation Methods for Electromagnetics, Artech House, 1990.

2. Chew, W. C., M. S. Tong, and B. Hu, Integral Equation Methods for Electromagnetic and Elastic Waves, Morgan and Claypool, 2007.

3. Qian, Z. G. and W. C. Chew, "A quantitative study of the low frequency breakdown of EFIE," Microw. Opt. Tech. Lett., Vol. 50, No. 5, 1159-1162, May 2008.
doi:10.1002/mop.23324

4. Wilton, D. R. and A. W. Glisson, "On improving the stability of the electric field integral equation at low frequencies," Proc. URSI Radio Sci. Meeting, 24, Los Angeles, CA, Jun. 1981.

5. Wu, W., A. W. Glisson, and D. Kajfez, "Study of two numerical solution procedures for the electric field integral equation at low frequency," Appl. Computat. Electromagn. Soc. J., Vol. 10, No. 3, 69-80, Nov. 1995.

6. Burton, M. and S. Kashyap, "A study of a recent, moment-method algorithm that is accurate to very low frequencies," Appl. Computat. Electromagn. Soc. J., Vol. 10, No. 3, 58-68, Nov. 1995.

7. Zhao, J. S. and W. C. Chew, "Integral equation solution of Maxwell's equations from zero frequency to microwave frequencies," IEEE Trans. Antennas Propag., Vol. 48, 1635-1645, Oct. 2000.

8. Vecchi, G., "Loop-star decomposition of basis functions in the discretization of EFIE," IEEE Trans. Antennas Propag., Vol. 47, 339-346, Feb. 1999.
doi:10.1109/8.761074

9. Lee, J. F., R. Lee, and R. J. Burkholder, "Loop star basis functions and a robust preconditioner for EFIE scattering problems," IEEE Trans. Antennas Propag., Vol. 51, No. 8, 1855-1863, Aug. 2003.
doi:10.1109/TAP.2003.814736

10. Andriulli, F. P., K. Cools, H. Bagci, F. Olyslager, A. Buffa, S. Christiansen, and E. Michielssen, "A multiplicative Calderon preconditioner for the electric field integral equation," IEEE Trans. Antennas Propag., Vol. 56, 2398-2412, Aug. 2008.
doi:10.1109/TAP.2008.926788

11. Stephanson, M. B. and J. F. Lee, "Preconditioner electric field integral equation using Calderon identities and dual loop/star basis functions," IEEE Trans. Antennas Propag., Vol. 57, No. 4, 1274-1279, Apr. 2009.
doi:10.1109/TAP.2009.2016173

12. Yan, S., J. M. Jin, and Z. Nie, "EFIE analysis of low-frequency problems with loop-star decomposition and Calderon multiplicative preconditioner," IEEE Trans. Antennas Propag., Vol. 58, No. 3, 857-867, Mar. 2010.
doi:10.1109/TAP.2009.2039336

13. Sun, S., Y. G. Liu, W. C. Chew, and Z. Ma, "Calderon multiplicative preconditioned EFIE with per-turbation method," IEEE Trans. Antennas Propag., Vol. 61, No. 1, 247-255, Jan. 2013.
doi:10.1109/TAP.2012.2220099

14. Qian, Z. G. and W. C. Chew, "An augmented EFIE for high speed interconnect analysis," Micro. Opt. Technol. Lett., Vol. 50, No. 10, 2658-2662, Oct. 2008.
doi:10.1002/mop.23736

15. Qian, Z. G. and W. C. Chew, "Fast full-wave surface integral equation solver for multiscale structure modeling," IEEE Trans. Antennas Propag., Vol. 50, 3594-3601, Nov. 2009.

16. Xia, T., H. Gan, M. Wei, W. C. Chew, H. Braunisch, Z. Qian, K. Aygun, and A. Aydiner, "An integral equation modeling of lossy conductors with the enhanced augmented electric field integral equation," IEEE Trans. Antennas Propag., Vol. 65, No. 8, 4181-4190, Jun. 2017.
doi:10.1109/TAP.2017.2718587

17. Qian, Z. G. and W. C. Chew, "Enhanced A-EFIE with perturbation method," IEEE Trans. Antennas Propag., Vol. 58, No. 10, 3256-3264, Oct. 2010.
doi:10.1109/TAP.2010.2055795

18. Xia, T., H. Gan, M. Wei, W. C. Chew, H. Braunisch, Z. Qian, K. Aygun, and A. Aydiner, "An enhanced augmented electric field integral equation formulation for dielectric objects," IEEE Trans. Antennas Propag., Vol. 64, No. 6, 2339-2347, Jun. 2016.
doi:10.1109/TAP.2016.2537389

19. Meng, L. L., X. Y. Xiong, T. Xia, and L. J. Jiang, "The error control of mixed-form fast multipole algorithm based on the high-order multipole rotation," IEEE Antenn. Wireless Propag. Lett., Vol. 16, 1655-1658, Jan. 2017.
doi:10.1109/LAWP.2017.2660880

20. Wu, J. W., Z. G. Qian, J. E. Schutt-Aine, and W. C. Chew, "Fast solution of low-frequency complex problems over a frequency band using enhanced A-EFIE and FMM," Micro. Opt. Technol. Lett., Vol. 56, No. 9, 2153-2158, 2014.
doi:10.1002/mop.28528

21. Dai, Q. I., J. W. Wu, H. Gan, Q. S. Liu, W. C. Chew, and W. E. I. Sha, "Large-scale characteristic mode analysis with fast multipole algorithms," IEEE Trans. Antennas Propag., Vol. 64, No. 7, 2608-2616, Jul. 2016.
doi:10.1109/TAP.2016.2526083

22. Hackbusch, W. and B. Khoromaskij, "A Sparse Matrix arithmetic based on H-matrices. Part I: Introduction to H-matrices," Computing, Vol. 62, 89-108, 1999.
doi:10.1007/s006070050015

23. Grasedyck, L. and W. Hackbusch, "Construction and arithmetics of H-matrices," Computing, Vol. 70, No. 4, 295-344, Aug. 2003.
doi:10.1007/s00607-003-0019-1

24. Borm, S., L. Grasedyck, and W. Hackbusch, "Introduction to hierarchical matrices with applications," Engineering Analysis with Boundary Elements, No. 27, 405-422, 2003.
doi:10.1016/S0955-7997(02)00152-2

25. Bebendorf, M. and Leipzig, "Hierarchical LU decomposition-based preconditioners for BEM," Computing, Vol. 74, 225-247, 2005.
doi:10.1007/s00607-004-0099-6

26. Chai, W. and D. Jiao, "An H2-matrix-based integral-equation solver of reduced complexity and controlled accuracy for solving electrodynamic problems," IEEE Trans. Antennas Propag., Vol. 57, No. 10, 3147-3159, Oct. 2009.
doi:10.1109/TAP.2009.2028665

27. Chai, W. and D. Jiao, "A complexity-reduced H-matrix based direct integral equation solver with prescribed accuracy for large-scale electrodynamic anaysis," Proc. IEEE Int. Symp. Antennas Propag..
doi:10.1109/TAP.2009.2028665

28. Chai, W. and D. Jiao, "Direct matrix solution of linear complexity for surface integral-equationbased impedance extraction of complicated 3-D structures," Proceedings of the IEEE, Vol. 01, No. 2, 372-388, Feb. 2013.
doi:10.1109/JPROC.2012.2190577

29. Wan, T., Z. N. Jiang, and Y. J. Sheng, "Hierarchical matrix techniques based on matrix decomposition algorithm for the fast analysis of planar layered structures," IEEE Trans. Antennas Propag., Vol. 59, No. 11, 4132-4141, Nov. 2011.

30. Zhao, J. S. and W. C. Chew, "Three dimensional multilevel fast multipole algorithm from static to electrodynamic," Microw. Opt. Technol. Lett., Vol. 26, No. 1, 43-48, 2000.
doi:10.1002/(SICI)1098-2760(20000705)26:1<43::AID-MOP14>3.0.CO;2-8

31. Zhao, J. S. and W. C. Chew, "Integral equation solution of Maxwell’s equations from zero frequency to microwave frequencies," IEEE Trans. Antennas Propag., Vol. 48, No. 10, 1635-1645, Oct. 2000.
doi:10.1109/8.899680