Geoscience Reference
In-Depth Information
Figure 4.7
Boundary-fitted coordinate transformation.
the coordinate transformation between physical domain (
x
,
y
) and logical domain
(
ξ
,
η
) is governed by the Poisson equations:
⎧
⎨
⎩
2
2
∂
x
2
+
∂
ξ
ξ
=
P
(ξ
,
η)
∂
∂
y
2
(4.74)
2
2
∂
x
2
+
∂
η
η
=
Q
(ξ
,
η)
y
2
∂
∂
where
P
and
Q
are source terms, which essentially determine the grid density
and smoothness.
Because the grid in the physical domain is unknown, it is inconvenient to solve the
equation set (4.74). Exchanging the independent and dependent parameters yields the
corresponding equations for
x
and
y
with respect to
ξ
and
η
:
⎨
J
2
P
∂
2
x
∂ξ
2
x
∂ξ∂η
+
2
x
∂η
A
∂
2
B
∂
C
∂
x
∂ξ
+
Q
∂
x
∂η
−
2
+
=
0
2
J
2
P
∂
(4.75)
2
y
∂ξ
2
y
∂ξ∂η
+
2
y
∂η
⎩
A
∂
2
B
∂
C
∂
y
∂ξ
+
Q
∂
y
∂η
2
−
2
+
=
0
2
2
,
B
2
where
A
=
(∂
x
/∂η)
+
(∂
y
/∂η)
=
∂
x
/∂ξ
·
∂
x
/∂η
+
∂
y
/∂ξ
·
∂
y
/∂η
, and
C
=
(∂
x
/∂ξ)
+
2
. Because the grid in the logical domain is prescribed, the two equations in
(4.75) can be solved conveniently.
The Jacobian determinant
J
of the coordinate transformation is
(∂
y
/∂ξ)
=
∂
∂
∂η
−
∂
∂
x
∂ξ
y
x
∂η
y
∂ξ
J
(4.76)
Under coordinate transformation (4.74), the first derivatives of
f
are
∂
f
1
∂
f
∂ξ
+
α
1
∂
f
∂η
1
2
x
=
α
(4.77)
∂
∂
f
2
∂
f
∂ξ
+
α
2
∂
f
∂η
1
2
y
=
α
(4.78)
∂
