Optimization and Computational Fluid Dynamics - Department of ...

6 Numerical Optimization for Advanced Turbomachinery Design 155
population size. The solution quality is maximum for N = 11 to 20. Small
populations (N 25) have a sluggish convergence to the optimal geometry because less
generations are allowed.
Crossover probability
In a single-point crossover operator, both parent strings are cut at a random
place and the right-side portions of both strings are swapped with the probability
pc (Fig. 6.3). In case of uniform crossover, the value of pc defines the
probability that crossover is applied per bit of the complete parent string.
High values of pc increase mixing of string parts but at the same time, increase
the disruption of good string parts. Low values limit the search to
combinations of samples in the existing design space. Experiments confirm
that a single point crossover with probability pc =0.5isoptimal.
Mutation probability
The mutation operator creates new individuals by changing in the offspring
strings a “1” to a “0” or vice versa. The mutation probability pm is defined
as the probability that a bit of a string is flipped. Systematic numerical
experiments confirm that the optimum setting for the mutation probability
is pm =1/(N × l) for all optimizations. This corresponds to changing on
average one bit at every generation.
Figure 6.5 shows how an optimization of the GA parameter settings can
lead to an improved and smoother GA convergence.
Creep mutation and Gray coding
Changing one digit in a binary code may result in a large variation of the
corresponding digital value: i.e., the small difference between 0111 and 1111
corresponds to a doubling of the digital value. Small variations of the digital
value may require a large number of binary digits to be changed: i.e., 0111
and 1000 are adjacent digital values but all four digits are different. This discontinuous
relation between the digital value and binary string may confuse
the GA optimizer. Creep mutation tries to avoid this by limiting the change
of the real value to a binary step length [5]. Gray coding uses an algorithm in
which similar binary strings correspond to adjacent digital values. In contrast
to what could be expected, no acceleration of convergence was obtained with
either one of these approaches.