Quality Triangulations with Locally Optimal Steiner Points

Abstract

We propose two novel ideas to improve the performance of Delaunay refinement algorithms which are used for computing quality triangulations. The first idea is an effective use of the Voronoi diagram and unifies previously suggested Steiner point insertion schemes (circumcenter, sink, off-center) together with a new strategy. The second idea is the integration of a new local smoothing strategy into the refinement process. These lead to two new versions of Delaunay refinement, where the second is simply an extension of the first. For a given input domain and a constraint angle α, Delaunay refinement algorithms aim to compute triangulations that have all angles at least α. The original Delaunay refinement algorithm of Ruppert is proven to terminate with size-optimal quality triangulations for α ≤ 20.7 • . In practice, the original and the consequent Delaunay refinement algorithms generally work for α ≤ 34 • and fail to terminate for larger constraint angles. Our algorithms provide the same theoretical guarantees as the previous Delaunay refinement algorithms. The second of the proposed algorithms generally terminates for constraint angles up to 42 • . Experiments also indicate that our algorithm computes significantly (about by a factor of two) smaller triangulations than the output of the previous Delaunay refinement algorithms. Moreover, the new algorithms are experimentally shown to outperform the previous algorithms even in the existence of additional constraints, such as the maximum area triangle constraint which is commonly used for computing uniform triangulations. Introduction.Triangulations are geometric discretization tools essential in many scientific applications, such as engineering simulations, medical imaging, visualizations, geological analysis, and meteorological forecasting. The shape, the size, as well as the number of triangles in a triangulation are key in its efficient and effective use in these and other applications. While the preferred shape of a triangle is not exactly the same for all applications, theoretical and experimental analysis (of numerical methods that are used in conjunction with triangulations) suggests that triangles with no large angles and/or small angles serve well in most applications [2]. In general, the better the shape of the triangles, the smaller the interpolation and approximation errors are in their use. A triangle is usually considered to have a good shape if its smallest angle is bounded from below (by a user-specified constraint angle α), which implies an upper bound on its largest angle also (π − 2α). Unless new vertices/points are added into the domain, even using Delaunay triangulations, which are optimum in maximizing the smallest angle [10], may result in output that involve bad quality triangles. Hence, we are allowed to introduce points (called the Steiner points) in addition to input points in order to compute good quality triangulations. This, however, might dramatically increase the number of points and triangles in a triangulation, which is a key fact...

Optimizing Voronoi Diagrams for Polygonal Finite Element Computations

“…To this end, a number of very successful mesh improvement approaches have been proposed, being based on either Delaunay refinement [28,29,9] or some variational optimization [1,38,22,39,10,41]. So-called pliant methods, which combine local topological changes (e.g., Delaunay refinement) and vertex relocation (e.g., Laplacian smoothing or Lloyd relaxation) have been found to be superior over methods involving one of the techniques only [5,22,38].…”

Nonobtuse Triangulations of PSLGs

“…Conforming Delaunay triangulations for Γ are also called Delaunay refinements of Γ . There are numerous papers discussing Delaunay refinements including [18,21,[39][40][41]45]. The argument in this paper does not seem to give a better estimate for Delaunay triangulations than for nonobtuse triangulations, nor does the proof appear to simplify in the Delaunay case.…”

Computing Triangulations without Small and Large Angles

“…However, the smallest angles in the output of this heuristic tend to be quite small and the largest angles are only slightly smaller than 90 • . No small angle triangulations: There are many algorithms for computing triangulations with no small angles, e.g., the quadtree-based methods [4], the Delaunay refinement methods [10], [13], [17], [20], [21], and the edge refinement methods [19]. Among these arguably the best performance is due to Delaunay refinement methods.…”

“…This definition combines the objectives of the no small angle triangulation problem [4], [10], [20], [21] and the no large angle triangulation problem [3], [5], [11]. Hence, it is harder than either one of these heavily studied problems.…”