Biomedical Engineering Reference
In-Depth Information
Following the same procedure for the other elements and adding up the contribu-
tion for the internal source term for all (three) elements gives
∼
=
f
1
+
f
∼
2
+
f
∼
f
3
.
(14.54)
∼
This leads to
⎡
⎣
⎤
⎦
f
1
w
1
w
2
w
3
w
4
f
2
+
f
1
T
f
∼
int
=
∼
.
(14.55)
f
2
+
f
1
f
2
What remains is the term
B
in Eq. (
14.25
). The effect of this boundary term
B
may
be included via
T
f
∼
B
=−
w
(
a
)
p
a
+
w
(
b
)
p
b
=
∼
ext
,
(14.56)
where
f
∼
ext
contains
p
a
and
p
b
at the appropriate positions, according to
⎡
⎣
⎤
⎦
−
p
a
0
0
p
b
w
1
w
2
w
3
w
4
B
=
.
(14.57)
This finally leads to an equation of the form:
T
K
∼
=
∼
T
(
f
∼
∼
int
+
f
∼
ext
) .
(14.58)
Using the fact that Eq. (
14.58
) must hold for 'all'
∼
, this results in the so-called
discrete set of equations
:
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
K
11
K
12
f
1
0
0
u
1
u
2
u
3
u
4
−
p
a
K
21
K
22
+
K
11
K
12
f
2
f
1
0
+
.
(14.59)
K
21
K
22
+
K
11
K
12
f
2
f
1
0
+
K
21
K
22
f
2
0
0
+
p
b
Assume that in the example of Fig.
14.4
at
x
=
x
1
, an essential boundary condi-
tion is prescribed
u
(
x
1
)
=
U
. This means that
p
a
=
p
u
is unknown beforehand.
At
x
=
x
4
a natural boundary condition is prescribed, so
p
b
=
P
is known
beforehand.
Then
⎡
⎤
⎡
⎤
⎡
⎤
K
11
K
12
f
1
0
0
U
u
2
u
3
u
4
−
p
u
⎣
⎦
⎣
⎦
=
⎣
⎦
K
21
K
22
+
K
11
K
12
f
2
+
f
1
0
.
(14.60)
K
21
K
22
+
K
11
K
12
f
2
+
f
1
0
K
21
K
22
f
2
0
0
+
P
