Biomedical Engineering Reference
In-Depth Information
Fig. 5.1.
Ellipsoidal domains for scaled speedup Test 2 (Table 5.1), decomposed in 8, 16, ...,2048
subdomains, each one composed of 32
×
32
×
32 finite elements (not shown)
Table 5.1.
Test 1. Scaled speedup test on ellipsoidal domains, 3 time steps with
τ
=0.05ms.
Condition number
κ
2
, extreme eigenvalues
λ
,
λ
min
, iteration counts
it
and cpu timings in seconds
max
(
time
) of the PCG solver with AS preconditioner
procs.
T
T
=
16
T
d.o.f.
κ
=
λ
/
λ
it
t ime
0
1
0
2
max
min
8
2
×
4
×
8
32
×
64
×
128
0.5M
336.50=20.19/0.06
50.33
14.25
16
2
×
8
×
8
32
×
128
×
128
1.1M
196.86=13.78/0.07
44.33
13.12
32
2
×
16
×
8
32
×
256
×
128
2.2M
134.12=10.73/0.08
40.67
11.38
64
2
×
32
×
8
32
×
512
×
128
4.4M
93.89=8.45/0.09
37.00
11.95
128
4
×
32
×
8
64
×
512
×
128
8.6M
109.87=8.79/0.08
38.00
15.06
256
8
×
32
×
8
128
×
512
×
128
17.1M
89.89=8.09/0.09
36.33
17.73
512
16
×
32
×
8
256
×
512
×
128
34.1M
87.89=7.91/0.09
36.00
17.57
1024
32
×
32
×
8
512
×
512
×
128
67.9M
88.11=7.93/0.09
36.00
21.03
2048
64
×
32
×
8
1024
×
512
×
128
135.7M
88.67=7.98/0.09
36.00
26.54
,
λ
Table 5.1 reports the condition number
κ
2
, the extreme eigenvalues
λ
min
,
max
and the iteration counts
it
and cpu timings of the PCG solver with AS precondi-
tioner. The results clearly show the scalability of the AS operator, since all the re-
ported quantities appear to approach constant values as the number of subdomains
(hence processors) increases.
.
5.5 Numerical results: Bidomain simulations
The previous Bidomain solver has been used for studying the effects of subendocar-
dial ischemic regions on the epicardial surface [31], for assessing the information
content of the controversial hybrid monophasic action potentials [32], for study-
ing the effects of cellular heterogeneities on the spatial distributions of repolariza-
tion time [35, 37] and for validating extracellular markers of repolarization time
[122, 123]. We have recently applied this parallel Bidomain solver to study the three
dimensional details of the mechanisms of cardiac excitation elicited by unipolar ex-
tracellular stimuli of anodal and cathodal type, see [33, 34].
