5. Bifurcation of Grids on Curves
S. Steinberg, Patrick J. Roache · Society for Industrial and Applied Mathematics eBooks · 1991
5.1. Background Attempts to use variational grid-generation methods to generate grids on certain surfaces of modest shape have failed to produce suitable grids. There were sufficient points in the grids to well-resolve the surface, so the failures were not easily explained. Similar difficulties were found for the analogous problem of variational grid generation on curves; those problems are caused by multiple solutions of the underlying nonlinear algebraic equations, as shown in this paper. Thus the difficulty is intrinsic to the discrete approximation of the variational problem used to generate the grids. For a description of the variational techniques, more details concerning the anomalous behavior of the grid-generation algorithm for both curves and surfaces, and a discussion of how symmetry of the difference equations affects the grid generator, see papers [65], [66], and [67], by Steinberg and Roache. A significantly improved surface grid generator is currently being developed by Knupp [43]. The difficulties with the grid generator are best described in terms of bifurcations, thus, we choose a family of curves that depend on a parameter. Throughout this paper we use the family of simple parabolas y=αx (1−x), 0≤x≤1,α≥0, 5.1 all of which depend on the parameter α. For the curve is a straight line that is trivial to grid; as α grows, the curvature increases and the curve becomes more difficult to grid. Several other curves were tested with similar results.