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Large-size conductive targets or coated targets are difficult problems in computational electromagnetics. In general, these problems can be classified as multi-scale problems. Multi-scale problems usually consume a large quantity of computational resources. A lot of efforts have been devoted to seeking for fast methods for these problems. When the skin depth is less than the size of a conductive target, the tangential component of the electric field and magnetic field over the surface of the target can be correlated by the surface impedance $ \tilde Z $ . The$ \tilde Z $ is usually a complex function of the frequency, and it can be used to formulate an impedance boundary condition (IBC) to describe iterative equations in time domain methods, avoiding the volumetric discretization of the target and improving computational efficiency. This condition is commonly known as the surface impedance boundary condition (SIBC). Similarly, for a conductor whose thickness is in the order of skin depth or less, it also has high resource requirements, if the target is of direct volume discretization. The transmission impedance boundary condition (TIBC) can be utilized instead of a coated object to reduce resource requirements. Therefore, there is no need to discretize volume.There are few studies on the IBC scheme by using the discontinuous Galerkin time-domain (DGTD) method. Li et al. (Li P, Shi Y, Jiang L J, Bağcι H 2015 IEEE Trans. Antennas Propag. 63 5686 ; Li P, Jiang L J, Bağcι H2015 IEEE Trans. Antennas Propag. 63 3065 ; Li P, Jiang L J, Bağcι H2018 IEEE Trans. Antennas Propag. 66 3590 ) discussed the IBC scheme by using the DGTD, which involves complex matrix operations in the processing of IBC. In the DGTD method, numerical flux is used to transmit data between neighboring elements, and the key to the IBC scheme in DGTD is how to handle numerical flux. We propose a DGTD method with a simple form and matrix-free IBC scheme. The key to dealing with IBC in DGTD is numerical flux. Unlike the way in the literature, the impedance$ \tilde Z $ is not approximated by rational functions in our study. A specfic function$ {\tilde Z_R} $ obtained after the derivation in this work is approximated by rational functions in the Laplace domain through using the vector-fitting (VF) method, and its time-domain iteration scheme is given. This approach avoids matrix operations. The TIBC and SIBC processing schemes are also given. The advantage of the proposed method are that the upwind flux’s standard coefficients are retained and the complex frequency-time conversion problem is implemented by the vector-fitting method. The one-dimensional and three-dimensional examples also show the accuracy and effectiveness of our proposed method in this work.-
Keywords:
- impedance boundary condition /
- vector-fitting method /
- discontinuous Galerkin time domain method
[1] Senior T 1960 Appl. Sci. Res. 8 418
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Google Scholar
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Google Scholar
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Google Scholar
[18] Li P, Jiang L J, Bağcι H 2015 IEEE Trans. Antennas Propag. 63 3065
Google Scholar
[19] Li P, Jiang L J, Bağcı H 2018 IEEE Trans. Antennas Propag. 66 3590
Google Scholar
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Google Scholar
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图 1 半空间示意图
Figure 1. Plane wave incident on a lossy dielectric half-space.
图 2 虚拟单元
Figure 2. Virtual element.
图 3 SIBC(
$ {\tilde Z_R} $ )Figure 3. SIBC (
$ {\tilde Z_R} $ ).
图 4 薄涂层示意图
Figure 4. A conductive object with thin sheet.
图 5 TIBC (
$ {\tilde Z_R} $ )Figure 5. TIBC (
$ {\tilde Z_R} $ ).
图 6 对于一维SIBC算例, 反射波及入射波时域数据
Figure 6. Incident wave and reflected wave for one-dimensional SIBC example.
图 7 对于一维SIBC算例, 反射系数幅值
Figure 7. Magnitude of the reflection coefficient for one-dimensional SIBC example.
图 8 对于一维SIBC算例, 反射系数相位
Figure 8. Phase of the reflection coefficient for one-dimensional SIBC example.
图 9 对于一维TIBC算例, 反射波及入射波时域数据
Figure 9. Incident wave and reflected wave for one-dimensional TIBC example.
图 10 对于一维TIBC算例, 反射系数幅值
Figure 10. Magnitude of the reflection coefficient for one-dimensional TIBC example.
图 11 对于一维TIBC算例, 反射系数相位
Figure 11. Phase of the reflection coefficient for one-dimensional TIBC example.
图 12 单站RCS结果
Figure 12. Monostatic RCS.
图 13 涂覆球
Figure 13. Coated sphere.
图 14 涂覆球单站RCS结果
Figure 14. Monostatic RCS.
表 1 数值通量系数表达式
Table 1. Coefficients of numerical flux.
通量系数 $ k_E^e $ $ k_H^e $ $ v_H^e $ $ v_E^e $ 迎风通量表达式(upwind flux) $\dfrac{{{{\tilde Y}^{e + }}}}{{{Y^e} + {{\tilde Y}^{e + }}}}$ $\dfrac{{{{\tilde Z}^{e + }}}}{{{Z^e} + {{\tilde Z}^{e + }}}}$ $\dfrac{1}{{{Y^e} + {{\tilde Y}^{e + }}}}$ $\dfrac{1}{{{Z^e} + {{\tilde Z}^{e + }}}}$ 
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表 2 计算参数
Table 2. Calculation parameters.
离散尺度/m 四面体数量 时间步/(10–11 s) 计算时间/min 均方根误差/dB DGTD (SIBC) 0.05 402094 0.328972 54 0.6237 DGTD (standard) 0.05 487677 0.328447 61 0.5836
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表 3 计算参数
Table 3. Calculation parameters.
离散尺度/m 四面体数量 时间步/(10–11 s) 计算时间/min 均方根误差/dB 0.05 402094 0.328972 55 0.8782
DownLoad: CSV
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[1] Senior T 1960 Appl. Sci. Res. 8 418
Google Scholar
[2] Feliziani M 2011 IEEE Trans. Electromagn. Compat. 54 299
Google Scholar
[3] Nayyeri V, Soleimani M, Ramahi O M 2013 IEEE Trans. Electromagn. Compat. 56 385
Google Scholar
[4] Hanson G W 2008 J. Appl. Phys. 103 064302
Google Scholar
[5] Nayyeri V, Soleimani M, Ramahi O M 2013 IEEE Trans. Antennas Propag. 61 6107
Google Scholar
[6] Shapoval O V, Gomez-Diaz J S, Perruisseau-Carrier J, Mosig J R, Nosich A I 2013 IEEE Trans. Terahertz Sci. Technol. 3 666
Google Scholar
[7] da Costa K Q, Dmitriev V, Nascimento C M, Silvano G L 2014 Microwave Opt. Technol. Lett. 56 1019
Google Scholar
[8] Beggs J H, Luebbers R J, Yee K S, Kunz K S 1992 IEEE Trans. Antennas Propag. 40 49
Google Scholar
[9] oh K S, Schutt-Aine J E 1995 IEEE Trans. Antennas Propag. 43 660
Google Scholar
[10] Kobidze G 2010 IEEE Trans. Antennas Propag. 58 2394
Google Scholar
[11] Gustavsen B, Semlyen A 1999 IEEE Trans. Power Delivery 14 1052
Google Scholar
[12] Yi M, Ha M, Qian Z, Aydiner A, Swaminathan M 2013 IEEE Trans. Microwave Theory Tech. 61 4029
Google Scholar
[13] Glisson A W 1992 Radio Sci. 27 935
Google Scholar
[14] Yan S, Jin J M 2013 IEEE Trans. Antennas Propag. 61 5533
Google Scholar
[15] Ylä-Oijala P, Kiminki S P, Järvenpää S 2010 IEEE Trans. Antennas Propag. 58 3997
Google Scholar
[16] Gyselinck J, Dular P, Geuzaine C, Sabariego R 2009 IEEE Trans. Magn. 45 1280
Google Scholar
[17] Li P, Shi Y, Jiang L J, Bağcι H 2015 IEEE Trans. Antennas Propag. 63 5686
Google Scholar
[18] Li P, Jiang L J, Bağcι H 2015 IEEE Trans. Antennas Propag. 63 3065
Google Scholar
[19] Li P, Jiang L J, Bağcı H 2018 IEEE Trans. Antennas Propag. 66 3590
Google Scholar
[20] Wang P, Shi Y, Tian C Y, Li L 2018 IEEE Antennas Wirel. Propag. Lett. 17 2169
Google Scholar
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