Biomedical Engineering Reference
In-Depth Information
Nevertheless, notice that the problem is always three-dimensional, and that hy-
potheses are introduced in certain parts of the system to simplify the problem. When
performing such reduction to a 1D or 0D model we are introducing, at the coupling
interfaces, incompatibilities between the corresponding fields. The present work pro-
poses a different perspective:
to explore the role of the variational formulation in
providing an appropriate extended variational formulation from which the coupling
conditions are naturally derived
. Then, we will see a
N
D(
N
0; 1; 2) simplified
model as the reduction of a complete 3D model through the introduction of suitable
kinematical restrictions. As mentioned earlier, due to the dissimilarity between the
underlying kinematics, we generated discontinuities in the involved fields. In this
new context, the original variational formulation, valid for fields that are continuous
in the sense of the trace of the functions over such a coupling interface, needs to be
reformulated in order to accommodate such discontinuities.
The goal of the present work is to give an account of the state-of-the-art in the
realm of extended variational formulations for problems where fields can become
discontinuous as the result of kinematical considerations introduced at some arti-
ficial internal boundary (coupling interface). This follows the ideas introduced in
[3, 6]. This formulation is applied in the modelling of fluid flow and more specifi-
cally in the hemodynamics field. However, notice that these ideas can be extended
straightforwardly to a great variety of problems. A theoretical account including the
extended variational formulation for such models was introduced by the authors with
regard to solid mechanics, [3, 7].
This work is organized as follows. In Sect. 2 an abstract extended variational
principle is formulated so as to highlight the basic ideas and mechanical concepts
which are behind the extended variational formulation for coupling dimensionally-
heterogeneous models. In Sect. 3 we apply the extended formulation to the problem
of the flow of an incompressible fluid and discuss several situations of potential
interest. In Sect. 4 we construct a heterogeneous model of the CVS for which we may
consider two different topological descriptions: an open-loop systemic network or a
closed-loop model of the entire system. In these two alternative global configurations
we embed 3D models of arterial vessels to model blood flow in specific districts.
Finally, in Sect. 5 numerical experiments are presented, illustrating the capabilities
of this approach.
=
9.2 Abstract Extended Variational Formulation
As pointed out in the introduction, the coupling of dimensionally-heterogeneous
models for modelling and simulation of the CVS will be dealt with an extended
variational formulation from which the coupling conditions are naturally derived.
The equivalent approach from the differential viewpoint is known in the literature
as
geometrical multi-scale modelling
[16, 17, 18, 50].
In fact, as it will be seen, the coupling conditions between the models correspond
to the so-called
Weierstrass-Erdmann Corner Conditions (WECCs)
associated to
this extended variational formulation at the coupling interfaces. Furthermore, since
