Biomedical Engineering Reference
In-Depth Information
For a simpler homogeneous material case, the solid density is obviously constant,
q
ð
q
I
Þ
¼q
; 8
q
I
2
Q.
The diagonal shape function matrix H
I
of the interest point q
I
is determined
using the meshless shape function obtained for the interest point q
I
considering only
the n field nodes belonging to the influence-domain of q
I
. Following the same
procedure described previously for the deformation matrixB
I
, firstly the local
diagonal shape function matrix is determined, H
loca
I
, and then it is assembled to the
global diagonal shape function matrix H
I
. The process is similar with the example
shown in Fig.
3.17
. The local diagonal shape function matrix is defined as,
00
0 0
0 0
1
2
n
local
H
0
0
0
0
0
0
ð
3
:
23
Þ
I
1
2
n
00
0 0
0 0
[3 3 ]
n
1
2
n
The mass matrix obtained with the RPI or the MLS meshless methods is sparse,
banded and symmetric.
3.4.3 Body Force Vector
In
Sect. 2.3.4
it was shown how the body force vector could be obtained for the
continuous solid domain. Using the background integration points discretizing the
continuous solid domain, it is possible to obtain the body force vector with,
¼
Z
H
T
b dX ¼
X
Q
_
I
H
I
f
b
|{z}
½3N
1
q
ð
q
I
Þ
g
|{z}
scalar
g
|{z}
½3
1
ð
3
:
24
Þ
|{z}
½3N
3
I¼1
X
being g the magnitude of the acceleration due the gravity and g ¼
f
g
x
;
g
y
;
g
z
g
a
non-dimensional unit vector,
jj
g
jj
¼ 1, defining the gravity direction. The con-
struction of the global diagonal shape function matrix H
I
is described in the
previous
Sect. 3.4.2
.
3.4.4 External Force Vector
In addition to the body forces, other forces can be applied in the solid boundary.
Consider the external forces
t
C
ð
x
Þ
applied along the boundary curve C
t
ð
C
Þ
2
C,
Fig.
3.15
. First, the field nodes on the boundary curve C
t
ð
C
Þ
2
C are identified:
