Meshing, geometric modeling and numerical simulation 1. Form functions, triangulations

✍ Scribed by Borouchaki H., George P.L

• Publisher: ISTE Press
• Year: 2017
• Tongue: English
• Leaves: 379
• Category: Library

✦ Table of Contents

Half-Title Page......Page 1
Dedication......Page 2
Title Page......Page 3
Copyright Page......Page 4
Contents......Page 5
Foreword......Page 9
Introduction......Page 11
1.1. Basic concepts......Page 15
1.2.1. Generic expression of shape functions......Page 18
1.2.2. Explicit expression for degrees 1?3......Page 22
1.3. Shape functions, reduced elements......Page 26
1.3.1. Simplices, triangles and tetrahedra......Page 27
1.3.2. Tensor elements, quadrilateral and hexahedral elements......Page 31
1.3.3. Other elements, prisms and pyramids......Page 48
1.4.1. Rational triangle with a degree of 2 or arbitrary degree......Page 49
1.4.3. General case, B-splines or Nurbs elements......Page 50
Chapter 2: Lagrange and Bézier Interpolants......Page 52
2.1. Lagrange?Bézier analogy......Page 53
2.2.1. The case of tensors, natural coordinates......Page 54
2.2.2. Simplicial case, barycentric coordinates......Page 62
2.4. Application to curves......Page 65
2.4.1. Bézier expression for a Lagrange curve......Page 66
2.4.2. Lagrangian expression for a Bézier curve......Page 69
2.5.1. Bézier expression for a patch in Lagrangian form......Page 70
2.5.2. Lagrangian expression for a patch in Bézier form......Page 72
2.6.1. The tensor case, Bézier expression for a reduced Lagrangian patch......Page 73
2.6.2. The tensor case, definition of reduced Bézier patches......Page 81
2.6.3. The tensor case, Lagrangian expression of a reduced Bézier patch......Page 89
2.6.4. The case of simplices......Page 91
Chapter 3: Geometric Elements and Geometric Validity......Page 94
3.1. Two-dimensional elements......Page 95
3.3. Volumetric elements......Page 104
3.4. Control points based on nodes......Page 110
3.5.1. Simplices, triangles and tetrahedra......Page 114
3.5.2. Tensor elements, quadrilaterals and hexahedra......Page 115
3.5.3. Other elements, prisms and pyramids......Page 119
3.6.2. Degree 2, working on the arc of a circle......Page 120
3.6.3. Application to the analysis of rational elements......Page 122
3.6.4. On the use of rational elements or more......Page 137
Chapter 4: Triangulation......Page 139
4.1.1. Definitions and basic concepts First-order simplices......Page 140
4.1.4. A shell of a k-face......Page 143
4.2. Topology and local topological modifications......Page 144
4.2.3. Flipping an edge in three dimensions......Page 146
4.2.4. Other flips?......Page 148
4.3.1. Minimal structure......Page 149
4.3.2. Enriched structure......Page 150
4.4. Construction of natural entities......Page 151
4.5. Triangulation, construction methods......Page 154
4.6. The incremental method, a generic method......Page 157
4.6.1. Naive triangulation......Page 158
4.6.2. Delaunay triangulation......Page 161
Chapter 5: Delaunay Triangulation......Page 163
5.1. History......Page 164
5.2. Definitions and properties......Page 166
5.3. The incremental method for Delaunay......Page 173
5.4. Other methods of construction......Page 179
5.5. Variants......Page 184
5.6. Anisotropy......Page 186
Chapter 6: Triangulation and Constraints......Page 191
6.1. Triangulation of a domain......Page 192
6.1.1. Triangulation of a domain in two dimensions......Page 193
6.1.2. Triangulation of a domain in three dimensions......Page 200
6.2. Delaunay Triangulation “Delaunay admissibility?......Page 212
6.3. Triangulation of a variety......Page 217
6.4. Topological invariants triangles and tetrahedra......Page 220
Chapter 7: Geometric Modeling: Methods......Page 231
7.1.1. Modeling an implicit curve, continuous ℿ discrete......Page 232
7.1.2. Modeling a parametric curve......Page 235
7.1.3. Modeling an implicit surface......Page 236
7.1.4. Modeling of a parametric surface......Page 240
7.2. Starting from a discretization or triangulation, discrete ℿ continuous......Page 244
7.2.1. Case of a curve......Page 245
7.2.2. The case of a surface......Page 251
7.3.1. The case of a curve in two dimensions......Page 276
7.3.2. The case of a surface......Page 281
7.4. Extraction of characteristic points and characteristic lines......Page 300
Chapter 8: Geometric Modeling: Examples......Page 302
8.1. Geometric modeling of parametric patches......Page 303
8.3. Parametrization of a surface patch through unfolding......Page 308
8.4. Geometric simplification of a surface triangulation......Page 321
8.5. Geometric support for a discrete surface......Page 322
8.6. Discrete reconstruction of a digitized object or environment......Page 327
Chapter 9: A Few Basic Algorithms and Formulae......Page 340
9.1.1. Subdivision of a curve......Page 341
9.1.2. Subdivision of a patch......Page 342
9.2. Computing control coefficients higher order elements......Page 345
9.3. Algorithms for the insertion of a point Delaunay......Page 348
9.3.1. Classic algorithm......Page 349
9.3.2. Modified algorithms......Page 352
9.4.1. Neighboring relationships......Page 354
9.4.2. Construction of the ball of a vertex......Page 356
9.4.3. Construction of the shell of an edge......Page 358
9.5.1. Triangulations or simplicial meshes......Page 360
9.6. Some formulae......Page 364
Conclusions and Perspectives......Page 366
Other titles from iSTE in Numerical Methods in Engineering......Page 368
Bibliography......Page 371
Index......Page 377