广义完全匹配层在二维电磁散射中应用
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摘要
电磁散射数值分析一般分为微分方程法和积分方程法。微分方程法又分为频域和时域微分方程法,频域微分法主要是有限元法(FEM)和频域有限差分法(FDFD),时域微分法主要是时域有限差分法(FDTD),本文是采用频域有限差分法结合完全匹配层吸收边界条件来解决电磁散射问题。
首先介绍了有限差分法的基本原理,以及它的差分格式和离散化,然后详细说明了利用有限差分法解决电磁散射问题的一般步骤,介绍了应用超松弛迭代法和共轭梯度法求解差分方程组的过程,并给出了具体的数值算例。利用有限差分法求解电磁散射问题的关键之一是吸收边界问题,其次具体介绍了完全匹配层(GPML)吸收边界条件的基本原理和公式,并且对广义完全匹配层的性能做了具体的分析。最后研究了广义完全匹配层在金属圆柱、金属方柱中的应用,证明了广义完全匹配层吸收边界对任意极化、任意频率、任意入射角的电磁波可以实现无反射吸收。本文给出了在直角坐标和柱坐标系下广义完全匹配层的优化参数,GPML吸收边界可紧靠散射体去截断差分网格从而能够获得比较精确的数值结果,并对复空间变量参数进行了详细的分析,得出一定的规律。
Electromagnetic scattering numerical analysis generally contains differential equation method and integral equation method. Differential equation method is divided into frequency domain differential equation and time domain differential equation. Frequency domain differential method mainly includes the finite element method (FEM) and frequency domain finite difference method (FDFD). Time domain differential method mainly includes the finite difference time domain method (FDTD). This paper used frequency domain finite difference method in conjunction with perfectly matched layer absorbing boundary conditions to solve electromagnetic scattering problems.
Firstly, the basic principles, difference format and the discretization of finite difference method is introduced, then basic process for analyzing electromagnetic scattering problems using finite difference methods and the course for calculating differential equations using SOR and conjugate gradient method are introduced and then several numerical examples are calculated.One key point of analyzing electromagnetic scattering problems using finite difference method is absorbing boundary condition. Secondly, the basic principles and formulas of generalized perfectly matched layer (GPML) absorbing boundary conditions are introduced, and the capability of generalized perfectly matched layer is analyzed.Finally, applying generalized perfectly matched layer in the metal cylinder and metal square columns is studied, which proved that the generalized perfectly matched layer absorbing boundary can absorb electromagnetic waves of any polarization and any frequency and incident angle Unconditionally In this thesis, the optimization parameters of generalized perfectly matched layer in rectangular coordinates and cylindrical coordinates are given, GPML absorbing boundary can be close to the truncated differential scattering grid in order to obtain accurate numerical results.The parameters of complex space variables are analyzed detailedly and the characteristics are obtained.
首先介绍了有限差分法的基本原理,以及它的差分格式和离散化,然后详细说明了利用有限差分法解决电磁散射问题的一般步骤,介绍了应用超松弛迭代法和共轭梯度法求解差分方程组的过程,并给出了具体的数值算例。利用有限差分法求解电磁散射问题的关键之一是吸收边界问题,其次具体介绍了完全匹配层(GPML)吸收边界条件的基本原理和公式,并且对广义完全匹配层的性能做了具体的分析。最后研究了广义完全匹配层在金属圆柱、金属方柱中的应用,证明了广义完全匹配层吸收边界对任意极化、任意频率、任意入射角的电磁波可以实现无反射吸收。本文给出了在直角坐标和柱坐标系下广义完全匹配层的优化参数,GPML吸收边界可紧靠散射体去截断差分网格从而能够获得比较精确的数值结果,并对复空间变量参数进行了详细的分析,得出一定的规律。
Electromagnetic scattering numerical analysis generally contains differential equation method and integral equation method. Differential equation method is divided into frequency domain differential equation and time domain differential equation. Frequency domain differential method mainly includes the finite element method (FEM) and frequency domain finite difference method (FDFD). Time domain differential method mainly includes the finite difference time domain method (FDTD). This paper used frequency domain finite difference method in conjunction with perfectly matched layer absorbing boundary conditions to solve electromagnetic scattering problems.
Firstly, the basic principles, difference format and the discretization of finite difference method is introduced, then basic process for analyzing electromagnetic scattering problems using finite difference methods and the course for calculating differential equations using SOR and conjugate gradient method are introduced and then several numerical examples are calculated.One key point of analyzing electromagnetic scattering problems using finite difference method is absorbing boundary condition. Secondly, the basic principles and formulas of generalized perfectly matched layer (GPML) absorbing boundary conditions are introduced, and the capability of generalized perfectly matched layer is analyzed.Finally, applying generalized perfectly matched layer in the metal cylinder and metal square columns is studied, which proved that the generalized perfectly matched layer absorbing boundary can absorb electromagnetic waves of any polarization and any frequency and incident angle Unconditionally In this thesis, the optimization parameters of generalized perfectly matched layer in rectangular coordinates and cylindrical coordinates are given, GPML absorbing boundary can be close to the truncated differential scattering grid in order to obtain accurate numerical results.The parameters of complex space variables are analyzed detailedly and the characteristics are obtained.
引文
[1].盛剑霓.电磁场数值分析[M].北京：科学出版社,1984：1-53F
[2]. J. P. Berenger, "Three-Dimensional perfectly matched layer for the absorption of electromagnetic waves," J.Comput. Phys., vol.127, pp.363-379, Sep.1996.
[3]. D. S. Katz, E. T. Thiele, A. Talflove, "Validation and extension to three dimension of the Berenger PML absorbing boundary condition for FDTD meshes," IEEE Microwave and Guided Wave Lett., vol.4, pp.268-270, Aug. 1994.
[4]. J. P. Berenger, "Perfectly matched layer for the FDTD solution of wave structure interaction problems," IEEE Trans. Antennas Propagat, vol.44, pp. 110-117, Jan.1996.
[5]. D. T. Prescott, N. V. Shuley, "Refletion analysis of FDTD boundary conditions —partⅡ:Berenger's PML absorbing layers," IEEE Trans. Microwave Theory Tech., vol.45, no.8, pp.1171-1178, Aug.1997.
[6]. C. E. Reuter, R. M. Joseph, E. T. Thiele, D. S. Katz, A. Talflove, "ultrawideband absorbing boundary condition for termination of waveguide structures in FDTD simulations," IEEE Microwave and Guided Wave Lett., vol.4, pp.344-346, Oct.1994.
[7]. B. Chen, D. G. Fang, B. H. Zhou, "Modified Berenger PML absorbing boundary condition for FDTD meshes," IEEE Microwave and Guided Wave Lett., vol.5, pp.399-401, Nov.1995.
[8]. J. D. Moerloose, M. A. Strchly, "Behavior of Berenger's ABC for evanescent waves," IEEE Microwave and Guided Wave Lett., vol.5, pp.344-346, Oct. 1995.
[9]. J. P. Berenger, "Evanescent waves in PML's:origin of the numerical reflection in wave structure interaction problems," IEEE Trans. Antennas Propagat., vol. 47, pp.1497-1503, Oct.1999.
[10]. J. P. Berenger, "Numerical reflection of evanescent waves from perfectly matched layers," in IEEE Antennas and Propagation symp Dig., Montreal, Canada, pp.1497-1503, July 13-17,1997.
[11]. J. P. Berenger, "An effective PML for the Absorption of evanescent waves in waveguides",IEEE Microwave and Guided Wave Lett., vol.8, pp.188-190, May.1998.
[12]. G. Lazzi,O. P. Gandhi, "On the optimal design of the PML absorbing boundary condition for the FDTD code," IEEE Trans. Antennas Propagat., vol. 45, pp.914-916, May.1997.
[13]. J. P. Berenger, "Improved PML for FDTD solution of wave structure interaction problems," IEEE Trans. Antennas Propagat., vol.45, pp.466-473, Mar.1997.
[14]. Z. S. sacks, D. M. Kingsland, R. Lee, 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.
[15]. S. D. Gedney, "An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices," IEEE Trans. Antennas Propagat., vol.44, pp. 1630-1639, Dec.1996.
[16]W. C. Chew, W. Weedon, "A 3D perfectly matched medium from modified Maxwell'sequations with stretched coordinates," Microwave Opt. Tech. Lett., vol 7, no.13, pp.599-604,1994.
[17]. R. Mitrra, U. Pekel, "A new look at the perfectly matched layer(PML) concept for the reflectionless absorption of electromagnetic waves," IEEE Microwave and Guided Wave Lett., vol.5, pp.84-86, Mar.1995.
[18]. M. Kuzuoglu, R. Mittra, "Frequency dependence of the constitutive parameters of causal perfectly matched anisotropic absorbers," IEEE Microwave and Guided Wave Lett., vol.6, pp.447-449, Dec.1996.
[19]. J. P. Berenger, "Application of the CFS PMLto the Absorption of evanescent waves in waveguide," IEEE Microwave and Wireless components Lett, vol.12, no.6, pp.218-220,2002.
[20]. E. A. Marengo, C. M. Rappaport, E. L. Miller, "Optimum PML ABC conductivity profile in FDFD," IEEE Transactions on magnetics, vol.35, no.3, pp.1506-1509, May,1999.
[21]. C.M. Rappaort, "Interpreting and improving the PML absorbing boundary condition using anisotropic lossy mapping of space," IEEE Transactions on magnetics, vol.32, no.3, pp.968-974, May,1996.
[22]. J. Y. Fang, Z. H. Wu, "Closed-form expression on numerical reflection coefficient at PML interfaces and optimization of PML performance," IEEE Microwave and Guided Wave Lett., vol.6, pp.332-334, Sep,1996.
[23]. Z. H. Wu, J. Y. Fang, "High-performance PML algorithms," IEEE Microwave and Guided Wave Lett., vol.6, pp.335-337, Sep,1996.
[24]. A. Mitcheu, D. M. Kokotoff, M. W. Austin, "Improved to the PML boundary condition in the FEM using mesh compression," IEEE Trans. Microwave Theory Tech., vol.50, no.5, pp.1297-1302, Aug,2002.
[25]. P. G. Petropoulos, " An analytical study of the discrete perfectly mathed layer for the Time-Domain Maxwell equations in cylindrical coordinate," IEEE Trans. Antennas Propagat., vol.51, pp.1671-1675, Jul.2003.
[26]. S. C. Winton, C. M. Rappaort, " Specifying PML conductivities by considering numerical reflection dependencies," IEEE Trans. Antennas Propagat., vol.48, pp.1055-1063, Jul.2000.
[27]. Y. S. Richard, N. K. Georgiva, "Problem-independent enhancement of PML ABC for the FDTD method," IEEE Trans. Antennas Propagat., vol.51, pp. 3002-3006, Oct.2003.
[28]. N. V. Kantartzis, etal "Nondiagonally Anisotropic PML:A generalized unsplit wide-angle absorber for the treatment of the near-grazing effect in FDTD meshes," IEEE Transactions on magnetics, vol.36, no.4, pp.907-911, July,2000.
[29]. H. A. Jamid, "Enhanced PML performance using higher order approximation,' IEEE Trans. Microwave Theory Tech., vol.52, no.4, pp.11667-1174, Apr, 2004.
[30]. S. Kim, J. Choi, "Optimal design of PML absorbing boundary condition for improving wide-angle reflection performance," Electronics Letters, vol. No.2, 2004.
[31]. D. Correia, J. M. JIN, "On the develepment of a high-order PML," IEEE Trans. Antennas Propagat., vol:53, pp.4157-4163, Dec.2005.
[2]. J. P. Berenger, "Three-Dimensional perfectly matched layer for the absorption of electromagnetic waves," J.Comput. Phys., vol.127, pp.363-379, Sep.1996.
[3]. D. S. Katz, E. T. Thiele, A. Talflove, "Validation and extension to three dimension of the Berenger PML absorbing boundary condition for FDTD meshes," IEEE Microwave and Guided Wave Lett., vol.4, pp.268-270, Aug. 1994.
[4]. J. P. Berenger, "Perfectly matched layer for the FDTD solution of wave structure interaction problems," IEEE Trans. Antennas Propagat, vol.44, pp. 110-117, Jan.1996.
[5]. D. T. Prescott, N. V. Shuley, "Refletion analysis of FDTD boundary conditions —partⅡ:Berenger's PML absorbing layers," IEEE Trans. Microwave Theory Tech., vol.45, no.8, pp.1171-1178, Aug.1997.
[6]. C. E. Reuter, R. M. Joseph, E. T. Thiele, D. S. Katz, A. Talflove, "ultrawideband absorbing boundary condition for termination of waveguide structures in FDTD simulations," IEEE Microwave and Guided Wave Lett., vol.4, pp.344-346, Oct.1994.
[7]. B. Chen, D. G. Fang, B. H. Zhou, "Modified Berenger PML absorbing boundary condition for FDTD meshes," IEEE Microwave and Guided Wave Lett., vol.5, pp.399-401, Nov.1995.
[8]. J. D. Moerloose, M. A. Strchly, "Behavior of Berenger's ABC for evanescent waves," IEEE Microwave and Guided Wave Lett., vol.5, pp.344-346, Oct. 1995.
[9]. J. P. Berenger, "Evanescent waves in PML's:origin of the numerical reflection in wave structure interaction problems," IEEE Trans. Antennas Propagat., vol. 47, pp.1497-1503, Oct.1999.
[10]. J. P. Berenger, "Numerical reflection of evanescent waves from perfectly matched layers," in IEEE Antennas and Propagation symp Dig., Montreal, Canada, pp.1497-1503, July 13-17,1997.
[11]. J. P. Berenger, "An effective PML for the Absorption of evanescent waves in waveguides",IEEE Microwave and Guided Wave Lett., vol.8, pp.188-190, May.1998.
[12]. G. Lazzi,O. P. Gandhi, "On the optimal design of the PML absorbing boundary condition for the FDTD code," IEEE Trans. Antennas Propagat., vol. 45, pp.914-916, May.1997.
[13]. J. P. Berenger, "Improved PML for FDTD solution of wave structure interaction problems," IEEE Trans. Antennas Propagat., vol.45, pp.466-473, Mar.1997.
[14]. Z. S. sacks, D. M. Kingsland, R. Lee, 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.
[15]. S. D. Gedney, "An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices," IEEE Trans. Antennas Propagat., vol.44, pp. 1630-1639, Dec.1996.
[16]W. C. Chew, W. Weedon, "A 3D perfectly matched medium from modified Maxwell'sequations with stretched coordinates," Microwave Opt. Tech. Lett., vol 7, no.13, pp.599-604,1994.
[17]. R. Mitrra, U. Pekel, "A new look at the perfectly matched layer(PML) concept for the reflectionless absorption of electromagnetic waves," IEEE Microwave and Guided Wave Lett., vol.5, pp.84-86, Mar.1995.
[18]. M. Kuzuoglu, R. Mittra, "Frequency dependence of the constitutive parameters of causal perfectly matched anisotropic absorbers," IEEE Microwave and Guided Wave Lett., vol.6, pp.447-449, Dec.1996.
[19]. J. P. Berenger, "Application of the CFS PMLto the Absorption of evanescent waves in waveguide," IEEE Microwave and Wireless components Lett, vol.12, no.6, pp.218-220,2002.
[20]. E. A. Marengo, C. M. Rappaport, E. L. Miller, "Optimum PML ABC conductivity profile in FDFD," IEEE Transactions on magnetics, vol.35, no.3, pp.1506-1509, May,1999.
[21]. C.M. Rappaort, "Interpreting and improving the PML absorbing boundary condition using anisotropic lossy mapping of space," IEEE Transactions on magnetics, vol.32, no.3, pp.968-974, May,1996.
[22]. J. Y. Fang, Z. H. Wu, "Closed-form expression on numerical reflection coefficient at PML interfaces and optimization of PML performance," IEEE Microwave and Guided Wave Lett., vol.6, pp.332-334, Sep,1996.
[23]. Z. H. Wu, J. Y. Fang, "High-performance PML algorithms," IEEE Microwave and Guided Wave Lett., vol.6, pp.335-337, Sep,1996.
[24]. A. Mitcheu, D. M. Kokotoff, M. W. Austin, "Improved to the PML boundary condition in the FEM using mesh compression," IEEE Trans. Microwave Theory Tech., vol.50, no.5, pp.1297-1302, Aug,2002.
[25]. P. G. Petropoulos, " An analytical study of the discrete perfectly mathed layer for the Time-Domain Maxwell equations in cylindrical coordinate," IEEE Trans. Antennas Propagat., vol.51, pp.1671-1675, Jul.2003.
[26]. S. C. Winton, C. M. Rappaort, " Specifying PML conductivities by considering numerical reflection dependencies," IEEE Trans. Antennas Propagat., vol.48, pp.1055-1063, Jul.2000.
[27]. Y. S. Richard, N. K. Georgiva, "Problem-independent enhancement of PML ABC for the FDTD method," IEEE Trans. Antennas Propagat., vol.51, pp. 3002-3006, Oct.2003.
[28]. N. V. Kantartzis, etal "Nondiagonally Anisotropic PML:A generalized unsplit wide-angle absorber for the treatment of the near-grazing effect in FDTD meshes," IEEE Transactions on magnetics, vol.36, no.4, pp.907-911, July,2000.
[29]. H. A. Jamid, "Enhanced PML performance using higher order approximation,' IEEE Trans. Microwave Theory Tech., vol.52, no.4, pp.11667-1174, Apr, 2004.
[30]. S. Kim, J. Choi, "Optimal design of PML absorbing boundary condition for improving wide-angle reflection performance," Electronics Letters, vol. No.2, 2004.
[31]. D. Correia, J. M. JIN, "On the develepment of a high-order PML," IEEE Trans. Antennas Propagat., vol:53, pp.4157-4163, Dec.2005.