## 4 Experimental and Discussion
### Performance of the Simplification Algorithm  
After the construction algorithm of Delaunay mesh via Algorithm 1, any triangle mesh can be transformed to the Delaunay mesh. However, the inserted vertices may lead to the increase of the mesh complexity and the data storage. Algorithm 2 provides a Delaunay mesh simplification algorithm that can maintain the local Delaunay property of the Delaunay mesh and transform the non-Delaunay mesh to the Delaunay mesh. A model of \"bumpy\" is used to show the effect of the simplification algorithm on the preservation of the geometric information. The original mesh with 1250 vertices is the Delaunay mesh, and the improved vertex collapsing operation is used to simplify it. As shown in Figure 10, the un-simplified mesh and the simplified meshes with 80% vertices, 40% vertices, and 20% vertices are displayed, respectively. Both the visualization and the  
Figure 9: The comparison of the construction algorithms with numerical simulation results, where regions A and B are significant changes. ratio \(r\), surface-area-to-volume, demonstrate that there is better preservation of geometric information as the number of vertices of the simplified mesh is reduced. Similarly, the simplification algorithm also works for non-Delaunay mesh. The variation trends of the number of vertices and the number of NLD edges with the iteration steps are shown in Figure 11. The blue curve indicates the trend of vertices versus iteration steps, and the red curve represents the trend of NLD edges versus iteration steps. The input mesh of Algorithm 2 is a non-Delaunay mesh with 9397 vertices and 1082 NLD edges, which are illustrated as A. The processing of algorithm can be divided into two parts. According to the simplification algorithm described in Section 3.3, for a non-Delaunay mesh, it firstly classifies all vertices into two types. For all of the non-collapsible vertices, it can be found that the number of the mesh vertices increased with the number of iterations of the algorithm in region-I of Figure 11a. Subsequently, when all the vertices in the mesh are collapsible vertices, which is illustrated as B, the number of vertices start to decrease. As the iterative step proceeds, the Delaunay mesh based on minimal volume destruction can be obtained, which is illustrated as C. In the region-II of Figure 11a, the NLD edges disappear, and the simplification algorithm is executed repeatedly until the user defined value is satisfied. Figure 11: The results after the hybrid construction and simplification algorithm with the model of “rocker”. Figure 10: The simplification results for a Delaunay mesh using the model of “bumpy”. The visualization of the meshes of A, B, C and D is shown in Figure 11b, respectively. As mentioned above, A is the original mesh with 9397 vertices and 1082 NLD-edges, B is a non-Delaunay meshes processed by the vertices inserting operation, C is the Delaunay mesh obtained by the Algorithm 2 with 8109 vertices, and D is a simplified mesh with 3000 vertices. Especially, the thermal diffusion simulation results for C and D are also shown in Figure 11c, respectively. As the number of vertices reduce, the geometric features of the original mesh can still be effectively displayed. These results demonstrate the effectiveness of the simplification algorithm in preserving geometric information. In addition, the thermal diffusion rendering of the two simplified meshes also show that the constructed Delaunay meshes can also achieve a good performance in numerical simulation. For the case of D, it is possible to obtain similar numerical simulation results in the case of C with fewer vertices. It will enable the simplification of both the Delaunay mesh and the non-Delaunay mesh without destroying the performance of the thermal diffusion simulation. In order to further illustrate the generality of Algorithm 2, three more experiments are conducted and shown in Figure 12. The processed models are \"elephant\", \"horse\", and \"head\" of non-Delaunay mesh, respectively.