Biomedical Engineering Reference
In-Depth Information
Domain geometry. We solve the Bidomain system on a domain
that is either a
slab or the image of a Cartesian slab using ellipsoidal coordinates, yielding a portion
of a truncated ellipsoid (see Fig. 5.1). This second choice has been used in cardiac
simulations with idealized ventricular geometries, see e. g. [29]. These two choices
also allow us to test the performance of our multilevel preconditioner in presence of
severe domain deformations.
Ω
Conductivity coefficients and parameter calibration. The values of the conduc-
tivity coefficients used in the numerical tests are
i
10 3
t
10 4
n
10 5
σ
l =
3
·
, σ
=
3
.
1525
·
, σ
=
3
.
1525
·
,
e
l
10 3
t
10 3
n
10 4
Siemens cm 1
σ
=
2
·
, σ
=
1
.
3514
·
, σ
=
6
.
757
·
(
) .
1mFcm 3 . The values of the
coefficients and parameters in the LR1 model are given in the original paper [74].
=
and the capacitance per unit volume is set to c m
Fine and coarse meshes. For both types of domains (Cartesian slabs and truncated
ellipsoids), we denote the Cartesian mesh used by
T = T i ×T j ×T k , indicating the
number of elements in each coordinate direction. This notation applies to both fine
and coarse meshes. When we scale up the mesh by a factor c , for brevity we define
c
is c 3
T =
c
T i ×
c
T j ×
c
T k , i.e. the number of elements in c
T
times the number
of elements in
T
.
Stimulation site, initial and boundary conditions. The depolarization process is
started by applying a stimulus of I i app =
200 mA/cm 3 lasting
1 ms on the face of the domain modelling the endocardial surface. The initial condi-
tions are at resting values for all the potentials and LR1 gating variables, while the
boundary conditions are for insulated tissue.
200 mA/cm 3 , I app =
Linear solver. At each time step, we solve iteratively the discrete Bidomain sys-
tem (5.27) by PCG with the two-level AS preconditioner. The PCG's initial guess
is the discrete solution of the previous time step and the stopping criterion is a 10 4
reduction of the residual norm on all levels except the coarse one, where it is 10 8
instead. In all runs, we report the PCG condition number, extreme eigenvalues, itera-
tion counts and cpu timings (all in seconds). The extreme eigenvalues are computed
by the standard Lanczos procedure during the PCG iteration (see e.g. [140]), yielding
close approximations of the exact extreme eigenvalues.
Scaled speedup on ellipsoidal domains. We consider a scaled speedup test on de-
formed ellipsoidal domains, that represent a severe test for the two-level AS solver.
The local size of each subdomain on the finest mesh is kept fixed at the value
32
32 (before adding the overlap) and the number of subdomains (hence pro-
cessors) is increased from 8 to 1024, forming increasing portions of ellipsoidal do-
mains
×
32
×
Ω
as shown in Fig 5.1. The fine mesh is chosen proportionally to the coarse
mesh as
T 1 =
16
T 0 so as to keep the local mesh size on each processor fixed at
32
×
32
×
32, and the overlap size is
δ =
h . The simulation is run for 3 time steps of
0
.
05 ms during the depolarization phase.
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