Biomedical Engineering Reference
In-Depth Information
Notice that the dynamic term was neglected and it was included an additional
term to enforce the displacement constrain u
u ¼ 0 by the penalty method.
Matrix a is defined as,
2
4
3
5
a
ð
x
Þ
1
0
0
0
a
ð
x
Þ
2
0
a ¼
ð
3
:
67
Þ
.
.
.
.
.
.
0
0
a
ð
x
Þ
m
where a
ð
x
Þ
i
are penalty factors and m is the number of degrees of freedom in each
field node x
2
d
required by the studied formulation. The magnitudes of the
penalty factors a
ð
x
Þ
i
may be assumed as a functional of the spatial location of the
interest point. Additionally, for the same interest point, distinct a
ð
x
Þ
i
can be
considered for distinct degrees of freedom. Nevertheless, generally the penalty
factors a
ð
x
Þ
i
are large positive values, constant along the solid discretized domain
and equal for all degrees of freedom: a
ð
x
Þ
i
¼ a. The magnitude of the penalty
factor is addressed in Section ''
Penalty Method
'' .
The last term of Eq. (
3.66
) represents an integral over the essential boundary
C
u
2
C, thus once again it will be necessary to numerically integrate the subdo-
main C
u
. Therefore the essential boundary subdomain has to be discretized with
integration points. The procedure is fully described in ''
Lagrange multipliers
'' .
First
R
the
NC
field
nodes
on
the
essential
boundary
are
identified:
X
Cu
¼ x
1
;
x
2
;
...
;
x
NC
f
g 2
C
u
, being X
Cu
2
X
and X
the complete nodal set,
3
, Fig.
3.15
. Afterwards, new
Fig.
3.16
a, discretizing the problem domain X
2
R
integration
points
are
determined
on
the
boundary
subdomain
C
u
:
n o
2
C
u
, being QC the number of integration points dis-
cretizing the essential boundary C
u
. To each integration point q
I
2
Q
Cu
it is
associated
Q
Cu
¼ q
1
;
q
2
;
...
;
q
QC
weight
_
C
I
, representing the dimensional size of
q
I
2
Q
Cu
. In addition, for each integration point q
I
a new influence-domain is
determined, considering only field nodes x
i
belonging to X
Cu
2
X. Meaning that
the meshless shape functions of the integration points q
I
2
Q
Cu
are constructed
just using field nodes on the essential boundary: X
Cu
2
X.
The last term of Eq. (
3.66
), included in the expression to ensure: u
u ¼ 0, can
be developed,
an
integration
