Adaptive 3-D triangulations for environmental problems

Abstract

In the finite element simulation of environmental processes that occur in a three-dimensional domain defined over an irregular terrain, a mesh generator capable of adapting itself to the topographic characteristics and to the numerical solution is essential. The objective of this work is to present our recent results in these topics. We construct an unstructured tetrahedral mesh of a region bounded in its lower part by the terrain and in its upper part by a horizontal plane. The main ideas for the construction of this initial mesh combine the use of a refinement/derefinement algorithm for two-dimensional domains and a tetrahedral mesh generator based on Delaunay triangulation. The points density over the terrain increases with the complexity of the orography and the point generation in the domain is done attending to a vertical spacing function over difierent layers defined from the terrain to the upper part of the domain. To avoid conforming problems between mesh and orography, the mesh will be designed with the help of an auxiliary parallelepiped. Once the 3-D Delaunay triangulation of the set of points has been constructed on the parallelepiped, points are replaced on their real positions keeping the mesh topology. In this last stage there can be occasional low quality elements, or even inverted elements. For this reason, we have developed a simultaneous untangling and smoothing procedure to optimise the resulting mesh. Once we have constructed the adapted mesh in accordance with the geometrical characteristics of our domain, we use an adaptive local refinement of tetrahedral meshes in order to improve the numerical solution. Finally, all these techniques are applied to the generation of adapted meshes for realistic and test problems.