Handbook of Turbomachinery Second Edition Revised - Ventech!

promising technique capable of addressing multiple objectives and
constraints without relying on user input is the Kreisselmeier–Steinhauser
(K–S) function approach [10, 11]. In this approach, the multiple objective
functions and constraints are combined using the K–S function to form a
single envelope function, which is then optimized. The optimization
procedure presented in this chapter uses the K–S function technique. The
following section briefly describes the K–S function technique.
Kreisselmeier-Steinhauser (K–S) Function Technique
In this technique, the original objective functions and constraints are
modified into reduced objective functions. Depending on whether the
individual objective functions are to be minimized or maximized, these
reduced objective functions assume one of the two following forms:
F*kðFÞ ¼ FkðFÞ
Fk0
F*kðFÞ ¼1:0
1:0 gmax40; k ¼ 1; ...; NOBJmin
FkðFÞ
Fk0
gmax40; k ¼ 1; ...; NOBJmax ð1Þ
where Fk0 represents the original value of the kth objective function ðFkÞ
calculated at the beginning of each optimization cycle and F is the design
variable vector. gmax represents the largest constraint in the original
constraint vector, gjðFÞ, and is held constant during each cycle. NOBJmin
and NOBJmax represent the number of objective functions that are to be
minimized or maximized, respectively. The reduced objective functions are
analogous to constraints. Therefore, a new constraint vector, fmðFÞ
(m ¼ 1; 2; ...; M, where M ¼ NC þ NOBJ), that includes the original
constraints and the reduced objective functions is introduced. Here NC is
the total number of original constraints. The new objective function to be
minimized is defined as
FKSðFÞ ¼fmax þ 1
r log X
e
M
e
m¼1
rðfmðFÞ fmaxÞ
where fmax is the largest constraint in the new constraint vector fmðFÞ. The
composite function FKSðFÞ, which represents an envelope function of the
original objective functions and constraints, can now be minimized using a
suitable unconstrained optimization technique. The parameter r is a
drawdown factor that may vary between optimization cycles. Large values
of r ‘‘draw down’’ the K–S function closer to the value of the largest
constraint. Typically, r is progressively increased such that the K–S function
Copyright © 2003 Marcel Dekker, Inc.
ð2Þ