# Planar Delaunay Mesh Smoothing Method Based on Angle and a Deep Q-Network
## 4 Results and Discussion
### Comparative Analysis of Experiments
#### 4.4.1 Example 1  
We selected the L-shaped flat plate [29] as Example 1. The area near the right angle inside is prone to stress concentration. Therefore, various methods were used to smooth Example 1, and the smoothed mesh was calculated with the finite element calculation method. The geometric shape and load form of Example 1 are shown in Figure 25. The maximum length and width of the L-shaped flat plate are 100 mm, with a mass density of \(\rho=7800\) kg/m\({}^{3}\), elastic modulus E = 206 Gpa, Poisson's ratio \(\mu=0.3\), and uniformly distributed load \(q=1\) N/mm\({}^{2}\). The initial, overall refined, and optimized mesh of the proposed method for Example 1 is shown in Figure 26. The comparison results of different methods are shown in Table 5. It can be seen that the proposed method is significantly better than the other methods for angle control. Since there is no analytical solution for the L-shaped plate, we used the finite element calculation results of the refined mesh, as shown in Figure 26b, as the reference solution for comparison. The stress nephograms for the initial and refined meshes of Example 1 are shown in Figure 27. The stress nephograms after optimization using the remaining optimization methods are shown in Figure 28.  
Figure 24: Distribution of quality between 0.5 and 0.8. Figure 25: L-shaped flat plate. The maximum stress selected is the stress at point A in Figure 25. The results of the finite element calculations are shown in Table 6. It can be seen that the finite element calculation results of the optimized mesh of the proposed method are closer to the reference solution with minimum error. Compared with the initial mesh, the proposed method reduces the error by about 7% and further reduces the error by about 1% compared to other methods. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Mesh** & **Method** & **Time/(s)** & **Nodes** & **Min. 0** & **Max. 0** & **(a)** & **(b)** & **Min. q** & **Avg. q** \\ \hline \multirow{8}{*}{2} & Initial & / & 26.93 & 108.78 & 45 & 86 & 0.7003 & 0.8635 \\  & Laplacian & 0.1024 & & 28.62 & 102.80 & 3 & 2 & 0.7139 & 0.9417 \\  & Angle-based & 0.2045 & & 28.82 & 100.26 & 2 & 1 & 0.7517 & 0.9475 \\  & ODT & 0.1377 & 788 & 28.62 & 104.52 & 3 & 7 & 0.7139 & 0.9401 \\  & CVT & 0.1249 & & 28.62 & 99.80 & 3 & 0 & 0.7139 & 0.9478 \\  & Unconstrained DQN & 0.3175 & & 30.06 & 103.39 & 0 & 3 & 0.7403 & 0.9262 \\  & Constrained DQN & 0.3261 & & 33.03 & 98.44 & 0 & 0 & 0.7954 & 0.9357 \\ \hline \hline \end{tabular}  
* Nodal SOLUTION STEP -1 TIME -1 ISOV (AVG) DANG-0.9023 SSMX-0.520*10  
\end{table} 
Table 5: Comparison table of the experimental results. Figure 27: Finite element calculation results of Example 1. (**a**) Initial mesh; (**b**) refined mesh. Figure 28: Example 1: finite element calculation results after smoothing for each optimization method. (**a**) Laplacian; (**b**) angle-based; (**c**) ODT; (**d**) CVT; (**e**) unconstrained DQN; and (**f**) constrained DQN. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Nodes** & **Elements** & **Stress (Mpa)** & **Error (\%)** \\ \hline Initial & & & 5.1531 & 12.5939 \\ Laplacian & & & 5.4915 & 6.8524 \\ Angle-based & & & 5.5006 & 6.6984 \\ ODT & 435 & 788 & 5.5068 & 6.5934 \\ CVT & & & 5.4987 & 6.7310 \\ Unconstrained DQN & & & 5.4453 & 7.6354 \\ Constrained DQN & & & 5.5564 & 5.7523 \\ Refinement & 804 & 1496 & 5.8955 & / \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Errors in maximum stress of different optimized meshes. Since the stress concentration problem occurs at point A in Figure 29 in actual engineering, the stress values in the x- and y-direction at point A are extracted, and the calculation results of the proposed method are compared with other methods; the data statistics are shown in Table 8. According to Table 8, it is easy to find that the computational results of the optimized mesh of the proposed method for finite element analysis are closer to the results in [30] with minimum errors.