Composite Materials Research Progress

66
Michaël Bruyneel
relies on using a convex design space with respect to the lamination parameters, while
keeping in the problem’s definition the physical variables in terms of thickness and
orientation. This iterative procedure – between both design spaces – consists in determining
a first (local) solution in terms of thicknesses and orientations. A new search direction
towards the global optimum is then computed by evaluating the first order derivative of the
objective function at the local solution with respect to the lamination parameters. The
global optimum is reached when this sensitivity is close to zero. Otherwise a new design
point is calculated in the space of the fibers orientations, and the process continues, usually
by adding new plies in the laminate. As seen in Figure 3.8, the structural response is not
convex with respect to θ while it is convex in terms of the lamination parameter ξ. With
this technique the knowledge of the feasible regions of the lamination parameters is not
mandatory.
Although efficient, this solution procedure can only be used for global structural
responses like the stiffness, the vibration frequencies and the buckling load.
f
1
2
3
4
Figure 3.8. Illustration of the optimization process after Foldager et al. (1998) in both spaces of the
lamination parameters ξ and the fibers orientation θ.
3.2.5. Alternative Parameterization
In order to decrease the non linearities introduced by the fibers orientation variables,
Fukunaga and Vanderplaats (1991a) proposed to parameterize the laminated composite
membranes with the following intermediate variables:
i = sin 2 i or xi = cos2θ
i
x θ
based on the relation (3.12) and (3.13). This formulation was tested by Vermaut et al. (1998)
for the optimal design of laminates with respect to strength and weight restrictions. As in the
previous section, the main difficulty is to compute the orientations corresponding to the
optimal intermediate variables values xi.
f(θ)
θ, ξ
f(ξ)