Many physical phenomena in science and engineering can be modeled by partial differential equations. When these equations have complicate boundary conditions or are posed on irregularly shaped objects or domains, they usually do not admit closed-form solutions. A numerical approximation of the solution is thus necessary. Numerical methods for solving partial differential equations include the finite element and the finite volume method. They are used to model disparate phenomena such as dopant diffusion, mechanical deformation, heat transfer, fluid flow, electromagnetic wave propagation, and quantum mechanics. These methods numerically approximate the solution of a linear or nonlinear partial differential equation by replacing the continuous system with a finite number of coupled linear or nonlinear algebraic equations. An essential step in these methods is to find a proper discretization of a continuous domain. This is the problem of mesh generation.
Due to the shrinking of semiconductor devices effects arise which cannot be handled by a two dimensional simulation, especially in the region of corners or vertical devices. Problems also occur in very small devices, which are already used in standard technologies, where all regions of the devices are located close to corners. Exactly these critical devices require three dimensional process and device simulations to be able to understand their behavior and the impact of processing steps. This results in three-dimensional mesh generation.
With this growing importance of three dimensional simulation mesh generation has become a critical factor and a lot of research has to be done. The amount of data in three dimensions requires efficient and more sophisticated algorithms and data structures than in two dimensions. The different requirements from the device and process simulation place particulary difficult demands on mesh generation too. If one can generate meshes that are completely satisfying for process simulation, the device simulations will definitively fail. Also the increasing topographical complexity of the devices and of the simulation domain pose a great challenge for meshing schemes. Up to nowadays, there has not been found any satisfying solution which is best for all requirements. Historically, the automation of mesh generation has proved to be as challenging as the entire remainder of the simulation process.