Civil Engineering Reference
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which have to be solved for obtaining vector
^`
c
t
and the two rows of matrix
K
and
A
.
It is obvious that for the first problem to be solved for region
I
, where for all nodes
u
=0,
^`
c
t
will also be zero. Following the procedure in chapter 7 and referring to the
element numbering of Figure 11.5 we obtain the following integral equations for the
second and third problem for region
I
.
2
I
2
I
4
I
4
I
1
2
u
' '
1
U
t
U
t
' '
U
t
U
t
''
T
T
1
2
1 1
i
2 1
i
1 1
i
2 1
i
2
i
1
i
(11.13)
22
2
I
4
I
4
I
3
2
u
' '
1
U
t
U
t
' '
U
t
U
t
''
T
T
1
3
1 2
i
2 2
i
1 2
i
2 2
i
1
i
2
i
for
i=1,2,3,4
. In Equation 11.13, two subscripts have been introduced for
t
: the first
subscript refers to the node number where
t
is computed and the second to the node
number where the unit value of
u
is applied. The roman superscript refers to the region
number. The notation for
' ' is the same as defined in Chapter 7, i.e. the first
subscript defines the node number and the second the collocation point number; the
superscript refers to the boundary element number (in square areas in Figure 11.5).
This gives the following system of equations with two right hand sides
and
UT
2
11
2
21
4
11
4
21
I
I
1
21
2
11
2
21
3
11
ª
º
-
½
-
½
'
U
'
U
'
U
'
U
t
t
'
T
'
T
'
T
'
T
11
12
«
»
°
®
°
¾
°
®
°
¾
2
12
2
22
4
12
4
22
I
I
1
22
2
12
1
22
2
12
'
U
'
U
'
U
'
U
t
t
'
T
'
T
'
T
'
T
21
22
«
»
(11.14)
2
13
2
23
4
13
4
23
I
I
1
23
2
13
1
23
2
13
'
U
'
U
'
U
'
U
°
t
t
°
°
'
T
'
T
'
T
'
T
°
31
32
«
»
°
°
°
°
2
14
1
24
4
14
4
24
I
I
1
24
2
14
1
24
2
14
'
U
'
U
'
U
'
U
t
t
'
T
'
T
'
T
'
T
¬
¼
¯
¿
¯
¿
41
42
After solving the system of equations we obtain
(11.15)
I
I
I
I
I
^`
I
^`
^`
^`
t
K
u
;
x
A
u
c
c
f
c
where
I
I
I
ª
º
- ½
-
°°
I
t
t
u
t
° °
^`
I
^`
I
I
11
12
1
1
t
;
K
«
»
;
u
®¾
® ¾
c
c
t
I
I
I
t
t
u
°°
«
»
¯¿
°°
¬
¼
¯ ¿
2
21
22
2
(11.16)
-½
I
ª
I
I
º
t
t
t
°°
I
^`
3
31
32
I
x
;
A
«
»
®¾
f
I
I
I
t
«
t
t
»
°
¯¿
¬
¼
4
41
42
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