Biomedical Engineering Reference
In-Depth Information
X
n
u
i
ð
x
Þ
¼1
ð
4
:
171
Þ
i¼1
if at least a constant is included in the basis. Consider a constant field u
ð
x
Þ
¼c,
being c
2
. The polynomial form presented in Eq. (
4.151
) can be used to rep-
resent the constant field, however in this example only the constant term of the
polynomial is different from zero, t = 1,
R
u
ð
x
Þ
¼
X
r
i
ð
x
i
x
I
Þ
0
þ
X
p
j
ð
x
Þ
c
j
ð
x
Þþ
X
n
t
m
p
k
ð
x
Þ
0¼c
1
þ
0 ¼ c
ð
4
:
172
Þ
i¼1
j¼1
k¼t
þ
1
Once again, the non-constants coefficients b
i
ð
x
Þ
of the approximation function
u
h
ð
x
Þ
defined with Eq. (
4.102
), can be obtained imposing the interpolation,
u
h
ð
x
i
Þ
¼u
ð
x
i
Þ
. Thus, the substitution of the n nodal values u
ð
x
i
Þ
, obtained with
Eq. (
4.152
), back in the equation system of Eq. (
4.153
) allow to obtain the
coefficients a
ð
x
I
Þ
and b
ð
x
I
Þ
. It is perceptible that Eq. (
4.153
) is true only if
a
i
ð
x
I
Þ
¼0
for
i = {1, 2, …, n},
and
b
1
ð
x
I
Þ
¼c
1
ð
x
I
Þ
,
and
b
k
ð
x
I
Þ
¼0
for
k ¼
f
2
;
3
;
...
;
m
g
. Therefore,
u
h
ð
x
Þ
¼b
1
ð
x
I
Þ
1 ¼ c
1
ð
x
I
Þ
1 ¼ c
ð
4
:
173
Þ
The approximated field variable value of an interest point x
I
is determined
using the shape function values obtained at the nodes within the support-domain of
x
I
,
u
h
ð
x
I
Þ
¼
X
u
i
ð
x
I
Þ
u
ð
x
i
Þ
¼
X
u
i
ð
x
I
Þ
c ¼ c
X
n
n
n
u
i
ð
x
I
Þ
ð
4
:
174
Þ
i¼1
i¼1
i¼1
being the approximation function defined as u
h
ð
x
Þ
¼c it is possible to write,
c ¼ c
X
u
i
ð
x
I
Þ,
X
n
n
u
i
ð
x
I
Þ
¼1
ð
4
:
175
Þ
i¼1
i¼1
Therefore, if a functional basis contains a constant term, then the RPI shape
function is of the partition unit. Notice that in the case of absence of the poly-
nomial basis, if another functional basis is used to construct the RPI shape function
and it contains a constant term, the previous demonstration is still valid and the
RPI shape function will continue to possess the partition unit property.
If in the RPI formulation the only basis function used is the RBF, the partition
of unit depends on the RBF. If compactly supported RBFs are used, the con-
structed RPI shape functions will possess the partition of unit property, since all
the presented compactly supported RBFs, Eq. (
4.145
)to(
4.149
), clearly present a
constant term. However some non-compactly supported RBFs, such as the MQ-
