Biomedical Engineering Reference
In-Depth Information
Fig. 3.20
a Essential boundary curve. b Essential boundary surface
U q
I
T
¼ U
1
q
I
U
2
q
I
... U
n
k
q
I
ð
3
:
50
Þ
The local diagonal interpolation matrix N
loca
I
can be assembled in a global
diagonal interpolation matrix N
I
using the process in Fig.
3.17
.
dk
ð
q
I
Þ
¼ N
I
dk
ð
x
Þ
|{z }
½3N
k
1
;
x
2
C
u
ð
3
:
51
Þ
|{z}
½3
3N
k
Notice that the dimension of the global diagonal interpolation matrix N
I
is not
½3
3N
as the global diagonal shape function matrix H
I
, but instead it is
½3
3N
k
, being N
k
the total number of nodes belonging to the essential boundary
domain C
u
.
As already mentioned, the interpolation functionU
ð
q
I
Þ
can be obtained using
the Lagrange interpolants used in the conventional FEM [
3
,
46
,
47
,
50
]. However
the author prefer to use a simpler radial point interpolation to obtain U
ð
q
I
Þ
[
43
-
45
].
The works available in the literature refer that the first order Lagrange inter-
polation function is sufficient [
3
,
46
,
47
,
50
]. That is to say that: the essential
boundary curve is discretized using line segments and each integration point
possesses only two nodes on the influence-domain (n
k
¼ 2); and the essential
boundary surface is discretized using triangular patches and each integration point
possesses three nodes on the influence-domain (n
k
¼ 3).
In order to obtain the interpolation function on the essential boundary curve
C
u
ð
C
Þ
using the radial point interpolation, first it is necessary to identify the two
closest field nodes from the integration point q
I
. Notice that the two field nodes
f
x
1
;
x
2
g2
C
u
ð
C
Þ
q
I
,
form
a
segment
line,
containing
the
integration
point
3
Fig.
3.20
a. The radial distance r
ij
2
R
between to spatial points
f
x
i
;
x
j
g2
R
is
determined by the Euclidean norm,
q
ð
x
j
x
i
Þ
2
þð
y
j
y
i
Þ
2
þð
z
j
z
i
Þ
2
¼
r
ij
¼ x
j
x
i
ð
3
:
52
Þ
