Civil Engineering Reference
In-Depth Information
x
3
C
3
x
4
C
4
C
2
C
1
x
2
x
1
Figure 3.5
Region bounded by four arbitrarily shaped curves.
As a unit square can be readily discretised along the co-ordinate lines into a rectangular
mesh of a specified element size, a mesh of the region can be generated by means of the fol-
lowing transfinite mapping function, which maps the unit square onto the region bounded
by the curves
C
1
,
C
2
,
C
3
and
C
4
.
x
(ξ, η) = (1 − η)
C
1
(ξ) + ξ
C
2
(η) + η
C
3
(ξ) + (1 − ξ)
C
4
(η) − [(1 − ξ)(1 − η)
x
1
+ ξ(1 − η)
x
2
+ ξη
x
3
+ (1 − ξ)η
x
4
]
ξ, η ∈ [0, 1]
where
x
1
,
x
2
,
x
3
and
x
4
are the corner points of the region where two boundary curves meet.
Prior to MG, one has to verify if the four curves meet at the corner points, i.e.
C
1
(0) =
C
4
(0) =
x
1
;
C
2
(0) =
C
1
(1) =
x
2
;
C
3
(1) =
C
2
(1) =
x
3
;
C
4
(1) =
C
3
(0) =
x
4
Figure 3.6 shows some regions meshed with the aid of transfinite mapping. Figure 3.6a is
a mesh of a quadrilateral region bounded by straight edges, which could as well be meshed
(a)
(b)
(c)
(d)
Figure 3.6
Regions meshed by means of transfinite mapping: (a) region bounded by straight edges; (b-d) region
with curved boundaries.
