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Move
to the boundary, and use the procedure given in Section 9.2.1 to show that the
contribution of the second term of the right-hand side of the above equation is
ξ
−
απσ(ξ)
.
Thus Eq. (9.46) becomes
q
∗
dS
bq
∗
dV
2
απ
q
(ξ )
=−
απσ(ξ)
+
S
σ(
x
)
−
(9.47)
V
appear both inside and outside
the integrals. This kind of integral equation is called the
Fredholm
2
Note that in Eqs. (9.45) and (9.47), the unknowns
σ
and
µ
integral equation of the
second kind
. In Eq. (9.43b) the unknown
appears only inside the integral sign, so that it
belongs to the class of integral equations called
Fredholm integral equations of the first kind
.
In this indirect formulation, the unknown variables
σ
in these equations are not the
variables
u
and
q
that are of primary interest. The variables
u
and
q
are evaluated after
σ
and
µ
σ
and
are found.
For Dirichlet problems, which are solved using Eqs. (9.43b) and (9.45),
u
on the left-hand
sides of these equations is known because the point
µ
ξ
is on the boundary. These relations
can then be solved to evaluate
. For Neumann problems based on Eq. (9.47), the
known boundary conditions of
q
at point
σ
and
µ
ξ
on the boundary are substituted into Eq. (9.47)
and then the equation is solved for
are found, they are
substituted into Eqs. (9.43b) and (9.44), as appropriate, to find
u
inside the domain
V
.
The integral equations of Eqs. (9.43b), (9.45), and (9.47) can be solved using the boundary
element procedure given in Section 9.2.2.
µ
. After the values of
σ
and
µ
EXAMPLE 9.4 Indirect Formulation of the Torsional Bar
Study the indirect boundary element formulation for the pure torsion of a prismatic bar.
The governing differential equation and boundary condition for the warping function
on the cross-section of a bar under pure torsion are given in Eq. (3) of Example 9.2. The
governing equation is a two-dimensional partial differential equation with a Neumann
boundary condition. For this problem, the warping function can be represented in the
same fashion as Eq. (9.43b) with
b
=
0
,
i.e., for a point
ξ
inside the cross-section,
u
∗
(ξ
2
π ω(ξ )
=
S
σ
,x
)
dS
(1)
is a point inside the cross-section,
x
is a point on the boundary, and
u
∗
=
where
is
the Green's function given in Eq. (9.22b) for two-dimensional problems. Take the derivative
of (1), with respect to the outer normal of the cross-section at point
ξ
ln
(
1
/
r
)
ξ
,
to obtain
n
ln
1
dS
u
∗
(ξ
π
∂ω(ξ )
∂
S
σ
∂
,x
)
∂
∂
2
=
2
π
q
=
dS
=
S
σ
n
∂
n
r
a
2
∂
∂
ln
1
r
ln
1
r
dS
a
3
∂
∂
=
S
σ
+
x
2
x
3
r
i
a
i
r
2
=−
S
σ
dS
i
=
2
,
3
(2)
where
a
i
is the direction cosine of the outer normal with respect to direction
x
i
,i
=
2
,
3 and
the bar over
q
indicates a prescribed condition.
2
Erik Ivar Fredholm (1866-1927) was a Swedish professor of mathematics and physics at Stockholm University.
He initiated the modern theory of integral equations, which are named after him.
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