Optimization and Computational Fluid Dynamics - Department of ...

6 Numerical Optimization for Advanced Turbomachinery Design 151
Fig. 6.2 Gradient method (- - -) and zero-order sweep of the design space (o)
To minimize the OF, most numerical algorithms require a large number of
performance evaluations and are often very expensive in terms of computer
resources. Zero order methods may require even more evaluations than gradient
methods but the latter may get stuck in a local minimum. The method
presented here uses a zero order search mechanism based on an evolutionary
theory.
6.2.1.1 Zero-order Search
A systematic sweep of the design space, defining v values between the minimum
and maximum limits of each of the n design parameters, requires v n
function evaluations. Figure 6.2 illustrates how such a sweep, calculating the
OF for 3 different values of X1 and X2, provides a very good estimation of
where the optimum is located with only 9 function evaluations. The risk of
converging to a local minimum is low and such a systematic sweep is a valid
alternative for analytical search methods for small values of n. However it
requires more than 14 × 10 6 evaluations for n = 15.
Evolutionary strategies such as GA and SA can accelerate the procedure
by replacing the systematic sweep with a more intelligent selection of new
geometries using the information obtained during previous calculations in a
stochastic way.
SA is derived from the annealing of solids [1]. At a given temperature, the
state of the system varies randomly. It is immediately accepted if the new
state has a lower energy level. If however, the variation results in a higher
state, it is only accepted with a probability Pr that is a function of the
temperature.