Meshing, Geometric Modeling and Numerical Simulation 3

Introduction

Triangulations and, more precisely, meshes, with the subtle difference that separates these two entities, lie at the heart of any number of problems that arise from varied scientific disciplines. A triangulation or a mesh is a discrete representation, using simple geometric elements (triangle, quadrilateral, tetrahedron, etc., any arbitrary polygon or polyhedron), of a domain that may be an object or a region of which we want a discrete, spatial description. There are, thus, many applications, including numerical simulations of any kind of physical problem, though not restricted to these. In particular, a discrete representation of a (volume) object or a surface may simply be seen as a geometric modeling problem as is. This book adopts a double point of view, as indicated by its title, and we will look both at the use of meshes in numerical simulations (the finite element method, especially) with, of course, the underlying constraints, as well as the use of these meshes for the (discrete) modeling of geometry.

The literature on triangulations and meshes may be classified in two chief categories: one more purely mathematical and geometric, the other more oriented toward industrial applications (numerical simulations) with, of course, though not always, relations between these categories.

The first point of view is covered by the computation geometry community, which studies (among others) Delaunay triangulations in all cuts, definitions, properties, construction ...