The Use of ADI-FDTD Subgrids in FDTD Ground Penetrating Radar Modeling

Realistic numerical modeling of ground penetrating radar (GPR) using the traditional finite-difference time-domain (FDTD) method could benefit from the implementation of subgrids, especially when fine details of targets need to be modeled. In this paper, we investigate the use of the alternating-direction-implicit (ADI) FDTD technique as a subgrid into the conventional FDTD method. As ADI-FDTD is unconditionally stable, it is envisaged that a common time-step could be used in both the main FDTD grid and in the subgird. We examine the performance of ADI-FDTD subgrids when implemented into the standard FDTD method, using different communication schemes for the information exchange between the two grids. In addition, results from the comparison between the proposed algorithms and a commonly employed purely FDTD subgridding technique are presented. One of the major limitations of the FDTD approach to the numerical solution of Maxwell's equations (1), (2), is its conditionally stable nature. The FDTD stability condition, known as the CFL condition, after the initials of Courant, Freidrichs, and Lewy, states that a maximum allowable time-step is limited by a minimum cell size in the computational domain. When structures of fine geometry need to be modeled, the spatial-step should be small enough so as all the objects in the computational domain are adequately resolved. For this class of models, an FDTD solution using a uniform small spatial-step would result in substantial computer memory requirements and an increase in execution time. To alleviate this problem a possible solution is the introduction of subgrids to model fine details into the FDTD method and therefore, economize on computational resources. Most studies on the introduction of subgrids into the FDTD method employ the standard explicit method in order to calculate the fields in both the coarse and the fine grids (3). As a result, the different spatial-steps used in the two grids lead to the requirement for different time-steps, as the CFL condition must be satisfied in both grids. Therefore, apart from spatial interpolation of the fields at the boundary between the coarse and the fine grid, time interpolation has to be also carried out. In addition, implementation could become cumbersome when trying to keep the calculations synchronized in time. To overcome the time interpolation requirement and make our calculations simpler, the ADI technique, which was initially introduced for solving parabolic differential equations (4), is implemented in the subgrid (5), instead of the explicit FDTD method. As this technique is used for the calculations in the subgrid and it is unconditionally stable, it could operate using the same time-step as the one that the traditional FDTD method uses in the main grid. A dispersion analysis (6), has proven that for the same temporal and spatial step the FDTD method is more accurate than the ADI-FDTD method. But for spatially over-resolved problems, known as ADI class problems, when a part or the whole computational domain needs to be resolved in great detail, the ADI-FDTD method provides satisfactory results while reducing the simulation run times. The focus of our research is the FDTD simulation of GPR responses from small targets for the non-destructive testing of structures and of near surface features. Therefore, we present different implementation procedures of ADI-FDTD subgrids into the standard FDTD grid and discuss the communication between the two methods, for a two-dimensional TMz GPR modeling case. In all our numerical tests, all media are modeled as low-loss dielectrics with constitutive parameters independent of frequency. Furthermore, all the media are considered to be non-magnetic.