Biomedical Engineering Reference
In-Depth Information
Fig. 4.16 Comparison between the second order partial derivatives of the MLS shape functions
obtained with the cubic spline (W3) and the quartic spline (W4)
1. The MLS shape function support-domain is defined based on the influence-
domain of the interest point x
I
, d
s
¼ max x
i
x
k k; 8
x
i
2
X
I
.
2. The polynomial vector for interest point x
I
is defined: p(x
I
). The dimension of
p(x
I
) depends on the monomials, [m 9 1]. For example, in a three-dimensional
space,
the
linear
polynomial
vector
for
interest
point
x
I
is
defined
as:
p
ð
x
I
Þ
¼
f
1 x
I
y
I
z
I
g
T
.
3. Next, the weight of each node can be determined using a weight function,
W
ð
x
I
Þ
¼
f
W
ð
x
1
x
I
Þ
W
ð
x
2
x
I
Þ
W
ð
x
n
x
I
Þg
.
4. With the nodal weight determined it is possible to construct the weighted
polynomial matrix B
ð
x
I
Þ
as indicated in Eq. (
4.24
). This matrix has a dimen-
sion [m 9 n].
5. The momentum matrix A
ð
x
I
Þ
is constructed as in Eq. (
4.22
) and then A
ð
x
I
Þ
1
is
computed. The momentum matrix is a square matrix with [m 9 m].
6. Finally, the interpolation function u
ð
x
I
Þ
is obtained with Eq. (
4.31
).
The MLS shape function first order partial derivatives can be obtained applying
Eq. (
4.33
) and the second order partial derivatives are obtained with Eq. (
4.41
).
4.3.6 Influence of the Size of the Support-Domain
In this subsection the importance of the support-domain size, which was referred
in
Sect. 4.2
, is shown with a simple example.
Consider a one-dimensional domain X
R
with x
2
R
: x
2
½0
;
1
, being X ¼
1
the set of nodes discretizing the problem domain.
Two distinct nodal distributions are considered in this example: a uniform nodal
distribution and an irregular nodal distribution. The objective is to adjust an
approximation function u
h
ð
x
Þ
to the N discrete nodal values u
ð
x
i
Þ
. The spatial
f
x
1
;
x
2
;
...
;
x
11
g 2
X
^
x
i
2
R
