# Optimization of Composite Structures with Thin Rigid Fibers Using Bioinspired Algorithms
## 5 Numerical Examples
### Optimization of Fiber Distribution in the Composite
Figure 13. The number of objective function evaluations for the maximization problem was 8550 for AIS and 9260 for PSO, while for the minimization problem it was 20,235 for AIS and 17,450 for PSO. On the basis of the obtained results, it can be concluded that the stiffest composite is obtained when the fibers are placed in several distant lines (Figure 13a). Conversely, the worst solution is when all the fibers are close together (Figure 13b). In the last example, two additional design variable (\(d_{5}\) and \(d_{6}\)) are added (Figure 14). The purpose of this modification is to find a better solution than the previous ones. The aim of optimization is to maximize the Young's modulus. The decision variables determine the location of fibers described by the parameters \(d_{1}\), \(d_{2}\), \(d_{3}\), \(d_{4}\), and \(d_{5}\). The last decision variable \(d_{6}\) determines the fibers' length. The fibers in the square plate are distributed symmetrically in the horizontal and vertical direction. Geometrical constraints are imposed on the decision variables. In addition, the condition for the total length of all fibers has to be fulfilled. The total length of all fibers should be constant. The relation between the length of the fibers is  
\[2\frac{d_{6}}{a}+\frac{d_{7}}{a}=3. \tag{17}\]  
Thus, during the optimization process, the position and length of the rigid fibers are changed. Figure 14. Dimensions and boundary conditions and type of fibers for the composite. Figure 13. The optimal distribution of rigid fibers for (**a**) maximization of the Young’s modulus and (**b**) minimization of the Young’s modulus. A description of the optimized parameters with bounds for box constraints is presented in Table 16.  
The parameters of AIS and PSO are included in Table 17.  
The results of the optimization of the composite are presented in Table 18.  
The result of maximizing the Young's modulus using AIS and PSO was the same. The number of objective function evaluations for the maximization problem was 13,500 for AIS and 12,850 for PSO. The result of the optimization after 30 iterations in the form of the optimal distribution of rigid fibers is presented in Figure 15.  
The introduction of two additional decision variables allowed us to obtain a stiffer composite. For the first example, the normalized effective Young's modulus is 1.1683, but for the second, stiffer one, it is 1.2812. The conclusion is that a stiffer composite is obtained when the fibers are placed in several distant lines. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Meaning** & **Lower Bound** & **Upper Bound** \\ \hline \(g_{1}\) & \(d_{1}\)/\(w\) & 0.3125 & 0.7500 \\ \hline \(g_{2}\) & \(d_{2}\)/\(w\) & 0.1875 & 0.7500 \\ \hline \(g_{3}\) & \(d_{3}\)/\(w\) & 0.0625 & 0.7500 \\ \hline \(g_{4}\) & \(d_{4}\)/\(w\) & 0.0625 & 0.8750 \\ \hline \(g_{5}\) & \(d_{5}\)/\(w\) & 0.1250 & 1.5000 \\ \hline \(g_{6}\) & \(d_{6}\)/\(w\) & 0.0625 & 0.1563 \\ \hline \hline \end{tabular}
\end{table} 
Table 16: Description of the decision parameters. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Decision Variable** & **Iteration 3** & **Iteration 10** & **Iteration 30** \\ \hline \(g_{1}\) & 0.3899 & 0.4239 & 0.5634 \\ \hline \(g_{2}\) & 0.2836 & 0.2640 & 0.2626 \\ \hline \(g_{3}\) & 0.1694 & 0.1678 & 0.2136 \\ \hline \(g_{4}\) & 0.5650 & 0.6181 & 0.6945 \\ \hline \(g_{5}\) & 0.8476 & 0.8476 & 0.9108 \\ \hline \(g_{6}\) & 0.0633 & 0.0625 & 0.0625 \\ \hline Normalized effective Young modulus \(E/E_{0}\) & 1.2261 & 1.2493 & 1.2812 \\ \hline \hline \end{tabular}
\end{table} 
Table 18: Results of the optimization process.