## 4 Experimental and Discussion
### Performance of the Construction Algorithm  
Algorithm 1 is supposed to transform an arbitrary triangular mesh into a Delaunay mesh by inserting a finite number of vertices. For each NLD edge in the original mesh, the inserting operation may change the local Delaunay property of the edge. When there are narrow triangles connected to the NLD edge, it will result in more inserting vertices. As mentioned in Section 3.2, the control factor \(\delta\) is defined to identify the narrow triangles and the improved face collapsing operation is used to handle this situation. Thus, for the construction algorithm of Delaunay mesh, \(\delta\) is the most important parameter that determines the space complexity, represented by the number of vertices of the constructed Delaunay mesh. The theoretical lower bound of \(\delta\) is \(l_{min}\cdot\sin(\theta_{min})\). In this experiment, \(\delta=2\cdot Average\_length\cdot\sin 15^{\circ}\) was selected as the suitable empirical parameter for the algorithms, which was able to achieve good performance for all the experimental meshes. In Figure 8, a triangle mesh \"dinosaur\" is used to demonstrate the effects of variable \(\delta\). We take \(l_{min}\cdot\sin(\theta_{min})\) as the smallest value and \(2*Average\_length*\sin 15^{\circ}\) as the largest value, and select five data in that range to displace the relationship between the number of vertices and \(\delta\). The rendering results of the smallest and largest values are shown. As shown in Figure 8a, the red curve depicts the relation between the control factor \(\delta\) and the number of vertices of the constructed Delaunay mesh. With the increase of control factor \(\delta\), the judgment criterion of the narrow triangles is amplified. Fewer vertices will be inserted because the face collapsing operation will be used before that. The geometric feature information of the triangle mesh is represented by the parameter \(r\), which is the ratio of surface area to volume of the triangular mesh. As the control factor \(\delta\) increases, the change of the parameter \(r\) remains almost constant. This indicates that the variation of the control factor \(\delta\) within a reasonable range has little effect on the geometric feature information of the Delaunay mesh. The constructed Delaunay meshes obtained by the two special control factors are rendered as shown in Figure 8b. When the control factor is set as \(\delta=l_{min}\cdot\sin(\theta_{min})\), more vertices will be inserted because the narrow triangles generated from the face collapsing operation are too small. If the control factor is set as \(\delta=2*Average\_length*\sin 15^{\circ}\), the density of vertices in local areas can be effectively improved. Figure 8: The results of parameter \(\delta\) selection with the model of “dinosaur”. The mesh renderings demonstrate the preservation of the geometric feature information. The thermal diffusion renderings are performed to illustrate the calculation effectiveness in numerical simulations. In Figure 9a, the original mesh \"refinement\" has 510 vertices. Due to the presence of many NLD edges in the original mesh, the results of its thermal diffusion simulation have many errors. For example, in the region A, the isotherms represented by the black line are not consistent with uniform diffusion. In the region B, necessary data information is missing because of too few vertices. The above two conditions show that a non-Delaunay mesh has the lower accuracy in numerical simulation. Figure 9b shows the constructed Delaunay mesh and the thermal diffusion rendering of it. After the processing of Algorithm 1, the Delaunay mesh with 688 vertices is obtained. It can be demonstrated that the constructed Delaunay mesh with 688 vertices can avoid the errors in the thermal diffusion simulation compared to results of the original mesh. In addition, we can find by the geometric feature parameter \(r\) that the algorithm proposed in this paper has a better preservation of the geometric feature information of the mesh.