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.

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.

Foreword 9Introduction 11Chapter 1 Finite Elements and Shape Functions 151.1 Basic concepts 151.2 Shape functions, complete elements 181.3 Shape functions, reduced elements 261.4 Shape functions, rational elements 49Chapter 2 Lagrange and Bézier Interpolants 532.1 Lagrange-Bézier analogy 542.2 Lagrange functions expressed in Bézier forms 552.3 Bézier polynomials expressed in Lagrangian form 662.4 Application to curves 662.5 Application to patches 712.6 Reduced elements 74Chapter 3 Geometric Elements and Geometric Validity 953.1 Two-dimensional elements 963.2 Surface elements 1053.3 Volumetric elements 1053.4 Control points based on nodes 1113.5 Reduced elements 1153.6 Rational elements 121Chapter 4 Triangulation 1414.1 Triangulation, definitions, basic concepts and natural entities 1424.2 Topology and local topological modifications 1464.3 Enriched data structures 1514.4 Construction of natural entities 1534.5 Triangulation, construction methods 1564.6 The incremental method, a generic method 159Chapter 5 Delaunay Triangulation 1655.1 History 1665.2 Definitions and properties 1685.3 The incremental method for Delaunay 1755.4 Other methods of construction 1815.5 Variants 1865.6 Anisotropy 188Chapter 6 Triangulation and Constraints 1936.1 Triangulation of a domain 1946.2 Delaunay Triangulation "Delaunay admissibility" 2146.3 Triangulation of a variety 2196.4 Topological invariants (triangles and tetrahedra) 222Chapter 7 Geometric Modeling: Methods 2337.1 Implicit or explicit form (CAD), starting from an analytical definition 2347.2 Starting from a discretization or triangulation, discrete--> continuous 2467.3 Starting from a point cloud, discrete--> discrete 2787.4 Extraction of characteristic points and characteristic lines 302Chapter 8 Geometric Modeling: Examples 3058.1 Geometric modeling of parametric patches 3068.2 Characteristic lines of a discrete surface 3118.3 Parametrization of a surface patch through unfolding 3118.4 Geometric simplification of a surface triangulation 3248.5 Geometric support for a discrete surface 3258.6 Discrete reconstruction of a digitized object or environment 330Chapter 9 A Few Basic Algorithms and Formulae 3439.1 Subdivision of an entity (De Casteljau) 3449.2 Computing control coefficients (higher order elements) 3489.3 Algorithms for the insertion of a point (Delaunay) 3519.4 Construction of neighboring relationships, balls and shells 3579.5 Localization problems 3639.6 Some formulae 367Conclusions and Perspectives 369Bibliography 371Index 377