时域积分方程及其混合算法在电磁脉冲效应中的研究与应用
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摘要
随着电子、电气设备的微电子化,电子系统在电磁脉冲照射下的敏感性和易损性日趋严重。与此同时,越来越多的天线装载于电大尺寸平台上,也更易遭受电磁干扰(Electromagnetic interference,EMI)。为了应对现代战争的电子对抗,尤其是超大功率超宽带电磁脉冲(EMP)武器以及电磁脉冲弹等新概念电子战武器,各国争相开展研究电磁兼容(Electromagnetic compatibility,EMC)、电磁干扰及电磁防护技术。正是本着这样的目标,本文系统地研究了时域积分方程的精确和高效求解及其混合数值方法,以及在外界电磁脉冲(EMP)作用下电磁效应的仿真计算和作用机理。
首先,本文从显式时间步进和隐式时间步进递推算法求解时域电场积分方程(TD-EFIE)的计算公式出发,给出了EMP照射下线面结构的TDIE-TDPO-MOT方法,推导了显式时间步进和隐式时间步进递推算法求解时域电场积分方程的计算公式。与TDIE-MOT相比,在精度变化不大的情况下,TDIE-TDPO-MOT减少了计算时间。同时也编程仿真了典型结构在EMP照射下散射体上感应电流的瞬态响应,并比较了数值结果。
第二,为了提高算法稳定性,提出了自适应阶数递推法和确定拉盖尔(Laguerre)多项式尺度因子的通用规则。为了消除中心近似和提高精度,又提出了准确时域伽辽金(Galerkin)检验的时域磁场(TD-MFIE)和混合场积分方程(TD-CFIE)。同时利用快速傅立叶变换(FFT),提出了加速MOD方法中卷积过程的分块算法,把MOD的计算复杂度从O(N2ON2S)降低至O(N2SNOlog2(NO))。
第三,基于对影响积分收敛速度的详细分析,提出了处理弱奇异和近奇异积分的径向积分法,这一方法可以看作是对极坐标积分法的本质改进。通过对被积函数进行光滑处理的多项式变换,相比传统的弱奇异处理技术,本文方法的积分收敛速度明显加快。文中也给出了数值算例以验证本章所提出方法的精度和收敛速度。
第四,为了精确快速地计算时域电场、磁场和混合场积分方程的矩阵元素,本文提出了一种新的四维(4-D)奇异积分方法,包括用以处理内层二维(2-D)奇异和高阶近奇异积分的径向积分法。然后仔细分析了外层2-D积分被积函数导数的性质,提出了处理外层2-D积分的被基函数平滑技术。相比传统高斯积分规则,本文方法的效率和精度大幅提高,同时也给出了数值算例以验证本章所提出方法的精度和收敛速度。因为本文提出的方法保留了处理各类被积函数的灵活性,所以其用途不仅仅限于时域积分方程,也同样适用于频域积分方程(frequency-dOmain integral equations, FDIE)、
第五,基于加权Laguerre多项式和阶数步进法(MOD),提出了具有高度稳定性的TDIE-TDPP混合方法,这一方法可以处理混合三维PEC结构的时域响应问题,其中TDPP近似应用于表面光滑的电大部分。为了正确实现混合方法,把问题的几何结构分为两个部分：TDIE区域和TDPO区域。在TDPO区域中,使用MOD方法求解一组时域电场积分方程来获得未知电流。在TDPO区域中,PO电流的求解通过入射场以及TDIE电流的辐射场得到。本文中提出的混合算法有两点优势。第一个优点是,只需要改变TDIE区域在整体中所占的比例,就可以控制近似误差,这优于单纯的TDPP近似。第二个优点是,提出的混合方法只依靠TDPP近似来节省内存需求和CPU时间,而不需要诸如PWTD这一类算法的加速。因此本文的方法可以很容易地推广到MOT以及FDDM的时间步进架构下。
最后,本文提出了基于RWG函数的投影方法,这一方法可以将系数矩阵的TDPP-TDPO部分变为单位块矩阵。因此可以基于这一投影方法对系数矩阵进行降维压缩处理。系数矩阵的维数等于NIES,而与NPOS无关,其中NIES和NPOP分别是TDIE和TDPO区域中空间基函数的数目。对于大多数混合问题,TDIE区域并不是电大尺寸,因此就可以使用直接法求解线性方程组,也避免了迭代法可能产生的收敛问题。同时,本文提出的TDIE-TDPO-MOD方法使用了自适应阶数递推,以及FFT加速MOD卷积过程的分块算法。
With the miniaturization of electronic and electrical devices, systems are increasingly sensitive and susceptible to electromagnetic interference (EMI). Similarly, many antennas are mounted on electrically large platforms, which will suffer much more EMI problems than before. In modern electronic countermeasures, especially, after the emergence of some new-style weapons such as the high-power wideband EMP launcher and EM bomb, countries around the world initiate a new round competition in the research of electromagnetic interference (EMI) and protection techniques. This dissertation is focused on the development of time-domain integral equations and its hybrid methods with high accuracy and efficiency to study electromagnetic pulse effects and interaction mechanism in some typical systems illuminated by a high-power electromagnetic pulse (EMP).
Firstly, time-domain integral equations with explicit and implicit marching-on-in-time (MOT) schemes are presented. An efficient hybrid method, TDIE-TDPO-MOT, is proposed for studying electromagnetic responses of several groups of wire and surface structures illuminated by an electromagnetic pulse (EMP), respectively. In comparison with the full TDIE-MOT method, computational complexity can be reduced drastically using our developed TDIE-TDPO-MOT method, and computational accuracy is still maintained.
Secondly, a time-domain magnetic (TD-MFIE) and combined field integral equations (TD-CFIE) for analyzing scattering from3-D PEC objects are presented. The surface current density is directly used as the unknown, and an exact temporal Galerkin testing procedure is performed with no central approximation used. An adaptive marching-on-in-degree method with FFT-based blocking scheme is proposed, with the approach to determine the optimal scaling factor also given. It is indicated that the proposed scheme can guarantee the accuracy of MOD scheme without increasing the computational cost. Our numerical results also show that the O(NO2NS2) dependence of the MOD method can be reduced to O(NS2NOlog2(NO)) now.
Thirdly, a radial integration scheme is proposed for handling weakly singular and near-singular potential integrals, which is an essentialimprovement over the previous polar integration method. We at first carryout some careful investigations on the problem of slow convergence. Then,a new technique of smoothing integrand based on polynomialtransformation is presented, which results in much faster convergencerate than that of the polar integration. Finally, some numerical resultsare given to demonstrate both accuracy and convergence rate of ourproposed scheme.
Fourthly, a new integration approach is proposed for accurate andefficient calculation of time-domain EFIE, MFIE, and CFIE matrixelements over triangular domains. It mainly consists of a radialintegration scheme for handling weakly singular and near-hypersingularinner integrals as well as some new smoothing techniques for treatingthe outer surface integrals. As the proposed method preserves theflexibility for handling different integral kernels, they are applicablefor both time-domain and frequency-domain integral equations.
Fifthly, an efficient hybrid TDIE-TDPO method with high stability,based on the weighted Laguerre polynomials and MOD scheme, is proposedfor investigating transient electromagnetic responses of3-D compositePEC objects, while the TDPO approximation is applicable to the large PECobjects with smooth surface. These objects are partitioned into tworegions: TDIE and TDPO ones. In the TDIE region, currents are updatedby solving a set of time-domain electric field integral equation (TD-EFIE)using the MOD. In the TDPO region, PO currents are obtained accordingto the incident fields as well as the ones radiated by all currents inthe TDIE one. The hybrid TDIE-TDPO method has advantages over TDPOapproximation alone, as it can control the approximation errors bychanging the percentage of TDIE regions. On the other hand, the proposedmethod does not require any specific acceleration algorithm such as PWTD,and its saving of both memory and CPU time is obtained by TDPOapproximation. Therefore, the hybrid formula can be directly applied forMOT scheme as well as finite difference delay modeling method with nodifficulty.
Finally, the TDPO-TDPO projection part of the system matrix is madeto be an identity block matrix by a projection procedure for theRao–Wilton–Glisson (RWG) basis functions. Thus, the reduction of dimension of system matrix can be obtained by this projection procedure. In other words, the reduced dimension of system matrix is equal to NIES and irrespective of NPOS, where NIES and NPOS is the number of spatial unknowns in TDIE and TDPO regions, respectively. Since the TDIE region is not electrically large for most hybrid problems, we can solve the linear equations by direct solver without convergence problem introduced by iterative methods. The method for adaptively determining the number of temporal basis functions and the fast Fourier transform (FFT)-based blocking scheme for accelerating the temporal convolutions, is also employed.
首先,本文从显式时间步进和隐式时间步进递推算法求解时域电场积分方程(TD-EFIE)的计算公式出发,给出了EMP照射下线面结构的TDIE-TDPO-MOT方法,推导了显式时间步进和隐式时间步进递推算法求解时域电场积分方程的计算公式。与TDIE-MOT相比,在精度变化不大的情况下,TDIE-TDPO-MOT减少了计算时间。同时也编程仿真了典型结构在EMP照射下散射体上感应电流的瞬态响应,并比较了数值结果。
第二,为了提高算法稳定性,提出了自适应阶数递推法和确定拉盖尔(Laguerre)多项式尺度因子的通用规则。为了消除中心近似和提高精度,又提出了准确时域伽辽金(Galerkin)检验的时域磁场(TD-MFIE)和混合场积分方程(TD-CFIE)。同时利用快速傅立叶变换(FFT),提出了加速MOD方法中卷积过程的分块算法,把MOD的计算复杂度从O(N2ON2S)降低至O(N2SNOlog2(NO))。
第三,基于对影响积分收敛速度的详细分析,提出了处理弱奇异和近奇异积分的径向积分法,这一方法可以看作是对极坐标积分法的本质改进。通过对被积函数进行光滑处理的多项式变换,相比传统的弱奇异处理技术,本文方法的积分收敛速度明显加快。文中也给出了数值算例以验证本章所提出方法的精度和收敛速度。
第四,为了精确快速地计算时域电场、磁场和混合场积分方程的矩阵元素,本文提出了一种新的四维(4-D)奇异积分方法,包括用以处理内层二维(2-D)奇异和高阶近奇异积分的径向积分法。然后仔细分析了外层2-D积分被积函数导数的性质,提出了处理外层2-D积分的被基函数平滑技术。相比传统高斯积分规则,本文方法的效率和精度大幅提高,同时也给出了数值算例以验证本章所提出方法的精度和收敛速度。因为本文提出的方法保留了处理各类被积函数的灵活性,所以其用途不仅仅限于时域积分方程,也同样适用于频域积分方程(frequency-dOmain integral equations, FDIE)、
第五,基于加权Laguerre多项式和阶数步进法(MOD),提出了具有高度稳定性的TDIE-TDPP混合方法,这一方法可以处理混合三维PEC结构的时域响应问题,其中TDPP近似应用于表面光滑的电大部分。为了正确实现混合方法,把问题的几何结构分为两个部分：TDIE区域和TDPO区域。在TDPO区域中,使用MOD方法求解一组时域电场积分方程来获得未知电流。在TDPO区域中,PO电流的求解通过入射场以及TDIE电流的辐射场得到。本文中提出的混合算法有两点优势。第一个优点是,只需要改变TDIE区域在整体中所占的比例,就可以控制近似误差,这优于单纯的TDPP近似。第二个优点是,提出的混合方法只依靠TDPP近似来节省内存需求和CPU时间,而不需要诸如PWTD这一类算法的加速。因此本文的方法可以很容易地推广到MOT以及FDDM的时间步进架构下。
最后,本文提出了基于RWG函数的投影方法,这一方法可以将系数矩阵的TDPP-TDPO部分变为单位块矩阵。因此可以基于这一投影方法对系数矩阵进行降维压缩处理。系数矩阵的维数等于NIES,而与NPOS无关,其中NIES和NPOP分别是TDIE和TDPO区域中空间基函数的数目。对于大多数混合问题,TDIE区域并不是电大尺寸,因此就可以使用直接法求解线性方程组,也避免了迭代法可能产生的收敛问题。同时,本文提出的TDIE-TDPO-MOD方法使用了自适应阶数递推,以及FFT加速MOD卷积过程的分块算法。
With the miniaturization of electronic and electrical devices, systems are increasingly sensitive and susceptible to electromagnetic interference (EMI). Similarly, many antennas are mounted on electrically large platforms, which will suffer much more EMI problems than before. In modern electronic countermeasures, especially, after the emergence of some new-style weapons such as the high-power wideband EMP launcher and EM bomb, countries around the world initiate a new round competition in the research of electromagnetic interference (EMI) and protection techniques. This dissertation is focused on the development of time-domain integral equations and its hybrid methods with high accuracy and efficiency to study electromagnetic pulse effects and interaction mechanism in some typical systems illuminated by a high-power electromagnetic pulse (EMP).
Firstly, time-domain integral equations with explicit and implicit marching-on-in-time (MOT) schemes are presented. An efficient hybrid method, TDIE-TDPO-MOT, is proposed for studying electromagnetic responses of several groups of wire and surface structures illuminated by an electromagnetic pulse (EMP), respectively. In comparison with the full TDIE-MOT method, computational complexity can be reduced drastically using our developed TDIE-TDPO-MOT method, and computational accuracy is still maintained.
Secondly, a time-domain magnetic (TD-MFIE) and combined field integral equations (TD-CFIE) for analyzing scattering from3-D PEC objects are presented. The surface current density is directly used as the unknown, and an exact temporal Galerkin testing procedure is performed with no central approximation used. An adaptive marching-on-in-degree method with FFT-based blocking scheme is proposed, with the approach to determine the optimal scaling factor also given. It is indicated that the proposed scheme can guarantee the accuracy of MOD scheme without increasing the computational cost. Our numerical results also show that the O(NO2NS2) dependence of the MOD method can be reduced to O(NS2NOlog2(NO)) now.
Thirdly, a radial integration scheme is proposed for handling weakly singular and near-singular potential integrals, which is an essentialimprovement over the previous polar integration method. We at first carryout some careful investigations on the problem of slow convergence. Then,a new technique of smoothing integrand based on polynomialtransformation is presented, which results in much faster convergencerate than that of the polar integration. Finally, some numerical resultsare given to demonstrate both accuracy and convergence rate of ourproposed scheme.
Fourthly, a new integration approach is proposed for accurate andefficient calculation of time-domain EFIE, MFIE, and CFIE matrixelements over triangular domains. It mainly consists of a radialintegration scheme for handling weakly singular and near-hypersingularinner integrals as well as some new smoothing techniques for treatingthe outer surface integrals. As the proposed method preserves theflexibility for handling different integral kernels, they are applicablefor both time-domain and frequency-domain integral equations.
Fifthly, an efficient hybrid TDIE-TDPO method with high stability,based on the weighted Laguerre polynomials and MOD scheme, is proposedfor investigating transient electromagnetic responses of3-D compositePEC objects, while the TDPO approximation is applicable to the large PECobjects with smooth surface. These objects are partitioned into tworegions: TDIE and TDPO ones. In the TDIE region, currents are updatedby solving a set of time-domain electric field integral equation (TD-EFIE)using the MOD. In the TDPO region, PO currents are obtained accordingto the incident fields as well as the ones radiated by all currents inthe TDIE one. The hybrid TDIE-TDPO method has advantages over TDPOapproximation alone, as it can control the approximation errors bychanging the percentage of TDIE regions. On the other hand, the proposedmethod does not require any specific acceleration algorithm such as PWTD,and its saving of both memory and CPU time is obtained by TDPOapproximation. Therefore, the hybrid formula can be directly applied forMOT scheme as well as finite difference delay modeling method with nodifficulty.
Finally, the TDPO-TDPO projection part of the system matrix is madeto be an identity block matrix by a projection procedure for theRao–Wilton–Glisson (RWG) basis functions. Thus, the reduction of dimension of system matrix can be obtained by this projection procedure. In other words, the reduced dimension of system matrix is equal to NIES and irrespective of NPOS, where NIES and NPOS is the number of spatial unknowns in TDIE and TDPO regions, respectively. Since the TDIE region is not electrically large for most hybrid problems, we can solve the linear equations by direct solver without convergence problem introduced by iterative methods. The method for adaptively determining the number of temporal basis functions and the fast Fourier transform (FFT)-based blocking scheme for accelerating the temporal convolutions, is also employed.
引文
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[1] S. M. Rao and D. R. Wilton,“Transient scattering by conducting surfaces ofarbitrary shape,” IEEE Trans. Antennas Propag.,1991,39(1):56-61.
[2] S. M. Rao and T. K. Sarkar,“An alternative version of the time-domain electricfield integral equation for arbitrarily shaped conductors,” IEEE Trans. AntennasPropag.,1993,41(6):831-834.
[3] G. Manara, A. Monorchio, and R. Reggiannini,“A space-time discretizationcriterion for a stable time-marching solution of the electric field integralequation,” IEEE Trans. Antennas Propag.,1997,45(3):527–532.
[4] M. J. Bluck and S. P. Walker,“Time-domain BIE analysis of large threedimensional electromagnetic scattering problems,” IEEE Trans. AntennasPropag.,1997,45(5):894–901.
[5] D. S. Weile, G. Pisharody, N. W. Chen, B. Shanker, and E. Michielssen,“Anovel scheme for the solution of the time-domain integral equations ofelectromagnetics,” IEEE Trans. Antennas Propag.,2004,52(1):283–295.
[6] X. Wang, R.A. Wildman, D.S. Weile, and P. Monk,“A finite difference delaymodeling approach to the discretization of the time domain integral equations ofelectromagnetics”, IEEE Trans. Antennas Propag.,2008,56(8):2442–2452.
[7] F. P. Andriulli, K. Cools, F. Olyslager, and E. Michielssen,“Time domainCalderón identities and their application to the integral equation analysis ofscattering by PEC objects part II: stability”, IEEE Trans. Antennas Propag.,2009,57(8):2365–2375.
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[11]B. H. Jung, Z. Ji, T. K. Sarkar, M. Salazar-Palma, and M. Yuan,“A comparisonof marching-on in time method with marching-on in degree method for theTDIE solver,” Progress Electromagn. Res.,2007,70:281–296.
[12]B. Shanker, A. A. Ergin, and E. Michielssen,“Plane-wave-time-domain-enhanced marching-on-in-time scheme for analyzing scattering fromhomogeneous dielectric structures,” J. Opt. Soc. Amer. A,2002,19(4):716–726.
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[1] S. M. Rao and D. R. Wilton,“Transient scattering by conducting surfaces ofarbitrary shape,” IEEE Trans. Antennas Propag.,1991,39(1):56-61.
[2] S. M. Rao and T. K. Sarkar,“An alternative version of the time-domain electricfield integral equation for arbitrarily shaped conductors,” IEEE Trans. AntennasPropag.,1993,41(6):831-834.
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[4] M. J. Bluck and S. P. Walker,“Time-domain BIE analysis of large threedimensional electromagnetic scattering problems,” IEEE Trans. AntennasPropag.,1997,45(5):894–901.
[5] D. S. Weile, G. Pisharody, N. W. Chen, B. Shanker, and E. Michielssen,“Anovel scheme for the solution of the time-domain integral equations ofelectromagnetics,” IEEE Trans. Antennas Propag.,2004,52(1):283–295.
[6] X. Wang, R.A. Wildman, D.S. Weile, and P. Monk,“A finite difference delaymodeling approach to the discretization of the time domain integral equations ofelectromagnetics”, IEEE Trans. Antennas Propag.,2008,56(8):2442–2452.
[7] F. P. Andriulli, K. Cools, F. Olyslager, and E. Michielssen,“Time domainCalderón identities and their application to the integral equation analysis ofscattering by PEC objects part II: stability”, IEEE Trans. Antennas Propag.,2009,57(8):2365–2375.
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