基于时域积分方程的金属目标时域电磁散射数值分析
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摘要
从六十年代后期开始,对于与电磁脉冲有关的时域电磁场的研究引起了人们极大的重视和兴趣,时域电磁场的相关研究迅速发展。近年来,更是由于对模拟超宽带信号和非线性系统的需求日益增加,急需寻求一种能够快速、精确和稳定的模拟和分析时域问题的有效算法。时域积分方程方法应运而生。目前时域积分方程方法的研究热点集中在稳定精确求解和求解电大尺寸目标的散射辐射等相关问题。本文主要从稳定精确求解时域积分方程着手进行研究和探索。
本文首先从时变场的麦克斯韦方程组出发,推导出时域电场积分方程(TDEFIE)、时域磁场积分方程(TDMFIE)以及时域混合场积分方程(TDCFIE)的相关方程式,并介绍了在求解时域积分方程中如何使用强有力的工具:矩量法(MoM)和时间步进算法(MOT)。另外,在基函数的选取上,一定程度左右着求解时域积分方程的精度和效率,本文详细介绍了几种常见、完备的时间、空间展开基函数,并就如何选取时间、空间展开基函数的依据和方法进行了分析和阐述;随后介绍了两种时间步进算法—显式、隐式时间步进算法和各自的优缺点和适用情形以及矩阵方程求解方法等。
时间步进算法出现后期时间不稳定性的主要原因之一是由于在离散时域积分方程时采用了不精确的数值计算方法和不恰当的解析近似,因此精确计算时域积分方程的时域阻抗矩阵元素可以大幅度改善用时间步进算法求解时域积分方程的精度和后时不稳定性。本文首先提出了一种Duffy坐标变换法处理奇异性积分和自适应积分处理非奇异性积分相结合的方法;其次阐述一种使用时域积分方程的推迟位解析表达式求解时域积分方程的方法,此方法可以避免奇异性求解困难和不精确的尴尬,使得后时不稳定性得到改善;再次推导出将时间展开基函数和矢量、标量推迟位的卷积解析表达,精确求解时域阻抗矩阵元素的方法。
Electromagnetic in time domain has been paid great attention and studied extensively since 1960s. In recent years, the urgent need for the simulation of ultra wideband signal and nonlinear circuit demands an efficient algorithm that can deal with time domain electromagnetic problems efficiently, accurately and stably. Apparently, numerical methods based on the time domain integral equation can well meet these requirements. Particularly, most studies in time domain integral equation are focused on the stable and accurate simulation of scattering and radiation problems of electrically large object. Therefore, the work in this paper is aimed at the stability and accuracy in the solving of time domain integral equation.
This paper begins with the derivation of time domain electric field integral equation (TDEFIE), time domain magnetic field integral equation (TDMFIE), and time domain combined field integral equation (TDCFIE) from the well known Maxwell’s equation in time domain. Numerical methods such as method of moment (MoM) and marching-on-in-time (MOT) algorithm are also described to illustrate their effectiveness in the solution of time domain integral equation. Subsequently, several kinds of commonly used temporal and space basis functions are introduced. Due to the fact that the basis function chosen will have a strong influence on the solving efficiency and accuracy, the selection rule of various temporal and space basis function are presented based on the analysis. Finally, limitations and advantages of the two MOT methods, i.e., explicit and inexplicit MOT, are analyzed and demonstrated. Solving methods for matrix equation are also presented.
One of the fatal problems in time domain integral equation is the late-time instability. The main ingredient can be due to the inaccurate numerical computation method and inexact approximation introduced in the discretization process of the time domain integral equation, thus we expect that an accurate impedance matrix computation method will significantly improve the accuracy of the time domain integral equation and effectively avoid the late-time instability. In this paper, a hybrid method based on the Duffy Coordinate Transformations method (for singularity terms) and adaptive integration method (for nonsingularity terms) are proposed to accurately compute impedance matrix. The second section describes a method of solving time-domain integral equation though analytizing a part of retarded-time potential integrals. This method can well eliminate the difficulties in the accurate computation of the singularity terms, which will improve the stability in late-time. Finally, the convolution of the time basis functions with the vector and scalar potential are derived to an analytical expression.
本文首先从时变场的麦克斯韦方程组出发,推导出时域电场积分方程(TDEFIE)、时域磁场积分方程(TDMFIE)以及时域混合场积分方程(TDCFIE)的相关方程式,并介绍了在求解时域积分方程中如何使用强有力的工具:矩量法(MoM)和时间步进算法(MOT)。另外,在基函数的选取上,一定程度左右着求解时域积分方程的精度和效率,本文详细介绍了几种常见、完备的时间、空间展开基函数,并就如何选取时间、空间展开基函数的依据和方法进行了分析和阐述;随后介绍了两种时间步进算法—显式、隐式时间步进算法和各自的优缺点和适用情形以及矩阵方程求解方法等。
时间步进算法出现后期时间不稳定性的主要原因之一是由于在离散时域积分方程时采用了不精确的数值计算方法和不恰当的解析近似,因此精确计算时域积分方程的时域阻抗矩阵元素可以大幅度改善用时间步进算法求解时域积分方程的精度和后时不稳定性。本文首先提出了一种Duffy坐标变换法处理奇异性积分和自适应积分处理非奇异性积分相结合的方法;其次阐述一种使用时域积分方程的推迟位解析表达式求解时域积分方程的方法,此方法可以避免奇异性求解困难和不精确的尴尬,使得后时不稳定性得到改善;再次推导出将时间展开基函数和矢量、标量推迟位的卷积解析表达,精确求解时域阻抗矩阵元素的方法。
Electromagnetic in time domain has been paid great attention and studied extensively since 1960s. In recent years, the urgent need for the simulation of ultra wideband signal and nonlinear circuit demands an efficient algorithm that can deal with time domain electromagnetic problems efficiently, accurately and stably. Apparently, numerical methods based on the time domain integral equation can well meet these requirements. Particularly, most studies in time domain integral equation are focused on the stable and accurate simulation of scattering and radiation problems of electrically large object. Therefore, the work in this paper is aimed at the stability and accuracy in the solving of time domain integral equation.
This paper begins with the derivation of time domain electric field integral equation (TDEFIE), time domain magnetic field integral equation (TDMFIE), and time domain combined field integral equation (TDCFIE) from the well known Maxwell’s equation in time domain. Numerical methods such as method of moment (MoM) and marching-on-in-time (MOT) algorithm are also described to illustrate their effectiveness in the solution of time domain integral equation. Subsequently, several kinds of commonly used temporal and space basis functions are introduced. Due to the fact that the basis function chosen will have a strong influence on the solving efficiency and accuracy, the selection rule of various temporal and space basis function are presented based on the analysis. Finally, limitations and advantages of the two MOT methods, i.e., explicit and inexplicit MOT, are analyzed and demonstrated. Solving methods for matrix equation are also presented.
One of the fatal problems in time domain integral equation is the late-time instability. The main ingredient can be due to the inaccurate numerical computation method and inexact approximation introduced in the discretization process of the time domain integral equation, thus we expect that an accurate impedance matrix computation method will significantly improve the accuracy of the time domain integral equation and effectively avoid the late-time instability. In this paper, a hybrid method based on the Duffy Coordinate Transformations method (for singularity terms) and adaptive integration method (for nonsingularity terms) are proposed to accurately compute impedance matrix. The second section describes a method of solving time-domain integral equation though analytizing a part of retarded-time potential integrals. This method can well eliminate the difficulties in the accurate computation of the singularity terms, which will improve the stability in late-time. Finally, the convolution of the time basis functions with the vector and scalar potential are derived to an analytical expression.
引文
[1]盛新庆著,计算电磁学要论,北京:科学出版社,2004
[2] S. M. Rao, Time domain electromagnetics, Academic Press, 1999
[3]王秉中.计算电磁学,科学出版社,2001
[4] Jianming Jin, The Finite Element Method in Electromagnetics, Second Edition, A Wiley Interscience Publication, 2002
[5] B P Rynne , Smith P D. Stability of time marching algorithms for the electric field integral equation. J ournal of Electromagnetic Waves and Applications ,1990 ,4 (12) :1181 - 1205.
[6] D S Weile ,G Pisharody ,N W Chen ,et al . A novel scheme for the solution of the time-domain integral equations of electromagnetics. IEEE Trans Antennas and Propagat , 2004 , 52 (1) :283 - 295.
[7] Harrington R F. Field Computation by Moment Methods.New York: Macmillan, 1968.
[8] S M Rao , T K Sarkar. An alternative version of the time domain electric field integral equation for arbitrarily shaped conductors. IEEE Trans Antennas and Propagat , 1993 , 41 (6) : 831 - 834.
[9] M J Bluck , S P Walker. Time2domain bie analysis of large three dimensional electromagnetic scattering problems. IEEE Trans Antennas and Propagat ,1997 ,45 (5) :894 - 901.
[10] B Shanker ,A A Ergin , K Aygün. Analysis of transient electromagnetic scattering from closed surfaces using a combined field integralequation. IEEE Trans Antennas and Propagat ,2000 ,48 (7) :1064 - 1074.
[11] S.M. Rao, D. R.Wilton, and A.W. Glisson, Electromagnetic scattering by surfaces of arbitrary shape, IEEE Trans. Antennas Propagat., vol. AP-30, pp. 409–418, May 1982
[12] E.K.Miller. Time domain measurements in electromagnetics, Van Nostrand-Reinhold, New York, 1986
[13] N. Sachdeva, S.M. Rao and N. Balakrishnan, A Comparison of FDTD-PML with TDIE, IEEE Trans. Antennas and Propagat., vol. 50, pp. 1609-1614, Nov. 2002
[14] P.J.Davies, On the stability of time-marching schemes for the general surface electric-field integral equation, IEEE Trans Antennas Propagat 44(1996), 1467-1473
[15] A. A. Ergin, B. Shanker, E.Michielssen. The Plane-Wave Time-Domain Algorithm for the FastAnalysis of Transient Wave Phenomena. IEEE Antennas and Propagation Magazine, 1999, 41(4):39-52
[16] S.P. Walker, M.J. Bluck and I. Chatzis, The Stability of Integral Equation Time-Domain Computaions for Three-Dimentional Scattering; Similarities and Differences beteween Electrodynamic and Elastodynamic Comutations, Int. J. Numer. Model.,2002 ,pp.459-473
[17] Young-seek Chung, T.K. Sarkar and Baek Ho Jung, Solution of A Time-Domain Magnetic Field Integral Equation For Arbitrarily Closed Conducting Bodies Using An Unconditionally Stable Methodology, Microwave and Opti. Tech. Letters, Vol. 35,pp. 493-499, Dec. 2002
[18] Daniel S. Weile, G. Pisharody, Nan-wei Chen, B.Shanker and Eric Michielssen, A Novel Scheme for the Solution of The Time-Domain Integral Equations of Electromagnetics, IEEE Trans. Antennas and Propagat., vol. 52, pp.283-294, January 2004
[19] Zhong Ji,Tapan K. Sarkar A Stable Solution of Time Domain Electric Field Integral Equation for Thin-Wire Antennas Using the Laguerre Polynomials,IEEE Transactions on antennas and propagation, 2004, 52(10):2641- 2649
[20] Hui Feng Li Zhi .Investigation of time-domain characteristics of thin-wire antennas. Microwave Optical Technology Letters, 2004,43(3)253-258
[21] S. M. Rao and D. R. Wilton,“E-field, H-field, and combined field so-lution for arbitrary shaped three dimensional dielectric objects,”Elec-tromagn., vol. 10, pp. 407–421, Oct.–Dec. 1990
[22] R. J. Adams and G. S. Brown,“Stabilization procedure for electric field integral equation,”Electron. Lett., vol. 35, no. 23, pp. 2015–2016, 1999
[23] R. J. Adams,“Physical and analytical properties of a stabilized electric field integral equation,”IEEE Trans. Antennas Propagat., vol. 52, pp. 362–372, Feb. 2004
[24] H. Contopanagos, B. Dembart, M. Epton, J. J. Ottusch, V. Rokhlin, J. L. Visher, and S. M. Wandzura,“Well-conditioned boundary inte-gral equations for three-dimensional electromagnetic scattering,”IEEE Trans. Antennas Propagat., vol. 50, pp. 1824–1830, 2002
[25] D. L. Colton and R. Kress, Integral Equation Methods in Scattering Theory. New York: Wiley, 1983
[26] Jin-Lin Hu, Chi H. Chan, Senior Member, IEEE, and Yuan Xu. A New Temporal Basis Function for the Time-Domain Integral Equation Method. IEEE Microwave And Wireless Components Letters, Vol. 11, No. 11, November 2001
[27] Rao S M ,Wilton D R. Transient scattering by conducting surfaces of arbitrary shape. IEEE Trans Antennas and Propagat ,1991 ,39 (1) :56 - 61.
[28] S M Rao ,D R Wilton ,A W Glisson. Electromagnetic scattering by surfaces of arbitrary shape. IEEE Trans Antennas and Propagat ,1982 ,30(3) :409 - 418.
[29] R. F. Harrington, Field Computation byMomentMethods. Piscataway, NJ: IEEE Press, 1993
[30]钟尔杰,黄廷祝.数值分析,高等教育出版社,2004
[31] R D Graglia. On the numerical integration of the linear shape functions times the 32D green’s function or its gradient on a plane triangle. IEEE Trans Antennas and Propagat , 1993 , 41 (10) :1448 - 1455.
[32] Wilton D R, Rao S M,Glisson A W, et a1. Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains. IEEE Transactions Antennas and Propagat, 1984, 32(3): 276-281
[33] M G Duffy. Quadrature over a pyramid or cube of integrands with a singularity at a vertex. SIAM J ournal on Numerical Analysis ,1982 ,19 (6) :1260 - 1262.
[34]赵延文,等.精确快速计算时域积分方程中奇异性积分的新方法.电子与信息学报,2005,27(11):1821-1824
[35]赵延文,聂在平,徐建华,等.精确稳定求解时域电场积分方程的一种新方法.电子学报,2006,34(6).
[36]赵延文,聂在平,徐建华,武胜波.基于RWG基函数的伽略金法中奇异性积分的精确快速计算.电子学报Vol. 33
[37] Ozg¨ur Salih Erg¨ul. Fast multipole method for the solution of electromagnetic scattering problems. June 2003
[38] Abdulkadir C. Yücel and A. Arif Ergin. Exact Evaluation of Retarded-Time Potential Integrals for the RWG Bases, IEEE Transactions on Antennas And Propagation. Vol. 54, No. 5, May 2006
[39] Ozg¨ur Salih Erg¨ul. Fast multipole method for the solution of electromagnetic scattering problems. June 2003
[40] M. Lu and E. Michielssen, Closed form evaluation of time domain fields due to Rao-Wilton-Glisson sources for use inmarching-on-in-time based EFIE solvers, VOL. 1, San Antonio, TX, Jun. 16–21, 2002, pp. 74–77.
[41]赵庆广,赵延文,毕海燕,聂在平.利用时间步进算法精确稳定求解时域积分方程,电子学报Vol.36
[2] S. M. Rao, Time domain electromagnetics, Academic Press, 1999
[3]王秉中.计算电磁学,科学出版社,2001
[4] Jianming Jin, The Finite Element Method in Electromagnetics, Second Edition, A Wiley Interscience Publication, 2002
[5] B P Rynne , Smith P D. Stability of time marching algorithms for the electric field integral equation. J ournal of Electromagnetic Waves and Applications ,1990 ,4 (12) :1181 - 1205.
[6] D S Weile ,G Pisharody ,N W Chen ,et al . A novel scheme for the solution of the time-domain integral equations of electromagnetics. IEEE Trans Antennas and Propagat , 2004 , 52 (1) :283 - 295.
[7] Harrington R F. Field Computation by Moment Methods.New York: Macmillan, 1968.
[8] S M Rao , T K Sarkar. An alternative version of the time domain electric field integral equation for arbitrarily shaped conductors. IEEE Trans Antennas and Propagat , 1993 , 41 (6) : 831 - 834.
[9] M J Bluck , S P Walker. Time2domain bie analysis of large three dimensional electromagnetic scattering problems. IEEE Trans Antennas and Propagat ,1997 ,45 (5) :894 - 901.
[10] B Shanker ,A A Ergin , K Aygün. Analysis of transient electromagnetic scattering from closed surfaces using a combined field integralequation. IEEE Trans Antennas and Propagat ,2000 ,48 (7) :1064 - 1074.
[11] S.M. Rao, D. R.Wilton, and A.W. Glisson, Electromagnetic scattering by surfaces of arbitrary shape, IEEE Trans. Antennas Propagat., vol. AP-30, pp. 409–418, May 1982
[12] E.K.Miller. Time domain measurements in electromagnetics, Van Nostrand-Reinhold, New York, 1986
[13] N. Sachdeva, S.M. Rao and N. Balakrishnan, A Comparison of FDTD-PML with TDIE, IEEE Trans. Antennas and Propagat., vol. 50, pp. 1609-1614, Nov. 2002
[14] P.J.Davies, On the stability of time-marching schemes for the general surface electric-field integral equation, IEEE Trans Antennas Propagat 44(1996), 1467-1473
[15] A. A. Ergin, B. Shanker, E.Michielssen. The Plane-Wave Time-Domain Algorithm for the FastAnalysis of Transient Wave Phenomena. IEEE Antennas and Propagation Magazine, 1999, 41(4):39-52
[16] S.P. Walker, M.J. Bluck and I. Chatzis, The Stability of Integral Equation Time-Domain Computaions for Three-Dimentional Scattering; Similarities and Differences beteween Electrodynamic and Elastodynamic Comutations, Int. J. Numer. Model.,2002 ,pp.459-473
[17] Young-seek Chung, T.K. Sarkar and Baek Ho Jung, Solution of A Time-Domain Magnetic Field Integral Equation For Arbitrarily Closed Conducting Bodies Using An Unconditionally Stable Methodology, Microwave and Opti. Tech. Letters, Vol. 35,pp. 493-499, Dec. 2002
[18] Daniel S. Weile, G. Pisharody, Nan-wei Chen, B.Shanker and Eric Michielssen, A Novel Scheme for the Solution of The Time-Domain Integral Equations of Electromagnetics, IEEE Trans. Antennas and Propagat., vol. 52, pp.283-294, January 2004
[19] Zhong Ji,Tapan K. Sarkar A Stable Solution of Time Domain Electric Field Integral Equation for Thin-Wire Antennas Using the Laguerre Polynomials,IEEE Transactions on antennas and propagation, 2004, 52(10):2641- 2649
[20] Hui Feng Li Zhi .Investigation of time-domain characteristics of thin-wire antennas. Microwave Optical Technology Letters, 2004,43(3)253-258
[21] S. M. Rao and D. R. Wilton,“E-field, H-field, and combined field so-lution for arbitrary shaped three dimensional dielectric objects,”Elec-tromagn., vol. 10, pp. 407–421, Oct.–Dec. 1990
[22] R. J. Adams and G. S. Brown,“Stabilization procedure for electric field integral equation,”Electron. Lett., vol. 35, no. 23, pp. 2015–2016, 1999
[23] R. J. Adams,“Physical and analytical properties of a stabilized electric field integral equation,”IEEE Trans. Antennas Propagat., vol. 52, pp. 362–372, Feb. 2004
[24] H. Contopanagos, B. Dembart, M. Epton, J. J. Ottusch, V. Rokhlin, J. L. Visher, and S. M. Wandzura,“Well-conditioned boundary inte-gral equations for three-dimensional electromagnetic scattering,”IEEE Trans. Antennas Propagat., vol. 50, pp. 1824–1830, 2002
[25] D. L. Colton and R. Kress, Integral Equation Methods in Scattering Theory. New York: Wiley, 1983
[26] Jin-Lin Hu, Chi H. Chan, Senior Member, IEEE, and Yuan Xu. A New Temporal Basis Function for the Time-Domain Integral Equation Method. IEEE Microwave And Wireless Components Letters, Vol. 11, No. 11, November 2001
[27] Rao S M ,Wilton D R. Transient scattering by conducting surfaces of arbitrary shape. IEEE Trans Antennas and Propagat ,1991 ,39 (1) :56 - 61.
[28] S M Rao ,D R Wilton ,A W Glisson. Electromagnetic scattering by surfaces of arbitrary shape. IEEE Trans Antennas and Propagat ,1982 ,30(3) :409 - 418.
[29] R. F. Harrington, Field Computation byMomentMethods. Piscataway, NJ: IEEE Press, 1993
[30]钟尔杰,黄廷祝.数值分析,高等教育出版社,2004
[31] R D Graglia. On the numerical integration of the linear shape functions times the 32D green’s function or its gradient on a plane triangle. IEEE Trans Antennas and Propagat , 1993 , 41 (10) :1448 - 1455.
[32] Wilton D R, Rao S M,Glisson A W, et a1. Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains. IEEE Transactions Antennas and Propagat, 1984, 32(3): 276-281
[33] M G Duffy. Quadrature over a pyramid or cube of integrands with a singularity at a vertex. SIAM J ournal on Numerical Analysis ,1982 ,19 (6) :1260 - 1262.
[34]赵延文,等.精确快速计算时域积分方程中奇异性积分的新方法.电子与信息学报,2005,27(11):1821-1824
[35]赵延文,聂在平,徐建华,等.精确稳定求解时域电场积分方程的一种新方法.电子学报,2006,34(6).
[36]赵延文,聂在平,徐建华,武胜波.基于RWG基函数的伽略金法中奇异性积分的精确快速计算.电子学报Vol. 33
[37] Ozg¨ur Salih Erg¨ul. Fast multipole method for the solution of electromagnetic scattering problems. June 2003
[38] Abdulkadir C. Yücel and A. Arif Ergin. Exact Evaluation of Retarded-Time Potential Integrals for the RWG Bases, IEEE Transactions on Antennas And Propagation. Vol. 54, No. 5, May 2006
[39] Ozg¨ur Salih Erg¨ul. Fast multipole method for the solution of electromagnetic scattering problems. June 2003
[40] M. Lu and E. Michielssen, Closed form evaluation of time domain fields due to Rao-Wilton-Glisson sources for use inmarching-on-in-time based EFIE solvers, VOL. 1, San Antonio, TX, Jun. 16–21, 2002, pp. 74–77.
[41]赵庆广,赵延文,毕海燕,聂在平.利用时间步进算法精确稳定求解时域积分方程,电子学报Vol.36