Composite Materials Research Progress

76
Michaël Bruyneel
The symbols ∑ ( +) and ∑ ( −) in (7.3) denote the summations over terms having positive
and negative first order derivatives. When the first order derivative of the considered
structural response is positive a linear approximation in terms of the direct variables is built,
while a reciprocal approximation is used on the contrary.
145
140
135
130
125
120
115
110
105
100
g(x
)
~ ( )
g k
l
( x)
Strain energy density
(N/mm)
~ ( )
g k
r
( x)
45 90 (k )
(k )
180
xr
xl
Figure 7.2. The Conlin approximation.
Conlin can only work with positive design variables since an asymptote is imposed at
xi=0. On top of that, the curvature of this approximation is imposed by the derivative at the
current design point and can not be adapted to better fit the problem.
The Method of Moving Asymptotes or MMA (Svanberg 1987) generalizes Conlin by
introducing two sets of new parameters, the lower and upper asymptotes, Li and Ui, that can
take positive or negative values, in order to adjust the convexity of the approximation in
accordance with the problem under consideration. The asymptotes are updated following
some rules provided by Svanberg (1987). The parameters pij and qij are built with the first
order derivatives.
Strain energy
145 density (N/mm)
140
135
130
125
120
115
110
105
g(x
)
~ ( )
g ( x)
k
100
(k )
L
100
(k )
U
45 90 (k ) 135 (k )* 180 45 90 (k )*
(k )
180
x
x
Strain energy
145 density (N/mm)
140
135
130
125
120
115
110
105
g(x
)
Figure 7.3. The MMA approximation.
x
~ ( )
g ( x)
k
x