Biomedical Engineering Reference
In-Depth Information
Therefore, it is possible to identify the interpolation function vector u
ð
x
I
Þ
on
Eq. (
4.121
),
¼
u
ð
x
I
Þ
T
ð
4
:
123
Þ
M
1
T
u
s
z
u
s
z
u
h
ð
x
I
Þ
¼
r
ð
x
I
Þ
T
p
ð
x
I
Þ
T
w
ð
x
I
Þ
T
The interpolation function vector u
ð
x
I
Þ
and the byproduct vector w
ð
x
I
Þ
are
defined as,
u
ð
x
I
Þ
T
¼
f
u
1
ð
x
I
Þ
u
2
ð
x
I
Þ
... u
n
ð
x
I
Þg
ð
4
:
124
Þ
w
ð
x
I
Þ
T
¼
f
w
1
ð
x
I
Þ
w
2
ð
x
I
Þ
... w
m
ð
x
I
Þg
ð
4
:
125
Þ
Notice that the byproduct vector w
ð
x
I
Þ
only exists if a polynomial basis is
considered, otherwise it does not appears. The components of vector w
ð
x
I
Þ
do not
possess any relevant physical meaning. Additionally, to obtain the interpolation
field variable, the byproduct vector w
ð
x
I
Þ
is multiplied to the null vector z,
therefore w
ð
x
I
Þ
can be completely neglected.
u
h
ð
x
I
Þ
¼
X
u
i
ð
x
I
Þ
u
i
þ
X
¼
X
n
m
n
w
i
ð
x
I
Þ
z
i
|{z}
0
u
i
ð
x
I
Þ
u
i
ð
4
:
126
Þ
i¼1
i¼1
i¼1
being u
i
ð
x
I
Þ
the shape function value of the interest point x
I
for the ith node
obtained considering the nodes n inside the support-domain of interest point x
I
.
Consider now two distinct interest points
d
, being x
J
6
¼
x
I
.
The shape functions of both interest points possess the exactly same support-
domain nodal set N
I
¼ N
J
¼ n
1
;
n
2
;
...
;
n
f g
N, being N the complete nodal set
discretizing the problem domain. Notice that for the interest point x
I
, the total
moment matrix M
I
T
constructed using the N
I
nodal set will be equal to the total
moment matrix M
T
of interest point x
J
obtained for the N
J
nodal set, leading to
a
i
ð
x
I
Þ
¼a
i
ð
x
J
Þ
and b
i
ð
x
I
Þ
¼b
i
ð
x
J
Þ
. Therefore the obtained coefficients a
i
ð
x
I
Þ
and
b
j
ð
x
I
Þ
are in fact constant as long as the same n nodes inside the support-domain of
the interest point x
I
are maintained. The most important conclusion is that the total
moment matrix, and also the RBF and the polynomial moment matrices, are not
directly dependent on the spatial position of the interest point x
I
. Consequently,
x
I
;
x
J
g
2
X
^
x
i
2
f
R
¼ 000
o
R
ox
o
P
ox
o
M
T
ox
f
g
ð
4
:
127
Þ
In order to compute the partial derivatives of the interpolated field function,
Eq. (
4.123
), it is necessary to obtain the respective RPI shape functions partial
derivatives.
