## 5 Conclusions  
In this paper, the Delaunay construction and simplification algorithms were proposed to improve the model complexity and simulating accuracy. Firstly, we proposed two improved remeshing operations of the vertex inserting and the vertex collapsing. Then, the improved algorithms for the construction and simplification of Delaunay mesh based on minimal volume destruction were presented. The Delaunay construction algorithm can transform the arbitrary mesh into the Delaunay mesh by inserting finite numbers of vertices. The simplification algorithm can classify the vertices into collapsible and non-collapsible vertices, which will be performed differently according to the types of vertices. Moreover, the renderings of triangular meshes demonstrated the accuracy of the proposed algorithms. Finally, the experimental results showed that the algorithm can improve the quality of the arbitrary meshes. It is also noteworthy that, when the original model with fewer vertices is simplified, the vertex collapsing may result in the undesired volume destruction. The similar limitation occurs when the meshes with a large number of vertices were simplified to a high level, for example to 20% of the original vertices. Then, the cost of the algorithm \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Meshes** & **Vertices** & **Volume** & **Surface Area** & \(r\) & \(\Delta r\) \\ \hline rocker\_original & 9397 & 0.0425 & 1.2965 & 30.4965 & 0 \\ rocker\_our & 3000 & 0.0423 & 1.28926 & 30.4283 & 0.00223 \\ rocker\_qem [3] & 3000 & 0.0424 & 1.30145 & 30.6283 & 0.00432 \\ rocker\_liu [13] & 3000 & 0.0423 & 1.28805 & 30.4141 & 0.00270 \\ \hline elephant\_original & 2775 & 0.046201 & 1.24496 & 26.9465 & 0 \\ elephant\_our & 1377 & 0.045656 & 1.22927 & 26.925 & 0.00079 \\ elephant\_qem [3] & 1377 & 0.04609 & 1.24725 & 27.061 & 0.00424 \\ elephant\_liu [13] & 1377 & 0.045757 & 1.24068 & 27.11 & 0.00606 \\ \hline horse\_original & 2000 & 0.157444 & 2.49807 & 15.8664 & 0 \\ horse\_our & 878 & 0.153055 & 2.44337 & 15.964 & 0.00615 \\ horse\_qem [3] & 878 & 0.156998 & 2.50834 & 15.9769 & 0.00696 \\ horse\_liu [13] & 878 & 0.153371 & 2.46727 & 16.087 & 0.01390 \\ \hline head\_original & 664 & 5.62153 & 0.183606 & 0.032661 & 0 \\ head\_our & 516 & 5.60071 & 0.181661 & 0.032435 & **0.00691** \\ head\_qem [3] & 516 & 5.61976 & 0.183612 & 0.032673 & 0.00035 \\ head\_liu [13] & 516 & 5.5796 & 0.18225 & 0.032664 & **0.00007** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The geometric feature information of the meshes. will increase significantly and the geometric feature information of the simplified mesh may degenerate. To minimize the destruction to the original mesh, the control factor \(\delta\) was designed to judge the narrow triangle. At present, the value of the control factor \(\delta\) should be based on the user defined values. We will try to realize the automatic and real-time optimization of the control factor in the future study. In our future work, the proposed algorithms will be applied to 3D model processing for stereo-lithography additive manufacturing and experiments will be conducted to evaluate the performance. The paper's initial idea was proposed by [PERSON], [PERSON], [PERSON] and [PERSON]. The conceptualization is by [PERSON] and [PERSON]; the software work and analysis were performed by [PERSON], [PERSON], [PERSON] and [PERSON]; Supervision by [PERSON] and [PERSON]. The original draft preparation was by [PERSON] and [PERSON] and the review and editing were performed by [PERSON] and [PERSON]. All authors have read and agreed to the published version of the manuscript. The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (Grant No. 52005479), the Technology and Engineering Center for Space Utilization, Chinese Academy of Sciences (Project No. CSU-JJKT-2020-6) and the Scientific Instrument Developing Project of the Chinese Academy of Sciences (Grant No. YJKYYQ20200068).