Analyzing Ultra-Wideband Electromagnetic Problems by Time-domain Integral Equation Formulations

The surface integral equations (SIEs) are widely employed to address scattering, radiation, and circuit simulation problems. However, both the frequency domain integral equations (FDIE) and time domain integral equations (TDIE) encounter issues related to low-frequency breakdown [1]. Specifically, there is a dominance of the scalar potential in the impedance matrix at low frequencies (ω → 0 in the frequency domain and Δt → ∞ in the time domain). This phenomenon leads to a significant increase in the condition number of the matrix, which notably affects the solution of the SIE. Moreover, the presence of numerical errors and ill-conditioned matrix often leads to inaccurate solutions at low frequencies. On the other hand, the characteristic of the $\mathcal{L}$ operator makes it susceptible to the mesh density, which results in large condition number compared to the $\mathcal{K}$ operator [2]. The commonly used method to address low frequency problems is quasi-Helmholtz decomposition [3], [4]. This approach helps eliminating the impact of numerical errors at low frequencies. Other strategy involves using additional equations to reduce the numerical imbalance [5], [6].