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 article 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 two 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 multistart optimization strategy revealed multiple local optima with distinct geometric characteristics, highlighting the multimodal nature of the optimization problem. These findings demonstrate the computational efficiency and effectiveness of the proposed equation-oriented framework, representing an advancement in meanline performance analysis and design optimization.