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The first term of Eq. (9.14) can be written as
V
δ
2
u
V
δ
u
(
∇
)
dV
=
uu
,ii
dV
∂
∂
dV
u
∂
u
∂
∂
u
∂
u
∂
u
∂
u
=
δ
+
δ
+
δ
x
1
∂
x
1
x
2
∂
x
2
∂
x
3
∂
x
3
V
∂
u
dV
u
∂
+
∂
u
∂
+
∂
u
∂
−
x
1
δ
u
x
2
δ
u
x
3
δ
∂
x
1
∂
∂
x
2
∂
∂
x
3
∂
V
=
[
(δ
uu
,i
)
,i
−
(
u
,i
δ
u
,i
)
]
dV
(9.15)
V
From Gauss' integral theorem [Eq. (II.8) of Appendix II], the first term in the final integral
on the right-hand side of Eq. (9.15) can be written as
V
(δ
uu
,i
)
,i
dV
=
S
(δ
uu
,i
)
a
i
dS
=
S
δ
u
(
u
,i
a
i
)
dS
where
a
i
is the direction cosine of the outer normal of the boundary with respect to the
x
i
axis. Since
u
,i
a
i
=
∂
u
/∂
n
[e.g., Gipson, 1987]
u
∂
u
S
δ
u
(
u
,i
a
i
)
dS
=
S
δ
n
dS
(9.16)
∂
The second term in the final integral of Eq. (9.15) can be processed as
2
u
,i
δ
u
,i
dV
=
V
(
u
δ
u
,i
)
,i
dV
−
u
(
∇
δ
u
)
dV
V
V
=
V
(
u
δ
u
,i
)
,i
dV
−
u
δ
u
,ii
dV
(9.17)
V
Again using Gauss' theorem on the first term on the right-hand side of Eq. (9.17) and follow-
ing the same procedure as that for manipulating Eq. (9.15) results in
u
∂
∂
δ
∂
u
V
(
u
δ
u
,i
)
,i
dV
=
n
δ
udS
=
u
n
dS
(9.18)
∂
S
S
Substitute Eqs. (9.16), (9.17), and (9.18) into Eq. (9.15) to find
u
dS
u
∂
u
u
∂
∂
2
u
2
V
δ
u
(
∇
)
dV
=
δ
n
−
n
δ
+
u
(
∇
δ
u
)
dV
(9.19)
∂
S
V
Substitute Eq. (9.19) into Eq. (9.14) to obtain
u
∂
∂
u
∂
∂
2
u
(
∇
δ
u
)
dV
=
b
δ
udV
−
q
δ
udS
−
q
δ
udS
+
n
δ
udS
+
n
δ
udS
V
V
S
q
S
u
S
q
S
u
(9.20a)
or
u
∂
∂
2
u
(
∇
δ
u
)
dV
=
b
δ
udV
−
q
δ
udS
+
n
δ
udS
(9.20b)
V
V
S
S
where
q
and
u
in the second and third integrals of the right-hand side of Eq. (9.20b) repre
s
ent
the variable
s
u
and
q
on the boundary, and they satisfy the boundary conditions
u
=
u
on
S
u
and
q
=
q
on
S
q
.
Let
2
∇
δ
=−
απ δ(ξ
)
u
2
,x
(9.21)
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