Composite Materials Research Progress

Optimization of Laminated Composite Structures… 79
optimization which includes a large amount of design variables (Bruyneel and Duysinx
2005). It has been made available in the BOSS Quattro optimization toolbox (Radovcic and
Remouchamps, 2002). In the following this solution procedure based on a mixed
approximation scheme is called Self Adaptive Method (SAM). Based on this approximation
scheme, it is possible to resort to the other ones (GBMMA, MMA, Conlin and the linear
approximation) by setting specific values to the asymptotes and by limiting the
approximations to the sets A or B in (7.6).
145
140
135
130
125
120
115
110
105
100
Strain energy
density
(N/mm)
(k )
L
g(θ
)
θ
(k ) *
MMA
g GCMMA
~
g MMA
~
(k )
U
45 90 (k ) 135 (k ) * 180
θ θGCMMA
400
350
300
250
200
150
g MMA
~
g(t
)
Strain energy
density
(N/mm)
1.2 1.3 1.4 1.5 (k ) 1.7
Figure 7.5. The mixed SAM approximation.
t
g GCMMA
~
(k )*
GCMMA
A summary of the approximations that will be compared in the following is presented in
Table 7.1.
Table 7.1. Summary of the approximations that will be compared in the numerical tests
Approximation Author Behavior
MMA Svanberg (1987) Monotonous
GCMMA Svanberg (1995) Non monotonous
SAM Bruyneel (2006) Mixed monotonous/non monotonous
7.3. Solution Procedure for Mono and Multi-objective Optimizations
Since the approximations are convex and separable the solution of each optimization subproblem
(Figure 2.3) is achieved by using a dual approach. Based on the theory of the duality,
solving the problem (2.2) in the space of the primal variables xi is equivalent to maximize a
function (7.10) that depends on the Lagrangian multipliers λ j , also called dual variables:
max minL
( x, λ)
λ
x
t
(k ) *
MMA
0 0,...,
( 0 1)
=
= ≥ λ
λ
m j j (7.10)
t