时域积分方程MOT算法的推迟位时间卷积数值积分新方法
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摘要
通过变量代换平滑三角形上推迟位(标量位函数和矢量位函数)并消除推迟矢量位旋度的奇异性,使得采用数值积分法就能够精确快速地计算任意正则时间基函数与推迟位函数及推迟矢量位旋度之间的时间卷积运算,可用于基于任意类型时间基函数的时域电场、时域磁场及其混合场积分方程时间步进(MOT)算法.与时间卷积运算的解析法对比分析表明,该时间卷积数值积分方法能够精确快速地计算基于任意类型时间基函数和不同时间步长条件下时域积分方程MOT算法的阻抗矩阵元素;而具体的计算实例也表明,阻抗矩阵的精确计算显著地提升了时域积分方程MOT算法的后时稳定性和求解精度.
A novel variable transformation is presented to smooth and eliminate the singularity of the retarded potential( scalar and vector potential) and the curl of the vector potential by variable substitution. So the convolution betw een any regular time basis function and retarded potential( or its curl) can be calculated quickly and accurately using the numerical integration method,the advantage is that it can be used in the M OT algorithm of the time-domain field integral equations,no matter how the time basis functions are. Compared to the analytical time convolution method,this numerical integration method can accurately and quickly calculate the impedance matrix elements of M OT algorithm w ith any type of time basis functions and different time-step,and as several numerical results w ill demonstrate,this novel numerical method can largely improve the accuracy and the stability of the M OT algorithm.
A novel variable transformation is presented to smooth and eliminate the singularity of the retarded potential( scalar and vector potential) and the curl of the vector potential by variable substitution. So the convolution betw een any regular time basis function and retarded potential( or its curl) can be calculated quickly and accurately using the numerical integration method,the advantage is that it can be used in the M OT algorithm of the time-domain field integral equations,no matter how the time basis functions are. Compared to the analytical time convolution method,this numerical integration method can accurately and quickly calculate the impedance matrix elements of M OT algorithm w ith any type of time basis functions and different time-step,and as several numerical results w ill demonstrate,this novel numerical method can largely improve the accuracy and the stability of the M OT algorithm.
引文
[1]Shanker B,Ergin A A,et al.Fast analysis of transient electromagnetic scattering phenomena using the multilevel plane w ave time domain algorithm[J].IEEE Transactions on Antennas Propagation,2003,51(3):628-641.
[2]Yilmaz A E,Jin Jian-Ming,Eric Michielssen.Time domain adaptive integral method for surface integral equations[J].IEEE Transactions on Antennas and Propagation,2004,52(10):2692-2708.
[3]任仪,赵延文,聂在平,马文敏.基于高阶叠层矢量基函数的时域电磁场积分方程方法[J].电子学报,2008,36(3):516-519.Ren Yi,Zhao Yanw en,Nie Zaiping,M a Wenmin.Time-domain integral equations using higher order hierarchical vector basis functions[J].Acta Electronica Sinica,2008,36(3):516-519.(in Chinese)
[4]Rao S M,Wilton D R.Transient scattering by conducting surfaces of arbitrary shape[J].IEEE Transactions on Antennas and Propagation,1991,39(1):56-61.
[5]Vechinski D,Rao S M.A stable procedure to calculate the transient scattering by conducting surfaces of arbitrary shape[J].IEEE Transactions on Antennas and Propagation,1992,40(6):661-665.
[6]Davies P J,Duncan D B.Averaging techniques for timemarching schemes for retarded potential integral equations[J].Applied Numerical Mathematics,1997,23(3):291-310.
[7]Dodson S J,Walker S P,Bluck M J.Implicit and stability of time domain integral equation scattering analysis[J].Applied Computational Electromagnetics Society Journal,1997,13(1):291-301.
[8]Manara G,et al.A space-time discretization criterion for a stable time-marching solution of the electric field integral equation[J].IEEE Transactions on Antennas and Propagation,1997,45(3):527-532.
[9]Weile D S,Pisharody G,Chen N W,Shanker B,Michielssen E.A novel scheme for the solution of the time-domain integral equations of electromagnetics[J].IEEE Transactions on Antennas and Propagation,2004,52(1):283-295.
[10]Hu J L,Chan C H,Xu Y.A new temporal basis function for the time-domain integral equation method[J].IEEE M icrow ave Wireless Components Letters,2001,11(1):465-466.
[11]Wang P,Xia M Y,Jin J M,Zhou L Z.Time-domain integral equation solvers using quadratic B-spline temperoral basis functions[J].M icrow ave Optical Technology Letter,2007,49(5):1154-1159.
[12]Pingenot J,chakraborty S,Jandhyala V.Polar integration for exact space-time quadrature in time-domain integral equations[J].IEEE Transactions on Antennas and Propagation,2006,54(10):3037-3042.
[13]Shanker B,Lu M,Michielssen E.Time domain integral equation analysis of scattering from composite bodies via exact evaluation of radiation fields[J].IEEE Transactions on Antennas and Propagation,2009,57(5):1506-1520.
[14]Shi Y F,Xia M Y,Chen R S,Michielssen E,Lu Mingyu.Stable electric field TDIE solvers via quasi-exact evaluation of M OT matrix elements[J].IEEE Transactions on Antennas and Propagation,2011,59(2):574-585.
[15]Yücel A C,Ergin A A.Exact evaluation of retarded-time potential integrals for the RWG bases[J].IEEE Transactions on Antennas and Propagation,2006,54(5):1496-1502.
[16]lküH A,Ergin A A.Analytical evaluation of transient magnetic fields due to RWG current bases[J].IEEE Transactions on Antennas and Propagation,2007,55(12):3565-3575.
[17]赵庆广,赵延文,毕海燕,聂在平.利用时间步进算法精确稳定求解时域积分方程[J].电子学报,2008,36(6):1135-1139.Zhao Qingguang,Zhao Yanw en,Bi Haiyan,Nie Zaiping.Accurate and stable solution of time-domain integral equation using marching on in time method[J].Acta Electronica Sinica,2008,36(6):1135-1139.(in Chinese)
[18]lküH A,Ergin A A.Application of analytical retardedtime potential expressions to the solution of time domain integrals equations[J].IEEE Transactions on Antennas and Propagation,2011,59(11):4123-4131
[19]lküH A,Ergin A A.On the singularity of the closedform expression of the magnetic field in time domain[J].IEEE Transactions on Antennas and Propagation,2011,59(2):691-694.
[20]Pray A J,Nair N V,Shanker B.Stability properties of the time domain electric field integral equation using a separable approximation for the convolution w ith the retarded potential[J].IEEE Transactions on Antennas and Propagation,2012,60(8):3772-3781.
[21]Zhu M D,Zhou X L,Yin W Y.Radial integration scheme for handling w eakly singular and near-singular potential integrals[J].IEEE Antennas and Wireless Propagation Letters,2011,10(1):792-795.
[22]Rao S M,Wilton D R,Glisson A W.Electromagnetic scattering by surfaces of arbitrary shape[J].IEEE Transactions on Antennas and Propagation,1982,30(3):408-418.
[23]Ma J,Rokhlin V,Wandzura S.Generalized Gaussian quadrature rules for systems of arbitrary functions[J].SIAM Journal on Numerical Analysis,1996,33(3):971-996.
[2]Yilmaz A E,Jin Jian-Ming,Eric Michielssen.Time domain adaptive integral method for surface integral equations[J].IEEE Transactions on Antennas and Propagation,2004,52(10):2692-2708.
[3]任仪,赵延文,聂在平,马文敏.基于高阶叠层矢量基函数的时域电磁场积分方程方法[J].电子学报,2008,36(3):516-519.Ren Yi,Zhao Yanw en,Nie Zaiping,M a Wenmin.Time-domain integral equations using higher order hierarchical vector basis functions[J].Acta Electronica Sinica,2008,36(3):516-519.(in Chinese)
[4]Rao S M,Wilton D R.Transient scattering by conducting surfaces of arbitrary shape[J].IEEE Transactions on Antennas and Propagation,1991,39(1):56-61.
[5]Vechinski D,Rao S M.A stable procedure to calculate the transient scattering by conducting surfaces of arbitrary shape[J].IEEE Transactions on Antennas and Propagation,1992,40(6):661-665.
[6]Davies P J,Duncan D B.Averaging techniques for timemarching schemes for retarded potential integral equations[J].Applied Numerical Mathematics,1997,23(3):291-310.
[7]Dodson S J,Walker S P,Bluck M J.Implicit and stability of time domain integral equation scattering analysis[J].Applied Computational Electromagnetics Society Journal,1997,13(1):291-301.
[8]Manara G,et al.A space-time discretization criterion for a stable time-marching solution of the electric field integral equation[J].IEEE Transactions on Antennas and Propagation,1997,45(3):527-532.
[9]Weile D S,Pisharody G,Chen N W,Shanker B,Michielssen E.A novel scheme for the solution of the time-domain integral equations of electromagnetics[J].IEEE Transactions on Antennas and Propagation,2004,52(1):283-295.
[10]Hu J L,Chan C H,Xu Y.A new temporal basis function for the time-domain integral equation method[J].IEEE M icrow ave Wireless Components Letters,2001,11(1):465-466.
[11]Wang P,Xia M Y,Jin J M,Zhou L Z.Time-domain integral equation solvers using quadratic B-spline temperoral basis functions[J].M icrow ave Optical Technology Letter,2007,49(5):1154-1159.
[12]Pingenot J,chakraborty S,Jandhyala V.Polar integration for exact space-time quadrature in time-domain integral equations[J].IEEE Transactions on Antennas and Propagation,2006,54(10):3037-3042.
[13]Shanker B,Lu M,Michielssen E.Time domain integral equation analysis of scattering from composite bodies via exact evaluation of radiation fields[J].IEEE Transactions on Antennas and Propagation,2009,57(5):1506-1520.
[14]Shi Y F,Xia M Y,Chen R S,Michielssen E,Lu Mingyu.Stable electric field TDIE solvers via quasi-exact evaluation of M OT matrix elements[J].IEEE Transactions on Antennas and Propagation,2011,59(2):574-585.
[15]Yücel A C,Ergin A A.Exact evaluation of retarded-time potential integrals for the RWG bases[J].IEEE Transactions on Antennas and Propagation,2006,54(5):1496-1502.
[16]lküH A,Ergin A A.Analytical evaluation of transient magnetic fields due to RWG current bases[J].IEEE Transactions on Antennas and Propagation,2007,55(12):3565-3575.
[17]赵庆广,赵延文,毕海燕,聂在平.利用时间步进算法精确稳定求解时域积分方程[J].电子学报,2008,36(6):1135-1139.Zhao Qingguang,Zhao Yanw en,Bi Haiyan,Nie Zaiping.Accurate and stable solution of time-domain integral equation using marching on in time method[J].Acta Electronica Sinica,2008,36(6):1135-1139.(in Chinese)
[18]lküH A,Ergin A A.Application of analytical retardedtime potential expressions to the solution of time domain integrals equations[J].IEEE Transactions on Antennas and Propagation,2011,59(11):4123-4131
[19]lküH A,Ergin A A.On the singularity of the closedform expression of the magnetic field in time domain[J].IEEE Transactions on Antennas and Propagation,2011,59(2):691-694.
[20]Pray A J,Nair N V,Shanker B.Stability properties of the time domain electric field integral equation using a separable approximation for the convolution w ith the retarded potential[J].IEEE Transactions on Antennas and Propagation,2012,60(8):3772-3781.
[21]Zhu M D,Zhou X L,Yin W Y.Radial integration scheme for handling w eakly singular and near-singular potential integrals[J].IEEE Antennas and Wireless Propagation Letters,2011,10(1):792-795.
[22]Rao S M,Wilton D R,Glisson A W.Electromagnetic scattering by surfaces of arbitrary shape[J].IEEE Transactions on Antennas and Propagation,1982,30(3):408-418.
[23]Ma J,Rokhlin V,Wandzura S.Generalized Gaussian quadrature rules for systems of arbitrary functions[J].SIAM Journal on Numerical Analysis,1996,33(3):971-996.