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Due to the limitation of the stability condition, the finite-difference time-domain (FDTD) method cannot efficiently deal with electromagnetic problems containing fine structures. The explicit and unconditionally stable (EUS) FDTD method can eliminate the constraint of the stability condition and improve the simulation efficiency of fine structures by filtering out the unstable modes for the system matrix. However, the EUS-FDTD method needs to solve the eigenvalues of the numerical system matrix, and the symmetry of the numerical system matrix needs to be ensured when the subgridding scheme is used to discretize targets containing fine structures. The existing EUS-FDTD subgridding method encounters some problems such as complex implementation and insufficient accuracy. In order to solve the above problems, in this work, the hanging variables subgridding (HVS) algorithm is applied to the EUS-FDTD algorithm. Starting from the symmetry of the system matrix, the stability of the hanging variables subgridding algorithm is proven, and a high-precision, stable, and easy-to-implement HVS-EUS-FDTD scheme is proposed. Numerical examples of the radiation of linear magnetic currents in free space, electromagnetic scattering of multiple dielectric objects, and a three-dimensional cavity containing a medium demonstrate the stability, high accuracy and efficiency of the proposed method. Numerical experiments show that the computational efficiency of the HVS-EUS-FDTD algorithm can be improved hundreds of times compared with that of the uniform fine grid FDTD algorithm, and the highest computational efficiency can be improved up to ratio (the size ratio of coarse grid to fine grid) times compared with that of the HVS-FDTD algorithm.
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Keywords:
- finite-difference time-domain /
- explicit and unconditionally stable /
- hanging variables /
- fine structures
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Google Scholar
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Google Scholar
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Google Scholar
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图 1 悬挂变量亚网格技术
Figure 1. The subgridding technique of hanging variables.
图 2 编号重排后的悬挂变量亚网格技术
Figure 2. The subgridding technique of hanging variables after numbering rearrangement.
图 3 计算模型示意图
Figure 3. The schematic diagram of calculation model.
图 4 直线AB上采样点的磁场强度
Figure 4. Magnetic field intensity of sampling points on line AB
图 5 特征值分布
Figure 5. The distribution of eigenvalues.
图 6 多个介质目标计算模型
Figure 6. Calculation model for multi medium targets.
图 7 悬挂变量亚网格离散模型
Figure 7. Subgridding discrete model with hanging variables.
图 8 监测点时域波形 (a) P1; (b) P2
Figure 8. Time domain waveform of monitoring points: (a) P1; (b) P2.
图 9 大时间步情形Uniform-FDTD与HVS-FDTD算法的时域波形 (a) Uniform-FDTD; (b) HVS-FDTD
Figure 9. Time domain waveforms of Uniform-FDTD and HVS-FDTD algorithms with large time steps: (a) Uniform-FDTD; (b) HVS-FDTD.
图 10 三维含介质PEC腔体计算模型
Figure 10. The computational model for 3-D PEC cavity containing a dielectric.
图 11 监测点时域波形
Figure 11. Time domain waveform of monitoring the point.
表 1 三种数值算法的计算时间比较
Table 1. Comparison of calculation times of three numerical algorithms.
方法 uniform FDTD HVS-FDTD HVS-EUS-FDTD 时间/s 2295.5 17.45 0.05+5.3 时间步长/(10–11 s) 1.67 1.67 8.33 迭代步数 50000 50000 10000 内存/MB 136.1 3.2 4.3 
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表 2 三种数值算法的计算参数比较
Table 2. Comparison of calculation parameters of three numerical algorithms.
Uniform FDTD HVS-FDTD HVS-EUS-FDTD 时间/s 91.36 2.19 0.29+0.78 时间步长/(10–12 s) 3.33 3.33 16.7 迭代步数 5000 5000 1000 内存/MB 86 7.2 11.6
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表 3 三种数值算法的计算参数比较
Table 3. Comparison of calculation parameters of three numerical algorithms.
Uniform FDTD HVS-FDTD HVS-EUS-FDTD 时间/s 28816.8 314.2 0.01+64.7 时间步长/(10–11 s) 1.67 1.67 8.33 迭代步数 5000 5000 1000 内存/MB 9773 321 342
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[1] Tan E 2020 Prog. Electromagn. Res. 168 39
Google Scholar
[2] He X B, Wei B, Fan K H, Li Y W, Wei X L 2019 Chin. Phys. B 28 074102
Google Scholar
[3] Gedney S 2011 Introduction to the Finite-difference Time-domain (FDTD) Method for Electromagnetics (Berlin: Springer Nature) pp1–11
[4] Namiki T 1999 IEEE Trans Microwave Theory Tech. 47 2003
Google Scholar
[5] Zheng F H, Chen Z Z 2001 IEEE Trans Microwave Theory Tech. 49 1006
Google Scholar
[6] Chen J, Hao G C, Liu Q H 2017 IEEE Trans. Electromagn. Compat. 59 1218
Google Scholar
[7] Shibayama J, Muraki M, Yamauchi J, Nakano H 2005 Electron. Lett. 41 1
Google Scholar
[8] Ahmed I, Chua E K, Li E P 2010 IEEE Trans. Antennas Propag. 58 3983
Google Scholar
[9] Ahmed I, Khoo E H, Li E P 2013 IEEE Microwave Wireless Compon. Lett. 23 306
Google Scholar
[10] Sun C, Trueman C W 2003 Electron. Lett. 39 595
Google Scholar
[11] Jiang H L, Cui T J 2019 IEEE Antennas Wireless Propag. Lett. 18 698
Google Scholar
[12] Chen J, Li J X, Liu Q H 2017 IEEE Trans. Microwave Theory Tech. 65 3689
Google Scholar
[13] Mai H X, Chen J, Yu X M, Zhang A X 2020 Int. J. RF Microwave Comput. Aided Eng. 30 e22166
Google Scholar
[14] Gaffar M, Jiao D 2015 IEEE Trans Microwave Theory Tech. 63 4215
Google Scholar
[15] Yan J, Jiao D 2017 IEEE Trans Microwave Theory Tech. 65 5084
Google Scholar
[16] Yan J, Jiao D 2018 IEEE Trans. Antennas Propag. 66 4137
Google Scholar
[17] Yan J, Jiao D 2017 IEEE Trans. Microwave Theory Tech. 65 2698
Google Scholar
[18] Zeng K Y, Jiao D 2020 IEEE Trans. Antennas Propag. 68 3047
Google Scholar
[19] Bekmambetova F, Zhang X Y, Triverio P 2017 IEEE Trans. Antennas Propag. 65 751
Google Scholar
[20] Zhang X Y, Bekmambetova F, Triverio P 2018 IEEE Trans. Antennas Propag. 66 827
Google Scholar
[21] Lee W 2018 IEEE J. Multiscale Multiphys. Comput. Tech. 3 16
Google Scholar
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