For the development of today's highly-integrated electrical devices, the use of numerical simulation tools has become indispensable. Details in the manufacturing procedure of these devices cannot be described by simple design rules and the layout of device or interconnect structures cannot be predicted by simple considerations. Electrical characteristics, the influence of the interconnect structures on the capacitances and inductances between the metal layers, delay times of the signals, and further effects have to be accounted for during the development process. The detailed simulation of etching, deposition, implantation, and diffusion processes have to be analyzed. Finally, the electrical characteristics of the devices have to be investigated.
Coming along with the rapidly rising complexity and down-scaling of electronic devices, many effects have to be considered. Effects near corners of the devices become more dominant and physical simulation models have to be expanded. On the one hand more complex models have to be used, on the other hand the existing models, respectively their numerical solutions, have to be expanded from two spatial dimensions to their three-dimensional representations.
Usually the applied models have an analytical representation, for instance in the form of partial differential equations. Only in the most trivial cases, it is possible to find an analytical solution for those problems. Even for simple linear partial differential equations closed solutions cannot be found, because of the complex geometries and resulting boundary conditions.