The Delaunay Triangulation which will be discussed in detail in Chapter 5 can be efficiently utilized as robust tetrahedralization engine for practical meshing applications. A Delaunay based meshing approach is a concept which consists of two tasks. One addresses the mesh topography which is defined through the placement of mesh points. The other task is to create the mesh topology by performing the Delaunay Triangulation for a known point set. The sequence in which the two tasks are carried out is a matter of choice and therefore two classes can be distinguished.

- The mesh points are first created by a variety of techniques. The Delaunay Triangulation is performed afterwards on the complete set of points.
- The Delaunay Triangulation is first computed for the boundary without internal points (boundary mesh). The mesh points are then inserted incrementally into the triangulation/tetrahedralization and the topology is updated according to the Delaunay definition.

A tetrahedralization engine based on the Delaunay Triangulation allows a
fast generation of elements without for instance the expensive intersection
tests of advancing front methods. The reason can be seen in the global
definition of the topology^{4.4}.
It is therefore not necessary to check and compare one region of the mesh
with every other region to verify the mesh consistency. Instead only a
locally confined test of the Delaunay definition is required and the
knowledge of other regions is implicit.

Fully unstructured meshes with anisotropic capabilities can be generated. Figure 4.10 depicts a detail of the Flash EEPROM in comparison to Fig. 4.7.

However, the concept where the mesh topography is dealt with separately from the mesh topology is for anisotropic applications not always advantageous. The topology for a given set of mesh points might not be as expected and might destroy the desired anisotropy. For the last of the following anisotropic schemes the construction of protection layers depends on the ability of the Delaunay method to cope with thin layers.- The mesh points are for example distributed in an advancing front style with a small stepsize compared to the lateral distances. The tetrahedralization engine is supplied with the point set. Adverse effects can occur when additional points are inserted and the topology is updated.
- In two dimensions the triangulation engine is supplied with grid
lines to enforce a certain topology. It can be advantageous if these grid
lines are not split. A Delaunay Triangulation can be employed without the
insertion of additional points
^{4.5}. The Delaunay criterion for elements near those grid lines and at the boundaries is not necessarily fulfilled. - The triangulation/tetrahedralization engine is supplied with grid lines/facets. The Delaunay Triangulation results in a refinement and the Delaunay criterion is fulfilled. An anisotropic refinement technique which can handle thin layers is mandatory.

Another characteristic of Delaunay methods which is important for meshing
applications is the global optimization of the element quality^{4.6}.
Unfortunately it is not always identical to the desired mesh quality, e.g. non-obtuse dihedral angles or more specific mesh requirements as described
in Section 3.2. However, interesting results have been
obtained which show that a three-dimensional Delaunay Triangulation
combined with a local optimization of the required mesh quality is
superior to methods which only perform local optimizations without
concern of the Delaunay criterion [71,100,73].
Local improvements of the dihedral angle results in a topology which
can be globally far from optimal. Alternating between the Delaunay
criterion and a minimum maximum dihedral angle criterion can achieve
better results.

2000-01-20