一种简化的时域非连续伽略金阻抗边界算法

杨谦; 魏兵; 李林茜; 邓浩川

doi:10.7498/aps.72.20222104

摘要

针对时域非连续伽略金(discontinuous Galerkin time-domain, DGTD)算法中的阻抗边界条件问题开展研究. 阻抗边界条件中的频域算符${\rm{j}}\omega $一般在根号内部, 其在时域数值算法中的实现有一定难度. 另一方面, DGTD算法中数值通量表达式也含有阻抗边界条件, 这也进一步增加了频时转换难度. 为了能给出简化的DGTD阻抗边界算法, 本文首先针对数值通量表达式进行推导, 得到一个特定函数$ {\tilde Z_R} $, 该函数包含频域算符${\rm{j}}\omega $, 函数以外表达式不含频域算符${\rm j}\omega$, 这样就可以仅处理$ {\tilde Z_{\rm{R}}} $的频时转换问题. 由于$ {\tilde Z_{\rm{R}}} $形式复杂, 对$ {\tilde Z_{\rm{R}}} $进行矢量匹配处理, 得到关于${\rm{j}}\omega$的一阶有理分式, 进而得到其时域迭代式. 这一过程简明、易于实施, 还可避开矩阵计算. 本文方案经一维及三维算例验证, 精度很好, 针对特定电磁问题如涂覆层问题可大幅降低计算时间.

关键词:

Abstract

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 H 2015 IEEE Trans. Antennas Propag. 63 3065 ; Li P, Jiang L J, Bağcι H 2018 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

作者及机构信息

1.
西安电子科技大学物理学院, 西安　710071

2.
北京环境特性研究所, 电磁散射重点实验室, 北京　100854

通信作者: 杨谦, qyang@xidian.edu.cn

基金项目: 国家自然科学基金(批准号: 61901324, 62001345)、中国博士后科学基金(批准号: 2019M653548, 2019M663928XB)、电波环境特性及模化技术国防科技重点实验室基金(批准号: 201903002)、电磁散射重点实验室基金(批准号: 61424090111)和中央高校基本科研业务费(批准号: XJS200501, XJS200507, JB200501)资助的课题.

Authors and contacts

1.
School of Physics, Xidian University, Xi’an 710071, China

2.
Science and Technology on Electromagnetic Scattering Laboratory, Beijing Institute of Environmental Features, Beijing 100854, China

Corresponding author: Yang Qian, qyang@xidian.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 61901324, 62001345), the China Postdoctoral Science Foundation (Grant Nos. 2019M653548, 2019M663928 XB), the Foundation of National Key Laboratory of Electromagnetic Environment, China (Grant No. 201903002), the Foundation of the Science and Technology on Electromagnetic Scattering Laboratory, China (Grant No. 61424090111), and the Fundamental Research Funds for the Central Universities of China (Grant Nos. XJS200501, XJS200507, JB200501).

文章全文

参考文献

[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

施引文献

图 1 半空间示意图

Fig. 1. Plane wave incident on a lossy dielectric half-space.

下载: 全尺寸图片幻灯片

图 2 虚拟单元

Fig. 2. Virtual element.

下载: 全尺寸图片幻灯片

图 3 SIBC($ {\tilde Z_R} $)

Fig. 3. SIBC ($ {\tilde Z_R} $).

下载: 全尺寸图片幻灯片

图 4 薄涂层示意图

Fig. 4. A conductive object with thin sheet.

下载: 全尺寸图片幻灯片

图 5 TIBC ($ {\tilde Z_R} $)

Fig. 5. TIBC ($ {\tilde Z_R} $).

下载: 全尺寸图片幻灯片

图 6 对于一维SIBC算例, 反射波及入射波时域数据

Fig. 6. Incident wave and reflected wave for one-dimensional SIBC example.

下载: 全尺寸图片幻灯片

图 7 对于一维SIBC算例, 反射系数幅值

Fig. 7. Magnitude of the reflection coefficient for one-dimensional SIBC example.

下载: 全尺寸图片幻灯片

图 8 对于一维SIBC算例, 反射系数相位

Fig. 8. Phase of the reflection coefficient for one-dimensional SIBC example.

下载: 全尺寸图片幻灯片

图 9 对于一维TIBC算例, 反射波及入射波时域数据

Fig. 9. Incident wave and reflected wave for one-dimensional TIBC example.

下载: 全尺寸图片幻灯片

图 10 对于一维TIBC算例, 反射系数幅值

Fig. 10. Magnitude of the reflection coefficient for one-dimensional TIBC example.

下载: 全尺寸图片幻灯片

图 11 对于一维TIBC算例, 反射系数相位

Fig. 11. Phase of the reflection coefficient for one-dimensional TIBC example.

下载: 全尺寸图片幻灯片

图 12 单站RCS结果

Fig. 12. Monostatic RCS.

下载: 全尺寸图片幻灯片

图 13 涂覆球

Fig. 13. Coated sphere.

下载: 全尺寸图片幻灯片

图 14 涂覆球单站RCS结果

Fig. 14. Monostatic RCS.

下载: 全尺寸图片幻灯片

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)	$\dfrac{{{{\tilde Y}^{e + }}}}{{{Y^e} + {{\tilde Y}^{e + }}}}$	$\dfrac{{{{\tilde Z}^{e + }}}}{{{Z^e} + {{\tilde Z}^{e + }}}}$	$\dfrac{1}{{{Y^e} + {{\tilde Y}^{e + }}}}$	$\dfrac{1}{{{Z^e} + {{\tilde Z}^{e + }}}}$

下载: 导出CSV

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

下载: 导出CSV

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

下载: 导出CSV

map

[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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