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FIGURE 8.10
The triangular mesh used in Example 8.5.
FIGURE 8.11
The numbering of the nodes of a triangle.
As in Example 8.3, the governing differential equation is given by
2
2
∂
y
2
+
∂
ψ
ψ
z
2
=−
2
with
ψ
=
0
on the boundary
(1)
∂
∂
in which
is the Prandtl stress function.
The corresponding global form can be based on a complementary principle such as the
principle of complementary virtual work which, using the normalized version of the stress
function of Chapter 1, Eq. (1.155), has the form (Eq. (3) of Chapter 2, Example 2.9)
ψ
∂ψ
∂
dy dz
∂
∂
z
δψ
+
∂ψ
∂
∂
W
∗
=
δ
y
δψ
−
2
δψ
=
0
(2)
z
∂
y
Use the gridwork of Fig. 8.9b, but now divide the squares into triangles as indicated in
Fig. 8.10, and number the nodes as shown in Fig. 8.11. The triangles occur in two orientations
(Figs. 8.11a and b). Equation (2) involves an integration over an area, and it is desirable that
the integrand be constant. From the simple central difference scheme, the derivatives of
ψ
in
a triangle are constant, and
δψ
in the triangle can be taken to be the average of the values of
δψ
at the corner nodes. Then the integrand in (2) becomes constant, which helps facilitate the
integration. Other kinds of meshes also achieve this [Pian, 1971], but are more complicated
than the triangles. From the central difference scheme in Table 8.1, the derivatives
∂ψ/∂
y
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