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where
J
−
1
∂
y
∂η
J
−
1
∂
y
∂ξ
J
−
1
∂
x
∂η
J
−
1
∂
x
∂ξ
1
1
2
1
1
2
2
2
α
=
,
α
=−
,
α
=−
,
α
=
(4.79)
The second derivatives are
2
f
2
f
∂ξ
2
f
∂ξ∂η
+
α
2
f
∂η
1
1
∂ξ
+
α
1
1
∂η
∂
1
∂
1
∂
1
∂
1
∂α
1
∂α
f
∂ξ
∂
1
1
1
1
1
2
2
1
2
1
2
=
α
α
2
+
2
α
α
α
2
+
α
∂
x
2
2
1
2
1
∂η
1
∂α
1
∂α
∂
f
∂η
1
2
+
α
∂ξ
+
α
(4.80)
2
f
2
f
∂ξ
2
f
∂ξ∂η
+
α
2
f
∂η
∂
2
∂
)
∂
2
∂
1
1
1
1
1
2
2
2
1
1
2
2
1
2
y
=
α
α
2
+
(α
α
+
α
α
α
∂
x
∂
2
α
1
2
∂ξ
+
α
1
2
∂η
2
2
∂ξ
+
α
2
2
∂η
1
∂α
1
∂α
∂
f
∂ξ
+
1
∂α
1
∂α
∂
f
∂η
1
2
1
2
+
α
(4.81)
α
2
f
2
f
∂ξ
2
f
∂ξ∂η
+
α
2
f
∂η
1
2
1
2
∂η
2
∂α
2
∂α
∂
2
∂
2
∂
2
∂
∂
f
∂ξ
1
2
1
1
2
2
2
2
2
1
2
=
α
α
2
+
2
α
α
α
2
+
∂ξ
+
α
∂
y
2
2
2
2
2
∂η
2
∂α
2
∂α
∂
f
∂η
1
2
+
α
∂ξ
+
α
(4.82)
Local coordinate transformation on fixed grids
The previous boundary-fitted coordinate transformation can provide high-quality
numerical grids with global properties, such as orthogonality and smoothness. How-
ever, two partial differential equations need to be solved in the entire domain.
A simpler method for handling irregular boundary problems is the local coordinate
transformation that is based on only individual elements.
Suppose a 2-D physical domain is represented by a quadrilateral grid, and the nine-
point quadrilateral isoparametric element shown in Fig. 4.8 is used as the basic element
(Wu and Li, 1992; Wang and Hu, 1993). At each element all nine points are numbered
1 through 9 and the 5th point is the central point. This irregular element is converted
into a rectangle by the following coordinate transformation:
9
9
x
=
x
k
ϕ
(ξ
,
η)
,
y
=
y
k
ϕ
(ξ
,
η)
(4.83)
k
k
k
=
1
k
=
1
where
x
k
and
y
k
are the coordinate values of the
k
th point in the (
x
,
y
) coordinate
system; and
ϕ
k
(
k
=
1, 2,
...
, 9) are the interpolation functions, which are quadratic
