# Optimization of Composite Structures with Thin Rigid Fibers Using Bioinspired Algorithms
## 5 Numerical Examples
### Optimization of Fiber Orientation in Composite
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Meaning** & **Lower Bound** & **Upper Bound** \\ \hline \(g1\) [rad] & \(d_{1}\) & 0 & 3.1416 \\ \hline \(g2\) [rad] & \(d_{2}\) & 0 & 3.1416 \\ \hline \(g3\) [rad] & \(d_{3}\) & 0 & 3.1416 \\ \hline \(g4\) [rad] & \(d_{4}\) & 0 & 3.1416 \\ \hline \(g5\) [rad] & \(d_{5}\) & 0 & 3.1416 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Description of the decision parameters. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**AIS**} \\ \hline
**Decision** & **Memory Cells** & **Number of** & **Ctones** & **Crowding** & **Gaussian** \\
**Variables** & & & & \\ \hline
5 & 10 & 10 & 0.5 & 0.5 \\ \hline \hline \multicolumn{5}{c}{**PSO**} \\ \hline
**Decision** & **Variables** & **Number of** & & \\
**Variables** & & & & \\ \hline
5 & 20 & 0.73 & 1.47 & 1.47 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of AIS and PSO. In the next step, the composite is optimized according to the Poisson's ratio (maximization and minimization). The results of the maximization of the Poisson's ratio of the composite are presented in Table 6.  
The orientation of fibers for the best composite is between 0.5642 [rad] and 0.5845 [rad] (for the obtuse angle, the acute angle is calculated using \(\pi\)-a). During this optimization example, the bioinspired algorithms found the optimal solution by taking into consideration the values from Table 1. The results of the minimization of the Poisson's ratio of the composite are presented in Table 7.  
The orientation of the fibers for the best composite is about 1.5708 [rad]. For the last example, it can be concluded that both algorithms have found the optimal solution (Table 1). The results of the maximization after 34 iterations and minimization after 145 iterations in the form of the optimal orientation of rigid fibers are presented in Figure 9. The number of objective function evaluations for the maximization problem was 2210 for AIS and 1650 for PSO, while for the minimization problem it was 9425 for AIS and 6580 for PSO. The second example of composite optimization is presented in Figure 10. For this example, the loading scheme is changed. In this example, a square plate is simply supported and loaded by shear traction P. The objective function for this type of boundary condition is described by the effective shear modulus:  
\[J_{3}(x)=G. \tag{16}\]  
Before the optimization process, several variants for the characteristic orientation of the fibers are analyzed (Table 8). For this example, all fibers with the same angle are changed. Five different angles of fiber orientation are considered: \(\alpha=0\), \(\pi/6\), \(\pi/4\), \(\pi/3\), \(\pi/2\). For these variants, the effective shear modulus of the composites is calculated. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Decision Variable** & **Iteration 3** & **Iteration 16** & **Iteration 34** \\ \hline \(g_{1}\) [rad] & 2.6850 & 0.5331 & 0.5845 \\ \hline \(g_{2}\) [rad] & 0.5668 & 2.5744 & 2.5701 \\ \hline \(g_{3}\) [rad] & 0.3588 & 2.5463 & 2.5572 \\ \hline \(g_{4}\) [rad] & 2.5142 & 0.5880 & 0.5642 \\ \hline \(g_{5}\) [rad] & 2.5555 & 2.6008 & 2.5769 \\ \hline Normalized effective Poisson’s ratio \(v/v_{0}\) & 1.1013 & 1.1106 & 1.1129 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results of maximization of Poisson’s ratio. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Decision Variable** & **Iteration 4** & **Iteration 21** & **Iteration 145** \\ \hline \(g_{1}\) [rad] & 1.5815 & 1.5424 & 1.5672 \\ \hline \(g_{2}\) [rad] & 1.5542 & 1.5635 & 1.5684 \\ \hline \(g_{3}\) [rad] & 1.6112 & 1.5634 & 1.5677 \\ \hline \(g_{4}\) [rad] & 1.5875 & 1.5620 & 1.5734 \\ \hline \(g_{5}\) [rad] & 3.1182 & 1.5871 & 1.5691 \\ \hline Normalized effective Poisson’s ratio \(v/v_{0}\) & 0.706953 & 0.650957 & 0.64629 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Results of minimization of Poisson’s ratio. In this example, the composite is optimized according to the shear modulus (maximization and minimization).