Smoothing or surface triangulation using optimal local projection

Abstract

In environmental modeling (air pollution, wind field [1], etc.) that occurs in three-dimensional domains defined over complex terrains, a mesh generator capable of adapting itself to topographical data, to chimney geometry and to the numerical solution is essential. For this purpose, in past works we have developed anautomatic unstructured tetrahedral mesh generator [2], by using a refinement/derefinement algorithm for two-dimensional domains and a version of 3-D Delaunay triangulation. Occasionally in this process, low qualityor even inverted elements may appear. For this reason, we also introduced a simultaneous untangling andsmoothing procedure to optimize the resulting meshes [3]. Besides, a new problem arose with the quality ofsurface triangulation of irregular terrains, since, in order to prevent a loss of details of the original surfaceinformation, we did not allow the movement of nodes placed over the terrain. This problem motivates theintroduction in this paper of a new procedure to improve the quality of triangular meshes defined on surfaces.The improvement is obtained by an iterative process in which each node of the mesh is moved to a newposition that minimizes certain objective function. This objective function is derived from an algebraic quality measures of the local mesh [4,5] (the set of triangles connected to the adjustable or free node). If we allow thefree node to move on the surface without imposing any restriction, only guided by the improvement of thequality, the optimization procedure can construct a high-quality local mesh, but with this node in an unacceptable position. To avoid this problem the optimization is done in the parametric mesh, where thepresence of barriers in the objective function maintains the free node inside the feasible region. In this way,the original problem on the surface is transformed into a two-dimensional one on the parametric space. In ourcase, the parametric space is an optimal plane, chosen in terms of the local mesh, in such a way that this meshcan be projected performing a valid mesh, that is, without inverted elements. Several examples andapplications presented in this work show how this technique is capable to improve the quality of surfacemeshes maintaining their topologies.