# Planar Delaunay Mesh Smoothing Method Based on Angle and a Deep Q-Network
## 4 Results and Discussion
### Comparative Analysis of Experiments  
In this section, we use four examples for an experimental comparison analysis, as shown in Figure 18, and the number of nodes are 2027, 926, 2987, and 5115, respectively. The red circle is the location for local comparison in the following text. It should be noted that Example 2 is a mesh in the dataset, while the other three examples have not undergone DQN model training. The results of the four examples after optimization in this paper are shown in Figure 19. Comparing Figures 18 and 19, it can be seen that the proposed method can show good optimization effects in meshes with different element sizes and boundary shapes and can effectively reduce the number of poor elements. Figure 15: Effectiveness of Example 4 comparison before and after smoothing. (**a**) Before smoothing; (**b**) after smoothing. Figure 16: Generalizability analysis of constrained DQN models. (**a**) Minimum angle contrast; (**b**) maximum angle contrast; (**c**) minimum quality contrast; (**d**) and average quality contrast. Figure 17: Model generalization analysis chart. In addition, Laplacian [8], angle-based [14], ODT [23], CVT [24], and unconstrained-DQN [27] are used for comparison in this paper. The experimental results are shown in Table 4, where Min. \(\theta\) is the minimum angle, Max. \(\theta\) is the maximum angle, (a) is the number of elements with angles less than 30 degrees, (b) is the number of elements with angles greater than 100 degrees, q is the quality of the elements calculated from the aspect ratio, Min. q is the minimum quality, and Avg. q is the average quality. In the following, (a) and (b) are collectively called the number of poor elements. Figure 19: The mesh after smoothing using the method in this paper. (**a**) Example 1; (**b**) Example 2; (**c**) Example 3; (**d**) and Example 4. Figure 18: Initial mesh. (**a**) Example 1; (**b**) Example 2; (**c**) Example 3; and (**d**) Example 4. As can be seen from Table 4, Laplacian smoothing has the shortest running time and the highest average quality. Still, it can lead to ineffective element quality improvement due to the geometric shape of some regions. Based on the minimum quality and the number of poor elements of Example 3, it can be found that Laplacian smoothing is prone to over-optimization for higher-quality meshes like Example 3, resulting in an increase in the number of poor elements. The angle-based smoothing method has a significantly higher running time due to the need to calculate the top angle of the polygon and, except for Example 3, provides good control of the maximum angle compared to Laplacian smoothing. The CVT algorithm has less running time than the ODT algorithm; in terms of average quality, the CVT algorithm is slightly better than the ODT algorithm. The ODT algorithm is not significant for the optimization of the maximum angle. Constraint DQN has a longer running time compared to the above methods. However, due to the autonomous learning nature of the agent in reinforcement learning, the number of poor elements can be effectively reduced compared to the other methods mentioned above when optimizing a mesh in a training set like Example 2. In addition, the method fails to produce better optimization results for three examples outside the training set, demonstrating that the generalization ability of the constrained DQN model needs to be improved. The algorithm in this paper has an increased running time relative to the first four methods because it improves the constrained DQN. However, the optimized mesh has the largest minimum angle, the smallest maximum angle, the smallest number of poor elements, and the highest minimum quality compared to the meshes optimized using the other five algorithms. Additionally, the average quality of the four examples optimized with the proposed method has improved compared to the constrained DQN and reached a higher level than other methods. However, since the algorithm in this paper improves