Civil Engineering Reference
In-Depth Information
Γ Γ
be the surface curves corresponding to line segments AC
Γ
and BC
Γ
,
respectively. The second condition can be expressed as
Let
AC
andBC
{
}
{
}
Γ
AC
∩
Γ
AC
,{
AC
,
}
andBC
∩
Γ
BCBC
,{ ,
}
Γ
Γ
Γ
Γ
Γ
For analytical surfaces, we can assume that the surface is given by the parametric equa-
tion S(u,v), and a surface curve
C()
on surface S(u,v) can be written as
Ct
()
=
Sutv(t)
((),
)
Thus, two surface curves
Cand CSus vs
= ((),
())
intersect with each other if
u(t) = u′(s)
and
v(t) = v′(s)
for some s and t within the range of the curves
Depending on how the surface is defined, intersection check can be done by a projection
of the line segments on a local tangent plane. For sufficiently smooth surfaces, check for
frontal intersections can be conveniently done between line segments on the parametric
domain.
4.4.2.2 Locate interior node
Apart from taking a node from the generation front, very often, a new node created at the
surface interior can form a well-shaped triangular element with the base segment. Given
the nodal spacing specification at each point on surface S, the position of the interior point
forming the best element with the base line segment can be determined. Let h be the required
element height for the base segment AB. As shown in Figure 4.61, from the mid-point of AB,
M, an approximate spatial position C
I
for the new interior node is given by
C
I
= M + h
e
2
The spatial point C
I
has to be relocated back to the curved surface S by means of nearest
point (normal) projection. Furthermore, the distance from surface point
C
I
on the curved
surface to the generation front Γ should be at least half the required element height, i.e.
h
(
)
>
DistanceC E
,
Edge E
∈
Γ
I
k
k
2
This ensures that poor-shaped triangles will not be formed with the frontal segments in
the subsequent mesh generation. In case any one of the conditions cannot be satisfied by the
candidate node, the required element height has to be reduced progressively until the quality
n
e
2
C
I
e
*
M
C
I
*
C´
Figure 4.61
Closest point projection.
