Biomedical Engineering Reference
In-Depth Information
5. The stiffness of the coupled system exhibited in the case
γ
=
1 is lower than the
case
0. Indeed, the constraint in the former case is far weaker than in the
latter. For instance, in the example of Sect. 9.3.2 below, the constraint (9.51a)
(obtained for
γ
=
γ
=
1) is weaker than (9.52a) (obtained for
γ
=
0).
6. From the computational point of view, the parameter
γ
also has an interesting
role.Inthissense,when
1, less unknowns are required as a result of the
weaker condition to be imposed, reducing the computational cost necessary to
obtain the solution. In fact, for the same example of Sect. 9.3.2, it is easy to see
that for
γ
=
γ
=
1 in the discrete problem we have to obtain an approximation of
t
1
which is a real number, while for
γ
=
0 we have to compute an approximation of
t
2
which is a field in
H
−
1
/
2
(
Γ
)
.
a
A last comment regarding the definition of
γ
is in order. In the formulation of
Problem 3
. The purpose of this was to in-
troduce a convex combination of the two duality products involved in the problem,
namely
γ
ranges over the closed interval
[
0
,
1
]
·,·
T
Γ
a
(
Q
1
)
×T
Γ
a
(
Q
1
)
and
·,·
T
Γ
a
(
Q
2
)
×T
Γ
a
(
Q
2
)
. Nonetheless, it can be shown
that the result achieved with
0, that is, its value
is actually immaterial in that case. Moreover, it could have been sufficient to define
Problem 3 for
γ
∈
(
0
,
1
)
is the same as with
γ
=
γ
∈{
0
,
1
}
.
9.3 Application to the coupling of fluid flow models
This section describes some common situations encountered when trying to cou-
ple incompatible models in fluid dynamics. In Sect. 9.2 we discussed the coupling
between two models with different kinematics. The differences in the kinematics
arise from differences in the nature of the models. As a result of the two underly-
ing kinematics it was possible to identify two different senses in which the coupling
equations, naturally derived from the extended variational Problem 3, are satisfied.
Let us give some examples of scenarios of interest in which formulation 3 provides
coupling equations. For the sake of brevity we will skip the presentation of the entire
flow models, assuming that the reader is aware of the physical elements present in
each case.
In the dimensional reduction process of a mathematical model we can find two
different situations concerning the kinematical restrictions: (i) a simple dimensional
reduction and (ii) a dimensional reduction plus augmented regularity. The two ex-
amples presented in the forthcoming sections for fluid flow problems correspond
to situation (i). In such cases the finite-dimensional problem does not entail further
complications from the point of view of the mathematical problem. On the other
hand, situation (ii) demands a careful manipulation of the trace spaces over the cou-
pling interfaces due to the mismatch in the regularity conditions. Examples of this
class can be found in the analysis of beams, plates and shells in solid mechanics [7].
In such cases, the finite-dimensional analysis should be carried out
ad hoc
.
