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DOI:10.6342/NTU.2004.02087

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 在三角形和四面體網格生成方法中，Delaunay準則為最廣泛應用之基準原則；然而，網格生成之元素品質仍尚未獲得解決。鑑於前人提出之refinement方法，主要乃以插入形狀低劣之三角形外接圓之圓心，以提高網格品質；本文提出一新的思考觀點，乃試圖插點於其影響區域之中心，以提高其品質。在二維方面，插點之有效區域乃指網格區域內之所有三角形外接圓或四面體外接球包含此點之三角形或四面體集合。此方法將使插點對應於其有效區域所構成之各個新元素之角度具等分之性質，使得相較於其他插入點而言較不易產生形狀低劣之三角形。本文提出之方法已實作於二維且經數值結果顯示，於二維之三角形元素具有品質提昇之能力。
 The Delaunay criterion is the most popular criterion for developing triangular and tetrahedral meshes, but the quality of a mesh has not been totally resolved. In this thesis, a novel method is proposed to raise the quality of triangular mesh generation. Whereas previous refinement methods mainly focus on inserting a new vertex into the circumcenter of a skinny triangle or tetrahedron, the proposed method is striving to insert a vertex near the geometric center of an effective region. The effective region is combined by all the effective triangles or tetrahedra whose circumcircle contains the inserted vertex. Using the proposed method, all the newly generated triangles and tetrahedra will have the property of equally divided angles. The proposed method is implemented for two dimensions. Numerical results using the proposed method show its capability to generate superior quality of triangles in two dimensions.
 Chapter 1: Introduction to Mesh Generation 11.1 Background and Motivation 11.1.1 Mesh Generation in Finite Element Analysis Procedure 11.1.2 Structured and Unstructured Mesh 21.2 Objectives and Thesis Organization 4Chapter 2: Triangulation Methods Using Delaunay Criterion 52.1 Overview 52.2 Delaunay Triangulation Methods 72.3 Delaunay Refinement Methods 92.3.1 Chew's Delaunay Refinement Method 92.3.2 Ruppert's Delaunay Refinement Method 102.3.3 Discussion 132.4 Summary 13Chapter 3: Mesh Generation and Quality Assurance 153.1 Delaunay Triangulation Implementation 153.2 Mesh Quality Measurement and Assessment 173.3 Slivers in Three Dimensions 193.3.1 The Sliver Problem 193.3.2 Solving the Sliver Problem 213.3.2.1 Concept 213.3.2.2 Implementation 243.4 Summary 31Chapter 4: Case Studies 334.1 Numerical Examples 334.2 Discussion 41Chapter 5: Conclusions and Future Studies 435.1 Conclusions 435.2 Future Studies 43References 45Appendix: Coordinates of Numerical Examples Used in Chapter 4 49
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