Environmental Engineering Reference
In-Depth Information
ter what cell model is chosen, comes from the conductivity in Eq. ( 1 ), computed as
ʔ
h 2
=
/
2 D , where D is the largest component of the D ij , which usually corre-
sponds to the fiber direction. After many numerical experiments we have found that
both time steps are approximately of the same order, being the solid mechanics time
step usually the smallest one. We have observed that due to the small time steps, there
is usually no need for subiterations. On the blood part, both fluid and mesh defor-
mation are solved implicitly, with no time restriction due to stability issues. Papers
describing the parallelization scheme are given by Vázquez et al. ( 2011 ), Lafortune
et al. ( 2012 ) and Houzeaux et al. ( 2009 ).
For the tissue problem, the scheme proposed here has some specific features
that have not been the standard in electromechanical cardiac coupling. Firstly, both
electrophysiology and mechanical problems are solved in the same mesh, avoiding
stability issues and interpolation errors but paying the price of a high computational
cost for the mechanical problem. As we program and solve all problems in the same
code, supported on the same mesh and with the same parallelization scheme, we
find our approach very natural. Providing the high parallel efficiency of the code,
it has proven both an efficient and accurate option. Consider also the increasingly
higher resolution obtained from the clinical images, which results in high fidelity
simulations. Secondly, the mechanical problem is dynamically solved so the tran-
sient effects are taken into account. Therefore, no quasi-static approximation is used.
Considering the fully coupled problem, references are scarce. For instance, Hosoi
et al. ( 2010 ) proposed an electro-mechanical-blood flowmodel. Themain differences
with our model is that they use a different electrical propagation model, a monolithic
approach for the fluid-structure interaction and an homogeneization procedure for
the tissue mechanical properties.
The two parts (tissue deformation and blood flow) of the fluid-structure interaction
(FSI) scheme run in an overlapping manner, concurring when forces and displace-
ments are interchanged for the wet surface. In turn, each part runs in parallel. To
assure good load-balance each Alya instance creates its partition taking into account
the relative weight of the four problems. Therefore, contrary to many other strategies
for FSI, both fluid and solid sides are equally parallelized.
The wet surface is shared between the two instances. The boundary conditions
are an important issue, with different aspects to be considered. Firstly, the nodes on
both sides need not be coincident. Both fluxes and unknowns can be interpolated in
a conservative way. However, to avoid the interpolation errors, we prefer coincident
surface meshes. Secondly, to avoid so-called added mass effect when fluid and solid
densities are similar, we use the method proposed by Wall et al. ( 2007 ), where an
Aitken scheme relax the wet surface motion on the fluid side. This relaxation scheme
is not always required, but it gives additional robustness.
t
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