## 6 Numerical examples
### Example 1  
This is the case of the search of fully isotropic laminates with twelve plies. It is a very difficult problem in laminate designs, already treated in the past by the author and co-workers, [21], and by other researchers, [22]; actually, it seems, up to now, that it is impossible to obtain a fully isotropic laminate composed by less than 12 identical plies; this example is a sort of benchmark in the laminates design field. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline Test & \(c_{0.0}\) & \(c_{0.s}\) & \(e_{0}\) & \(c_{1.0}\) & \(c_{1.s}\) & \(e_{1}\) & \(c_{2.0}\) & \(c_{2.s}\) & \(e_{2}\) \\ \hline T1 & 0.729 & 0.729 & 0 & 1.490 & 1.490 & 0 & 1.490 & 1.490 & 0 \\ T2 & 0.600 & 0.600 & 0 & 1.700 & 1.700 & 0 & 1.700 & 1.700 & 0 \\ T3 & 1.000 & 0.500 & 0.5 & 5.000 & 1.500 & 0.5 & 1.000 & 1.800 & 2 \\ T4 & 0.800 & 0.000 & 0.5 & 2.000 & 1.400 & 0.2 & 1.400 & 2.000 & 0.2 \\ T5 & 0.400 & 0.000 & 0.5 & 2.000 & 1.400 & 0.2 & 1.400 & 2.000 & 0.2 \\ \hline \end{tabular}
\end{table} 
Table 1: Numerical data for the tuning of parameters \(c_{0}\), \(c_{1}\), \(c_{2}\) in the tests. Figure 4: Trajectories in the Trelea’s convergence domain of tests T1 to T5. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \(E_{1}\) & \(E_{2}\) & \(G_{12}\) & \(\
u_{12}\) & \(Q_{11}\) & \(Q_{22}\) & \(Q_{66}\) & \(Q_{12}\) & \(T_{0}\) & \(T_{1}\) & \(R_{0}\) & \(R_{1}\) & \(\varPhi_{0}\) & \(\varPhi_{1}\) \\ \hline
181.00 & 10.30 & 7.17 & 0.28 & 18 1.81 & 10.35 & 7.17 & 2.89 & 26.88 & 24.74 & 19.71 & 21.43 & 0\({}^{\circ}\) & 0\({}^{\circ}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Technical, Cartesian and polar constants of carbon-epoxy T300-5208 plies (GPa). The case of fully isotropy, that comprehends also bending-extension uncoupling, is determined by the choice \(H_{ii}=1,\ i=3,4,9,10,15,16,\) the other components being null, so that function \(I(\mathbf{P})\) reduces to  
\[I(\mathbf{P})=P_{3}^{2}+P_{4}^{2}+P_{9}^{2}+P_{10}^{2}+P_{15}^{2}+P_{16}^{2}=0. \tag{25}\]  
Quantities \(P_{i}\) must be computed through eq. (18), the problem is hence strongly non-convex, the design variables being the angles \(\delta_{k}\). The results of the numerical tests are summarised in Tab. 3. For each tests, composed, as already said, by 100 runs of ALE-PSO with the same initial swarm, two quantities, obtained in a run for the objective function, are considered: the best value of the average on the swarm and the value of the best particle in the swarm (_i.e._ the particle having the lowest value of the objective function). For both these quantities, the statistics data relative to all the 100 runs, are: the mean value, the standard deviation, the best value, the worst value and the number of evaluations \(n_{eval}\) of the objective function to obtain the quantity's mean value. Of course, the exact solution being valued 0, the data reported in columns 3, 5 and 6 of Tab. 3 are to be considered as numerical residual of the computation. Under a mechanical point of view, a residual of the magnitude \(10^{-2}\) for this problem can be considered as technically acceptable, _i.e._ the final laminate with a value of \(I(\mathbf{P})\)\(\sim\)\(10^{-2}\) can be considered very close to be fully isotropic, under a practical point of view. In Fig. 5, for instance, the case of a laminate with \(I(\mathbf{P})\)= 0.023 is shown; it corresponds to the best particle in test T3, whose stacking sequence is [\(-16.43^{\circ}\), \(-70.38^{\circ}\), \(66.70^{\circ}\), \(31.97^{\circ}\), \(31.76^{\circ}\), \(-59.02^{\circ}\), \(78.65^{\circ}\), \(-12.13^{\circ}\), \(-12.34^{\circ}\), \(-47.06^{\circ}\), \(-90.00^{\circ}\), \(36.06^{\circ}\)].