三维导体的瞬态电磁散射特性研究
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摘要
本文研究了用于分析三维目标瞬态电磁散射特性的时域积分方程方法。全文内容包括了时域积分方程的稳定精确求解方法和时域积分方程的快速算法，以及对其时域基函数的关键参数的优化选取做了研究。
首先阐述了时域计算的主要的两大类方法：基于时域差分方程的时域有限差分法(FDTD)和时域有限元法(TDFEM)，和基于时域积分方程(TDIE)的矩量法(MOM)算法：时间步进法(MOT)和阶数步进法(MOD)。介绍了FDTD，TDFEM的这两种方法的原理，求解步骤，技术难点，并与TDIE方法进行比较，分析了各自的优缺点，介绍了MOM算法在求解时域积分方程时所面临的两大难题：时间步进法(MOT)后期响应不稳定，而阶数步进法(MOD)虽然可以克服后期响应不稳定但是其计算复杂度高。对MOD算法求解时域积分方程所涉及的细节进行了分析，并介绍了求解矩阵方程的数值算法。
其次对MOD算法中的加权拉盖尔函数的重要参数的选取进行了优化，给出了优化以后对原始信号的复原，得到了想要的结果，并推导出由时域响应转换到频域的快速转换公式，并引出了在时域电磁计算领域的应用。
最后，为提高MOD算法的计算效率和降低内存需求，研究了时域积分方程的快速算法。基于MOD算法提出了电偶极矩等效的快速算法，计算实例表明在同等的计算精度情况下，该算法明显地提高了形成MOM的矩阵元素的计算速度。
In this paper, the time domain integral equation method (TDIE) and its application for transient scattering of arbitrarily shaped 3D conducting objects are studied. The main works include: the research of the stability solution of TDIE, the optimization of the parameter of time domain basis function set that is derived from the Laguerre functions, and its application is also presented.
In the first part, the fundamental theory of time-domain electromagnetic is introduced. The time-domain methods include the time-domain differential equation (TDDE) methods and the time-domain integral equation (TDIE) methods. The general concepts of FDTD, TDFEM, FD-MoM and TD-MoM are discussed respectively, and the advantage and disadvantage between TDIE and TDDE are also pointed out. A traditional method to solve the TDIE has two fatal weaknesses, one is the late-time instability; the other is the computation complexity which make it impossible to accurately analysis for the electrically large object subjects.
A stable method called MOD algorithm to solve the TD-CFIE is introduced. In this algorithm, the weighted Laguerre basis functions are chosen as the time domain basis function, while the RGW functions as spatial basis. The characteristics of weighted laguerre basis functions are discussed, and the emphasis is how to optimize its parameters.
Next, a general solution method of MOD is introduced, and its numerical result is compared with the Inverse Discrete Fourier Transform (IDFT) of frequency domain combined integral equation (FD-CFIE).
Finally, a method for efficient generation of MOM matrices based on equivalent dipole to solve the TD-CFIE coupled with the MOD algorithm is presented.
首先阐述了时域计算的主要的两大类方法：基于时域差分方程的时域有限差分法(FDTD)和时域有限元法(TDFEM)，和基于时域积分方程(TDIE)的矩量法(MOM)算法：时间步进法(MOT)和阶数步进法(MOD)。介绍了FDTD，TDFEM的这两种方法的原理，求解步骤，技术难点，并与TDIE方法进行比较，分析了各自的优缺点，介绍了MOM算法在求解时域积分方程时所面临的两大难题：时间步进法(MOT)后期响应不稳定，而阶数步进法(MOD)虽然可以克服后期响应不稳定但是其计算复杂度高。对MOD算法求解时域积分方程所涉及的细节进行了分析，并介绍了求解矩阵方程的数值算法。
其次对MOD算法中的加权拉盖尔函数的重要参数的选取进行了优化，给出了优化以后对原始信号的复原，得到了想要的结果，并推导出由时域响应转换到频域的快速转换公式，并引出了在时域电磁计算领域的应用。
最后，为提高MOD算法的计算效率和降低内存需求，研究了时域积分方程的快速算法。基于MOD算法提出了电偶极矩等效的快速算法，计算实例表明在同等的计算精度情况下，该算法明显地提高了形成MOM的矩阵元素的计算速度。
In this paper, the time domain integral equation method (TDIE) and its application for transient scattering of arbitrarily shaped 3D conducting objects are studied. The main works include: the research of the stability solution of TDIE, the optimization of the parameter of time domain basis function set that is derived from the Laguerre functions, and its application is also presented.
In the first part, the fundamental theory of time-domain electromagnetic is introduced. The time-domain methods include the time-domain differential equation (TDDE) methods and the time-domain integral equation (TDIE) methods. The general concepts of FDTD, TDFEM, FD-MoM and TD-MoM are discussed respectively, and the advantage and disadvantage between TDIE and TDDE are also pointed out. A traditional method to solve the TDIE has two fatal weaknesses, one is the late-time instability; the other is the computation complexity which make it impossible to accurately analysis for the electrically large object subjects.
A stable method called MOD algorithm to solve the TD-CFIE is introduced. In this algorithm, the weighted Laguerre basis functions are chosen as the time domain basis function, while the RGW functions as spatial basis. The characteristics of weighted laguerre basis functions are discussed, and the emphasis is how to optimize its parameters.
Next, a general solution method of MOD is introduced, and its numerical result is compared with the Inverse Discrete Fourier Transform (IDFT) of frequency domain combined integral equation (FD-CFIE).
Finally, a method for efficient generation of MOM matrices based on equivalent dipole to solve the TD-CFIE coupled with the MOD algorithm is presented.
引文
[1] C.L.Bennet, "A technique for computing approximate electromagnetic impulse response of conducting bodies", Ph.D.Purdue University, 1968
[2] C.L.Bennet, "Transient scattering from comducting cylinders"[J], IEEE T-Ap,vol.18,No.9 Sept. 1970
[3] A.J.Plggio,E.K.Miller,R.Mittra, "Computer techniques in electromagnetic: Integral equation solution of 3D scattering problems"[M], Oxford, 1973
[4] C.L.Bennet,GF.Ross, "The time electrmoagnetics and its appIications"[J], Proc. IEEE,Vol.66,1978
[5] N.J.Damaskos, "Transient scattering by Resistive Cylinders"[J], IEEE T-AP, Vol.33,Jan.1985
[6] A.G.Tijhuis,"Toward a stable marching-on-in-time method for two dimensional electromagnetic scattering problems"[J], Radio Sci.,vol.19, no.5,pp.l311-1317, Sept.Oct.1984
[7] B.P.Rynne, "Stability and convergence of time marching methods in scattering problems"[J], HVIA J. Appl. Math., vol. 35, pp. 297-310,1985.
[8] B. P. Rynne, "Instabilities in time marching methods for scattering problems"[J],Electromagn., vol. 6, pp. 129-144, Apr./June 1986
[9] S.M.Rao, D.R.Wilton, "Transient scattering by conducting surfaces of arbitrary shape"[J], IEEE Transactions on Antennas and Propagation, vol, 39, NO, 1, January,1991
[10] B. P. Rynne and P. D. Smith, "Stability of time marching algorithms for the electric field integral equation"[J], J. Electromagn. Waves Appl, vol. 4, no. 12, pp. 1181-1205,Dec. 1990
[11] B.P.Rynne, " Time domain scattering from arbitrary surfaces using the electric field integral equation" [J], Electromagnetic waves Appl., Jan. 1991,vol.5, pp.93-112.
[12] S. M. Rao and T. K. Sarkar, "Time domain modeling of two dimensional conducting ylinders utilizing an implicit scheme-TM incident"[J], Microwave Opt. Technol. Lett.,vol. 15, no. 6, pp. 342-347, Aug. 1997
[13] S. M. Rao, D. A. Vechinski, and T. K. Sarkar, "Transient scattering by conducting cylinders-implicit solution for the transverse electric case"[J], Microwave Opt. Technol. Lett., vol. 21, no. 2, pp. 129-134, Apr. 1999
[14] M. D. Pocock, M. J. Bluck, and S. P. Walker, "Electromagnetic scattering from 3-D curved dielectric bodies using time-domain integral equations"[J], IEEE Trans.Antennas Propagat., vol. 46, pp. 1212-1219, Aug. 1998
[15] S. J. Dodson, S. P. Walker, and M. J. Bluck, "Implicitness and stability of time domain integral equation scattering analysis"[J], ACES J., vol. 13, no. 3, pp. 291-301, Nov.1998.
[16] S.M.Rao, "Time Domain Electromagnetic" [M], ACADEMIC, 1999[17] P. D. Smith,"Instability in time marching methods for scattering: Cause and rectification"[J],Electromagnetics, vol. 10, pp. 439-451, 1990
[18] D. A. Vechinski and S. M. Rao, "Transient scattering from two-dimensional dielectric cylinders and arbitrary shape"[J], IEEE Trans. Antennas Propagat., vol. 40, pp.1054-1060, Sept. 1992
[19] D.A.Vechinski and S.M.Rao, "A stable procedure to calculate the transient scattering by conducting surface of arbitrary shape "[J], IEEE Transactions on Antennas and Propagation, 1992,40(6): 661-665
[20] Ali Sadigh and Ercument Arvas, "Treating the instabilities in marching-on-in-time method from a different perspective "[J], IEEE Transactions on Antennas and Propagation, vol, December 1993,41,No. 12
[21] P.J.Dadvies, "Stability of time-marching numerical schemes for the electric field integral equation"[J], Journal of Electromagnetic waves Appl., 1994, 8(1):85-114
[22] P. J. Davies, "On the stability of time-marching schemes for the general surface electric-field integral equation"[J], IEEE Trans. Antennas Propagat., vol. 44, pp. 1467-1473, Nov. 1996.
[23] J. Garrett,A. Ruehli, and C. Paul, "Stability improvements of integral equations''[J], in Proc. IEEE AP-S Int. Symp. Dig., vol. 3, Montreal, PQ, Canada, Julyl997, pp.1810-1813.
[24] W. Pinello, A. Ruehli, and A. Cangellaris, "Stabilization of time domain solutions of EFIE based on partial element equivalent circuit models" [J],in Proc. IEEE AP-S Int. Symp. Dig., vol. 2, Montreal, PQ, Canada, July 1997, pp. 966-969.
[25] S. Kashyap, M. Burton, and A. Louie, "A stabilizing scheme for the explicit time-domain integral-equation algorithm "[J], ACES J., vol. 13, no. 3, pp. 226-233,Nov. 1998.
[26] Jin-Lin Hu,Chi H. Chan, and Yuan Xu, "A new temporal basis function for the time-domain integral equation method"[J],IEEE Microwave and wireless components letters, 2001,vol.11, No.11, November.
[27] B. H. Jung, T. K. Sarkar, "Time-domain CFEE for the analysis of transient scattering from arbitrarily shaped 3D conducting objects"[J], Microwave and Optical Technology Letters 2002; 34(4):289-296.
[28] B. H.Jung, T. K. Sarkar, "An accurate and stable implicit solution for transient scattering and radiation from wire structures "[J], Microwave and Optical Technology Letters 2002; 34(5):354-359.
[29] B.H.Jung, T. K. Sarkar, Young-seek Chung, "Analysis of transient electromagnetic scattering from dielectric objects using a combined-field integral equation"[J], Microwave and Optical Technology Letters Volume40, Issue6, Date:20 March 2004,Pages:476-481
[30] B. H. Jung, T. K. Sarkar, Young-seek Chung, "Time-domain combined field integral equation using Laguerre polynomials as temporal basis functions "[J], International Journal of Numerical Modelling: Electronic Networks, Devices and Fields Volumel7,Issue3, Date:May/June 2004, Pages:251-268
[31] A.A.Ergin, B. Shanker, and E. Michielssen, "The plane wave timedomain algorithm for the fast analysis of transient wave phenomena"[J], IEEE Antennas Propagat. Mag., vol. 41, pp. 39-52, Aug. 1999.
[32] B. Shanker, A. A. Ergin, K. Aygun, and E. Michielssen, "Analysis of transient electromagnetic scattering phenomena using a two-level plane wave time domain algorithm"[J], IEEE Trans. Antennas Propagat., vol. 48, pp. 510-523, 2000.
[33] B. Shanker, A. A. Ergin, K. Aygun, and Mingyu Lu, "Fast Analysis of Transient Electromagnetic Scattering Phenomena Using the Multilevel Plane Wave Time Domain Algorithm" [J], IEEE Trans. Antennas Propagat, Vol. 51, No. 3, March 2003.
[34] W. Cai, Yu T, Wang H, Yu Y. "Higher-order mixed RWG basis functions for electromagnetic applications"[J], IEEE Transactions on Microwave Theory and Techniques 2001; 49(7): 1295-1303.
[35] W. Cai, Yu Y, Yuan XC. "Singularity treatment and higher-order RWG basis functions for integral equations of electromagnetic scattering" [J], International Journal for Numerical Methods in Engineering, 2002, 53: 31-47.
[36] 谢处方，吴先良电磁散射理论与计算[M]安徽大学出版社2002
[37] 武胜波，三维目标电磁散射的时域积分方程方法，[硕士学位论文]，成都：电子科技大学，2005
[38] Mautz, J. R.., and R. F. Harrington, "H-field, E-field, and combined field solutions for conducting bodies of revolution" [J], AEU, Vol.32, No.4, pp.157-164, April 1978
[39] R. F. Harrington, "Field computation by Method of Moments"[M], New York: Macmillan, 1968
[40] C. H. Golub and C.E Vanloan, "Matrix Computations"[M], Third Edition, The Johns Hopkins University Press, London, 2003
[41] Xue. Guan Liu, Hui.Ping Guo, "The Application of the Equivalent Dipole-moment Method to Electromagnetic Scattering of 3D Objects" [J], Vol 3 p 1830-1832, Proc. Of APMC2005, Suzhou China. 2005
[2] C.L.Bennet, "Transient scattering from comducting cylinders"[J], IEEE T-Ap,vol.18,No.9 Sept. 1970
[3] A.J.Plggio,E.K.Miller,R.Mittra, "Computer techniques in electromagnetic: Integral equation solution of 3D scattering problems"[M], Oxford, 1973
[4] C.L.Bennet,GF.Ross, "The time electrmoagnetics and its appIications"[J], Proc. IEEE,Vol.66,1978
[5] N.J.Damaskos, "Transient scattering by Resistive Cylinders"[J], IEEE T-AP, Vol.33,Jan.1985
[6] A.G.Tijhuis,"Toward a stable marching-on-in-time method for two dimensional electromagnetic scattering problems"[J], Radio Sci.,vol.19, no.5,pp.l311-1317, Sept.Oct.1984
[7] B.P.Rynne, "Stability and convergence of time marching methods in scattering problems"[J], HVIA J. Appl. Math., vol. 35, pp. 297-310,1985.
[8] B. P. Rynne, "Instabilities in time marching methods for scattering problems"[J],Electromagn., vol. 6, pp. 129-144, Apr./June 1986
[9] S.M.Rao, D.R.Wilton, "Transient scattering by conducting surfaces of arbitrary shape"[J], IEEE Transactions on Antennas and Propagation, vol, 39, NO, 1, January,1991
[10] B. P. Rynne and P. D. Smith, "Stability of time marching algorithms for the electric field integral equation"[J], J. Electromagn. Waves Appl, vol. 4, no. 12, pp. 1181-1205,Dec. 1990
[11] B.P.Rynne, " Time domain scattering from arbitrary surfaces using the electric field integral equation" [J], Electromagnetic waves Appl., Jan. 1991,vol.5, pp.93-112.
[12] S. M. Rao and T. K. Sarkar, "Time domain modeling of two dimensional conducting ylinders utilizing an implicit scheme-TM incident"[J], Microwave Opt. Technol. Lett.,vol. 15, no. 6, pp. 342-347, Aug. 1997
[13] S. M. Rao, D. A. Vechinski, and T. K. Sarkar, "Transient scattering by conducting cylinders-implicit solution for the transverse electric case"[J], Microwave Opt. Technol. Lett., vol. 21, no. 2, pp. 129-134, Apr. 1999
[14] M. D. Pocock, M. J. Bluck, and S. P. Walker, "Electromagnetic scattering from 3-D curved dielectric bodies using time-domain integral equations"[J], IEEE Trans.Antennas Propagat., vol. 46, pp. 1212-1219, Aug. 1998
[15] S. J. Dodson, S. P. Walker, and M. J. Bluck, "Implicitness and stability of time domain integral equation scattering analysis"[J], ACES J., vol. 13, no. 3, pp. 291-301, Nov.1998.
[16] S.M.Rao, "Time Domain Electromagnetic" [M], ACADEMIC, 1999[17] P. D. Smith,"Instability in time marching methods for scattering: Cause and rectification"[J],Electromagnetics, vol. 10, pp. 439-451, 1990
[18] D. A. Vechinski and S. M. Rao, "Transient scattering from two-dimensional dielectric cylinders and arbitrary shape"[J], IEEE Trans. Antennas Propagat., vol. 40, pp.1054-1060, Sept. 1992
[19] D.A.Vechinski and S.M.Rao, "A stable procedure to calculate the transient scattering by conducting surface of arbitrary shape "[J], IEEE Transactions on Antennas and Propagation, 1992,40(6): 661-665
[20] Ali Sadigh and Ercument Arvas, "Treating the instabilities in marching-on-in-time method from a different perspective "[J], IEEE Transactions on Antennas and Propagation, vol, December 1993,41,No. 12
[21] P.J.Dadvies, "Stability of time-marching numerical schemes for the electric field integral equation"[J], Journal of Electromagnetic waves Appl., 1994, 8(1):85-114
[22] P. J. Davies, "On the stability of time-marching schemes for the general surface electric-field integral equation"[J], IEEE Trans. Antennas Propagat., vol. 44, pp. 1467-1473, Nov. 1996.
[23] J. Garrett,A. Ruehli, and C. Paul, "Stability improvements of integral equations''[J], in Proc. IEEE AP-S Int. Symp. Dig., vol. 3, Montreal, PQ, Canada, Julyl997, pp.1810-1813.
[24] W. Pinello, A. Ruehli, and A. Cangellaris, "Stabilization of time domain solutions of EFIE based on partial element equivalent circuit models" [J],in Proc. IEEE AP-S Int. Symp. Dig., vol. 2, Montreal, PQ, Canada, July 1997, pp. 966-969.
[25] S. Kashyap, M. Burton, and A. Louie, "A stabilizing scheme for the explicit time-domain integral-equation algorithm "[J], ACES J., vol. 13, no. 3, pp. 226-233,Nov. 1998.
[26] Jin-Lin Hu,Chi H. Chan, and Yuan Xu, "A new temporal basis function for the time-domain integral equation method"[J],IEEE Microwave and wireless components letters, 2001,vol.11, No.11, November.
[27] B. H. Jung, T. K. Sarkar, "Time-domain CFEE for the analysis of transient scattering from arbitrarily shaped 3D conducting objects"[J], Microwave and Optical Technology Letters 2002; 34(4):289-296.
[28] B. H.Jung, T. K. Sarkar, "An accurate and stable implicit solution for transient scattering and radiation from wire structures "[J], Microwave and Optical Technology Letters 2002; 34(5):354-359.
[29] B.H.Jung, T. K. Sarkar, Young-seek Chung, "Analysis of transient electromagnetic scattering from dielectric objects using a combined-field integral equation"[J], Microwave and Optical Technology Letters Volume40, Issue6, Date:20 March 2004,Pages:476-481
[30] B. H. Jung, T. K. Sarkar, Young-seek Chung, "Time-domain combined field integral equation using Laguerre polynomials as temporal basis functions "[J], International Journal of Numerical Modelling: Electronic Networks, Devices and Fields Volumel7,Issue3, Date:May/June 2004, Pages:251-268
[31] A.A.Ergin, B. Shanker, and E. Michielssen, "The plane wave timedomain algorithm for the fast analysis of transient wave phenomena"[J], IEEE Antennas Propagat. Mag., vol. 41, pp. 39-52, Aug. 1999.
[32] B. Shanker, A. A. Ergin, K. Aygun, and E. Michielssen, "Analysis of transient electromagnetic scattering phenomena using a two-level plane wave time domain algorithm"[J], IEEE Trans. Antennas Propagat., vol. 48, pp. 510-523, 2000.
[33] B. Shanker, A. A. Ergin, K. Aygun, and Mingyu Lu, "Fast Analysis of Transient Electromagnetic Scattering Phenomena Using the Multilevel Plane Wave Time Domain Algorithm" [J], IEEE Trans. Antennas Propagat, Vol. 51, No. 3, March 2003.
[34] W. Cai, Yu T, Wang H, Yu Y. "Higher-order mixed RWG basis functions for electromagnetic applications"[J], IEEE Transactions on Microwave Theory and Techniques 2001; 49(7): 1295-1303.
[35] W. Cai, Yu Y, Yuan XC. "Singularity treatment and higher-order RWG basis functions for integral equations of electromagnetic scattering" [J], International Journal for Numerical Methods in Engineering, 2002, 53: 31-47.
[36] 谢处方，吴先良电磁散射理论与计算[M]安徽大学出版社2002
[37] 武胜波，三维目标电磁散射的时域积分方程方法，[硕士学位论文]，成都：电子科技大学，2005
[38] Mautz, J. R.., and R. F. Harrington, "H-field, E-field, and combined field solutions for conducting bodies of revolution" [J], AEU, Vol.32, No.4, pp.157-164, April 1978
[39] R. F. Harrington, "Field computation by Method of Moments"[M], New York: Macmillan, 1968
[40] C. H. Golub and C.E Vanloan, "Matrix Computations"[M], Third Edition, The Johns Hopkins University Press, London, 2003
[41] Xue. Guan Liu, Hui.Ping Guo, "The Application of the Equivalent Dipole-moment Method to Electromagnetic Scattering of 3D Objects" [J], Vol 3 p 1830-1832, Proc. Of APMC2005, Suzhou China. 2005