Environmental Engineering Reference
In-Depth Information
In this equation,
C
50
,
˃
max
and
ʻ
f
are model parameters. We have introduced a para-
meter 0
<ʳ <
1 to calibrate the amount of active stress and measure its sensitivity.
2.1.4 Blood Flow
In this model, blood is considered as incompressible and Newtonian. The time
discretization is a second order backward differentiation scheme. Linearization
is done using the Picard method. The space discretization is based on the finite
element method, combined with a variational multiscale method (VMS) described
by Houzeaux and Principe (
2008
). At each time and linearization iteration, the fol-
lowing system is solved:
A
uu
A
up
A
pu
A
pp
u
p
b
u
b
p
=
(6)
where
u
and
p
are velocity and pressure nodal unknowns. Avoiding the use of com-
plex preconditioners to account for the velocity-pressure coupling for the monolithic
system, an algebraic fractional scheme is used (see Houzeaux et al.
2011
). Through
this scheme, we segregate velocity and pressure systems at the algebraic level, solving
the pressure Schur complement using an iterative method (herein the Orthomin(1)).
This strategy offers two main advantages. Therefore, one shot of the method involves
the solution of the momentum equation and the solution of a symmetric system for
the pressure (Laplacian) representing the continuity equation. The momentum equa-
tions usually converge very well, even with a simple diagonal preconditioner. On
the contrary, the continuity equation is much stiffer, so it is solved with the Deflated
Conjugate Gradient solver (DCG) (Löhner et al.
2011
), together with a linelet precon-
ditioner when anisotropic boundary layers (Soto et al.
2003
) are present. Besides and
with respect to classical fractional step methods, no fractional errors are introduced
and the solution converges to the same as the monolithic one.
2.1.5 Arbitrary Lagrangian-Eulerian ALE Scheme
for the Fluid Mesh Deformation
Coupling between the blood flow and the ventricles is done by the Arbitrary
Lagrangian-Eulerian (ALE) method. While the blood transmits the force to the ven-
tricle, the ventricle transmits a deformation to the mesh where the moving blood is
simulated. This has a double effect. On one hand, as the Navier-Stokes equations
are solved on a moving mesh, they include a correction term taking into account the
relative velocity of the fluid and the mesh. On the other hand, the volume fluid mesh
must be smoothly deformed following the surface mesh solidary to the ventricle
wall (see Donea et al. (
2004
) for a comprehensive description of the ALE method).
There are several methods to perform the volume mesh deformation. In this paper we
follow the method proposed by Calderer and Masud (
2010
). This method has proven
as extremely robust and specially well-suited for a parallel multi-physics code like
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