This work started with the attend to develop a new kind of
simulation tool avoiding general well known problems concerning local
oxidation. A particle dynamic algorithm based on *cellular
automata* [Zan93] should be used to solve the problem of local
oxidation instead of applying standard numeric methods such as *finite
elements* or *finite boxes*. Since first attempts on this area concerning
diffusion processes have been very promising [Rad94] further
investigations seemed to be justified.

The big difference between diffusion and local oxidation is the type of differential equation that describes the physical phenomena of both effects. In case of diffusion the differential equation can be classified as a parabolic type and, we have, among others, an elliptic type for oxidation. In other words, whereas the parabolic differential equation implicitly acts like a cellular automata, which means, that a local change in concentration can just be realized in the very near neighborhood and does not effect any other areas, oxidation causes an immediate effect across the whole simulation domain in case of stress changes. Since the temporal distribution of shock-waves (which for instance implies a local physical law that can be handled by cellular automata) is not of primary interest for oxidation simulations the dissemination of mechanical stresses has to be accelerated to get a fast steady state solution. Therefore, a hierarchical cellular automata has been developed, where in upper regions (big cells) low frequent solutions and in lower regions (small cells) high frequent ones are smoothed - similar to a multi-grid solver [Jop93]. Although first success was in sight it turned out that it is hardly possible to fulfill all requirements on a professional simulation tool using cellular automata techniques. The most striking points are:

- Natural extensions to an existing model developed on basics of cellular automata are only possible with great effort.
- Physical modeling can not be extracted from the code since the algorithm itself implicitly describes the physical characteristics.
- The well experienced mechanical behavior of materials used in a lot of disciplines, can not be transferred immediately to cellular automata strategies, since often no particle dynamic description exists - just the macrodynamic behavior is well known.
- The explicit time integration method (forward Euler integration) forces to use short time steps to keep the error within acceptable bounds.
- Even if it is possible to describe an oxidation process with cellular automata all existing mechanical theories have to be verified again based on molecular dynamic aspects.

All these arising problems using a cellular automata let us come to the decision that a well established technique, such as a simulator based on finite elements, finite boxes or finite differences, would be more satisfying than using a new methodology that lacks of physical descriptions as well as real progress in comparison to standard discretization methods. Therefore AMIGOS has been developed which offers the possibility for rapid prototyping of several physical problems. Although this thesis was devoted to the simulation of oxidation processes in three dimensions, several other models, such as pair-diffusion, thermodynamic processes, segregation and grid adaptation are also presented to show the flexibility of the developed simulation system.

The thesis is organized as follows: In Chapter 2 a short description of the physical basics of diffusion and oxidation is presented. Chapter 3 provides an introduction into general numerical discretization methods, such as finite elements and finite boxes with the intend to introduce the mathematical basics that are necessary to understand the rapid prototyping abilities of AMIGOS. In Chapter 4 a detailed description about the implementation and features of AMIGOS is given, where the possibilities of AMIGOS are presented with the help of several examples. Chapter 5 provides a detailed description of modeling local oxidation, especially the mechanical behavior of the material. Furthermore, a new model is presented that combines diffusion, chemical reaction and mechanical deformation in a single coupled numerical model that is used to solve several examples of typical oxidation processes. The thesis concludes with a brief discussion of the achieved results and gives some outlook for future work.

1998-12-11