Civil Engineering Reference
In-Depth Information
4.3.1 Ellipse-packing algorithm
The idea of anisotropic mesh generation is to connect the centres of ellipses of variable
size and orientation packed together by the ADF approach. Unlike the conventional frontal
technique, the procedure does not start from the boundary of the object but starts from a
convenient point within an open domain. The initial front consists of three tangent ellipses,
which are to be expanded towards the exterior and has to be updated whenever a new ellipse
is introduced. Triangular elements are subsequently formed when the centres of the ellipse
are connected systematically along the generation front.
4.3.1.1 Data structure
The structure of the generation front is very simple, which is a closed loop of line segments.
As a result, in the packing process, only those ellipses on the generation front need to be
considered. In the implementation, the generation front is represented by an ordered list
of ellipses on the boundary of the pack. For each ellipse, the centre, the length of the two
principal axes, the direction of the major principal axis and the distance from the origin are
stored. The process of front updating is just to apply the operations of inserting (deleting) a
point to (from) the generation front. It is interesting to note that the ellipses along the gen-
eration front are tightly packed and in contact with each other.
4.3.1.2 Three criteria for ellipse packing
if
Densest - Ellipses are to be packed as close to one another as possible.
Ideally all
ellipses are packed closely together so that the gaps between them are minimised.
However, it is sometimes difficult to pack them together as the size and the orientation
of a given ellipse may not always fit its neighbours. Hence, not only the position of the
inserted ellipse but also the size and the orientation of the ellipse have to be adjusted
to fit the local site in order to achieve a dense packing.
ii.
No overlapping - There is no or little overlapping between any two ellipses.
It is best
that any new ellipse added on the generation front does not overlap with the existing
ellipses. The overlapping of ellipses leads to perhaps not only bad mesh quality but also
faulty connections such as mesh intersection, overlapping and holes.
iii.
Nearest - A new ellipse is always generated at the location closest to the origin.
The
front nearest to the origin is often the most concave part. Packing new ellipses at the
concave part can fill gaps, which reduces the chance of forming large holes or voids.
By packing ellipses nearest to the origin, the shape of the generation front is basically
convex like a circle with minor concave parts.
4.3.1.3 Unit metric
The unit metric field M for a curved surface has been discussed in detail in Section 4.2.6,
and the unit length requirement for arbitrary vector
v
in any direction is given by
v
T
M
v
≤ 1.
The length of vector
v
on the 2D plane is given by the Euclidean distance, i.e. h = ‖
v
‖.
Alternatively, if metric M is represented by (λ, μ, θ) as described in Section 4.2.8, then the
length of unit vector
v
making an angle α with the x-axis is given by (Figure 4.45)
h
2
= λ
2
cos
2
(α − θ) + μ
2
sin
2
(α − θ)
Given the unit length metric, the above equation allows us to compute the required dis-
tance of a proposed point in any direction.
