Biomedical Engineering Reference
In-Depth Information
Notice that we have enforced the kinematic coupling condition both in the thin
structure acceleration term and in the viscous part of the thin structure equation.
We are now ready to split the problem. For this purpose, observe that the portion
K
h@
t
v C
L
vis
v DJ
n
e
r
of the dynamic coupling condition is all given
in terms of the trace v of the fluid velocity on (recall, depends on v). We
can, therefore, use this as a lateral boundary condition on for the fluid sub-
problem. This observation is crucial because keeping the structure inertia term
K
h@
t
v together with the inertia of the fluid in the fluid sub-problem is of paramount
importance for designing a stable and convergence scheme.
This is different from the classical loosely coupled schemes. In classical
Dirichlet-Neumann loosely coupled scheme, the boundary condition for the fluid
sub-problem is the Dirichlet condition for the fluid velocity v on given in terms
of the structure velocity @=@t, namely v D @=@t,where@=@t is calculated at
theprevioustimestep! This inclusion of the structure inertia from the previous time
step (explicitly) makes the fluid sub-problem unstable for certain parameters values
[
30
]. The main reason for this is that the kinetic energy at this time step includes
only the fluid kinetic energy from the current time step, and not the thin structure
kinetic energy, since the thin structure velocity enters from the previously calculated
time step. For strong geometric nonlinearities, which often happen when the fluid
and structure densities are comparable, this mismatch between the kinetic energy of
the discretized problem (where only the fluid kinetic energy appears in the current
time step) and the kinetic energy of the continuous problem (where both the fluid
and structure kinetic energy are tied together in a strongly coupled FSI problem)
gives rise to an unstable numerical scheme [
30
].
Therefore, the strategy of our splitting, mentioned above, to keep the thin
structure inertia together with the fluid inertia in the fluid sub-step will give rise
to the kinetic energy of the discretized problem that approximates well the kinetic
energy of the continuous problem, giving rise to a scheme that is unconditionally
stable for all the parameters in the problem [
29
]. In Sect.
2.6
we prove that the
scheme converges to a weak solution to the underlying FSI problem.
We therefore define the operators A
1
and A
2
as follows:
Problem A1
W
STRUCTURE
S
@
t
V
Dr
S
; in
S
@
t
d
D
V
; in
S
d
D
e
r
; on
@
t
D v; on
K
h@
t
v D
L
el
./ C R
Se
r
e
r
on
Here, of course, the PDE system in
S
can be solved just as a single second-order
PDE for
d
:
S
d
tt
Dr
S
. Problem A1 is solved with the initial data .
d
;
V
;;v/
given by the solution from the previous time step. This means, in particular, that
the thin structure velocity v is set to be equal to the trace of the fluid velocity on
, calculated in the previous time step. Thus, we are solving the elastodynamics
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