Meshing, Geometric Modeling and Numerical Simulation 1

Triangulations, and more precisely meshes, are at the heart of many problems relating to a wide variety of scientific disciplines, and in particular numerical simulations of all kinds of physical phenomena. In numerical simulations, the functional spaces of approximation used to search for solutions are defined from meshes, and in this sense these meshes play a fundamental role. This strong link between the meshes and functional spaces leads us to consider advanced simulation methods in which the meshes are adapted to the behaviors of the underlying physical phenomena. This book presents the basic elements of this meshing vision.

Auteur

Houman Borouchaki, University of Technology of Troyes, France.

Paul-Louis George, French Institute for Research in Computer Science and Automation, France.

Contenu

Foreword 9

Introduction 11

Chapter 1 Finite Elements and Shape Functions 15

1.1 Basic concepts 15

1.2 Shape functions, complete elements 18

1.3 Shape functions, reduced elements 26

1.4 Shape functions, rational elements 49

Chapter 2 Lagrange and Bézier Interpolants 53

2.1 Lagrange-Bézier analogy 54

2.2 Lagrange functions expressed in Bézier forms 55

2.3 Bézier polynomials expressed in Lagrangian form 66

2.4 Application to curves 66

2.5 Application to patches 71

2.6 Reduced elements 74

Chapter 3 Geometric Elements and Geometric Validity 95

3.1 Two-dimensional elements 96

3.2 Surface elements 105

3.3 Volumetric elements 105

3.4 Control points based on nodes 111

3.5 Reduced elements 115

3.6 Rational elements 121

Chapter 4 Triangulation 141

4.1 Triangulation, definitions, basic concepts and natural entities 142

4.2 Topology and local topological modifications 146

4.3 Enriched data structures 151

4.4 Construction of natural entities 153

4.5 Triangulation, construction methods 156

4.6 The incremental method, a generic method 159

Chapter 5 Delaunay Triangulation 165

5.1 History 166

5.2 Definitions and properties 168

5.3 The incremental method for Delaunay 175

5.4 Other methods of construction 181

5.5 Variants 186

5.6 Anisotropy 188

Chapter 6 Triangulation and Constraints 193

6.1 Triangulation of a domain 194

6.2 Delaunay Triangulation "Delaunay admissibility" 214

6.3 Triangulation of a variety 219

6.4 Topological invariants (triangles and tetrahedra) 222

Chapter 7 Geometric Modeling: Methods 233

7.1 Implicit or explicit form (CAD), starting from an analytical definition 234

7.2 Starting from a discretization or triangulation, discrete continuous 246

7.3 Starting from a point cloud, discrete discrete 278

7.4 Extraction of characteristic points and characteristic lines 302

Chapter 8 Geometric Modeling: Examples 305

8.1 Geometric modeling of parametric patches 306

8.2 Characteristic lines of a discrete surface 311

8.3 Parametrization of a surface patch through unfolding 311

8.4 Geometric simplification of a surface triangulation 324

8.5 Geometric support for a discrete surface 325

8.6 Discrete reconstruction of a digitized object or environment 330

Chapter 9 A Few Basic Algorithms and Formulae 343

9.1 Subdivision of an entity (De Casteljau) 344

9.2 Computing control coefficients (higher order elements) 348

9.3 Algorithms for the insertion of a point (Delaunay) 351

9.4 Construction of neighboring relationships, balls and shells 357

9.5 Localization problems 363

9.6 Some formulae 367

Conclusions and Perspectives 369

Bibliography 371

Index 377