# Optimization of Composite Structures with Thin Rigid Fibers Using Bioinspired Algorithms
## 5 Numerical Examples
### Optimization of Fiber Orientation in Composite  
The first example of composite optimization is presented in Figure 6. The square plate is simply supported and uniformly loaded by traction P in the horizontal direction. The square plate contains 25 uniformly distributed inclined fibers, as shown in Figure 6. The decision variables for this example determine the orientation of fibers described by several parameters: \(d_{1}\), \(d_{2}\), \(d_{3}\), \(d_{4}\), and \(d_{5}\). The length of fibers is \(2a\), where \(a\)/\(w=1/8\). For this example, each decision variable (from \(d_{1}\) to \(d_{5}\)) is responsible for changing all the fibers in one column by changing the angle from \(d_{1}\) to \(d_{5}\). The composite is discretized using 330'boundary elements (80 for the external boundary and 10 for each fiber). The objective function is described by the effective elastic Young's modulus in the direction of loading:  
\[J_{1}(x)=E_{1}, \tag{14}\]  
and the effective Poisson's ratio:  
\[J_{2}(x)=\
u_{12}. \tag{15}\]  
Before the optimization process, several variants for the characteristic orientation of parallel fibers are studied (Table 1). Five different angles of fiber orientation are considered: \(\alpha=d_{1}=d_{2}=d_{3}=d_{4}=d_{5}=0\), \(\pi/6\), \(\pi/4\), \(\pi/3\), and \(\pi/2\). For these variants, the effective properties of the composites are calculated. The effective Young's moduli and Poisson's ratios are calculated using the average strains. The normalized effective properties--the Young's modulus \(E_{1}/E_{o}\) and the Poisson's ratio \(\
u_{12}/\
u_{o}\)--are given in Table 1, where \(E_{o}\) and \(\
u_{o}\) are properties of the matrix in plane strain conditions. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\alpha\)[rad] & 0.000 & 0.524 & 0.785 & 1.047 & 1.571 \\ \hline \(E_{1}\)/\(E_{o}\) & 1.667 & 1.132 & 1.024 & 1.001 & 1.079 \\ \hline \(\
u_{12}\)/\(\
u_{o}\) & 1.000 & 1.098 & 1.079 & 0.971 & 0.640 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Effective properties of composites. Figure 6: Dimensions and boundary conditions and type of fibers for the composite. The effective properties of the composites are the same for the fibers in two orientations: for the acute angle \(\alpha\) and the obtuse angle \(\pi\)-\(\alpha\). A description of the optimized parameters with bounds for box constraints is presented in Table 2.  
The parameters of AIS and PSO are included in Table 3. The selection of the parameters of the optimization algorithms for AIS and PSO are tested and verified to ensure that a global solution was found for each optimization. In this example, the composite due to the [PERSON]'s modulus is optimized (maximization and minimization). The results of the maximization of the [PERSON]'s modulus of the composite are presented in Table 4.  
The orientation of the fibers for the best composite is 0 [rad]. Comparing these values with the values from Table 1, it can be concluded that both algorithms have found the same optimal solution. The results of the minimization of the [PERSON]'s modulus of the composite are presented in Table 5. The change in the value of the objective function for the best individual is presented in Figure 7. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Decision Variable** & **Iteration 6** & **Iteration 8** & **Iteration 12** \\ \hline \(g1\) [rad] & 0.0663 & 0.0663 & 3.1416 \\ \hline \(g2\) [rad] & 0.0000 & 0.0000 & 3.1416 \\ \hline \(g3\) [rad] & 3.1416 & 3.1416 & 3.1416 \\ \hline \(g4\) [rad] & 3.1293 & 3.1293 & 0.0000 \\ \hline \(g5\) [rad] & 0.4593 & 0.0000 & 3.1416 \\ \hline Normalized effective & & & \\ Young modulus \(E/E_{0}\) & 1.5273 & 1.6570 & 1.6669 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of maximization of [PERSON]’s modulus.