Biomedical Engineering Reference
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FS interface displacement at t = 1 ms
FS interface displacement at t = 6 ms
FS interface displacement at t = 12 ms
0.52
0.52
0.52
fluid−thick structure
fluid−thin−thick structure
0.51
0.51
0.51
0.5
0.5
0.5
0
2
4
6
0
2
4
6
0
2
4
6
z axis (cm)
z axis (cm)
z axis (cm)
FS interface velocity at t = 1 ms
FS interface velocity at t = 12 ms
FS interface velocity at t = 6 ms
10
10
10
5
5
5
0
0
0
−5
−5
−5
−10
−10
−10
0
2
4
6
0
2
4
6
0
2
4
6
z axis (cm)
z axis (cm)
z axis (cm)
Fig. 2.20
Fluid-structure interface displacement (
top
) and velocity (
bottom
) obtained using a
multi-layered wall model and the single-layered model from [
22
], shown at times t
D
1 ms,
t
D
6 ms, and t
D
12 ms
Fig. 2.21
Regularizing effects of point mass. These figures originally appeared as Fig. 1 and Fig. 2
(left) in [
107
] and are reprinted by permission of c
American Mathematical Society 2014. The
initial data (
left panel
) is smoothed out as the transmitted wave traveling to the right passes through
the point mass (
right panel
)
form. The transmitted wave, which passes through the point mass, gets smoothed
out to H
2
.0;L/ regularity in displacement, and H
1
.0;L/ regularity in velocity. A
numerical simulation of this phenomenon was shown in [
107
]. Figure
2.21
shows
one of the results from [
107
]. The panel on the left shows the initial displacement
in H
1
.0;L/ with zero initial velocity, located just left from the point mass. The
panel on the right shows the solution at time T D 10s at which the reflected
and transmitted waves have formed, with the displacement of the reflected wave
on the left of the point mass still in H
1
.0;L/, but with the displacement of the
transmitted wave, shown to the right of the point mass, belonging to H
2
.0;L/.For
a reader with further interest in the area of simplified coupled problems we mention
[
91
,
123
,
134
,
145
].
2.8
Conclusions
This chapter addresses an operator splitting approach to study multi-physics prob-
lems related to FSI with application to hemodynamics. The methodology is based on
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