Civil Engineering Reference
In-Depth Information
n
A
M
h
e
2
B
C
i
e
1
C´
Figure 4.60
Local reference frame and estimated position of a new node.
The unit vector
n
is a vector normal to the surface and perpendicular to base segment AB.
In the actual implementation, the average of the surface normals at A and B is sufficient for
the purpose of mesh generation. The interior of the region is bounded by the current genera-
tion front where new elements are to be generated in the direction
e
2
.
4.4.2.1 Find the best node on the generation front
An extension of the method discussed in Section 3.6.2 for planar domain is used to find
the best node on the generation front. Let Λ be the set of node points on the generation
front Γ. A node C
Γ
∈ Λ is said to be a candidate node of AB if the following conditions are
satisfied.
ii. The triangle ABC
Γ
lies on the interior bounded by the generation front.
ii. The triangle ABC
Γ
does not cut into the generation front.
The first condition can be ensured by comparing the normal
n
and normal
n
Γ
of triangle
ABC
Γ
, which is defined as
AB
×
×
AC
Γ
n
Γ
=
AB
AC
Γ
If the angle between
n
and
n
Γ
is less than 90°, triangle ABC
Γ
is considered to be lying at
the interior of the unmeshed region. Mathematically, this can be expressed as the dot prod-
uct of the vectors
n
and
n
Γ
.
AB
×
×
AC
Γ
nn
⋅ =⋅
n
>
0
or simply
n
⋅
(
AB
×
AC
)
>
0
Γ
Γ
AB
AC
Γ
The second condition involves the intersection check over a curved surface, which is
slightly more complicated than a similar process on a planar domain. The standard intersec-
tion test of line segments using the spatial nodal points is not sufficient, as the line segments
forming the generation front are not necessarily lying on the curved surface (in fact, only the
end points of the line segments are on the surface). Two line segments may not intersect in
space even if their corresponding surface curves are crossing each other.
