Geoscience Reference
In-Depth Information
Figure 4.9
3-D local coordinate transformation.
where
x
k
,
y
k
, and
z
k
are the coordinate values of the
k
th point in the (
x
,
y
,
z
) coordinate
system, and
ϕ
k
(
k
=
1, 2,
...
,27
)
are interpolation functions (Wu, 1996b):
⎧
⎨
⎩
2
2
2
(ξξ
+
ξ
)(ηη
+
η
)(ζζ
+
ζ
)/
8
k
=
1, 3, 7, 9, 19, 21, 25, 27
k
k
k
2
2
2
(
1
−
ξ
)(ηη
+
η
)(ζζ
+
ζ
)/
4
k
=
2, 8, 20, 26
k
k
2
2
2
(ξξ
+
ξ
)(
1
−
η
)(ζζ
+
ζ
)/
4
k
=
4, 6, 22, 24
k
k
2
2
2
(ξξ
k
+
ξ
)(ηη
k
+
η
)(
1
−
ζ
)/
4
k
=
10, 12, 16, 18
(4.88)
ϕ
=
k
2
2
2
(
1
−
ξ
)(
1
−
η
)(ζζ
k
+
ζ
)/
2
k
=
5, 23
2
2
2
(
1
−
ξ
)(ηη
k
+
η
)(
1
−
ζ
)/
2
k
=
11, 17
2
2
2
(ξξ
k
+
ξ
)(
1
−
η
)(
1
−
ζ
)/
2
k
=
13, 15
2
2
2
(
1
−
ξ
)(
1
−
η
)(
1
−
ζ
)
k
=
14
Note that the local coordinate transformations (4.83) and (4.87) do not specify
how to generate the computational grid. The grid can be generated by either the
boundary-fitted coordinate method or another more arbitrary method. However, to
ensure a monotonic coordinate transformation, the angles between
ξ
,
η
, and
ζ
grid
lines should be away from 0
◦
and 180
◦
in the physical space. It is preferable that the
angles are between 45
◦
and 135
◦
.
Local coordinate transformation on moving grids
The local coordinate transformations (4.83) and (4.87) on fixed grids can be extended
to moving grids. For a 2-D case, for example, the physical domain is represented
by a boundary-fitted quadrilateral grid at each time or iteration step. Because the
grid adapts to the changing boundaries, the coordinate values of each grid point are
functions of time, i.e.,
x
k
=
, as shown in Fig. 4.10. Therefore,
the local coordinate transformation at each element reads (Wu, 1996a)
x
k
(τ)
and
y
k
=
y
k
(τ)
9
9
x
=
x
k
(τ)ϕ
k
(ξ
,
η)
,
y
=
y
k
(τ)ϕ
k
(ξ
,
η)
,
t
=
τ
(4.89)
k
=
1
k
=
1
where
ϕ
k
(
k
=
1, 2,
...
,9
)
are the interpolation functions expressed in Eq. (4.84).
