有限元/边界积分方程混合方法在电磁散射问题中的应用
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摘要
本文将有限元(FEM)／边界积分方程(BI)混合方法应用于电磁散射问题。在二维和三维散射问题中，分别应用二维结点FEM／BI混合方法和三维矢量FEM／BI混合方法。有限元单元使用二阶曲边四边形单元和四面体矢量单元。探讨和研究应用该混合方法分析复杂目标电磁散射特性的实现方法以及该混合方法的精度、效率和通用性的提高等问题。讨论了数值实现技术。提出了一种优化的有限元剖分方案，这种方法能够减少剖分出的未知量。开发了应用程序。计算结果与精确结果或文献结果吻合得很好。
Application of hybrid finite element method/boundary integral method (FEM/BI) to EM scattering problems is studied in this paper. Nodal FEM/BI is employed for two-dimensional problems, and vector FEM/BI for three-dimensional problems. Second-order quadrilateral nodal finite elements and tetrahedral edge finite elements are used for computing scattering from two- and three-dimensional structures, respectively. Schemes on implementing the hybrid method to analyze EM scattering properties of complex objects is discussed, and improvement of the accuracy, efficiency and versatility of the hybrid method is also discussed. An optimum scheme for constructing finite element meshes is proposed, which can reduce the sum of the unknowns. Numerical technique is discussed, and application program is developed for EM scattering problems. Numerical results are in excellent agreement with those presented by other papers or exact solutions.
Application of hybrid finite element method/boundary integral method (FEM/BI) to EM scattering problems is studied in this paper. Nodal FEM/BI is employed for two-dimensional problems, and vector FEM/BI for three-dimensional problems. Second-order quadrilateral nodal finite elements and tetrahedral edge finite elements are used for computing scattering from two- and three-dimensional structures, respectively. Schemes on implementing the hybrid method to analyze EM scattering properties of complex objects is discussed, and improvement of the accuracy, efficiency and versatility of the hybrid method is also discussed. An optimum scheme for constructing finite element meshes is proposed, which can reduce the sum of the unknowns. Numerical technique is discussed, and application program is developed for EM scattering problems. Numerical results are in excellent agreement with those presented by other papers or exact solutions.
引文
[1] P.P. Silvester, "A general high-order finite-element waveguide analysis program," IEEE Trans. Microwave Theory Tech., vol. 17, pp. 204-210, Apr. 1969.
[2] P.P. Silvester, "Finite-element solution of homogeneous waveguide problems," Alta Frequenza, vol. 38, pp. 313-317, May 1969.
[3] R. Coccioli, T. ltoh, G. Pelosi, and P. P. Silvester, "Finite-element methods in microwaves: a selected bibliography," IEEE Antennas Propagat. Mag., vol. 38, pp. 34-47, Dec. 1996.
[4] J.L. Yao Bi, L. Nicolas, A. Nicolas, "Vector absorbing boundary conditions for nodal or mixed finite elements," IEEE Trans. On Mag., vol. 32, pp. 848-853, May. 1996.
[5] P.P. Silvester, G. Pelosi, Finite Eleraentsfor Wave Electromagnetics, New York: IEEE Press, 1994.
[6] B. Stupfel, "Numerical implementation of second- and third-order conformal absorbing boundary for the vector-wave equation," IEEE Trans. Antennas Propagat., vol. 45, pp. 487-492, Mar. 1997.
[7] Omar M. Ramahi, "Stability of Absorbing Boundary Conditions," IEEETrans. Antennas Propagat., vol. 47, pp.593-599, Apr. 1999.
[8] J.P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J Comput. Phys., vol. 114, pp. 185-200, Oct. 1994.
[9] Z.S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, "A perfectly matched anisotropic absorber for use as an absorbing boundary condition," IEEE Trans. Antennas Propagat., vol. 43, pp. 1460-1463, Dec. 1995.
[10] S. D. Gedney, "An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices," IEEE Trans. Antennas Propagat., vol. 44, pp. 1630-1639, Dec.1996.
[11] J.Y. Wu, D. M. Kingsland, J. F. Lee, and R. Lee, "A comparison of anisotropic PML to Berenger's PML and its application to the finite-element method for EM scattering," IEEE Trans. Antennas Propagat., vol. 45, pp. 40～50, Jan. 1997.
[12] F.L. Teixeira and W. C. Chew, "Analytical derivation ofa conformal perfectly matched absorber for electromagnetic waves," Microwave Opt. Technol. Lett., vol. 17, no. 4, pp. 231-236, 1998.
[13] P.P. Silvester and M.S.Hsieh, "Finite-element solution of two-dimensional exterior field problems," IEEE Proc. ,vol. 118, pp. 1743-1747, Dec. 1971.
[14] B.H. McDonald and A. Wexler, "Finite-element solution of unbounded field problems," IEEE Trans. Microwave Theory Tech., vol. MTT-20, pp. 841-847, Dec. 1972.
[15] 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.
[16] X. Q Sheng, E. K. N. Yung, C. H. Chan, J. M. Jin, and W. C. Chew, "Scattering from a large body 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.
[17] 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.
[18] D.R. Lynch and K. D. Paulsen, "Origin of parasites in numerical Maxwell solutions," IEEE Trans. Microwave Theory Tech., vol. 39, pp. 383-393, Mar. 1991.
[19] D. Sun, J. Manges, X. Yuan, and Z. Cendes, "Spurious modes in finite-element methods," IEEE Antennas Propagat. Mag., vol. 37, pp. 12-24, Oct. 1995.
[20] J.B. Anderson, "Field behavior near a dielectric wedge," IEEE Trans. Antennas Propagat., vol. 26, pp. 598-602, July 1978.
[21] J.V. Bladel, "Field singularities at the tip of a dielectric cone," IEEE Trans. Antennas Propagat., vol. 33, pp. 893-895, Aug. 1985.
[22] R.D. Smedt and J. G. V. Bladel, "Field singularities at the tip of a metallic cone of arbitrary cross section," IEEE Trans. Antennas Propagat., vol. 34 pp. 865-870, July 1986.
[23] S.P. Marin, "Computing scattering amplitude for arbitrary cylinders under incident plane waves," IEEE Trans. Antennas Propagat., vol. 30, pp. 1045-1049, Nov. 1982.
[24] J.M. Jin and V. V. Liepa, "Application of hybrid finite element method to electromagnetic scattering from coated cylinders," IEEE Trans. Antennas Propagat., vol. 36, pp. 50-54, Jan. 1988.
[25] Z.Gong and A. W. Glisson, "A hybrid equation approach for the solution of electromagnetic scattering problems involving two-dimensional inhomogeneous dielectric cylinders," IEEE Trans. Antennas Propagat., vol. 38, pp. 60-68, Jan. 1990.
[26] K.L. Wu, G. Y. Delisle and D. G. Fang, "EM scattering of an arbitrary multiple dielectric coated conducting cylinder by coupled finite boundary element method," IEE Proc., vol.137, pp. 1-4, Feb. 1991.
[27] W.E. Boyse and K. D. Paulsen, "Accurate solutions of Maxwell's equations around PEC comers and highly curved surfaces using nodal finite elements," IEEE Trans. Antennas Propagat., vol. 45, pp. 1758-1767, Dec. 1997.
[28] J. D. Collins, J. M. Jin, and J. L. Volakis, "Eliminating interior resonances in finite element-boundary integral methods for scattering," IEEE Trans. Antennas Propagat., vol. 40, pp. 1583-1585, Dec. 1992.
[29] 曾余庚，徐国华，宋国乡著。电磁场有限单元法，科学出版社，1982年6月。
[30] H. Whitney, "Geometric integration theory," Princeton, NJ: Princeton University Press, 1957.
[31] J.Y. Wu and R. Lee, "The advantages of triangular and tetrahedral edge elements for electromagnetic modeling with the finite-element method," IEEE Trans. Antennas Propagat., vol. 45, pp. 1431-1437, Sep. 1997.
[32] R.D. Graglia, D. R. Wilton, A. F. Peterson, and I. L. Gheorma, "Higher order interpolatory vector bases on prism elements," IEEE Trans. Antennas Propagat., vol. 46, pp. 442-450, Mar. 1998.
[33] A. Freni, C. Mias, and R. L. Ferrari, "Hibrid finite-element analysis of electromagnetic plane wave scattering from axially periodic cylindrical structures," IEEE Trans. Antennas Propagat., vol. 46, pp. 1859-1866, Dec. 1998.
[34] L. Vegni, F. Bilotti, and A. Toscano, "Analysis of cavity backed rectangular patch antennas with inhomogeneous chiral substrates via a FEM-BEM formulation," IEEE Trans. Magn., vol. 37, pp. 3260-3263, Sep. 2001.
[35] 金建铭(美)著，王建国译，葛德彪校，电磁场有限元方法，西安电子科技大学出版社，1998年1月．
[36] 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.
[37] J. P. Webb, "Hierarchal vector bases functions of arbitrary order for triangular and tetrahedral finite elements," IEEE Trans. Antennas Propagat., vol. 47, pp. 1244-1253, Aug. 1999.
[38] O.C. Zienkiewicz and R. L. Taylor, The Finite Element Method (4th edition), New York: McGraw-Hill, 1989.
[39] P. P. Silvester and R. L. Ferrari, Finite Element for Electrical Engineers (2nd editon), Cambridge: Cambridge University Press, 1990.
[40] 杨菊生，揽生瑞，《有限元法程序设计》，西安交通大学出版社，1990年6月。
[41] 陈和群，彭宣茂，《有限元法微机程序与图形处理》，河海大学出版社，1992年6月．
[42] R.C. Baucke, "Scattering by two-dimensional lossy, inhomogeneous dielectric and magnetic cylinders using linear pyramid basis functions and point matching," IEEE Trans. Antennas Propagat., vol. 39, pp. 255-259.
[43] M.D. Deshpande, C. R. Cockrell, and C. J. Reddy, NASA Technical Paper 3575, July, 1996.
[44] E. Arvas, Y. Qian, T. K. Sarkar, and F. Asian, "TE scattering from a conduting cylinder of arbitrary cross-section covered by multiple layers of lossy dielectrics," IEE Proc. vol. 136, pp. 425-430, Dec, 1989.
[45] X. Yuan, "Three-dimensional electromagnetic scattering from inhomogeneous objects by the hybrid moment and finite element method," IEEE Trans. Microwave Theory Tech., vol. 38 pp. 1053-1058, Aug. 1990.
[2] P.P. Silvester, "Finite-element solution of homogeneous waveguide problems," Alta Frequenza, vol. 38, pp. 313-317, May 1969.
[3] R. Coccioli, T. ltoh, G. Pelosi, and P. P. Silvester, "Finite-element methods in microwaves: a selected bibliography," IEEE Antennas Propagat. Mag., vol. 38, pp. 34-47, Dec. 1996.
[4] J.L. Yao Bi, L. Nicolas, A. Nicolas, "Vector absorbing boundary conditions for nodal or mixed finite elements," IEEE Trans. On Mag., vol. 32, pp. 848-853, May. 1996.
[5] P.P. Silvester, G. Pelosi, Finite Eleraentsfor Wave Electromagnetics, New York: IEEE Press, 1994.
[6] B. Stupfel, "Numerical implementation of second- and third-order conformal absorbing boundary for the vector-wave equation," IEEE Trans. Antennas Propagat., vol. 45, pp. 487-492, Mar. 1997.
[7] Omar M. Ramahi, "Stability of Absorbing Boundary Conditions," IEEETrans. Antennas Propagat., vol. 47, pp.593-599, Apr. 1999.
[8] J.P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J Comput. Phys., vol. 114, pp. 185-200, Oct. 1994.
[9] Z.S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, "A perfectly matched anisotropic absorber for use as an absorbing boundary condition," IEEE Trans. Antennas Propagat., vol. 43, pp. 1460-1463, Dec. 1995.
[10] S. D. Gedney, "An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices," IEEE Trans. Antennas Propagat., vol. 44, pp. 1630-1639, Dec.1996.
[11] J.Y. Wu, D. M. Kingsland, J. F. Lee, and R. Lee, "A comparison of anisotropic PML to Berenger's PML and its application to the finite-element method for EM scattering," IEEE Trans. Antennas Propagat., vol. 45, pp. 40～50, Jan. 1997.
[12] F.L. Teixeira and W. C. Chew, "Analytical derivation ofa conformal perfectly matched absorber for electromagnetic waves," Microwave Opt. Technol. Lett., vol. 17, no. 4, pp. 231-236, 1998.
[13] P.P. Silvester and M.S.Hsieh, "Finite-element solution of two-dimensional exterior field problems," IEEE Proc. ,vol. 118, pp. 1743-1747, Dec. 1971.
[14] B.H. McDonald and A. Wexler, "Finite-element solution of unbounded field problems," IEEE Trans. Microwave Theory Tech., vol. MTT-20, pp. 841-847, Dec. 1972.
[15] 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.
[16] X. Q Sheng, E. K. N. Yung, C. H. Chan, J. M. Jin, and W. C. Chew, "Scattering from a large body 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.
[17] 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.
[18] D.R. Lynch and K. D. Paulsen, "Origin of parasites in numerical Maxwell solutions," IEEE Trans. Microwave Theory Tech., vol. 39, pp. 383-393, Mar. 1991.
[19] D. Sun, J. Manges, X. Yuan, and Z. Cendes, "Spurious modes in finite-element methods," IEEE Antennas Propagat. Mag., vol. 37, pp. 12-24, Oct. 1995.
[20] J.B. Anderson, "Field behavior near a dielectric wedge," IEEE Trans. Antennas Propagat., vol. 26, pp. 598-602, July 1978.
[21] J.V. Bladel, "Field singularities at the tip of a dielectric cone," IEEE Trans. Antennas Propagat., vol. 33, pp. 893-895, Aug. 1985.
[22] R.D. Smedt and J. G. V. Bladel, "Field singularities at the tip of a metallic cone of arbitrary cross section," IEEE Trans. Antennas Propagat., vol. 34 pp. 865-870, July 1986.
[23] S.P. Marin, "Computing scattering amplitude for arbitrary cylinders under incident plane waves," IEEE Trans. Antennas Propagat., vol. 30, pp. 1045-1049, Nov. 1982.
[24] J.M. Jin and V. V. Liepa, "Application of hybrid finite element method to electromagnetic scattering from coated cylinders," IEEE Trans. Antennas Propagat., vol. 36, pp. 50-54, Jan. 1988.
[25] Z.Gong and A. W. Glisson, "A hybrid equation approach for the solution of electromagnetic scattering problems involving two-dimensional inhomogeneous dielectric cylinders," IEEE Trans. Antennas Propagat., vol. 38, pp. 60-68, Jan. 1990.
[26] K.L. Wu, G. Y. Delisle and D. G. Fang, "EM scattering of an arbitrary multiple dielectric coated conducting cylinder by coupled finite boundary element method," IEE Proc., vol.137, pp. 1-4, Feb. 1991.
[27] W.E. Boyse and K. D. Paulsen, "Accurate solutions of Maxwell's equations around PEC comers and highly curved surfaces using nodal finite elements," IEEE Trans. Antennas Propagat., vol. 45, pp. 1758-1767, Dec. 1997.
[28] J. D. Collins, J. M. Jin, and J. L. Volakis, "Eliminating interior resonances in finite element-boundary integral methods for scattering," IEEE Trans. Antennas Propagat., vol. 40, pp. 1583-1585, Dec. 1992.
[29] 曾余庚，徐国华，宋国乡著。电磁场有限单元法，科学出版社，1982年6月。
[30] H. Whitney, "Geometric integration theory," Princeton, NJ: Princeton University Press, 1957.
[31] J.Y. Wu and R. Lee, "The advantages of triangular and tetrahedral edge elements for electromagnetic modeling with the finite-element method," IEEE Trans. Antennas Propagat., vol. 45, pp. 1431-1437, Sep. 1997.
[32] R.D. Graglia, D. R. Wilton, A. F. Peterson, and I. L. Gheorma, "Higher order interpolatory vector bases on prism elements," IEEE Trans. Antennas Propagat., vol. 46, pp. 442-450, Mar. 1998.
[33] A. Freni, C. Mias, and R. L. Ferrari, "Hibrid finite-element analysis of electromagnetic plane wave scattering from axially periodic cylindrical structures," IEEE Trans. Antennas Propagat., vol. 46, pp. 1859-1866, Dec. 1998.
[34] L. Vegni, F. Bilotti, and A. Toscano, "Analysis of cavity backed rectangular patch antennas with inhomogeneous chiral substrates via a FEM-BEM formulation," IEEE Trans. Magn., vol. 37, pp. 3260-3263, Sep. 2001.
[35] 金建铭(美)著，王建国译，葛德彪校，电磁场有限元方法，西安电子科技大学出版社，1998年1月．
[36] 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.
[37] J. P. Webb, "Hierarchal vector bases functions of arbitrary order for triangular and tetrahedral finite elements," IEEE Trans. Antennas Propagat., vol. 47, pp. 1244-1253, Aug. 1999.
[38] O.C. Zienkiewicz and R. L. Taylor, The Finite Element Method (4th edition), New York: McGraw-Hill, 1989.
[39] P. P. Silvester and R. L. Ferrari, Finite Element for Electrical Engineers (2nd editon), Cambridge: Cambridge University Press, 1990.
[40] 杨菊生，揽生瑞，《有限元法程序设计》，西安交通大学出版社，1990年6月。
[41] 陈和群，彭宣茂，《有限元法微机程序与图形处理》，河海大学出版社，1992年6月．
[42] R.C. Baucke, "Scattering by two-dimensional lossy, inhomogeneous dielectric and magnetic cylinders using linear pyramid basis functions and point matching," IEEE Trans. Antennas Propagat., vol. 39, pp. 255-259.
[43] M.D. Deshpande, C. R. Cockrell, and C. J. Reddy, NASA Technical Paper 3575, July, 1996.
[44] E. Arvas, Y. Qian, T. K. Sarkar, and F. Asian, "TE scattering from a conduting cylinder of arbitrary cross-section covered by multiple layers of lossy dielectrics," IEE Proc. vol. 136, pp. 425-430, Dec, 1989.
[45] X. Yuan, "Three-dimensional electromagnetic scattering from inhomogeneous objects by the hybrid moment and finite element method," IEEE Trans. Microwave Theory Tech., vol. 38 pp. 1053-1058, Aug. 1990.