Biomedical Engineering Reference
In-Depth Information
with
u
n
C
1
r;h
j
.t
n
/
D
u
n
C
2=3
j
.t
n
/
D v
n
C
2=3
r;h
:
r;h
It was shown in [
24
] that an energy estimate associated with unconditional
stability of this scheme holds for the full nonlinear FSI problem. Therefore, we
expect that this scheme is unconditionally stable for all the parameters in the
problem.
2.7.7
Numerical Examples
We present two numerical examples. One is a simplified problem for which there
exists an exact solution against which we can test our numerical scheme. The
other one is a fully nonlinear FSI problem with a thin and thick structural layer.
Since there are no numerical results in literature on FSI problems with multiple
structural layers against which we could test our solution, in this second example
we calculated solutions to a sequence of problems for which the thickness of the
thin structure converges to zero, and showed that the limiting solution is the same
as the solution of the FSI problem in which the structure consists of only one thick
structural layer. This was proved using analytical methods in [
24
]. The solution of
the limiting problem was then numerically tested against the solution of the FSI
problem with only one thick structural layer, which was obtained using a different
solver. We show below that the two solutions, obtained with two different solvers,
are in good agreement.
Example 1
We consider a simplified FSI problem with multiple structural layers that satisfies
the following simplifying assumptions:
1.
The fluid problem is defined on the fixed, reference domain of width R,and
length L (the coupling is linear).
2.
The fluid problem is driven by the constant inlet and outlet pressure data p
in
and
p
out
D 0 (the pressure drop is constant).
3.
Only radial displacement of the thin and thick structure is assumed to be different
from zero.
Assumption 3 implies that the thin structure membrane model takes the form:
K
h
@
2
r
@t
C C
0
r
D f
r
;
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