FEM-BI在电磁散射和辐射中的研究与应用
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摘要
有限元法作为计算电磁学的主要方法之一,由于在处理复杂形状、复杂介质目标中的优势,近年来在波导传输、目标RCS计算、微波电路计算、天线设计等问题中得到了广泛利用。
本文首先从变分原理出发,介绍了矢量有限元法的基本原理,并计算了几种腔体的谐振值。通过数值结果可以看出矢量有限元可以避免标量有限元中出现的伪解问题。
其次把有限元法用于开域问题的计算中,在开域问题中必须添加边界条件把无限大区域截断为有限大区域。其中截断边界条件有局部边界条件和全域边界条件,吸收边界条件(ABC)和完全匹配层(PML)属于局部边界条件,边界积分属于全域边界条件。局部边界条件可以保持有限元矩阵的稀疏性,但由于是近似条件,对最后结果会产生误差,本文分析了这种误差产生的原因。
再次本文把边界积分方程用于有限元边界的截断,首先推导了有限元边界积分方法(FEM-BI),并得出了最后矩阵的组成和矩阵元素的表达式,再介绍了有限元稀疏矩阵的储存、有限元边界积分方程的求解和边界积分方程中奇异性的处理。最后把FEM-BI用于介质、涂覆目标和各向异性介质雷达散射截面(RCS)的计算,并与解析结果和文献进行比较,验证了程序的正确性。最后再将FEM-BI用于简单天线模型辐射问题中,计算了天线辐射方向图,并考察和分析了天线罩对天线方向图的影响。
As one of the main methods for computational electromagnetics, Finite Element Method (FEM) possesses the advantages in dealing with complex shapes and complex media. So it has been widely used in the waveguide transmission, the target RCS calculation, the microwave circuits calculation,antenna design and other issues in recent years.
Firstly, based on the variational principle, the basic principle of vector finite element method is introduced, and the eigenvalues of several cavity resonaces are calculated. The numerical results show that the vector finite element avoids the problem of pseudo-solution that appears in the scalar finite element method.
Secondly, the finite element method is utilized to analyze the open boundary problems, in which an artifical boundary must be added to truncate the infinite region to a finite area. Truncated boundary conditions contain the local boundary conditions and the global boundary conditions. Absorbing boundary condition (ABC) and the perfectly matched layer (PML) belong to the local boundary conditions, while boundary integral equation belongs to the global boundary conditions. The local boundary conditions can keep finite element matrix sparse, but because it is approximate, it will arouse numerical errors in the final results, this essay analyzes the causes of these errors. In this paper, the boundary integral equation is utilized to truncate computational region.
Thirdly, the principle of the finite element boundary integral method (FEM-BI) is derivated, and the form of the matrix and the expression of matrix elements are provided. Then sparse storage for the finite element matrix, the solution of the finite element boundary integral equations, and the treatment of the singularity of boundary integral equation are introduced. Next, radar cross section (RCS) for the dielectric, coating target and anisotropic media are calculated with FEM-BI, and compared with the analytical results and the results from literatures to verify the validity of the program. Finally, the FEM-BI is emlpoyed for some simple antenna models, the antenna radiation pattern is calculated, and the influence of radome on the antenna pattern is investigated and analyzed.
本文首先从变分原理出发,介绍了矢量有限元法的基本原理,并计算了几种腔体的谐振值。通过数值结果可以看出矢量有限元可以避免标量有限元中出现的伪解问题。
其次把有限元法用于开域问题的计算中,在开域问题中必须添加边界条件把无限大区域截断为有限大区域。其中截断边界条件有局部边界条件和全域边界条件,吸收边界条件(ABC)和完全匹配层(PML)属于局部边界条件,边界积分属于全域边界条件。局部边界条件可以保持有限元矩阵的稀疏性,但由于是近似条件,对最后结果会产生误差,本文分析了这种误差产生的原因。
再次本文把边界积分方程用于有限元边界的截断,首先推导了有限元边界积分方法(FEM-BI),并得出了最后矩阵的组成和矩阵元素的表达式,再介绍了有限元稀疏矩阵的储存、有限元边界积分方程的求解和边界积分方程中奇异性的处理。最后把FEM-BI用于介质、涂覆目标和各向异性介质雷达散射截面(RCS)的计算,并与解析结果和文献进行比较,验证了程序的正确性。最后再将FEM-BI用于简单天线模型辐射问题中,计算了天线辐射方向图,并考察和分析了天线罩对天线方向图的影响。
As one of the main methods for computational electromagnetics, Finite Element Method (FEM) possesses the advantages in dealing with complex shapes and complex media. So it has been widely used in the waveguide transmission, the target RCS calculation, the microwave circuits calculation,antenna design and other issues in recent years.
Firstly, based on the variational principle, the basic principle of vector finite element method is introduced, and the eigenvalues of several cavity resonaces are calculated. The numerical results show that the vector finite element avoids the problem of pseudo-solution that appears in the scalar finite element method.
Secondly, the finite element method is utilized to analyze the open boundary problems, in which an artifical boundary must be added to truncate the infinite region to a finite area. Truncated boundary conditions contain the local boundary conditions and the global boundary conditions. Absorbing boundary condition (ABC) and the perfectly matched layer (PML) belong to the local boundary conditions, while boundary integral equation belongs to the global boundary conditions. The local boundary conditions can keep finite element matrix sparse, but because it is approximate, it will arouse numerical errors in the final results, this essay analyzes the causes of these errors. In this paper, the boundary integral equation is utilized to truncate computational region.
Thirdly, the principle of the finite element boundary integral method (FEM-BI) is derivated, and the form of the matrix and the expression of matrix elements are provided. Then sparse storage for the finite element matrix, the solution of the finite element boundary integral equations, and the treatment of the singularity of boundary integral equation are introduced. Next, radar cross section (RCS) for the dielectric, coating target and anisotropic media are calculated with FEM-BI, and compared with the analytical results and the results from literatures to verify the validity of the program. Finally, the FEM-BI is emlpoyed for some simple antenna models, the antenna radiation pattern is calculated, and the influence of radome on the antenna pattern is investigated and analyzed.
引文
[1] A. Taflove and S. C. Hagness. Computational Electrodynamics: The Finite-Difference Time-Domain Method. Artech House, Norwood, MA, USA, 2ed., 2000
[2] R. F. Harrington. Field Computation by Moment Methods. New York: Macmillan, 1968; reprinted by New York: IEEE Press, 1993
[3] S. C. Lee, J. F. Lee,and R. Lee. Hierarchical vector finite elements for analyzing waveguiding structures. IEEE Trans. on Microwave Theory Tech., 2003, 51(8): 1897-1905
[4] J. Liu, J. M. Jin, A novel hybridization of higher order finite element and boundary integral methods for electromagnetic scattering an dradiation problems, IEEE Trans. on Antennas Propagt., 2001, 49 (12): 1794-1806
[5]金建铭.电磁场有限元方法.西安电子科技大学出版社,1998
[6] D. K. Sun, Z. Cendes. Fast high-order FEM solutions of dielectric wave guiding structures, IEE Proc. Microw. Antennas Propag., 2003, 150(4): 230-236
[7] P. Borys, P. Yuriy. Tunable filters based on metal-dielectric resonators 2010 18th International Conference on Microwave Radar and Wireless Communications. MIKON: 2010 : 1– 3
[8] L. K. Yeh, C. Y. Chen, H. R. Chuang. A Millimeter-Wave CPW CMOS On-Chip Bandpass Filter Using Conductor-Backed Resonators. Digital Object Identifier: 2010,31(5): 399– 401E. H. Newman and D. M. Poza. Electromagnetic modeling of composite wire and surface geometries. IEEE Trans. Antennas Propagat, 1978, AP-26(6):784–789
[9] B. Stupfel, R. Mittra. A theoretical study of numerical absorbing boundary conditions.IEEE Trans.Antenna propagation, 1995, 43(5):478-487
[10] J. P . Berenger. A perfertly matched layer for the absorption of electromagnetic waves. Comput. Phys., 1994, 114(1):185-200
[11]胡俊.复杂目标矢量电磁散射的高效算法——快速多极子算法及其应用.电子科技大学博士学位论文,2000
[12] J. M. Song and W. C. Chew. Fast multipole solution of three dimensional integral equation. IEEE Antennas Propagation Symposium, pp.1528-1531, 1995
[13] J.M.Song, C.C.Lu, and W.C.Chew. Multilevel fast multipole algorithm for electromagneticscattering by large complex objects. IEEE Transactions on Antennas and Propagation. Vol. 45, No. 10, pp. 1488-1493, 1997
[14] J. M. Song, C. C. Lu, W. C. Chew and S.W. Lee. Introduction to Fast Illinois Solver Code (FISC). IEEE Antennas Propagation Symposium. pp. 48-51, 1997
[15] S. Velamparambil, W. C. Chew. Analysis and Performance of a Distributed Memory Multilevel Fast Multipole Algorithm. IEEE Transactions on Antennas and Propagation, Vol.53, No. 8 II, pp. 2719-2727, August 2005
[16] X. Q. Sheng, S. J. Xu. An efficient high-order mixed-edge rectangular-element method for lossy anisotropic dielectric waveguides, IEEE Trans. Microwave Theory Tech., 1997, 45(7): 1009-1013
[17] M. M. IIic, A. Z. Ilic, B. M. Notaros. Efficient large-domain 2-D FEM solution of arbitrary waveguides using p-refinement on generalized quadrilaterals. IEEE Trans. Microwave Theory Tech., 2005, 53(4): 1377-1383
[18] R. S. Chen, X. W. Ping, E. K. N. Yung, et al. Application of diagonally perturbed Incomplete Factorization preconditioned conjugate gradient algorithms for edge finite-element analysis of Helmholtz equations. IEEE Trans. Antennas Propagat. May 2006vol. 54, no. 5:1604-1608
[19] J. M. Jin, J. Liu, Z. Lou, C. S. T. Liang, A fully high-order finite-element simulation of scattering by deep cavities, IEEE Trans. Antennas Propagt., 2003, 51(9): 2420-2429.
[20] J. P. Webb. Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements. IEEE Trans. on Antennas Propagat., 1999, 47 (8): 1244-1253
[21] R. D. Graglia, D. R. Wilton, A. F. Peterson, Higher order interpolatory vector bases for computational electromagnetics, IEEE Trans. Antennas Propagat., 1997, 45 (3): 329-342
[22]张文.复杂目标电磁散射的有限元/边界积分法.南京理工大学硕士学位论文,2009
[23] M. Abramowitz, I. A. Stegun. Handbook of Mathematical Functionswith Formulas. Graphs, and Mathematical Tables. New York:Dover, 1972
[24] Z. Shao, Z. Shen, Q. He, G. Wei, A generalized higher order finite-difference time-domain method and its application in guided-wave problems, IEEE Trans. Microwave Theory Tech., IEEE Trans. Antennas Propagat., 2003, 51(3): 856-861
[25] D.-K. Sun, Z. Cendes, Fast high-order FEM solutions of dielectric wave guiding structures, IEE Proc. Microw. Antennas Propag., 2003, 150(4): 230-236
[26]蒿正伟,王勇,朱德才.矢量边界元法在三维电磁场本征值问题中的应用.科学技术与工程, 2008, 8(16): 4654-4657
[27] S. C. Lee, J. F. Lee, R. Lee. Hierarchical vector finite elements for analyzing waveguiding structures, IEEE Trans. Microwave Theory Tech., 2003, 51(8): 1897-1905
[28] X. Y. Zhang, T. Zhang, A. M. Hu. Tunable microring resonator based on dielectric-loaded surface plasmon-polariton waveguides 2010 3rd International Digital Object Identifier Nanoelectronics Conference, 2010 :1355– 1356
[29] S. M. Razavizadeh. A Band-Notched UWB Microstrip Antenna with a Resonance Back C-Shaped Ring 2010 Second International Conference on Digital Object Identifier: 2010: 37– 41
[30]班永灵.高阶矢量有限元方法及其在三维电磁散射与辐射问题中的应用.电子科技大学博士学位论文,2006
[31] S. Rao, D. Wilton, A. Glisson. Electromagnetic scattering by surfaces of arbitrary shape. IEEE Transactions on Antennas and Propagation, Vol. 30, pp. 409-418, 1982
[32] J. Hu, Z. P. Nie ,L. Lei. Solving 3-D electromagnetic scattering from conducting object by MLFMA with curvilinear RWG basis. IEEE Antennas and Propagation Society International Symposium, pp: 460-463, 2003
[33]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
[34] E. Arvas, R. Harrington, J. Mautz. Radiation and scattering from electrically small conducting bodies of arbitrary shape. IEEE Trans. Antennas Propag., vol. 34, pp. 66–77, Jan. 1986
[35] J. Liu. Higher-order finite element-boundary integral methods for electromagnetic scattering and radiation analysis: [Ph. D Thesis], Urbana, Illinois, U. S. A., Univ. of Illinois Urbana-Champaign, 2002
[36] J. Liu, J. M. Jin. A highly effective preconditioner for solving the finite element-boundary integral matrix equation of 3-D scattering. IEEE Trans. Antennas Propagat. , Sept. 2002, vol. 50, no. 9:1212-1221
[37] X. Q. Sheng, J. M. Jin, W. C. Chew and C. C. Lu. Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies. IEEE Trans. Antennas Propagat. , Nov. 1998, vol.46, no. 11:1718-1726
[38]聂在平.目标与环境电磁散射特性建模.北京:国防工业出版社,2009
[39]杨法.三维金属/介质复合结构电磁散射的有限元/边界积分方法.电子科技大学硕士学位论文,2008
[40]王浩刚,聂在平,王军.曲面参数二次模拟结合积分奇异降阶方法的矩量法数值计算.电子与信息学报. Jan. 2002, vol. 24, no. 1:83-89
[41]谭云华.含各向异性材料的复杂目标电磁散射的边棱元---快速多级子混合算法研究:【博士论文】,北京,北京大学,2003
[42]李世智.电磁辐射与散射问题的矩量法.北京:电子工业出版社,1985
[43] A. Khebir, J. D’Angelo, J. Joseph. A new Finite Element Formulation for RF Scattering by Complex Bodies of Revolution. IEEE Trans. Antennas and Propagtion, Vol.41, pp.534~541, 1993
[44] Y. L. Geng. Scattering of a plane wave by an anistropic ferrite-coated conducting sphere, IET Microw. Antenna Propag.,2008, 2(2):158-162
[45]耿友林,矢量波函数在各向异性介质电磁散射中的应用,西安电子科技大学博士论文,2006
[46] B. D. Popovic.金属天线与散射体分析.哈尔滨工业大学出版社,1999
[47] J. M. Jin, J. L. Volakis, J. D. Collins. A finite element-boundary integral method for scattering and radiation by two- and three-dimensional structures. IEEE Trans. on Antennas and Propagat. Mag. , June 1991, vol. 33, pp. 22-32
[2] R. F. Harrington. Field Computation by Moment Methods. New York: Macmillan, 1968; reprinted by New York: IEEE Press, 1993
[3] S. C. Lee, J. F. Lee,and R. Lee. Hierarchical vector finite elements for analyzing waveguiding structures. IEEE Trans. on Microwave Theory Tech., 2003, 51(8): 1897-1905
[4] J. Liu, J. M. Jin, A novel hybridization of higher order finite element and boundary integral methods for electromagnetic scattering an dradiation problems, IEEE Trans. on Antennas Propagt., 2001, 49 (12): 1794-1806
[5]金建铭.电磁场有限元方法.西安电子科技大学出版社,1998
[6] D. K. Sun, Z. Cendes. Fast high-order FEM solutions of dielectric wave guiding structures, IEE Proc. Microw. Antennas Propag., 2003, 150(4): 230-236
[7] P. Borys, P. Yuriy. Tunable filters based on metal-dielectric resonators 2010 18th International Conference on Microwave Radar and Wireless Communications. MIKON: 2010 : 1– 3
[8] L. K. Yeh, C. Y. Chen, H. R. Chuang. A Millimeter-Wave CPW CMOS On-Chip Bandpass Filter Using Conductor-Backed Resonators. Digital Object Identifier: 2010,31(5): 399– 401E. H. Newman and D. M. Poza. Electromagnetic modeling of composite wire and surface geometries. IEEE Trans. Antennas Propagat, 1978, AP-26(6):784–789
[9] B. Stupfel, R. Mittra. A theoretical study of numerical absorbing boundary conditions.IEEE Trans.Antenna propagation, 1995, 43(5):478-487
[10] J. P . Berenger. A perfertly matched layer for the absorption of electromagnetic waves. Comput. Phys., 1994, 114(1):185-200
[11]胡俊.复杂目标矢量电磁散射的高效算法——快速多极子算法及其应用.电子科技大学博士学位论文,2000
[12] J. M. Song and W. C. Chew. Fast multipole solution of three dimensional integral equation. IEEE Antennas Propagation Symposium, pp.1528-1531, 1995
[13] J.M.Song, C.C.Lu, and W.C.Chew. Multilevel fast multipole algorithm for electromagneticscattering by large complex objects. IEEE Transactions on Antennas and Propagation. Vol. 45, No. 10, pp. 1488-1493, 1997
[14] J. M. Song, C. C. Lu, W. C. Chew and S.W. Lee. Introduction to Fast Illinois Solver Code (FISC). IEEE Antennas Propagation Symposium. pp. 48-51, 1997
[15] S. Velamparambil, W. C. Chew. Analysis and Performance of a Distributed Memory Multilevel Fast Multipole Algorithm. IEEE Transactions on Antennas and Propagation, Vol.53, No. 8 II, pp. 2719-2727, August 2005
[16] X. Q. Sheng, S. J. Xu. An efficient high-order mixed-edge rectangular-element method for lossy anisotropic dielectric waveguides, IEEE Trans. Microwave Theory Tech., 1997, 45(7): 1009-1013
[17] M. M. IIic, A. Z. Ilic, B. M. Notaros. Efficient large-domain 2-D FEM solution of arbitrary waveguides using p-refinement on generalized quadrilaterals. IEEE Trans. Microwave Theory Tech., 2005, 53(4): 1377-1383
[18] R. S. Chen, X. W. Ping, E. K. N. Yung, et al. Application of diagonally perturbed Incomplete Factorization preconditioned conjugate gradient algorithms for edge finite-element analysis of Helmholtz equations. IEEE Trans. Antennas Propagat. May 2006vol. 54, no. 5:1604-1608
[19] J. M. Jin, J. Liu, Z. Lou, C. S. T. Liang, A fully high-order finite-element simulation of scattering by deep cavities, IEEE Trans. Antennas Propagt., 2003, 51(9): 2420-2429.
[20] J. P. Webb. Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements. IEEE Trans. on Antennas Propagat., 1999, 47 (8): 1244-1253
[21] R. D. Graglia, D. R. Wilton, A. F. Peterson, Higher order interpolatory vector bases for computational electromagnetics, IEEE Trans. Antennas Propagat., 1997, 45 (3): 329-342
[22]张文.复杂目标电磁散射的有限元/边界积分法.南京理工大学硕士学位论文,2009
[23] M. Abramowitz, I. A. Stegun. Handbook of Mathematical Functionswith Formulas. Graphs, and Mathematical Tables. New York:Dover, 1972
[24] Z. Shao, Z. Shen, Q. He, G. Wei, A generalized higher order finite-difference time-domain method and its application in guided-wave problems, IEEE Trans. Microwave Theory Tech., IEEE Trans. Antennas Propagat., 2003, 51(3): 856-861
[25] D.-K. Sun, Z. Cendes, Fast high-order FEM solutions of dielectric wave guiding structures, IEE Proc. Microw. Antennas Propag., 2003, 150(4): 230-236
[26]蒿正伟,王勇,朱德才.矢量边界元法在三维电磁场本征值问题中的应用.科学技术与工程, 2008, 8(16): 4654-4657
[27] S. C. Lee, J. F. Lee, R. Lee. Hierarchical vector finite elements for analyzing waveguiding structures, IEEE Trans. Microwave Theory Tech., 2003, 51(8): 1897-1905
[28] X. Y. Zhang, T. Zhang, A. M. Hu. Tunable microring resonator based on dielectric-loaded surface plasmon-polariton waveguides 2010 3rd International Digital Object Identifier Nanoelectronics Conference, 2010 :1355– 1356
[29] S. M. Razavizadeh. A Band-Notched UWB Microstrip Antenna with a Resonance Back C-Shaped Ring 2010 Second International Conference on Digital Object Identifier: 2010: 37– 41
[30]班永灵.高阶矢量有限元方法及其在三维电磁散射与辐射问题中的应用.电子科技大学博士学位论文,2006
[31] S. Rao, D. Wilton, A. Glisson. Electromagnetic scattering by surfaces of arbitrary shape. IEEE Transactions on Antennas and Propagation, Vol. 30, pp. 409-418, 1982
[32] J. Hu, Z. P. Nie ,L. Lei. Solving 3-D electromagnetic scattering from conducting object by MLFMA with curvilinear RWG basis. IEEE Antennas and Propagation Society International Symposium, pp: 460-463, 2003
[33]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
[34] E. Arvas, R. Harrington, J. Mautz. Radiation and scattering from electrically small conducting bodies of arbitrary shape. IEEE Trans. Antennas Propag., vol. 34, pp. 66–77, Jan. 1986
[35] J. Liu. Higher-order finite element-boundary integral methods for electromagnetic scattering and radiation analysis: [Ph. D Thesis], Urbana, Illinois, U. S. A., Univ. of Illinois Urbana-Champaign, 2002
[36] J. Liu, J. M. Jin. A highly effective preconditioner for solving the finite element-boundary integral matrix equation of 3-D scattering. IEEE Trans. Antennas Propagat. , Sept. 2002, vol. 50, no. 9:1212-1221
[37] X. Q. Sheng, J. M. Jin, W. C. Chew and C. C. Lu. Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies. IEEE Trans. Antennas Propagat. , Nov. 1998, vol.46, no. 11:1718-1726
[38]聂在平.目标与环境电磁散射特性建模.北京:国防工业出版社,2009
[39]杨法.三维金属/介质复合结构电磁散射的有限元/边界积分方法.电子科技大学硕士学位论文,2008
[40]王浩刚,聂在平,王军.曲面参数二次模拟结合积分奇异降阶方法的矩量法数值计算.电子与信息学报. Jan. 2002, vol. 24, no. 1:83-89
[41]谭云华.含各向异性材料的复杂目标电磁散射的边棱元---快速多级子混合算法研究:【博士论文】,北京,北京大学,2003
[42]李世智.电磁辐射与散射问题的矩量法.北京:电子工业出版社,1985
[43] A. Khebir, J. D’Angelo, J. Joseph. A new Finite Element Formulation for RF Scattering by Complex Bodies of Revolution. IEEE Trans. Antennas and Propagtion, Vol.41, pp.534~541, 1993
[44] Y. L. Geng. Scattering of a plane wave by an anistropic ferrite-coated conducting sphere, IET Microw. Antenna Propag.,2008, 2(2):158-162
[45]耿友林,矢量波函数在各向异性介质电磁散射中的应用,西安电子科技大学博士论文,2006
[46] B. D. Popovic.金属天线与散射体分析.哈尔滨工业大学出版社,1999
[47] J. M. Jin, J. L. Volakis, J. D. Collins. A finite element-boundary integral method for scattering and radiation by two- and three-dimensional structures. IEEE Trans. on Antennas and Propagat. Mag. , June 1991, vol. 33, pp. 22-32