Composite Materials Research Progress

Optimization of Laminated Composite Structures… 55
x2
(k)
*
X
* global
X
no
(k)
X
Initial design
End
yes
* local
X
g j (X)
Approximated optimization problem
Solution of the approximated problem
Optimal design ?
~ ( k)
g j ( X)
Figure 2.3. Definition of an approximated optimization problem based on the information at the current
design point x (k) . The corresponding feasible domain is defined by the constraints of (2.2).
Each function entering the problem (2.1) is replaced by a convex approximation
~ ( k)
g j ( X)
based on a Taylor series expansion in terms of the direct design variables x i or
intermediate ones as for example the inverse design variables 1 xi
. For a current design x (k)
at iteration k, the approximated optimization problem writes:
~
~ ( k)
min g0
( x)
( k)
max
g j ( x ) ≤ g j
j = 1,...,
m
(2.2)
( k)
( k)
xi ≤ xi
≤ xi
i = 1,...,
n
where the symbol ~ is related to an approximated function. The explicit and convex
optimization problem (2.2) is itself solved by dedicated methods of mathematical
x1