Civil Engineering Reference
In-Depth Information
Zone 2
Zone 0
Figure 5.73
Mesh of boundary nodes.
boundary protection scheme as described in Sections 5.3 and 5.4. The DT of the cube consists
of two zones separated by the boundary surface as shown in Figure 5.73; zone 0 consisting of
those tetrahedra within the boundary surface is a region to be meshed by the Delaunay-ADF
procedure, and zone 2 consists of the auxiliary tetrahedra outside the boundary surface. The
tetrahedral elements in zone 0 will be given a label 0, and those of zone 2 will be given a label
2 for easy identification. Indeed, the boundary can be implicitly defined once we have properly
labelled the tetrahedral elements of the two zones. It is important to keep all the tetrahedral ele-
ments within the cube as the search for the BASE tetrahedron of an insertion point merely by
the tetrahedra in zone 0 is rather difficult and time consuming, as zone 0 is non-convex and even
disconnected, whereas the union of zone 0 and zone 2, i.e. the entire cube, is always convex.
5.6.1.3 Zonal division and MG front
Similar to the classical frontal process, tetrahedral elements are created with frontal base tri-
angles one by one until zone 0 is filled up with newly created tetrahedral elements. It is noted
that the object has already been discretised into a valid tetrahedral mesh after boundary sur-
face recovery, and any element creation with the frontal triangles by means of point insertion
is just a mesh modification to improve the quality of the mesh to fulfil the mesh characteristic
requirements. From the last step, where is zone 1? The answer is that zone 0 will be split into
two zones, zone 0 and zone 1, in the process of Delaunay-ADF meshing. At the beginning of
Delaunay-ADF mesh, all tetrahedral elements within the boundary surface belong to zone 0,
and whenever a new tetrahedral element is formed with a boundary triangular facet, it will be
assigned to zone 1. Hence, the current generation front is given by the boundary between ele-
ments of zone 0 and elements in either zone 1 or zone 2. While the number of elements in zone 2
is fixed, in the progress of Delaunay-ADF meshing, the number of elements in zone 1 will keep
on increasing, whereas elements in zone 0 will decrease. At the end of Delaunay-ADF meshing,
all the tetrahedral elements in zone 0 will be converted to elements in zone 1.
5.6.1.4 Generation of tetrahedral elements on a frontal triangle
Let's talk about the basic mechanics for the creation of a tetrahedral element with a base
triangle on the generation front by means of Delaunay insertion. The strategic position of
such an insertion point and the shape optimisation of the elements will be discussed later
in Section 5.6.1.7. Let Δ be the base triangle and T be the associated tetrahedron in zone 0
