# Multi-Objective Optimization of Three-Stage Turbomachine
## 3 Optimization Problem
### Multi-Objective Rotordynamic Optimization
#### 3.1.2 Fitness Functions  
Turbomachines are designed by leveraging multiple engineering disciplines. Improvements in rotordynamics are made to enable main requirements such as flow, performance, and thrust. These main requirements ensure that the rotor architecture remains within a specific envelope. In this study, the design space was defined for this envelope with approximate constraints derived from the benchmark rotor, as shown in Figure 1. Our primary accuracy criterion is for the initially equal-length rotor geometry to resemble the benchmark after optimization. Additionally, individual evaluation of the desired objective functions and their collective assessment within the defined fitness function will provide insights into convergence. The objective of this design is to elevate the first natural frequency above the operational speed. Additionally, the rotordynamic solver, which operates as a black-box model, provides outputs for bearing forces and disk deflections. These outputs are utilized to develop new objective functions aimed at minimizing bearing forces and disk deflections for the first mode. To achieve this, the force and deflection values obtained by scanning the frequency response up to 20% above the natural frequency at 0 Hz operating speed were evaluated. Furthermore, a penalty function, denoted as \(g_{1}\), is introduced to reduce rotor mass and constrain the total rotor length. In this context, the subscript \(F_{i}\), represents the \(i\)th bearing and is used in the objective function focused on minimizing bearing loads. Similarly, the subscript \(\delta_{i}\) refers to the \(i\)th disk and is employed in the objective function aimed at minimizing disk deflections. \[f_{1}(x)=-\omega_{1} \tag{19}\]  
\[f_{2}(x)=\rho\pi\big{\{}{x_{10}}^{2}x_{1}+{x_{11}}^{2}x_{2}+{x_{12}}^{2}x_{3}+ {x_{13}}^{2}x_{4}+{x_{14}}^{2}x_{5}+{x_{15}}^{2}x_{6}+{x_{16}}^{2}x_{7} \tag{20}\]  
\[ \begin{array}{c}f_{3}(x)=\left\{\begin{array}{ll}0&if\;\;F_{1}\;\leq\;50 00\\ |5000-F_{1}|&if\;\;F_{1}>5000\end{array}\right.\end{array}  \tag{21}\]  
\[f_{4}(x)=\left\{ \begin{array}{ll}0&if\;\;F_{2}\;\leq\;5000\\ |5000-F_{2}|&if\;\;F_{2}>5000\end{array} \right. \tag{22}\]  
\[f_{5}(x)=\left\{ \begin{array}{ll}0&if\;\;F_{3}\;\leq\;5000\\ |5000-F_{3}|&if\;\;F_{3}>5000\end{array} \right. \tag{23}\]  
\[f_{6}(x)=\left\{ \begin{array}{ll}0&if\;\;\delta_{1}\;\leq\;1\\ |1-\delta_{1}|&if\;\;\delta_{1}>1\end{array} \right. \tag{24}\]  
\[f_{7}(x)=\left\{ \begin{array}{ll}0&if\;\;\delta_{2}\;\leq\;1\\ |1-\delta_{2}|&if\;\;\delta_{2}>1\end{array} \right. \tag{25}\]  
\[f_{8}(x)=\left\{ \begin{array}{ll}0&if\;\;\delta_{3}\;\leq\;1\\ |1-\delta_{3}|&if\;\;\delta_{3}>1\end{array} \right. \tag{26}\]  
\[g_{1}(x)= |1000-x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{7}+x_{8}+x_{9}| \tag{27}\]  
The fitness function was obtained by combining the given objective functions and the penalty function as in Equation (28). This multi-objective combination was carried out according to the global criterion method [33]. Here, \(A_{i}\) and \(B\) are the impact coefficients and only \(A_{1}=5\) is determined; all other impact coefficients have a factor of 1. The contribution of disk deflection and bearing force minimization to the objective function was weighted equally with a coefficient, as both are desired outcomes. However, the coefficient for increasing the natural frequency was assigned a higher value since this aspect is particularly crucial for rotordynamic criteria. Once a critical speed is present, there is no advantage in improving the disk deflection or bearing force. Another reason for choosing \(A_{1}=5\) is that, based on our preliminary optimization studies for this article, this value provided themost reasonable convergence than others. These coefficients remained constant across all optimization algorithms, ensuring a fair comparison.