Biomedical Engineering Reference
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is valid when the incompatibilities arise in just one physical quantity (the velocity
field in our case). A more general framework is needed when considering coupled
phenomena with multiple kinematical incompatibilities at coupling interfaces, such
as the case of the classical fluid-structure interaction problem. For the interested
reader, examples of incompatibilities in solid mechanics using this same framework
have been addressed in [7].
9.4 3D-1D-0D heterogeneous model of the cardiovascular system
Firstly we discuss the use of the variational formulation (9.24) in this context, and
then we succintly describe each of the models used in setting up either a 3D-1D-0D
open-loop model of the arterial network or a 3D-1D-0D closed-loop model of the
entire cardiovascular system.
9.4.1 On the use of the non-classical extended variational
formulation
In the present work 3D models of arterial vessels will always be embedded in the
1D arterial network. Therefore, we have two different situations: (a) coupling 1D
and 0D models and (b) coupling 3D and 1D models. Case (a) does not involve any
difficulty because the 0D model and the 1D model share the same kinematics (char-
acterized simply by a real number) at the coupling interfaces, which are denoted by
Γ
01
. Case (b) is the one that raises more questions. Although the machinery presented
in Sect. 9.2 applies to both cases, the latter brings a richer theoretical treatment as a
result of the lack of compatibility in the kinematics. The interfaces between 1D and
3D models will be denoted by
Γ
13
.
The value of
γ
is relevant for the kinematical incompatibility emerging at
Γ
13
,
while it is immaterial for the coupling at
01
(since both kinematics for the 0D and
the 1D models are governed by a single real number over the coupling interface).
We consider in the 1D model the kinematical hypothesis that led to the coupling
equations seen in Sect. 9.3.2. In particular, we take
Γ
1. As discussed in the previ-
ous section, this imposes the continuity of the velocity field in the sense of the mean
value (in the sense of
γ
=
0
implies the imposition of the velocity profile at the coupling interfaces in the 3D
model, as expressed by Eq. (9.52a), which is too strong in a fluid flow problem.
Hence with this choice the continuity of the traction force (dual variable) on
R
) over the coupling interface
Γ
13
. Otherwise, note that
γ
=
Γ
13
is satisfied pointwise (indeed, from the variational formulation we obtained a con-
stant force in the 3D model as given by (9.51b)-(9.51c)) and the velocity profile
is allowed to take quite arbitrary shapes in the 3D solution. This provides a richer
framework in which, for example, it is possible to capture Womersley-like velocity
profiles, that is, situations in which the velocities normal to the cross-section area of
the vessel can be positive and negative at the same time.
It must be said that other coupling techniques are based on the pointwise impo-
sition of the velocity profile (see [13, 27]), restricting the solution at the entrances
