复杂结构散射与辐射的特征基函数法及改进技术
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摘要
电大复杂目标电磁特性分析在工程中有着广泛的应用,人们对开发出更高效更准确的算法一直有极高的热情。矩量法作为重要的精确数值算法之一,数十年来得到了广泛应用,它具有计算精度高的优点,但其计算复杂度与内存消耗高,难以直接应用到复杂电大尺寸电磁特性分析。在矩量法基础上发展起来的多层快速多极子方法(MLFMA),自适应积分方法(AIM)等方法大大降低了矩阵向量相乘的计算复杂度和内存需求,能快速求解电大目标问题,但必须依赖于迭代求解法。对于多入射问题,其迭代过程仍然较费时。本文研究的特征基函数方法能显著减少未知量个数,该方法特别适合周期阵列结构的电磁特性分析。
本文首先介绍了积分方程方法的思想与步骤,并对其求解方法——矩量法的原理与求解过程进行了详细介绍。
然后,本文深入研究了特征基函数方法,并对该方法中扩展区宽度选取,奇异值分解(SVD)门限确定等参数进行了分析。并对建立特征基函数的平面波激励分布进行了优化。
接着,本文对特征基函数的快速算法进行了研究。通过引入快速多极子与多层快速多极子方法加速填充属于不同块的特征基函数阻抗矩阵填充,提高了传统方法矩阵填充速度。并采用了自适应交叉近似方法近似计算扩展块阻抗矩阵,提高了建立特征基函数速度。
其次,本文将特征基函数方法与快速多极子-快速傅里叶变换法结合用于分析大型平面阵列结构电磁特性。研究了特征基函数的设计方法,对结合特征基函数方法前后的未知量个数,迭代时间进行了比较,并对其原因进行了细致的分析。
最后,本文针对大型平面阵列结构对特征基函数方法中建立特征基函数的过程进行了改进,发展出了模型特征基函数方法,该方法采用两层特征基函数,能够进一步减少未知量数目,最后对其适用条件和效果进行了分析。
Analysis of the electromagnetic characteristic of complex targets in engineering has found plenty of applications, people has great enthusiasm in developing more efficient and more accurate algorithms for it. As an important accurate numerical algorithm, method of moments has been applied for several decades, it has the advantages of high accuracy, but has high computational complexity and high memory consumption, thus it is difficult to apply directly for very complicated targets with large electrical sizes. The multilevel fast multipole algorithm (MLFMA), adaptive integral method (AIM) and other MoM based methods can be employed to solve large objects quickly, but must rely on iterative process. In the case of multi-incidence problems, the iterative process is still very time-consuming. The characteristic basis function method studied in this paper can significantly reduce the number of unknowns, this method is very suitble for analysing EM properties of planar periodic structures.
Firstly, this paper introduces the idea of the integral equation methods and its procedures, and introduces the principle and steps of MoM which is used to solve the integral equation methods.
Secondly, this paper studies the characteristic basis function method .The standard for selecting the width of expansion area, singular value decomposition (SVD) threshold is analyzed. Distribution of plane wave excitations which are used in building up CBFs is optimized.
Thirdly, fast algorithms based on CBFM are studied. By introducing the FMM and the MLFMA to accelerate filling process of the CBFM matrix, the speed of filling CBFM matrix is improved greatly, compared to the traditional CBFM methods. In addition, the adaptive cross approximation method is used to realize fast building up CBFs for each extended blocks.
Then, the CBFM is combined with FMM-FFT method for the analysis of the electromagnetic properties of planar array. We investigate the manner of building up the CBFs, the number of unknowns and the time for iterations of this method is compared with traditional basis functions.
Finally, for large planar array structure, the process of building up CBFs in CBFM-FMM-FFT is modified, the model CBFM is developed. In this method two level CBFs are employed, thus the number of unknows is reduced, and the applicable conditions and effects are analyzed.
本文首先介绍了积分方程方法的思想与步骤,并对其求解方法——矩量法的原理与求解过程进行了详细介绍。
然后,本文深入研究了特征基函数方法,并对该方法中扩展区宽度选取,奇异值分解(SVD)门限确定等参数进行了分析。并对建立特征基函数的平面波激励分布进行了优化。
接着,本文对特征基函数的快速算法进行了研究。通过引入快速多极子与多层快速多极子方法加速填充属于不同块的特征基函数阻抗矩阵填充,提高了传统方法矩阵填充速度。并采用了自适应交叉近似方法近似计算扩展块阻抗矩阵,提高了建立特征基函数速度。
其次,本文将特征基函数方法与快速多极子-快速傅里叶变换法结合用于分析大型平面阵列结构电磁特性。研究了特征基函数的设计方法,对结合特征基函数方法前后的未知量个数,迭代时间进行了比较,并对其原因进行了细致的分析。
最后,本文针对大型平面阵列结构对特征基函数方法中建立特征基函数的过程进行了改进,发展出了模型特征基函数方法,该方法采用两层特征基函数,能够进一步减少未知量数目,最后对其适用条件和效果进行了分析。
Analysis of the electromagnetic characteristic of complex targets in engineering has found plenty of applications, people has great enthusiasm in developing more efficient and more accurate algorithms for it. As an important accurate numerical algorithm, method of moments has been applied for several decades, it has the advantages of high accuracy, but has high computational complexity and high memory consumption, thus it is difficult to apply directly for very complicated targets with large electrical sizes. The multilevel fast multipole algorithm (MLFMA), adaptive integral method (AIM) and other MoM based methods can be employed to solve large objects quickly, but must rely on iterative process. In the case of multi-incidence problems, the iterative process is still very time-consuming. The characteristic basis function method studied in this paper can significantly reduce the number of unknowns, this method is very suitble for analysing EM properties of planar periodic structures.
Firstly, this paper introduces the idea of the integral equation methods and its procedures, and introduces the principle and steps of MoM which is used to solve the integral equation methods.
Secondly, this paper studies the characteristic basis function method .The standard for selecting the width of expansion area, singular value decomposition (SVD) threshold is analyzed. Distribution of plane wave excitations which are used in building up CBFs is optimized.
Thirdly, fast algorithms based on CBFM are studied. By introducing the FMM and the MLFMA to accelerate filling process of the CBFM matrix, the speed of filling CBFM matrix is improved greatly, compared to the traditional CBFM methods. In addition, the adaptive cross approximation method is used to realize fast building up CBFs for each extended blocks.
Then, the CBFM is combined with FMM-FFT method for the analysis of the electromagnetic properties of planar array. We investigate the manner of building up the CBFs, the number of unknowns and the time for iterations of this method is compared with traditional basis functions.
Finally, for large planar array structure, the process of building up CBFs in CBFM-FMM-FFT is modified, the model CBFM is developed. In this method two level CBFs are employed, thus the number of unknows is reduced, and the applicable conditions and effects are analyzed.
引文
[1] V. Rokhlin. Rapid solution of integral equations of scattering theory in two dimensions.Journal of Computational physics, 1990, 86:414-439
[2] Song J M , Lu C C , et al . Multilevel fast multiplole algorithm for electromagnetic scattering by large complex objects [ J ] . IEEE Trans. AP, 1997, 45:1488-1492
[3] E. Bleszynski, M.Bleszynski, and T.Jaroszewicz. AIM: Adaptive integral method for solving large-scale electromagnetic scattering and radiation problems. Radio Sci, 1996, vol. 31, no.5, pp:1225-1252
[4] Phillips, J. R. Error and complexity analysis for a collocation-grid-projection plus precorrected-FFT algorithm for solving potential integral equations with Laplace or Helmholtz kernels. Proc. of 1995 Copper Mountain Conference on Multigrid Methods
[5] S. M. Seo and J.F. Lee. A fast IE-FFT algorithm for solving PEC scattering problems. IEEE Trans. Magn., 2005, 41(5):1476–1479
[6] Gonzalez-Ovejero. D,.Craeye. C. Fast computation of macro basis functions interactions in non uniform arrays. Antennas and Propagation Society International Symposium 2008, AP-S 2008. IEEE
[7] L. Matekovits, V. Laza, and G. Vecchi. Analysis of large complex structures with the synthetic-functions approach. IEEE Trans. Antennas Propagat, Sep. 2007, vol. 55, no. 9: 2509-2521
[8] V. V. S. Prakash and R. Mittra. Characteristic basis function method: A new technique for efficient solution of method of moments matrix equation. Microw.Opt.Technol.Lett, Jan. 2003, vol. 36, no. 2:95-100
[9] G. Tiberi, M. Degiorgi, A. Monorchio, G. Manara, R. Mittra.A Class of Physical Optics- SVD Derived Basis Functions for Solving Electromagnetic Scattering Problems. 2005 IEEE Antennas and Propagation Society International Symposium
[10] E. Garcia , C. Delgado, F Sáez de Adana , F. Cátedra, R. Mittra,Incorporating the Multilevel Fast Multipole Method into the Characteristic Basis Function Method to Solve Large Scattering and Radiation problems. Antennas and Propagation Society International Symposium, 2007 IEEE
[11] J. Laviada, R. Mittra, M. R. Pino, and F. Las-Heras.Generation of nested characteristic basis functions.in Proc. 2009 European Conference on Antennas and Propagation
[12] Li Hu, Le-Wei Li.CBFM-Based p-FFT Method: A New Algorithm for Solving Large-Scale Finite Periodic Arrays Scattering Problems. Asia Pacific Microwave Conference 2009
[13] Li Hu, Le-Wei Li, Mittra R. Electromagnetic Scattering by Finite Periodic Arrays Using the Characteristic Basis Function and Adaptive Integral Methods. IEEE Transactions on AP,2010, vol. 58, no. 9:3086-3090
[14]聂在平,徐利明.电磁散射数值分析中的特征基函数方法.电波科学学报, 2004,vol.19:45-49
[15]阙肖峰,聂在平.大型阵列结构电磁特性分析的特征基函数方法.系统工程与电子技术,2006,vol.28,no.11:1613-1616
[16] Jiangong Wei, Zaiping Nie.A wide band Scattering Analysis of Conformal FSS by CBFM and SVD method. APMC 2008
[17]韩国栋.基于特征基函数的高效算法及在电磁散射中的应用.南京航空航天大学博士学位论文,2008
[18]侯兆国,王超,殷红成.电大复杂目标电磁散射计算的特征基函数方法.制导与引信,2009,30(2):24-29
[19] K. Zhao, M.N. Vouvakis and J.F. Lee. The Adaptive Cross Approximation Algorithm for Accelerated MoM Computations of EMC problems. Trans. on Elec. Comp., November 2005, 47(4): 763-773
[20] Robert L.Wagner, Jiming Song, and Weng C.Chew. Monte carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces. IEEE Transactions on Antennas and Propagation, 1997, 45(2):235-245
[21] W. C. Chew, C. P Davis, K. F. Warnick, Z. P. Nie, J. Hu, S. Yan. L. Gürel. EFIE and MFIE, Why the Difference? IEEE International Symposium on Antennas and Propagation and USNC/URSI National Radio Science Meeting, APSURSI, 2008.
[22] R. F. Harrington. Field Computation by Moment Methods.New York, Macmilan, 1968
[23] S. Rao, D. Wilton, A. Glisson. Electromagnetic scattering by surfaces of arbitrary shape. IEEE Transactions on Antennas and Propagation, 1982 vol. 30:409-418
[24] Jun Hu, Zaiping Nie and Lei Lin. Solving 3-D electromagnetic scattering from conducting object by MLFMA with curvilinear RWG basis. 2003 IEEE Antennas and Propagation Society International Symposium, pp: 460-463
[25]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
[26]金建铭.电磁场有限元方法.西安电子科技大学出版社,1998
[27] Y. Saad. Iterative methods for sparse linear systems. PWS Publishing Company, Boston, 1996
[28] Sun, Y.F., Chan, C.H., Mittra. R, Tsang, L. Characteristic basis function method for solving large problems arising in dense medium scattering. Antennas and Propagation Society International Symposium, 2003, IEEE, vol. 2:1068 -1071
[29] Eliseo García, Carlos Delgado, Iván González Diego, and Manuel Felipe Cátedra. An Iterative Solution for Electrically Large Problems Combining the Characteristic Basis Function Method and the Multilevel Fast Multipole Algorithm. IEEE Transactions on Antennas and Propagation, 2008 ,vol. 56, no. 8: 2363-2371
[30] Rob Maaskant, Raj Mittra, Anton Tijhuis. Fast Analysis of Large Antenna Arrays Using the Characteristic Basis Function Method and the Adaptive Cross Approximation Algorithm..IEEE Transactions on Antennas and Propagation, 2008, vol. 56,no.11
[31] Jaime Laviada, Fernando Las-Heras, Marcos R. Pino, and Raj Mittra. Solution of Electrically Large Problems With Multilevel Characteristic Basis Functions. IEEE Transactions on Antennas and Propagation, 2009,vol.57, no.10 3189-3198
[32]胡俊,复杂目标矢量电磁散射的高效方法-快速多极子方法及其应用,电子科技大学博士论文,2000
[33] Eugenio Lucente, Agostino Monorchio and Raj Mittra. Generation of Characteristic Basis Functions by Using Sparse MoM Impedance Matrix to Construct the Solution of Large Scattering and Radiation Problems. Antennas and Propagation Society International Symposium 2006, IEEE : 4091-4094
[34] C. Pelletti, K. Panayappan, R. Mittra and A. Monorchio.On the Hybridization of RUFD Algorithm with the DM Approach for Solving Multiscale Problems.2010 URSI International Symposium on Electromagnetic Theory,pp:850-852
[35] Alessio Cucini, Matteo Albani, and Stefano Maci. Truncated Floquet Wave Full-Wave (T(FW)2) Analysis of Large Periodic Arrays of Rectangular Waveguides. IEEE Transactions on Antennas and Propagation,2003,vol.51,no.6a:1373-1385
[36] Li Hu, Le-Wei Li, and Raj Mittra.Electromagnetic Scattering by Finite Periodic Arrays Using the Characteristic Basis Function and Adaptive Integral Methods. IEEE Transactions on Antennas and Propagation, 2010, vol.58, no.9: 3086-3090
[37] Zuhui Ma, Lijun Jiang, Wengcho Chew, Maokun Li, Zhiguo Qian. Augmented EPA with Augmented EFIE Method for Packaging Analysis. Electrical Design of Advanced Packaging & Systems Symposium (EDAPS), 2010 IEEE
[38] Lu W. B, Cui T. J, Qian Z.G, Yin X. X, Hong W. Accurate Analysis of Large-Scale Periodic Structures Using an Efficient Sub-Entire-Domain Basis Function Method. IEEE Transactions on Antennas and Propagation, 2004, vol. 52, Part 11:3078-3085
[39] Anja K. Skrivervik and Juan R. Mosig. Analysis of Finite Phase Arrays of Microstrip Patches. IEEE Transactions on Antennas and Propagation, 1993, vol. 41, no. 8: 1105-1114
[40] Lawrence Carin and Leopold B. Felsen. Time Harmonic and Transient Scattering by Finite Periodic Flat Strip Arrays: Hybrid (Ray)-(Floquet Mode)-(MoM) Algorithm. IEEE Transactions on Antennas and Propagation, 1993, vol. 41, no. 4:412-421
[41] Renaud Chiniard, AndréBarka, Olivier Pascal. Fast and Accurate Modeling of Large Plane Array Antenna. Antennas and Propagation, EuCAP 2006, First European Conference
[42] Cai-Cheng Lu. A Fast Algorithm Based on Volume Integral Equation for Analysis of Arbitrarily Shaped Dielectric Radomes. IEEE Transactions on Antennas and Propagation, 2003, vol.51, no. 3:606-612
[43] C. C. Lu and W. C. Chew. A coupled surface-volume integral equation approach for the calculation of electromagnetic scattering from composite metallic and material targets. IEEE Trans. Antennas Propag, 2000, 48(12):1866-1868
[2] Song J M , Lu C C , et al . Multilevel fast multiplole algorithm for electromagnetic scattering by large complex objects [ J ] . IEEE Trans. AP, 1997, 45:1488-1492
[3] E. Bleszynski, M.Bleszynski, and T.Jaroszewicz. AIM: Adaptive integral method for solving large-scale electromagnetic scattering and radiation problems. Radio Sci, 1996, vol. 31, no.5, pp:1225-1252
[4] Phillips, J. R. Error and complexity analysis for a collocation-grid-projection plus precorrected-FFT algorithm for solving potential integral equations with Laplace or Helmholtz kernels. Proc. of 1995 Copper Mountain Conference on Multigrid Methods
[5] S. M. Seo and J.F. Lee. A fast IE-FFT algorithm for solving PEC scattering problems. IEEE Trans. Magn., 2005, 41(5):1476–1479
[6] Gonzalez-Ovejero. D,.Craeye. C. Fast computation of macro basis functions interactions in non uniform arrays. Antennas and Propagation Society International Symposium 2008, AP-S 2008. IEEE
[7] L. Matekovits, V. Laza, and G. Vecchi. Analysis of large complex structures with the synthetic-functions approach. IEEE Trans. Antennas Propagat, Sep. 2007, vol. 55, no. 9: 2509-2521
[8] V. V. S. Prakash and R. Mittra. Characteristic basis function method: A new technique for efficient solution of method of moments matrix equation. Microw.Opt.Technol.Lett, Jan. 2003, vol. 36, no. 2:95-100
[9] G. Tiberi, M. Degiorgi, A. Monorchio, G. Manara, R. Mittra.A Class of Physical Optics- SVD Derived Basis Functions for Solving Electromagnetic Scattering Problems. 2005 IEEE Antennas and Propagation Society International Symposium
[10] E. Garcia , C. Delgado, F Sáez de Adana , F. Cátedra, R. Mittra,Incorporating the Multilevel Fast Multipole Method into the Characteristic Basis Function Method to Solve Large Scattering and Radiation problems. Antennas and Propagation Society International Symposium, 2007 IEEE
[11] J. Laviada, R. Mittra, M. R. Pino, and F. Las-Heras.Generation of nested characteristic basis functions.in Proc. 2009 European Conference on Antennas and Propagation
[12] Li Hu, Le-Wei Li.CBFM-Based p-FFT Method: A New Algorithm for Solving Large-Scale Finite Periodic Arrays Scattering Problems. Asia Pacific Microwave Conference 2009
[13] Li Hu, Le-Wei Li, Mittra R. Electromagnetic Scattering by Finite Periodic Arrays Using the Characteristic Basis Function and Adaptive Integral Methods. IEEE Transactions on AP,2010, vol. 58, no. 9:3086-3090
[14]聂在平,徐利明.电磁散射数值分析中的特征基函数方法.电波科学学报, 2004,vol.19:45-49
[15]阙肖峰,聂在平.大型阵列结构电磁特性分析的特征基函数方法.系统工程与电子技术,2006,vol.28,no.11:1613-1616
[16] Jiangong Wei, Zaiping Nie.A wide band Scattering Analysis of Conformal FSS by CBFM and SVD method. APMC 2008
[17]韩国栋.基于特征基函数的高效算法及在电磁散射中的应用.南京航空航天大学博士学位论文,2008
[18]侯兆国,王超,殷红成.电大复杂目标电磁散射计算的特征基函数方法.制导与引信,2009,30(2):24-29
[19] K. Zhao, M.N. Vouvakis and J.F. Lee. The Adaptive Cross Approximation Algorithm for Accelerated MoM Computations of EMC problems. Trans. on Elec. Comp., November 2005, 47(4): 763-773
[20] Robert L.Wagner, Jiming Song, and Weng C.Chew. Monte carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces. IEEE Transactions on Antennas and Propagation, 1997, 45(2):235-245
[21] W. C. Chew, C. P Davis, K. F. Warnick, Z. P. Nie, J. Hu, S. Yan. L. Gürel. EFIE and MFIE, Why the Difference? IEEE International Symposium on Antennas and Propagation and USNC/URSI National Radio Science Meeting, APSURSI, 2008.
[22] R. F. Harrington. Field Computation by Moment Methods.New York, Macmilan, 1968
[23] S. Rao, D. Wilton, A. Glisson. Electromagnetic scattering by surfaces of arbitrary shape. IEEE Transactions on Antennas and Propagation, 1982 vol. 30:409-418
[24] Jun Hu, Zaiping Nie and Lei Lin. Solving 3-D electromagnetic scattering from conducting object by MLFMA with curvilinear RWG basis. 2003 IEEE Antennas and Propagation Society International Symposium, pp: 460-463
[25]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
[26]金建铭.电磁场有限元方法.西安电子科技大学出版社,1998
[27] Y. Saad. Iterative methods for sparse linear systems. PWS Publishing Company, Boston, 1996
[28] Sun, Y.F., Chan, C.H., Mittra. R, Tsang, L. Characteristic basis function method for solving large problems arising in dense medium scattering. Antennas and Propagation Society International Symposium, 2003, IEEE, vol. 2:1068 -1071
[29] Eliseo García, Carlos Delgado, Iván González Diego, and Manuel Felipe Cátedra. An Iterative Solution for Electrically Large Problems Combining the Characteristic Basis Function Method and the Multilevel Fast Multipole Algorithm. IEEE Transactions on Antennas and Propagation, 2008 ,vol. 56, no. 8: 2363-2371
[30] Rob Maaskant, Raj Mittra, Anton Tijhuis. Fast Analysis of Large Antenna Arrays Using the Characteristic Basis Function Method and the Adaptive Cross Approximation Algorithm..IEEE Transactions on Antennas and Propagation, 2008, vol. 56,no.11
[31] Jaime Laviada, Fernando Las-Heras, Marcos R. Pino, and Raj Mittra. Solution of Electrically Large Problems With Multilevel Characteristic Basis Functions. IEEE Transactions on Antennas and Propagation, 2009,vol.57, no.10 3189-3198
[32]胡俊,复杂目标矢量电磁散射的高效方法-快速多极子方法及其应用,电子科技大学博士论文,2000
[33] Eugenio Lucente, Agostino Monorchio and Raj Mittra. Generation of Characteristic Basis Functions by Using Sparse MoM Impedance Matrix to Construct the Solution of Large Scattering and Radiation Problems. Antennas and Propagation Society International Symposium 2006, IEEE : 4091-4094
[34] C. Pelletti, K. Panayappan, R. Mittra and A. Monorchio.On the Hybridization of RUFD Algorithm with the DM Approach for Solving Multiscale Problems.2010 URSI International Symposium on Electromagnetic Theory,pp:850-852
[35] Alessio Cucini, Matteo Albani, and Stefano Maci. Truncated Floquet Wave Full-Wave (T(FW)2) Analysis of Large Periodic Arrays of Rectangular Waveguides. IEEE Transactions on Antennas and Propagation,2003,vol.51,no.6a:1373-1385
[36] Li Hu, Le-Wei Li, and Raj Mittra.Electromagnetic Scattering by Finite Periodic Arrays Using the Characteristic Basis Function and Adaptive Integral Methods. IEEE Transactions on Antennas and Propagation, 2010, vol.58, no.9: 3086-3090
[37] Zuhui Ma, Lijun Jiang, Wengcho Chew, Maokun Li, Zhiguo Qian. Augmented EPA with Augmented EFIE Method for Packaging Analysis. Electrical Design of Advanced Packaging & Systems Symposium (EDAPS), 2010 IEEE
[38] Lu W. B, Cui T. J, Qian Z.G, Yin X. X, Hong W. Accurate Analysis of Large-Scale Periodic Structures Using an Efficient Sub-Entire-Domain Basis Function Method. IEEE Transactions on Antennas and Propagation, 2004, vol. 52, Part 11:3078-3085
[39] Anja K. Skrivervik and Juan R. Mosig. Analysis of Finite Phase Arrays of Microstrip Patches. IEEE Transactions on Antennas and Propagation, 1993, vol. 41, no. 8: 1105-1114
[40] Lawrence Carin and Leopold B. Felsen. Time Harmonic and Transient Scattering by Finite Periodic Flat Strip Arrays: Hybrid (Ray)-(Floquet Mode)-(MoM) Algorithm. IEEE Transactions on Antennas and Propagation, 1993, vol. 41, no. 4:412-421
[41] Renaud Chiniard, AndréBarka, Olivier Pascal. Fast and Accurate Modeling of Large Plane Array Antenna. Antennas and Propagation, EuCAP 2006, First European Conference
[42] Cai-Cheng Lu. A Fast Algorithm Based on Volume Integral Equation for Analysis of Arbitrarily Shaped Dielectric Radomes. IEEE Transactions on Antennas and Propagation, 2003, vol.51, no. 3:606-612
[43] C. C. Lu and W. C. Chew. A coupled surface-volume integral equation approach for the calculation of electromagnetic scattering from composite metallic and material targets. IEEE Trans. Antennas Propag, 2000, 48(12):1866-1868