ISBN 978-3-86853-928-8

978-3-86853-928-8, Reihe Ingenieurwissenschaften

Heinrich Büssing
Finite Integration auf Tetraedergittern

129 Seiten, Dissertation Technische Universität Berlin (2010), Softcover, B5

Zusammenfassung / Abstract

The method of finite differences (FD) is applied here on tetrahedral grids. This is because partial differential equations are easily implemented with the FD method. Tetrahedral grids are used to avoid staircase patterns along curved surfaces, for example in order to calculate the quality of spherical hollow resonators with higher precision. To this end, the tetrahedral grid is distorted in such a way that the triangles forming the edges of the tetrahedral grid cells are located directly on the surfaces of the bodies that are included in the calculation. Thus this tetrahedral grid is a kind of perfect mesh. There already exist many grid generators which generate such grids matched to the geometry, but these generators are only suitable for the more complex methods of finite elements (FE), finite volumes (FV) or finite differences on nonorthogonal grids (UFD). For the first time, here a grid generator is introduced, which generates distorted tetrahedral grids which have dual orthogonal grids, making them suitable for a simple finite integration scheme. The grid generation is based on the uniform, favorable placement of grid nodes in the volume using variational principles applied recursively on a set of grid points. Also for the first time, here the phase error of a structured tetrahedral grid is calculated and expressed analytically as a power series which - as with a rectangular grid - converges quadratically with the step size, but the phase error and the dispersion error are about one third to one half lower than with a rectangular grid at the same calculation effort. Resonant fields, frequencies and qualities of spherical hollow resonators calculated with distorted tetrahedral grids have the same power of convergence as with rectangular grids but are computed more precisely. But the distortion of the grid constrains one to decrease the maximum time step to approximately one quarter of that required for an undistorted grid. Thus the tetrahedral grid is better than the rectangular grid, but not as good as expected at the beginning of this research.

The effort of grid generation is linear with the number of grid points because of their geometric organization in a hash table. But the topology of the tetrahedral grid makes data organization and programming more complex than with a rectangular grid. The implementation of perfectly matched layers (PML) as absorbing boundary conditions becomes possible by attaching a prismatic grid to the absorbing surface