next up previous contents
Next: 6.3.2 Simulation Results Up: 6.3 Periodic Boundaries Previous: 6.3 Periodic Boundaries   Contents

6.3.1 Mesh Generation

Although periodic boundary conditions can be applied to an arbitrary pair of faces with unique bidirectional node mapping, we will restrict this paragraph to parallelepiped structures for the sake of clarity. The periodic boundary conditions are applied at two opposite parallel faces. The mesh generated has to guarantee that the surface meshes on these faces are identical. Our interconnect simulation software Smart Analysis Programs uses two different three-dimensional mesh generation approaches. The first one is a layered approach which extends two-dimensional meshes [124] into the third dimension by means of linear extrusion. The second approach is a fully unstructured mesh generation method based on the program delink [107,108]. Both approaches do not fulfill the above mentioned requirements for periodic boundaries a priori. To extend the mesh generation for periodic boundaries we use an iterative approach. In the first step the simulation domain is meshed without any special treatment for periodic boundaries. Afterwards the periodic boundary faces are checked for conformity. If they are not conform, they are merged and the newly generated points are fed into the mesh generator as additional input. After re-meshing of the geometry the periodic boundaries are again checked for conformity. These steps are repeated until conformity is reached. In the layer based meshing approach this iteration procedure must only be applied to the two-dimensional mesh generation process. The additional extrusion step preserves the conformity of the side walls. In the fully unstructured meshing method the conformity of the nodes on the periodic faces is not sufficient, because the same set of boundary nodes can lead to different boundary meshes (at least for cospherical points). Therefore also edge conformity has to be guaranteed during the iteration. Because of these additional difficulties the layer based mesh generation method is preferred for problems with periodic boundaries. Unfortunately, we were not able to prove theoretically that the iteration process necessary for fully unstructured meshes will always terminate, however we have not found any example so far, which took more than 11 iterations.


next up previous contents
Next: 6.3.2 Simulation Results Up: 6.3 Periodic Boundaries Previous: 6.3 Periodic Boundaries   Contents

A. Nentchev: Numerical Analysis and Simulation in Microelectronics by Vector Finite Elements