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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
=
[
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
uu ,i
)
,i dV
=
S
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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