# Planar Delaunay Mesh Smoothing Method Based on Angle and a Deep Q-Network
## 4 Results and Discussion
### Comparative Analysis of Experiments
\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}{*}{1} & Initial & / & & 17.62 & 130.12 & 1434 & 1115 & 0.5001 & 0.7715 \\  & Laplacian & **0.2886** & & 26.86 & 110.70 & 3 & 14 & 0.6495 & **0.9608** \\  & Angle-based & 0.4520 & & 26.29 & 105.47 & 2 & 7 & 0.6840 & 0.9537 \\  & ODT & 0.3890 & 2027 & 19.00 & 126.50 & 42 & 54 & 0.5125 & 0.9109 \\  & CVT & 0.3390 & & 27.90 & 106.34 & 6 & 12 & 0.7249 & 0.9308 \\  & Unconstrained DQN & 1.5944 & & 25.78 & 114.74 & 24 & 37 & 0.6479 & 0.9185 \\  & Constrained DQN & 1.6218 & & **32.499** & **96.28** & **0** & **0** & **0.7944** & 0.9426 \\ \hline \multirow{8}{*}{2} & Initial & / & & 10.49 & 146.45 & 158 & 117 & 0.2902 & 0.8481 \\  & Laplacian & **0.1172** & & 28.05 & 110.21 & 6 & 7 & 0.6713 & **0.9239** \\  & Angle-based & 0.1461 & & 21.67 & 106.68 & 28 & 11 & 0.5972 & 0.9096 \\  & ODT & 0.1367 & 926 & 22.70 & 124.73 & 26 & 30 & 0.5517 & 0.9019 \\  & CVT & 0.1211 & & 27.14805 & 114.34 & 4 & 12 & 0.6454 & 0.9154 \\  & Unconstrained DQN & 0.3721 & & 26.05 & 104.54 & 2 & 6 & 0.6871 & 0.9179 \\  & Constrained DQN & 0.3986 & & **29.25** & **100.25** & **1** & **1** & **0.7249** & 0.9208 \\ \hline \multirow{8}{*}{3} & Initial & / & & 33.31 & 101.55 & 0 & 3 & 0.7713 & 0.9499 \\  & Laplacian & **0.3666** & & 30.27 & 102.95 & 0 & 8 & 0.7166 & **0.9563** \\  & Angle-based & 0.7174 & & 25.05 & 120.23 & 13 & 11 & 0.5888 & 0.9474 \\ \cline{1-1}  & ODT & 0.5078 & 2987 & 33.07 & 101.55 & 0 & 2 & 0.7713 & 0.9481 \\ \cline{1-1}  & CVT & 0.4544 & & 32.15 & 99.94 & 0 & 0 & 0.7850 & 0.9492 \\ \cline{1-1}  & Unconstrained DQN & 2.2742 & & 34.68 & 98.56 & 0 & 0 & 0.7978 & 0.9487 \\ \cline{1-1}  & Constrained DQN & **2.3175** & & **36.18** & **94.92** & **0** & **0** & **0.8229** & 0.9496 \\ \hline \multirow{8}{*}{4} & Initial & / & & 17.67 & 130.17 & 3789 & 2908 & 0.5001 & 0.7686 \\ \cline{1-1}  & Laplacian & **1.1419** & & 26.00 & 109.88 & 3 & 14 & 0.6560 & **0.9630** \\ \cline{1-1}  & Angle-based & 1.6108 & & 23.23 & 103.44 & 16 & 17 & 0.6196 & 0.9560 \\ \cline{1-1}  & ODT & 1.5983 & 5115 & 19.56 & 128.79 & 64 & 69 & 0.5041 & 0.9367 \\ \cline{1-1}  & CVT & 1.4202 & & 24.03 & 105.90 & 11 & 24 & 0.6490 & 0.9486 \\ \cline{1-1}  & Unconstrained DQN & 3.6788 & & 23.97 & 117.88 & 29 & 41 & 0.6365 & 0.9127 \\ \cline{1-1}  & Constrained DQN & 3.7988 & & **32.31** & **100.77** & **0** & **2** & **0.7539** & 0.9465 \\ \hline \hline \end{tabular} Note: bold means the best result; (a): number of angles \(<30^{\circ}\); (b): number of angles \(>100^{\circ}\).  
\end{table} 
Table 4: Comparison table of the experimental results. the minimum quality significantly, it will be more difficult to improve the average quality again with smoothing. Moreover, due to the high quality of the meshes after Laplacian smoothing in Examples 1-4, the paper further analyzes the optimization effect of each method on the minimum angle of the element and the poor element shape. Figure 20a shows two distorted elements, element 1 and element 2, in Example 1 after Laplacian smoothing, indicating that the Laplacian cannot handle poor elements well. From Figure 20b,e, it can be seen that the shapes of these two poor elements are still not well improved after being optimized using other methods. Compared with Figure 20e,f, the constrained DQN optimization will result in two poor elements: element 3 and element 4. However, the algorithm proposed in this paper can effectively improve the shape of elements 1 and 2 without generating poor elements. As shown in Figure 21a-e, after optimization, element 1 still maintains its narrow and elongated shape. And, this algorithm can effectively improve the shape of element 1, as shown in Figure 21f. Figure 21: Example 2—local comparison of each method after smoothing. (**a**) Laplacian; (**b**) angle-based; (**c**) ODT; (**d**) CVT; (**e**) unconstrained DQN; and (**f**) constrained DQN.