在進行有限元素分析之前，需將CAD模型轉化為實體網格，以提供分析軟體進行計算。目前的實體網格技術建構技術中，六面體網格可相較於四面體網格以較少的網格數量完成建構，進而降低分析的運算時間。而六面體的實體網格是以四邊形表面網格為基礎建構，為此本實驗室進行四邊形網格自動化建構技術之開發，然而在產出高品質的四邊形網格上，該技術中仍有可優化之空間。本研究提出了多種考慮模型中節點相鄰網格狀況的方法進行網格結構的優化，並結合網格平滑化方法，以達到提高整體網格品質的成效。首先，本研究針對節點不理想的狀況，依品質優化流程的程序逐一偵測與優化，使整體網格結構趨近於理想，本研究亦將上述觀念應用於三角形網格品質優化方法的開發。最後，為了確保整體四邊形品質優化方法的成效，本研究對比數個品質優化流程之網格建構結果的品質指標，得到成效較佳之流程作為本研究採用的四邊形網格品質優化流程。本研究對建構完成的網格進行品質優化，測試的案例中，節點連接性異常的狀況已被優化，且各案例的網格結構均得到優化，其中四邊形網格中節點為連接性理想狀況之比例提升至86.80%至96.56%不等，並有效減少低品質網格比例，最小內角不高於60°之比例降低為0.21%至2.28%不等，證明本研究可有效優化整體網格結構及網格之品質。

Before finite element analysis, the CAD model must be converted into a solid mesh to provide analysis software for calculation. Among current solid mesh techniques, hexahedral meshes require fewer elements than tetrahedral meshes, thereby reducing the analysis computation time. The hexahedral solid mesh is constructed based on the quadrilateral surface mesh. For this reason, our lab has developed an automatic quadrilateral meshing method. However, this method still has possibility for quality improvement. This study introduces several improvement techniques that consider the number of near meshes on a node to improve mesh structure, combined with mesh smoothing to enhance overall quality. The improvement process identifies and improve undesirable node conditions to guide the mesh toward an ideal structure. This study also applies the above concept to the development of triangular mesh quality improvement methods.To evaluate effectiveness, multiple improvement workflows were compared, and the best-performing process was adopted. The results show significant improvements: abnormal node connections were corrected, and ideal connectivity in quadrilateral meshes increased to 86.80%–96.56%. Additionally, the proportion of minimum angles below 60° was reduced to 0.21%–2.28%, proving the method's ability to enhance both mesh structure and quality.

[1] T. D. Blacker and M. B. Stephenson, “Paving: a new approach to automated quadratic mesh generation”, International Journal for Numerical Methods in Engineering, Vol. 32, No. 4, pp. 811-847, 1991.
[2] J. Y. Lai, P. Putrayudanto, Q. Wang, J. Huang, P.P. Song and Y.C. Tsai “Development of automatic quadratic meshing technique with unequal spreading intervals on outer and inner contours for IC CAD models”, System Innovation for an Artificial Intelligence Era, CRC Press, pp. 11-18, 2024.
[3] A. V. Skovpen, “Modified algorithm for unstructured quadrilateral meshing”, RFNC-VNIITF, Sep. 2004.
[4] Y. Liu, H. L. Xing and Z. Guan, “An indirect approach for automatic generation of quadrilateral meshes with arbitrary line constraints”, International Journal for Numerical Methods in Engineering, Vol. 87, No. 9, pp. 906–922, 2011.
[5] H. Xu, B. He, C. Zhang, H. Lin, X. Kuai and R. Guo, “Quality-preserving multilevel mesh generation for building models”, International Journal of Digital Earth , Vol. 17, No. 1, 2024.
[6] X. Liang, M. S. Ebeida and Y. Zhang, “Guaranteed-quality all-quadrilateral mesh generation with feature preservation”, Proceedings of the 18th International Meshing Roundtable, Springer, Berlin Heidelberg, 2009.
[7] K. F. Tchon, J. Dompierre and R. Scamarero, “Automated refinement of conformal quadrilateral and hexahedral meshes”, International Journal for Numerical Methods in Engineering, Vol. 59, No. 12, pp. 1539-1562, 2004.
[8] P. M. Knupp, “Algebraic mesh quality metrics”, SIAM Journal on Scientific Computing, Vol. 23, No. 1, pp. 193-218, 2001.
[9] R. V. Garimella, M. J. Shashkov and P. M. Knupp, “Triangular and quadrilateral surface mesh quality optimization using local parametrization”, Computer Methods in Applied Mechanics and Engineering, Vol. 193, No. 9-11, pp. 913-928, 2004.
[10] D. Dinkler and U. Kowalsky, “Introduction to finite element methods”, Springer Vieweg, 2024.
[11] M. L. Brewer, L. F. Diachin, P. M. Knupp, T. Leurent and D. J. Melander,
“The mesquite mesh quality improvement toolkit”, Internatioal Meshing Roundtable, 2003.
[12] S. A. Canann, S. N. Muthukrishnan and R. K. Phillips, “Topological improvement procedures for quadrilateral finite element meshes”, Engineering with Computers , Vol. 14, No. 2, pp. 168-177, 1998.
[13] M. L. Staten and S. A. Canann, “Post refinement element shape improvement for quadrilateral meshes”, ASME Applied Mechanics Division-Publications-AMD, Vol. 220, pp. 9-16, 1997.
[14] J. Y. Lai, J. S. Su, S. C. Tseng, H. J. Liou, S. J. Jheng, J. H. Huang, P. P. Song and Y. C. Tsai, “Quality improvement for an automatic quadratic mesh generation method for IC CAD models”, 2024 International Conference on Machining, Materials and Mechanical Technologies, Phan Thiet City. Vietnam, Sep. 11-15, 2024
[15] P. Kinney, “Cleanup: Improving quadrilateral finite element meshes’’, Proceedings of the 6th International Meshing Roundtable, pp. 437-447. 1997.
[16] B. D. Anderson, S. E. Benzley and S. J. Owen, “Automatic all quadrilateral mesh adaption through refinement and coarsening”, Proceedings of the 18th International Meshing Roundtable. Springer, Berlin Heidelberg, pp. 557-574, 2009.
[17] J. Docampo-Sánchez and R. Haimes, “A regularization approach for automatic quad mesh generation”, Proceedings of the 28th International Meshing Roundtable. Zenodo, pp. 296-307, 2020.
[18] B. D. Anderson, J. F. Shepherd, J. Daniels and S. E. Benzley, “Quadrilateral mesh improvement”, Research Note, 17th International Meshing Roundtable , 2008.
[19] K. Xu, X. Gao and G. Chen, “Hexahedral mesh quality improvement via edge-angle optimization”, Computers & Graphics, Vol. 70, pp. 17-27, 2018.
[20] J. Kim, M. Shin and W. Kang, “A derivative‐free mesh optimization algorithm for mesh quality improvement and untangling”, Mathematical Problems in Engineering , Vol. 2015, No. 1, 2015.
[21] M. N. Akram, L. Si and G. Chen, “An embedded polygon strategy for quality improvement of 2D quadrilateral meshes with boundaries”, Proceedings of the 16th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2021), Vol. 1, pp. 177-184, 2021.
[22] M. S. Joun and M. C. Lee, “Quadrilateral finite-element generation and mesh quality control for metal forming simulation”, International Journal for Numerical Methods in Engineering, Vol. 40, No. 21, pp. 4059-4075, 1997.
[23] T. Liu, M. Chen, Y. Song, H. Li and B. Lu , “Quality improvement of surface triangular mesh using a modified Laplacian smoothing approach avoiding intersection”, PLoS One, Vol. 12, No. 9, 2017.
[24] Y. Zhang, C. Bajaj and G. Xu, “Surface smoothing and quality improvement of quadrilateral/hexahedral meshes with geometric flow’’, Communications in Numerical Methods in Engineering, Vol. 25, No. 1, pp. 1-18, 2009.
[25] N. Wang, L. Zhang and X. Deng, “Unstructured surface mesh smoothing method based on deep reinforcement learning ”, Computational Mechanics, Vol. 73, No. 2, pp. 341-364, 2024.
[26] 陳定輝，「應用於非結構化四邊形網格建構之輪廓撒點與網格品質改善技術發展」，國立中央大學碩士論文，2023。
[27] 李奇勳，「非結構化四邊形網格自動建構研究」，國立中央大學碩士論文，2024。
[28] 王齊，「四邊形網格自動建構之多尺寸輪廓撒點技術研究」，國立中央大學碩士論文，2024。
[29] 蔡敬倫，「四邊形網格自動建構之網格品質改善研究」，國立中央大學碩士論文，2024。
[30] 吳弈翰，「IC CAD模型之混合結構化與非結構化四邊形網格自動建構技術發展」，國立中央大學碩士論文，2024。
[31] 梁秉傑，「不同網格類型對於IC CAD模型模流分析之影響探討」，國立中央大學碩士論文，2024。
[32] Rhinoceros. Available:https://www.rhino3d.com. [Accessed 20 May 2025]
[33] Moldex3D. Available:https://www.moldex3d.com. [Accessed 20 May 2025]
[34] 劉弘堅，「四形網格自動建構之輪廓灑點方法與四邊形網格轉三角形網格之研究」，國立中央大學碩士論文，2025。
[35] 曾思齊，「IC CAD模型之混合結構化與非結構化四邊形網格自動建構技術優化」，國立中央大學碩士論文，2025。
[36] 鄭憲杰，「應用IC CAD模型之非結構化四邊形網格自動建構與測試模組研究」，國立中央大學碩士論文，2025。