Delaunay refinement, a versatile method of mesh generation, is plagued by memory thrashing when required to generate large output meshes. To address this space issue, a localized version of Delaunay refinement was proposed for generating meshes for smooth surfaces and volumes bounded by them. The method embodies a divide-and-conquer paradigm in that it maintains the growing set of sample points with an octree and produces a local mesh within each individual node, and stitches these local meshes seamlessly. The proofs of termination and global consistency for localized methods exploit recently developed sampling theory for smooth surfaces. Unfortunately, these proofs break down for a larger class called piecewise smooth complexes (PSCs) that allow smooth surface patches that are joined along ridges and corners. In this work, we adapt a recently developed sampling and meshing algorithm for PSCs into the localization framework. This requires revisiting the original algorithm, and more importantly re-establishing the correctness proofs to accommodate the localization framework. Our implementation of the algorithm exhibits that it can indeed generate large meshes with significantly less time and memory than the original algorithm without localization. In fact, it beats a state-of-the-art meshing tool of CGAL for generating large meshes.

Abstract Delaunay refinement, recognized as a versatile tool for meshing a variety of geometries, has the deficiency that it does not scale well with increasing mesh size. The bottleneck can be traced down to the memory usage of 3D Delaunay triangulations. Recently an approach has been suggested to tackle this problem for the specific case of smooth surfaces by subdividing the sample set in an octree and then refining each subset individually while ensuring termination and consistency. We extend this to localized refinement of volumes, which brings about some new challenges. We show how these challenges can be met with simple steps while retaining provable guarantees, and that our algorithm scales many folds better than a state‐of‐the‐art meshing tool provided by CGAL.

Abstract The technique of Delaunay refinement has been recognized as a versatile tool to generate Delaunay meshes of a variety of geometries. Despite its usefulness, it suffers from one lacuna that limits its application. It does not scale well with the mesh size. As the sample point set grows, the Delaunay triangulation starts stressing the available memory space which ultimately stalls any effective progress. A natural solution to the problem is to maintain the point set in clusters and run the refinement on each individual cluster. However, this needs a careful point insertion strategy and a balanced coordination among the neighboring clusters to ensure consistency across individual meshes. We design an octtree based localized Delaunay refinement method for meshing surfaces in three dimensions which meets these goals. We prove that the algorithm terminates and provide guarantees about structural properties of the output mesh. Experimental results show that the method can avoid memory thrashing while computing large meshes and thus scales much better than the standard Delaunay refinement method.