有限元—边界积分混合方法在电磁散射问题中的应用
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摘要
有限元法特别适合于解决具有非均匀媒质的复杂几何结构的电磁问题。在电磁学领域内,有限元法己广泛用于解决辐射、散射、波导传输及谐振腔等问题。由于对许多目标的电磁现象的辐射及散射分析都涉及到无限区域,有限元方法需要在离开目标物体一段距离的地方设置吸收边界条件,这就自然增加了计算量。而基于积分方程的边界积分方法尽管可以直接分析目标问题,但其不利的一面是产生的矩阵是一个满阵,因此受到计算机内存的限制,只适合于分析电小尺寸问题。为了避开这两种方法的不利一面同时保留其优点,发展了一种混合算法:有限元-边界积分混合方法(FE-BI)。该方法的基本原理是引入一个包围所研究目标的虚构边界,在虚构边界内部用有限元方法来分析,在边界上用边界元来处理。这两个区域中的场在虚构边界上通过场的连续性耦合起来,从而得到一个内部和边界场解的耦合方程组。
推导了基于四面体剖分的矢量基函数的有限元公式,编写了有限元程序,并用该程序计算了波导以及传输线内的场,程序计算结果与数值结果吻合良好。根据有限元-边界积分混合方法的原理,推导了有限元-边界积分混合方法的公式,其中内部区域采用基于四面体剖分的矢量基函数,边界部分采用的三角形剖分的RWG基函数。根据这些公式编写了程序,应用该程序计算了三维目标的电磁散射问题,程序计算结果与传统矩量法和有限元法计算结果吻合良好。
The finite element method (FEM) suits specially to solve the electromagnetic problems of the structures consisting of an inhomogeneous dielectric body of arbitrary shape. In the electromagnetic domain, the finite element method has widely used in solving problems such as radiation, scattering, wave guide transmission and resonant cavity problem. Since the electromagnetic radiation and scattering analysis of many problems is in open space, the absorbing boundary elements at the outer surface of the meshed region must be employed when the finite element method is used, thus the computation burden is increased out of question. Although the boundary integral method based on the integral equation is very efficient at solving the open radiation problems, the matrix gained by this method is full and requires too much memory. So the boundary integral method is only suitable to solve the problem of the electrically small object. To take advantage the strengths of the both methods and avoid their disadvantages, a hybrid method is presented, viz. the hybrid finite element-boundary integral method. The principle of this hybrid method is that the FEM is employed to handle the interior domain of bodies and the boundary integrate is used to develop surface integrals that relate the field quantities on boundary surfaces with the equivalent surface currents. These integral equations are then coupled to the finite element equations through the continuity of the tangential magnetic fields across the hybrid boundaries.
Based on the edge-based vector basis functions defined within tetrahedrons, the finite element formulae are deduced and the program is written to compute the fields within the wave-guide and transmission line. The computed results are in excellent agreement with the exact solutions. Then, based on the principle of the hybrid finite element-boundary integral method, the formulae of this hybrid method are deduced. In the interior region the edge-based vector finite elements are used and on the surface the RWG basis functions are used. Based on these formulae, another program is also written. The electromagnetic scattering programs of the three-dimensional objects are computed using our hybrid method program and the computed results agree well with the results computed by the moment method and the finite element method.
推导了基于四面体剖分的矢量基函数的有限元公式,编写了有限元程序,并用该程序计算了波导以及传输线内的场,程序计算结果与数值结果吻合良好。根据有限元-边界积分混合方法的原理,推导了有限元-边界积分混合方法的公式,其中内部区域采用基于四面体剖分的矢量基函数,边界部分采用的三角形剖分的RWG基函数。根据这些公式编写了程序,应用该程序计算了三维目标的电磁散射问题,程序计算结果与传统矩量法和有限元法计算结果吻合良好。
The finite element method (FEM) suits specially to solve the electromagnetic problems of the structures consisting of an inhomogeneous dielectric body of arbitrary shape. In the electromagnetic domain, the finite element method has widely used in solving problems such as radiation, scattering, wave guide transmission and resonant cavity problem. Since the electromagnetic radiation and scattering analysis of many problems is in open space, the absorbing boundary elements at the outer surface of the meshed region must be employed when the finite element method is used, thus the computation burden is increased out of question. Although the boundary integral method based on the integral equation is very efficient at solving the open radiation problems, the matrix gained by this method is full and requires too much memory. So the boundary integral method is only suitable to solve the problem of the electrically small object. To take advantage the strengths of the both methods and avoid their disadvantages, a hybrid method is presented, viz. the hybrid finite element-boundary integral method. The principle of this hybrid method is that the FEM is employed to handle the interior domain of bodies and the boundary integrate is used to develop surface integrals that relate the field quantities on boundary surfaces with the equivalent surface currents. These integral equations are then coupled to the finite element equations through the continuity of the tangential magnetic fields across the hybrid boundaries.
Based on the edge-based vector basis functions defined within tetrahedrons, the finite element formulae are deduced and the program is written to compute the fields within the wave-guide and transmission line. The computed results are in excellent agreement with the exact solutions. Then, based on the principle of the hybrid finite element-boundary integral method, the formulae of this hybrid method are deduced. In the interior region the edge-based vector finite elements are used and on the surface the RWG basis functions are used. Based on these formulae, another program is also written. The electromagnetic scattering programs of the three-dimensional objects are computed using our hybrid method program and the computed results agree well with the results computed by the moment method and the finite element method.
引文
[1] J. S. Wang and R. Mittra, "Finite element analysis of MMIC structures and electronic packages using absorbing boundary conditions," IEEE Trans. Microwave Theory Tech., vol.42, pp.441-449, Mar.1994.
[2] K. Ise, K. Inoue, and M. Koshiba, "Three-dimensional finite-element method with edge elements for electromagnetic waveguide discontinuities," IEEE Trans. Microwave Theory Tech., vol.39, pp.1289-1295, Aug.1991.
[3] J. F Lee and R .Mittra, "A note on the application of edge-element for modeling three-dimensional inhomogeneously filled cavities," IEEE Trans. Microwave Theory Tech., vol.40, pp.1767-1773, Sept.1992.
[4] M. Koshiba, K. Hayata, and M. Suzuki, "Finite-element method analysis of microwave and optical waveguides: Trends in countermeasures to spurious solutions." Electronics and Communications in Japan, part2, vo1.70, pp.96-108, 1987.
[5] B. M, A. Rahman,E. A. Fernandez, and J. B. Davies, "Review of finite element method for Microwave and optical waveguides," Proc. IEEE, vol.79, pp.1442-1448, Oct.1991.
[6] Z. J. Cendes and P. Silvester, "Numerical solution of dielectric loaded waveguide: 1–Finite element analysis," IEEE Trans. Microwave Theory Tech., vol.18, pp.1124-1131, 1970.
[7] B. M. A. Rahman and J. B. Davies, "Penalty function improvement of waveguide solution by Finite elements," IEEE Trans. Microwave Theory Tech., vol.32, pp.922-928, Aug.1984.
[8] I. P .Webb, "Finite element analysis of dispersion in waveguides with sharp metal edges," IEEE Trans. Microwave Theory Tech., vo1.36, no.12, pp.1819-1824, Dec.1988.
[9] A .Bossavit, "A rationale for ‘edge-elements’ in 3–D fields computations," IEEE Trans. Magn., vol.24, no.I, pp.74 -79, Jan.1988.
[10] X. Q. Sheng, J. M. Jin, J. Song, C. C. Lu, and W. C. Chew, "On the formulation of hybrid finite-element and boundary-integral methods for 3-D scattering," IEEE Trans. Antennas Propagat., vol.46, pp.303-311, Mar.1998.
[11] X. Q. Sheng, E. K. N. Yung, C. H. Chan, and W.C. Chew, "Scattering from a largebody with cracks and cavities by the fast and accurate finite-element boundary-integral method," IEEE Trans. Antennas Propagat., vol.48, pp.1153-1160 Aug.2000.
[12] Paul Soudasis, Perre Leca, Jerome Simon, and Thibault Volpert, "Computation of the Scattering from inhomogeneous objects with a discrete rotational symmetry and a nonsymmetric part," IEEE Trans, Antennas Propagat, vol.50, pp.168-174, Feb.2002.
[13] Jian Liu and Jian-Ming Jin,“A Special Higher Order Finite-Element Method for Scattering by Deep Cavities,” IEEE Transactions on Antennas and Propagation, vol.48, no.5, pp.694-703, May.2000.
[14] Jian Liu and Jian-Ming Jin,“Scattering Analysis of a Large Body with Deep Cavities,” IEEE Antennas Propagation Society International Symposium, Salt Lake City, Utah, USA, pp.556–559, July 16-21, 2000.
[15] Jian-Ming Jin,Jian Liu,Zheng Lou,and Charles S.T.Liang,“A Fully High-Order Finite-Element Simulation of Scattering by Deep Cavities,” IEEE Transactions on Antennas and Propagation, vol.51, no.9, pp.2420-2429, Sep.2003.
[16] Aihua Wood, “Analysis of electromagnetic scattering from an overfilled cavity in the ground plane,” Journal of Computational Physics, Vol.215, pp630–641, 2006.
[17] 金建铭,电磁场有限元方法,西安电子科技大学出版社,2001.
[18] Mohammad W. Ali, Todd H. Hubing, and James L. Drewniak “A Hybrid FEM/MOM Technique for Electromagnetic Scattering and Radiation from Dielectric Objects with Attached Wires,” IEEE Transactions on electromagnetic compatibility, vol.39, pp.304-314, Nov.1997.
[19] Hu, F.-G., Wang, C.-F, Gan, Y.-B., “Efficient Calculation of Interior Scattering From Large Three-Dimensional PEC Cavities,” Antennas and Propagation,IEEE Transactions on Volume 55, Issue1, Page(s):167–177, Jan.2007.
[20] Burkholder, R.J, Lundin T, “Forward-backward iterative physical optics algorithm for computing the RCS of open-ended cavities” Antennas and Propagation,IEEE Transactions on Volume 53, Issue 2, Page(s):793 – 799, Feb 2005.
[21] 聂小春,葛德彪,袁宁. 导电平板上任意孔缝的 TM 波散射及传输特性分析.电子与信息学报,23 (2),pp.168-174,2001.
[22] 聂小春,葛德彪,袁宁. 边界积分法及连接算法分析任意腔体的散射. 微波学报,15 (4),pp.334-338,1999.
[23] 聂小春,葛德彪,袁宁. 导电平面上三维任意腔体的散射分析[J ]. 微波学报, 16 (4),pp.440-444,2000.
[24] 魏兵,葛德彪.二维金属凹槽填充各向异性介质时散射的边界元方法: TE 情形. 西安电子科技大学学报,29 (6),pp.737-740,2002.
[25] 何小祥,徐金平. 改进的 IPO 与 FEM 混合法分析复杂电大腔体电磁散射.电波科学学报,Vol.19,No.5,pp.607-612,October,2004.
[26] 丁卫平,徐金平. 带有腔体或槽缝的电大尺寸目标电磁散射特性分析. 电子学报,Vol.30,No. 6,pp.815-818,June,2002.
[27] 盛新庆,彭朕. 合元极技术再认识 —— 一种电大复杂目标散射混合计算技术的考察. 电子学报,34(1),pp.93-98,2006.
[28] 陈杰夫,郑长良,钟万勰. 电磁波导的辛分析与对偶棱边元. 物理学报,Vol.55,No.5,pp.2340-2346,May,2006.
[29] 陈杰夫 郑长良 钟万勰. 电磁对偶元的子区域分析. 微波学报,Vol.22,No.2,pp.7-10,Apr.2006.
[30] 杜永兴 席晓莉. 2450MHz 医用微波辐射天线的设计. 微波学报,Vol.22,Supplement,pp.86-89,Jun.2006.
[31] 邱兆杰,侯新宇,许家栋,万伟. 三维目标电磁散射矢量有限元/边界元法的公式研究. 电子学报,Vol.34,No.9,Sep.2006.
[32] 年丰 董硕 周乐柱 夏明耀. 三维电磁辐射问题混合阶矢量基有限元完全匹配层方法的研究. 北京大学学报(自然科学版)网络版,第 1 卷,第 3 期,pp.1-5,2006-09-30.
[33] A. Bossavit and J. C. Verite, “A mixed FEM-BIEM method to solve 3-D eddy current problems,” IEEE Trans. Magnetics, vol.MAG-18, pp.431-435, Mar.1982.
[34] M. L. Barton and Z. J. Cendes, “New vector finite elements for three-dimensional magnetic field computation,” J. Appl. Phys., vol.61, no.8, pp.3919-3921, Apr.1987.
[35] C. W. Crowley, “Mixed order covariant projection finite elements for vector fields,” PH. D. dissertation, McGill University, Montreal, 1988.
[36] A. W. Glisson and D. R. Wilton, “Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces,” IEEE Trans. Antennas Propagat., vol.AP-28, pp.593-603, Sept.1980.
[37] 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.
[38] D. H. Schaubert, D. R. Wilton, and A. W. Glisson, “A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies,” IEEE Trans. Antennas Propagat., vol.AP-32, pp.77-85, Jan. 1984.
[39] M. R. Hestenes and E. Stiefel, “Method of conjugate gradients for solving linearsystems,” J. Res. Nat. Bur. Standards, vol.49, pp.409-436, Dec. 1952.
[40] B. J. Rubin and S. Daijavad, "Radiation and Scattering from Structures Involving Finite-Size Dielectric Regions", IEEE Trans. Antenna Propagat., vol.38, pp.1863-1873, Nov.1990.
[41] T. K. Sarkar, S. M. Rao, and A. R. Djordjevic, “Electromagnetic scattering and radiation from finite microstrip structures,” IEEE Trans. Microwave Theory Tech., vol.38, pp.1568-1575, Nov.1990.
[42] M. L. Barton and Z. J. Cendes, “New vector finite elements for three-dimensional magnetic field computation,” J. Appl. Phys., vol.61, no.8, pp.3919-3921, Apr.1987.
[43] A. Chatterjee, J. M. Jin, and J. L. Volakis, “Computation of cavity resonance using edge elements,” IEEE Trans. Microwave Theory Tech., vol.40, pp.2106-2108, Nov.1992.
[2] K. Ise, K. Inoue, and M. Koshiba, "Three-dimensional finite-element method with edge elements for electromagnetic waveguide discontinuities," IEEE Trans. Microwave Theory Tech., vol.39, pp.1289-1295, Aug.1991.
[3] J. F Lee and R .Mittra, "A note on the application of edge-element for modeling three-dimensional inhomogeneously filled cavities," IEEE Trans. Microwave Theory Tech., vol.40, pp.1767-1773, Sept.1992.
[4] M. Koshiba, K. Hayata, and M. Suzuki, "Finite-element method analysis of microwave and optical waveguides: Trends in countermeasures to spurious solutions." Electronics and Communications in Japan, part2, vo1.70, pp.96-108, 1987.
[5] B. M, A. Rahman,E. A. Fernandez, and J. B. Davies, "Review of finite element method for Microwave and optical waveguides," Proc. IEEE, vol.79, pp.1442-1448, Oct.1991.
[6] Z. J. Cendes and P. Silvester, "Numerical solution of dielectric loaded waveguide: 1–Finite element analysis," IEEE Trans. Microwave Theory Tech., vol.18, pp.1124-1131, 1970.
[7] B. M. A. Rahman and J. B. Davies, "Penalty function improvement of waveguide solution by Finite elements," IEEE Trans. Microwave Theory Tech., vol.32, pp.922-928, Aug.1984.
[8] I. P .Webb, "Finite element analysis of dispersion in waveguides with sharp metal edges," IEEE Trans. Microwave Theory Tech., vo1.36, no.12, pp.1819-1824, Dec.1988.
[9] A .Bossavit, "A rationale for ‘edge-elements’ in 3–D fields computations," IEEE Trans. Magn., vol.24, no.I, pp.74 -79, Jan.1988.
[10] X. Q. Sheng, J. M. Jin, J. Song, C. C. Lu, and W. C. Chew, "On the formulation of hybrid finite-element and boundary-integral methods for 3-D scattering," IEEE Trans. Antennas Propagat., vol.46, pp.303-311, Mar.1998.
[11] X. Q. Sheng, E. K. N. Yung, C. H. Chan, and W.C. Chew, "Scattering from a largebody with cracks and cavities by the fast and accurate finite-element boundary-integral method," IEEE Trans. Antennas Propagat., vol.48, pp.1153-1160 Aug.2000.
[12] Paul Soudasis, Perre Leca, Jerome Simon, and Thibault Volpert, "Computation of the Scattering from inhomogeneous objects with a discrete rotational symmetry and a nonsymmetric part," IEEE Trans, Antennas Propagat, vol.50, pp.168-174, Feb.2002.
[13] Jian Liu and Jian-Ming Jin,“A Special Higher Order Finite-Element Method for Scattering by Deep Cavities,” IEEE Transactions on Antennas and Propagation, vol.48, no.5, pp.694-703, May.2000.
[14] Jian Liu and Jian-Ming Jin,“Scattering Analysis of a Large Body with Deep Cavities,” IEEE Antennas Propagation Society International Symposium, Salt Lake City, Utah, USA, pp.556–559, July 16-21, 2000.
[15] Jian-Ming Jin,Jian Liu,Zheng Lou,and Charles S.T.Liang,“A Fully High-Order Finite-Element Simulation of Scattering by Deep Cavities,” IEEE Transactions on Antennas and Propagation, vol.51, no.9, pp.2420-2429, Sep.2003.
[16] Aihua Wood, “Analysis of electromagnetic scattering from an overfilled cavity in the ground plane,” Journal of Computational Physics, Vol.215, pp630–641, 2006.
[17] 金建铭,电磁场有限元方法,西安电子科技大学出版社,2001.
[18] Mohammad W. Ali, Todd H. Hubing, and James L. Drewniak “A Hybrid FEM/MOM Technique for Electromagnetic Scattering and Radiation from Dielectric Objects with Attached Wires,” IEEE Transactions on electromagnetic compatibility, vol.39, pp.304-314, Nov.1997.
[19] Hu, F.-G., Wang, C.-F, Gan, Y.-B., “Efficient Calculation of Interior Scattering From Large Three-Dimensional PEC Cavities,” Antennas and Propagation,IEEE Transactions on Volume 55, Issue1, Page(s):167–177, Jan.2007.
[20] Burkholder, R.J, Lundin T, “Forward-backward iterative physical optics algorithm for computing the RCS of open-ended cavities” Antennas and Propagation,IEEE Transactions on Volume 53, Issue 2, Page(s):793 – 799, Feb 2005.
[21] 聂小春,葛德彪,袁宁. 导电平板上任意孔缝的 TM 波散射及传输特性分析.电子与信息学报,23 (2),pp.168-174,2001.
[22] 聂小春,葛德彪,袁宁. 边界积分法及连接算法分析任意腔体的散射. 微波学报,15 (4),pp.334-338,1999.
[23] 聂小春,葛德彪,袁宁. 导电平面上三维任意腔体的散射分析[J ]. 微波学报, 16 (4),pp.440-444,2000.
[24] 魏兵,葛德彪.二维金属凹槽填充各向异性介质时散射的边界元方法: TE 情形. 西安电子科技大学学报,29 (6),pp.737-740,2002.
[25] 何小祥,徐金平. 改进的 IPO 与 FEM 混合法分析复杂电大腔体电磁散射.电波科学学报,Vol.19,No.5,pp.607-612,October,2004.
[26] 丁卫平,徐金平. 带有腔体或槽缝的电大尺寸目标电磁散射特性分析. 电子学报,Vol.30,No. 6,pp.815-818,June,2002.
[27] 盛新庆,彭朕. 合元极技术再认识 —— 一种电大复杂目标散射混合计算技术的考察. 电子学报,34(1),pp.93-98,2006.
[28] 陈杰夫,郑长良,钟万勰. 电磁波导的辛分析与对偶棱边元. 物理学报,Vol.55,No.5,pp.2340-2346,May,2006.
[29] 陈杰夫 郑长良 钟万勰. 电磁对偶元的子区域分析. 微波学报,Vol.22,No.2,pp.7-10,Apr.2006.
[30] 杜永兴 席晓莉. 2450MHz 医用微波辐射天线的设计. 微波学报,Vol.22,Supplement,pp.86-89,Jun.2006.
[31] 邱兆杰,侯新宇,许家栋,万伟. 三维目标电磁散射矢量有限元/边界元法的公式研究. 电子学报,Vol.34,No.9,Sep.2006.
[32] 年丰 董硕 周乐柱 夏明耀. 三维电磁辐射问题混合阶矢量基有限元完全匹配层方法的研究. 北京大学学报(自然科学版)网络版,第 1 卷,第 3 期,pp.1-5,2006-09-30.
[33] A. Bossavit and J. C. Verite, “A mixed FEM-BIEM method to solve 3-D eddy current problems,” IEEE Trans. Magnetics, vol.MAG-18, pp.431-435, Mar.1982.
[34] M. L. Barton and Z. J. Cendes, “New vector finite elements for three-dimensional magnetic field computation,” J. Appl. Phys., vol.61, no.8, pp.3919-3921, Apr.1987.
[35] C. W. Crowley, “Mixed order covariant projection finite elements for vector fields,” PH. D. dissertation, McGill University, Montreal, 1988.
[36] A. W. Glisson and D. R. Wilton, “Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces,” IEEE Trans. Antennas Propagat., vol.AP-28, pp.593-603, Sept.1980.
[37] 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.
[38] D. H. Schaubert, D. R. Wilton, and A. W. Glisson, “A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies,” IEEE Trans. Antennas Propagat., vol.AP-32, pp.77-85, Jan. 1984.
[39] M. R. Hestenes and E. Stiefel, “Method of conjugate gradients for solving linearsystems,” J. Res. Nat. Bur. Standards, vol.49, pp.409-436, Dec. 1952.
[40] B. J. Rubin and S. Daijavad, "Radiation and Scattering from Structures Involving Finite-Size Dielectric Regions", IEEE Trans. Antenna Propagat., vol.38, pp.1863-1873, Nov.1990.
[41] T. K. Sarkar, S. M. Rao, and A. R. Djordjevic, “Electromagnetic scattering and radiation from finite microstrip structures,” IEEE Trans. Microwave Theory Tech., vol.38, pp.1568-1575, Nov.1990.
[42] M. L. Barton and Z. J. Cendes, “New vector finite elements for three-dimensional magnetic field computation,” J. Appl. Phys., vol.61, no.8, pp.3919-3921, Apr.1987.
[43] A. Chatterjee, J. M. Jin, and J. L. Volakis, “Computation of cavity resonance using edge elements,” IEEE Trans. Microwave Theory Tech., vol.40, pp.2106-2108, Nov.1992.