Biomedical Engineering Reference
In-Depth Information
These results, which are the adaptation of [
1
,
64
], to our setting are to be
commented. First of all, concerning the stability, it is obtained under a sufficient
condition that requires that the time steps is small enough with respect to the mesh
size. In the case where one considers a more general setting, for instance a 3D
fluid coupled with a 3D structure then, to obtain the same kind of stability property,
one has to consider different space discretizations for the fluid and for the structure.
Nevertheless this restriction is not seen in practice. Note moreover that the sufficient
condition depends on the physical parameters of the problem and on the ratio of the
densities. Once again this restriction has not been observed for practical applications
such as blood flow in arteries.
This sufficient condition comes from the fact that we have to control the viscous
residual
R
†
h
@
t
d
n
C
h
/ since the viscous effect is treated explicitly
(see Remark
1.16
). It will be the same sufficient condition if one considers a heat-
wave (parabolic-hyperbolic) explicit coupling. To prove these stability result, one
would like to choose as test functions
.
u
n
C
1
/.
u
n
C
1
D
d
n
C
1
h
d
h
t
u
n
C
1
h
;
u
n
C
1
h
;
:
/ ¤
h
.
d
n
C
1
D
h
.
d
h
d
n
1
d
h
t
Butattheinterfacewehave
u
n
C
1
h
/. Thus we choose
h
h
t
h
.
d
n
C
1
/
!
;
u
n
C
1
h
2d
h
C d
n
1
d
n
C
1
h
d
h
t
h
u
n
C
1
h
h
C Ext
h
;
;
t
as test functions, where Ext
h
is an extension of the structure test function in the fluid
domain such that the L
2
norm is of order h and the H
1
norm blows up as
h
. The dif-
ficulty here is to control the extra terms coming from Ext
h
h
.
d
n
C
1
/
.
2d
h
C
d
n
1
h
h
t
Note that these terms are easily controlled by the dissipation coming from the
Leap-Frog chosen scheme for the time discretization of the structure. In the case
of Newmark scheme, which is not dissipative, the proof does not hold any more.
Nevertheless the same stability estimate can be obtained provided that the kinematic
boundary conditions are imposed in a weak way thanks to Nitsche's method [
3
],
since Nitsche's method introduces some dissipation in the coupled scheme.
Concerning the convergence result, the time error is at least of order
p
t.
It is well known that a non-incremental Chorin-Temam scheme has time rate of
convergence that is less than one, for the pressure in norm L
1
.0;T I L
2
.
O
f
//,and
for the velocity in norm L
1
.0;T I H
1
.
O
f
// (see [
5
,
47
,
95
,
96
,
143
] for details). Here
it is also the case for the fluid velocity in L
1
.0;T I L
2
.
O
f
// norm. The reasons for
this may be due to our proof technique, that may not be optimal, or due to the fact
that the error for the fluid pressure propagates through the structure to the whole
coupled system.
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