Biomedical Engineering Reference
In-Depth Information
Table 12.1.
Left: Comparison of the results of a regularized DA vs a non-regularized interpolated
DA. Right:
Relative
errors of the WSS computed with the DA procedure and a forward Navier-
Stokes noisy simulation in a 2D carotid bifurcation for different values of the SNR
N
in
interpolation
a
E
U
it
14
no
0
0.068
14
SNR
E
WSS
,
DA
E
WSS
,
FW
14
no
0.021
0.061
15
14
yes
0
0.059
18
100
0.2536
0.2667
14
yes
0.021
0.056
16
20
0.2591
0.3030
8
no
0
0.199
11
10
0.2738
0.3861
8
no
0.038
0.137
18
5
0.3149
0.6114
8
yes
0
0.139
17
8
yes
0.038
0.129
17
12.2.2.3 Assimilated derived quantities in nontrivial geometries
In view of real hemodynamics applications, we present a demonstrative test case in
non-trivial geometries (representing a 2D simplified model of the aortic arch and
an arterial bifurcation). Since in these cases we do not have an analytical solution,
we have computed a “reference” solution on an extremely fine mesh grid (using
parabolic inflow conditions and homogeneous Neumann conditions at the outflow)
in both cases. Successively, a noise generated with a uniform distribution has been
added to the solution. We consider several values of Signal-to-Noise Ratio (SNR),
defined as the ratio between the maximum magnitude of the velocity data and the
noise standard deviation.This generates a set of noisy data to be assimilated rep-
resented by the black vector field in Fig. 12.5, left. Results of the assimilation are
significantly close to the reference solution. As a matter of fact, we consider as an
index of the accuracy for the solution the ratio
E
U
=
U
re f
2
U
re f
2
. To test the com-
petitiveness of the DA procedure we compare the relative error of the assimilated
velocity with the one of the velocity obtained from a forward simulation where noisy
data on
U
−
Γ
in
are prescribed as a Dirichlet condition; in this case we obtain
E
U
=
8.1e-2
and
E
U
forward
=
16.0e-2. This pinpoints the role of DA as a process for de-noising
the available data thanks to mathematical models. The DA procedure in fact corrects
the measurements according to the physical principles underlying the mathematical
model. This is evident not only for the primitive variable, but also checking non-
primitive interesting quantities. In Fig. 12.5, right, we report the vorticity map re-
covered from an assimilated velocity field on a geometry approximating an arterial
bifurcation. For the same simulation, we check the accuracy of the WSS. Accuracy
results are reported in Table 12.1, right. The WSS is retrieved in two ways. In the
first case, we perform the DA procedure and use the assimilated velocity field for ex-
tracting the WSS. In the second case, we use again the inflow noisy data as boundary
conditions for a forward computation of the incompressible Navier-Stokes equations
on the same mesh where DA is performed. In particular, in the table we report the
relative errors, i.e. the difference of the WSS compared with the noise-free reference
