Three-Dimensional Boundary-Constrained Triangulations

We discuss the problem of generating a boundary-constrained tetrahedral mesh in a convex polyhedron, where the triangular meshes on the boundary faces of the polyhedron are given. These boundary-constrained triangulations are used in our nite element tetrahedral mesh generation method (which includes a convex poly-hedron decomposition step), since a boundary face may be common to two convex polyhedra in the decomposition. A boundary-constrained triangulation always exists if there is at least one mesh vertex given in the interior of the polyhedron, and in this case, we present an algorithm which uses local transformations to construct a boundary-constrained triangulation based on the local sphere or max-min solid angle (or other) criterion. We present an example to show that a boundary-constrained triangulation may not exist if there are no mesh vertices given in the interior of the polyhedron; in this case, we describe our approach for constructing a boundary-constrained triangulation, which may sometimes require the addition of an interior mesh vertex.