Biomedical Engineering Reference
In-Depth Information
2
4
3
5
¼
Z
Z
dk
T
ð
u
u
Þ
dC
u
þ
Z
Cu
k
T
ð
u
u
Þ
dC
u
k
T
d
ð
u
u
Þ
dC
u
d
Cu
Cu
¼
Z
dk
T
ð
u
u
Þ
dC
u
þ
Z
Cu
k
T
ð
du
du
Þ
dC
u
ð
3
:
39
Þ
|{z}
0
Cu
¼
Z
dk
T
ð
u
u
Þ
dC
u
þ
Z
Cu
du
T
kdC
u
Cu
Permitting to re-write the Eq. (
2.65
) as,
Z
d L
ð
T
cL
ð
dX
Z
du
T
b dX
Z
du
T
t dC
t
Z
Cu
dk
T
ð
u
u
Þ
dC
u
X
X
Ct
Z
Cu
ð
3
:
40
Þ
du
T
kdC
u
¼ 0
From Eq. (
3.1
) it is possible to determine the virtual displacement approxi-
mation of the interest point q
I
2
Q,
du
ð
q
I
Þ
¼
X
n
u
i
ð
q
I
Þ
du
ð
x
i
Þ
¼H
local
du
ð
x
Þ
| {z }
½3n
1
ð
3
:
41
Þ
I
|{z}
½3
3n
i¼1
being n the number of nodes inside the influence-domain of the interest point q
I
.
The virtual displacement on node x
i
2
X is defined by du
ð
x
i
Þ
and the local
diagonal shape function matrix H
loca
I
defined by Eq. (
3.23
). Using the procedure
described in Fig.
3.17
it is possible to present Eq. (
3.41
) using the global arrays,
du
ð
q
I
Þ
¼ H
I
du
ð
x
Þ
| {z }
½3N
1
ð
3
:
42
Þ
|{z}
½3
3N
In the classic three-dimensional deformation theory assuming small strains, the
linear differential operator L is defined as,
2
4
3
5
o
ox
00
oy
o
oz
0
L
T
¼
o
oy
o
ox
o
oz
0
0
0
ð
3
:
43
Þ
00
oz
o
oy
o
ox
0
Thus, using Eqs. (
3.41
) and (
3.43
) it is possible to present the first term of
Eq. (
3.40
) as,
