Biomedical Engineering Reference
In-Depth Information
Fig. 5
Finite element mesh for the bypass optimization problem
For the optimization problem the graft/artery ratio diameter varies from 0.6 to
1.2, the height of the sinus curve varies from 10 to 20 mm and a circular anastomosis
is set accordingly. Only symmetric geometries were considered since removing the
symmetry constrain does not have a major effect [ 30 ]. So asymmetry is not requisite
for the design of the bypass under the given flow conditions.
The search space is not known in absolute terms and simulations of 100
random possible graft designs have been conducted in order to get an indication
of the objective space distribution. Figure 6 presents results for maximum pressure
variation in the whole domain . b /and for the longitudinal velocity in the critical
domain . b /where reversed flow and residence times are enhanced. The optimal
solutions in the decision space are in general denoted as the Pareto set and its image
in the objective space as Pareto front. Results shown in Fig. 6 allow identifying the
likely presence of a Pareto front in the design problem. The shape optimization will
allow at least a 20% decrease on the pressure variation.
Since objective values are distributed in different ranges normalizing objective
values by the fittest in the generation before the weighted-sum operation has been
proposed [ 31 ]. For the optimization example, ( 23 ) becomes:
w 1 ® 1 . b /
w 2 ® 2 . b /
® 2 . b /
F . b /
D
® 1 . b / C
(28)
where ' 1 and ' 2 are the fittest values for ' 1 and ' 2 , respectively, in the generation.
The weight parameters w 1 and w 2 are random values calculated as in ( 24 ). The
fitness function to be maximized by the GA is then defined as:
F . b /
FIT
D
A
P
(29)
being A a positive integer to ensure positiveness and P a value to penalize design
vectors that do not conform with constraints. As a compromise between computer
time and population diversity, parameters for the genetic algorithm were taken as
N pop D
5 for the population and elite group size, respectively. The
number of bits in binary codifying for the design variables was N bit D
12 and N e D
5. Optimal
bypass geometries were obtained setting the maximum number of generations as
200. One optimal graft with design parameters given as graft diameter of 11.7 mm
and height 18.7 mm is discussed here. The simulated longitudinal velocity values for
the optimal graft solution are given in Fig. 7 . The longitudinal velocity distribution
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