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where
ξ
is a source point inside the domain
V, x
is the integration point,
α
=
1 for two-
dimensional problems, and
2 for three-dimensional problems. As is common in the
boundary element literature, the symbols
α
=
and
x
are used to indicate points, but not
coordinates for a multi-dimensional problem. The weighting function
ξ
δ
u
is now a function
relating two points
for the beam problem. From the properties
of the Dirac delta function of Eq. (9.6), substitution of Eq. (9.21) into Eq. (9.20b) will result
in the expression of
u
at the source point
ξ
and
x
. This is the same as
δw
ξ
ξ
. It should be noted that the positions of
and
x
ξ
in Eq. (9.21) can be interchanged, meaning that
x
will be the source point and
will be the
integration point, and
should be involved in
dV
and
dS
of Eq. (9.20) instead of
x
.From
partial differential equation theory [e.g., Haberman, 1987], the expressions for
ξ
δ
u
satisfying
Eq. (9.21) are
1
δ
u
=
δ
u
(ξ
,x
)
=
(9.22a)
r
(ξ
,x
)
for three-dimensional problems and
1
δ
u
=
δ
u
(ξ
,x
)
=
ln
(9.22b)
r
(ξ
,x
)
[
(
=
√
r
j
r
j
with
r
j
2
]
1
/
2
for two-dimensional problems, where
r
(ξ
,x
)
=
x
j
−
ξ
)
=
x
j
−
ξ
j
is
j
the distance from the point
,
respectively.
The quantities in Eqs. (9.22) are the
fundamental solutions (Green's Functions)
for Poisson's
and Laplace's problems.
From the properties of the delta function
ξ
to
x,
and
x
j
and
ξ
j
are the coordinates of
x
and
ξ
u
is replaced by
u
∗
δ(ξ
,x
)
of Eq. (9.6), and if
δ
u
∗
∂
∂
∂
∂
q
∗
,
Eq. (9.20) can be written as
and
n
δ
u
by
n
=
u
∗
(ξ
u
∗
(ξ
q
∗
(ξ
2
απ
u
(ξ )
=−
b
(
x
)
,x
)
dV
(
x
)
+
q
(
x
)
,x
)
dS
(
x
)
−
u
(
x
)
,x
)
dS
(
x
)
V
S
S
(9.23)
where
1
r
a
i
a
i
u
∗
∂
u
∗
∂
q
∗
=
∂
n
=
∂
∂
∂
∂
∂
1
(
a
i
=
=
x
i
x
i
x
i
x
j
−
ξ
)
2
j
1
2
∂
(
=−
(
x
j
−
ξ
)
2
a
i
j
x
j
−
ξ
)
∂
x
i
j
(
−
ξ
)
1
x
i
i
=−
(
(
2
a
i
2
x
j
−
ξ
)
x
j
−
ξ
)
j
j
r
i
a
i
r
3
=−
for three-dimensional problems and
ln
1
r
a
i
=
ln
a
i
u
∗
∂
u
∗
∂
q
∗
=
∂
n
=
∂
∂
∂
∂
∂
1
a
i
=
(
x
i
x
i
x
i
x
j
−
ξ
j
)
2
−
1
2
∂
=
(
(
x
j
−
ξ
j
)
2
a
i
∂
x
i
x
j
−
ξ
j
)
−
1
x
i
−
ξ
i
)
(
(
r
i
a
i
r
2
=
(
2
a
i
=−
x
j
−
ξ
)
2
x
j
−
ξ
)
j
j
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