Multidisciplinary Design Optimization of Cooling Turbine Blade: An Integrated Approach with R/ICSM

Abstract

This paper presents an efficient integrated multidisciplinary design optimization method for shaping a high-pressure cooling turbine blade in aero engines. This approach utilizes a novel regression/interpolation combination surrogate model (R/ICSM), facilitating comprehensive design optimization through collaborative coupling feature parameterization modeling and numerical simulation analysis across various disciplines. The optimized blade adjusts the load distribution on its surface, effectively eliminating flow separation at the tip and trailing edge. Notably, the optimized blade achieves a 0.69% increase in isentropic efficiency while satisfying aerodynamic, strength, and structural constraints. This highlights the effectiveness and progressiveness of the multidisciplinary design optimization method for a cooling turbine blade based on the R/ICSM in enhancing overall performance. It offers a novel and feasible approach for turbine blade design optimization and provides valuable insights for future research and applications.

1. Introduction

High-pressure cooling turbine blades play a critical role in aero engines, operating under harsh conditions characterized by high temperature, high pressure, and high rotational speeds, which significantly influence the overall performance of the aero engine [1]. During the service life of the blade, several disciplines such as aerodynamics, heat transfer, and structural strength are intricately interrelated and heavily influenced. Therefore, the design of the high-pressure cooling turbine blade is a typical multidisciplinary and complex work of system engineering. In high-temperature environments, conventional blade materials often struggle to maintain optimal performance. Employing high-temperature materials alongside thermal barrier coatings can enhance the thermal protection capabilities of blades to some extent. Furthermore, utilizing cooling mediums like low-temperature gas for heat exchange or employing insulation film cooling represent highly effective cooling strategies. However, the adoption of such cooling methods can lead to heightened string stress conditions [2,3]. Therefore, achieving an advanced turbine blade design requires comprehensive consideration of the interplay and synergistic effects between multiple disciplines, including aerodynamics, heat transfer, structure, and strength.

Traditional methods for designing a cooled turbine blade typically involve sequential analyses of aerodynamics, heat transfer, and structural strength. These methods often require multiple iterations to meet comprehensive performance criteria. However, relying on iterative design and verification, traditional methods have limited exploration capabilities for performance and struggle to identify optimal blade configurations. In contrast, Multidisciplinary Design Optimization (MDO) technology provides a solution by synchronizing considerations of various disciplines. It fully utilizes advanced optimization techniques to coordinate conflicts between discipline indicators in real time, thereby obtaining the most comprehensive blade design scheme [4]. In recent years, MDO technology has found widespread application in turbine blade design optimization. For example, Li et al. [5] conducted a multidisciplinary analysis of a cooling turbine blade using time-sequential conjugate heat transfer analysis and temperature–pressure interpolation strength analysis. Their study significantly improved the performance of the cooling turbine blade by integrating multiple disciplines and optimizing the blade design. Xiao et al. [6] and Qian et al. [7] established reliability models for a cooling turbine blade using fluid–thermal–structural coupled analysis methods. Compared to traditional models with failure-independent assumptions, such models demonstrate higher efficiency and accuracy. Yang et al. [4] proposed a sensitivity analysis method for a cooling turbine blade considering surface thickness uncertainty, combining mesh deformation, neural network models, and multidisciplinary analysis. Li et al. [8] established a multidisciplinary coupled analysis model for aerodynamics, heat transfer, and thermodynamics, effectively enhancing cooling performance through internal heat transfer reinforcement and external membrane cooling improvements. These studies offer compelling evidence supporting the feasibility and effectiveness of MDO–based optimization in engineering applications for cooled turbine blades, underscoring its substantial value.

The analysis of cooling turbine blades within a single discipline is already quite complex. When considering comprehensive optimization across multiple disciplines, challenges arise in terms of model complexity, computational complexity, and information exchange complexity. To address these challenges in practical engineering, surrogate models are often employed to replace intricate multidisciplinary system analyses. The surrogate model, which is also known as the metamodel, is one of the core technologies of MDO. Utilizing a limited amount of sample data generated by physical model tests and numerical analysis, it can establish mathematical models to improve computational efficiency under the premise of a certain accuracy level [9]. In recent years, the surrogate model has received extensive research attention. Numerous surrogate models, including Polynomial Response Surface (PRS) models [10], Kriging models [11], Support Vector Regression (SVR) models [12], Radial Basis Function (RBF) models [13], and Artificial Neural Network models [14], were proposed. Huang et al. [15] proposed a new failure-related analysis strategy based on the Kriging model, conducting reliability assessments on high-pressure turbine rotors and enhancing computational efficiency. Qian et al. [7] established a stress probability analysis model for aviation engine turbine rotors’ critical locations using response surface methods, demonstrating high efficiency and accuracy. Pietro et al. [16] proposed a Multi-Fidelity Successive Response Surface (MF-SRS) model for collision testing, demonstrating superior accuracy and effectiveness in predictions when compared to the Successive Response Surface Method (SRSM). Xiao et al. [17] introduced a machine learning-based method for predicting the lifespan of a turbine blade and predicted the remaining useful life through five start–stop cycles of a gas turbine. Xing et al. [17] used a Kriging surrogate model to capture input data uncertainty and successfully obtained better performance variability in a robust optimization considering design uncertainties for pulse-corrected ballistic trajectories. Yang et al. [4] constructed a neural network model for conducting sensitivity analysis on the multidisciplinary system of a turbine blade, reducing computational workload. The application of surrogate models in the optimization of a high-pressure cooling turbine blade addresses challenges posed by the complexity of multidisciplinary interactions, computational demands, and information exchange.

While the surrogate model offers various advantages, the suitability of different surrogate models for diverse problems varies. Given the absence of prior knowledge regarding the multidisciplinary design optimization of the cooling turbine blade, finding appropriate and highly precise surrogate models for practical engineering problems proves challenging. A common approach to address this challenge involves constructing an ensemble surrogate model, which integrates two or more independent surrogate models by introducing weight factors. The concept of the ensemble surrogate model traces back to the 1990s and has become a current research focus. In a study by Huang et al. [18], a robust combination surrogate model was proposed through weighted combination, resulting in enhanced model accuracy. Yin et al. [19] introduced a new weighting factor and established a new ensemble surrogate model named EM-MROWF, demonstrating superior performance compared to a single-surrogate model. Xue, et al. [20] conducted a study on composite surrogate models from the aspects of selecting the model, calculating weight, and compensating residuals, proposing the OSF-TLPE composite surrogate model method and demonstrating its more flexible and reliable performance through testing. Combining the advantages of regression surrogate models and interpolation surrogate models, the authors of [21] proposed a regression/interpolation combination surrogate model to leverage the global trend-fitting capability of regression surrogate models and the local accuracy prediction advantage of interpolation surrogate models, which can effectively enhance the accuracy and robustness of surrogate models. However, this surrogate model has not yet been applied in practical engineering optimization to demonstrate its utility.

This study focuses on the design optimization of a two-stage high-pressure turbine blade in a specific aeroengine. It establishes a parameterized model for a complex cooling blade and employs a regression/interpolation combination surrogate model to enhance efficiency in design optimization of the cooling turbine blade. The uniqueness of the proposed method lies in the establishment of a parameterized model that comprehensively considers the aerodynamic shape and internal cooling features of the cooling turbine blade. This parameterized model allows for the coupling of disciplines such as aerodynamics and strength in the design process, resulting in a comprehensive improvement in blade performance during the design optimization. Furthermore, this research offers certain advantages in improving the isentropic efficiency of the turbine stage under design conditions and enhancing the surface load distribution of the blade.

2. Turbine Stage Blade Model

The structure diagram of a two-stage high-pressure turbine rotor structure is shown in Figure 1. The first-stage rotor blade is a cooling blade, which contains complex air-cooled structures such as partitions, turbulence ribs, internal cooling root segments, and trailing edge holes. Through the performance analysis of the initial scheme, it is known that the first-stage cooling turbine blade has a significant impact on the performance of the high-pressure turbine. Operating at high speeds in high-temperature gas environments, the blade endures complex forces such as centrifugal, aerodynamic, and temperature loads, leading to notable strength challenges. Therefore, the design of the cooling turbine blade needs to comprehensively consider multidisciplinary factors such as cooling efficiency and aerodynamics.

To enhance turbine performance, the emphasis is on the multidisciplinary design optimization of the cooling blade in the first-stage turbine. The aerodynamic performance of a high-pressure turbine is assessed based on four parameters: mass flow rate $W_{41}$, total expansion ratio $\pi$, 41-section efficiency ${\eta }_{41}$, and outlet flow angle $\mathsf{\alpha }$. Specifically, section 41 efficiency ${\eta }_{41}$ is defined as follows:

$${\eta }_{41}=\frac{P_W}{W_{41}\frac{k}{k-1}R{T}_{41}\left(1-\frac{1}{{\pi }^{\frac{k-1}{k}}}\right)}$$

(1)

In the formula, $k$ represents the adiabatic index of gas at section 41; $P_W$ represents the total power of the high-pressure turbine, units kW; $T_{41}$ represents the total gas temperature at the outlet of the first-stage guide vane of the high-pressure turbine, units K; and $R$ represents the universal gas constant.

To enhance turbine performance, this article considers the optimization of the 41-section efficiency ${\eta }_{41}$ as the objective function. The emphasis is on the multidisciplinary design optimization of the cooling blade in the first-stage turbine.

3. Multidisciplinary Design and Optimization of Turbine Stage Blade

This study focuses on optimizing the turbine blade. To enhance isentropic efficiency within the constraints of aerodynamic and strength performance, an efficient multidisciplinary integrated design optimization method for the cooling turbine blade is proposed. This method combines a parametric modeling approach for the cooling turbine blade with cooperative coupling characteristics, numerical simulation analysis methods from various disciplines, and a regression/interpolation combination surrogate model (Figure 2). The specific steps involved in this method are as follows:

• Select appropriate design variables and determine their ranges.
• Select a suitable Design of Experiments (DOE) technique to generate a design matrix representing a set of sample points. Considering the precision required for the surrogate model and the time constraints for aerodynamic and strength analysis, an initial set of 200 sample points was generated using the “lhsdesign” program in MATLAB (v.2011).
• Employ the blade modeling approach outlined in Section 3.1.1. A custom MATLAB (v.2011) program was utilized to produce blade profile curves for the tip, middle, and root sections of the first-stage rotor blade across the 200 samples from step (2). The scatter coordinates of these blade profile curves were exported in various formats to facilitate blade modeling within the aerodynamic domain and parameterized modeling of the cooling blade within the structural domain.
• Utilize the parameterized modeling approach outlined in Section 3.1.2 for the cooling blade. The UG macro command was employed to automatically generate the geometric model of the first-stage rotor blade for strength analysis across the 200 samples from step (2).
• Use the aerodynamic analysis method detailed in Section 3.2.1. A high-fidelity Computational Fluid Dynamics (CFD) simulation of the two-stage high-pressure turbine blade was conducted using ANSYS (2020 R1) CFX software across all samples. Subsequently, pertinent response parameters were extracted, namely W₄₁, π, η and α.
• Employ the strength analysis method outlined in Section 3.2.2. Finite element simulation analyses of the first-stage rotor blade were conducted across all samples using the APDL language within ANSYS (2020 R1) software. Subsequently, the relevant response parameters were extracted.
• Combine sampling points with their respective response parameters to create the initial sample database.
• Utilizing existing sample data, integrate the advantages of two different surrogate models, and construct a regression/interpolation combination surrogate model.
• Select a suitable optimization algorithm for the optimization process. The optimization objectives and constraint parameters are approximated by the regression/interpolation combination surrogate model developed in step (8). Details of the constructed optimization model will be presented in Section 3.4. A Genetic Algorithm (GA) was employed to solve the constrained optimization problem in this study.
• Generate a new blade profile curve for the tip, middle, and root sections of the first-stage rotor blade based on the optimized design obtained in step (9). Output the scatter coordinates of the curve in different formats.
• Employ UG macro commands to automatically generate the geometric models of the first-stage rotor blade for the new design scheme obtained in step (9).
• Utilizing ANSYS CFX software, conduct high-fidelity CFD simulations for the two-stage high-pressure turbine blade based on the new design scheme. Then extract relevant response parameters.
• Using the APDL language within ANSYS (2020 R1) software, conduct finite element simulation analyses of the first-stage rotor blade for the new design scheme. Extract relevant response parameters σ_eq,max.
• Verify the fulfillment of convergence criteria. If satisfied, conclude the loop and acquire an improvement plan. If not satisfied, update the existing sample database and iterate through steps (8) to (14) again.

3.1. Parameterized Modeling Method of Cooling Turbine Blade

3.1.1. Blade Shape Parameterization

The planar cascade design forms the fundamental basis of the turbine blade design. Its objective is to shape the aerodynamics of the blade across various radial cross-sections and provide blade profile data essential for the radial stacking of the blade. Typically, turbine blade cascades comprise four profiles: the leading edge, trailing edge, blade suction surface, and blade pressure surface. While the leading and trailing edge profiles are commonly represented by circular arc curves, the blade suction and pressure surface curves tend to be more complex. To ensure the smoothness of the blade profiles, this study employs a direct analytical approach for shaping the flat blade grid. Figure 3 illustrates the geometric parameters and key control points necessary for the adopted planar cascade modeling method. Based on 11 primary parameters—including the leading edge radius $R_q$, trailing edge radius $R_h$, inlet geometric angles ${\beta }_q$, outlet geometric angles ${\beta }_h$, inlet wedge angles $W_q$, outlet wedge angles $W_h$, turning angle $\delta$, blade stagger angle $\gamma$, axial chord length $S$, throat width $a$, and blade pitch $t$—the positions of the five blade control points and their tangent directions can be uniquely determined (as shown in Figure 3). With the positions of control points and tangent directions established, a third-order Bezier curve is employed to delineate the entire blade bowl profile and the frontal section profile at the back of the blade. Additionally, a fourth-order polynomial curve is utilized to depict the rear section profile, ensuring curvature continuity between the two curves at the throat point and mitigating aerodynamic losses arising from curve connections.

Radial stacking of the blade aims to interpolate blade profiles across various radial sections, utilizing the design output from the planar cascade shape. This interpolation is used to perform radial stacking according to the chosen stacking law, thereby achieving a smoothly contoured blade body. Given that typical aerodynamic parameters of the turbine blade exhibit linear deviation along the blade height, a parabolic stacking law was employed to shape the blade body after obtaining profile data at the root, middle, and tip sections, as illustrated in Figure 4.

3.1.2. Parametric Modeling of Cooling Characteristics

MDO requires the automatic updating of geometric models based on design parameters. This involves collaboratively integrating feature parameterization modeling for both blade air cooling characteristics and adjustments to blade shape.

To mitigate the bending moment caused by centrifugal force in the turbine rotor blade, it is essential to distribute the center of gravity of the blade section radially along its height. The incorporation of typical cooling features can significantly alter the center of gravity of the blade section. Therefore, it is necessary to develop a parametric model for the hollow blade with variable wall thickness, which allows for swift adjustment of the center of gravity by altering the wall thickness distribution. Variable wall thickness encompasses three aspects: (1) variation of wall thickness dependent on radial height, (2) variation of wall thickness in the axial direction, and (3) independent variation of thickness at the pressure and suction sides. To accomplish this, the parametric modeling approach for the blade cavity in this study includes the following steps:

• Establish eight feature control points on the suction side profile, as illustrated in Figure 5.
• Determine the corresponding eight feature control points on the pressure side profile using geometric constraints based on inscribed circles at the feature control points.
• Connect the corresponding feature control points on the pressure and suction side profile and control the wall thickness at the pressure and suction sides using line length percentages.
• Ensuring cavity smoothness by incorporating the wall thickness values at the first, third, and eighth feature control points on the pressure and suction sides as variables. These values are then interpolated through quadratic interpolation to determine the wall thickness at the remaining feature control points.
• Generate cavity curves on the pressure and suction using spline curves.
• Repeat steps (1) to (5) to obtain the cavity design sections corresponding to the external blade shape at the root, middle, and tip. The blade cavity is obtained using the UG curve group command.

Expanding on the parametric model for the variable wall thickness hollow blade, a feature-based parametric modeling approach was adopted to separately design the leading edge and dovetail structures. Subsequently, typical cooling features, including baffles, turbulence ribs, internal cooling root segments, and trailing edge holes, were iteratively enhanced. Through this iterative process, a comprehensive parametric model for the cooling blade was developed, as illustrated in Figure 6.

3.2. Subject Analysis of Turbine Stage Blade

3.2.1. Fidelity CFD Simulation Analysis

The ANSYS (2020 R1) CFX software, renowned for its high accuracy in solving turbine aerodynamic problems and demonstrating robust convergence stability, was employed in this study for high-fidelity CFD simulation [22]. It resolved the steady-state compressible Reynolds Average Navier-Stokes (RANS) equation via the finite volume method, utilizing the Shear Stress Transport (SST) turbulence model to simulate the viscous term. Each blade row’s rotor static interface was configured as “Stage”, and the Advance Scheme and Turbulence Numbers were set to “High Resolution”.

A single blade channel served as the computational domain, with periodic boundary conditions applied on both sides. The flow working fluid in the main channel of this article was gas, with a molar mass of 28.94 kg/kmol. The Dynamic Viscosity was set to 1.831 × 10⁻⁵ kg/(m * s). The inlet boundary conditions encompassed total temperature, total pressure, and airflow angle direction. Inlet turbulence was moderated at 5%, while outlet boundary conditions were specified in terms of average static pressure. Solid wall boundaries were treated as adiabatic and no slip wall.

Grid partitioning was performed using ANSYS (2020 R1) TurboGrid software, with the grid topology structure set to ATM Optimized. The total number of grids and the number of spanwise grids in the meridian channel were separately adjusted to regulate overall grid quality, ensuring a value less than or approximately equal to 1 (as depicted in Figure 7) to meet the SST turbulence model grid quality requirements. The rotor blade tip clearance was set at 0.3 mm, with over 13 grid layers at the clearance to capture flow details. To verify grid independence and mitigate the flow field calculation result dependency on grid accuracy, five grid partitioning schemes with varying grid numbers were established: 0.6 million, 1.2 million, 2.4 million, 4.8 million, and 9.6 million. Figure 8 illustrates the radial distribution of the total expansion ratio of high-pressure turbines across these five grid schemes, revealing the following: (1) A trend towards consistent radial distribution with increasing grid numbers. (2) When the grid number reaches 2.4 million, the impact on flow field calculation results is relatively minor, indicating grid independence.

Considering both computational accuracy and efficiency, this paper ultimately selected a grid partitioning scheme comprising 2.4 million grids (illustrated in Figure 9) for numerical calculations and optimization design.

3.2.2. Strength Analysis of Rotor Blade Considering Internal Cooling Structure

The cooling turbine blade in service should withstand significant mechanical strain caused by centrifugal loads and coupled with complex loads such as geometric stress concentration, airflow pressure, and temperature load. The stress state is extremely complex, so it is necessary to analyze the stress distribution of the turbine blade and evaluate the strength through the finite element method, providing a basis for the MDO of the blade.

This section analyzes the stress on the first-stage rotor blade with linear elastic finite element analysis by ANSYS. The unit type is 10 nodes tetrahedral unit SOLID187. The temperature field distribution of the cooling turbine blade affects both thermal stress and material mechanical properties, thereby affecting blade strength. The use of a three-dimensional high-fidelity numerical simulation for the flow heat coupling calculation of the blade needs a long cycle, resulting in low efficiency in multidisciplinary design optimization iterations. Therefore, in the optimization analysis of this study, a simplified temperature field was adopted, which is to calculate the average temperature of the blade along the radial section and obtain the distribution of blade temperature along the radial direction. Figure 10 shows the grid and temperature field used in the optimized design.

Based on a simplified temperature field and applying loads such as rotational speed, the equivalent stress cloud map of the blade is analyzed, as shown in Figure 11.

3.3. Regression/Interpolation Combination Surrogate Model

Generally, different individual surrogate models are suitable for different types of prediction tasks. For instance, the regression surrogate model captures the global trend effectively while the interpolation surrogate model exhibits higher local accuracy at sampling points [23,24]. To amalgamate the strengths of common individual surrogate models and bolster their accuracy and robustness, the authors introduced a regression/interpolation combination surrogate model [21]. This model integrates the advantages of regression and interpolation surrogate models.

The PRS model is one of the most widely used regression surrogate models in MDO. It primarily employs simple low-order polynomials (usually first- or second-order polynomials) to approximate relationships between weak nonlinear responses and variables in actual physical systems. This approach offers notable advantages in terms of modeling efficiency and transparency [25]. The RBF model serves as an interpolation surrogate model that transforms multidimensional prediction problems into one-dimensional problems by considering Euclidean distance as a variable. It demonstrates excellent flexibility, high computational efficiency, and outstanding adaptability to high-dimensional nonlinear problems [26]. Therefore, this paper selects the PRS model and RBF model to construct an ensemble surrogate model named the regression/interpolation combination surrogate model.

3.3.1. Construct a Regression Surrogate Model

The second-order PRS model is chosen as the regression surrogate model. The PRS can be written as

$${\widehat{y}}_{\mathrm{prs}}(x)=z^T\beta ={\mathsf{\beta }}_0+{\displaystyle \sum }_{i=1}^k{\mathsf{\beta }}_i{x}_i+{\displaystyle \sum }_{i=1}^k{\displaystyle \sum }_{j=i}^k{\mathsf{\beta }}_{\frac{2j+i\left(2k-i+1\right)}{2}}{x}_i{x}_j$$

(2)

where ${\widehat{y}}_{\mathrm{prs}}(x)$ denotes the second-order polynomial response surface model, $z={\left(1,x_1,x_2,\dots ,x_k,x_1{x}_2,\dots ,x_{k-1}{x}_k,x_k{x}_k\right)}^T$ denotes a polynomial basis-function vector, and $\beta ={\left({\mathsf{\beta }}_0,{\mathsf{\beta }}_1,\dots ,{\mathsf{\beta }}_{(k^2+3k)/2}\right)}^T$ denotes a coefficient vector.

3.3.2. Update Training Sample Information

By using Equation (2), the responses of the established PRS $\left({\widehat{y}}_{\mathrm{prs},1},{\widehat{y}}_{\mathrm{prs},2},\dots ,{\widehat{y}}_{\mathrm{prs},n}\right)$ at the initial sampling locations $\left(x_1,x_2,\dots ,x_n\right)$ can be calculated. The updated PRS training dataset can be obtained accordingly:

$$\begin{array}{l}\left\{\left(x_1,y_{\mathrm{d},\mathrm{prs},1}\right),\left(x_2,y_{\mathrm{d},\mathrm{prs},2}\right),\cdots ,\left(x_n,y_{\mathrm{d},\mathrm{prs},n}\right)\right\}\\ =\left\{\left(x_1,y_1-{\widehat{y}}_{\mathrm{prs},1}\right),\left(x_2,y_2-{\widehat{y}}_{\mathrm{prs},2}\right),\cdots ,\left(x_n,y_n-{\widehat{y}}_{\mathrm{prs},n}\right)\right\}\end{array}$$

(3)

3.3.3. Construct the Interpolation Class Surrogate Model

The RBF model, by choosing the multiquadric-form basis function, is selected as the interpolation-type surrogate model. The general form of the RBF can be expressed as [13]

$${\widehat{y}}_{\mathrm{rbf}}(x)={\displaystyle \sum }_{i=1}^n{\mathsf{\lambda }}_i\mathsf{\varphi }\left(‖x-x_i‖\right)$$

(4)

where ${\mathsf{\lambda }}_i$ denotes an interpolation coefficient, $\mathsf{\varphi }\left(r\right)$ denotes a radially symmetric basis function, and $r=‖x-x_i‖=\sqrt{{\left(x-x_i\right)}^T\left(x-x_i\right)}$ denotes the distance between points $x$ and $x_i$.

The multiquadric-form basis function can be expressed as $\mathsf{\varphi }\left(r\right)={\left(r^2+c^2\right)}^{\frac{1}{2}}$.

By using the given training dataset $(x_i,y_i)\left(i=1,\dots ,n\right)$, it is easy to calculate the interpolation coefficient using the following equation:

$$\lambda =A^{-1}y$$

(5)

where

$$\begin{array}{l}\lambda ={\left[{\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_n\right]}^{\mathrm{T}}\\ y={\left[y_1,y_2,\cdots ,y_n\right]}^{\mathrm{T}}\\ A=\left[\begin{array}{cccc}\mathsf{\varphi }\left(‖x_1-x_1‖\right)& \mathsf{\varphi }\left(‖x_1-x_2‖\right)& \cdots & \mathsf{\varphi }\left(‖x_1-x_n‖\right)\\ \mathsf{\varphi }\left(‖x_2-x_1‖\right)& \mathsf{\varphi }\left(‖x_2-x_2‖\right)& \cdots & \mathsf{\varphi }\left(‖x_2-x_n‖\right)\\ ⋮& ⋮& \ddots & ⋮\\ \mathsf{\varphi }\left(‖x_n-x_1‖\right)& \mathsf{\varphi }\left(‖x_n-x_2‖\right)& \cdots & \mathsf{\varphi }\left(‖x_n-x_n‖\right)\end{array}\right]\end{array}$$

(6)

After choosing the multiquadric-form basis function, the RBFM (RBF with multiquadric-form basis function, ${\widehat{y}}_{\mathrm{RBFM}}(x)$) can be constructed to approximate the actual model by substituting ${\widehat{y}}_{\mathrm{RBFM}}(x)$ and ${\lambda }_{i,\mathrm{rbfm}}$ into Equation (4). The coefficient ${\lambda }_{\mathrm{rbfm}}$ can be calculated using Equation (5).

Moreover, by choosing the multiquadric-form basis function, a model ${\widehat{y}}_{\mathrm{rbfm}}(x)$ can be established to approximate ${\widehat{y}}_{\mathrm{d},\mathrm{prs}}\left(x\right)$, the deviation function of the PRS, which is obtained by subtracting the PRS(${\widehat{y}}_{\mathrm{prs}}(x)$) from the actual model. The coefficient ${\lambda }_{\mathrm{rbfm}}$ of ${\widehat{y}}_{\mathrm{rbfm}}(x)$ can be calculated by substituting the updated training dataset of the PRS $\left(x_i,y_{\mathrm{d},\mathrm{prs},i}\right)\left(i=1,\dots ,n\right)$ into Equation (5).

3.3.4. Construct a Composite Surrogate Model

By adding the established ${\widehat{y}}_{\mathrm{prs}}\left(x\right)$ and ${\widehat{y}}_{\mathrm{rbfm}}\left(x\right)$ together, R/ICSM(${\widehat{y}}_{\mathrm{R}/\mathrm{ICSM}}\left(x\right)$) can be subsequently constructed as follows:

$${\widehat{y}}_{\mathrm{R}/\mathrm{ICSM}}\left(x\right)={\widehat{y}}_{\mathrm{prs}}\left(x\right)+{\widehat{y}}_{\mathrm{rbfm}}\left(x\right)$$

(7)

The established R/ICSM ${\widehat{y}}_{\mathrm{R}/\mathrm{ICSM}}\left(x\right)$ can be used to predict the response at any point within the design space by using Equation (7).

3.4. Optimization Model Description

This article aims to enhance the efficiency of the turbine 41 cross-section through multidisciplinary design optimization of the cooling blade in the first-stage turbine. Drawing on previous design experience, this study identified design variables, including the radius at the leading edge radius $R_q$, inlet geometric angles ${\beta }_q$, outlet geometric angle ${\beta }_h$, inlet wedge angles $W_q$, turning angle $\delta$, blade stagger angle $\gamma$, and axial chord length $S$, for each section. It is crucial to mention that, due to installation considerations involving blade and tenons, the axial chord length of the blade root section was excluded from the scope of design variables. To mitigate potential adverse effects on downstream components caused by the optimization of the high-pressure turbine blade, it is imperative to improve the efficiency of the high-pressure turbine 41 section while maintaining a constant gas flow rate $W_{41}$ and total expansion ratio $\pi$. Additionally, it is essential to ensure that the outlet flow angle α of the high-pressure turbine falls within a reasonable range. Simultaneously, it is imperative to ensure that the maximum von Mises stress ${\sigma }_{\mathrm{eq},\max }$ on the blade remains below a specified threshold $0.9{\sigma }_{0.1}$ to guarantee the blade’s safe and reliable operation. Therefore, parameters $W_{41}$, $\pi$, α and ${\sigma }_{\mathrm{eq},\max }$ were selected as constraint parameters.

In summary, the optimization model is represented by the following formula:

$$\begin{array}{ll}\max :\hfill & \eta \hfill \\ \mathrm{w}.\mathrm{r}.\mathrm{t}.:\hfill & \left\{\begin{array}{l}{R}_{q1},{\beta }_{q1},{\beta }_{h1},W_{q1},{\delta }_1,{\gamma }_1,S_1\\ R_{q2},{\beta }_{q2},{\beta }_{h2},W_{q2},{\delta }_2,{\gamma }_2,S_2\\ R_{q3},{\beta }_{q3},{\beta }_{h3},W_{q3},{\delta }_3,{\gamma }_3\end{array}\right.\hfill \\ \mathrm{s}.\mathrm{t}.:\hfill & \left\{\begin{array}{l}80\le \mathsf{\alpha }\le 90\hfill \\ \left|{\sigma }_{\mathrm{eq},\max }\right|\le 0.9{\sigma }_{0.1}\hfill \\ \left|\frac{W_{41}-W_{41,\mathrm{bas}}}{W_{41,\mathrm{bas}}}\right|\le e_1\hfill \\ \left|\frac{\pi -{\pi }_{\mathrm{bas}}}{\pi }\right|\le e_2\hfill \\ {\mathit{x}}^l\le \mathit{x}\le {\mathit{x}}^u\hfill \end{array}\right.\hfill \end{array}$$

(8)

In the formula, $e_i\left(i=1,2\right)$ represents a small constant, $x$ denotes the design variable vector, and $l$ and $u$ denote the lower and upper limits of the design variable $x$, respectively. Subscripts 1, 2, and 3 denote the blade tip section, blade center section, and blade root section, respectively, while the subscript $bas$ denotes the relevant parameters of the initial scheme of the high-pressure turbine.

4. Optimization Results and Discussion

4.1. Comparison of Optimization Results

Table 1. Comparison of typical performance parameters of turbine blade before and after optimization.

Parameters	$\frac{{\eta }_{\mathrm{opt}}-{\eta }_{\mathrm{bas}}}{{\eta }_{\mathrm{bas}}}$	$\frac{{\pi }_{\mathrm{opt}}-{\pi }_{\mathrm{bas}}}{{\pi }_{\mathrm{bas}}}$	$\frac{W_{41,\mathrm{opt}}-W_{41,\mathrm{bas}}}{W_{41,\mathrm{bas}}}$	$\frac{{\mathsf{\alpha }}_{\mathrm{opt}}-{\mathsf{\alpha }}_{\mathrm{bas}}}{{\mathsf{\alpha }}_{\mathrm{bas}}}$	$\frac{{\sigma }_{\mathrm{eq},\max ,\mathrm{opt}}-{\sigma }_{\mathrm{eq},\max ,\mathrm{bas}}}{{\sigma }_{\mathrm{eq},\max ,\mathrm{bas}}}$
Value	+0.69%	+0.08%	−0.04%	+0.24	+20.38%

presents a comparison of performance parameters between the initial and enhanced turbine schemes. In contrast to the initial scheme, the improved scheme exhibits notable enhancements: a 0.69% increase in efficiency for the 41st section, a 0.08% boost in total expansion ratio, a 0.04% reduction in gas flow rate, and a 0.24% increase in outlet airflow angle, aligning it closer to the axial direction. However, it is worth noting that the maximum von Mises stress on the first stage rotor blade experienced a significant 20.38% increase, nearing the boundary of the strength constraint. These results underscore the improved aerodynamic performance of the optimized turbine blade while still adhering to strength constraints.

Table 2. Comparison of 20 blade profile design variables of the first-stage rotor blade in the initial and improved scheme.

Tip Section	Value	Middle Section	Value	Root Section	Value
$\frac{R_{q1,\mathrm{opt}}-R_{q1,\mathrm{bas}}}{R_{q1,\mathrm{bas}}}$	−1.09%	$\frac{R_{q2,\mathrm{opt}}-R_{q2,\mathrm{bas}}}{R_{q2,\mathrm{bas}}}$	+23.05%	$\frac{R_{q3,\mathrm{opt}}-R_{q3,\mathrm{bas}}}{R_{q3,\mathrm{bas}}}$	−3.30%
$\frac{{\beta }_{q1,\mathrm{opt}}-{\beta }_{q1,\mathrm{bas}}}{{\beta }_{q1,\mathrm{bas}}}$	−4.50%	$\frac{{\beta }_{q2,\mathrm{opt}}-{\beta }_{q2,\mathrm{bas}}}{{\beta }_{q2,\mathrm{bas}}}$	−4.66%	$\frac{{\beta }_{q3,\mathrm{opt}}-{\beta }_{q3,\mathrm{bas}}}{{\beta }_{q3,\mathrm{bas}}}$	−7.42%
$\frac{{\beta }_{h1,\mathrm{opt}}-{\beta }_{h1,\mathrm{bas}}}{{\beta }_{h1,\mathrm{bas}}}$	+6.27%	$\frac{{\beta }_{h2,\mathrm{opt}}-{\beta }_{h2,\mathrm{bas}}}{{\beta }_{h2,\mathrm{bas}}}$	−8.72%	$\frac{{\beta }_{h3,\mathrm{opt}}-{\beta }_{h3,\mathrm{bas}}}{{\beta }_{h3,\mathrm{bas}}}$	−7.13%
$\frac{W_{q1,\mathrm{opt}}-W_{q1,\mathrm{bas}}}{W_{q1,\mathrm{bas}}}$	+7.32%	$\frac{W_{q2,\mathrm{opt}}-W_{q2,\mathrm{bas}}}{W_{q2,\mathrm{bas}}}$	−27.88%	$\frac{W_{q3,\mathrm{opt}}-W_{q3,\mathrm{bas}}}{W_{q3,\mathrm{bas}}}$	−2.88%
$\frac{{\delta }_{1,\mathrm{opt}}-{\delta }_{1,\mathrm{bas}}}{{\delta }_{1,\mathrm{bas}}}$	−3.96%	$\frac{{\delta }_{2,\mathrm{opt}}-{\delta }_{2,\mathrm{bas}}}{{\delta }_{2,\mathrm{bas}}}$	+20.03%	$\frac{{\delta }_{3,\mathrm{opt}}-{\delta }_{3,\mathrm{bas}}}{{\delta }_{3,\mathrm{bas}}}$	+10.19%
$\frac{{\gamma }_{1,\mathrm{opt}}-{\gamma }_{1,\mathrm{bas}}}{{\gamma }_{1,\mathrm{bas}}}$	+16.63%	$\frac{{\gamma }_{2,\mathrm{opt}}-{\gamma }_{2,\mathrm{bas}}}{{\gamma }_{2,\mathrm{bas}}}$	+8.61%	$\frac{{\gamma }_{3,\mathrm{opt}}-{\gamma }_{3,\mathrm{bas}}}{{\gamma }_{3,\mathrm{bas}}}$	−1.19%
$\frac{S_{1,\mathrm{opt}}-S_{1,\mathrm{bas}}}{S_{1,\mathrm{bas}}}$	+13.25%	$\frac{S_{2,\mathrm{opt}}-S_{2,\mathrm{bas}}}{S_{2,\mathrm{bas}}}$	+12.23%	---	---

compares the 20 design variables of the first-stage rotor blade between the initial and enhanced schemes of the high-pressure turbine. Significant changes are evident in the blade design variables. Specifically, in the middle section, the leading edge radius $R_{q2}$ increased by 23.05%, while the inlet wedge angle $W_{q2}$ decreased by 27.88%. Additionally, the turning angle ${\delta }_2$ of the middle section increased by 20.03%, the turning angle ${\delta }_3$ of the root section increased by 10.19%, the blade stagger angle ${\gamma }_1$ of the tip section rose by 16.63%, the axial chord length $S_1$ of the tip section increased by 13.25%, and the axial chord length $S_2$ of the middle section increased by 12.23%.

After optimization, the axial chord length of the blade root section remains unchanged, with minimal alteration to the blade root profile. However, there are substantial variations in the axial chord length of the middle and tip sections, leading to notable changes in the tip and middle sections, as depicted in Figure 12.

Figure 13 contrasts the three-dimensional models of the first-stage rotor blade between the initial and enhanced schemes. After optimization, there is an amplified discrepancy in installation angles between the blade tip, middle, and root sections, accompanied by significant blade body twisting. Moreover, except for the root section, the axial length of the blade is notably extended.

4.2. Detail Analysis of Disciplines

Figure 14 presents a comparison of the von Mises stress distribution of the first stage rotor blade between the initial and enhanced schemes. Compared to the initial design, the improved blade exhibits an increased maximum von Mises stress, with its location shifting from the root extension near the trailing edge of the blade pressure surface to the root of the blade pressure surface. The underlying cause of this change is the expansion of the cross-sectional area of the blade above the root post-optimization, whereas the cross-sectional area of the blade root remains unaltered. Consequently, there is a higher mass situated above the blade root, resulting in amplified centrifugal force and subsequently elevated stress levels at the blade root.

Table 3. The energy loss coefficient of each row blade before and after optimization.

Blade	First-Stage Stator Blade	First-Stage Rotor Blade	Second-Stage Stator Blade	Second-Stage Rotor Blade
Initial	0.0593	0.1221	0.0709	0.0690
Optimized	0.0595	0.1043	0.0689	0.0685
Rate	0.42%	−14.57%	−2.71%	−0.78%

presents the energy loss coefficients for each row of the blade in both the initial and optimized schemes. The data indicate that the energy loss for the first-stage stator blade experienced a slight increase, whereas the energy loss for the subsequent blade rows decreased. Notably, the energy loss for the first-stage rotor blade saw the most significant reduction, with a decrease of 14.57%. Consequently, the subsequent discussion will concentrate on analyzing the flow field of the first-stage rotor blade.

Figure 15 illustrates the spanwise distribution of the energy loss coefficient for the first-stage rotor blade in both the initial and enhanced schemes. Comparing the two plans, it is evident that the enhanced scheme shows minimal changes in losses in areas below 50% blade height, slightly increased losses in areas between 50% and 58% blade height, and notably reduced losses in areas above 58% blade height.

Figure 16 presents the entropy distribution at the outlet of the first-stage rotor blade in the initial and enhanced schemes. The following observations can be made: (1) the improved scheme significantly reduces the area of high energy losses; (2) consistent with the results in Figure 15, losses remain relatively unchanged below 50% blade height, while there is a significant decrease in losses above 58% blade height; and (3) the improved plan slightly shifts the high-loss areas downwards, resulting in a slight increase in losses between 50% and 58% of the blade height.

Figure 17, Figure 18 and Figure 19 display the relative Mach number distribution at the 10%, 55%, and 90% blade height cross-sections for the first-stage rotor blade in the initial and improved schemes. Key observations include the following: (1) the improved scheme reduces the highest Mach number at each blade height section; (2) the initial scheme exhibits a large low-speed zone at the trailing edge of the suction surface at the 90% blade height section, which is eliminated in the improved scheme; and (3) at the 55% blade height section, there is a slight increase in the low-speed zone at the trailing edge in the improved scheme.

Figure 20 depicts the velocity vector distribution at the 90% blade height section of the first-stage rotor blade in both the initial and improved schemes. Notably, the initial scheme shows significant flow separation at the trailing edge of the suction surface, while the improved scheme eliminates this flow separation.

Figure 21 compares the static pressure distribution on the surface of the first stage rotor blade at the 10%, 55%, and 90% blade height cross sections between the initial and improved schemes. The findings show that the surface static pressure distribution in the improved scheme is more reasonable, with a significant reduction in the reverse pressure gradient on the suction surface. In particular, at the 90% blade height section, the improved scheme remarkably enhances flow acceleration on the blade surface. This indicates that optimizing the blade profile can adjust the pressure gradient distribution on the blade surface, thereby weakening or even eliminating the flow separation on the suction surface.

In summary, the improvement in turbine aerodynamic performance can be attributed to the optimization of the first-stage rotor blade profile, which adjusts the surface load distribution. This adjustment effectively eliminates flow separation near the tip of the first-stage rotor blade’s suction surface, thereby reducing energy loss.

5. Conclusions

This study employs a regression/interpolation combination surrogate model for a multidisciplinary design optimization of a turbine blade, offering an efficient approach for optimizing the cooling turbine blade. The methodology integrates collaborative coupling feature parameterization modeling for the cooling turbine blade, simulation analyses focusing on aerodynamics and structural strength, and a regression/interpolation combination surrogate model. By optimizing the shape of the cooling turbine blade while considering various disciplinary constraints, the study achieves improved aerodynamic efficiency within the turbine stage. The main findings of this research are as follows:

• The optimized turbine blade exhibits enhanced aerodynamic performance while maintaining a relatively constant total expansion ratio and gas flow rate under strength constraints. Additionally, the 41-section efficiency improves by 0.69%.
• The increased chord length at the tip of the improved first-stage rotor blade effectively improves flow conditions in over 58% of the blade height region, leading to a reduction in low-speed flow area on the blade suction side.
• The optimization of the first-stage rotor blade profile adjusts the load distribution on the blade surface, which helps eliminate flow separation at the trailing edge of the suction side near the blade tip, consequently reducing energy loss.

These findings highlight the significant potential and prospects of the innovative regression/interpolation combination surrogate model in addressing challenging issues across various practical engineering applications. For future research, it is advisable to integrate additional disciplines such as vibration, heat transfer, and lifespan based on the insights gained from this study. This expansion will further enrich the multidisciplinary analysis model and fully exploit the advantages of the regression/interpolation combination surrogate model-based multidisciplinary design optimization approach for turbine blades in practical engineering applications.