Parametrization-independent elliptic surface grid generation

Abstract

The generation of computational grids on surfaces of three-dimensional configurations is an important component of many areas of computational research, both as a boundary grid for volume grid generation or to perform computations on the surface. For reasons of computational efficiency, structured grids remain widely used in areas such as the computation of aerodynamic flows. Elliptic grid generation is the leading structured grid generation method and creates high-quality grids for simple geometries. However, when applied to surfaces in three dimensions, problems are encountered among other things with the redistribution of grid nodes that need to remain on the surface and with maintaining the high quality of the grid. The present work introduces a new approach to elliptic surface grid generation which is based on intrinsic properties of the surface and not the parameterization that describes the surface. Previous work has based the surface grid generator on the parametric space of the specific surface parameterization at hand, but this often leads to low-quality grids, specifically for surfaces with large variations in curvature. This work aims to address this issue by developing a method which performs elliptic smoothing on an initial grid created using any parameterization, but redistributes the grid nodes according to their location in physical space, as opposed to in parametric space. This ensures that the final grid will always be of high quality in the physical space. Furthermore, a set of control functions which control grid lines and node bunching over the surface are developed. To be consistent with the parameterization independent elliptic method, these functions are to be based on intrinsic properties of the surface. In the approach proposed here, we specifically use the curvatures in the directions of the grid lines and the length of the grid lines.