具有精细结构目标电磁散射分析的快速算法研究
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摘要
具有精细结构目标的电磁散射问题在实际工程中有着广泛的应用,一些军事目标如飞机、导弹和雷达天线等,这些目标除了具有较大的电尺寸,复杂的外形也增加了实际计算的困难度。本文研究了一种新型算法—快速笛卡尔展开算法,并与快速多极子方法相结合,在分析具有精细结构目标的电磁散射问题时有较高的计算效率和精确度。
首先,本文系统阐述了积分方程矩量法和快速多极子方法的基本原理和关键技术。对于矩量法,重点介绍了基函数和权函数的选取、平面三角基函数和曲面三角基函数以及四面体基函数和曲四面体基函数,这是分析实际算例的基础。然后,我们把矩量法应用到表面积分方程、体积分方程和体表积分方程的分析中。对于快速多极子方法,详细分析了单层和多层快速多极子算法的基本原理和步骤,并应用在求解各类积分方程中,为后面章节打下了基础。
接着,本文重点研究理想导体目标电磁散射的快速笛卡尔展开算法及其与快速多极子方法相结合的混合算法。介绍了笛卡尔张量的定义及基本运算方式并将其引入到格林函数的展开中,从电场积分方程和磁场积分方程两个方面阐述了快速笛卡尔展开算法的基本原理。然后,把这种新型算法和快速多极子方法相结合,用这种混合算法处理理想导体目标的电磁散射问题。
最后,在分析了理想导体目标电磁散射问题的基础上,对混合算法进一步扩展,研究了混合算法在分析介质体和金属介质混合目标电磁散射问题时的应用。实际算例表明,混合算法在处理基于体积分方程和体表积分方程的介质体和金属介质混合目标电磁散射问题时有较好的稳定性和高效性。
The fast analysis of electromagnetic scattering from multiscale targets has a wide application in engineering.The targets include aircraft, missiles, radar detection and so on. Not only the electrically large size but also the complex geometrical structure make it more difficult when calculating the targets. In this paper,we propose a novel scheme which is called accelerated Cartesian expansion(ACE) algorithm to accelerate integral equation solvers when applied to multiscale targets.The algorithm has showed efficiency and accuracy when it is combined with the fast multipole method.
Firstly, the paper elaborates method of moment and the fast multipole method when applied to solve integral equations. The standard of choosing the base fuction and the test fuction for the method of moment is demonstrated also with the RWG and CRWG base fuction which are the base of the numerical calculate. Then we apply this method analysising the surface integral equation, volume integral equation and the volume-surface hybrid integral equation. For fast multipole method, the principles and base steps are detailed.
Secondly, the accelerated Cartesian expansion algorithm and the hybid algorithm with fast multipole method are researched to solve the electromagnetic scattering problems of the perfect metal targets. The conception and the basic arithmetic of Cartesian tensor are stated so the algorithm can be used to expand the green fuction. In both electric field integral equation and magnetic field integral equation, the basic principles of accelerated Cartesian expansion are formulated. Then we propose a hybrid algorithm which is combined with accelerated Cartesian expansion algorithm and fast multipole method to solve the electromagnetic scattering problems of the perfect metal targets.
Finally, we make further efforts, which is the extanded application of the hybrid algorithm, on the electromagnetic scattering problems of the dielectric targets and the metal-dielectric combined targets. The numerical results demonstrate the stability and efficiency of the algorithm.
首先,本文系统阐述了积分方程矩量法和快速多极子方法的基本原理和关键技术。对于矩量法,重点介绍了基函数和权函数的选取、平面三角基函数和曲面三角基函数以及四面体基函数和曲四面体基函数,这是分析实际算例的基础。然后,我们把矩量法应用到表面积分方程、体积分方程和体表积分方程的分析中。对于快速多极子方法,详细分析了单层和多层快速多极子算法的基本原理和步骤,并应用在求解各类积分方程中,为后面章节打下了基础。
接着,本文重点研究理想导体目标电磁散射的快速笛卡尔展开算法及其与快速多极子方法相结合的混合算法。介绍了笛卡尔张量的定义及基本运算方式并将其引入到格林函数的展开中,从电场积分方程和磁场积分方程两个方面阐述了快速笛卡尔展开算法的基本原理。然后,把这种新型算法和快速多极子方法相结合,用这种混合算法处理理想导体目标的电磁散射问题。
最后,在分析了理想导体目标电磁散射问题的基础上,对混合算法进一步扩展,研究了混合算法在分析介质体和金属介质混合目标电磁散射问题时的应用。实际算例表明,混合算法在处理基于体积分方程和体表积分方程的介质体和金属介质混合目标电磁散射问题时有较好的稳定性和高效性。
The fast analysis of electromagnetic scattering from multiscale targets has a wide application in engineering.The targets include aircraft, missiles, radar detection and so on. Not only the electrically large size but also the complex geometrical structure make it more difficult when calculating the targets. In this paper,we propose a novel scheme which is called accelerated Cartesian expansion(ACE) algorithm to accelerate integral equation solvers when applied to multiscale targets.The algorithm has showed efficiency and accuracy when it is combined with the fast multipole method.
Firstly, the paper elaborates method of moment and the fast multipole method when applied to solve integral equations. The standard of choosing the base fuction and the test fuction for the method of moment is demonstrated also with the RWG and CRWG base fuction which are the base of the numerical calculate. Then we apply this method analysising the surface integral equation, volume integral equation and the volume-surface hybrid integral equation. For fast multipole method, the principles and base steps are detailed.
Secondly, the accelerated Cartesian expansion algorithm and the hybid algorithm with fast multipole method are researched to solve the electromagnetic scattering problems of the perfect metal targets. The conception and the basic arithmetic of Cartesian tensor are stated so the algorithm can be used to expand the green fuction. In both electric field integral equation and magnetic field integral equation, the basic principles of accelerated Cartesian expansion are formulated. Then we propose a hybrid algorithm which is combined with accelerated Cartesian expansion algorithm and fast multipole method to solve the electromagnetic scattering problems of the perfect metal targets.
Finally, we make further efforts, which is the extanded application of the hybrid algorithm, on the electromagnetic scattering problems of the dielectric targets and the metal-dielectric combined targets. The numerical results demonstrate the stability and efficiency of the algorithm.
引文
[1] Weng Cho Chew, Jian-Ming Jin, Cai-Cheng Lu, Eric Michielssen and Ji-Ming Song. Fast Solution Methods in Electromagnetics. IEEE Transactions on Antennas and Propagation, Vol. 45, No. 3, pp. 533-543, March, 2007.
[2]盛新庆.计算电磁学要论.科学出版社,2004.
[3] K. S. Yee, J. S. Chen, and A. H. Chang. Conformal finite-difference time-domain(FDTD) with overlapping grids. IEEE Transactions on Antennas and Propagation, 1992, Vol. 40, No.9, pp. 1068-1075, 1992.
[4] A. Taflve. Computational electrodynamics: the finite-difference time-domain Method. Norwood, MA: Artech House, 1995.
[5]金建铭.电磁场有限元方法.西安电子科技大学出版社,1998.
[6] M. A. Morgan and K. K. Mei, Finite-element computation of scattering by inhomogeneous penetrable bodies of revolution. IEEE Transactions on Antennas and Propagation, 1979, AP-27, No.2, pp. 202-214, Mar. 1979.
[7] R. F. Harrington. Field Computation by Moment Methods. New York, MacMillan, 1968.
[8]李世智.电磁辐射与散射中的矩量法.电子工业出版社,1985年.
[9] A. A. Becker. The Boundary Element Method in Engineering. McGrall Hill International (UK), 1992.
[10] V. Rokhlin. Rapid solution of integral equation of scattering theory in two dimensions. Journal of Computational physics, pp. 414-439. 1990。
[11] J. M. Song and W. C. Chew. Fast multipole solution of three dimensional integral equation. IEEE Antennas Propagation Symposium, pp.1528-1531, 1995.
[12] J.M.Song, C.C.Lu, and W.C.Chew. Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects. IEEE Transactions on Antennas and Propagation. Vol. 45, No. 10, pp. 1488-1493, 1997.
[13] J. M. Song, C. C. Lu, W. C. Chew and S.W. Lee. Introduction to Fast Illinois Solver Code (FISC). IEEE Antennas Propagation Symposium. pp. 48-51, 1997.
[14] S. Velamparambil, J. M. Song, W. C. Chew and K. Gallivan. ScaleME: a portable scaleablemultipole engine for electromagnetic and acoustic integral equation. IEEE Antennas Propagation Symposium, 1998.
[15] S. Velamparambil, W. C. Chew, and Jiming Song. 10 Million Unknowns: Is It That Big? IEEE Transaction on Antennas Propagation Magazine, Vol. 45, pp. 43-58, April 2003.
[16] B.Shanker and H.Huang, Accelerated Cartesian expansions—A fast method for computing of potentials of the form R for all real r v, J.Computat.Phys. , vol.226, pp.732–753, 2007.
[17] M.Vikram and B.Shanker, Fast evaluation of time domain fields in sub-wavelength source/observer distributions using accelerated Cartesian expansions (ACE), J. Computat.Phys., vol.227, pp.1007–1023, 2007.
[18] J.G.Simmonds, A Brief on Tensor Analysis, Springer-Verlag, 1982.
[19] H.Jeffreys, Cartesian Tensors,Cambridge Press, 1931.
[20] M.Vikram, B. Shanker, T. Van. A Novel Wideband FMM for Fast Integral Equation Solution of Multiscale Problems in Electromagnetics, IEEE Trans. Antennas and Propagation, 57, 2094-2104, 2009.
[21] S. Velamparambil, W. C. Chew. Analysis and Performance of a Distributed Memory Multilevel Fast Multipole Algorithm. IEEE Transactions on Antennas and Propagation, Vol.53, No. 8 II, pp. 2719-2727, August 2005.
[22] S. Rao, D. Wilton, A. Glisson. Electromagnetic scattering by surfaces of arbitrary shape. IEEE Transactions on Antennas and Propagation, Vol. 30, pp. 409-418, 1982.
[23] Jun Hu, Zaiping Nie and Lei Lin. Solving 3-D electromagnetic scattering from conducting object by MLFMA with curvilinear RWG basis. IEEE Antennas and Propagation Society International Symposium, pp: 460-463, 2003.
[24]阙肖峰.导体介质组合目标电磁辐射与散射分析的精确建模和快速算法研究.电子科技大学博士学位论文,2008.
[25] M. R. Hestenes and E. Stiefel. Method of conjugate gradienis for solving linear Systems. J. Res. Nat. Bur. Standards,Vol.49,pp.409-436, Dec. 1952.
[26] Y. Saad and M. Schultz,“Gmres: a generalized minimal residual algorithm for solving non-symmetric linear systems,”SIAM J. Sci. Statist. Comput., Vol.7, No.3, pp.856-869, 1986.
[27] M. Eiermann, O. G. Ernst, and O. Schneider. Analysis of acceleration strategies for restarted minimum residual methods. Journal of Computational and Applied Mathematics, Vol.123, pp.261-292, 2000.
[28] M.Vikram, H.Griffith, He Huang, and B.Shanker. Accelerated Cartesian Harmonics for Fast Computation of Time and Frequency Domain Low-Frequency Kernels. IEEE APS, pp.5607-5610, 2007.
[29] Jon Applequist. Traceless Cartesian tensor forms for spherical harmonic functions: new theorems and applications to electrostatics of dielectric media. J.Phy, pp, 4303-4330, 1989.
[30] C.A.Balafoutis, P.Misrat, and R.V.Patel, A Cartesian Tensor Approach For Fast Computation of Manipulator Dynamics. IEEE APS, pp.1348-1353, 1988.
[31] E.Bleszynski. AIM: adaptive integral method for solving large-scale electromagnetic scatting and radiation problems. Radio Science, pp.1225-1251, 1996.
[32] T.F.Eibert, J.L.Volakis. Adaptive integral method for hybrid FE/BI modeling of 3-D doubly periodic structures. IEE Proc-Microw. Antennes Propag., pp. 17-22, 1999.
[33] Eric Michielssen and Amir Boag. Multilevel evaluation of electromagnetic fields for the rapid solution of scatting problems. Microwave and Optical Technology Letters, pp. 790-795, 1994.
[34] Nie Xiaochun, Ning Yuan, Le-Wei Li, Yeow-Beng Gan, Tat Soon Yeo. A Fast Combined Field Volume Integral Equation Solution to EM Scattering by 3-D Dielectric Objects of Arbitrary Permittivity and Permeability. IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, Vol.54, NO.3, pp.961-969. 2006.
[35] J.L.Rodriguez, J.M.Taboada, Marta G.Araujo, Fernando Obelleiro Basteiro, Luis Landesa, and Ines Garefa-Tunon. On the Use of the Singular Value Decomposition in the Fast Multipole Method. IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, Vol.56, NO.8, pp.2325-2334. 2008.
[36] J. C. Maxwell. A Treatise on Electricity and Magnetism. Calrendon, U.K.: Oxford, 1873.
[37] A.Peterson, S. Ray, and R.Mittra. Computational Methods for Electromagnetics. Piscataway, NJ: IEEE Press, 1997.
[38] W. C. Chew. Waves and Fields in Inhomogeneous Media. New York:IEEE Press, 1995.
[39] M.Vikram and B.Shanker. An incomplete review of fast multipole methods—from static to wideband—as applied to problems in computational electromagnetics. ACES, 2008.
[40] L.Gurel and O.Ergul. Fast and accurate solutions of extremely large integral-equation problems discretised with tens of millions of unknowns. Electron. Lett., vol. 43, no. 9, pp. 499–500, Apr. 2007.
[41] J.-S.Zhao and W.C.Chew. Integral equation solution to Maxwell’s equations from zerofrequency to microwave frequency. IEEE Trans. Antennas Propag., vol. 48, pp. 1635–1645, 2000.
[42] L.Greengard, J.Huang, V.Rokhlin, and S.Wandzura. Accelerating fast multipole methods for the Helmholtz equation at low frequencies. IEEE Computat. Sci. Eng., vol. 5, pp. 32–38, 1998.
[43] J.Mautz and R.Harrington. An E-field solution for a conducting surface small or comparable to the wavelength. IEEE Trans. Antennas Propag., vol. 32, pp. 330–339, Apr. 1984.
[44] E.Arvas, R.Harrington, and J.Mautz. Radiation and scattering from electrically small conducting bodies of arbitrary shape. IEEE Trans. Antennas Propag., vol. 34, pp. 66–77, Jan. 1986.
[45] W.Wu, W.Glisson, and D.Kajfez. A study of two numerical solution procedures for the electric field integral equation at low frequency. Appl. Computat. Electromagn. Soc. J., vol. 10, pp. 69–80, 1995.
[46] M.Burton 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, pp. 69–80, 1995.
[47] L.J.Jiang and W.C.Chew. Low-frequency fast inhomogeneous plane-wave algorithm (LF-FIPWA). Microw. Opt. Technol. Lett., vol. 20, pp. 117–122, 2004.
[48] E.Darve and P.Have. Efficient fast multipole method for low-frequency scattering. J. Computat. Phys., vol. 197, pp. 341–363, Jan. 2004.
[49] I.Bogaert, J.Peeters, J.Fostier, and F.Olyslager. NSPWMLFMA: A low frequency stable formulation of the MLFMA in three dimensions.in Proc. IEEE Antennas and Propag. Society Int. Symp. AP-S, pp. 1–4, Jul. 2008.
[50] M.Abramowitz and I.A.Stegun. Handbook of Mathematical Functionswith Formulas. Graphs, and Mathematical Tables. New York:Dover, 1972.
[51] M. R. Hestenes and E. Stiefel. Method of conjugate gradienis for solving linear Systems. J. Res. Nat. Bur. Standards,Vol.49,pp.409-436, Dec. 1952.
[52] Hu Jun, Nie Zaiping. Solution of electromagnetic scatting from two-dimensional conductor by using FMM with BiCG method. Chinese Journal of Electronics, Vol 6, No.4,pp.40-42, 1997.
[53] Y. Saad and M. Schultz,“Gmres: a generalized minimal residual algorithm for solving non-symmetric linear systems,”SIAM J. Sci. Statist. Comput., Vol.7, No.3, pp.856-869, 1986.
[54] M. Eiermann, O. G. Ernst, and O. Schneider. Analysis of acceleration strategies for restarted minimum residual methods. Journal of Computational and Applied Mathematics, Vol.123, pp.261-292, 2000.
[55] A. C. Woo, H. T. G. Wang, M. J. Schuh and M. L. Sander. Benchmark radar targets for the validation of computational electromagnetics programs. IEEE Trans. Antennas and Propagation Magazine, 35(1): 84-89. 1993.
[56] S. Rao, D.Wilton, and A. Glisson. A Tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies. IEEE Trans. Antennas Propag., vol. 30, pp. 409–418, May 1982.
[57] S. Wandzura. Electric current basis functions for curved surfaces. Electromagrietics, vol. 12, pp. 77-91, 1992.
[2]盛新庆.计算电磁学要论.科学出版社,2004.
[3] K. S. Yee, J. S. Chen, and A. H. Chang. Conformal finite-difference time-domain(FDTD) with overlapping grids. IEEE Transactions on Antennas and Propagation, 1992, Vol. 40, No.9, pp. 1068-1075, 1992.
[4] A. Taflve. Computational electrodynamics: the finite-difference time-domain Method. Norwood, MA: Artech House, 1995.
[5]金建铭.电磁场有限元方法.西安电子科技大学出版社,1998.
[6] M. A. Morgan and K. K. Mei, Finite-element computation of scattering by inhomogeneous penetrable bodies of revolution. IEEE Transactions on Antennas and Propagation, 1979, AP-27, No.2, pp. 202-214, Mar. 1979.
[7] R. F. Harrington. Field Computation by Moment Methods. New York, MacMillan, 1968.
[8]李世智.电磁辐射与散射中的矩量法.电子工业出版社,1985年.
[9] A. A. Becker. The Boundary Element Method in Engineering. McGrall Hill International (UK), 1992.
[10] V. Rokhlin. Rapid solution of integral equation of scattering theory in two dimensions. Journal of Computational physics, pp. 414-439. 1990。
[11] J. M. Song and W. C. Chew. Fast multipole solution of three dimensional integral equation. IEEE Antennas Propagation Symposium, pp.1528-1531, 1995.
[12] J.M.Song, C.C.Lu, and W.C.Chew. Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects. IEEE Transactions on Antennas and Propagation. Vol. 45, No. 10, pp. 1488-1493, 1997.
[13] J. M. Song, C. C. Lu, W. C. Chew and S.W. Lee. Introduction to Fast Illinois Solver Code (FISC). IEEE Antennas Propagation Symposium. pp. 48-51, 1997.
[14] S. Velamparambil, J. M. Song, W. C. Chew and K. Gallivan. ScaleME: a portable scaleablemultipole engine for electromagnetic and acoustic integral equation. IEEE Antennas Propagation Symposium, 1998.
[15] S. Velamparambil, W. C. Chew, and Jiming Song. 10 Million Unknowns: Is It That Big? IEEE Transaction on Antennas Propagation Magazine, Vol. 45, pp. 43-58, April 2003.
[16] B.Shanker and H.Huang, Accelerated Cartesian expansions—A fast method for computing of potentials of the form R for all real r v, J.Computat.Phys. , vol.226, pp.732–753, 2007.
[17] M.Vikram and B.Shanker, Fast evaluation of time domain fields in sub-wavelength source/observer distributions using accelerated Cartesian expansions (ACE), J. Computat.Phys., vol.227, pp.1007–1023, 2007.
[18] J.G.Simmonds, A Brief on Tensor Analysis, Springer-Verlag, 1982.
[19] H.Jeffreys, Cartesian Tensors,Cambridge Press, 1931.
[20] M.Vikram, B. Shanker, T. Van. A Novel Wideband FMM for Fast Integral Equation Solution of Multiscale Problems in Electromagnetics, IEEE Trans. Antennas and Propagation, 57, 2094-2104, 2009.
[21] S. Velamparambil, W. C. Chew. Analysis and Performance of a Distributed Memory Multilevel Fast Multipole Algorithm. IEEE Transactions on Antennas and Propagation, Vol.53, No. 8 II, pp. 2719-2727, August 2005.
[22] S. Rao, D. Wilton, A. Glisson. Electromagnetic scattering by surfaces of arbitrary shape. IEEE Transactions on Antennas and Propagation, Vol. 30, pp. 409-418, 1982.
[23] Jun Hu, Zaiping Nie and Lei Lin. Solving 3-D electromagnetic scattering from conducting object by MLFMA with curvilinear RWG basis. IEEE Antennas and Propagation Society International Symposium, pp: 460-463, 2003.
[24]阙肖峰.导体介质组合目标电磁辐射与散射分析的精确建模和快速算法研究.电子科技大学博士学位论文,2008.
[25] M. R. Hestenes and E. Stiefel. Method of conjugate gradienis for solving linear Systems. J. Res. Nat. Bur. Standards,Vol.49,pp.409-436, Dec. 1952.
[26] Y. Saad and M. Schultz,“Gmres: a generalized minimal residual algorithm for solving non-symmetric linear systems,”SIAM J. Sci. Statist. Comput., Vol.7, No.3, pp.856-869, 1986.
[27] M. Eiermann, O. G. Ernst, and O. Schneider. Analysis of acceleration strategies for restarted minimum residual methods. Journal of Computational and Applied Mathematics, Vol.123, pp.261-292, 2000.
[28] M.Vikram, H.Griffith, He Huang, and B.Shanker. Accelerated Cartesian Harmonics for Fast Computation of Time and Frequency Domain Low-Frequency Kernels. IEEE APS, pp.5607-5610, 2007.
[29] Jon Applequist. Traceless Cartesian tensor forms for spherical harmonic functions: new theorems and applications to electrostatics of dielectric media. J.Phy, pp, 4303-4330, 1989.
[30] C.A.Balafoutis, P.Misrat, and R.V.Patel, A Cartesian Tensor Approach For Fast Computation of Manipulator Dynamics. IEEE APS, pp.1348-1353, 1988.
[31] E.Bleszynski. AIM: adaptive integral method for solving large-scale electromagnetic scatting and radiation problems. Radio Science, pp.1225-1251, 1996.
[32] T.F.Eibert, J.L.Volakis. Adaptive integral method for hybrid FE/BI modeling of 3-D doubly periodic structures. IEE Proc-Microw. Antennes Propag., pp. 17-22, 1999.
[33] Eric Michielssen and Amir Boag. Multilevel evaluation of electromagnetic fields for the rapid solution of scatting problems. Microwave and Optical Technology Letters, pp. 790-795, 1994.
[34] Nie Xiaochun, Ning Yuan, Le-Wei Li, Yeow-Beng Gan, Tat Soon Yeo. A Fast Combined Field Volume Integral Equation Solution to EM Scattering by 3-D Dielectric Objects of Arbitrary Permittivity and Permeability. IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, Vol.54, NO.3, pp.961-969. 2006.
[35] J.L.Rodriguez, J.M.Taboada, Marta G.Araujo, Fernando Obelleiro Basteiro, Luis Landesa, and Ines Garefa-Tunon. On the Use of the Singular Value Decomposition in the Fast Multipole Method. IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, Vol.56, NO.8, pp.2325-2334. 2008.
[36] J. C. Maxwell. A Treatise on Electricity and Magnetism. Calrendon, U.K.: Oxford, 1873.
[37] A.Peterson, S. Ray, and R.Mittra. Computational Methods for Electromagnetics. Piscataway, NJ: IEEE Press, 1997.
[38] W. C. Chew. Waves and Fields in Inhomogeneous Media. New York:IEEE Press, 1995.
[39] M.Vikram and B.Shanker. An incomplete review of fast multipole methods—from static to wideband—as applied to problems in computational electromagnetics. ACES, 2008.
[40] L.Gurel and O.Ergul. Fast and accurate solutions of extremely large integral-equation problems discretised with tens of millions of unknowns. Electron. Lett., vol. 43, no. 9, pp. 499–500, Apr. 2007.
[41] J.-S.Zhao and W.C.Chew. Integral equation solution to Maxwell’s equations from zerofrequency to microwave frequency. IEEE Trans. Antennas Propag., vol. 48, pp. 1635–1645, 2000.
[42] L.Greengard, J.Huang, V.Rokhlin, and S.Wandzura. Accelerating fast multipole methods for the Helmholtz equation at low frequencies. IEEE Computat. Sci. Eng., vol. 5, pp. 32–38, 1998.
[43] J.Mautz and R.Harrington. An E-field solution for a conducting surface small or comparable to the wavelength. IEEE Trans. Antennas Propag., vol. 32, pp. 330–339, Apr. 1984.
[44] E.Arvas, R.Harrington, and J.Mautz. Radiation and scattering from electrically small conducting bodies of arbitrary shape. IEEE Trans. Antennas Propag., vol. 34, pp. 66–77, Jan. 1986.
[45] W.Wu, W.Glisson, and D.Kajfez. A study of two numerical solution procedures for the electric field integral equation at low frequency. Appl. Computat. Electromagn. Soc. J., vol. 10, pp. 69–80, 1995.
[46] M.Burton 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, pp. 69–80, 1995.
[47] L.J.Jiang and W.C.Chew. Low-frequency fast inhomogeneous plane-wave algorithm (LF-FIPWA). Microw. Opt. Technol. Lett., vol. 20, pp. 117–122, 2004.
[48] E.Darve and P.Have. Efficient fast multipole method for low-frequency scattering. J. Computat. Phys., vol. 197, pp. 341–363, Jan. 2004.
[49] I.Bogaert, J.Peeters, J.Fostier, and F.Olyslager. NSPWMLFMA: A low frequency stable formulation of the MLFMA in three dimensions.in Proc. IEEE Antennas and Propag. Society Int. Symp. AP-S, pp. 1–4, Jul. 2008.
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