Environmental Engineering Reference
In-Depth Information
Alya, because mesh deformation is computed as an additional low-cost problem. The
method consists of solving a Laplacian equation:
D
ij
∂
∂
u
i
=
0in
t
(7)
∂
x
i
∂
x
j
t
u
i
=
g
i
(
x
j
,
t
)
on
∂
f
u
i
=
0on
∂
m
t
f
where
t
is the time changing domain,
∂
is the moving boundary,
∂
is the fix
boundary and
−
V
min
/
1
V
max
D
ij
=
(
1
+
)ʴ
ij
(8)
V
e
/
V
max
is the diffusion parameter which is a function of the local Volume element. The effect
of this parameter is to progressively make stiffer the smaller elements, preserving
boundary layers and refined zones, thus allowing for large deformations in larger
elements.
2.1.6 Solution Scheme and Computational Aspects
A fully coupled problem comprises two material parts (tissue and blood) and four
coupled simulation problems (electrophysiology and mechanical deformation on
the tissue and mesh deformation and fluid mechanics on the blood). Two different
instances of Alya deals with each of the parts, running linked together in a staggered
fashion. At each time step, coupled problems can be solved either monolithically or
by blocks, grouping the unknowns. We use here a by-block structure, being elec-
trophysiology, mechanical, blood flow and mesh deformation separated problems
solved in a staggered and paired way.
On one hand, one instance of Alya solves the governing equations of both the
electrical and mechanical parts on a spatially discretized mesh of the heart volume
using the finite element method, being the same mesh used for both problems, thus
avoiding instabilities and interpolation errors. Given the always increasing resolution
and accuracy of anatomical information obtained from clinical imaging, Alya Red is
conceived fromdesign to deal with large, high-definition unstructuredmeshes. On the
other hand, a second instance of Alya solves the Navier-Stokes and the mesh deform-
ing equations. In turn, each of the two coupled Alya instances run in parallel thanks to
an automatic mesh partition of the unstructured meshes usingMetis andMPI tasks. A
careful grouping of each problem's respectiveMPI tasks allows an efficient communi-
cation thanks to MPI communicators. This strategy is reported by Cajas et al. (
2014
).
On the tissue part, both problems are solved explicitly using the same time step
allowing for a straight synchronization. The time step is the smallest one computed
for each of the two problems based on stability criteria. Solid mechanics time step
is related to the approximate sound speed propagation (Belytschko et al.
2000
). For
the electrophysiology problem, we found that the most stringent criterion, no mat-
Search WWH ::
Custom Search