Abstract：With the improvement of processing technology, many devices are becoming more integrated and their sizes are developing towards sub-millimeter or nanometer scales, which makes electrically small systems, such as chips, an area of increasing interest. Therefore, it is significant to develop an efficient numerical method with a large time step size which is not strictly limited by the Courant Friedrichs Lewy (CFL) stability condition defined by the minimum spatial step size. The precise integration time domain (PITD) method is a full-wave electromagnetic numerical method with a relaxed numerical stability condition, especially its numerical dispersion error is almost independent of the time step size, which provides it with a natural advantage in enlarging the time step size and improving computational efficiency. As a consequence, it has attracted a lot of attention in the field of electromagnetism in recent years. However, the PITD method needs to calculate the matrix integral associated with the inhomogeneous term introduced by the excitation source when simulating in the active region. At present, the methods used to calculate the matrix integral are either easy to be unstable due to the existence of matrix inverse operation or difficult to maintain the accuracy of the calculation results at large time step sizes, which has become an urgent problem in the popularisation and application of the PITD method. To address these issues, this paper introduces response matrix method into the PITD method. Firstly, the response matrix method transforms the computation of matrix integrals into the computation of a series of elementary form functions' response matrices. Then, the values of these response matrices will be calculated based on the addition theorem and the incremental storage idea. Finally, the value of the matrix integral can be easily obtained. Considering the common types of excitation sources, the calculation formulas for the response matrix method are given in the form of exponential function, polynomial function and raised cosine function. Based on these calculation formulas, it can be seen that the important advantage of the response matrix method is the avoidance of matrix inversion operation, which is computationally intensive, easily unstable and sometimes unsolvable. This makes it convenient to extend the PITD method's application and improve its numerical stability. Actually, the numerical integral methods commonly used at present, such as the Gaussian integral method and Simpson integral method, can also avoid the matrix inverse operation. However, these methods have a low computational accuracy of the results at large time step sizes. On the contrary, the response matrix method has a good accuracy and can ensure the accuracy of the simulation results at large time step sizes. Simulations are carried out and the results show that the response matrix method is feasible and effective in ensuring the computational accuracy of the results under a large time step size which is 60 times as large as that limited by CFL stability condition. Compared with the commonly used numerical integral method, the response matrix method can obviously increase the time step size determined by the computational accuracy. This is beneficial to improve the computational efficiency of PITD simulation. The results of the two-dimensional biological tissue simulation model show that the simulation time of the response matrix method is only 14.51% of the Gaussian integral method and 49.26% of the Simpson integral method.
迟明珺, 马西奎, 马亮, 朱晓杰. 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.
Chi Mingjun, Ma Xikui, Ma Liang, Zhu Xiaojie. A Large Time Step Size Matrix Numerical Integral Method for Precise Integration Time Domain Simulation of Electromagnetic Waves in Active Regions. Transactions of China Electrotechnical Society, 2024, 39(21): 6604-6613.
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2024, Vol. 39 
