时域积分方程后时不稳定性的研究与应用
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摘要
由于积分方程有着自动满足辐射边界条件,并且只需要在目标表面进行离散等优点,因而受到了广泛地关注。随着计算机的发展,其研究范围和应用领域也随之不断扩大。成为分析电磁场问题的一个重要手段。
本文首先介绍了矩量法(MOM)的基本原理和频域积分方程的基本情况,推导了一维情况下的海伦方程,接着分析了二维情况下的电场、磁场以及组合场积分方程,并对三种积分方程的计算效果进行了比较。然后简单地介绍了矩量法结合离散傅里叶逆变换求解瞬态响应的问题。
其次,介绍了时域积分方程基本情况,通过对一维细直导线的分析介绍了时间步进(marching on in time)算法的基本求解原理,接着,我们着重介绍了二维无限长导体情况下的时域电场、磁场以及组合场积分方程,分别分析了TM波和TE波入射下的三种积分方程的显式与隐式算法,并将结果进行了比较。另外,本文还将三种积分方程通过数值算例进行了比较。针对MOT(marching on in time)算法在求解时域积分方程中的后时不稳定性,我们通过对均值方法的进一步的研究,提出了一种改善方案,在一定程度上改善、延缓了后时振荡的发生。
然后,介绍了二维无限长介质体的计算,我们采用PMCHW方程来计算介质体,推导了两种形式的用于求解二维介质体的PMCHW方程。
最后,因为分析不同时间基函数在MOT算法中的计算效果也是MOT算法中研究的一个方面,所以,我们探讨了两个时间基函数对MOT算法稳定性的影响与时间基函数自身频谱成分之间的关系。接着,我们把二阶子域时间基函数加入到PMCHW方程的数值求解之中,其数值结果验证了其可行性。
The integral equation attracted extensive attention since it can satisfy the radiation boundary conditions automatically and only the target surface is required to be discretized. With the development of computer, the field of research and application of integral equation is expanded. And the integral equation method has being an important tool to analysis the electromagnetic problems.
Firstly, we introduce the theory of moment of methods (MOM) to study the basis functions of frequency domain integral equation. And then the solution for Hallen's equation, and the electrical field integral equation (EFIE), magnetic field integral equation (MFIE) and the combined field equation (CFIE) for two-dimensional are deduced. The inverse discrete Fourier transform in conjugation with MOM for analysis the transient scattering problems is also introduced.
Secondly, time domain integral equation (TDIE) method based on marching on in time (MOT) is introduced to analysis the thin straight wire. And the explicit and implicit scheme of the TD-EFIE、TD-MFIE and TD-CFIE is deduced, which is used to scattering analysis of two-dimensional objects illuminated by a Gaussian plane wave pulse. Both the TM and TE mode are discussed. For the later-time oscillation problems, an improved solution scheme based on further study of averaging techniques is proposed.
Thirdly, for the transient field analysis of dielectric objects, PMCHW equations of two forms are deduced for the calculation of two-dimensional dielectric objects.
Finally, considering analyzing the effect for different kinds of the temporal basis functions is also an important aspect of MOT in time scheme optimization, the relationship between the frequency distributions of temporal basis with the stability of MOT is discussed for two kinds of temporal basis. Based on the discussion, the second order sub-domain time basis is proposed for the solution of PMCHW equation, and numerical results and analysis shows that the proposed method is practicable.
本文首先介绍了矩量法(MOM)的基本原理和频域积分方程的基本情况,推导了一维情况下的海伦方程,接着分析了二维情况下的电场、磁场以及组合场积分方程,并对三种积分方程的计算效果进行了比较。然后简单地介绍了矩量法结合离散傅里叶逆变换求解瞬态响应的问题。
其次,介绍了时域积分方程基本情况,通过对一维细直导线的分析介绍了时间步进(marching on in time)算法的基本求解原理,接着,我们着重介绍了二维无限长导体情况下的时域电场、磁场以及组合场积分方程,分别分析了TM波和TE波入射下的三种积分方程的显式与隐式算法,并将结果进行了比较。另外,本文还将三种积分方程通过数值算例进行了比较。针对MOT(marching on in time)算法在求解时域积分方程中的后时不稳定性,我们通过对均值方法的进一步的研究,提出了一种改善方案,在一定程度上改善、延缓了后时振荡的发生。
然后,介绍了二维无限长介质体的计算,我们采用PMCHW方程来计算介质体,推导了两种形式的用于求解二维介质体的PMCHW方程。
最后,因为分析不同时间基函数在MOT算法中的计算效果也是MOT算法中研究的一个方面,所以,我们探讨了两个时间基函数对MOT算法稳定性的影响与时间基函数自身频谱成分之间的关系。接着,我们把二阶子域时间基函数加入到PMCHW方程的数值求解之中,其数值结果验证了其可行性。
The integral equation attracted extensive attention since it can satisfy the radiation boundary conditions automatically and only the target surface is required to be discretized. With the development of computer, the field of research and application of integral equation is expanded. And the integral equation method has being an important tool to analysis the electromagnetic problems.
Firstly, we introduce the theory of moment of methods (MOM) to study the basis functions of frequency domain integral equation. And then the solution for Hallen's equation, and the electrical field integral equation (EFIE), magnetic field integral equation (MFIE) and the combined field equation (CFIE) for two-dimensional are deduced. The inverse discrete Fourier transform in conjugation with MOM for analysis the transient scattering problems is also introduced.
Secondly, time domain integral equation (TDIE) method based on marching on in time (MOT) is introduced to analysis the thin straight wire. And the explicit and implicit scheme of the TD-EFIE、TD-MFIE and TD-CFIE is deduced, which is used to scattering analysis of two-dimensional objects illuminated by a Gaussian plane wave pulse. Both the TM and TE mode are discussed. For the later-time oscillation problems, an improved solution scheme based on further study of averaging techniques is proposed.
Thirdly, for the transient field analysis of dielectric objects, PMCHW equations of two forms are deduced for the calculation of two-dimensional dielectric objects.
Finally, considering analyzing the effect for different kinds of the temporal basis functions is also an important aspect of MOT in time scheme optimization, the relationship between the frequency distributions of temporal basis with the stability of MOT is discussed for two kinds of temporal basis. Based on the discussion, the second order sub-domain time basis is proposed for the solution of PMCHW equation, and numerical results and analysis shows that the proposed method is practicable.
引文
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[2]聂在平,方大纲.目标与环境电磁散射特性建模-理论、方法与实现[M].国防工业出版社,2009.
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[4]武胜波.三维目标电磁散射的时域积分方程法[D].电子科技大学,2005.
[5]李丽娟.时域积分方程法在细线导体瞬态响应计算中的应用[D].华北电力大学,2007.
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[9]姜光兴,曹伟,朱洪波.TD-EFIE分析任意几何形状导体的瞬态特性[J].电子学报,2006,34(4)：741-745.
[10]姜兴光.基于积分方程的复杂导体目标瞬态分析[D].南京邮电大学,2007.
[11]张乖红,夏明耀.介质目标瞬态散射分析的时域积分方程方法与并行计算[J].北京大学学报(自然科学版),2008,44(3)：353-358.
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[15]Z. B. Ye, R. S. Chen, and S. S. Li.The scattering and radiation of arbitrary thin wire structures using TDIE with several different temporal basis functions, IEEE Antennas Propagat. Int. Symp. Albuquerque,2006,2951-2954.
[16]A.Geranmayeh, W.Ackermann, T.Weiland. Temporal discretization choices for stable boundary element methods in electromagnetic scattering problems[J]. Applied Numerical Mathematics.59 (2009) 2751-2773.
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[19]B.H.Jung,T.K.SarKar.Time-Domain Electric-Field Integral Equation with Central Finite Difference[J]. Microwave and optical technology Letters.2001, Vol.31.429-435.
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[28]D. A. Vechinski,S. M. Rao. Transient Scattering from Two-Dimensional Dielectric Cylinders of Arbitrary Shape[J].1992,40(9):1054-1060.
[29]G. H. Zhang, M. Y. Xia, G. L. Dai and C. H. Chan.Time Domain Integral Equation Methods for Analyses of Transient Radiation and Scattering Problems IEEE Antennas Propagat.2007,4589-4592.
[30]王生水,毛钧杰,柴舜连.时域积分方程中不同时间基函数的比较[C].全国微波毫米波会议论文集.2005,948-951.
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[38]P.J. Davies, D.B. Duncan. Averaging techniques for time-marching schemes for retarded potential integral equations[J]. Applied Numerical Mathematics 23(1997)291-310.
[39]张春媛.目标与粗糙面电磁散射的时域积分方程法[D].西安电子科技大学,2006.
[40]蔡明娟.时域积分方程在分析介质问题中的算法研究与应用[D].国防科学技术大学,2006.
[41]A. Bondeson, T. Rylander, P. Ingelstrom.Computational Electromagnetics[M]. Springer Science+Business Media, Inc,2005.
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[2]聂在平,方大纲.目标与环境电磁散射特性建模-理论、方法与实现[M].国防工业出版社,2009.
[3]赵庆广.电磁散射与辐射的时域积分方程法[D].电子科技大学,2008.
[4]武胜波.三维目标电磁散射的时域积分方程法[D].电子科技大学,2005.
[5]李丽娟.时域积分方程法在细线导体瞬态响应计算中的应用[D].华北电力大学,2007.
[6]卢万铮.天线理论与技术[M].西安电子科技大学出版社,2004.
[7]N.J.Damaskos,R.T.Brown,J.R.Jameson. Transient Scattering by Resistive Cylinders[J]. IEEE Transactions On AP.1985,33(1):21-25.
[8]吴振森,秦三团,代少玉.一种改善TDIE时间步进算法稳定性的新平均方法[J].电波科学学报,2004,Vo1.19,102-109.
[9]姜光兴,曹伟,朱洪波.TD-EFIE分析任意几何形状导体的瞬态特性[J].电子学报,2006,34(4)：741-745.
[10]姜兴光.基于积分方程的复杂导体目标瞬态分析[D].南京邮电大学,2007.
[11]张乖红,夏明耀.介质目标瞬态散射分析的时域积分方程方法与并行计算[J].北京大学学报(自然科学版),2008,44(3)：353-358.
[12]毕海燕.用时域积分方程法分析金属介质目标的时域电磁散射[D].电子科技大学,2008.
[13]Jaroslav Lacik, Zbynek Lukes, Zbynek Raida.Analysis of Stability of MOT Scheme with Different Temporal Basis Functions[C]. the 38th European Microwave Conference.2008,1370-1373.
[14]J.L.Hu,C.H.Chan.A New Temporal Basis Function for the Time- Domain Integral Equation Method[J].IEEE Microwave And Wireless Components Letters.2001,11 (11):465-466.
[15]Z. B. Ye, R. S. Chen, and S. S. Li.The scattering and radiation of arbitrary thin wire structures using TDIE with several different temporal basis functions, IEEE Antennas Propagat. Int. Symp. Albuquerque,2006,2951-2954.
[16]A.Geranmayeh, W.Ackermann, T.Weiland. Temporal discretization choices for stable boundary element methods in electromagnetic scattering problems[J]. Applied Numerical Mathematics.59 (2009) 2751-2773.
[17]S.M.Rao,T.K.Sarkar.Time domain Modeling of Two-Dimensional Conducting Cylinders Utilizing An Implicit Scheme-TM Incidence[J].Microwave and optical technology Letters.1997,Vol.15:342-347.
[18]S.M.Rao,D.A.Vechinshi,T.K.Sarkar. Transient Scattering by Conducting Cylinders-Implicit Solution for the Transverse Electric Case[J]. Microwave and optical technology Letters.1999, Vol.21:129-134.
[19]B.H.Jung,T.K.SarKar.Time-Domain Electric-Field Integral Equation with Central Finite Difference[J]. Microwave and optical technology Letters.2001, Vol.31.429-435.
[20]唐煜,顾长青.一种改进稳定性的时域电场积分方程法[J].微波学报,2008,24(4)：31-35.
[21]谢处方.近代天线理论[M].成都电讯工程学院出版社,1987.
[22]关鑫璞,郭建明,吴铁平等.求解任意形状目标时域电磁散射的隐式算法分析[J].微波学报,2008,24(6)：1-7.
[23]S.M.Rao,D.R.Wilton.Transient Scattering by Conducting Surfaces of Arbitrary Shape[J].IEEE Transactions and Propagation.1991,39(1):56-61.
[24]D.A.Vechinshi, S.M.Rao.A Stable Procedure to Calculate the Transient Scattering buy Conducting Surfaces of Arbitrary Shape[J]. IEEE Transactions On Antennas and Propagation.1992,40(6):661-665.
[25]李小勇.FDTD和TDIE中几个关键问题研究[D].西安：西安电子科技大学理学院,2010.
[26]任猛,周东明,刘锋等.时域磁场积分方程迭代求解[J].国防科技大学学报,2007,29(4)：77-81.
[27]Giuliano Manara, Agostino Monorchio, Ruggero Reggiannini.A Space-Time Discretization Criterion for a Stable Time-Marching Solution of the Electric Field Integral Equation[J].IEEE Transactions On Antennas and Propagation.1997,45(3):527-532.
[28]D. A. Vechinski,S. M. Rao. Transient Scattering from Two-Dimensional Dielectric Cylinders of Arbitrary Shape[J].1992,40(9):1054-1060.
[29]G. H. Zhang, M. Y. Xia, G. L. Dai and C. H. Chan.Time Domain Integral Equation Methods for Analyses of Transient Radiation and Scattering Problems IEEE Antennas Propagat.2007,4589-4592.
[30]王生水,毛钧杰,柴舜连.时域积分方程中不同时间基函数的比较[C].全国微波毫米波会议论文集.2005,948-951.
[31]哈林顿著,王尔杰等译.计算电磁学中的矩量法[M].国防工业出版社,1981.
[32]秦三团.电磁散射的时域积分方程法研究[D].西安：西安电子科技大学理学院,2005.
[33]R. Gomez Martin, A. Salinas,A. Rubio Bretones. Time-Domain Integral Equation Methods For Transient Analysis[J]. IEEE Antennas and Propagation. 1992,34(3):15-23.
[34]A. Geranmayeh, W.Ackermann, T.Weiland.Survey of Temporal Basis Functions for Integral Equation Methods. in:Proc.7th IEEE Workshop on Computational Electromanetics in Time-Domain, CEM-TD'07, vol.533, Perugia, Italy,2007 IEEE.2007
[35]J. A. Stratton, Electromagnetic Theory[M]. New York:McGraw-Hill,1941.
[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]Walton C.Gibson.The Method of Moment in Electromagnetics[M]. Taylor&Francis Group,2008
[38]P.J. Davies, D.B. Duncan. Averaging techniques for time-marching schemes for retarded potential integral equations[J]. Applied Numerical Mathematics 23(1997)291-310.
[39]张春媛.目标与粗糙面电磁散射的时域积分方程法[D].西安电子科技大学,2006.
[40]蔡明娟.时域积分方程在分析介质问题中的算法研究与应用[D].国防科学技术大学,2006.
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