Computer-aided Geometric Design Techniques for Surface Grid Generation

This chapter focuses on three computer-aided geometric design (CAGD) techniques that are often needed to \prepare" a complex geometry for grid generation. Standard grid generation methods, as discussed in [George '91], [Knupp & Steinberg '93], and [Thompson et al. '85], assume that parametric surfaces are well parametrized and free of undesired discontinuities. We describe CAGD techniques that are extremely helpful for the preparation of complex geometries for the grid generation process. Surface re nement and reparametrization. One problem that has plagued most grid generation systems is that poorly parametrized surfaces create a poorly distributed grid. This is due to the fact that the grid distributions are performed in the parametric domain and then mapped back to physical space. It is desirable that the parametrization re ect the geometry of the surface in physical space, i.e., the parametrization should mimic the surface in physical space. The more a distribution of points in parametric space resembles the corresponding distribution of points in physical space the better the parametrization is. Imagine using a uniform parametrization on a sample of points that are not distributed evenly; if one distributes points evenly in parameter space, they are not uniform in physical space. There are ways to achieve a uniform distribution in physical space, but most approaches are based on iterative procedures. To eliminate the need for such iteration procedures, a chord length parametrization can be used { it best represents the surface in physical space. Approximation of discontinuous geometries. Grid generation is concerned with discretizing surfaces and surrounding volumes in three-dimensional (3D) space { in this context, it is important that a given geometry is continuous. Before one can generate grids for geometries containing discontinuities, one must approximate the geometry by a set of surface patches which are continuous. The most common problems