# Efficient Parallel FDTD Method Based on Non-Uniform Conformal Mesh
## 3 Results
### Performance Analysis on Multi-Node  
We conducted experiments utilizing two distinct scenarios: a 100-period folded waveguide and a 16 \(\times\) 16 phased array antenna array, depicted in Figure 8a,b, respectively. These experiments were aimed at evaluating the algorithm's parallel efficacy across multiple computational nodes. The operating frequencies were set at 219-221 GHz and 4-5 GHz correspondingly, with excitation applied at the waveguide ports. Gaussian signals were adopted, and the computational domains were discretized into non-uniform conformal grids, sized at 41,277,600 (70 \(\times\) 91 \(\times\) 6480) and 33,808,320 (540 \(\times\) 728 \(\times\) 86), respectively. What is more, boundary conditions were enforced by using electric boundary and CPML absorbing techniques. Figure 9 shows the S-parameter results of the folded waveguide, where S11 is completely below \(-\)15 dB and S21 is completely within \(-\)0.12 dB. Both our algorithm and CST show good transmission characteristics. Figure 10a,b depict the far-field patterns for the antenna array. The serial and parallel results are entirely consistent, and basically consistent with the CST results, also indicating the correctness of the parallel algorithm results. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Processors**} & \multicolumn{3}{c}{**Unmanned Helicopter**} & \multicolumn{3}{c}{**Helical Antenna**} \\ \cline{2-7}  & **Time(s)** & **Scalability** & **Efficiency** & **Time(s)** & **Scalability** & **Efficiency** \\ \hline
1 & 148,714 & 1 & 100\% & 133,379 & 1 & 100\% \\
2 & 75,172.8 & 1.98 & 98.9\% & 69,023 & 1.93 & 96.6\% \\
4 & 38,031.1 & 3.91 & 97.8\% & 33,983.5 & 3.92 & 98.1\% \\
8 & 19,596.8 & 7.59 & 94.9\% & 17,899.5 & 7.45 & 93.1\% \\
12 & 13,397.6 & 11.10 & 92.5\% & 12,158.5 & 10.97 & 91.4\% \\
16 & 10,313.0 & 14.42 & 90.1\% & 9192.2 & 14.51 & 90.7\% \\ \hline \end{tabular}
\end{table} 
Table 1: Single-node parallel testing results table. Figure 7: (**a**) The parallel scalability and efficiency of an unmanned helicopter on a single node; (**b**) the parallel scalability and efficiency of helical antenna on a single node. On the supercomputing platform, the parallel non-uniform conformal FDTD algorithm was executed with different numbers of processors ranging from 1 to 1024. The number of processors for domain decomposition was selected from 1 to 1024, including 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, and 1024. Multi-node parallelism started with 64 processors, and 1024 processors were run on 33 nodes. The specific test data are provided in Table 2. From Figure 11, it can be observed that the computational time for both cases decreased  
Figure 8: (a) A 100-period folded waveguide; (b) a 16 \(\times\) 16 phased array antenna. Figure 10: (a) A 16 \(\times\) 16 phased array antenna far-field pattern at phi = 90\({}^{\circ}\); (b) a 16 \(\times\) 16 phased array antenna far-field pattern at theta = 90\({}^{\circ}\). Figure 9: The S-parameter results of the 100-period folded waveguide. significantly from 4,672,834 s and 996,042 s for single processors to 6264.5 s and 1316.3 s for 1024 processors, respectively, resulting in a substantial reduction in computation time. The scalability and efficiency of the parallel non-uniform conformal FDTD algorithm are illustrated in Figure 12. The overall parallel efficiency for both cases is above 70%, reaching a maximum of 99.3% (with 2 processors) and a minimum of 72.8% (with 1024 processors). The efficiency can reach 90% with a small number of processors, and the scalability approaches linearity. However, as the number of processors increases, especially when cross-node communication is involved, the parallel efficiency gradually decreases, which is consistent with general parallel trends.