Biomedical Engineering Reference
In-Depth Information
With this choice of test functions we have for ">0
Z
T
Z
Z
T
Z
L
CC
2
2
.@
t
N
0
.t/@
t
N
0
.th//
2
C
N
0
h
.t/
j
u
.t/
u
.th/j
C
N
0
CC";
0
0
0
8LJ
2
small enough;
with s<
4
and for some 0 < < 1=4. It implies finally
Z
T
Z
Z
T
Z
L
CC
2
2
.@
t
.t/@
t
.th//
2
C
N
0
h
.t/
j
u
.t/
u
.th/j
C
N
0
CC";
0
0
0
8LJ
2
small enough:
This result together with the energy estimates enable to obtain the desired
convergences, compactness and to pass to the limit in the weak formulation. We
then obtain the existence of at least one weak solution until the elastic structure
touches the bottom of the fluid cavity. Note that recently Muha and Canic [
129
]
prove the same kind of result for a 2D=1D coupled problem with >0. The proof
is based on a time discretization and a splitting of the structure equations, inspires
from a numerical strategy, that enables to have stability for an explicit coupling
independently of the added mass effect [
56
,
98
].
In the case D 0 since we do not have a lot regularity of the structure
displacement at the interface, the proof relies strongly on the only transverse motion.
In the general case where we do not neglect the longitudinal displacement and if
.;/ satisfies only the energy estimates, the displacement of the structure is not
regular enough to properly define the problem. Thus there is a need to work with
smooth solutions.
In the next subsection we then explain how one can obtain existence of a unique
strong solution and review some results of Lequeurre that considers also the case
where D 0. It is nevertheless a first step to further be able to consider the full
coupled problem.
1.2.3
Existence of Strong Solutions
In this subsection we will give the general steps of the results obtained by Lequeurre
[
119
,
120
], which are a generalization of the one obtained in [
10
]. In these studies,
two-dimensional as well as three-dimensional problems with >0or D 0
and with LJ
2
>0have been considered, but the general scheme of proof is the
same in any case. The main difference comes from the regularity of the structure
displacement . We will focus on the general steps without detailing the proofs
which are rather technical. The goal is to underline the possible links between
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