SOME TOPOLOGICAL ASPECTS OF THE (UN) STRUCTURED GENERATION OF MESHES: A POSSIBLE ENHANCEMENT OF MESHMAKER IN TOUGH2

The non-linear partial differential equations solved in TOUGH by the Integral Finite Difference Technique need meshes with diverse degrees of sophistication to model the geometry of physical systems in 2D or 3D domains with boundaries that can be of complex shapes. The main mathematical tools required to create efficient grids are differential geometry, tensor analysis and topology. We present several results from topology and geometry applied to practical grid generation. Structured mesh generation is the first step in the solution of problems with boundary conforming meshes. Structured meshes deal with the construction of coordinate curves in 2D and of coordinate surfaces in 3D. The intersection of these curves and surfaces produces mesh points and cells inside the solution domain. The grid cells are generally four sided geometric objects in 2D and finite volumes with six curved faces in 3D. The connectivity of points is the manner in which grid points are connected to each other in the solution domain. This connectivity depends on the overall generation scheme used. The Cartesian coordinates of every point can be stored in specific matrices with geometric and topological information. A variational approach is used for grid properties (orthogonality, longitude, area and smoothness) that can be controlled by the minimization of a functional. In unstructured meshes the connectivity between grid points can vary from point to point and it has to be described explicitly by an appropriate and particular data structure. This characteristic makes the unstructured solution algorithms more expensive in computational cost but more flexible and useful when employed in adaptive solutions of transient flows and moving boundary problems. This scheme is widely used in many applications of the Finite Element Method and in the Galerkin Discontinuous approach. We introduce an unstructured mesh generation using the Delaunay triangulation in 2D. The Delaunay tetrahedrization holds in 3D. In the first part we work in two dimensions using classic constructions from Euclidean geometry. In the second part we introduce topological concepts to generate meshes in three dimensions. We developed Fortran and Visual C codes to show some results in 2D. The practical aspects of this work could be useful as enhanced options for the TOUGH2 Meshmaker module.