Equation-oriented and black-box design optimization of axial turbines using gradient-based and gradient-free algorithms

Abstract Meanline modeling provides an effective tool for preliminary turbomachinery design, yet, its integration with Black-Box optimization frameworks is often hindered by high computational costs and potential convergence issues as the meanline equations must be solved at each optimization iteration. To address these challenges, this paper introduces a unified Equation-Oriented framework for turbomachinery design and analysis, representing the first meanline methodology seamlessly integrating performance prediction and design optimization. By directly incorporating the meanline equations into the optimization process and leveraging gradient-based solvers, the proposed Equation-Oriented framework significantly reduces computational cost while enhancing solution robustness compared to traditional Black-Box formulations. To quantify these benefits, a comparative study between Equation-Oriented and Black-Box optimization utilizing various gradient-free as well as gradient-based solvers was conducted. The results suggest that the Equation-Oriented approach requires orders of magnitude fewer model evaluations while achieving similar or superior design performance. Additionally, the Equation-Oriented approach exhibits less sensitivity to problem dimensionality, maintaining similar convergence rates in both single-stage and two-stage turbine cases. A multi-start optimization strategy identified several local optima with minimal variation in efficiency, indicating a flat optimization landscape near the global optimum. These findings demonstrate the computational efficiency and effectiveness of the proposed Equation-Oriented framework, representing a substantial advancement in meanline performance analysis and design optimization.