Biomedical Engineering Reference
In-Depth Information
It possible to write Eq. (
4.67
) as,
u
h
ð
x
Þ
¼g
ð
x
Þ
T
A
ð
x
Þ
1
A
ð
x
Þ
e
ð
x
Þ
¼g
ð
x
Þ
T
e
ð
x
Þ
¼u
ð
x
Þ
ð
4
:
69
Þ
Proving that any functional appearing in the basis can be exactly reproduced by
the MLS approximation. This property permits to construct MLS shape functions
to deal with specific practical problems. The singular stress field at a crack tip can
be obtained with the inclusion of singular functions in the basis combined with the
polynomial basis [
19
,
20
] as shown in Eq. (
4.63
). This technique permits to
increase the solution accuracy, since using only a normal complete polynomial
basis usually lead to significant errors in the analysis of crack tip problems.
Nevertheless, if these enriched basis functions are included in the basis of the MLS
approximation, it is important to certify the non-singularity of the weighted
moment matrix A(x).
4.3.3.3 Partition of Unity
The MLS shape function u
i
(x) satisfies the partition of unity,
X
n
u
i
ð
x
Þ
¼1
ð
4
:
70
Þ
i¼1
if a constant is included in the basis. The argument used to prove the consistency
property of the MLS shape function can be used to prove the partition of unity
property. Consider the constant field u
ð
x
Þ
¼c,withc
2
, which can assume the
same polynomial form as in Eq. (
4.58
). However in this case only the constant
term of the polynomial exists, k = 1,
R
u
ð
x
Þ
¼
X
k
p
i
ð
x
Þ
a
i
ð
x
Þþ
X
m
p
j
ð
x
Þ
0 ¼ a
1
þ
0 ¼ c
ð
4
:
71
Þ
i¼1
j¼k
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. (
4.6
), can be obtained with the minimization of the qua-
dratic norm presented in Eq. (
4.11
). Then, substituting in Eq. (
4.11
) the approxi-
mation function, u
h
ð
x
Þ
, from Eq. (
4.6
), and the field function, u
ð
x
Þ
, defined by
Eq. (
4.71
), it is possible to write,
"
!
p
1
ð
x
Þ
a
1
ð
x
Þþ
X
!
#
2
J ¼
X
p
1
ð
x
Þ
b
1
ð
x
Þþ
X
n
m
m
W
ð
x
Þ
p
j
ð
x
Þ
b
j
ð
x
Þ
p
j
ð
x
Þ
0
i¼1
j¼2
j¼2
ð
4
:
72
Þ
