Biomedical Engineering Reference
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a
b
3e+06
2.5e+06
2.2e+06
2e+06
1.8e+06
2e+06
1.6e+06
1.5e+06
1.4e+06
1e+06
1.2e+06
500000
1e+06
0
800000
0
0.01
0.02
0.03
0.04
0.05
0.06
0
0.01
0.02
0.03
0.04
0.05
0.06
Time [s]
Time [s]
c
2.2e+06
2e+06
1.8e+06
1.6e+06
1.4e+06
1.2e+06
1e+06
800000
0
0.01
0.02
0.03
0.04
0.05
0.06
Time [s]
Fig. 6.15
History of the Young modulus estimates for SNR
D
5 for different values of the POD
threshold for the aortic arch test case. (
a
)
D
0:9 (N
u
D
5; N
D
5), (
b
)
D
0:95 (N
u
D
8; N
D
7), (
c
)
D
0:99 (N
u
D
22; N
D
12). The
colors
refer to the different components of
the vector E (E
1
(
blue
), E
2
(
red
), E
3
(
green
))
Finally, in Fig.
6.15
we show the history of the Young modulus estimates for
different choices of the POD threshold in the case of SNR D 5. It is interesting
to notice that, despite the fact that the level of the noise is as large as 20 % of the
intensity of the signal, the average estimates are still close to the correct values.
In particular, even when using a low dimensional size for the reduced model, the
optimization procedure clearly detects that the Young modulus in the second region
is larger than in the other two regions.
A Kalman-Based Parameter Estimation Approach
Let us consider the FSI system after time-space discretization and linearization that
we write as
U
k
D A
k
1
U
k
1
C F
k
1
;
where U
k
N
is the vector of velocity and pressure degrees of freedom. In order
to estimate the parameter E 2
R
2
R
p
the augmented state approach is used. Define
X
.k/
WD ŒU
.k/
;E
.k/
, then the system becomes
A
.k/
0
0I
;
F
.k/
0
:
D A
.k
1
X
X
.k
1/
C F
.k
1
X
;
.k/
X
F
.k/
X
X
.k/
D
D
(6.81)
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