SUMMARYGuaranteed-quality unstructured meshing algorithms facilitate the development of automatic meshing tools. However, these algorithms require domains discretized using a set of linear segments, leading to numerical errors in domains with curved boundaries. We introduce an extension of Ruppert's Delaunay reÿnement algorithm to two-dimensional domains with curved boundaries and prove that the same quality bounds apply with curved boundaries as with straight boundaries. We provide implementation details for two-dimensional boundary patches such as lines, circular arcs, cubic parametric curves, and interpolated splines. We present guaranteed-quality triangular meshes generated with curved boundaries, and propose solutions to some problems associated with the use of curved boundaries.

Unstructured mesh quality, as measured geometrically, has long been known to influence solution accuracy and efficiency for finite-element and finite-volume simulations. Recent guaranteed-quality unstructured meshing algorithms are therefore welcome tools. However, these algorithms allow no explicit control over mesh resolution or grading. We define a geometric length scale, similar in principle to the local feature size, which allows automatic global control of mesh resolution and grading. We describe how to compute this length scale efficiently and modify Ruppert's two-dimensional and Shewchuk's three-dimensional meshing algorithms to produce meshes matching our length scale. We provide proofs of mesh quality, good grading, and size optimality for both two-and three-dimensions, and present examples, including comparison with existing schemes known to generate good-quality meshes.

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