Abstract

We introduce a new concept for the geometry description called contravariant geometry description (or shortly cogeometry). This concept is dual to the standard way to describe the geometry. It is based on functions which compute intersections with boundary segments. We consider different algorithms which allow to create and modify cogeometries for arbitrary space dimension. We consider also problems of grid generation for diffusion-reaction equations. We consider the connection between optimal grids and discretization and remark that for finite volume discretization (in contrast to FEM) the Delaunay grid is optimal even in 3D. We emphasize the necessity of anisotropic grids. We consider the problems related with anisotropic Delaunay grid generation and the usage of the contravariant geometry description. We describe a combined octree/Delaunay method for grid generation which allows to solve most of the problems. The concept has been implemented in the grid generation and geometry description package IBG. We show some examples for an application of IBG.

Preface

The aim of the thesis is to derive a concept for geometry description and grid generation which allows to meet the (very high) requirements of the simulation of semiconductor technology. We hope, that these algorithms may be applied in a wide range of other applications. They have been implemented in the general purpose grid generation and geometry description package IBG, which will be also described here.

This thesis consists of two main parts --- a geometry description and a grid generation part. In an introduction we describe our motivation to consider these two questions. We describe our reference application --- the simulation of semiconductor technology --- and list the resulting requirements for grid generation and geometry description.

In the geometry description part we introduce a new concept of geometry description which is dual to the usual concept. In this concept, the geometry description consists of a set of functions. The first function allows to find the region containing a point, the other allow to compute intersections of simplices with boundary segments.

We show that this concept allows to create and modify geometry descriptions in a much more natural way than the standard geometry description based on a cell complex. Because of the natural functional behaviour of this alternative geometry description we call it contravariant geometry description or (shortly) cogeometry. The concept may be used for arbitrary space dimension.

In the grid generation part we consider different questions connected with 3D grid generation. We give a short overview over usual grid generation techniques, consider the requirements connected with the discretization of partial differential equations and the problems connected with anisotropy. Then we describe in detail the grid generation algorithm used in the package IBG.

At last, we consider the results of some example applications of the IBG package. They especially show the simplicity of creating complicate geometries using the contravariant geometry description.