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Difference stars for equations containing summations of derivatives are also readily formed.
For example, for
h
x
=
h
y
=
h,
1
2
u
2
u
∂
x
2
+
∂
h
2
h
2
y
2
=
u
=
(
1
/
)
1
−
2
1
u
+
(
1
/
)
−
2
1
u
∂
∂
1
0
1
0
h
2
h
2
=
(
1
/
)
1
−
2
−
2
1
u
=
1
/
1
−
4
1
u
(8.26)
1
0
1
0
4
u
4
u
4
u
∂
∂
y
2
+
∂
x
4
+
2
y
4
=
u
=
∂
∂
x
2
∂
∂
1
2
−
4
2
−
4
6
h
4
h
4
h
4
(
1
/
)
1
−
4
6
−
41
u
+
(
1
/
)
u
+
(
1
/
)
u
−
4
8
−
4
2
−
4
2
−
4
1
001 00
02
−
8
2
0
h
4
=
(
1
/
)
u
(8.27)
1
−
8
20
−
81
02
8 2 0
001 00
−
The difference operation of Eq. (8.27) can also be obtained with the help of a two-
dimensional Taylor series. Improved difference quotients can be established, including
multiple position differences (see, for example, Zurm uhl (1957)).
EXAMPLE 8.3 Torsional Stresses on the Cross-Section of a Bar
Consider the torsion of a prismatic bar with the square cross-section shown in Fig. 8.9a.
The force form of the governing equation is [Chapter 1, Eq. (1.159)]
∂
2
y
2
+
∂
ψ
2
ψ
z
2
=−
ψ
=
2
with
0
on the boundary
(1)
∂
∂
in which
is the Prandtl stress function defined in Eq. (1) of Example 7.8. This is a normal-
ized version of the stress function of Eq. (1.155).
First establish the grid system in the form shown in Fig. 8.9b. Let the grid points be
equally spaced in the
j
and
k
directions with
h
y
=
ψ
h
z
=
1
/
3
.
Let
ψ
j,k
,j,k
=
1
,
2
,
3
,
4
,
be the
values of
ψ
at the nodal points
j, k
. The boundary conditions for
ψ
are
ψ
j,
1
=
0
ψ
1
,k
=
0
ψ
j,
4
=
0
and
ψ
4
,k
=
0
(2)
j
=
1
,
2
,
3
,
4
k
=
1
,
2
,
3
,
4
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