Biomedical Engineering Reference
In-Depth Information
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Numerical Experiments
The aim of this section is to illustrate performance of numerical methods presented
above, the semi-implicit schemes presented in Sect.
1.3.2
as well as the kinemati-
cally coupled schemes presented in Sect.
1.3.3
.
We start with the semi-implicit schemes and illustrate by numerical experiments
performance of the semi-implicit scheme for strong added mass effect. It is worth
mentioning that in the Newmark scheme is used in every numerical test without any
stability issue. In every test case the projection step is solved by a Poisson equation
on the pressure (see (
1.37
), (
1.38
)), for which one has really performant algorithms,
and not the mixed Darcy problem (see (
1.30
), (
1.31
)).
First of all we have compared for D 0 and D 0 three difference schemes
relying on
explicit
,
semi-implicit
,and
implicit
strategies (Fig.
1.2
.)
The second test case deals with a three-dimensional problem, where the structure
satisfies a shell equation. It is the benchmark case proposed in [
69
] (for the used
parameters we refer to [
64
]). The fluid is discretized by
Q
1
finite elements and
the structure by shell element MITC4 [
27
]. The implicit step is solved by a Newton
algorithm (Fig.
1.3
).
These results are in agreement with [
61
,
69
,
81
]. Note moreover that this scheme
has the same precision as an implicit scheme with a computational cost which
is much lower. Moreover, the theoretical convergence rate in time has not been
recovered easily in practice and only on well-chosen cases [
1
].
In the following experiment we compare the accuracy of the first and second
order kinematically coupled fluid-structure interaction schemes. To derive the first
or second order methods we can apply, respectively, the first order Marchuk-
Q
1
=
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