Biomedical Engineering Reference
In-Depth Information
The discrete system of equations represented in Eq. (
3.64
) can be organized in
the following matrix form,
2
3
8
<
9
=
8
<
9
=
K
G
k
f
|{z}
½3N
1
g
k
u
|{z}
½3N
3N
|{z}
½3N
3N
k
|{z}
½3N
1
k
4
5
¼
ð
3
:
65
Þ
G
k
0
:
;
|{z}
½
ð
3N
þ
3N
k
Þ
1
:
;
|{z}
½
ð
3N
þ
3N
k
Þ
1
|{z}
½
3N
k
3N
k
|{z}
½3N
k
1
|{z}
½3N
k
1
|{z}
½3N
k
3N
|{z}
½
ð
3N
þ
3N
k
Þð
3N
þ
3N
k
Þ
which is the obtained final discrete system of equations when the Lagrange
multipliers method is used to enforce the essential boundary conditions. The
displacement nodal parameters vector u is determined with Eq. (
3.65
) and then,
the displacement of any interest point x
I
2
X can be obtained with Eq. (
3.1
).
In structural analysis it is very common to use the Lagrange multipliers to
impose the essential boundary conditions when approximation meshless methods
are used, such as the EFGM. This technique is accurate, permitting to impose
exactly the essential boundary conditions. However, as Eq. (
3.65
) shows, the
Lagrange multipliers method increases the number of unknowns of the initial
system of equations from 3N to 3N
þ
3N
k
. If the essential boundary contains a
high percentage of field nodes, then the efficiency of the meshless analysis is
certainly compromised. Additionally, the linear system of equations is no longer
positive definite [
49
] and it loses the banded property, increasing the computa-
tional cost of the analysis. Nevertheless, the final discrete system of equations
preserves the symmetry.
Penalty Method
The penalty method is an alternative efficient numerical technique capable to
impose the essential boundary conditions in approximation meshless methods [
49
].
Since approximation shape functions, such as the MLS shape functions, do not
possess the Kronecker delta property, the Galerkin weak form expression can be
presented as,
Z
d L
ð
T
cL
ð
dX
Z
du
T
b dX
Z
du
T
t dC
t
X
2
4
X
3
5
Ct
Z
1
2
ð
u
u
Þ
T
a
ð
u
u
Þ
dC
u
þ
d
Cu
¼ 0
ð
3
:
66
Þ
