Biomedical Engineering Reference
In-Depth Information
Notice that r
j
are the three principal stresses obtained for the interest point x
I
and
are the inverse functions of r
j
ð
q
I
Þ
defined in Eqs. (
6.9
) and (
6.10
). Therefore it is
possible to obtain the following expression for the principal stress r
1
,
axial
ð
q
I
Þ
¼q
axial
¼ 8
:
14
10
4
f
1
ð
r
1
Þþ
235
:
3
r
1
1
ð
q
I
Þ
¼r
1
f
1
ð
r
1
Þ
0
:
439
ð
6
:
18
Þ
and for the remaining principal stresses r
j
with j ¼
f
2
;
3
g
,
trans
ð
q
I
Þ
¼
q
trans
¼
1
:
34
10
3
f
2
ð
r
j
Þþ
3
:
34
10
4
r
1
j
ð
q
I
Þ
¼
r
1
0
:
669
ð
6
:
19
Þ
f
2
ð
r
j
Þ
the functions f
1
ð
r
1
Þ
: C
7!
R and f
2
ð
r
j
Þ
: C
7!
R are defined as,
f
1
ð
r
1
Þ
¼
Re
1
:
54
10
8
þ
4
:
47
10
7
r
1
þ
2
:
44
10
3
q
2
:
31
10
9
r
1
þ
3
:
35
10
9
r
1
ð
6
:
20
Þ
f
2
ð
r
j
Þ
¼Re
1
:
25
10
11
þ
1
:
68
10
8
r
j
þ
1
:
49
10
4
q
1
:
88
10
11
r
j
þ
1
:
26
10
8
r
j
ð
6
:
21
Þ
The expressions presented in Eqs. (
6.18
) and (
6.19
) are applied to the interest
points with SED values belonging to the following interval,
U
ð
x
I
Þ2
U
m
;
U
m
þ
a
DU
½
½
[
U
M
b
DU
;
U
M
;
8
U
ð
x
I
Þ2
R
ð
6
:
22
Þ
being U
m
¼ min
ð
U
Þ
, U
M
¼ max
ð
U
Þ
and DU ¼ U
M
U
m
. It is possible to define
the SED field of the problem domain by: U ¼
f
U
ð
x
1
Þ
U
ð
x
2
Þ
U
ð
x
Q
Þ g
.
The parameters a and b define the growth rate and the decay rate of the apparent
density. The remodelling equilibrium is achieved when,
Dq
Dt
¼ 0
_ð
q
model
Þ
t
j
¼ q
control
ð
6
:
23
Þ
app
app
The values of parameters a and b and the value of the control apparent density
q
control
app
vary with the analysed problem.
6.3.5.2 Remodelling Algorithm
In this subsection the implemented iterative remodelling process, a forward Euler
scheme, is described with detail. The presented algorithm is shown in Fig.
6.9
.
First the NNRPIM pre-processing phase is initiated, in which the problem domain
is discretized in a nodal mesh and the respective Voronoï Diagram is constructed.
