Civil Engineering Reference
In-Depth Information
3.7 MESHING BY A COMBINED SCHEME
OF DT AND ADF APPROACH
3.7.1 Introduction
The AFT and the DT are, so far, the most popular approaches developed for the generation
of unstructured meshes on a planar domain, on curved surfaces and over volumes for con-
strained meshing. However, it is interesting to note that the two approaches are not contra-
dicting to each other and can be put together quite naturally in harmony to form an overall
more efficient scheme, in which the speed and the robustness of the DT and the quality in
the generation of interior nodes of the AFT can all be retained in the combined scheme.
There are two ways in merging these two methods, which are to be referred to as the
ADF-Delaunay scheme and the Delaunay-ADF scheme. In the ADF-Delaunay scheme, the
set of interior points is already known before MG. The segments on the generation front are
classified into Delaunay and non-Delaunay segments depending on whether the circumcircle
of the supporting triangle of the segment contains any points in its interior. Given a base seg-
ment on the generation front, a point on the generation front or at the interior of the domain
is selected to form a triangle based on the Delaunay criterion. As segments on the generation
front are classified into Delaunay and non-Delaunay segments, intersection check is reduced
only to those non-Delaunay segments, which are relatively few on the generation front.
For the Delaunay-ADF scheme, interior points are not known in advance, which are to be
generated one by one as new elements are created at the generation front. At a typical stage
including the initial stage, the unmeshed part of the domain is a constrained DT of the gen-
eration front. A point is created following the same procedure of the AFT, which is inserted
into the constrained DT by means of the Delaunay-insertion kernel. The Delaunay triangle
formed with the base segment and the inserted point is taken away from the unmeshed
region, and the generation front is updated accordingly. In case no interior points can be
generated over a given frontal segment, the supporting Delaunay triangle associated with
this segment is deleted from the unmeshed part.
3.7.2 Advancing-front-Delaunay scheme
3.7.2.1 DT of non-convex planar domains
Let ∂Ω be the boundary of planar domain Ω and Λ be the set of interior nodes. A con-
strained DT for the domain Ω and interior nodes Λ is a collection of triangles {T
k
} satisfying
the following properties:
ii. Each ii.
k
is formed by three nodes from ∂Ω ∪ Λ.
ii. Each ii.
k
lies completely inside the domain Ω.
iii. The circumcircle associated with each T
k
contains, in its interior, no other node point,
which forms a valid triangle with any edge of (ii).
k
satisfying (i) and (ii).
iv. Ω is totally covered by {T
k
}, and no two triangles of {T
k
} overlap.
3.7.2.2 Delaunay and non-Delaunay triangles
i. Delaunay triangles are those triangles whose associated circumcircles contain no node
points on their circumferences or in their interiors.
iii. Triangle if
k
will be said to be
semi-Delaunay
if there are node(s) lying on the circum-
ference of the circumcircle of T
k
.
iii. Triangle if
k
will be said to be
non-Delaunay
if there are node(s) lying inside the cir-
cumcircle associated with T
k
.
