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Meshing, Geometric Modeling and Numerical Simulation 1

Book Description

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.

Table of Contents

  1. Cover
  2. Dedication
  3. Title
  4. Copyright
  5. Foreword
  6. Introduction
  7. Chapter 1: Finite Elements and Shape Functions
    1. 1.1. Basic concepts
    2. 1.2. Shape functions, complete elements
    3. 1.3. Shape functions, reduced elements
    4. 1.4. Shape functions, rational elements
  8. Chapter 2: Lagrange and Bézier Interpolants
    1. 2.1. Lagrange–Bézier analogy
    2. 2.2. Lagrange functions expressed in Bézier forms
    3. 2.3. Bézier polynomials expressed in Lagrangian form
    4. 2.4. Application to curves
    5. 2.5. Application to patches
    6. 2.6. Reduced elements
  9. Chapter 3: Geometric Elements and Geometric Validity
    1. 3.1. Two-dimensional elements
    2. 3.2. Surface elements
    3. 3.3. Volumetric elements
    4. 3.4. Control points based on nodes
    5. 3.5. Reduced elements
    6. 3.6. Rational elements
  10. Chapter 4: Triangulation
    1. 4.1. Triangulation, definitions, basic concepts and natural entities
    2. 4.2. Topology and local topological modifications
    3. 4.3. Enriched data structures
    4. 4.4. Construction of natural entities
    5. 4.5. Triangulation, construction methods
    6. 4.6. The incremental method, a generic method
  11. Chapter 5: Delaunay Triangulation
    1. 5.1. History
    2. 5.2. Definitions and properties
    3. 5.3. The incremental method for Delaunay
    4. 5.4. Other methods of construction
    5. 5.5. Variants
    6. 5.6. Anisotropy
  12. Chapter 6: Triangulation and Constraints
    1. 6.1. Triangulation of a domain
    2. 6.2. Delaunay Triangulation “Delaunay admissibility”
    3. 6.3. Triangulation of a variety
    4. 6.4. Topological invariants (triangles and tetrahedra)
  13. Chapter 7: Geometric Modeling: Methods
    1. 7.1. Implicit or explicit form (CAD), starting from an analytical definition
    2. 7.2. Starting from a discretization or triangulation, discrete → continuous
    3. 7.3. Starting from a point cloud, discrete → discrete
    4. 7.4. Extraction of characteristic points and characteristic lines
  14. Chapter 8: Geometric Modeling: Examples
    1. 8.1. Geometric modeling of parametric patches
    2. 8.2. Characteristic lines of a discrete surface
    3. 8.3. Parametrization of a surface patch through unfolding
    4. 8.4. Geometric simplification of a surface triangulation
    5. 8.5. Geometric support for a discrete surface
    6. 8.6. Discrete reconstruction of a digitized object or environment
  15. Chapter 9: A Few Basic Algorithms and Formulae
    1. 9.1. Subdivision of an entity (De Casteljau)
    2. 9.2. Computing control coefficients (higher order elements)
    3. 9.3. Algorithms for the insertion of a point (Delaunay)
    4. 9.4. Construction of neighboring relationships, balls and shells
    5. 9.5. Localization problems
    6. 9.6. Some formulae
  16. Conclusions and Perspectives
  17. Bibliography
  18. Index
  19. End User License Agreement