Biomedical Engineering Reference
In-Depth Information
5.2 RPI Shape Function Patch Test
The patch test [
8
] was originally designed to prove the convergence in non-
conforming finite element formulation. Generally, the test consists on the impo-
sition of a known displacement field in the boundary of the patch. In this topic
linear patch tests are used. If the prescribed field is reproduced in the interior of the
patch then the test is verified. Although being a benchmark for the evaluation and
validation of non-conforming elements, in the context of the meshless methods the
relevance of the patch test, from the convergence point of view, is still an open
issue.
To perform the patch test first the problem domain, X
d
, is discretized with
R
d
. Then the procedure described in
Table
5.2
is performed and in step 7 a known displacement field is imposed in the
essential boundary of the patch. Afterwards, in step 8, the displacement field
U ¼
f
u
1
;
u
2
;
...
;
u
N
g
is obtained. Since in this topic only the two-dimensional
deformation theories (plane strain and plane stress) and the classical three-
dimensional deformation theory are considered, the number of degrees of freedom
in
a nodal distribution, X ¼
f
x
1
;
x
2
;
...
;
x
N
g2
R
each
node
is
equal
to
the
domain
dimensional
space,
being
u
i
¼
f
u
1
;
u
2
;
...
;
u
d
g
T
2
d
. The patch test exact solution is compared with the
meshless solution using the following three-dimensional medium error expression,
R
q
(u
i
)
meshless
(u
i
)
exact
Þ
2
þ
(v
i
)
meshless
(v
i
)
exact
Þ
2
þ
(w
i
)
meshless
(w
i
)
exact
Þ
2
X
N
ð
ð
ð
E
med
¼
1
N
q
(u
i
)
exact
þ
(v
i
)
exact
þ
(w
i
)
exact
i¼1
ð
5
:
1
Þ
Being N the total number of nodes discretizing the problem domain. In this
topic only the MQ-RBF is considered to construct the RPI shape functions. The
MQ-RBF expression, Eq. (
4.108
), require two shape parameters: c and p. In order
to obtain both shape parameters an optimization test will be performed using the
linear patch test.
In the following subsection it is reproduced the patch test of the radial point
interpolation method (RPIM) [
9
] in order to clearly explain the major differences
between the RPIM and the NNRPIM and to introduce the NNRPIM procedure for
the elastostatic and elastodynamic analysis.
5.2.1 RPIM Patch Test
In this topic, the considered RPIM formulation is analogous with the one suggested
in the literature [
9
,
10
]. Consider a two-dimensional domain x
2
2
: x
2
½0
;
1
;
y
2
½0
;
1
discretized with two distinct nodal distributions: an irregular distribution,
R
