辛算法在时域电磁散射计算中的应用
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摘要
近年来，随着计算机性能的飞速发展和计算数学、计算物理中各种新型算法的出现，计算电磁学呈现出空前繁荣的局面。各种电磁场数值方法层出不穷，但这些方法面临计算时间、存储空间及计算精度等方面的困难，而且随着人们对问题的物理本质认识的深入，意识到在追求算法高精度的同时，还应力求保持原系统的内在性质。由于电磁场方程可以转化为一无穷维Hamilton系统，而Hamilton系统具有一系列的内在性质，因而在对Hamilton系统的数值求解时，保持其内在性质就显得尤为重要。辛算法正是保持Hamilton系统内在性质的一种新型数值方法，该算法在长时间的数值计算中，具有常见数值方法无可比拟的计算优势。本文将辛算法引入到电磁计算中，针对辛算法在时域电磁散射计算中的应用，具体展开了以下几方面的工作：
(1)．介绍了辛算法的数学理论基础，包括辛算法常见的构造方法：辛Runge-Kutta法，辛传播子技术及生成函数法，利用辛传播子技术结合误差最小化及稳定性Courant-Fredrichs-Lewy(CFL)条件数最大化的两种优化方案，构造了新的传播子系数；
(2)．建立了二维电磁散射问题的辛算法理论，主要包括：基于辛传播子理论建立二维可分Hamilton系统的辛算法；基于辛PRK方法首次建立了二维不可分Hamilton系统的高阶辛算法；探讨了二维辛算法的稳定性及数值色散性，通过计算实例进一步证实了辛算法在二维电磁散射计算中的优势；
(3)．引入辛时域有限差分法(SFDTD)计算三维电磁散射问题，建立了各阶SFDTD法，首次对各阶SFDTD法的稳定性及数值色散性进行了系统的分析。数值结果表明SFDTD法较标准的FDTD法及高阶FDTD的稳定性及数值色散性等方面都有较大改进，尤其是高阶SFDTD法的引入，为计算三维电大尺寸目标的散射提供了新的解决方案和思路；
(4)．详细探讨了SFDTD法在三维电磁散射计算中实现的技术细节，包括SFDTD法中各类激励源的引入；SFDTD法的吸收边界条件及改进的高阶PML吸收边界条件；SFDTD法的高阶近场—远场转换技术，为SFDTD法在目标的散射计算方面提供了技术途径；
(5)．利用SFDTD法计算了三维典型散射体的近场分布及远场的雷达散射截面(RCS)，就算法的稳定性、复杂度、精度等方面，与常用的时域数值方法如FDTD及高阶FDTD方法作了详细比较，进一步表明了辛算法的正确性及高效性。
The computational electromagnetics have been developed rapidly with the emergence of new schemes in numerical mathematics and physics in recent years. Various numerical methods for electromagnetics simulations are proposed, but these methods have difficulties in the aspect of simulations time, required memory and accuracy. In additional, with the depth of the knowledge of the physical characters of the problems, people realized that it is also important to keep some features of the original system while pursuing the high accuracy of the schemes. Since the electromagnetics field equations can be rewritten as an infinite dimensional Hamiltonian system, which has some insight characters, the proper solution should be obtained using symplectic schemes. The symplectic schemes have the ability to preserve the global symplectic structure of the phase space for a Hamiltonian system. They have substantial benefits in numerical computation for Hamiltonian system, especially in long-term simulations. The dissertation introduced symplectic schemes for electromagnetics simulations, mainly focus on the application of the schemes in time-domain computation.The main studies are as follows:
(1).Introduced the mathematics theories foundation of the symplectic schemes, include the commonly established methods for construction symplectic schemes, which are symplectic Runge- Kutta method, symplectic propagation techniques and generating function method. A minimization of the truncation error-function and optimal Courant-Friedrichs-Levy (CFL) number schemes are well established using sympletic propagation techniques.
(2).The symplectic schemes for two-dimensional electromagnetics scattering problems are well established, including the schemes for separable Hamiltonian system using symplectic propagation techniques, the symplectic partitioned Runge-Kutta(PRK) schemes for non-separable Hamiltonian system for the first time. The stability and numerical dispersion were analyzed; numerical results show the efficiency of the schemes.
(3).The symplectic finite-difference time-domain (SFDTD) schemes were introduced in numerical solution of three-dimensional electromagnetics. The stability and numerical dispersion for SFDTD schemes were analyzed for the first time. Numerical results show SFDTD schemes are superior to standard FDTD and high order FDTD. Especially, the introduction of the high order SFDTD schemes provides a new way for solution of three-dimensional electric largely objects scattering problems.
(4).The total field and scattered field (TF-SF) technique is derived for the SFDTD method to provide the incident wave source conditions, the absorbing boundary conditions (ABC) and the improved high order perfectly matched layers (PML) ABC, the high order near field-far field transform, are all well established. All those provided the necessary techniques for three-dimensional electric largely objects scattering computations.
(5).The symplectic schemes are applied to the time-domain simulation of benchmark objects scattering problems. The stability and accuracy are compared with the commonly used methods such as FDTD and high order FDTD. The results further confirmed the accuracy and efficiently of the symplectic schemes.
(1)．介绍了辛算法的数学理论基础，包括辛算法常见的构造方法：辛Runge-Kutta法，辛传播子技术及生成函数法，利用辛传播子技术结合误差最小化及稳定性Courant-Fredrichs-Lewy(CFL)条件数最大化的两种优化方案，构造了新的传播子系数；
(2)．建立了二维电磁散射问题的辛算法理论，主要包括：基于辛传播子理论建立二维可分Hamilton系统的辛算法；基于辛PRK方法首次建立了二维不可分Hamilton系统的高阶辛算法；探讨了二维辛算法的稳定性及数值色散性，通过计算实例进一步证实了辛算法在二维电磁散射计算中的优势；
(3)．引入辛时域有限差分法(SFDTD)计算三维电磁散射问题，建立了各阶SFDTD法，首次对各阶SFDTD法的稳定性及数值色散性进行了系统的分析。数值结果表明SFDTD法较标准的FDTD法及高阶FDTD的稳定性及数值色散性等方面都有较大改进，尤其是高阶SFDTD法的引入，为计算三维电大尺寸目标的散射提供了新的解决方案和思路；
(4)．详细探讨了SFDTD法在三维电磁散射计算中实现的技术细节，包括SFDTD法中各类激励源的引入；SFDTD法的吸收边界条件及改进的高阶PML吸收边界条件；SFDTD法的高阶近场—远场转换技术，为SFDTD法在目标的散射计算方面提供了技术途径；
(5)．利用SFDTD法计算了三维典型散射体的近场分布及远场的雷达散射截面(RCS)，就算法的稳定性、复杂度、精度等方面，与常用的时域数值方法如FDTD及高阶FDTD方法作了详细比较，进一步表明了辛算法的正确性及高效性。
The computational electromagnetics have been developed rapidly with the emergence of new schemes in numerical mathematics and physics in recent years. Various numerical methods for electromagnetics simulations are proposed, but these methods have difficulties in the aspect of simulations time, required memory and accuracy. In additional, with the depth of the knowledge of the physical characters of the problems, people realized that it is also important to keep some features of the original system while pursuing the high accuracy of the schemes. Since the electromagnetics field equations can be rewritten as an infinite dimensional Hamiltonian system, which has some insight characters, the proper solution should be obtained using symplectic schemes. The symplectic schemes have the ability to preserve the global symplectic structure of the phase space for a Hamiltonian system. They have substantial benefits in numerical computation for Hamiltonian system, especially in long-term simulations. The dissertation introduced symplectic schemes for electromagnetics simulations, mainly focus on the application of the schemes in time-domain computation.The main studies are as follows:
(1).Introduced the mathematics theories foundation of the symplectic schemes, include the commonly established methods for construction symplectic schemes, which are symplectic Runge- Kutta method, symplectic propagation techniques and generating function method. A minimization of the truncation error-function and optimal Courant-Friedrichs-Levy (CFL) number schemes are well established using sympletic propagation techniques.
(2).The symplectic schemes for two-dimensional electromagnetics scattering problems are well established, including the schemes for separable Hamiltonian system using symplectic propagation techniques, the symplectic partitioned Runge-Kutta(PRK) schemes for non-separable Hamiltonian system for the first time. The stability and numerical dispersion were analyzed; numerical results show the efficiency of the schemes.
(3).The symplectic finite-difference time-domain (SFDTD) schemes were introduced in numerical solution of three-dimensional electromagnetics. The stability and numerical dispersion for SFDTD schemes were analyzed for the first time. Numerical results show SFDTD schemes are superior to standard FDTD and high order FDTD. Especially, the introduction of the high order SFDTD schemes provides a new way for solution of three-dimensional electric largely objects scattering problems.
(4).The total field and scattered field (TF-SF) technique is derived for the SFDTD method to provide the incident wave source conditions, the absorbing boundary conditions (ABC) and the improved high order perfectly matched layers (PML) ABC, the high order near field-far field transform, are all well established. All those provided the necessary techniques for three-dimensional electric largely objects scattering computations.
(5).The symplectic schemes are applied to the time-domain simulation of benchmark objects scattering problems. The stability and accuracy are compared with the commonly used methods such as FDTD and high order FDTD. The results further confirmed the accuracy and efficiently of the symplectic schemes.
引文
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[1]黄志祥，吴先良．辛算法的稳定性及数值色散性分析[J]．电子学报，Vol．34.No.3， pp.535—538．
[2]HUANG Zhi-xiang, WU Xian-liang. Symplectic partitioned Runge-Kutta scheme for Maxwell's equations [J].Int.J.of Quantum Chem.,2006,106:839-842.
[3]HUANG Zhi-xiang,WU Xian-liang.A second order symplectic partitioned Runge-Kutta scheme for Maxwell's Equations[J]. Applied Computational Electromagnetics Society Journal.,Vol.20, pp.235-239.
[4]E.Forest,R.D.Ruth. Fourth-order symplectic integration [J]. Physica D., 1990, 43:105-117.
[5]G.Mur. Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations[J]. IEEE Trans. Electromagn. Compt.,1981, EMC-23 (4): 377-382.
[6]T.Hirono,W.W.Lui, K.Yokoyanma.Time-domain simulation of electromagnetic field using a symplectic integrator [J]. IEEE Microwave and Guide Wave Letters., 1997,7(9):279-281.
[7]T.Hirono, W.W.Lui, K.Yokoyanma, S.seki. Stability and numerical dispersion of symplectic fourth-order time-domain schemes for optical field simulation [J]. Journal of Lightwave Technology.,1998, 16(10): 1915-1920.
[1]K.S.Yee. Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media[J].IEEE Transactions on Antennas and Propagation.,1966, 14: 302-307.
[2]Karl S. Kunz, Raymond J. Luebbers. The Finite Difference Time Domain Method for Electromagnetics [M].CRC Press: Boca Raton, FL, 1993.
[3]Allen Taflove.Computational Electrodynamics:The Finite-Difference Time-Domain Method [M]. Boston, MA: Artech House, 1995.
[4]Allen Taflove.Advances in Computational Electrodynamics:The Finite-Difference Time-Domain Method [M]. Boston, MA: Artech House, 1998.
[5]葛德彪，闫玉波．电磁波时域有限差分方法[M]．西安：西安电子科技大学出版社，2001．
[6]王秉中．计算电磁学[M]．北京：科学出版社，2002．
[7]余文华，苏涛，Raj Mittra，刘永峻．并行时域有限差分[M]．北京：北京广播学院出版社，2005．
[8]J. Fang.Time domain finite difference computation for Maxwell"s equations [D]. USA: EECS University of California,Berkeley,CA,1989.
[9]C. W. Manry, S.L. Broschat, J.B. Schneider.Higher-order FDTD methods for large problems [J]. J. Appl. Comp. Electromagn. Soc., 1995, 10:17-29.
[10]M.F.Hadi,M. Piket-May.A modified FDTD (2,4) scheme for modeling electrically large structures with high-phase accuracy[J].IEEE Transactions on Antennas and Propagation.,1997,45:254-264.
[11]T.Hirono,W.Lui,S. Seki,Y.Yoshikuni.A three-dimensional fourth-order finite-difference time-domain scheme using a symplectic integrator propagator[J]. IEEE Transactions on Microwave Theory and Techniques.,2001, 49:1640-1648.
[12]Sha. Wei, Huang. Zhi-xiang, Wu. Xian-liang,Chen.Ming-shen.Total field and scattered field technique for fourth-order symplectic finite difference time domain method[J].Chin.Phy.Lett.,2006, 23:103-105.
[13]R.D. Ruth. A canonical integration technique[J]. IEEE Trans. Nucl. Sci., 1983,NS-30: 2669-2671.
[1]Karl S. Kunz, Raymond J. Luebbers. The Finite Difference Time Domain Method for Electromagnetics [M].CRC Press: Boca Raton,FL,1993.
[2]Allen Taflove.Computational Electrodynamics:The Finite-Difference Time-Domain Method [M]. Boston, MA: Artech House, 1995.
[3]Allen Taflove. Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method [M]. Boston, MA: Artech House, 1998.
[4]葛德彪，闫玉波．电磁波时域有限差分方法[M]．西安：西安电子科技大学出版社，2001．
[5]高本庆，时域有限差分法[M]．北京：国防工业出版社，1995．
[6]黄志祥，吴先良，高阶PE算法及其在非局部边界条件中的应用[J]．合肥工业大学学报，2004,27：1558-1561．
[7]黄志祥，吴先良，PML技术及非局部边界条件在抛物线方程中的应用[J]．合肥工业大学学报，2005，02：176-179．
[8]J.R Bergener,A perfectly matched layer for the absorption of electromagnetic waves[J].Journal of computational Physics.,1994, 114:185-200.
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[11]T. Hirono, W. Lui, S. Seki, Y. Yoshikuni. A three-dimensional fourth-order finite-difference time-domain scheme using a symplectic integrator propagator[J]. IEEE Transactions on Microwave Theory and Techniques., 2001, 49: 1640-1648.
[1]Allen Taflove.Advances in Computational Electrodynamics:The Finite-Difference Time-Domain Method[M]. Boston, MA: Artech House, 1998.
[2]葛德彪，闫玉波．电磁波时域有限差分方法[M]．西安：西安电子科技大学出版社，2001．
[3]Sha. Wei, Huang. Zhi-xiang, Wu. Xian-liang, Chen. Ming-shen. Total field and scattered field technique for fourth-order symplectic finite difference time domain method[J]. Chin. Phy. Lett.,2006, 23:103-105.
[4]Huang. Zhi-xiang, Sha. Wei, Wu. Xian-liang, Chen. Ming-shen. Symplectic finite-difference time-Domain scheme for electromagnetic simulations[J]. (Under review)
[5]Sha. Wei, Huang. Zhi-xiang, Wu. Xian-liang, Chen. Ming-shen. Application of the symplectic finite-difference time-domain scheme to electromagnetic simulation[J]. Journal of Computation Physics, 2007.(accepted and Available online at www.sciencedirect.com)
[6]Huang. Zhi-xiang, Sha. Wei, Wu. Xian-liang, Chen. Ming-shen. A novel high-order time-domain scheme for three-dimensional Maxwell's equations[J]. Microwave and Optical Technology Letters.,2006,48 (6): 1123-1125.
[7]Jian Ming Jin. The Finite Element Method in Electromagnetics[M]. New York: John Wiley & Sons, 1993.
[2]K. S. Yee. Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media[J]. IEEE Transactions on Antennas and Propagation., 1966, 14: 302-307.
[3]Karl S. Kunz, Raymond J. Luebbers. The Finite Difference Time Domain Method for Electromagnetics[M]. CRC Press: Boca Raton, FL, 1993.
[4]Allen Taflove.Computational Electrodynamics:The Finite-Difference Time-Domain Method[M]. Boston, MA: Artech House, 1995.
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