Biomedical Engineering Reference
In-Depth Information
(
)
(
)
10
20
T
T
140.021
140.499
1
1
(
)
(
)
10
20
T
217.737
T
218.499
2
2
=
and
=
(
)
(
)
292.944
293.499
10
20
T
T
3
3
365.315
365.500
(
)
(
)
10
20
T
T
4
4
The temperature values obtained through the Gauss-Siedel method at this present
stage are comparable to the values obtained by the Jacobi method at 20 iterations.
We can infer from this example that the Gauss-Siedel iteration is twice as fast as
the Jacobi iteration. Convergence is achieved quicker by the Gauss-Siedel method
because of the
immediate substitution
of the temperatures to the right-hand side of
the equations when they are made available. Thus far, we have not discussed the
issue of terminating the iteration process for this particular problem. The degree to
which convergence is achieved is set by the user. If the absolute maximum differ-
ence
+
−
is chosen as the condition for termination, the accuracy of the
solution depends on the targeted number of significant figures you wish to obtain
for the temperatures. The smaller the acceptable error, the higher the number of
iterations but this will achieve greater accuracy.
(
k
1)
(
k
)
φ
φ
j
j
5.6
Fluid-Structure Interactions (FSI)
5.6.1
FSI in Computational Haemodynamics
The fluid structure interaction approach can provide advanced haemodynamic mod-
elling of diseased vessel structure. A number of computational approaches to simu-
late the cardiovascular flow can be categorised into three types as follows (Cheng
et al. 2005):
•
Geometry-prescribed CFD
constitutes prescribed moving meshes of a cardio-
vascular structure that is reconstructed from medical imaging modalities such as
magnetic resonance imaging and computed tomography based on different time
phases of a cardiac cycle. It does not consider the feedback from the structure to
the fluid medium, and hence is considered as a one-way FSI approach.
•
Fictitious Fluid-Structure Interaction
method is based on the immersed bound-
ary method that simplifies the structure to elastic fibers represented as chains of
points immersed in a fluid. Then, a Dirac delta based interpolation function is
used to compute the interaction of the fluid with the structure.
•
Realistic FSI
models the structural mesh using Finite Element Method and then
simulates the fluid flow by CFD and the two models coupled via algorithms to
achieve a fluid-structure interaction. By resolving the complex properties of the
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