Composite Materials Research Progress

Optimization of Laminated Composite Structures… 77
⎛
⎞ ⎛
⎞
~ ( k)
( k)
( k)
⎜ 1
1 ⎟ ( k)
⎜ 1 1
g
⎟
j ( x ) =g j ( x ) + ∑ pij
−
+ ∑q−(7.4)
⎜ ( k)
( k)
( k)
⎟ ij ⎜ ( k)
( k)
( k)
⎟
+ ⎝U
i − xi
U i − xi
⎠ − ⎝ xi
− Li
xi
− Li
⎠
As it will be seen later those monotonous schemes are not efficient for optimizing
structural functions presenting non monotonous behaviors, as in Figure 3.4.
7.2.2. Non Monotonous Approximations
Based on MMA, Svanberg (1995) developed the Globally Convergent MMA approximation
(GCMMA). As illustrated in Figure 7.4 it is non monotonous and still only based on the
information at the current design point (functions values, first order derivatives, asymptotes
values). Here both Ui and Li are used simultaneously. It was not the case in (7.4).
k
g
~ ( )
j
⎛
⎞ ⎛
⎞
( k)
( k)
∑
⎜ 1 1 ⎟ ( k)
+ ∑
⎜ 1 1
( x ) =g +
−
− ⎟
j ( x ) pij
q
(7.5)
⎜ ( k)
( k)
( k)
⎟ ij ⎜ ( k)
( k)
( k)
⎟
i ⎝U
i − xi
Ui
− xi
⎠ i ⎝ xi
− Li
xi
− Li
⎠
Using this method can lead to slow convergence given that it can generated too
conservative approximations of the design functions (Figure 7.1a).
145
140
135
130
125
120
115
110
105
100
Strain energy
density (N/mm)
(k )
L
g(x
)
(k )
U
45 90 (k ) 135 (k )* 180
x
~ ( )
g ( x)
k
Figure 7.4. The GCMMA approximation.
In order to improve the quality of this approximation it was proposed in Bruyneel and
Fleury (2002) and Bruyneel et al. (2002) to use the gradients at the previous iteration to
improve the quality of the approximation, leading to the definition of the Gradient Based
MMA approximations (GBMMA). In those methods the pij and qij parameters of (7.5) are
computed based on the function value and gradient at the current design point and on the
gradient at the previous iteration. The rules defined by Svanberg (1995) for updating the
asymptotes are used.
x