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STRUCTURAL ENGINEERING Lo Finite Finite F i Element n Element i t Mesh e Mesh E Generation l e Generation m This book provides a concise and comprehensive guide to the application e of finite element mesh generation over 2D domains, curved surfaces, n and 3D space. Organised according to the geometry and dimension of the problem domains, it develops from the basic meshing algorithms to the t most advanced schemes to deal with problems with specific requirements such as boundary conformity, adaptive and anisotropic elements, shape M qualities, and mesh optimization. e It sets out the fundamentals of popular techniques, including: • Delaunay triangulation s • Advancing-front (ADF) approach h • Quadtree/Octree techniques • Refinement and optimization-based strategies G From the geometrical and the topological aspects and their associated operations and inter-relationships, each approach is vividly described and e illustrated with examples. Beyond the algorithms, the book also explores the practice of using metric tensor and surface curvatures for generating n anisotropic meshes on parametric space. It presents results from research e including 3D anisotropic meshing, mesh generation over unbounded domains, meshing by means of intersection, re-meshing by Delaunay-ADF r approach, mesh refinement and optimization, generation of hexahedral a meshes, and large scale and parallel meshing, along with innovative unpublished meshing methods. The author provides illustrations of major t meshing algorithms, pseudo codes, and programming codes in C++ i o or FORTRAN. DANIEL S.H. LO n Y122197 ISBN: 978-0-415-69048-5 90000 A SPON BOOK 9 780415 690485 Y122197_Cover_Sheridan.indd All Pages 10/20/14 9:44 AM Finite Element Mesh Generation Finite Element Mesh Generation DANIEL S.H. LO A SPON BOOK CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2015 by S.H. 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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Contents Preface xvii Acknowledgements xix 1 Introduction 1 1.1 Finite element method 1 1.2 What is finite element mesh generation? 1 1.3 Why finite element mesh generation? 2 1.4 Problem definition, scope and philosophy: Science or art? 3 1.5 General strategies, robustness, difficulties and methodologies 4 1.6 Mathematics 4 1.7 Historical development 5 1.8 So far achieved and what lies ahead 7 1.9 Topics discussed in the chapters 8 2 Fundamentals 11 2.1 Introduction 11 2.2 Notations, symbols and abbreviations 12 2.2.1 Notations 12 2.2.2 Symbols 12 2.2.3 Abbreviations 13 2.3 Terminologies and data structures 14 2.3.1 Triangulation 14 2.3.2 Delaunay triangulation 14 2.3.3 Constrained triangulation 14 2.3.4 Mesh and FE mesh 14 2.3.5 Structured and unstructured meshes 15 2.3.6 Mixed and hybrid meshes 15 2.3.7 Discretised manifold 16 2.3.8 Control space 16 2.3.9 Adaptive mesh 16 2.3.10 Data structure 16 2.3.10.1 Nodal points 17 2.3.10.2 Boundary of a planar domain 17 v vi Contents 2.3.10.3 Boundary of a 3D domain 17 2.3.10.4 Node labelling of FEs 18 2.4 Geometrical operations and formulas 19 2.4.1 Distance from a point P to a line segment AB, d(P, AB) 19 2.4.2 Distance from a point P to a triangular facet ABC, d(P, ABC) 20 2.4.3 Distance between line segments in space, d(AB, CD) 20 2.4.4 Intersection between two line segments on a plane 21 2.4.4.1 Analytical method 21 2.4.4.2 Vectorial method 22 2.4.4.3 Parametric method 22 2.4.4.4 The max–min method 23 2.4.5 Solid angle 23 2.4.6 Normal at a node 25 2.4.7 Intersection between a line segment and a triangular facet 25 2.4.8 Distance between a line segment and a triangular facet in space, d(PQ, ABC) 26 2.4.9 Dividing an edge into segments 27 2.4.9.1 Element size is specified at nodal points 27 2.4.9.2 Element size is specified along the edge 28 2.4.10 γ Value of a tetrahedron cannot exceed the α value of its face 30 2.4.11 Determine whether a point is inside or outside of the problem domain 32 2.4.11.1 Two-dimensional domain 32 2.4.11.2 Three-dimensional domain 32 2.5 Topological operations and algorithms 33 2.5.1 Find the neighbouring elements of a triangular mesh 33 2.5.2 Find the neighbouring elements of a tetrahedral mesh 34 2.5.3 Find the elements connected to each node in a mesh 35 2.5.4 Find the edges (unique line segments) of a triangular mesh 37 2.5.5 Find the faces (unique triangular facets) of a tetrahedral mesh 38 2.5.6 Find the edges (unique line segments) of a tetrahedral mesh 38 2.5.7 Retrieve the boundary (loop of line segments) of a triangular mesh 39 2.5.8 Retrieve the boundary (triangular facets) of a tetrahedral mesh 40 2.5.9 Find the tetrahedral elements connected to an edge 41 2.5.10 Delete flagged elements from a tetrahedral mesh 41 2.5.11 Find the tetrahedral elements within the boundary surface 42 2.6 Sorting 43 2.6.1 Bubble sort 43 2.6.2 Insertion sort 44 2.6.3 Quick sort 45 2.6.4 Bin sort 47 2.6.5 Comparison of the sorting methods 50 2.7 Background grid 53 2.7.1 Regular (uniform) grid (2D) 53 2.7.2 Regular (uniform) grid (3D) 54 2.7.3 Searching for general objects by means of a background grid 56 Contents vii 2.7.3.1 Method 1: Search by neighbourhood 56 2.7.3.2 Method 2: By checking the distance 57 2.7.3.3 Method 3: By elimination 57 2.7.4 Determine the cells intersected by a triangular facet 58 2.7.5 Irregular grid 59 2.7.6 Quadtree 61 2.7.7 Octree 66 2.7.8 Kd-tree 69 2.7.8.1 Construction of 2-d tree 69 2.7.8.2 Construction of 3-d tree 73 3 Mesh generation on planar domain 77 3.1 Introduction 77 3.2 Structured mesh on planar domain 78 3.2.1 FE interpolation 78 3.2.2 Transfinite mapping 81 3.2.3 Drag method and sweeping method 83 3.3 Unstructured mesh on planar domain 83 3.3.1 MG using contour 84 3.3.2 Coring method 84 3.3.3 Mesh refinement by subdivision 85 3.4 Meshing by quadtree decomposition 86 3.4.1 Boundary specification 86 3.4.2 Spatial partition of the bounding box 86 3.4.3 Creation of internal points and elements 90 3.4.4 Connection of the interior elements with the boundary segments 90 3.5 Delaunay triangulation 91 3.5.1 Introduction 91 3.5.1.1 The convex hull of a given point set 92 3.5.2 Properties of DT 92 3.5.3 Time complexity in the construction of DT 93 3.5.4 FE meshing by DT 93 3.5.4.1 Fundamentals and strategy 95 3.5.4.2 Point insertion algorithm 97 3.5.4.3 Determination of the CORE 98 3.5.4.4 Searching for the BASE 98 3.5.4.5 Steps in locating the BASE 100 3.5.4.6 Circumcentre and circumcircle 101 3.5.4.7 Procedure for the creation of the CORE 102 3.5.4.8 Correction of the CORE 105 3.5.4.9 Construction of triangles in the CORE and establishment of the adjacency relationship 106 3.5.5 Details in computer programming 106 3.5.6 Generation of interior points 108 3.5.6.1 Specification of nodal spacing 108 viii Contents 3.5.6.2 Control space 108 3.5.6.3 Element size based on domain boundary 109 3.5.6.4 Element size based on a previous analysis 110 3.5.6.5 Creation of interior points 111 3.5.7 Boundary recovery in two dimensions 117 3.5.7.1 Determination of the pipe 118 3.5.7.2 Divide-and-conquer 118 3.5.7.3 Swapping of diagonals 119 3.5.8 Closure 120 3.6 Advancing front approach 121 3.6.1 Introduction 121 3.6.2 Adaptive meshing by the AFT 123 3.6.3 Use of background grid 128 3.6.3.1 Construction of the background grid 129 3.6.3.2 Setting the size of each cell in the grid 129 3.6.3.3 Marking and unmarking cells intersected by a line segment 131 3.6.3.4 Marking cells intersected by a line segment L 132 3.6.3.5 Unmarking cells intersected by a line segment L 133 3.6.3.6 Search for nearby line segments with the help of the background grid 133 3.6.3.7 Updating boundary segments 133 3.6.4 Test examples 134 3.6.5 Closure 139 3.7 Meshing by a combined scheme of DT and ADF approach 140 3.7.1 Introduction 140 3.7.2 Advancing-front–Delaunay scheme 140 3.7.2.1 DT of non-convex planar domains 140 3.7.2.2 Delaunay and non-Delaunay triangles 140 3.7.2.3 Delaunay and non-Delaunay segments 141 3.7.2.4 Triangulation process 141 3.7.2.5 Updating Γ and Γ 142 1 2 3.7.2.6 Existence and Delaunay property of the triangulation 142 3.7.2.7 Delaunay property of triangulation 143 3.7.3 Delaunay–advancing-front scheme 144 3.8 Enhanced quadtree meshing 147 3.8.1 Quadtree partition of the bounding box 148 3.8.2 Removal of quadrilaterals near domain boundary 148 3.8.3 Boundary recovery for triangulation 149 3.8.4 Advancing-front MG 149 3.9 Quadrilateral mesh 151 3.9.1 Direct method 151 3.9.2 Indirect method 152 3.9.3 Quadrilateral-dominated mesh 153 3.9.3.1 Distortion coefficient β of a quadrilateral 154 3.9.3.2 Merging of triangles to form quadrilaterals 155