# Design and Multi-Objective Optimization of Fiber-Reinforced Polymer Composite Flywheel Rotors
## 4 Optimization
### Formulation and Characterization of the Optimization Problem
The rim number and the material choice are the most challenging optimization variables as they tremendously increase the complexity of the optimization problem. The optimization of material properties is discrete. Since a gradient cannot be calculated for a discrete problem, this excludes all local optimization algorithms. Local optimization algorithms are more effective than global algorithms,therefore, the discretization of the problem negatively effects the computation time. To solve discrete optimization problems, they can be transferred into integer problems. For the considered problem, the tmaterial choice \(x_{\text{material},i}\) has to be an integer variable:  
\[x_{\text{material},i}\in\mathbb{Z}. \tag{26}\]  
Each integer represents a material of a given material set. Hence, \(x_{\text{material},i}\) can be interpreted as the material number. To specify the range of this variable, \(x_{\text{material},i}\) is bounded as:  
\[1\leq x_{\text{material},i}<k, \tag{27}\]  
where \(k\) is the number of considered materials. To specify \(x_{\text{material},i}\) as taking discrete values from a set \(S=v_{1},\dots,v_{k}\) a function \(S(x_{\text{material},i})\) is used that assigns a specific property to each material. Herein, \(S\) denotes a single material property, e.g., the material density \(\rho\). Therefore, the function \(S_{\rho}\) would return the density of material \(x_{\text{material},i}\):  
\[S_{\rho}(x_{\text{material},i})=\rho_{x_{\text{material},i}}. \tag{28}\]  
Since the material choice is the only integer variable in the optimization vector, the entire problem is called a mixed-integer optimization problem. The number of optimization variables depends on the number of rims. For this reason, the addition of the rim number as an optimization variable leads to a dynamically sized optimization vector. There are no algorithms that can directly solve an optimization problem with a dynamically sized optimization vector. Therefore, the optimization problem must be solved for each number of rims separately. The mathematical formulation of the optimization problem is given by Equation (29). \[ \begin{array}{c}\text{Maximize }E_{\text{kin}}(\mathbf{x})\\ \text{Minimize }C_{\text{total}}(\mathbf{x})\\ \text{Maximize }P(\mathbf{x})\\ \text{with respect to }\mathbf{x}=\left[\begin{array}{c}N_{\text{rim}}\\ \omega\\ t_{1}\\ \dots\\ t_{N_{\text{rim}}}\\ x_{\text{material},1}\\ \dots\\ \delta_{1}\\ \delta_{N_{\text{rim}}}\\ \end{array}\right]\\ \text{subject to }R_{\text{residual}}(\mathbf{x})-1\leq 0,\\ R_{\text{presfti}}(\mathbf{x})-1\leq 0,\\ R_{\text{operation}}(\mathbf{x})-1\leq 0,\\ 1\leq x_{\text{material},i}\leq k,x_{\text{material},i}\in\mathbb{Z},\\ t_{1}+\dots+t_{N_{\text{rim}}}=r_{\text{o}}-r_{i},\\ t_{1}+\dots+t_{N_{\text{rim}}}=r_{\text{o}}-r_{i},\\ 0\leq\delta_{i}\leq\delta_{\text{ub}},\text{ and}\\ 0\leq\omega\leq\omega_{\text{ub}},\end{array}  \tag{29}\]  
where \(R_{\text{residual}}\), \(R_{\text{presfti}}\) and \(R_{\text{operation}}\) denote the strength ratios for the three load cases. The last three boundary conditions help to accelerate the convergence procedure by limiting the problem space. Herein \(t_{\text{lb}}\) is the lower bound for the rim thickness which can be determined by the production accuracy. \(t_{\text{ub}}\), \(\delta_{\text{ub}}\) and \(\omega_{\text{ub}}\) are the upper bounds for the rim thickness, the radial displacement and the angular velocity, respectively. They are to be set to reasonable values. The sum of all rim thicknesses must be equal to the difference between outer and inner radius. This is the only equality constraint and further increases the problem complexity. In summary, the problem can be classified as a multi-objective, mixed-integer optimization problem with nonlinear equality and inequality constraints and a dynamically dimensioned optimization vector.