Biomedical Engineering Reference
In-Depth Information
•
Step 2
: Projection step (velocity correction)
-
Step 2.1:
8
<
u
n
C
1
t
f
u
n
C
1
Crp
n
C
1
O
f
;
D 0; in
O
f
;
div
u
n
C
1
D 0; in
(1.122)
:
n
C
1
n
t
†:
u
n
C
1
n D
n; on
-
Step 2.2: Structure equation on
.0;L/
<
s
e
n
C
1
2
n
C
n
1
Ǜ
1
@
xx
n
C
1
D .
f
.
u
n
C
1
;p
n
C
1
/ n/
1
;
t
2
:
s
e
n
C
1
2
n
C
n
1
C @
x
n
C
1
Ǜ
2
@
xx
n
C
1
D .
f
.
u
n
C
1
;p
n
C
1
/ n/
2
:
(1.123)
t
2
Here only the projection step is implicitly coupled with the structure equation.
Remark 1.14.
When dealing with the full nonlinear problem all the nonlinearity
(convection terms, geometrical nonlinearities) are treated at the explicit advection-
diffusion step.
Remark 1.15.
The projection step is written as a Darcy system but we could also
have written it as a Poisson problem on the pressure as we did it for the toy problem
(see (
1.37
)). This scheme is based on the same kind of splitting idea summarized in
Sect.
1.2.3
.
Remark 1.16.
Note that if one computes the residual of the fluid equation on the
interface it is not equal to zero but to
!
:
Z
d
n
C
1
d
n
t
.
u
n
C
1
/
u
n
C
1
†
D
It involves only the viscous stress and does not involve the pressure so one could
hope to control it.
Stability and Convergence
Now we are going to see why this scheme is stable and convergent for the coupled
system. We are considering where a 2D fluid is interacting with a 1D structure.
Note that in [
64
]and[
1
] more general coupled systems are considered in particular
3D=3D coupling. For the purpose of the analysis we assume that the coupled
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