# Modeling GPR Wave Propagation in Complex Underground Structures Using Conformal ADI-FDTD Algorithm
## 3 Numerical Simulations
### Complex Multi-Pipe Model  
This model used the conformal ADI-FDTD algorithm to simulate the complex multi-tube model. Figures 12 and 13 show the schematic diagrams of complex multi-pipe models. Figure 12 shows an underground concrete multi-pipe model, with pipe diameters of 0.03 m, 0.06 m, and 0.1 m. Figure 13 shows an underground multi-pipe model with pipe diameters of 0.4 m. The pipe materials were concrete, metal, and PVC plastic. The space step and time step were set as 0.005 m and 0.01 ns, and simulation iterations were 5000. The model had two layers: the upper layer was an air layer, and the lower layer was a clay layer. Table 2 provides the relative permittivity and conductivity. The excitation source and GPR system were the same as for the single-pipe model.  
Figure 11: GPR B-scan image of a disease model with an irregular cavity below pipeline: (**a**) Obtained by non-conformal method; (**b**) Obtained by the conformal method.  
Figure 12: Underground multi-pipe model with different pipe diameters.  
Figures 14 and 15 show the GPR B-scan images obtained by the underground multi-pipe model using the conformal ADI-FDTD algorithm. Figure 14, G1, G2, and G3 indicate the intersection points of the hyperbolic asymptotes at the tops of concrete pipes with diameters of 0.3 m, 0.6 m, and 1.0 m, respectively. Reflected waves are evident at the top and bottom of the pipeline; the more reflective the diameter, the more pronounced the reflected waves are. There are multiple reflected and stray waves at the bottom of the profile image. In Figure 15, the PVC plastic pipe on the right side of the profile has weak reflected waves, and the concrete pipeline located on the left side has distinct hyperbolic reflected waves. The reflected waves are evident at the top of the middle metal tube, while not at the bottom. In addition, some interfering stray waves and multiple reflected waves can be observed at the bottom of the image.  
Figure 14: Simulation results of underground multi–pipe model with different pipe diameters.  
Figure 13: Underground multi-pipe model with different pipe materials.  
As shown in Figures 14 and 15, for an underground multi-pipe model with the same depth and materials but different diameters, an increase in diameter causes the intersection position of the hyperbolic asymptote at the top of the pipe to become higher, and the reflection waves at the bottom and top of the pipe will be more obvious. As the thickness of the pipe wall is related to the pipe's diameter, two obvious hyperbolas with larger diameters can be seen at the bottom and top of the concrete pipe. For underground multi-pipe models with the same diameter and composed of different materials, the closer the dielectric properties of the pipe material are to those of the surrounding clay, the weaker its response to the electromagnetic waves and the less pronounced the diffraction hyperbolic features are at the top and bottom. Due to the strong response of electromagnetic waves to metals, the diffraction hyperbolic characteristics at the top of the metal pipe were very obvious, while there were no hyperbolic characteristics at the bottom of the pipe and fewer multiple reflections occurred inside.