Biomedical Engineering Reference
In-Depth Information
x
I
obtained with the expression,
Z
h
i
U(x
I
) ¼
1
2
1
e
T
r dX
I
¼
_
I
2
r(x
I
)
T
e(x
I
)
ð
6
:
24
Þ
X
I
Then, it is possible to determine the complete SED field U
i
for the considered load
case j: U
i
¼
f
U(x
1
)
i
U(x
2
)
i
U(x
Q
)
i
g
.
The described process is repeated for each load case considered in the analysis.
In the end, the variable fields obtained for each load case are properly weighted.
Considering a generic variable field n, the final weighted field value is determined
by an appropriate superposition of a number of relevant discrete load cases, l,
weighted according to the corresponding number of load cycles, m.
n
i
¼
X
l
m
ð
j
Þ
n
i
P
l
k¼1
m
ð
k
Þ
ð
6
:
25
Þ
j¼1
The variables field weighted with Eq. (
6.25
) are: the displacement field, n
i
¼ u
i
;
the strain field, n
i
¼ e
i
; the stress field, n
i
¼ r
i
; the principal stresses field,
n
i
¼ r(n)
i
; the principal directions field, n
i
¼ n
i
; and the SED field, n
i
¼ U
i
.
Afterwards, the interest points presenting the SED values indicated in the
Eq. (
6.22
) are identified and subjected to a density remodelling process, all the
other interest points maintain the previous density. The weighted principal stress
field of the interest points presenting lower SED values is determined using
Eq. (
6.25
), then the individual new apparent density of these interest points is
obtained with Eq. (
6.17
). With the new apparent density defined, the process
moves forward the next iteration step.
Using the new apparent density field, in each interest point the material prop-
erties are actualized with Eqs. (
6.7
) and (
6.8
). Then, the constitutive elasticity
matrix, defined for each interest point, is oriented using the principal directions
obtained in the previous iteration step. Thus, the constitutive matrix is obtained
with (2.45). This procedure permits to align iteratively the material properties with
the actualized load path. The process stops when the medium apparent density of
the model, q
model
app
app
, or if two consecutive iteration
steps present the same medium apparent density, Dq
=
Dt ¼ 0. The control value
can be determined by the user, based on clinical observations.
Notice that the presented remodelling algorithm is in fact a topology optimi-
zation based model for bone adaptation. In each step not all the interest points
optimize the density in relation to the mechanical stimulus. Although for every
interest point the material is oriented with the principal directions obtained in each
iteration step, the density on each interest point is not. Only the interest points with
lower SED or higher SED, Eq. (
6.22
), are subject to the density remodelling
, reaches a control value, q
control
