Composite Materials Research Progress

88
Michaël Bruyneel
In Figure 8.8 the evolution of the vertical displacement under the load is drawn with
respect to the fibers orientation in the case of the homogeneous membrane (Figure 8.4, n=1).
The global minimum displacement is obtained for a value of 170°. When the starting point of
the optimization process of the problem (8.2) is close to 45°, 0° fibers orientation is found as
a local optimum. As -45° is chosen here for the initial design (i.e. 135°), the global optimum
can be reached. This illustrates the fact that a gradient based method is not able to reach the
global optimum, unless the starting point is in its vicinity. In Figure 8.8, the influence of the
mesh refinement on the solution is presented, as well.
8.3. Multi-objective Optimization
A symmetric laminate made of 4 plies and subjected to 2 in-plane load cases is considered.
N 2
3
N 1
1
θ
x
N 6
N 1
Figure 8.9. Laminate subjected to in-plane loads.
The applied loads and the initial configuration are reported in Table 8.2. The load case
(2) is variable : the factor k takes the values 0,1,2,…,8. The extreme load cases are, on one
hand (1000,0,0) and on the other hand the combination of (1000,0,0) and (0,2000,0) N/mm.
Table 8.2. Definition of the problem: load case and starting point
Load case (1) Load case (2) Initial orientations Initial thickness
( N 1,
N 2 , N6
) ( N 1,
N 2 , N6
) θ = ( θ1,
θ 2 )
t = ( t1,
t2
)
in N/mm
in N/mm
in degrés
en mm
(1000,0,0) (0, k × 250 ,0) (30,120) (1,2)
The performance of three approximation schemes are compared : GCMMA, MMA and
SAM. The optimization problem writes :
1
min max ε( ) ( j)
2 j Aε
j 1,2
T
θ, t =
TW ( j)
( θ i , ti
) ≤1
i
, j = 1,
2
2
N 2