A High-Order Hybrid FDTD-PITD Method of Electromagnetic Waves for Embedded Thin Conductive Layers
Ma Liang1, Ma Xikui1, Chi Mingjun1, Xiang Ru1, Zhu Xiaojie2
1. State Key Laboratory of Electrical Insulation and Power Equipment Xi'an JiaotongUniversity Xi'an 710049 China; 2. Dipartimento di Elettronica Informazione e Bioingegneria Politecnico di Milano Milan 20133 Italy
Abstract：For wideband characteristic analysis, frequency-domain numerical methods need to calculate complicated equations at each frequency point repeatedly. However, time-domain numerical methods only need to perform the Fourier transform on time-domain results of one calculation to obtain the wideband frequency-domain information. The finite-difference time-domain (FDTD) method is the most widely used time-domain method, but its time step size is limited by the Courant-Friedrichs-Lewy (CFL) stability condition. To overcome this shortcoming, the precise-integration time-domain (PITD) method is proposed to be free from the CFL restriction by discretizing the spatial derivatives with difference and solving the ordinary differential equations about time by using the precise integration technique, designated as the second-order PITD [PITD(2)]. For the improvement of the numerical dispersion characteristics, the fourth-order PITD [PITD(4)] method is proposed by using the fourth-order spatial central difference scheme. The difference in memory requirements between PITD(2) and PITD(4), merely reflected in the size and sparsity of the matrix exponential, is not significant. Thin conductive layers, as typical multiscale problems, exist widely in electromagnetic (EM) systems for shielding, posing challenges to any single numerical algorithm. For such problems, the cell size of FDTD must be sufficiently small to capture the thickness and skin effect of the thin conductive layer, requiring significant resources. To describe their EM properties more effectively, the subgridding technology is introduced to employ high-density sampling only for the local thin layer and coarse-grid divisions for the remaining areas. Since the waves inside the good conductor propagate nearly perpendicular to its surfaces, the thin conductive layer is divided into fine grids only along the vertical direction and its internal EM fields can be found by one-dimensional (1-D) full-wave simulation. To relax the CFL restriction of fine grids, PITD can be used to synchronize the time step size for fine grids with that for coarse grids. Meanwhile, 1-D simulation can greatly reduce the memory requirement of PITD. Compared with PITD(2), PITD(4) has higher accuracy without additional memory burden. Therefore, the high-order hybrid FDTD-PITD method is proposed to model the embedded thin conductive layer, which is a synergetic combination of FDTD and 1-D PITD(4). FDTD is adopted for coarse grids outside the thin conductive layer, while 1-D PITD(4) is used for fine grids inside the thin conductive layer. Conventional FDTD is used to update the EM fields outside the thin conductive layer, and the magnetic fields adjacent to the boundaries of the thin conductive layer need to be specially treated by using the boundary electric fields obtained by 1-D PITD(4) inside the thin conductive layer. For the PITD domain, the transition region is introduced between the high-order region and the interface of different grids. The fourth-order spatial central difference is used for the high-order region and the second-order for the transition region. The thickness of the transition region is selected following the principle that all unknowns used to solve the high-order region are located inside the thin conductive layer. The tangential electric field components at the interface are updated by using effective permittivity and conductivity, thereby connecting coarse and fine grids. Finally, the numerical stability and numerical reflection of the hybrid method are analyzed, and several canonical numerical examples are presented to verify the effectiveness and accuracy of the proposed method.
[1] 赵志刚, 张学增. LLC平面变压器绕组损耗与漏感改进有限元计算方法[J]. 电工技术学报, 2022, 37(24): 6204-6215. Zhao Zhigang, Zhang Xuezeng.Improved finite element method of winding loss and leakage inductance for planar transformer used in LLC converter[J]. Transactions of China Electrotechnical Society, 2022, 37(24): 6204-6215. [2] 魏鹏, 陈龙, 贲彤, 等. 一种考虑动态磁滞效应的高效稳定时域有限元计算方法[J]. 电工技术学报, 2023, 38(21): 5661-5672. Wei Peng, Chen Long, Ben Tong, et al.An efficient and stable time domain finite element method considering dynamic hysteresis effect[J]. Transactions of China Electrotechnical Society, 2023, 38(21): 5661-5672. [3] 胡亚楠, 包家立, 朱金俊, 等. 纳秒电脉冲对肝脏组织不可逆电穿孔消融区分布的时域有限差分法仿真[J]. 电工技术学报, 2021, 36(18): 3841-3850. Hu Yanan, Bao Jiali, Zhu Jinjun, et al.The finite difference time domain simulation of the distribution of irreversible electroporation ablation area in liver tissue by nanosecond electrical pulse[J]. Transactions of China Electrotechnical Society, 2021, 36(18): 3841-3850. [4] 迟明珺, 马西奎, 马亮, 等. 一种适用于有源区域电磁波时程精细积分仿真的大时间步长矩阵数值积分方法[J]. 电工技术学报, 2024, 39(21): 6604-6613+6895. Chi Mingjun, Ma Xikui, Ma Liang, et al. A large Time step size matrix numerical integral method for precise integration time domain simulation of electromagnetic waves in active regions [J]. Transactions of China Electrotechnical Society, 2024, 39(21): 6604-6613+6895. [5] Kang Zhen, Ma Xikui, Yang Fang, et al.Stability condition and numerical dispersion of fourth-order compact precise-integration time-domain method[J]. IEEE Transactions on Magnetics, 2019, 55(6): 7200804. [6] 孟雪松, 张瀚, 鲍献丰, 等. 碳纤维增强复合材料薄层高效建模方法研究[J]. 电波科学学报, 2019, 34(1): 19-26. Meng Xuesong, Zhang Han, Bao Xianfeng, et al.High efficient modeling techniques of carbon fiber reinforced composite thin layer[J]. Chinese Journal of Radio Science, 2019, 34(1): 19-26. [7] Wang Yong, Langdon S.A time-domain method for analyzing arbitrarily shaped thin resistive sheets[J]. IEEE Antennas and Wireless Propagation Letters, 2019, 18(7): 1424-1428. [8] Shao Jinghui, Ma Xikui, Yin Shuli, et al.Conformal precise-integration time-domain method for analyzing electromagnetic fields in fine-structured device with moving 3-D part[J]. IEEE Transactions on Microwave Theory and Techniques, 2018, 66(7): 3200-3209. [9] Lei Qi, Wang Jianbao, Shi Lihua.Shielding analysis of uniaxial anisotropic multilayer structure by impedance network boundary condition method[J]. IEEE Transactions on Antennas and Propagation, 2019, 67(4): 2462-2469. [10] Wang Yuhui, Deng Langran, Liu Hanhong, et al.Toward the 2-D stable FDTD subgridding method with SBP-SAT and arbitrary grid ratio[J]. IEEE Transactions on Microwave Theory and Techniques, 2024, 72(3): 1591-1605. [11] Ruiz Cabello M, Diaz Angulo L, Alvarez J, et al.A hybrid crank-Nicolson FDTD subgridding boundary condition for lossy thin-layer modeling[J]. IEEE Transactions on Microwave Theory and Techniques, 2017, 65(5): 1397-1406. [12] Van Londersele A, De Zutter D, VandeGinste D. A new hybrid implicit-explicit FDTD method for local subgridding in multiscale 2-D TE scattering problems[J]. IEEE Transactions on Antennas and Propagation, 2016, 64(8): 3509-3520. [13] 康祯, 杨方, 张瑞祥, 等. 用于电磁波多尺度问题求解的高效FDTD-PITD混合算法[J]. 微波学报, 2022, 38(3): 46-52. Kang Zhen, Yang Fang, Zhang Ruixiang, et al.An efficient hybrid FDTD-PITD formulation for solving multiscale electromagnetic wave problems[J]. Journal of Microwaves, 2022, 38(3): 46-52. [14] Chi Mingjun, Ma Xikui, Ma Liang.A switched-Huygens-subgridding-based combined FDTD-PITD method for fine structures[J]. IEEE Microwave and Wireless Technology Letters, 2023, 33(7): 947-950. [15] Ma Liang, Ma Xikui, Chi Mingjun, et al.A nonoverlapping hybrid FDTD-PITD method using upwind conditions for multiscale problems[J]. IEEE Transactions on Microwave Theory and Techniques, 2024, 72(5): 2961-2972. [16] Ma Liang, Ma Xikui, Chi Mingjun, et al.Toward the modeling of thin conductive layer with hybrid FDTD-PITD method[J]. IEEE Transactions on Magnetics, 2024, 60(12): 1-4. [17] Zhou Longjian, Yang Feng, Long Rui, et al.A hybrid method of higher-order FDTD and subgriddingtechnique[J]. IEEE Antennas and Wireless Propagation Letters, 2016, 15: 1261-1264. [18] Xiang Ru, Ma Xikui, Ma Liang, et al.A stable discontinuous Galerkin time-domain method with implicit-explicit time-marching for lossy media[J]. IEEE Transactions on Magnetics, 2024, 60(12): 1-4.
2025, Vol. 40 
