Civil Engineering Reference
In-Depth Information
T
P
∆
Figure 5.74
Front triangle and its associated tetrahedron.
Insertion cavity (CORE)
P
∆
Tetrahedron ∆P
Figure 5.75
Insertion cavity associated with point P.
as shown in Figure 5.74 (such a tetrahedron T exists and is unique). A point P is inserted at
the vicinity of triangular facet Δ, which can be at the centroid of tetrahedron T or on the
normal to the triangle Δ. The BASE tetrahedron is determined using tetrahedra in all the
zones 0, 1 and 2. Although T is likely to be the BASE tetrahedron in most cases, other tetra-
hedron in zone 0 can also be the BASE depending on where point P has been inserted. The
basic conditions that point P could be accepted are that (i) the BASE tetrahedron is in zone
0 and (ii) tetrahedron T is non-Delaunay with respect to point P, i.e. ΔP is a tetrahedron to
be constructed in the insertion process, as shown in Figure 5.75. Starting from the BASE tet-
rahedron, all the non-Delaunay tetrahedra in zone 0 are identified, which form an insertion
cavity (CORE) for the insertion point P. The tetrahedral elements in the CORE are deleted
from zone 0, and tetrahedral elements formed with the boundary triangles of the CORE are
added to zone 0 except tetrahedron ΔP, which is assigned to zone 1 as it is a new tetrahedral
element generated at the base facet Δ. As only non-Delaunay tetrahedra in zone 0 are taken
to form the CORE, it is a form of restricted or constrained DT.
5.6.1.5 Updating the generation front
The generation front is implicitly and consistently defined by the moving boundary between the
elements in zone 0 and the elements in other zones 1 and 2. However, for easy monitoring of
the progress of MG and the strategic placement of insertion points, a list of frontal triangular
facets, Γ = {Δ
i
, i = 1, N
Δ
}, is explicitly maintained. The generation front Γ has to be updated
whenever a new tetrahedral element is created with a frontal triangle in Γ. Let A, B and C be the
vertices of triangle Δ; updating the generation front can be easily done, as shown in Figure 5.76.
