Biomedical Engineering Reference
In-Depth Information
Fig. 3.1
An idealized sketch of the domains
f
;
s
and of the corresponding partition of
their boundaries is reported on the
left
. The actual configuration of the domains as well as the
computational grids used in the simulations is shown on the
right
data space, we should be able to find (at least) one particular force f and a sequence
ff
N
g with the following property: The total energy of the unique T -periodic solution
corresponding to f
N
should increase as N !1. However, our finding does not
seem to corroborate this view, at least for the type of forces that we have considered
(squared profile). The only (expected) phenomenon observed is that the oscillation
energy of the solid may increase as the viscosity of the liquid decreases.
For the sake of clarity, we recall below the precise problem formulation,
including initial, boundary, and interface conditions. For practical reasons, we
slightly modify the definition of the fluid Cauchy stress using the dynamic viscosity,
namely T.v;p/D
f
.rvC.rv/
>
p
1
. Then, the problem solved in the numerical
simulation is the following,
s
u
tt
s
u
.
s
C
s
/rdiv
u
D
s
f in
s
.0; 1/
f
v
t
rT.v;p/D 0; divv D 0
in
f
.0; 1/
T.v;p/ n D .
u
/ n; v D
u
t
on .0; 1/
u
D 0
on
s;2
[
s;4
.0; 1/
(3.125)
.
u
/ n D 0
on
s;3
.0; 1/
T.v;p/ n D
0
on
f;2
[
f;4
.0; 1/
v n D 0; t T.v;p/ n D 0
on
f;1
.0; 1/
u
D
u
t
D 0
in
s
f0g
v D 0
in
f
f0g
The fluid and solid domains are represented by two rectangular adjacent regions,
of vertical and horizontal sides a; b, respectively. We refer to Fig.
3.1
for a sketch
of the domains and a precise definition of the boundaries
f;i
;
s;i
;iD 1;:::;4.
We observe that the interface coincides with
s;1
and
f;3
. These domains are
embedded in
2
, endowed with a Cartesian coordinate system x; y with origin in
the bottom left corner of
f
.
As discussed in Sect.
3.2
, resonance effects emerge only when some precise
vibration frequencies of the system are excited. For the particular case of the
linear elasticity equation on a rectangular domain, the characteristic frequencies
R
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