# Multi-Objective Optimization of Three-Stage Turbomachine
###### Abstract  
This study presents the complex transfer matrix method (CTMM) as an advanced mathematical model, providing significant advantages over the finite element method (FEM) by yielding rapid solutions for complex optimization problems. In order to design a more efficient structure of a three-stage turbomachine rotor, we integrated this method with various optimization algorithms, including genetic algorithm (GA), differential evolution (DE), simulated annealing (SA), gravitational search algorithm (GSA), black hole (BH), particle swarm optimization (PSO), Harris hawk optimization (HHO), artificial bee colony (ABC), and non-metaheuristic pattern search (PS). Thus, the best rotor geometry can be obtained fast with minimum bearing forces and disk deflections within design limits. In the results, the efficiency of the CTMM for achieving optimized designs is demonstrated. The CTMM outperformed the FEM in both speed and applicability for complex rotordynamic problems. The CTMM was found to deliver results of comparable quality much faster than the FEM, especially with higher element quality. The use of the CTMM in the iterative optimization process is shown to be highly advantageous. Furthermore, it is noted that among the different optimization algorithms, ABC provided the best results for this multi-objective optimization problem. multi-objective; optimization; transfer matrix method; rotordynamics +
Footnote †: journal: Applied sciences  
for non-continuous rotors [8], swirl brake optimization for rotordynamic performance [9], and stability analysis of rotor-bearing systems under the influence of misalignment [10]. These efforts reflect the multifaceted requirements of rotordynamics that are essential for achieving optimal system performance. When examining these optimization studies, it becomes clear that different optimization algorithms are employed, depending on the objectives because optimization algorithms can exhibit varying performance based on the specific problem type. Examples include genetic algorithm (GA) for design optimization [11], differential evolution (DE) for reducing vibration levels [12], simulated annealing (SA) for multi-objective optimization [13], particle swarm optimization (PSO) for modeling twin rotor systems [14], and Harris hawk optimization (HHO) for addressing unbalance characteristics [15]. Each of these methods presents distinct advantages depending on the complexity of the rotordynamic problem being tackled. A shared characteristic of these studies is their reliance on the finite element method (FEM). Key results, such as natural frequencies, stability, and bearing forces--set as objective functions--are typically derived through FEM-based black-box solvers, either developed by researchers or accessed via commercial software. However, the FEM can be computationally intensive, and in certain cases, alternative mathematical models may offer more efficient and faster solutions. An alternative mathematical model to the FEM is the transfer matrix method (TMM), which is widely used in diverse intricate system such as rotor-bearing systems [16; 17], multibody systems [18], acoustic systems [19], and optical systems [20]. The advantage of fast solutions makes this method suitable for optimization and has led to its application across a diverse range of areas [21; 22; 23]. Despite these benefits, there is currently no study highlighting the TMM's superiority over the FEM in structural applications or exploring its advantages in optimization. The TMM delivers results that closely align with analytical methods and provides high-quality, efficient solutions regardless of the number of elements. Given this capability, it follows that the TMM will offer significant advantages in optimization studies, provided that a mathematical model of a system is established. To address this gap, we applied the complex transfer matrix method (CTMM) for rotordynamic optimization with modern metaheuristic optimization algorithms.