On Delaunay refinement for curved geometries
Purpose - The purpose of this paper is to investigate the extension of Delaunay refinement algorithms to work directly with a curved geometry in arbitrary dimensional spaces, which is also able to refine geometry pieces of different dimensions altogether. Design/methodology/approach - The extension of Delaunay refinement is based on ideas of the Bowyer-Watson algorithm and Ruppert algorithm. Findings - The attempt to extend the fundamental ideas of Delaunay refinement to cope with curved geometries led to an algorithm whose performance in practice, regarding speed and mesh quality, is comparable to classical Delaunay refinement for flat geometries. Unfortunately, there are only theoretical guarantees that the refinement itself works under some conditions. No theoretical mesh quality bounds are provided. Research limitations/implications - A mesh refinement algorithm that deals with curved geometries is a key feature for adaptive mesh generators, so that points are inserted properly in the curved pieces instead of in linear approximations of them. For instance, it is well known that sharp edges are singular points of finite element formulations. This singularity fulfills in practice as mesh is refined around them. Those corners can be rounded up to avoid singularities. Furthermore, with this kind of tool, for instance, a user could start to mesh a disc from a single triangle representing it. Points would be efficiently inserted in the circle as needed during refinement. Originality/value - This paper introduces the concept of manifold complex and also an extension of Delaunay refinement algorithm to deal with curved geometries.
mesh generation programming and algorithm theory geometric planes and solids