Biomedical Engineering Reference
In-Depth Information
Fig. 6.16
Left
: Idealized abdominal aortic aneurysm geometry with subregions and fluid velocity
field; displacement and pressure fields at the outlet as a function of time.
Right
: Noise compared
to the typical wall displacements in the five regions and signal to noise ratios. Adapted from [
6
],
with permission of
c
J. Wiley & Sons, 2012
The initial state is assumed to be X
.0/
D ŒU
.0/
;E
ref
C
.0/
,andF
.k/
D 0, i.e., the
initial velocity and displacement are assumed to be known without uncertainty and
the model is considered exact. The variables to be estimated are denoted by .The
measures of the displacement are affected by a white noise
.k/
, i.e.,
.k/
meas
D H
k
X
.k/
C
.k/
:
Since the problem is nonlinear, an UKF approach (see Sect.
6.2.5
)isusedwhere
.N C p C 1/ sample points (for details see [
6
]) are needed to approximate the
average and the covariance of the evolving state. As explained in Sect.
6.2.5
,the
predictor phase consists in evaluating X
.k
i
for each sample X
.k
1
i
, which requires
the solution of the FSI problem .N C p C 1/ times at each time step. This is
computationally prohibitive, therefore a model reduction is performed. The idea
is to exploit the fact that the initial covariance is given by
"
0
#
;
0
0 Cov
.0/
ƒ
.0/
D
and to use a factorized formulation of the UKF. In this way [
6
] it is possible to use
only p C 1 sample points, which significantly reduce the computational cost of the
method when p N, i.e., when the number of parameters is much smaller than
the dimension of the state.
Consider the idealized 3D geometry of an abdominal aortic aneurysm showed in
Fig.
6.16
, left. The structure is divided, a priori, into five regions featuring different
values of the Young modulus E, corresponding to different colors in Fig.
6.16
.The
typical displacements and noise recorded in the five regions are shown in Fig.
6.16
,
right.
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