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9.8 For a Poisson's equation
∇
1
u
=−
b
with
k
1
∂
∂
k
2
∂
∂
2
1
∇
=
x
1
+
for two-dimensional problems
x
2
k
1
∂
∂
k
2
∂
∂
k
3
∂
∂
2
1
∇
=
x
1
+
x
2
+
for three-dimensional problems
x
3
derive a boundary integral equation.
/
√
k
i
and then use Eqs. (9.22) and
=
Hint:
Begin with the transformation
x
i
x
i
(9.29).
Answer:
The boundary integral equation is the same as Eq. (9.29), with the defi-
nitions
k
1
∂
u
∗
k
2
∂
u
∗
1
√
k
1
k
2
ln
1
r
u
∗
=
q
∗
=
x
1
+
∂
∂
x
2
1
k
1
(
1
k
2
(
r
=
x
1
−
ξ
)
2
+
x
2
−
ξ
)
2
for two-dimensional problems
1
2
2
b
9.9 Show that when the homogeneous term
b
of a Poisson's equation satisfies
∇
≡
0,
there exists a relationship
b
∂v
∗
∂
dS
n
−
v
∗
∂
b
bu
∗
dV
=
∂
n
V
S
where
v
∗
=
r
2
/
4[ln 1
/
r
+
1] for two-dimensional problems,
v
∗
=
r
/
4 for three-dimensional problems.
Start from
V
(
2
v
∗
−
v
∗
∇
2
b
2
v
∗
=
u
∗
.
Hint:
b
∇
)
dV
and recognize that
∇
9.10 Fill in the details of the derivation of the relations in Eqs. (9.48) to (9.51).
9.11 Verify that
∂ε
∂
∂ε
∂
lim
→
dS
=
0
x
i
x
j
0
S
Hint:
Use the relationship shown in Fig. 9.2 and obtain formulas for
ε
i
/ε
,
ϕ
and
2
sin
θ.
Also, note that
dS
=
ε
ϕ
d
θ
d
ϕ.
9.12 Find the coefficients
c
i
of
u
i
of Eq. (9.88) when
ξ
is located at the vertex of a cone.
Hint:
The range of the variation of
ϕ
is changed.
9.13 Calculate the elements
H
12
,H
13
,H
21
,H
31
of the
H
matrix of Eq. (9.94) for a square
plane strain region of 1
×
1
.
Discretize the boundary of the square region into 4 linear
elements.
9.14 Calculate the elements
G
12
,G
13
,G
21
,G
31
of the
G
matrix of Eq. (9.94) for the square
plane strain region for Problem 9.13. Let the boundary element discretization be the
same as in Problem 9.13.
9.15 Use two point Gauss quadrature to integrate the integral equation for the torsion
problem of a 1
1 square cross-section. Form the system matrix and calculate the
torsional constant.
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