Integrated Surrogate Model-Based Approach for Aerodynamic Design Optimization of Three-Stage Axial Compressor in Gas Turbine Applications

Abstract

The refined aerodynamic design optimization of multistage compressors is a typical high-dimensional and expensive optimization problem. This study proposes an integrated surrogate model-assisted evolutionary algorithm combined with a Directly Manipulated Free-Form Deformation (DFFD)-based parametric dimensionality reduction method, establishing a high-precision and efficient global parallel aerodynamic optimization platform for multistage axial compressors. The DFFD method achieves a balance between flexibility and low-dimensional characteristics by directly controlling the surface points of blades, which demonstrates a particular suitability for the aerodynamic design optimization of multistage axial compressors. The integrated surrogate model enhances prediction accuracy by simultaneously identifying optimal solutions and the most uncertain solutions, effectively addressing highly nonlinear design space challenges. A three-stage axial compressor in a heavy-duty gas turbine is selected as the optimization object. The results demonstrate that the optimization task takes less than 48 h and achieves an improvement of 0.6% and 4% in the adiabatic efficiency and surge margin, respectively, while maintaining a nearly unchanged flow rate and pressure ratio at the design point. The proposed approach provides an efficient and reliable solution for complex aerodynamic optimization problems.

1. Introduction

Aerodynamic design optimization of multistage axial compressors faces significant challenges, including high dimensionality, computational expense, and black-box behavior. Conventional optimization approaches often fail to achieve globally optimal solutions within practical engineering timeframes due to the curse of dimensionality [1,2,3].

To address the curse of dimensionality in multistage axial compressors, several strategies have been proposed. The first approach is dimensionality reduction through decreased design variables. A prototypical method involves blade shape parametrization; Wang et al. [4,5] applied Bézier surfaces to reduce dimensionality for semi-span and full-span blades by imposing intrinsic constraints on chordwise and circumferential control points. This ensures surface smoothness while effectively lowering dimensionality. Other researchers [6,7,8] employed the more flexible Free-Form Deformation (FFD) technique for aerodynamic redesign of multistage compressors. However, FFD’s typically large number of control points limits its dimensionality reduction effectiveness. Consequently, Huang et al. [9] developed Directly Manipulated Free-Form Deformation (DFFD) for enhanced dimensionality reduction, demonstrating promising outcomes in nine-stage compressor optimization. Beyond novel parametrization, machine learning techniques have recently addressed dimensionality challenges. Pretsch et al. [10] proposed a two-stage reduction method combining Principal Component Analysis (PCA) and Trust Regions (TRs), significantly improving high-dimensional compressor blade optimization efficiency. Bird et al. [11] found nonlinear techniques (Kernel PCA, Local Linear Embedding) superior to linear methods for complex compressor aerodynamic optimization spaces. Huang et al. [12] integrated PCA with improved Multi-Objective Particle Swarm Optimization (MOPSO) for wide-body aircraft multipoint optimization. Guo et al. [13] introduced an Efficient Sparse Surrogate Model (ESSM), fusing PCA with Gaussian processes to manage compressor blade design uncertainty while reducing variable dimensionality.

A second approach to mitigating the curse of dimensionality in multistage axial compressors employs surrogate models to replace computationally expensive CFD evaluations. Traditional surrogate modeling techniques [14,15,16] require high-fidelity approximations that typically demand large sample sizes. As design variable counts increase in multistage compressor optimization, the number of samples needed to construct sufficiently accurate surrogates grows exponentially. Surrogate-assisted evolutionary algorithms address this limitation by replacing costly CFD evaluations without requiring high-precision surrogate models, achieving superior optimization outcomes with fewer CFD evaluations. Lu et al. [17,18] investigated adaptive region-segmentation surrogate models using Support Vector Regression (SVR) and Kriging for transonic compressors, approximating geometric–performance relationships through varied sampling strategies. Ji et al. [19] developed a Multi-Task Particle Swarm Optimization algorithm incorporating surrogate models and autoencoders to decompose high-dimensional problems into low-dimensional sub-tasks, augmented by a bimodal local development strategy to enhance sub-task exploitation. This method demonstrates significant advantages in compressor blade optimization over conventional surrogate-assisted approaches. Zhou et al. [20] implemented Kriging-assisted evolutionary optimization for 3D compressor blades, effectively mitigating shock–boundary layer interactions, suppressing supersonic rotor radial migration, and reducing tip losses to enhance design-point performance. However, existing methods fail to fully leverage the unique nonlinear characteristics of diverse surrogate models, limiting exploration capability and predictive accuracy during model construction. Ensemble surrogate modeling addresses this limitation.

The concept of ensemble surrogate models can be applied to solve this problem. In the selection of ensemble surrogate models, Kriging models, quadratic polynomial models, and Radial Basis Function (RBF) models have attracted significant attention due to their respective characteristics and widespread applications in engineering fields. The most prominent advantage of the Kriging model lies in its high prediction accuracy and its ability to provide estimates of prediction error (i.e., prediction variance or confidence intervals) [21,22]. This capability to quantify prediction uncertainty is crucial for assessing the reliability of design schemes and guiding the selection of subsequent sampling points (such as the Expected Improvement criterion in active learning strategies), especially in the field of aero-engine blade design where extremely high precision and reliability are required [23]. By integrating with adaptive sampling strategies (like active learning), the Kriging model can build high-precision models with relatively few sample points, significantly reducing the number of expensive function evaluations and thereby lowering computational costs [22,24]. The RBF model excels particularly in handling high-dimensional and highly nonlinear problems, capable of performing precise interpolation or approximation of scattered (non-gridded) datasets [22,25]. This is highly useful for unstructured design-point sets obtained through CFD simulations in blade design. RBF models have been successfully applied to various engineering design optimization problems, such as centrifugal pump design, small rotorcraft UAV design, and transient control law design optimization for aero-engines [26,27]. In multi-objective optimization problems, the RBF model can effectively approximate complex nonlinear relationships, supporting the search for the Pareto front [22]. The quadratic polynomial model is a classic and simple form within the Response Surface Methodology (RSM), approximating the relationship between inputs and outputs by fitting a second-order polynomial function [28]. The quadratic polynomial model has a simple structure, is easy to understand and implement, and has relatively low computational costs for construction and evaluation. This gives it an advantage during the preliminary exploration of the design space, for rapid approximation of local regions, or when serving as a basis function for more complex models. In robust optimization, second-order polynomial RSM is often used to estimate the mean and variance of the response around an optimal design [29]. Integrating the characteristics of the aforementioned three surrogate models with the specific challenges of multistage axial compressor blade design clearly reveals their respective suitability and complementarity. Addressing high computational cost and complex nonlinearity: The high computational cost and strong nonlinear characteristics of blade design necessitate high-precision surrogate models. Kriging and RBF models, due to their excellent nonlinear fitting capabilities and high prediction accuracy, are highly suitable for this problem [30]. They can effectively replace expensive CFD simulations, accelerating the optimization process while precisely capturing the subtle effects of blade geometric variations on aerodynamic performance [23,30]. A three-dimensional blade shape optimization study for a transonic axial compressor rotor successfully achieved the global optimal design and significantly improved aerodynamic performance by introducing a Kriging surrogate model and a sequential sampling strategy. Multi-objective optimization and global search: Blade design is typically a multi-objective optimization problem aiming to simultaneously improve efficiency, the pressure ratio, and the stability margin. Kriging and RBF models can effectively construct multi-objective response surfaces and, combined with global optimization algorithms like evolutionary algorithms (e.g., NSGA-II), assist in finding the Pareto-optimal solution set. The prediction variance provided by the Kriging model can also serve as a basis for exploring unknown regions, helping the algorithm escape local optima and enhancing global search capability [31]. Although the quadratic polynomial model has limited global accuracy, its high computational efficiency gives it application value in the preliminary exploration stage of design, useful for quickly screening design variables or identifying broad design trends [27]. Within certain optimization frameworks, it can also serve as the global trend term (basis function) in a Kriging model, or as a low-fidelity model in a multi-fidelity model, complementing other high-precision models.

Leveraging parametric dimensionality reduction and ensemble surrogate model-assisted evolutionary algorithms, this study proposes an optimization framework integrating DFFD-based three-dimensional blade parametrization with ensemble surrogate-assisted evolutionary computation. The method is applied to conduct high-fidelity optimization of blade and flow-path components in a three-stage gas turbine axial compressor to demonstrate its enhanced effectiveness and reliability.

2. Optimization Methodology

2.1. Dimensionality Reduction Method via DFFD Parameterization

The DFFD parameterization method builds upon the FFD approach by incorporating inverse operations, allowing for the manipulation of blade surface points to directly influence the movement of control frame vertices, thereby enabling overall deformation of the blade surface. Figure 1 illustrates the principle of the DFFD parameterization method. As depicted in Figure 1, the DFFD principle is as follows: First, establish a control frame around the peripheral points on the blade surface. Second, normalize both the original blade surface points and the frame points, ensuring that each surface point falls within the [0, 1] range. Third, select several blade surface points as design variables for the optimization process. Fourth, sequentially move the control points on the blade surface. Fifth, calculate the offset of all vertices of the control frame to obtain a new control frame. Sixth, based on the new control frame and the FFD parameterization mapping relationship, derive the altered blade surface points.

The FFD parametric mapping equation in Step 6 is as follows:

$$Q_{i,j,k}={\sum }_{i=0}^n{\sum }_{j=0}^m{\sum }_{k=0}^l{P}_{i,j,k}{B}_{i,n}\left(u\right)B_{j,m}\left(v\right)B_{b,k}\left(w\right)$$

(1)

where $Q_{i,j,k}$ represents the coordinates of the actual blade surface points, and $\left(u,v,w\right)$ are the unitized computational domain local coordinates. $l,m,n$ are the fractions of the control frame divided along the three directions, and $P_{i,j,k}$ represents the coordinates of the vertices of the control frame. $B_{i,n}$ is the Bernstein basis function with the following equation:

$$B_{i,n}\left(u\right)={\displaystyle \frac{n!}{n!\left(n-i\right)!}}{u}^i{(1-u)}^{n-i}$$

(2)

The variation in the blade surface points can be expressed in the following matrix form:

$$∆Q=B∆P$$

(3)

where $∆P$ is the offset of the control frame vertices and $∆Q$ is the offset of the blade surface points.

The DFFD parameterization method needs to invert the offset of the control frame vertices by the offset of the blade surface points, so the above Equation (3) is transformed into the following Equation (4):

$$∆P=B^{+}∆Q$$

(4)

where $B^{+}$ is the generalized inverse matrix of $B$. When $B$ is a one-row non-zero matrix, $B^{+}$ is calculated as in Equation (5):

$$B^{+}={\displaystyle \frac{1}{{‖B‖}^2}}{B}^T$$

(5)

The advantage of the DFFD parameterization method is its capability to directly and precisely control the displacement of points on the blade surface. This direct control over blade surface modification allows for the incorporation of design expertise into the optimization process, significantly reducing the number of control variables and achieving a dimensional reduction in the parameterization of the optimization process.

2.2. Integrated Surrogate Model-Assisted Evolutionary Algorithm

The core of the integrated surrogate model-assisted evolutionary algorithm employed in this paper is the differential evolution (DE) algorithm. DE is a population-based optimization algorithm that steers the search direction through the exploitation of differences among individuals. It generates new individuals via crossover, mutation, and selection operations and progresses towards the optimal solution by iteratively selecting fitter individuals into the next generation based on fitness evaluation. For a detailed description of the differential evolution algorithm, readers are referred to refs. [32,33].

Figure 2 depicts the schematic of the integrated surrogate model-assisted evolutionary algorithm. The core principles are outlined below:

Step 1: The initial population is generated using the optimal Latin hypercube sampling method to maximize the exploration of the design space. The true values are evaluated via CFD simulations and subsequently stored in the sample library.

Step 2: The samples from the library are utilized to construct the initial generation of the integrated surrogate model.

Step 3: The algorithm enters the main structure of the differential evolution algorithm, optimizing for both maximum uncertainty and best fitness values. The top 50% of the population from each optimization goal are selected to form a new population.

Step 4: The new population undergoes parallel CFD evaluations and is then added to the sample library.

Step 5: The integrated surrogate model is updated with the new sample library, and the process re-enters the loop.

Step 6: The loop terminates when the number of CFD evaluations reaches the predefined limit, at which point the optimal solution from the sample library is output.

The integrated surrogate models in Steps 2 and 5 are the heart of the algorithm, comprising a weighted-sum-based surrogate model and a committee-based active learning surrogate model [34]. Both integrated surrogate models include three sub-models: the Kriging model [34], the quadratic polynomial model [35], and the RBF model [36]. The weighted-sum-based surrogate model is used to locate the optimal solution within the design space, while the committee-based active learning surrogate model is employed to identify the most uncertain solutions in the design space. Refs. [34,37] suggests that incorporating the most uncertain solutions into the sample library during optimization signifies areas where no sub-model can accurately predict, and thus, actual evaluations in these regions enhance the diversity of the sample library, thereby improving the accuracy of the surrogate models.

The weighted-summation-based surrogate model is constructed as follows:

$${\widehat{f}}_{sum}\left(\mathit{x}\right)={w_1\widehat{f}}_1+{w_2\widehat{f}}_2{+w_3\widehat{f}}_3$$

(6)

where ${\widehat{f}}_{sum}$ refers to the surrogate model based on weighted summation, ${\widehat{f}}_i$ is the i-th output of each number, 1 ≤ i ≤ 3, and $w_i$ is the weight of the i-th output defined by the following equation:

$$w_i=0.5-{\displaystyle \frac{e_i}{2(e_1+e_2+e_3)}}$$

(7)

Here, $e_i$ denotes the root mean square error of the i-th model. This approach accounts for the impact of model accuracy on optimization performance, with higher precision models receiving greater weights, while simultaneously satisfying the requirement that the weights are constrained to sum to unity. Further details can be found in ref. [27].

When the integrated surrogate model based on weighted summation is constructed, the differential evolutionary algorithm performs optimization with the objective of minimizing the fitness value of this integrated surrogate model, and this optimization task can be expressed as follows:

$$x_{\mathrm{s}\mathrm{u}\mathrm{m}}=arg \underset{\mathit{x}}{\min } {\widehat{f}}_{\mathrm{s}\mathrm{u}\mathrm{m}}(x,D_t)$$

(8)

where x sum refers to the solution with minimum fitness, and D_t refers to the subset selected from the sample pool. It is important to note that the process of finding the minimum fitness value retains the ${\displaystyle \frac{1}{2}}\mathit{N}$ optimal solutions as half of the new population. Here, N is the population scope.

Committee-based active querying [28,38] is a machine learning strategy that constructs a “committee” of diverse models to make predictions. These models work in concert, with their level of disagreement in predictions analyzed to identify the most uncertain samples in the dataset. These samples are then selected for real evaluation to ensure the optimal use of labeling resources, thereby enhancing the training efficiency and predictive performance of the models. In the method proposed in this paper, this active learning strategy is primarily used to identify the points with the greatest prediction disagreement among the three sub-surrogate models, ${\widehat{f}}_1, {\widehat{f}}_2,{\widehat{f}}_3$. This task is inherently an optimization problem and can be formulated as follows:

$$x_u=\arg \underset{\mathit{x}}{\max } U_{\mathrm{e}\mathrm{n}\mathrm{s}}\left(\mathit{x}\right)$$

(9)

where $x_u$ represents the solution with the maximum uncertainty, and $U_{ens}$ measures the disagreement (i.e., the difference in fitness values) among the predictions made by different sub-surrogate models for $\mathit{x}$. The goal is to find the x that maximizes $U_{ens}$, which is considered the most uncertain solution. It is important to note that during the process of finding the most uncertain solutions, the top $50\%\ast N$ optimal solutions are retained to constitute the other half of the new population.

$U_{\mathrm{e}\mathrm{n}\mathrm{s}}$ can be expressed by Equation (10):

$$U_{\mathrm{e}\mathrm{n}\mathrm{s}}\left(\mathit{x}\right)=\mathrm{m}\mathrm{a}\mathrm{x}({\widehat{f}}_i\left(\mathit{x},D_t\right)-{\widehat{f}}_j\left(\mathit{x},D_t\right))$$

(10)

Equation (10) indicates that the prediction uncertainty is measured by the maximum difference between the outputs of every pair of sub-surrogate models, where 1 ≤ i, j ≤ 3, as determined by the specific numerical values of the surrogate models.

The new population, obtained through optimization by two integrated surrogate models (each contributing 10% of the individuals), is subjected to real-value evaluation using CFD. The solutions obtained are used to update the sample library, thereby refining the surrogate models, and the process enters the next iteration.

The method proposed in this paper calculates multiple new solutions in each iteration without a significant increase in the time per iteration, primarily due to the adoption of a multi-task concurrent processing mechanism based on supercomputing. For details on this aspect, refer to ref. [39].

In each iteration, N new samples are added to the sample library, and as the number of samples increases, the cost of training the surrogate models also rises sharply. To reduce the training cost of the integrated surrogate models, the diversity-induced dissimilarity measurement method proposed in ref. [34] is employed. This involves selecting the most distinct 5d sample points from the total sample library D to train the surrogate models, which are then stored in the sample library $D_t$.

3. Aerodynamic Design Optimization of a Three-Stage Axial Flow Compressor

3.1. Optimization Object

The optimization target is a three-stage axial compressor for a land-based gas turbine, designed independently. The geometric model of this compressor is shown in Figure 3. It comprises a total of eight blade rows, including three stages of working blades (each stage consisting of one row of rotor blades and one row of stator vanes), an inlet guide vane (IGV, S0), and an outlet guide vane (OGV, S4). The blade counts for each row are detailed in

Table 1. Number of blades in each row of a three-stage axial compressor.

Blade row	S0	R1	S1	R2	S2	R3	S3	S4
Blade number	40	22	34	29	40	39	52	50

. The design rotational speed is 9000 rpm. The optimization objectives encompass the meridional flow-path contour and the blade airfoil shapes.

To significantly enhance optimization efficiency, an axisymmetric single-passage sector model (1/N periodic interval model) was employed in the subsequent numerical simulations and optimization process to represent this three-stage axial compressor. Periodic boundary conditions were applied rigorously across each blade row (S0, R1, S1, R2, S2, R3, S3, S4) within this model to replicate the full-annulus flow characteristics. This approach enables a reduction in computational cost by several orders of magnitude while maintaining adequate solution accuracy, forming a critical foundation for achieving efficient global optimization. Furthermore, the proposed Direct Free-Form Deformation (DFFD) parameterization method demonstrates excellent compatibility with this single-passage model, ensuring both geometric deformation flexibility and consistency across flow passages.

3.2. Numerical Method

CFD calculations for the optimization method are conducted using NUMECA-V8.9 software. The grid generation is handled by the IGG/Autogrid5 module, and the flow-field calculations are performed using the Fine_Turbo module. For the grid generation, both the blades and the flow passages are discretized with O4H-type grids. The specific grid information is shown in

Table 2. Mesh information.

Near-Wall Mesh Thickness	0.001 mm
Number of Grids	6,100,000
Minimum Orthogonal Angle	16°
Maximum Aspect Ratio	3.7

, which meets the requirement of y⁺ ≤ 5, and the tip clearances for all three stages of moving blades are set to 0.5 mm.

To ensure the accuracy of the numerical simulation, a grid independence study is conducted.

Table 3. Grid independence verification.

Number of Grids	Flow Rate (kg/s)	Efficiency (%)
6,100,000	102.3	88.12
10,200,000	102.5	88.23
14,400,000	102.4	89.31
18,600,000	102.4	89.32

presents a comparison of the design-point flow rate and efficiency at different grid counts. It can be observed that when the total number of grids reaches 14.4 million, the aerodynamic performance of the numerical simulation remains almost unchanged, indicating that the grid independence requirement is met. To further reduce the optimization time, the optimization process follows the strategy of “coarse grid optimization, fine grid verification”. The detailed mesh specifications are illustrated in Figure 4. The topology of the B2B face mesh is shown in Figure 4a, where the chordwise mesh density along the blade section is adaptively adjustable. Figure 4b,c display detailed features of the hub surface flow path and B2B face mesh, respectively.

For the flow-field calculations, the boundary conditions are set as follows: the inlet total temperature is 288.15 K, the inlet total pressure is 100,065 Pa, the inlet flow direction is tangential, and the rotational speed is 9000 rpm. The turbulence model used is the S-A model, time discretization is achieved with the fourth-order Runge–Kutta method, and spatial discretization is performed using the finite volume central difference scheme. The convergence is accelerated by employing multigrid convergence and implicit residual convergence techniques. Calculations are performed starting from the design point in two directions, reaching the choke point and the surge point, to obtain the complete compressor performance curve.

3.3. Construction of the Optimization Framework

3.3.1. DFFD Parameterization Setup

The primary advantage of employing the DFFD parametric method lies in its capability to directly manipulate blade surface points as design variables, which establishes a foundation for further reducing the number of variables in the optimization process. Based on preliminary flow-field analysis, the geometry of the rotor blades significantly influences the overall flow field. Consequently, this study selects three rows of rotor blades as the direct optimization targets. For each row, three sections (hub, mid-span, and tip) are chosen, with four control points distributed across each section, as illustrated in Figure 5a. Among these, two points are positioned at the leading and trailing edges, with their movement restricted to the axial direction. The remaining two points are located at the suction side (1/3 chord) and pressure side (2/3 chord), denoted by green dots, with their movement confined to the circumferential direction. This configuration enables a broad design space with a minimal number of control points while significantly reducing ineffective design regions. In total, 36 optimization variables are defined for the three rows of rotor blades, with each variable constrained to a range of [−5 mm, 5 mm].

Due to constraints on modifying the geometry of the outlet guide vanes, the optimization is conducted solely on the first seven rows of blades. As shown in Figure 5b, the DFFD (Direct Free-Form Deformation) control framework constructed for these seven rows employs a regularized cubic lattice structure. This framework incorporates a precise grid discretization along the blade’s key geometric directions: 40 equally spaced divisions are set along the x-axis (representing the axial direction), 10 divisions along the y-axis (representing the spanwise direction, i.e., blade height), and 15 divisions along the z-axis (representing the chordwise direction, perpendicular to blade height and axial flow, typically associated with thickness or stacking direction). This partitioning generates a three-dimensional cubic lattice framework comprising 40 × 10 × 15 = 6000 densely packed vertices. The primary rationale for selecting this cubic framework structure lies in its intuitive and efficient construction—parametric definition and generation of regular grids are straightforward. More critically, this high-density vertex distribution constitutes the core mechanism enabling the DFFD method to achieve smooth and continuous geometric deformations of the blade. Each framework vertex essentially functions as a control point, whose displacement defines a spatial deformation field via the embedded trivariate tensor-product B-spline volume. When the optimization algorithm adjusts the positions of these control points, this deformation field can be accurately and smoothly mapped onto the blade surface geometry. The high-density segmentation along the x-axis (axial direction, 40 divisions) ensures precise control over leading-edge and trailing-edge curvature variations, as well as bend–twist distributions along the span; the 10 divisions along the y-axis (spanwise direction) balance computational efficiency with the requirement to capture spanwise load distributions; while the fine segmentation along the z-axis (chordwise direction, 15 divisions) is crucial for accurately describing the complex aerodynamic profiles of the pressure and suction surfaces, maintaining surface continuity, and capturing subtle features such as secondary flows. Consequently, this specific-resolution cubic lattice framework configuration, while ensuring computational manageability, establishes a robust foundation for achieving high-fidelity, smooth, and controllable parametric optimization of blade geometry via DFFD.

It is noteworthy that although the direct control points are selected only on the rotor blade surfaces, the DFFD method inherently modifies the overall control framework during intermediate steps. As a result, the optimization process affects the shape of all seven rows of blades. This characteristic is one of the key reasons why the DFFD method achieves effective parametric dimensionality reduction.

To further enhance the design flexibility of the compressor, in addition to optimizing the three rows of rotor blades, the flow-path parameters were also optimized. As shown in Figure 6, there are a total of 26 control points on the casing and hub flow-path lines. Among these, 14 control points within the red dashed box are selected as optimization design variables, with 7 points each on the casing and hub (in red dashed box). These control points are allowed to move only in the radial direction, with a variation range of [−10 mm, 10 mm], resulting in a total of 14 optimization variables. The primary reason for focusing on this region is the presence of a spoon-shaped flow path, which has a significant impact on performance. Therefore, the three-stage axial compressor optimization has a total of 12 $\times$ 3 + 14 = 50 design variables.

3.3.2. Optimization Process and Optimization Objective

Figure 7 illustrates the aerodynamic optimization process for a three-stage axial compressor based on an integrated surrogate model. The process begins with the parameterization of the original three-dimensional compressor geometry, where the core step involves applying the DFFD parametric method to achieve dimensionality reduction. Subsequently, mesh generation and three-dimensional CFD flow-field calculations are performed to evaluate the compressor’s aerodynamic performance. If the exit criteria are met at this stage, the optimized compressor geometry is obtained. If not, the process proceeds to an evolutionary algorithm assisted by the integrated surrogate model to identify several optimal solutions, which are then carried forward to the next iteration.

The optimization algorithm selected is the differential evolution algorithm, with a population size of 100, 50 generations, a recombination probability of 0.7, and 20 CFD evaluations required per iteration. It is important to note that due to the parallel computing module within the algorithm, although 20 CFD evaluations are performed in each iteration, the associated computational time does not increase significantly. The exit criterion for the overall optimization process is set at a total of 1000 CFD evaluations.

The optimization objective is to maximize the aerodynamic efficiency of the three-stage axial compressor without imposing any constraints. The formula for isentropic efficiency ${\eta }_c$ is as follows:

$${\eta }_c={\displaystyle \frac{{\pi }^{(\gamma -1)/\gamma }-1}{T_2/T_1}}$$

(11)

where $\pi$ refers to the total pressure ratio between the outlet and the inlet, γ refers to the specific heat ratio, and $T_2$ and $T_1$ refer to the total temperatures at the outlet and the inlet, respectively. The calculation of the total pressure ratio $\pi$ is given by Equation (12).

$$\pi ={\displaystyle \frac{p_2}{p_1}}$$

(12)

Since the optimization algorithm seeks to minimize the objective function, the efficiency values are converted into their negative counterparts for the optimization process. The computational resources utilized include an AMD EPYC 9654P 96-core processor (Jutai Numerical Simulation Computing Platform, Xi’an, China). With the support of a distributed parallel computing strategy, the optimized geometry can be obtained within 48 h.

4. Optimization Results and Analysis

4.1. Comparative Analysis of Aerodynamic Performance

Figure 8 illustrates the comparison of the compressor’s aerodynamic performance before and after optimization. As shown in Figure 8a, the pressure ratio at the design point remains nearly unchanged, while the mass flow rate increases significantly. The entire mass flow–pressure ratio curve shifts to the right, indicating a substantial improvement in the compressor’s flow capacity after optimization. From Figure 8b, it is evident that the entire mass flow–efficiency curve shifts upward, suggesting enhanced efficiency across nearly the entire operating range. Although the optimized flow range is slightly narrower than before optimization, the mass flow rate near the surge point increases significantly, demonstrating superior flow performance near the surge point compared to the pre-optimized condition.

It is important to note that the optimization target in this study is the compressor of an actual ground-based gas turbine. To protect intellectual property, the mass flow rates have been normalized. The normalization is achieved by dividing the mass flow rate at each point by the maximum mass flow rate.

Table 4. Performance comparison at design point before and after optimization.

Performance Comparison	Relative Flow Rate	Total Pressure Ratio	Adiabatic Efficiency	Surge Margin
Before Optimization	0.982	3.28	88.12%	12%
After Optimization	0.992	3.29	88.72%	16%
Difference	+0.01	+0.01	+0.6%	+4%

compares the aerodynamic performance of the three-stage axial compressor at the design point before and after optimization. As shown in the table, the optimized design exhibits a 0.6% improvement in efficiency and a 4% enhancement in the surge margin.

4.2. Comparative Analysis of Geometries

Figure 9 illustrates a comparison of the flow-path geometry before and after optimzation. It can be observed that the flow-path geometry in the spoon-shaped region has shifted upward overall, with a particularly noticeable upward shift at the hub and a relatively smaller shift at the casing. This indicates that the flow area of the optimized flow path is smaller than that of the pre-optimized design.

Figure 10 illustrates the comparison of three-dimensional blade geometry before and after optimization, showing that all rows of blades exhibit full three-dimensional deformation after optimization. From the geometric comparison of the hub, mid-span, and tip sections of each row of blades in Figure 11, it can be observed that the optimized inlet guide vanes exhibit significant changes in the exit construction angles at the hub and tip, a reduction in the curvature of the mid-span section (S1), a notable decrease in the chord lengths of all S2 sections, and an increase in the curvature of the mid-span and hub sections of S3. For the rotor blades, R1 shows significant changes in the exit angle of the mid-span section and a reduction in curvature, R2 exhibits a decrease in the inlet angle at the tip section and reduced curvature, with significant changes in the stagger angle of the mid-span section, and R3 demonstrates noticeable changes in curvature at the tip and mid-span sections. Notably, the maximum thickness of the rotor blades remains largely unchanged after optimization, ensuring the structural strength of the blades is maintained.

4.3. Flow-Field Comparison Analysis at the Design Point

Figure 12 illustrates the comparison of meridional entropy distributions before and after optimization. It can be observed that, due to the upward shift of the spoon-shaped flow path at the hub (as shown in Figure 9), the axial velocity in the corresponding region increases, resulting in smoother airflow. This leads to a significant expansion of the low-entropy region on the meridional plane after optimization.

Figure 13 and Figure 14 present the comparisons of Mach number contours and static pressure distributions at the hub, mid-span, and tip sections before and after optimization, respectively. By analyzing these results in conjunction with the changes in blade geometry, the flow-field variations at each section can be further understood.

From Figure 13 and Figure 14, it is evident that for R1, the pre-shock Mach numbers at the mid-span and tip sections are significantly reduced, leading to a notable decrease in shock losses. Additionally, due to the reduced curvature of the optimized R1 mid-span section, the adverse pressure gradient behind the shock is weakened, resulting in a substantial reduction in flow separation losses near the trailing edge. As shown in Figure 13b,c, the optimized R2 tip section exhibits a reduced inlet angle, which decreases the positive incidence angle. This weakens the acceleration of the flow on the suction side, lowers the pre-shock Mach number, and consequently reduces the shock intensity and associated losses. For R3, the reduced curvature at the hub section after optimization significantly weakens the shock strength on the suction side, thereby reducing shock losses in this region.

For the stator blades, the S1 hub section shows a slight reduction in the inlet geometric angle, which reduces the supersonic region near the leading edge of the suction side and correspondingly decreases shock losses. As shown in Figure 14c, the optimized S1 tip section exhibits a reduced positive incidence angle, leading to more reasonable flow acceleration on the suction side. Furthermore, the longer chord length of the optimized S1 reduces the adverse pressure gradient at the trailing edge, thereby decreasing the flow separation region and associated losses.

Figure 15 illustrates the comparison of limiting streamlines on the suction surfaces of the blades before and after optimization. It can be observed that the limiting streamlines on the optimized R1 suction surface exhibit a significant rearward shift, and the distance between flow separation and reattachment in the upper-middle portion of the limiting streamlines is reduced. This indicates a substantial decrease in flow separation losses on the optimized R1 surface. For the optimized R2, the recirculation near the hub trailing edge is eliminated, while for the optimized R3, the recirculation region near the hub trailing edge is reduced, thereby decreasing the associated recirculation losses. Additionally, it is noteworthy that due to changes in the flow-path geometry and the stator blade geometry itself, the limiting streamlines near the hub of the S1 and S2 suction surfaces become smoother, reducing radial flow. This aligns with the conclusion drawn from Figure 11, which highlights an increase in axial flow velocity near the hub region.

In summary, the optimized rotor blades improve the flow conditions on the suction surface, reduce the shock intensity at the tip, and thereby decrease both flow separation losses and shock losses. The optimized stator blades primarily achieve performance improvements by modifying the inlet geometric angles, which optimizes the flow incidence angles and reduces flow separation losses on the suction surface.

5. Conclusions

An integrated surrogate model-assisted evolutionary algorithm is proposed, which leverages three sub-surrogate models to predict both the optimal solutions and the most uncertain solutions. This approach is integrated into the overall optimization process and combined with the DFFD dimensionality reduction parametric method to construct a high-precision and efficient global parallel aerodynamic optimization platform for compressors. The platform is applied to the global optimization of a three-stage axial compressor, yielding the following conclusions:

(1)

The DFFD method adopted in this study achieves a balance between flexibility and low-dimensional characteristics by directly controlling the surface points of the rotor blades, which in turn drives changes in the shape of all blades. This makes it particularly suitable for aerodynamic optimization of multistage axial compressors.

(2)

The proposed optimization method, based on the integrated surrogate model, incorporates predictions of both optimal and most uncertain solutions, enhancing the predictive accuracy of the surrogate model. It demonstrates excellent applicability to problems with highly nonlinear design spaces, such as the aerodynamic optimization of three-stage axial compressors.

(3)

The optimization results are obtained within 48 h, achieving a 0.6% improvement in adiabatic efficiency and a 4% expansion in the surge margin while maintaining a nearly unchanged flow rate and pressure ratio at the design point.

(4)

The optimized three-stage axial compressor exhibits improved flow conditions on the suction surface of the rotor blades, primarily reflected in the reduction in flow separation regions and the weakening of pre-shock intensity. Additionally, the stator blades achieve improved inlet incidence angles, thereby reducing separation losses near the trailing edge.

The proposed method demonstrates good applicability in solving aerodynamic optimization problems for multistage axial compressors characterized by high dimensionality, strong nonlinearity, and computationally expensive simulations. Its core advantage lies in the effective integration of DFFD’s low-dimensional yet flexible control with an intelligent sampling/search strategy driven by the ensemble surrogate model, significantly enhancing optimization efficiency and global search capability.

Although this method achieved significant success in the current application, its universal applicability as an “optimal” approach requires further validation and comparative studies across broader design spaces (e.g., more complex geometric constraints, multi-objective optimization) and diverse compressor configurations (e.g., transonic, high-load designs).

Future research should focus on algorithmic and intelligent enhancements:

-

Extending the framework to multi-objective optimization (e.g., trade-offs among efficiency, pressure ratio, surge margin);

-

Developing more robust ensemble surrogate modeling strategies and adaptive sampling criteria;

-

Exploring deep learning methods to build more accurate and generalizable surrogate models and enable feature extraction (e.g., flow-field data-based features).