Finite element time domain simulation and propagation characteristics analysis of GPR in Cole-Cole dispersive media based on adaptive Padé approximation algorithm

Time-domain numerical simulation is an essential approach for characterizing high frequency electromagnetic wave propagation in Cole-Cole dispersive media. Due to the fractional power function form of complex permittivity with respect to angular frequency, transforming the frequency domain into the time domain introduces fractional order derivatives, which cannot be discretized using conventional integer-order time-stepping schemes. Although the Padé approximation with a fixed characteristic frequency avoids direct discretization by introducing auxiliary integer-order differential equations, it results in significant estimation errors of complex permittivity, particularly when the relaxation time is short. To address this issue, an adaptive Padé approximation-based Finite Element Time Domain (FETD) algorithm is proposed for Ground Penetrating Radar (GPR) simulation in Cole-Cole dispersive media. Based on the time domain electromagnetic wave equations and their finite element formulations derived using the fixed characteristic frequency Padé approximation, the total simulation time window is uniformly divided into multiple segments. Within each segment, a set of characteristic frequencies selected from the bandwidth of the excitation wavelet are used to perform simulations, which are then compared against analytical solutions. The frequency corresponding to the minimum relative error is identified as the optimal characteristic frequency for that segment. The resulting time-domain solution is passed as the initial condition to the subsequent segment. This procedure is repeated iteratively until the final segment is completed, allowing adaptive selection of the Padé approximation frequency in each segment during time marching. Numerical experiments demonstrate that the proposed adaptive Padé approximation based FETD algorithm achieves higher simulation accuracy than the fixed-frequency Padé approach. Compared with non-dispersive media, high frequency electromagnetic wave in Cole-Cole dispersive media propagate at a lower velocity and exhibit stronger energy attenuation. In contrast to Debye dispersive media, wave propagation in Cole-Cole media is faster. The difference in energy attenuation between the two is determined by the intersection point of their respective attenuation coefficient curves as a function of frequency. On the left side of this intersection, Cole-Cole media shows stronger attenuation, while on the right side, the attenuation becomes weaker. As the relaxation factor increases, the wave propagation velocity in Cole-Cole media decreases. Energy attenuation is influenced jointly by propagation distance and relaxation factor. Specifically, attenuation becomes more pronounced either when the propagation distance is large and the relaxation factor is small, or when the distance is short and the relaxation factor is large.