4.1.2 PDE Method

The mapping is a transformation based on a system of partial differential equations. The system is solved on a reference mesh to generate the structured grid. Depending on the type of the equations two classes can be distinguished.

• Elliptic grid generation
• Hyperbolic grid generation

The latter uses hyperbolic operators and is suitable to grow a structured grid from a boundary. The first commonly utilizes the regularizing properties of the Laplace operator [168,183,103]. In two dimensions two systems with corresponding boundary conditions are solved.

$$u_{xx} + u_{yy} = 0$$

$$v_{xx} + v_{yy} = 0$$

Extra control of the grid spacing and orthogonality can be introduced by using Poisson equations. The resulting grids are of high quality, boundary-fitted, and possess good orthogonality. The application of conformal mapping techniques to semiconductor device simulation, where boundary-fitted and possibly orthogonal meshes are a great concern, has been tested so far in two dimensions [26]. Non-planar thin layers (geometrical anisotropy) and protection layers (physical anisotropy near boundaries) should be manageable in three dimensions. In practice the structured grid must often become a part of a larger mesh with regions of different anisotropic requirements. The region between the various structured grid parts could for example be filled with an unstructured mesh.