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Delaunay Mesh Generation

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Lecture Notes on Delaunay Mesh Generation Jonathan Richard Shewchuk September 20, 1999 Department of Electrical Engineering and Computer Science University of California at Berkeley Berkeley, CA 94720 Copyright 1999 Jonathan Richard Shewchuk Effort sponsored in part by the Advanced Research Projects Agency and Rome Laboratory, Air Force Materiel Command, USAF under agreement number F30602-96-1-0287, in part by the National Science Foundation under Grant CMS-9318163, in part by the Natural Sci
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  Lecture Notes onDelaunay Mesh Generation Jonathan Richard Shewchuk September 20, 1999 Department of Electrical Engineering and Computer ScienceUniversity of California at BerkeleyBerkeley, CA 94720 Copyright 1999 Jonathan Richard Shewchuk  Effort sponsored in part by the Advanced Research Projects Agency and Rome Laboratory, Air Force Materiel Command,USAF under agreement number F30602-96-1-0287, in part by the National Science Foundation under Grant CMS-9318163, in partby the Natural Sciences and Engineering Research Council of Canada under a 1967 Science and Engineering Scholarship, and inpart by a grant fromthe Intel Corporation. The U.S.Government isauthorized toreproduce and distributereprintsfor Governmentalpurposes notwithstanding any copyright annotation thereon. The views and conclusions contained herein are those of the authorsand should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of theAdvanced Research Projects Agency, Rome Laboratory, or the U.S. Government.  Keywords:  triangular mesh generation, tetrahedral mesh generation, Delaunay triangulation, Delaunayrefinement, computational geometry  Contents 1 Introduction 1 1.1 Meshes and Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Desirable Properties of Meshes and Mesh Generators . . . . . . . . . . . . . . . . . . . . . 31.3 Why Unstructured Meshes? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 The Delaunay Triangulation and Unstructured Mesh Generation 9 2.1 Delaunay Triangulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.1 The Delaunay Triangulation in Two Dimensions . . . . . . . . . . . . . . . . . . . 102.1.2 Planar Straight Line Graphs and Constrained Delaunay Triangulations . . . . . . . . 172.1.3 Higher-Dimensional Delaunay Triangulations . . . . . . . . . . . . . . . . . . . . . 192.1.4 Piecewise Linear Complexes and Higher-Dimensional Constrained Delaunay Trian-gulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.1.5 Incremental Insertion Algorithms for Constructing Delaunay Triangulations . . . . . 302.1.6 The Analysis of Randomized Incremental Insertion in Two Dimensions . . . . . . . 342.1.7 Gift-Wrapping Algorithms for Constructing Delaunay Triangulations . . . . . . . . 392.1.8 Other Algorithms for Constructing Delaunay Triangulations . . . . . . . . . . . . . 402.2 Research in Unstructured Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . . 402.2.1 Delaunay Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.2.2 Advancing Front Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.2.3 Grid, Quadtree, and Octree Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 432.2.4 Smoothing and Topological Transformations . . . . . . . . . . . . . . . . . . . . . 44 3 Two-Dimensional Delaunay Refinement Algorithms for Quality Mesh Generation 49 3.1 A Quality Measure for Simplices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2 The Key Idea Behind Delaunay Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . 513.3 Chew’s First Delaunay Refinement Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 523.4 Ruppert’s Delaunay Refinement Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 543.4.1 Description of the Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.4.2 Local Feature Sizes of Planar Straight Line Graphs . . . . . . . . . . . . . . . . . . 603.4.3 Proof of Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.4.4 Proof of Good Grading and Size-Optimality . . . . . . . . . . . . . . . . . . . . . . 663.5 Algorithm and Implementation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.5.1 Segment Recovery and Nonconvex Triangulations . . . . . . . . . . . . . . . . . . 683.5.2 Implementing Delaunay Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . 723.6 Improving the Quality Bound in the Interior of the Mesh . . . . . . . . . . . . . . . . . . . 743.7 A Negative Result on Quality Triangulations of PSLGs that Have Small Angles . . . . . . . 763.8 Practical Handling of Small Input Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . 78i  4 Three-Dimensional Delaunay Refinement Algorithms 85 4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.2 A Three-Dimensional Delaunay Refinement Algorithm . . . . . . . . . . . . . . . . . . . . 874.2.1 Description of the Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.2.2 Local Feature Sizes of Piecewise Linear Complexes . . . . . . . . . . . . . . . . . 964.2.3 Proof of Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.2.4 Proof of Good Grading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.3 Improving the Quality Bound in the Interior of the Mesh . . . . . . . . . . . . . . . . . . . 1064.4 Sliver Removal by Delaunay Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Bibliography 111 ii

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