Biomedical Engineering Reference
In-Depth Information
which can be explicitly defined as,
2
4
2
4
3
5
2
4
3
5
2
4
3
5
3
5
1
x
1
y
1
z
1
1
x
2
y
2
z
2
1
x
n
y
n
z
n
B
ð
x
I
Þ
;
n
¼
o
W
ð
x
1
x
I
Þ
o
W
ð
x
2
x
I
Þ
on
oW
ð
x
n
x
I
Þ
on
ð
4
:
36
Þ
on
and A
ð
x
I
Þ
1
is computed as,
;
n
A
ð
x
I
Þ
1
;
n
¼
A
ð
x
I
Þ
1
A
ð
x
I
Þ
;
n
A
ð
x
I
Þ
1
ð
4
:
37
Þ
being,
A
ð
x
I
Þ
;
n
¼
X
n
o
W
ð
x
i
x
I
Þ
on
p
ð
x
i
Þ
p
ð
x
i
Þ
T
ð
4
:
38
Þ
i¼1
The partial derivative of the moment matrix A
ð
x
I
Þ
can be explicitly defined as,
2
3
1
x
i
y
i
z
i
4
5
A
ð
x
I
Þ
;
n
¼
X
n
o
W
ð
x
i
x
I
Þ
on
½
1
x
i
y
i
z
i
i¼1
2
3
1
x
1
y
1
z
1
4
5
x
1
x
1
x
1
y
1
x
1
z
1
¼
o
W
ð
x
1
x
I
Þ
on
þ
ð
4
:
39
Þ
y
1
y
1
y
1
x
1
y
1
z
1
z
1
z
1
z
1
x
1
z
1
y
1
2
3
1
x
n
y
n
z
n
4
5
x
n
x
n
x
n
y
n
x
n
z
n
þ
o
W
ð
x
n
x
I
Þ
on
y
n
y
n
y
n
x
n
y
n
z
n
z
n
z
n
z
n
x
n
z
n
y
n
Following a similar methodology, it is possible to obtain the following second
order partial derivatives of the approximation field function with respect to g
(which is another generic variable representing x, y or z),
¼
X
o
2
u
h
ð
x
I
Þ
onog
n
o
2
u
i
ð
x
I
Þ
onog
u
i
¼ u
ð
x
I
Þ
;
ng
u
s
ð
4
:
40
Þ
i¼1
and the spatial second order partial derivatives of the MLS shape function with
respect to n and g are obtained with,
