Biomedical Engineering Reference
In-Depth Information
¼
of
ð
x
I
Þ
T
on
F
1
þ
f
ð
x
I
Þ
T
o
F
1
on
|{z}
0
u
ð
x
I
Þ
;
n
¼ f
ð
x
I
Þ
T
F
1
ð
4
:
95
Þ
;
n
Notice that, since the moment matrix F does not depend on the interest point x
I
spatial position, it is only require to obtain the partial derivative of the functional
f
ð
x
I
Þ
. The second order partial derivative of the interpolated field function, with
respect to the generic variables n and g, is obtained with,
¼
X
2
u
h
ð
x
I
Þ
onog
n
2
u
i
ð
x
I
Þ
onog
o
o
u
i
¼ u
ð
x
I
Þ
;
ng
u
s
ð
4
:
96
Þ
i¼1
As expected, the spatial second order partial derivative of the PIM shape
function with respect to n and g, is defined as,
2
f
ð
x
I
Þ
T
onog
¼
o
u
ð
x
I
Þ
;
ng
¼ f
ð
x
I
Þ
T
F
1
F
1
ð
4
:
97
Þ
;
ng
If the functional f
ð
x
I
Þ
is defined by a complete polynomial, being f
i
ð
x
I
Þ
monomial terms of a complete polynomial basis, it is possible that F becomes
singular or ill-conditioned, leading to the PIM failure. The most common reason
for the inexistence of F
-1
is the spatial collinearity of field nodes belonging to the
same support-domain, which is recurrent in uniformly distributed nodal meshes or
linear domain boundaries.
This drawback can be shown with the following simple example. Consider an
interest point x
I
2
X with a support-domain containing the following six nodes
defined in the two-dimensional space,
000111
012012
X
I
¼ x
1
f
x
2
x
3
x
4
x
5
x
6
g
¼
ð
4
:
98
Þ
It is possible to observe that nodes x
1
, x
2
and x
3
are collinear, as well as nodes
x
4
, x
5
and x
6
. Since the support-domain has six nodes, n = 6, in order to construct
a polynomial PIM shape function it is required a moment matrix F defined by a
complete polynomial basis with m = 6,
f
ð
x
Þ
¼ p
1
ð
x
Þ
f
p
2
ð
x
Þ
... p
6
ð
x
Þ
g
¼
1
xy
2
y
2
g
ð
4
:
99
Þ
xy
Therefore the moment matrix can be written as,
