Computational geometry techniques for two-dimensional and three-dimensional unstructured mesh generation with application to the solution of divergence form partial differential equations.

摘要

The numerical solution of partial differential equations requires an underlying network of computational points called a mesh. It is at these mesh points that all the physical variables are either known or solved for. The generation of this mesh has been, for too long, an ad hoc and painful procedure. The methods developed in this thesis go a long way toward simplifying the grid generation process.; This thesis develops new theoretical results in the field of computational geometry concerning the continuity of the Delaunay triangulation and the Voronoi diagram as functions of point placement. An identification theorem is introduced and an adaptive correction result in 2D is extended to 3D to allow for the existence of required features (edges in two dimensions; edges or triangles in three dimensions). The Delaunay and Voronoi tessellations are then used to automatically generate two and three dimensional unstructured grids. Graded triangulations in 2D and graded tetrahedralizations in 3D are developed by the introduction of weight values at the boundary points of the computational domain. Individual cell and local proximity metrics are described and combined to define a single measure to quantify mesh quality. Techniques are introduced to allow for mesh adaptation via local mesh refinement or local mesh coarsening. The control region discretization technique is described for a class of PDEs that naturally mates with the biorthogonal grids produced. Finally, results are shown to support the use of this gridding technique. Numerical solutions of divergence form PDEs discretized on the Delaunay-type unstructured grids are found and compared with known analytic results.