Meshing, Geometric Modeling and Numerical Simulation 1

Form Functions, Triangulations and Geometric Modeling

Houman Borouchaki, Paul Louis George

• Publisher: Wiley-ISTE
• Year: 2017
• Print ISBN:
  9781786300386
• Edition: 1
• eBook ISBN:
  9781119384038
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

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.

Chapter 1
Finite Elements and Shape Functions

There is a wide range of existing literature on finite elements, both on theoretical aspects (for example [Oden, Reddy-1977], [Ciarlet-1978], [Hughes-1987], [Ciarlet-1991]) and on practical aspects ([Zienkiewicz, Taylor-1989], [Bathe-1996], [Dhatt et al. 2007]). The purpose of this chapter, therefore, is not to provide yet another description of this method, but rather to introduce a point of view that is strongly oriented toward the underlying geometric aspects. Indeed, classically, these are all the aspects of approximations of functions (polynomial space or others, convergence, convergence rate, etc.) that are examined. We will, thus, only review basic definitions related to finite elements¹, as well as their shape functions. The classic case of finite elements whose degrees of freedom are nodal values of the considered functions (in other words, like Lagrange elements) is described for complete elements, reduced elements as well as rational elements. The less common finite elements such as Hermite elements, for example, where nodal or other derivatives are involved are not explicitly considered².

1.1. Basic concepts

The finite elements method allows us to calculate an approximate solution to a problem formulated in terms of a system of partial derivatives over a continuum Ω across two related approximations: a spatial approximation and an approximation for calculated solutions. The physical problem under study is modeled by a system of partial derivatives equations that constitutes a continuous problem with its operators, parameters, data and boundary conditions. The finite element method consists of searching for solutions in a particular space of functions (a Sobolev space) that is bu...

Table of contents

• Cover
• Table of Contents
• Dedication
• Title
• Copyright
• Foreword
• Introduction
• Chapter 1: Finite Elements and Shape Functions
• Chapter 2: Lagrange and Bézier Interpolants
• Chapter 3: Geometric Elements and Geometric Validity
• Chapter 4: Triangulation
• Chapter 5: Delaunay Triangulation
• Chapter 6: Triangulation and Constraints
• Chapter 7: Geometric Modeling: Methods
• Chapter 8: Geometric Modeling: Examples
• Chapter 9: A Few Basic Algorithms and Formulae
• Conclusions and Perspectives
• Bibliography
• Index
• End User License Agreement