A simplified impedance boundary algorithm in discontinuous Galerkin time-domain
Authors: Yang Qian , Wei Bing , Li Lin-Qian , Deng Hao-Chuan ,   School of Physics, Xidian University, Xi’an 710071, China,   Science and Technology on Electromagnetic Scattering Laboratory, Beijing Institute of Environmental Features, Beijing 100854, China
Published: January 2023

<sec>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 <inline-formula><tex-math id="M9">\begin{document}$ \tilde Z $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222104_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222104_M9.png"/></alternatives></inline-formula>. The <inline-formula><tex-math id="M10">\begin{document}$ \tilde Z $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222104_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222104_M10.png"/></alternatives></inline-formula> 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.</sec><sec>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 <ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="http://doi.org/10.1109/TAP.2015.2491969">2015 <i>IEEE Trans. Antennas Propag.</i> <b>63</b> 5686</ext-link>; Li P, Jiang L J, Bağcι H <ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="http://doi.org/10.1109/TAP.2015.2426198">2015 <i>IEEE Trans. Antennas Propag.</i> <b>63</b> 3065 </ext-link>; Li P, Jiang L J, Bağcι H <ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="http://doi.org/10.1109/TAP.2018.2826567">2018 <i>IEEE Trans. Antennas Propag.</i> <b>66</b> 3590 </ext-link>) 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 <inline-formula><tex-math id="M11">\begin{document}$ \tilde Z $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222104_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222104_M11.png"/></alternatives></inline-formula> is not approximated by rational functions in our study. A specfic function <inline-formula><tex-math id="M12">\begin{document}$ {\tilde Z_R} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222104_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20222104_M12.png"/></alternatives></inline-formula> 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.</sec>