时域积分方程法分析导体的瞬态散射和辐射问题
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摘要
相对于频域方法而言，时域积分方程法有这许多优势，因此，它是计算电磁学数值方法中一个重要的方法。本文全面系统地分析讨论了时域积分方程法。
本文中，基于电场方程形式的时域表面积分方程用来分析任意形状理想导体结构的瞬态散射和辐射问题。一种高斯平面波，或者时变的高斯电压源，被作为入射场或是激励源，用来分析它们的冲击响应。首先，对于时域电场积分方程，通过时间步进法(MOT，marching-on-in-time)，即空间变量部分用矩量法(MoM)离散，时间变量部分则采用差分法，求解任意形状导体结构的瞬态散射问题，其中，显式解法和隐式解法分别都被采用进行计算分析。但是，为了避免时间后期的振荡发散现象，时间步进法的时间步长的选取要受到Courant稳定条件的限制。然后，本文分析了使用基于Laguerre多项式的阶数步进法(MOO，marching-on-in-order)，它在求解时域电场积分方程时，可以将时间变量从中分离出来。这样，这种方法可以克服时间步进法的晚时不稳定的缺点，用于分析任意形状导体结构的瞬态散射问题，可以得到精确稳定的结果。此外，本文还分析了对原算法的改进，引入中心有限差分式的方案。考虑到阶数步进法的无条件稳定的特性，时域电场积分方程的阶数步进解法还在本文中用于分析一些导体天线的瞬态辐射问题。
本文推导分析了详细的算法过程，计算分析了一些典型的导体结构并且算法之间还做了比较分析。
The time domain integral equation (TDIE) method is one of the important methods in electromagnetic numerical simulations for several advantages over conventional method in frequency domain. In this thesis, the TDIE methodology has been studied and disused systematically.
A time domain surface integral equation based on the electric field formulation is utilized to calculation the transient scattering and radiation form arbitrary conducting structures in the thesis. The Gaussian plane wave or time-dependent Gaussian source voltage is used as an incident field or impressed generator to calculate impulse response. Firstly, the solution of the time domain electric flied integral equation (TD-EFIE) is based on the method of moments (MoM) and using the marching-on-in-time (MOT) technique, in which both explicit solution scheme and implicit solution scheme are used, to calculate the scattering form arbitrary shaped conducting structure. However, the time step in MOT is limited by the Courant condition to avoid the late-time oscillation. Secondly, by the marching-on-in-order (MOO) algorithm with Laguerre polynomials, time domain electric field integral equation can be solved without the time variables. Consequently this method, which can overcome the late-time instabilities in the MOT algorithm, is proposed to analysis the transient scattering from arbitrary shaped conducting structure to obtain the accurate and stable results. In addition, an improvement for the MOO methodology, using the central finite difference, is also presented in this thesis. Furthermore, considering the unconditional stable ability of the MOO method, TD-EFIE with MOO scheme is also used to calculate the transient radiation of some conducting antennas.
Detailed mathematical steps are included, and several representative numerical results are presented and compared in the thesis.
本文中，基于电场方程形式的时域表面积分方程用来分析任意形状理想导体结构的瞬态散射和辐射问题。一种高斯平面波，或者时变的高斯电压源，被作为入射场或是激励源，用来分析它们的冲击响应。首先，对于时域电场积分方程，通过时间步进法(MOT，marching-on-in-time)，即空间变量部分用矩量法(MoM)离散，时间变量部分则采用差分法，求解任意形状导体结构的瞬态散射问题，其中，显式解法和隐式解法分别都被采用进行计算分析。但是，为了避免时间后期的振荡发散现象，时间步进法的时间步长的选取要受到Courant稳定条件的限制。然后，本文分析了使用基于Laguerre多项式的阶数步进法(MOO，marching-on-in-order)，它在求解时域电场积分方程时，可以将时间变量从中分离出来。这样，这种方法可以克服时间步进法的晚时不稳定的缺点，用于分析任意形状导体结构的瞬态散射问题，可以得到精确稳定的结果。此外，本文还分析了对原算法的改进，引入中心有限差分式的方案。考虑到阶数步进法的无条件稳定的特性，时域电场积分方程的阶数步进解法还在本文中用于分析一些导体天线的瞬态辐射问题。
本文推导分析了详细的算法过程，计算分析了一些典型的导体结构并且算法之间还做了比较分析。
The time domain integral equation (TDIE) method is one of the important methods in electromagnetic numerical simulations for several advantages over conventional method in frequency domain. In this thesis, the TDIE methodology has been studied and disused systematically.
A time domain surface integral equation based on the electric field formulation is utilized to calculation the transient scattering and radiation form arbitrary conducting structures in the thesis. The Gaussian plane wave or time-dependent Gaussian source voltage is used as an incident field or impressed generator to calculate impulse response. Firstly, the solution of the time domain electric flied integral equation (TD-EFIE) is based on the method of moments (MoM) and using the marching-on-in-time (MOT) technique, in which both explicit solution scheme and implicit solution scheme are used, to calculate the scattering form arbitrary shaped conducting structure. However, the time step in MOT is limited by the Courant condition to avoid the late-time oscillation. Secondly, by the marching-on-in-order (MOO) algorithm with Laguerre polynomials, time domain electric field integral equation can be solved without the time variables. Consequently this method, which can overcome the late-time instabilities in the MOT algorithm, is proposed to analysis the transient scattering from arbitrary shaped conducting structure to obtain the accurate and stable results. In addition, an improvement for the MOO methodology, using the central finite difference, is also presented in this thesis. Furthermore, considering the unconditional stable ability of the MOO method, TD-EFIE with MOO scheme is also used to calculate the transient radiation of some conducting antennas.
Detailed mathematical steps are included, and several representative numerical results are presented and compared in the thesis.
引文
1．盛剑霓．电磁散射电磁场数值分析．北京：科学出版社，1984．
2．王长清．现代计算电磁学基础．北京：北京大学出版社，2005．
3．文舸一．电磁理论的新进展．北京：国防工业出版社，1999．
4. K. S. Yee. Numerical Solution of Initial Boundary Value Problems Involving Maxwell's Equations in Isotropic Media. IEEE Transactions on Antennas and Propagation, 1966, 14(3): 302-307.
5. R. F. Harrington. Field Computation by Moment Methods. New York: The Macmillan Company, 1968.
6．倪光正，杨仕友等．工程电磁场数值计算．北京：机械工业出版社，2004．
7．刘圣民．电磁场的数值方法．武昌：华中理工大学出版社，1991．
8. C. L. Bennett, W. L. Weeks. Electromagnetic Pulse Response of Cylindrical Scatters. IEEE G-AP International Symposium. Boston: Northeastern University, 1968: 176-183.
9. K. M. Mitzner. Numerical Solution from a Hard Surface of Arbitrary-Retarded Potential Technique. Journal of Acoustic Society of America, 1967, 42: 391-397.
10. R. P. Shaw. Diffraction of Acoustic Pluses by Obstacles of Arbitrary Shape with a Robin Boundary Condition. Journal of Acoustic Society of America, 1967, 41: 855-859.
11. S. M. Rao, D. R. Wilton. Transient Scattering by Conducting Surfaces of Arbitrary Shape. IEEE Transactions on Antennas and Propagation, 1991, 39(1): 56-61.
12. D. A. Vechinski, S. M. Rao. A Stable Procedure to Calculate the Transient Scattering by Conducting Surfaces of Arbitrary. IEEE Transactions on Antennas and Propagation, 1992, 40(6): 661-665.
13. P. J. Davies. On the Stability of Time-Marching Schemes for the General Surface Electric-Field Integral Equation. IEEE Transactions on Antennas and Propagation, 1996, 44(11): 1467-1473.
14. S. M. Rao, T. K. Sarkar. Time-Domain Modeling of Two-Dimensional Conducting Cylinders Utilizing an Implicit Scheme-TM Incidence. Microwave and Optical Technology Letters, 1997, 15(6): 342-347.
15. G. Manara, A. Monorchio, et al. A Space-Time Discretization Criterion for a stable Time-Marching Solution of the Electric Field Integral Equation. IEEE Transactions on Antennas and Propagation, 1997, 45(3): 527-532.
16. H. Mieras, C. L. Bennett. Space-Time Integral Equation Approach to Dielectric Targets. IEEE Transactions on Antennas and Propagation, 1982, 30(1): 2-9.
17. A. G Tijhuis. Towards a Stable Marching-on-in-Time Method for Two Dimensional Transient Electromagnetic Scattering Problems. Radio Science, 1984, 19: 1311-1317.
18. D. A. Vechinski, S. M. Rao. Transient Scattering from Two-Dimensional Dielectric Cylinders of Arbitrary Shape. IEEE Transactions on Antennas and Propagation, 1992, 40(9): 1054-1060.
19. A. A. Ergin, B. Shanker, and E. Michielssen. The plane wa(?)e time domain algorithm for fast analysis of transient wave Phenomena. IEEE Antennas and Propagation Magazine, 1999, 41(4): 39-51.
20. B. Shanker, A. A. Ergin, K. Aygun, E. Michielssen. Analysis of transient electromagnetic scattering phenomena using a two-level plane wave time domain algorithm. IEEE Transactions on Antennas and Propagation, 2000, 48(4): 510-523.
21. B. Shanker, A. A. Ergin, M. Y. Lu, E. Michielssen. Fast Analysis of Transient Electromagnetic Scattering Phenomena Using the Multilevel Plane Wave Time Domain Algorithm. IEEE Transactions on Antennas and Propagation, 2003, 51(3): 628-641.
22. A. E. Yilmaz, J. M. Jin, E. Michielssen. Time Domain Adaptive Integral Method for Surface Integral Equations. IEEE Transactions on Antennas and Propagation, 2004, 52(10): 2692-2708.
23. A. E. Yilmaz, J. M. Jin, E. Michielssen. A Fast Fourier Transform Accelerated Marching-on-in-Time Algorithm for Electromagnetic Analysis. Electromagnetics, 2001,21:181-197.
24. R. S. Adve, T. K. Sarkar, O. M. Pereira-Filho, S. M. Rao. Extrapolation of time domain responses from three dimensional conducting objects utilizing the matrix pencil technique. IEEE Transactions on Antennas and Propagation, 1997, 45(1): 147-156.
25. M. M. Rao, T. K. Sarkar, T. Anjali, R. S. Adve, Simultaneous extrapolation in time and frequency domains using Hermite expansion. IEEE Transactions on Antennas and Propagation, 1999,47(6): 1108-1115.
26. T. K. Sarkar, J. Koh. Generation of a Wide-Band Electromagnetic Response through a Laguerre Expansion Using Early-Time and Low-Frequency Data. IEEE Transactions on Antennas and Propagation, 2002, 50(5): 1408-1416.
27. B. H. Jung, Y. S. Chung, T. K. Sarkar. Time-Domain EFIE, MFIE, and CFIE Formulations using Laguerre Polynomials as Temporal Basis Functions for the analysis of Transient Scattering from Arbitrarily Shaped Conducting Structures. Progress in Electromagnetics Research, 2003, 39: 1-45.
28. Y. S. Chung, T. K. Sarkar, B. H. Jung, et al. Solution of Time Domain Electric Field Integral Equation Using the Laguerre Polynomials. IEEE Transactions on Microwave Theory and Techniques, 2004, 52: 2319-2328.
29. Z. Ji, T. K. Sarkar, B. H. Jung, et al. Solving Time Domain Electric Field Integral Equation Without the Time Variable. IEEE Transactions on Antennas and Propagation, 2006, 54(1): 258-262.
30．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．
31. S. M. Rao. Time Domain Electromagnetics. New York: Academic Press, 1999.
32. B. H. Jung, T. K. Sarkar, Y. S. Chung, Z. Ji. An Accurate and Stable Implicit Solution for Transient Scattering and Radiation from Wire Structure. Microwave and Optical Technology Letter, 2002, 34(5): 354-359.
33. S. M. Rao, D. R. Wilton, A. W. Glisson. Electromagnetic scattering by surface of arbitrary shape. IEEE Transactions on Antennas and Propagation, 1982, 30(3): 409-411.
34. A. Sadigh, E. Arvas. Treating the Instabilities in Marching-on-in-Time Method from a Different Perspective. IEEE Transactions on Antennas and Propagation, 1993, 41(12): 1695-1702.
35. B. P. Rynne, P. D. Smith. Stability of time marching algorithms for the electric field integral equation. Journal of Electromagnetic Waves and Applications, 1990, 4: 1181-1205.
36. W. W. Lee. Transient scattering from arbitrarily shaped composite conducting and dielectric structures. Ph. D. Thesis, Syracuse University, Syracuse, NY, 2001.
37. Y. Saad. Iterative Methods for Sparse Linear Systems. Boston: PWS Publishing Company, 1996.
38. B. Shanker, A. A. Ergin, K. Aygun, E. Michielssen. Analysis of Transient Electromagnetic Scattering form Close surface using a Combined Field Integral Equation. IEEE Transactions on Antennas and Propagation, 2000, 48(7): 1064-1074.
39. I. S. Gradshteyn, I. M. Ryzhik. Table of Integrals, Series, and Products. New York: Academic Press, 1980.
40. P. Arcioni, M. Bressan, L. Perregrini. On the Evaluation of the Double Surface Integrals arising in the Application of the Boundary Integral Method to 3-D Problems. IEEE Transactions on Microwave Theory and Techniques, 1997, 45(3): 436-439.
41. Y. S. Chung, T. K. Sarkar, B. H. Jung, et al. Solution of Time Domain Electric Field Integral Equation Using the Laguerre Polynomials. IEEE Transactions on Antennas and Propagation, 2004, 52(9): 2319-2328.
42. Z. Ji, T. K. Sarkar, B. H. Jung, M. S. Palma, M. T. Yuan. A Comparison of Marching-on-in-Time Method with Marching-on-in Degree Method for the TD-EFIE Solver. IEEE/ACES International Conference, 2005: 297-300.
43. S. Makarov. MoM Antenna Simulation with Matlab: RWG Basic Functions. IEEE Antennas and Propagation Magazine, 2001, 43(5): 100-107.
44. Z. Ji, T. K. Sarkar, B. H. Jung, M. S. Palma, M. T. Yuan. A Stable Solution of Time Domain Electric Field Integral Equation for Thin-Wire Antennas Using the Laguerre Polynomials. IEEE Transactions on Antennas and Propagation, 2004, 52(10): 2641-2649.
2．王长清．现代计算电磁学基础．北京：北京大学出版社，2005．
3．文舸一．电磁理论的新进展．北京：国防工业出版社，1999．
4. K. S. Yee. Numerical Solution of Initial Boundary Value Problems Involving Maxwell's Equations in Isotropic Media. IEEE Transactions on Antennas and Propagation, 1966, 14(3): 302-307.
5. R. F. Harrington. Field Computation by Moment Methods. New York: The Macmillan Company, 1968.
6．倪光正，杨仕友等．工程电磁场数值计算．北京：机械工业出版社，2004．
7．刘圣民．电磁场的数值方法．武昌：华中理工大学出版社，1991．
8. C. L. Bennett, W. L. Weeks. Electromagnetic Pulse Response of Cylindrical Scatters. IEEE G-AP International Symposium. Boston: Northeastern University, 1968: 176-183.
9. K. M. Mitzner. Numerical Solution from a Hard Surface of Arbitrary-Retarded Potential Technique. Journal of Acoustic Society of America, 1967, 42: 391-397.
10. R. P. Shaw. Diffraction of Acoustic Pluses by Obstacles of Arbitrary Shape with a Robin Boundary Condition. Journal of Acoustic Society of America, 1967, 41: 855-859.
11. S. M. Rao, D. R. Wilton. Transient Scattering by Conducting Surfaces of Arbitrary Shape. IEEE Transactions on Antennas and Propagation, 1991, 39(1): 56-61.
12. D. A. Vechinski, S. M. Rao. A Stable Procedure to Calculate the Transient Scattering by Conducting Surfaces of Arbitrary. IEEE Transactions on Antennas and Propagation, 1992, 40(6): 661-665.
13. P. J. Davies. On the Stability of Time-Marching Schemes for the General Surface Electric-Field Integral Equation. IEEE Transactions on Antennas and Propagation, 1996, 44(11): 1467-1473.
14. S. M. Rao, T. K. Sarkar. Time-Domain Modeling of Two-Dimensional Conducting Cylinders Utilizing an Implicit Scheme-TM Incidence. Microwave and Optical Technology Letters, 1997, 15(6): 342-347.
15. G. Manara, A. Monorchio, et al. A Space-Time Discretization Criterion for a stable Time-Marching Solution of the Electric Field Integral Equation. IEEE Transactions on Antennas and Propagation, 1997, 45(3): 527-532.
16. H. Mieras, C. L. Bennett. Space-Time Integral Equation Approach to Dielectric Targets. IEEE Transactions on Antennas and Propagation, 1982, 30(1): 2-9.
17. A. G Tijhuis. Towards a Stable Marching-on-in-Time Method for Two Dimensional Transient Electromagnetic Scattering Problems. Radio Science, 1984, 19: 1311-1317.
18. D. A. Vechinski, S. M. Rao. Transient Scattering from Two-Dimensional Dielectric Cylinders of Arbitrary Shape. IEEE Transactions on Antennas and Propagation, 1992, 40(9): 1054-1060.
19. A. A. Ergin, B. Shanker, and E. Michielssen. The plane wa(?)e time domain algorithm for fast analysis of transient wave Phenomena. IEEE Antennas and Propagation Magazine, 1999, 41(4): 39-51.
20. B. Shanker, A. A. Ergin, K. Aygun, E. Michielssen. Analysis of transient electromagnetic scattering phenomena using a two-level plane wave time domain algorithm. IEEE Transactions on Antennas and Propagation, 2000, 48(4): 510-523.
21. B. Shanker, A. A. Ergin, M. Y. Lu, E. Michielssen. Fast Analysis of Transient Electromagnetic Scattering Phenomena Using the Multilevel Plane Wave Time Domain Algorithm. IEEE Transactions on Antennas and Propagation, 2003, 51(3): 628-641.
22. A. E. Yilmaz, J. M. Jin, E. Michielssen. Time Domain Adaptive Integral Method for Surface Integral Equations. IEEE Transactions on Antennas and Propagation, 2004, 52(10): 2692-2708.
23. A. E. Yilmaz, J. M. Jin, E. Michielssen. A Fast Fourier Transform Accelerated Marching-on-in-Time Algorithm for Electromagnetic Analysis. Electromagnetics, 2001,21:181-197.
24. R. S. Adve, T. K. Sarkar, O. M. Pereira-Filho, S. M. Rao. Extrapolation of time domain responses from three dimensional conducting objects utilizing the matrix pencil technique. IEEE Transactions on Antennas and Propagation, 1997, 45(1): 147-156.
25. M. M. Rao, T. K. Sarkar, T. Anjali, R. S. Adve, Simultaneous extrapolation in time and frequency domains using Hermite expansion. IEEE Transactions on Antennas and Propagation, 1999,47(6): 1108-1115.
26. T. K. Sarkar, J. Koh. Generation of a Wide-Band Electromagnetic Response through a Laguerre Expansion Using Early-Time and Low-Frequency Data. IEEE Transactions on Antennas and Propagation, 2002, 50(5): 1408-1416.
27. B. H. Jung, Y. S. Chung, T. K. Sarkar. Time-Domain EFIE, MFIE, and CFIE Formulations using Laguerre Polynomials as Temporal Basis Functions for the analysis of Transient Scattering from Arbitrarily Shaped Conducting Structures. Progress in Electromagnetics Research, 2003, 39: 1-45.
28. Y. S. Chung, T. K. Sarkar, B. H. Jung, et al. Solution of Time Domain Electric Field Integral Equation Using the Laguerre Polynomials. IEEE Transactions on Microwave Theory and Techniques, 2004, 52: 2319-2328.
29. Z. Ji, T. K. Sarkar, B. H. Jung, et al. Solving Time Domain Electric Field Integral Equation Without the Time Variable. IEEE Transactions on Antennas and Propagation, 2006, 54(1): 258-262.
30．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．
31. S. M. Rao. Time Domain Electromagnetics. New York: Academic Press, 1999.
32. B. H. Jung, T. K. Sarkar, Y. S. Chung, Z. Ji. An Accurate and Stable Implicit Solution for Transient Scattering and Radiation from Wire Structure. Microwave and Optical Technology Letter, 2002, 34(5): 354-359.
33. S. M. Rao, D. R. Wilton, A. W. Glisson. Electromagnetic scattering by surface of arbitrary shape. IEEE Transactions on Antennas and Propagation, 1982, 30(3): 409-411.
34. A. Sadigh, E. Arvas. Treating the Instabilities in Marching-on-in-Time Method from a Different Perspective. IEEE Transactions on Antennas and Propagation, 1993, 41(12): 1695-1702.
35. B. P. Rynne, P. D. Smith. Stability of time marching algorithms for the electric field integral equation. Journal of Electromagnetic Waves and Applications, 1990, 4: 1181-1205.
36. W. W. Lee. Transient scattering from arbitrarily shaped composite conducting and dielectric structures. Ph. D. Thesis, Syracuse University, Syracuse, NY, 2001.
37. Y. Saad. Iterative Methods for Sparse Linear Systems. Boston: PWS Publishing Company, 1996.
38. B. Shanker, A. A. Ergin, K. Aygun, E. Michielssen. Analysis of Transient Electromagnetic Scattering form Close surface using a Combined Field Integral Equation. IEEE Transactions on Antennas and Propagation, 2000, 48(7): 1064-1074.
39. I. S. Gradshteyn, I. M. Ryzhik. Table of Integrals, Series, and Products. New York: Academic Press, 1980.
40. P. Arcioni, M. Bressan, L. Perregrini. On the Evaluation of the Double Surface Integrals arising in the Application of the Boundary Integral Method to 3-D Problems. IEEE Transactions on Microwave Theory and Techniques, 1997, 45(3): 436-439.
41. Y. S. Chung, T. K. Sarkar, B. H. Jung, et al. Solution of Time Domain Electric Field Integral Equation Using the Laguerre Polynomials. IEEE Transactions on Antennas and Propagation, 2004, 52(9): 2319-2328.
42. Z. Ji, T. K. Sarkar, B. H. Jung, M. S. Palma, M. T. Yuan. A Comparison of Marching-on-in-Time Method with Marching-on-in Degree Method for the TD-EFIE Solver. IEEE/ACES International Conference, 2005: 297-300.
43. S. Makarov. MoM Antenna Simulation with Matlab: RWG Basic Functions. IEEE Antennas and Propagation Magazine, 2001, 43(5): 100-107.
44. Z. Ji, T. K. Sarkar, B. H. Jung, M. S. Palma, M. T. Yuan. A Stable Solution of Time Domain Electric Field Integral Equation for Thin-Wire Antennas Using the Laguerre Polynomials. IEEE Transactions on Antennas and Propagation, 2004, 52(10): 2641-2649.