# Design and Multi-Objective Optimization of Fiber-Reinforced Polymer Composite Flywheel Rotors
## 4 Optimization
### Hybrid Optimization Strategy
Figure 3: Schematic of the hybrid optimization. To overcome these problems, a local improvement is used. The basic idea is to use the existing knowledge about the system to accelerate the convergence process. There are two key facts known about the system. First, the interferences \(\delta_{1}\) only affect stresses. Second, the angular velocity \(\omega\) only affects stresses and kinetic energy. After terminating, the GA returns the Pareto-front \(\mathbf{X^{\prime}_{\text{opt}}}\), \(\mathbf{F^{\prime}_{\text{opt}}}\), where \(\boldsymbol{x^{\prime}_{\text{s,opt}}}\) is a single solution on this front and is defined as:  
\[\boldsymbol{x^{\prime}_{\text{s,opt}}}=\left[ \begin{array}{c}\omega_{\text{ opt}}\\ t_{1,\text{opt}}\\ \ldots\\ t_{N_{\text{rim,opt}}}\\ x_{\text{material,1,opt}}\\ \ldots\\ x_{\text{material,N_{rim,opt}}}\\ \delta_{1,\text{opt}}\\ \ldots\\ \delta_{N_{\text{rim,opt}}}\end{array} \right] \tag{31}\]  
Applying the system knowledge, each solution \(\boldsymbol{x^{\prime}_{\text{s,opt}}}\) on the Pareto-front can be further improved by solving a single-objective sub-problem. This sub-problem only maximizes the kinetic energy by optimizing the angular velocity and the interferences, i.e.,  
\[ \begin{array}{c}\text{Maximize }E_{\text{kin}}(\boldsymbol{x})\\ \text{with respect to }\boldsymbol{x^{\prime\prime}}=\left[\begin{array}{c} \omega\\ \delta_{1}\\ \ldots\\ \delta_{N_{\text{rim}}}\end{array}\right]\\ \text{subject to }R_{\text{operation}}(\boldsymbol{x})-1\leq 0,\\ R_{\text{pressfit}}(\boldsymbol{x})-1\leq 0,\\ 0\leq\delta_{i}\leq\delta_{\text{ub}},\text{ and}\\ 0\leq\omega\leq\omega_{\text{ub}}.\end{array}  \tag{32}\]  
Since there is no relation between the angular velocity and the interferences on the one hand and the residual stresses on the other hand, the load case previous to press fitting is not relevant to this sub-problem. The sub-problem can be characterized as a continuous nonlinear constrained single-objective optimization problem for which the SQP algorithm is appropriate. Although the sub-problem has to be solved for each solution \(\boldsymbol{x^{\prime}_{\text{s,opt}}}\), the additional computation time is comparatively low. This is due to the fact that the sub-problem is less complex and gradient-based algorithms generally  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Solution** & **1** & **2** & **3** & **Unit** \\ \hline \(n\) & 20,000 & 25,000 & 25,000 & rpm \\ \(t_{1}\) & 100 & 100 & 100 & mm \\ \(t_{2}\) & 100 & 100 & 100 & mm \\ \(x_{\text{material,1}}\) & 1 & 1 & 1 & - \\ \(x_{\text{material,2}}\) & 3 & 3 & 3 & - \\ \(\  
converge faster. To further improve the optimization result, the GA and the local improvement is run a second time. Based on the results of the first optimization, the initial population of the GA is generated. To justify the use of the hybrid optimization strategy, its performance was compared to the genetic algorithm. It was shown that the solutions of the hybrid strategy always dominated the GA. In particular, the difference of performance between hybrid and genetic algorithm is larger for a greater number of rims. This is due to the fact that the number of variables increases with the number of rims. A high number of variables impedes the optimization procedure of the genetic algorithm. The local improvement of the hybrid strategy supports the genetic algorithm resulting in greatly enhanced performance.