Next: A Negative Result on Quality Triangulations Up: Triangle: Engineering a 2D Mesh Generator Previous: Ruppert's Delaunay Refinement Algorithm

Selected Implementation Issues

  Triangle removes extraneous triangles from holes and concavities before the refinement stage. This presents no problem for the refinement algorithm; the requirement that no segment be encroached and the Delaunay property together ensure that the circumcenter of every triangle lies within the mesh. (Roundoff error might perturb a circumcenter to just outside the mesh, but it is easy to identify the conflicting edge and treat it as encroached.) An advantage of removing triangles before refinement is that computation is not wasted refining triangles that will eventually be deleted.

A more important advantage is illustrated in Figure 13. If extraneous triangles remain during the refinement stage, overrefinement can occur if very small features outside the object being meshed cause the creation of small triangles inside the mesh. Ruppert suggests solving this problem by using the constrained Delaunay triangulation, and ignoring interactions that take place outside the region being triangulated. Early removal of triangles provides a nearly effortless way to accomplish this effect. Segments that would normally be considered encroached are ignored (Figure 13, right), because encroached segments are diagnosed by noticing that they occur opposite an obtuse angle in a triangle.


Figure 13: Two variations of the Delaunay refinement algorithm with a 20o minimum angle. Left: Mesh created using segment splitting and late removal of triangles. This illustration includes external triangles, just prior to removal, to show why overrefinement occurs. Right: Mesh created using constrained Delaunay triangulation and early removal of triangles.


Figure 14: Two meshes with a 33o minimum angle. The left mesh, with 290 triangles, was formed by always splitting the worst existing triangle. The right mesh, with 450 triangles, was formed by using a first-come first-split queue of bad triangles.

Another determinant of the number of triangles in the final mesh is the order in which bad triangles are split, especially when a strong angle constraint is used. Figure 14 demonstrates how sensitive the refinement algorithm is to the order. For this example with a 33o minimum angle, a heap of bad triangles indexed by their smallest angle confers a 35% reduction in mesh size over a first-in first-out queue. (This difference is typical for large meshes with a strong angle constraint, but thankfully disappears for small meshes and small constraints.) The discrepancy probably occurs because circumcenters of very bad triangles are likely to split more bad triangles than circumcenters of mildly bad triangles. Unfortunately, a heap is slow for large meshes, especially when small area constraints force all of the triangles into the heap. Delaunay refinement usually takes O(n) time in practice, but use of a heap increases the complexity to O(n log n).

Triangle's solution, chosen experimentally, is to use 64 FIFO queues, each representing a different interval of angles. It is counterproductive (in practice) to order well-shaped triangles by their worst angle, so one queue is used for well-shaped but too-large triangles whose angles are all roughly larger than 39o. Triangles with smaller angles are partitioned among the remaining queues. When a bad triangle is chosen for splitting, it is taken from the ``worst'' nonempty queue. This method yields meshes comparable with those generated using a heap, but is only slightly slower than using a single queue. During the refinement phase, about 21,000 new vertices are generated per second on a DEC 3000/700. These vertices are inserted using the incremental Delaunay algorithm, but are inserted much more quickly than Table 1 would suggest because a triangle's circumcenter can be located quickly by starting the search at the triangle.

Next: A Negative Result on Quality Triangulations Up: Triangle: Engineering a 2D Mesh Generator Previous: Ruppert's Delaunay Refinement Algorithm

Jonathan Richard Shewchuk
Mon Aug 12 10:28:49 EDT 1996