# Multi-Objective Optimization of Three-Stage Turbomachine
## 7 Conclusions  
The evaluation of nine different optimization algorithms highlighted the significant advantages of selecting the most suitable algorithm for accelerating the preliminary design phase and improving decision-making accuracy in complex optimization problems. Although the fitness function values of all optimization algorithms converged below 1, the multi-objective improvement was conducted through different parameters due to the differences in their behavior during the exploration and exploitation processes as stipulated by the NFL theorem. Among these, the metaheuristic ABC and pattern-based PS algorithms achieved commendable results by considering the requirements of all objective functions without focusing solely on improving a single point. Additionally, it should be noted that these two algorithms had relatively good optimization times compared to others, except for SA. The study successfully achieved a rotor geometry that operates within a 200 Hz margin, avoids any natural frequencies within this range, and minimizes bearing and disk forces as much as possible, all while maintaining a lightweight structure. The optimized rotor geometry, as shown in Figure 16, bears a close resemblance to the benchmark J85-GE, with similarity references drawn from the positions of the compressor and turbines due to the unclear bearing positions in the figure. Our optimization efforts utilized the CTMM-based rotordynamic solution, which proved to be efficient and effective. The CTMM approach not only matched benchmark \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **TMM** & **FEM—10 mm** & **FEM—1 mm** & **Diff. with 10 mm** & **Diff. with 1 mm** \\ \hline Time [s] & 0.099 & 4.1 & 214.4 & \(\times\)41.41 & \(\times\)2165.65 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: TMM and FEM FRF time consumption comparison for per iteration. Figure 19: FRF response at Bearing 3.  
geometries with satisfactory time efficiency but also demonstrated its superiority over the FEM in terms of computational time. This reinforces the CTMM's suitability for robust rotordynamic optimization. These results indicate that the use of the TMM in other engineering fields where it is already applied will provide an advantage in terms of solution time for optimization studies compared to the FEM. It is important to note that in the optimization problem addressed, the focus is on the force and displacement values on the system rather than the stress distribution at a specific point. While the FEM may maintain its superiority over the CTMM in this regard, significant optimization studies can be conducted using the CTMM. Future work will focus on further advancing the CTMM tool to tackle more intricate rotordynamic challenges and facilitate rapid iterations. These efforts aim to solidify the CTMM as a versatile and efficient solution for optimizing rotor systems across a wide range of engineering applications. Conceptualization, H.T.N. and A.Y. ; methodology, H.T.N. and A.Y. ; software, H.T.N. ; validation, H.T.N. ; formal analysis, H.T.N. ; investigation, H.T.N. and A.Y. ; resources, H.T.N. ; data curation, H.T.N. ; writing--original draft preparation, H.T.N. ; writing--review and editing, H.T.N. and A.Y. ; visualization, H.T.N. ; supervision, H.T.N. and A.Y. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. Data Availability Statement: Data are contained within the article. I would like to thank the Scientific and Technological Research Institution of Turkey (TUBITAK) for its support within the scope of the TUBITAK 2244 Industry Doctoral Program, \"Development of Gas Turbine Engine Technologies\" project in Turkey and TUSAS Engine Industry for their support within the scope of the project and providing the license for the programs analyzed within the scope of the article. Also, I would like to thank [PERSON], who was my industrial advisor, for their support. Author [PERSON] was employed by the company TUSAS Engine Industries Inc.