Automatic Construction of Quality Nonobtuse Boundary And/or Interface Delaunay Triangulations the Control Volume Methods

In this paper we discuss a new algorithm that for any input quality constrained Delaunay triangulation with minimum interior angle greater than or equal to 30 , produces a quality nonobtuse boundary and/or interface Delaunay triangulation by the Delaunay insertion of a nite number of boundary and/or interface points. A boundary (interface) obtuse triangle is a triangle that has an obtuse angle opposite to a boundary (interface) edge. The output mesh might have a small number of triangles with interior angles less than 30 in the neighborhood of the triangles with boundary constrained angles. The analysis of the algorithm considers two cases depending on the geometric complexity of the domain: (a) simple polygonal domains which may include holes and (b) polygonal domains with interfaces. In case (a) every obtuse triangle with one boundary edge is eliminated by the Delaunay insertion of one point, and every obtuse triangle having both medium size edge and longest edge (of respective lengths l and L) over the boundary and boundary constrained angle is eliminated by building an isosceles triangle of boundary edges of lengths l=2 (which maintains the Delaunay triangulation) followed by the Delaunay insertion of a nite number of points N, where N < K, and K = d2 + 2Ll l p 1cos() e. In case (b), there are interface obtuse triangles either isolated or arranged into a group of adjacent interface triangles. The isolated interface obtuse triangles are destroyed in a similar way to boundary obtuse triangles of case (a), and on the contrary, the grouped interface triangles are destroyed together by Delaunay insertion of a nite number of points. It is proved that the algorithm produces an almost (non-constrained) Delaunay triangulation in the sense that all pair of triangles sharing an interface edge satisfy the Delaunay condition. Examples of the practical behavior of the algorithm combined with a Lepp-Delaunay algorithm to produce the initial triangulation are also included.