Optimization and Computational Fluid Dynamics - Department of ...

6 Numerical Optimization for Advanced Turbomachinery Design 153
by means of a systematic study on two typical design cases: one geometry
defined by 7 parameters and one defined by 27 parameters [8]. Conclusions
are based on the solution quality q, i.e., the degree to where the GA optimum
approaches the real one within a given effort (5,000 function evaluations). It
is defined by:
q = OFAV − OFGA
· 100%
OFAV − OFmin
where OFAV is the average of the OF over the complete design space.
OFmin is the global minimum value of the OF obtained from a systematic
(numerically very expensive) scanning of the whole design space.
OFGA is the minimum value of the OF obtained from the GA optimization.
A q value of 100% indicates that the global minimum has been found. The
function evaluations for the numerical experiments are made by means of an
approximation of the NS solver based on Artificial Neural Networks (ANN,
explained in Sect. 6.3.1).
Optimum substring length
In a standard binary-coded GA, the n real-valued design parameters Xi,
defining a geometry, are jointly represented by one binary string:
1101...0 1001...1 0011...0 ... ... ... 0101...1
X1 X2X3 Xn
The substring length, denoted by l (number of bits per variable), determines
the total number of values (2 l ) that each design parameter can take.
The minimum substring length li for the ith design variable depends on
and Xmax i ,aswellasonthe
desired resolution (εi) for this variable:
the lower and upper bound respectively X min
i
li =log 2
X min
i
− Xmax
i
Very short substrings (l