Clustering feature driven hierarchical evolutionary design method for aerodynamic shape

Abstract

The application discloses a clustering feature driving-based aerodynamic shape hierarchical evolution design method, and relates to the technical field of aerodynamic shape hierarchical evolution. The specific implementation scheme is as follows: initial particle groups are generated according to the value range of a plurality of design optimization parameters of an air dynamic shape to be optimized, an optimal cluster of particles is selected according to the objective function value of the particles, principal component analysis is carried out, the main direction and the exploration step of the design optimization parameter iteration are determined, the trial points are generated according to the exploration step and the center point of the particles and the iteration is carried out, the offset of the mass center of the optimal cluster after the iteration and the mass center of the last generation is calculated, the exploration step of the next generation is adaptively adjusted, the final design optimization parameter is determined according to the exploration step size and the dispersion degree of the design optimization parameter, and the application realizes the significant improvement of the calculation efficiency and the convergence speed of the high-dimensional aerodynamic shape optimization under the premise of ensuring the optimization precision.

Description

Technical Field

[0001] This application relates to the field of aerodynamic shape layered evolution technology, and more specifically, to an aerodynamic shape layered evolution design method based on clustering features. Background Technology

[0002] The shape of an aircraft directly determines its aerodynamic performance; even minor changes in geometry can significantly impact the overall flow field and aerodynamic characteristics. Therefore, aerodynamic shape optimization has become a crucial aspect of modern aircraft design. A typical optimization process, combined with computational fluid dynamics (CFD), can be divided into four modules: shape parameterization, mesh generation, flow field solution, and optimization methods. To obtain accurate flow field simulations, the shape parameterization module often requires introducing a large number of design optimization parameters to comprehensively describe the geometric features. However, increasing the number of design optimization parameters leads to a sharp increase in the dimensionality of the design space. This not only makes the optimization problem extremely complex, but also the range of values ​​for the design optimization parameters in each iteration directly affects the evaluation results, thus significantly reducing optimization efficiency. To address this contradiction, existing technologies have developed optimization methods based on surrogate models and data dimensionality reduction techniques. However, surrogate models have limited accuracy in high-dimensional problems and are highly dependent on training data, while the dimensionality of the design space may still remain high after dimensionality reduction, making it difficult to significantly reduce computational complexity. While the hierarchical optimization strategy proposed in recent years has improved efficiency to some extent, the population size of each layer remains fixed and the search direction is still along the original coordinate axis. In the later stages of optimization, there is a lot of redundant computation, resulting in a high total number of CFD evaluations, which is difficult to meet the needs of engineering practice for rapid design iteration.

[0003] The information disclosed in the background section is only intended to enhance the understanding of the background of this disclosure, and therefore may include information that does not constitute prior art known to those skilled in the art. Summary of the Invention

[0004] In view of this, the purpose of this application is to provide a hierarchical evolution design method for aerodynamic shape based on clustering features, which can significantly improve the computational efficiency and convergence speed of high-dimensional aerodynamic shape optimization while ensuring optimization accuracy.

[0005] According to a first aspect of this application, a hierarchical evolution design method for aerodynamic shapes based on clustering features is provided, comprising: An initial particle swarm is generated based on the range of values ​​of multiple design optimization parameters for the aerodynamic shape to be optimized. CFD calculations are performed on each particle to obtain the objective function value. Based on the objective function value of each particle, clustering is performed, and the particle cluster with the optimal cluster center value is selected as the current optimal cluster. Principal component analysis is performed on the sample matrix composed of the design optimization parameters of the particles in the current optimal cluster. The direction corresponding to the eigenvector whose eigenvalues ​​meet the requirements is selected as the main search direction. The initial exploration step size is determined according to the variance contribution rate of each main search direction. Using the particles in the optimal cluster as the center point, test points are generated on both sides of the main direction of each center point according to the exploration step size. CFD is called for each test point to obtain the target function value after iteration. The iterative trial points are merged with the particles of the previous generation of optimal clusters, and the optimal clusters are re-clustered and iterated according to the objective function value. The offset between the centroid of the iterative optimal cluster and the centroid of the previous generation is calculated. Based on the relationship between the offset and the current exploration step size, the exploration step size of the next generation is adaptively adjusted. Repeat the above iterative process, determine the first convergence condition based on the size of the exploration step, and determine the second convergence condition based on the dispersion of the design optimization parameters of the optimal cluster. When the first and second convergence conditions are satisfied at the same time, select the particle with the best performance from the optimal cluster as the final design optimization parameters and output it.

[0006] In some embodiments, each particle represents a set of shape design optimization parameters; The step of generating an initial particle swarm based on the value range of multiple design optimization parameters of the aerodynamic shape to be optimized is as follows: Based on multiple design optimization parameters of the aerodynamic shape to be optimized, the value range of all design optimization parameter types is determined, and the initial number of particles to be generated is set. Then, the value range of each design optimization parameter type is evenly divided into small intervals with the same number of initial particles. Next, each design optimization parameter type is randomly sampled without replacement. From all the small intervals of each design optimization parameter type, one interval is selected, and any value within that interval is taken as the value of the design optimization parameter type on the corresponding particle. After all design optimization parameters have been independently sampled, the values ​​of each design optimization parameter type on each particle are combined to form a complete set of design optimization parameter vectors. Finally, the generated particles are boundary checked to ensure that all values ​​do not exceed the preset range. If the values ​​exceed the limit due to numerical error, they are corrected to the nearest boundary value, resulting in an initial particle swarm that is evenly distributed in the design space.

[0007] In some embodiments, the clustering step is as follows: Using the objective function value corresponding to each particle as the basis for clustering, the number of clusters is set, and an initial number of cluster centers is randomly preset, the same as the number of clusters. For each cluster feature, the absolute difference between it and each cluster center is calculated, and it is assigned to the cluster represented by the center with the smallest difference. After all values ​​are assigned, the arithmetic mean of all cluster features in each cluster is recalculated and used as the new center of that cluster. The above assignment and update steps are repeated until the cluster centers no longer change or the preset maximum number of iterations is reached. The clusters are sorted according to the size of the center value of each cluster, and the cluster with the smallest center value is selected as the cluster with the optimal objective function value.

[0008] In some embodiments, the step of obtaining the variance contribution rate of the main search direction is as follows: A matrix is ​​formed with the particle types in the optimal cluster as rows and the design optimization parameter types of each particle as columns. Take the average value of the design optimization parameter types for each column of the matrix; Subtract the mean of the design optimization parameter type from the value of each design optimization parameter type for each particle, so that the data is distributed around the origin, and a centered matrix is ​​obtained to eliminate the effect of translation. Using a centered matrix, the covariance between different types of design optimization parameters is calculated to obtain a covariance matrix. The diagonal of the matrix represents the variance of each variable, while the off-diagonal represents the covariance between the variables. Eigenvalue decomposition of the covariance matrix yields a set of eigenvalues ​​and corresponding eigenvectors. The magnitude of the eigenvalues ​​reflects the degree of dispersion of the data along the direction of the eigenvector. The top few eigenvectors with the largest eigenvalues ​​are selected as principal directions, and the variance contribution rate of each principal direction is calculated to measure the importance of each principal direction.

[0009] In some embodiments, the method for determining the initial exploration step size is as follows: Set a baseline step size value, and perform a linear mapping between the variance contribution rate of each principal direction and the baseline step size value to obtain the exploration step size corresponding to each principal direction.

[0010] In some embodiments, the strategy for generating test points along both the positive and negative sides of each main direction is as follows: Obtain the average distance between adjacent particles, determine the number of test points based on the average distance and the exploration step size, and based on the number of test points, move the center point along the positive and negative directions of the main direction according to the exploration step size to generate the corresponding number of test points.

[0011] In some embodiments, the method for calculating the offset between the centroid of the optimal cluster after iteration and the centroid of the previous generation is as follows: Take all particles from the previous generation's optimal cluster, calculate the average value of each design optimization parameter, and obtain the previous generation's centroid; Take all particles from the new optimal cluster obtained from the current iteration, calculate the average value of each design optimization parameter, and obtain the new centroid; Subtract the corresponding components of the new centroid from the old centroid, square them, sum them, and then take the square root to obtain the offset.

[0012] In some embodiments, the method for adaptively adjusting the exploration step size for the next generation is as follows: Based on the average step size of all main directions, calculate the ratio of the centroid displacement to the average value, and set an adjustment threshold. When the ratio is less than the set adjustment threshold, it is determined that the population has gathered in a stable region. Therefore, the exploration step size in each main direction is multiplied by a convergence factor less than 1 to achieve a more refined search. When the ratio is not less than the set adjustment threshold, the exploration step size remains unchanged, and the exploration continues at the same scale.

[0013] In some embodiments, the method for determining whether the first convergence condition is met is as follows: Obtain the maximum value of the exploration step size in all main directions, and set the step size convergence threshold. When this maximum value is less than the step size convergence threshold, it is determined that the current iteration number satisfies the first convergence condition.

[0014] In some embodiments, the method for determining whether the second convergence condition is met is as follows: For each design optimization parameter, calculate the difference between the maximum and minimum values ​​of the particles in the current optimal cluster for that design optimization parameter, and divide this difference by the corresponding range of the initial design space; When the relative range of all design optimization parameters is less than the preset ratio, the current iteration number is judged to meet the second convergence condition.

[0015] One embodiment of the above application has the following advantages or beneficial effects: Compared to existing hierarchical optimization methods, this invention first selects the optimal cluster from the initial population through performance clustering. Then, it automatically extracts a few principal directions within these clusters that are most sensitive to changes in the objective function. The exploration step size is adaptively allocated based on the variance contribution rate of each principal direction, ensuring that the search scale in each principal direction matches its sensitivity to performance. This dimensionality-reduction-guided search strategy fundamentally changes the traditional method of isotropically exploring along all original coordinate axes, effectively compressing the high-dimensional optimization problem into a low-dimensional sensitive subspace. The number of trial points required in each iteration is significantly reduced, and each step directly affects the direction with the greatest performance impact, significantly accelerating the convergence speed. Thus, the total number of CFD evaluations is reduced by more than an order of magnitude while maintaining the same optimization accuracy.

[0016] Furthermore, this application employs a multi-center, bidirectional step-by-step approach with taboo evaluation. Multiple center points are selected from the optimal cluster for parallel exploration, avoiding the risk of missing optimal regions due to a single center. Simultaneously, a taboo table is used to skip trial points too close to historical evaluation points, effectively eliminating redundant calculations. Based on this, by calculating the offset between the centroid of the new optimal cluster and the centroid of the previous generation, it automatically determines whether the population has stabilized at the current scale, thus adaptively reducing the step size and shrinking the design space, gradually focusing the search on the vicinity of the optimal solution. This series of adaptive mechanisms ensures that the entire optimization process requires no manual intervention, guaranteeing both global exploration capabilities and fine-grained positioning in the later stages, greatly improving the engineering practicality and automation level of high-dimensional aerodynamic shape optimization.

[0017] Other effects of the above-mentioned alternative methods will be described below in conjunction with specific embodiments. Attached Figure Description

[0018] The accompanying drawings are provided for a better understanding of this solution and do not constitute a limitation of this application. Wherein: Figure 1 This is a flowchart illustrating a hierarchical evolution design method for aerodynamic shape based on clustering features, provided in an embodiment of this application. Figure 2 This is a schematic diagram of a hierarchical evolution process of aerodynamic shape based on clustering features, provided in an embodiment of this application. Figure 3 This is a schematic diagram of main direction extraction and bidirectional stepping provided in an embodiment of this application. Detailed Implementation

[0019] In the following description, specific details such as particular system architectures and techniques are set forth for illustrative purposes and not for limitation, in order to provide a thorough understanding of the embodiments of this application. However, those skilled in the art will understand that this application may also be implemented in other embodiments without these specific details. In other instances, detailed descriptions of well-known systems, apparatuses, circuits, and methods have been omitted so as not to obscure the description of this application with unnecessary detail.

[0020] To facilitate understanding of this application, the embodiments of this application will be briefly described below: In aircraft design, airfoil shape parameters (such as maximum thickness location, camber, and trailing edge angle) directly affect the aerodynamic performance of the aircraft, including lift and drag. To obtain airfoils with high aerodynamic efficiency, designers need to repeatedly adjust these parameters and evaluate the performance of each candidate airfoil through computational fluid dynamics (CFD) simulations. However, airfoil optimization typically involves multiple design variables (e.g., 10–30), resulting in a high design space dimension and extremely high CFD computation costs. Traditional optimization methods (such as genetic algorithms and particle swarm optimization) often require thousands or even tens of thousands of CFD evaluations to converge to the optimal solution, making it difficult to meet the needs of rapid iteration in engineering. While some existing hierarchical optimization strategies introduce clustering and design space shrinkage, their population size at each layer is fixed, and the search direction still follows the original coordinate axes, leading to a large amount of redundant computation in later stages. This application aims to solve the above problems by rapidly screening superior regions through performance clustering, extracting sensitive directions through principal component analysis, conducting multi-center bidirectional step-by-step parallel exploration, and adjusting the adaptive step size. This compresses the high-dimensional search into a low-dimensional sensitive subspace, significantly reducing the number of CFD evaluations, thereby achieving rapid iterative optimization of airfoil shape while ensuring optimization accuracy.

[0021] See Figure 1 This is a flowchart illustrating a hierarchical evolution design method for aerodynamic shapes based on clustering features, provided in an embodiment of this application. Figure 1 The execution subject of the method shown can be a combination of software and / or hardware, specifically, it can be one or more of various types of terminals, hardware systems, cloud computing, etc.

[0022] Figure 1 The aerodynamic shape hierarchical evolution design method based on clustering features, as shown, includes steps 1 to 5, as detailed below: Step 1: Generate an initial particle swarm based on the value range of multiple design optimization parameters of the aerodynamic shape to be optimized, perform CFD calculations on each particle to obtain the objective function value, and perform clustering based on the objective function value of each particle, selecting the particle cluster with the optimal cluster center value as the current optimal cluster. This step obtains design optimization parameters and generates an initial particle swarm. Combined with CFD simulation, it obtains the objective function value for each particle, then clusters these objective function values, retaining only the optimal cluster. Its advantage lies in its ability to quickly eliminate a large number of poor-performing candidate shapes with extremely low computational cost, avoiding the computational waste caused by treating all particles equally after uniform sampling across the entire design space in traditional methods. Compared to existing technologies, this step uses performance clustering instead of indiscriminate sampling, allowing subsequent dimensionality reduction analysis and step search to focus only on the most promising regions, thus significantly reducing the number of particles needing to enter the fine-tuning stage and compressing the total number of CFD evaluations from the source. In this scheme, step 1 plays a dual role of "pre-screening" and "direction guidance." It not only provides high-quality data samples for the principal component analysis in the next step, ensuring that the extracted principal direction truly reflects the sensitive region of the objective function, but also forms a progressive optimization chain with subsequent steps such as multi-center bidirectional stepping and adaptive step size adjustment. This allows the overall scheme to significantly improve convergence efficiency and reduce computational complexity while maintaining global search capabilities.

[0023] In aerodynamic shape optimization, Computational Fluid Dynamics (CFD) is a simulation technique that uses computers to numerically solve fluid motion equations to simulate the flow state, pressure distribution, drag, lift, and other properties of air as it passes around an object. It's essentially building a "virtual wind tunnel" in a computer. A particle refers to a candidate design scheme generated in a single optimization iteration. Each particle corresponds to a complete set of variable values ​​in the design optimization parameters, representing a specific airfoil or shape. For example, if the design optimization parameters include "maximum thickness location (range 0.2–0.5)" and "camber (range 0.00–0.05)," then one particle could be "maximum thickness location = 0.26, camber = 0.035," and another particle could be "maximum thickness location = 0.30, camber = 0.020." The objective function value is a quantitative indicator calculated by CFD to measure the quality of the design scheme. For example, the drag coefficient (Cd) of an airfoil; a smaller value indicates better aerodynamic performance. In reverse engineering problems, the objective function value can also be the sum of squared errors in the pressure distribution between the current airfoil and the target airfoil. The goal of the optimization process is to find a set of design optimization parameters (i.e., a particle) that minimizes the objective function value.

[0024] Furthermore, each particle represents a set of design optimization parameters for the shape; The step of generating an initial particle swarm based on the value range of multiple design optimization parameters of the aerodynamic shape to be optimized is as follows: Based on multiple design optimization parameters of the aerodynamic shape to be optimized, the value range of all design optimization parameter types is determined, and the initial number of particles to be generated is set. Then, the value range of each design optimization parameter type is evenly divided into small intervals with the same number of initial particles. Next, each design optimization parameter type is randomly sampled without replacement. From all the small intervals of each design optimization parameter type, one interval is selected, and any value within that interval is taken as the value of the design optimization parameter type on the corresponding particle. After all design optimization parameters have been independently sampled, the values ​​of each design optimization parameter type on each particle are combined to form a complete set of design optimization parameter vectors. Finally, the generated particles are boundary checked to ensure that all values ​​do not exceed the preset range. If the values ​​exceed the limit due to numerical error, they are corrected to the nearest boundary value, resulting in an initial particle swarm that is evenly distributed in the design space.

[0025] This method for generating initial particles enables uniform coverage of the entire design space with a significantly smaller sample size than full factorial designs, even when design optimization parameters have high dimensionality or large differences in value range. It avoids sample clustering or blank areas that may occur with traditional pure random sampling, thus ensuring strong representativeness and space-filling properties in the initial particle swarm. Furthermore, since the value range of each design optimization parameter is forcibly divided into small intervals equal to the number of particles, and each interval is sampled only once, this ensures that sample points are present across the entire value range of each variable, without missing edge regions. This is crucial for potential boundary optima in aerodynamic shape optimization. In addition, this method independently performs random sampling without replacement on each variable and then randomly combines them into complete particles. This maintains the independence of each variable's value and avoids spurious correlations between variables, providing unbiased initial data for subsequent performance clustering and principal component analysis.

[0026] Taking the two design optimization parameters, "maximum thickness position (value range 0.2~0.5)" and "curvature (value range 0.00~0.05)", as an example, when four initial particles need to be generated, the thickness position interval [0.2,0.5] is first evenly divided into four smaller intervals: [0.2,0.275], [0.275,0.35], [0.35,0.425], and [0.425,0.5]. Similarly, the curvature interval [0.00,0.05] is divided into four smaller intervals: [0.00,0.0125], [0.0125,0.025], [0.025,0.0375], and [0.0375,0.05]. Then, random sampling without replacement is performed on the thickness positions, for example, sampling the 3rd, 1st, 4th, and 2nd intervals sequentially, and randomly selecting values ​​within each interval, resulting in 0.38, 0.22, 0.48, and 0.31. Similarly, random sampling without replacement is performed independently on the curvature, for example, sampling the 2nd, 4th, 1st, and 3rd intervals, resulting in 0.018, 0.048, 0.006, and 0.032. Finally, the values ​​in the corresponding order are combined into particles: particle 1 (0.38, 0.018), particle 2 (0.22, 0.048), particle 3 (0.48, 0.006), and particle 4 (0.31, 0.032). This yields four initial particles evenly distributed within the design space, covering the low, medium, and high regions of the thickness position and different ranges of curvature, avoiding the situation where all samples are crowded in the middle or at the edges and are missed.

[0027] Furthermore, the clustering step is as follows: Using the objective function value corresponding to each particle as the basis for clustering, the number of clusters is set, and an initial number of cluster centers is randomly preset, the same as the number of clusters. For each cluster feature, the absolute difference between it and each cluster center is calculated, and it is assigned to the cluster represented by the center with the smallest difference. After all values ​​are assigned, the arithmetic mean of all cluster features in each cluster is recalculated and used as the new center of that cluster. The above assignment and update steps are repeated until the cluster centers no longer change or the preset maximum number of iterations is reached. The clusters are sorted according to the size of the center value of each cluster, and the cluster with the smallest center value is selected as the cluster with the optimal objective function value.

[0028] The advantages of using K-means clustering to cluster objective function values ​​are: it automatically divides particles into different levels according to their performance, eliminating the need for manually setting performance thresholds, and it is computationally simple and converges quickly. By using the objective function value as a one-dimensional feature for clustering, particles with similar performance levels can be effectively grouped into the same cluster, thus accurately identifying the best-performing particle group and avoiding the possibility of containing particles with large performance differences in "good" clusters due to simply sorting and then selecting a fixed proportion. Furthermore, K-means clustering, through iterative optimization of cluster centers, can adapt to the distribution of objective function values ​​in different optimization problems. For example, when the optimal region has dense particles with similar performance, the cluster centers will naturally shift towards that region, making the selection of the "optimal cluster" more accurate. Compared to simply sorting and selecting the top 30%, this method can automatically remove outliers, improving the reliability of subsequent principal component analysis (PCA), which is sensitive to outliers, while clustering can filter out particles that have good performance by chance but whose design parameters are abnormally isolated.

[0029] The following example uses two design optimization parameters (maximum thickness position x1∈[0.2,0.5], curvature x2∈[0.00,0.05]) to generate 20 particles. A table of objective function values ​​(drag coefficient f) for each particle is provided, and the detailed process of K-means clustering (K=3) is demonstrated.

[0030] Step 2: Perform principal component analysis on the sample matrix composed of the design optimization parameters of the particles in the current optimal cluster, select the direction corresponding to the eigenvector whose eigenvalues ​​meet the requirements as the main search direction, and determine the corresponding initial exploration step size according to the variance contribution rate of each main search direction.

[0031] The advantage of this step lies in its ability to automatically identify the directions sensitive to changes in the design optimization parameters of all particles in the optimal cluster through principal component analysis. These directions, known as principal directions, quantify the importance of each principal direction using variance contribution rate, and then assign an initial exploration step size matching its sensitivity to each principal direction. Compared to existing methods that search along all original coordinate axes or rely solely on experience to set the step size, this dimensionality-reduction-guided adaptive step size setting strategy concentrates search resources on the key directions that truly affect performance, avoiding ineffective explorations in redundant or insensitive directions, thereby significantly reducing the number of CFD evaluations required per generation. In this scheme, step 2 serves as a dual link between "direction awareness" and "step size customization." It takes the high-quality optimal clusters selected in step 1 and transforms their distribution information into a clear direction and reasonable step size for the multi-center bidirectional stepping in step 3. Simultaneously, it provides an initial benchmark for the adaptive step size adjustment based on centroid displacement in step 4, enabling the entire optimization process to proceed efficiently within a low-dimensional sensitive subspace, significantly shortening the number of iterations required for convergence, and ultimately achieving a substantial reduction in total computational cost.

[0032] Among them, the principal direction refers to the principal direction obtained by principal component analysis of the coordinates of the design optimization parameters of the particles in the optimal cluster. Each principal direction is a specific linear combination in the original design optimization parameter space, representing the direction in which the objective function value fluctuates most violently with the change of the design optimization parameters, which is the search direction that is most worth exploring first in the optimization process.

[0033] For example, when the design optimization parameters include the location of the maximum thickness ( (range 0.2~0.5) and curvature ( The two parameters (range 0.00 to 0.05) are used to obtain an approximate eigenvector of the principal direction through principal component analysis. In mathematical notation, superscript... This represents "transpose". For example, a row vector becomes a column vector after being transposed, and this direction points to "" in the two-dimensional plane. and The 45° diagonal line that "increases or decreases simultaneously" has another principal direction's eigenvector as... , pointing to " Increase and The -45° slash that signifies "reduction".

[0034] Taking the first principal direction with the larger eigenvalue as an example, representing the direction most sensitive to changes in the drag coefficient, moving the design parameters along this direction will significantly alter the airfoil's aerodynamic performance. The second principal direction, however, has lower sensitivity; therefore, subsequent iterations only need to follow these principal directions instead of the original ones. shaft or By generating test points along the axis, the key changing characteristics of the objective function can be accurately captured with minimal computation.

[0035] Furthermore, the steps for obtaining the variance contribution rate of the main search direction are as follows: A matrix is ​​formed with the particle types in the optimal cluster as rows and the design optimization parameter types of each particle as columns. Take the average value of the design optimization parameter types for each column of the matrix; Subtract the mean of the design optimization parameter type from the value of each design optimization parameter type for each particle, so that the data is distributed around the origin, and a centered matrix is ​​obtained to eliminate the effect of translation. Using a centered matrix, the covariance between different types of design optimization parameters is calculated to obtain a covariance matrix. The diagonal of the matrix represents the variance of each variable, while the off-diagonal represents the covariance between the variables. Eigenvalue decomposition of the covariance matrix yields a set of eigenvalues ​​and corresponding eigenvectors. The magnitude of the eigenvalues ​​reflects the degree of dispersion of the data along the direction of the eigenvector. The top few eigenvectors with the largest eigenvalues ​​are selected as principal directions, and the variance contribution rate of each principal direction is calculated to measure the importance of each principal direction.

[0036] Furthermore, by obtaining each principal direction and its variance contribution rate through principal component analysis, the complex correlation structure in the high-dimensional design optimization parameter space can be compressed into a few orthogonal sensitive directions. This allows for a quantitative assessment of the influence of each direction on the objective function's variation, thus providing mathematically optimal guidance for subsequent adaptive step-size searches. Compared to searching directly along the original coordinate axes (e.g., along the maximum thickness axis or curvature axis respectively), this method avoids redundancy in trial directions caused by inter-variable coupling. For example, when the particle distribution in the optimal cluster is obliquely banded, searching along the original coordinate axes may require many steps to approach the optimum, while searching along the principal direction can cross the main variation region in a single step.

[0037] In addition, the variance contribution rate provides an objective weighting metric, which allows the search step size to be proportional to the importance of the direction. Sensitive directions can be advanced rapidly, while insensitive directions can be finely adjusted, thereby significantly improving the marginal benefit of each CFD evaluation and reducing the total number of iterations.

[0038] The design optimization parameters include the location of the maximum thickness ( (range 0.2~0.5) and curvature ( Taking the optimal cluster (containing particles 1, 7, 9, 13, and 16) obtained by setting the parameters (range 0.00 to 0.05) as an example, the calculation process of principal component analysis is explained in detail. The data of the optimal cluster particles are shown in Table 1: Table 1: Optimization Parameters for the Best Cluster Particles

[0039] A matrix is ​​formed with the particle types in the optimal cluster as rows and the design optimization parameter types for each particle as columns. ;

[0040] Specifically, a two-dimensional array is constructed with the particles in the optimal cluster as rows and the design optimization parameters as columns. Each element represents the specific value of a particle at a certain design optimization parameter (e.g., 0.26 at the maximum thickness position). Each row of the matrix corresponds to a candidate airfoil, and each column corresponds to a type of design parameter, serving as the raw data for all subsequent calculations.

[0041] Calculate the mean value of each design optimization parameter; Mean of the location of maximum thickness =(0.26+0.24+0.27+0.25+0.26) / 5=1.28 / 5=0.256; Mean curvature =(0.038+0.035+0.033+0.040+0.036) / 5=0.182 / 5=0.0364; Its mean vector =[0.256, 0.0364]; The arithmetic mean of the values ​​of all particles in each column of the matrix (i.e., the same design optimization parameter) is calculated to obtain the center value of the design optimization parameter in the optimal cluster. The mean of the maximum thickness position of the 5 particles is 0.256, which represents the average design level of the optimal cluster in this dimension.

[0042] Subtract the mean of the design optimization parameter type from the value of each design optimization parameter type for each particle, so that the data is distributed around the origin, and a centered matrix is ​​obtained to eliminate the effect of translation. The centering matrix is ​​a new matrix obtained by subtracting the mean of the corresponding column from each element of the original matrix. It eliminates the influence of the absolute position of the design optimization parameters, making the data distributed with the zero point as the center, thereby highlighting the relative differences and dispersion structure between particles.

[0043] The specific calculation formula is as follows: Calculate line by line: Particle 1: (0.26-0.256, 0.038-0.0364) = (0.004, 0.0016); Particle 7: (0.24-0.256, 0.035-0.0364) = (-0.016, -0.0014); Particle 9: (0.27-0.256, 0.033-0.0364) = (0.014, -0.0034); Particle 13: (0.25-0.256, 0.040-0.0364) = (-0.006, 0.0036); Particle 16: (0.26-0.256, 0.036-0.0364) = (0.004, -0.0004); Obtain the centralized matrix :

[0044] Using a centered matrix, the covariance between different types of design optimization parameters is calculated, resulting in a covariance matrix. The diagonal lines represent the variances of each variable, while the off-diagonal lines represent the covariances between the variables.

[0045] The covariance matrix is ​​a square matrix calculated based on the centered matrix, and its dimension is equal to the number of design optimization parameters. The diagonal elements represent the variance of each design optimization parameter (e.g., the variance of the maximum thickness location is 0.00013), reflecting the dispersion of the variable in the optimal cluster. The off-diagonal elements represent the covariance between two different design optimization parameters (e.g., -0.0000105), reflecting the linear correlation between them. Positive values ​​indicate changes in the same direction, and negative values ​​indicate changes in opposite directions.

[0046] Centralized matrix Multiplying the transpose of the matrix by itself yields the cross product of each design optimization parameter, which is then divided by the sample size minus 1 (i.e., degrees of freedom) to obtain the covariance matrix among the variables. ;

[0047] Eigenvalue decomposition of the covariance matrix yields a set of eigenvalues ​​and corresponding eigenvectors. The magnitude of the eigenvalues ​​reflects the degree of dispersion of the data along the direction of the eigenvector. Furthermore, eigenvalue decomposition is a mathematical operation that decomposes a square matrix into a set of eigenvalues ​​and eigenvectors. The covariance matrix C describes the correlation and dispersion of the design optimization parameters in the optimal cluster. Through eigenvalue decomposition, the direction with the largest data variance (i.e., the principal component direction) can be found.

[0048] Here, eigenvalues ​​are a set of scalars obtained by eigenvalue decomposition of the covariance matrix, sorted from largest to smallest. The magnitude of each eigenvalue represents the variance of the data along the corresponding eigenvector direction. The larger the eigenvalue, the more dispersed the data distribution along the corresponding eigenvector direction, which means that the objective function is more sensitive to changes in that direction. This indicates that moving the design optimization parameters along that direction will cause more drastic changes in the objective function value. The role of eigenvalue decomposition is to rotate the originally coupled design optimization parameter space to a set of orthogonal principal directions, thereby retaining only the first few principal directions with the largest eigenvalues ​​and ignoring directions with small variances, reducing the search dimension. The magnitude of the eigenvalues ​​is directly used as the basis for calculating the variance contribution rate, which is used for subsequent adaptive step size setting, making the principal directions mutually orthogonal, eliminating the linear correlation between variables, and facilitating independent step size setting.

[0049] Eigenvectors are unit vectors that correspond one-to-one with eigenvalues, indicating the direction of maximum variance in the data. Each eigenvector defines a new coordinate axis direction (principal direction) in the design space, for example... A 45° diagonal line indicating a decrease in curvature as the thickness increases.

[0050] The specific calculation process for eigenvalue decomposition of the covariance matrix to find eigenvalues ​​and eigenvectors is as follows: First, assume the covariance matrix is ​​a d×d square matrix (d is the number of design optimization parameters). To find the eigenvalues, we need to solve the characteristic equation, which is to subtract the determinant of the unknown multiplied by the identity matrix from the matrix and make it equal to zero, resulting in a high-order equation about the unknown. In the two-dimensional case (d=2), this equation is a quadratic equation; for higher dimensions, the degree is even higher. Solving this equation yields several solutions, which are the eigenvalues, and they are sorted in descending order. After obtaining each eigenvalue, substitute it back into the original matrix and subtract the eigenvalue multiplied by the identity matrix to obtain the matrix. Then, solve the homogeneous linear equation system to find a non-zero vector such that multiplying the matrix by this vector equals zero. This vector is the eigenvector corresponding to the eigenvalue. Usually, the eigenvector needs to be normalized to make its length 1. Since the solution to the homogeneous equation is not unique (any non-zero multiple is a solution), the unit vector is usually taken as the final eigenvector. In this way, each eigenvalue and its corresponding unit eigenvector constitute a feature pair. The magnitude of the eigenvalue reflects the variance of the data along the direction of that eigenvector, and the eigenvector indicates the orientation of that direction in the original design space. According to the design optimization parameters, the location of the maximum thickness is included. (range 0.2~0.5) and curvature ( Taking the optimal cluster (containing particles 1, 7, 9, 13, and 16) obtained by setting the parameters (range 0.00 to 0.05) as an example, we obtain the eigenvalues. =0.00013089, =0.00000641, and obtain the eigenvector based on the eigenvalue. , ; Eigenvalues =0.00013089 and =0.00000641 are two eigenvalues ​​of the covariance matrix, which represent the values ​​of particles in the optimal cluster along the first principal direction. Second main direction The variance; Feature vector and This gives the optimization parameters of the original design for these two main directions ( Location of maximum thickness The direction in the camber coordinate system.

[0051] Specifically, eigenvalues ​​and eigenvectors , In the relationship, correspond" Increase, The 45° diagonal line direction of "reduction" is because 0.7071 ≈ When moving in that direction, and The changes are equal in magnitude but opposite in sign. ,in, for and The change in thickness indicates that in the optimal cluster, the distribution of particles mainly unfolds along this oblique line, that is, there is a strong negative correlation between the location of maximum thickness and curvature. Particles with larger thickness tend to have smaller curvature, and vice versa.

[0052] Feature vector This represents a direction in a two-dimensional design space, signifying the movement along that direction from the origin. and The change in quantity satisfies That is, by how much the thickness increases at the maximum position, the curvature decreases by the same amount (because the negative sign indicates the reverse change). Since the vector magnitude is 1, it only indicates the direction and does not specify the step size.

[0053] Feature vector correspond" and Simultaneously increase or decrease the 45° diagonal direction ( The variance in this direction is very small (). Compare The small value indicates that particles in the optimal cluster are almost not dispersed along this direction, meaning that changing both variables simultaneously has a very weak impact on performance.

[0054] The magnitude of the eigenvalue directly measures the degree of dispersion of the data in the corresponding direction. Much larger This means that the shape of the optimal cluster is a shape along the eigenvector. An elongated ellipse, its minor axis direction ( The objective function (such as the drag coefficient) remains almost unchanged near the current optimal solution. increase, The combination of "reduction" is extremely sensitive, while the combination of "same-direction change" is not sensitive.

[0055] Therefore, the optimization search should prioritize following the feature vector. Explore the direction in large strides, as movement in that direction can significantly alter performance, while moving along the feature vector... Direction can be achieved with small steps or ignored, thus saving computational resources. Eigenvectors provide specific movement directions, while eigenvalues ​​quantify the importance of each direction.

[0056] The top few eigenvectors with the largest eigenvalues ​​are selected as principal directions, and the variance contribution rate of each principal direction is calculated to measure the importance of each principal direction.

[0057] The variance contribution rate is the proportion of each feature value to the sum of all selected features, usually expressed as a percentage. It quantitatively measures the importance of the corresponding principal direction in the overall data variation. The higher the variance contribution rate, the greater the influence of that direction on the change of the objective function, and therefore a larger exploration step size should be allocated in subsequent step searches.

[0058] The variance contribution rate is calculated as follows: The eigenvalues ​​corresponding to all selected principal directions are summed to obtain a total eigenvalue. Then, for each principal direction, its own eigenvalue is divided by this total. The result is the variance contribution rate for that principal direction. The contribution rate is usually expressed as a percentage to intuitively reflect the proportion of each principal direction in the overall data variation. A principal direction with a larger eigenvalue has a higher variance contribution rate, meaning that the data dispersion in that direction is greater, and the objective function value is more sensitive to changes in the design optimization parameters along that direction.

[0059] With the above eigenvalues =0.00013089 and Taking =0.00000641 as an example, the variance contribution rates are as follows:

[0060]

[0061] Therefore, the first principal direction, u1≈[0.707,-0.707]T, contributes 95.3% of the variance and is the most sensitive direction, meaning that moving the design optimization parameters along this direction will cause the objective function to change most drastically. The second principal direction contributes only 4.7% and is not sensitive; subsequent steps will be based on... =0.953 and =0.047 Set the exploration step size respectively; Furthermore, the method for determining the initial exploration step size is as follows: Set a baseline step size value, and perform a linear mapping between the variance contribution rate of each principal direction and the baseline step size value to obtain the exploration step size corresponding to each principal direction.

[0062] The initial exploration step size is set by linearly mapping the variance contribution rate to the baseline step size. Its core advantage is that it achieves an adaptive match between step size and direction sensitivity. The main direction with a high variance contribution rate has a greater impact on the objective function, and a large step size is needed to quickly traverse the direction to approach the optimal region. The main direction with a low variance contribution rate has a smaller impact on performance, and a small step size should be used to avoid excessive search in the direction and waste of computation. This data-driven method avoids the uncertainty of relying on experience or fixed step size, and can automatically concentrate computational resources on the most valuable direction in the early stage of optimization, thereby accelerating the convergence speed and reducing the total number of CFD evaluations.

[0063] The baseline step size is usually directly related to the range of values ​​for the design optimization parameters. It can be calculated by taking the width of the range for each parameter (i.e., the upper limit minus the lower limit) and then using a certain percentage of the average width of all parameters as the baseline step size. For example, if the maximum thickness range is 0.2–0.5 (width 0.3) and the curvature range is 0.00–0.05 (width 0.05), then the average width is (0.3 + 0.05) / 2 = 0.175. Taking 5%–10% of this average width as the baseline step size (e.g., 0.0175) is appropriate. Factors affecting the baseline step size include the physical scale of the design optimization parameters (parameters with larger ranges naturally require larger step sizes), the convergence accuracy requirements of the optimization problem (higher accuracy requires smaller step sizes), and the density of the initial sampling particles (denser particle density allows for a smaller step size). Therefore, setting the baseline step size is essentially normalizing the scale of the design space and mapping it proportionally to a reasonable movement step size, thereby ensuring that the initial exploration step size in each main direction is neither too large, causing skipping the optimal region, nor too small, causing slow convergence.

[0064] Step 3: Using the particles in the optimal cluster as the center point, generate trial points on both sides of the main direction of each center point according to the exploration step size, and iterate. Call CFD for each trial point to obtain the target function value after iteration.

[0065] This step generates trial points along each principal direction determined in step 2, using each particle in the optimal cluster as the center point and corresponding exploration step sizes on both sides. CFD is then used to evaluate the objective function values ​​of these trial points. The core advantage of this operation lies in transforming the abstract principal directions and adaptive step sizes into practically evaluable candidate shapes, achieving a crucial connection from "direction recognition" to "directional search." Specifically, this step fully utilizes the high-quality seeds selected in step 1 and the sensitive directions extracted in step 2, ensuring that the trial points generated in each iteration are concentrated in the most promising region and move along the most efficient path. This avoids the inefficient behavior of blind random perturbation or uniform sampling along the original coordinate axes in traditional optimization methods. Furthermore, since only two trial points (positive and negative) are generated along each principal direction from each center point, compared to the method of evaluating a fixed number of particles per generation in global particle swarm optimization, the number of CFD calls per iteration is significantly reduced. Furthermore, the multi-center parallel bidirectional stepping mechanism naturally introduces local competition and information sharing. The trial points generated by different center points will cover each other's optimal regions, so that even if a center point deviates from the true optimal, the trial points of other center points may still capture a better solution, thereby enhancing the robustness of the algorithm.

[0066] Furthermore, the strategy for generating test points on both the positive and negative sides of each main direction is as follows: Obtain the average distance between adjacent particles, determine the number of test points based on the average distance and the exploration step size, and based on the number of test points, move the center point along the positive and negative directions of the main direction according to the exploration step size to generate the corresponding number of test points.

[0067] The strategy of generating test points on both the positive and negative sides of each main direction has the core advantage of dynamically adjusting the number of test points based on the average distance between adjacent particles. This ensures that the sampling density matches the local dispersion of the current optimal cluster. When the particle distribution in the optimal cluster is dense, the average distance is small, and the number of test points generated is reduced accordingly, avoiding oversampling in fully explored areas. When the distribution is sparse, the average distance is large, so the number of test points is increased to more comprehensively cover uncertain areas, thereby achieving adaptive allocation of computational resources. Simultaneously, determining the specific movement distance based on the exploration step size ensures that the scale of each test step is consistent with the sensitivity of the current optimization stage. The simultaneous generation of trial points on both the positive and negative sides of each main direction is because, in an unknown objective function terrain, the optimal solution may lie on either side of the center point. Bidirectional stepping allows for symmetrical exploration in both directions, avoiding missing potential optimal regions by moving only to one side. Furthermore, simultaneous evaluation on both sides helps determine the current center point's position relative to peaks and valleys—if the positive trial point performs better, the optimum is in the positive direction; if the negative trial point performs better, it's in the negative direction; if both are poor, the current center point may be close to a local optimum, and the step size should be reduced. This bidirectional symmetric mechanism improves the efficiency of information acquisition in a single iteration and accelerates the convergence process.

[0068] Step 4: Merge the iterated trial points with the previous generation of optimal cluster particles, re-cluster the optimal clusters according to the objective function value, and calculate the offset between the centroid of the iterated optimal cluster and the centroid of the previous generation. Based on the relationship between the offset and the current exploration step size, adaptively adjust the exploration step size of the next generation.

[0069] This step involves merging all the trial points generated in this iteration with the particles in the previous generation's optimal cluster, and then re-clustering them based on the objective function value to update the optimal cluster. This achieves the accumulation of population information and the weeding out of inferior individuals, avoiding the loss of excellent individuals from the previous generation by only retaining trial points. Simultaneously, the offset between the centroids of the new and old optimal clusters is calculated and compared with the current exploration step size. This allows for a quantitative assessment of whether the population has stabilized at the current search scale. When the centroid offset is significantly smaller than the step size, it indicates that the optimal cluster has essentially converged to a local region. Continuing to use the same step size would cause ineffective minor perturbations. In this case, adaptively reducing the step size can make subsequent searches more refined, avoiding repeated oscillations near the optimal solution. Conversely, if the centroid offset is large, it indicates that the optimal cluster is still moving, and maintaining the same step size allows for continued rapid approximation. This dynamic adjustment mechanism enables the optimization process to automatically match the appropriate search scale at different convergence stages, ensuring both the ability to explore quickly with large step sizes in the early stages and to achieve precise local positioning in the later stages. This significantly reduces the total number of iterations and CFD evaluations required to reach convergence, thereby improving optimization efficiency and convergence stability.

[0070] Furthermore, the method for calculating the offset between the centroid of the optimal cluster after iteration and the centroid of the previous generation is as follows: Take all particles from the previous generation's optimal cluster, calculate the average value of each design optimization parameter, and obtain the previous generation's centroid; Take all particles from the new optimal cluster obtained from the current iteration, calculate the average value of each design optimization parameter, and obtain the new centroid; Subtract the corresponding components of the new centroid from those of the old centroid, square the results, sum them, and then take the square root to obtain the offset.

[0071] This method calculates the offset between the centroid of the optimal cluster after iteration and the centroid of the previous generation because the centroid, as the center point of the optimal cluster, directly reflects the overall movement trend of the population. By subtracting the corresponding components of the new and old centroids in the same design variable dimension (i.e., subtracting the first component of the new centroid from the first component of the old centroid, the second component from the second component, and so on), the differences in each dimension are obtained. These differences are then squared, summed, and the square root is taken to obtain the straight-line distance between the two centroids. The advantage of this method is that it integrates the change information of all design variable dimensions, intuitively measuring the spatial displacement of the optimal cluster with a single scalar value, and is unaffected by the coordinate axis direction, facilitating comparison with the current exploration step size. Corresponding components refer to dimensions at the same position in the design variable space; for example, maximum thickness corresponds to maximum thickness, curvature corresponds to curvature, ensuring that the subtraction is performed on the same type of parameter, thus guaranteeing consistent physical meaning. By comparing the displacement with the step size, it is possible to determine whether the population has stabilized at the current search scale, providing a reliable basis for adaptive adjustment of the step size. Furthermore, the method for adaptively adjusting the exploration step size for the next generation is as follows: Based on the average step size of all main directions, calculate the ratio of the centroid displacement to the average value, and set an adjustment threshold. When the ratio is less than the set adjustment threshold, it is determined that the population has gathered in a stable region. Therefore, the exploration step size in each main direction is multiplied by a convergence factor less than 1 to achieve a more refined search. When the ratio is not less than the set adjustment threshold, the exploration step size remains unchanged, and the exploration continues at the same scale.

[0072] The ratio of centroid displacement to average step size is used to drive the adaptive adjustment of the exploration step size. The core reason is that centroid displacement reflects whether the optimal cluster is still moving significantly at the current search scale, while the average step size represents the sensitivity of the current attempt. When the displacement is much smaller than the step size, it indicates that the population has stabilized at the current resolution, and continuing to maintain a large step size will only cause the trial points to bounce ineffectively around the optimal region. Therefore, the step size needs to be reduced for fine-tuning. Conversely, if the displacement is still large, the step size should be maintained to continue rapidly approaching the target. This judgment method based on relative ratios avoids the dimensional dependence of absolute values, making the adjustment criterion insensitive to scale changes in design variables and possessing good versatility. The adjustment threshold is usually set between 0.2 and 0.3. Its setting needs to consider the local curvature and noise level of the design variable space. An excessively large adjustment threshold will cause the step size to decay prematurely, potentially leading to local optima; an excessively small adjustment threshold will delay the reduction of the step size, resulting in wasted computation in later stages. The convergence factor is typically chosen between 0.5 and 0.8. Its selection requires a balance between convergence speed and accuracy. A convergence factor that is too small will cause the step size to decay too quickly, potentially skipping the optimal solution. Conversely, a convergence factor that is too large will result in slow step size reduction, with little improvement in convergence efficiency. In practical applications, the threshold and convergence factor can be initially set through offline testing or by estimating the second derivative of the objective function, and then fine-tuned during the optimization process based on the actual convergence curve. Typically, in practice, a threshold of 0.2 and a convergence factor of 0.6 are initially set as default values, and then adjusted through a small number of preliminary experiments or based on engineering experience.

[0073] Please see Figure 2 This figure illustrates the hierarchical evolution process of aerodynamic shape based on clustering features, as provided in this application embodiment. It comprehensively demonstrates the overall technical flow and core mechanism of the "hierarchical evolution design method for aerodynamic shape based on clustering features," covering the complete evolution process from initial sampling to final convergence. The figure uses design variables... (e.g., at the location of maximum thickness, the value ranges from 0.2 to 0.5) and design variables The background is a two-dimensional design space (such as curvature, with a value range of 0.00 to 0.05). The background presents the spatial distribution of the objective function value (such as the drag coefficient) in the form of contour lines. Warm-colored areas indicate higher objective function values ​​(poor performance), and cool-colored areas indicate lower objective function values ​​(better performance).

[0074] In the initial stage, several uniformly distributed initial particles (represented by red scatter dots in the figure) are generated within the design space. Each particle represents a complete set of design optimization parameter vectors. The value range of each design variable is uniformly divided into small intervals equal to the number of initial particles. Each variable is independently randomly sampled without replacement and then randomly combined, ensuring uniform spatial filling of the samples throughout the design space. After generating the initial particles, CFD is used to calculate the objective function value for each particle. K-means clustering (K=3) is then performed based on the objective function value, dividing the particles into clusters of different performance levels. The clusters are sorted according to their center values, and the cluster with the smallest center value is selected as the cluster with the optimal objective function value (highlighted by a gold dashed ellipse in the figure). The core function of this step is to quickly eliminate a large number of poor-performing candidate shapes with extremely low computational cost, allowing subsequent dimensionality reduction analysis and step search to focus only on the most promising regions.

[0075] Then, principal component analysis is performed on all particles in the optimal cluster. A matrix is ​​formed with the particle design optimization parameter type as the column and the particle type as the row. After centering, covariance matrix calculation, and eigenvalue decomposition, a set of eigenvalues ​​and corresponding eigenvectors are obtained. The first few eigenvectors are selected as principal directions according to the eigenvalues ​​from largest to smallest, and the variance contribution rate of each principal direction is calculated.

[0076] Please see Figure 3 This is a schematic diagram of main direction extraction and bidirectional stepping provided in the embodiments of this application. It shows in detail how to extract the sensitive search direction from the particle distribution of the optimal cluster, and generate test points based on the direction for directional exploration. The main part of the diagram shows a two-dimensional design space with design variables x1 (e.g., maximum thickness location) and x2 (e.g., curvature) as coordinate axes. In the diagram, solid blue dots represent all particles in the optimal cluster obtained after clustering in step 1. These particles exhibit a clear oblique banded distribution in the design space—that is, there is a strong negative correlation between the maximum thickness location and curvature; particles with larger thickness locations tend to have smaller curvature, and vice versa. This distribution characteristic is the data basis for principal component analysis to play a role.

[0077] The first principal direction, marked with a thick red arrow in the figure, corresponds to a relatively large eigenvalue (e.g., eigenvalue λ1 = 0.00013089) and a variance contribution rate of approximately 95.3%. This indicates that moving the design parameters along this direction causes the most drastic change in the objective function value (e.g., drag coefficient), making it the most sensitive direction in the optimization search. Geometrically, this direction corresponds to a 45° diagonal line where x1 increases and x2 decreases. The second principal direction, marked with a blue arrow in the figure, corresponds to a relatively small eigenvalue (e.g., λ2 = 0.00000641) and a variance contribution rate of only about 4.7%. This indicates that the data distribution along this direction remains almost unchanged, and the objective function is not sensitive to changes in combinations along this direction.

[0078] After determining the primary directions, the initial exploration step size is set according to the method described in the instruction manual: first, a baseline step size value related to the range of design variable values ​​is set (usually 5% to 10% of the average width of each design variable range), and then the variance contribution rate of each primary direction is linearly mapped to the baseline step size value. Since the first primary direction has a contribution rate of 95.3%, it is assigned a larger exploration step size (marked in the figure as "step size = 0.65", representing a large-step, fast-forward strategy); the second primary direction has a contribution rate of only 4.7%, and is assigned a smaller exploration step size (marked in the figure as "step size = 0.06", representing a small-step, fine-tuning strategy). This data-driven adaptive step size allocation method allows search resources to be concentrated on the critical directions that truly affect performance.

[0079] The diagram uses the centroid of the optimal cluster (marked with a red square) as one of the center points, illustrating the process of generating test points on both sides of the main directions: moving with large steps in both the positive and negative directions of the first main direction generates test points marked with orange rhombuses; moving with small steps in both the positive and negative directions of the second main direction generates test points marked with light blue triangles. According to the specification, the number of test points is determined by the average distance between adjacent particles and the exploration step size—fewer test points are generated when particles are densely distributed in the optimal cluster, and more test points are generated when they are sparsely distributed, thus achieving adaptive adjustment of the sampling density. Simultaneously, this bidirectional stepping mechanism, which symmetrically generates test points on both sides of each main direction, allows for simultaneous exploration in two directions when the objective function terrain is unknown, improving the efficiency of information acquisition in a single iteration and accelerating the convergence process. Furthermore, the diagram also illustrates the idea of ​​multi-center parallel exploration—selecting multiple particles from the optimal cluster as center points to generate test points separately avoids the risk of missing optimal regions due to a single center, enhancing the robustness of the algorithm.

[0080] Overall, the diagram clearly presents the complete technical chain from "data distribution analysis" to "sensitive direction identification", then to "adaptive step size allocation" and finally "bidirectional directional probing", which is the core visual expression of the dimensionality reduction acceleration in the invention method.

[0081] After merging the iteratively tested points with the previous generation's optimal cluster particles, the clusters are re-clustered, and the offset between the new optimal cluster centroid and the previous generation's centroid is calculated. Based on the average of the exploration step size in all principal directions, the ratio of the centroid displacement to this average value is calculated. When the ratio is less than a set adjustment threshold (usually 0.2–0.3), the exploration step size in each principal direction is multiplied by a convergence factor less than 1 (usually 0.5–0.8) to achieve a finer search. In the figure, the shrinking elliptical region represents the shrinkage process of the optimal cluster in the parameter space, and the step-like decreasing curve represents the adaptive decay of the exploration step size. This schematic diagram presents a four-layer progressive optimization evolution path from "global uniform coverage" to "clustering focused screening," then to "dimensionality reduction sensitive direction guidance," and finally "adaptive fine convergence," clearly demonstrating how this method can significantly reduce the total number of CFD evaluations while ensuring optimization accuracy.

[0082] Step 5: Repeat the above iterative process, determine the first convergence condition based on the size of the exploration step, and determine the second convergence condition based on the dispersion of the design optimization parameters of the optimal cluster. When the first and second convergence conditions are satisfied at the same time, select the particle with the best performance from the optimal cluster as the final design optimization parameters and output it.

[0083] This step, by repeatedly executing the aforementioned clustering, dimensionality reduction, bidirectional stepping, and adaptive step size adjustment processes, and checking the size of the exploration step size and the dispersion of the optimal cluster after each iteration, provides a clear convergence determination and termination mechanism for the entire optimization scheme. When the maximum value of the exploration step size in all main directions is less than the preset convergence threshold, and the ratio of the range of particles in the optimal cluster on each design variable to the corresponding range in the initial design space is less than a very small proportion, it indicates that the search has become so refined that the step size cannot be effectively reduced further, and the optimal cluster has shrunk to an extremely compact region. At this point, continuing the iteration will not produce meaningful improvement. The role of this step is to automatically identify the endpoint of the optimization process, avoiding the blindness of manually setting a fixed number of iterations. This ensures that the final solution meets the engineering accuracy requirements without wasting computational resources due to excessive iteration. At the same time, by combining the two independent indicators of step size and dispersion, it overcomes the misjudgment that may be caused by a single indicator (e.g., the optimal cluster is still scattered even though the step size is small, or the dispersion is small but the step size is still large), ensuring the robustness of the convergence judgment. Finally, step 5 outputs the globally optimal design optimization parameters and their corresponding objective function values, enabling the entire optimization process to deliver the optimal shape design usable by the project automatically, efficiently, and reliably.

[0084] Furthermore, the method for determining whether the first convergence condition is met is as follows: Obtain the maximum value of the exploration step size in all main directions, and set the step size convergence threshold. When this maximum value is less than the step size convergence threshold, it is determined that the current iteration number satisfies the first convergence condition. Furthermore, the method for determining whether the second convergence condition is met is as follows: For each design optimization parameter, calculate the difference between the maximum and minimum values ​​of the particles in the current optimal cluster for that design optimization parameter (i.e., the range), and divide the range by the corresponding range of the initial design space; When the relative range of all design optimization parameters is less than the preset ratio, the current iteration number is judged to meet the second convergence condition.

[0085] The setting of the preset ratio threshold needs to take into account the numerical accuracy of the computational fluid dynamics simulation itself. Typically, the numerical error of CFD calculations is on the order of one-thousandth to one-hundredth. If the preset ratio is too small (such as one-millionth), the convergence condition may never be met, because the simulation noise itself will mask such a small shape difference. The sensitivity of the objective function in the design space also needs to be considered. If the terrain of the objective function near the optimal cluster is very flat, a small preset ratio will not bring significant performance improvement. In this case, the threshold can be appropriately relaxed to terminate early. Conversely, if the terrain is steep, a smaller ratio is needed to accurately locate the valley bottom. In this scheme, the preset ratio is set to one ten-thousandth.

[0086] This method ensures the robustness and physical meaning of the convergence criterion by simultaneously monitoring two independent indicators: the exploration step size and the optimal cluster dispersion. Step size convergence reflects that the algorithm can no longer make finer moves at the current scale; that is, from the perspective of the "search step size," the preset resolution limit has been reached. The dispersion of the optimal cluster reflects the clustering range of candidate solutions in the parameter space—when the relative range of all design variables is less than a preset proportion, it indicates that the particles in the optimal cluster are highly concentrated in every dimension, and the differences between them are less than the tolerance required by engineering. Combining the two indicators can avoid misjudgments based on a single indicator. For example, if the step size is small but the optimal cluster is still dispersed, it may indicate that it has entered a flat region or local oscillation, and should not be terminated. Conversely, if the dispersion is small but the step size is still large, it may indicate that there are better regions that have not been explored finely. The step size convergence threshold is usually set between one-hundredth and one-thousandth of the initial range of design variables. Specifically, it needs to consider the sensitivity of the objective function, the accuracy of CFD calculation, and the tolerance required by actual engineering. The threshold should be smaller when the sensitivity or accuracy requirement is high, and larger when the sensitivity or accuracy requirement is low. Using the relative range (current range divided by the initial range) as the dispersion index makes the judgment criterion dimensionless, applicable to design variables of different dimensions. A preset ratio of 0.01% indicates that the size of the optimal cluster has been reduced to less than 0.01% of the initial design space. At this point, any further adjustments will typically result in performance improvements far less than the numerical errors of the CFD simulation itself or the allowable deviations in engineering. Therefore, when both conditions are met, it can be confident that the best particle in the current optimal cluster is very close to the true global optimum, and using its output as the final design optimization parameter is reliable and efficient.

[0087] Through a four-layer progressive strategy—performance clustering pre-screening, dimensionality reduction sensitive direction extraction, multi-center bidirectional adaptive stepping, dynamic step size adjustment, and convergence judgment—the global search of the high-dimensional design space is gradually focused on the low-dimensional sensitive subspace. At each stage, only a small amount of CFD evaluation is needed to complete direction localization and fine optimization. First, a uniform initial particle swarm is generated through sampling. Based on the objective function value, clustering quickly eliminates poorly performing particles, retaining only the optimal cluster as seeds for subsequent searches, reducing invalid computation at the source. Principal component analysis is performed on the particles in the optimal cluster, extracting the top few eigenvectors with the largest variance contribution as principal directions. The initial exploration step size for each direction is adaptively allocated according to the contribution rate, thus compressing the search dimension from the original design variables to the few most sensitive directions. This avoids blindly exploring along all coordinate axes. The number of CFD evaluations in a single iteration increases linearly with dimensionality rather than exponentially. Using multiple particles in the optimal cluster as centers, exploration points are generated along the positive and negative sides of each principal direction according to the current step size. By utilizing a tabu list to skip locations too close to historical evaluation points, each trial step falls within the most promising region. Parallel exploration by multiple centers enhances the ability to escape local optima. New trial points are merged with the previous generation's optimal clusters and re-clustered. The ratio of the old and new centroid displacements to the average step size is calculated. When the displacement is significantly smaller than the step size, the step size is automatically reduced and the design space shrinks; otherwise, the step size is maintained to continue exploration. This adaptive mechanism ensures that the search scale matches the current convergence stage, preventing premature localized searches and ineffective lingering later. The maximum step size and the relative range of the optimal cluster across various design variables are jointly monitored. When both are simultaneously less than a preset minimum threshold, convergence is determined, and the optimal solution is output. In summary, this scheme, through clustering dimensionality reduction to guide direction, multi-center bidirectional parallel exploration, adaptive step size shrinkage, and dual convergence criteria, ensures that each CFD evaluation generates maximum marginal benefit. The total number of evaluations is reduced by more than an order of magnitude compared to traditional hierarchical optimization methods, thus achieving rapid iterative optimization of aerodynamic shapes.

[0088] The above embodiments can be implemented, in whole or in part, by software, hardware, firmware, or any other combination thereof. When implemented in software, the above embodiments can be implemented, in whole or in part, as a computer program product. Those skilled in the art will recognize that the units and algorithm steps of the various examples described in conjunction with the embodiments disclosed herein can be implemented by electronic hardware, or a combination of computer software and electronic hardware. Whether these functions are implemented in hardware or software depends on the specific application and design constraints of the technical solution.

[0089] The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units; they may be located in one place or distributed across multiple network units. Some or all of the units can be selected to achieve the purpose of this embodiment, depending on actual needs.

[0090] The above description is merely a specific embodiment of this application, but the scope of protection of this application is not limited thereto. Any changes or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application.

Claims

1. A hierarchical evolution design method for aerodynamic shape based on clustering features, characterized in that, include: An initial particle swarm is generated based on the range of values ​​of multiple design optimization parameters for the aerodynamic shape to be optimized. CFD calculations are performed on each particle to obtain the objective function value. Based on the objective function value of each particle, clustering is performed, and the particle cluster with the optimal cluster center value is selected as the current optimal cluster. Principal component analysis is performed on the sample matrix composed of the design optimization parameters of the particles in the current optimal cluster. The direction corresponding to the eigenvector whose eigenvalues ​​meet the requirements is selected as the main search direction. The initial exploration step size is determined according to the variance contribution rate of each main search direction. Using the particles in the optimal cluster as the center point, test points are generated on both sides of the main direction of each center point according to the exploration step size. CFD is called for each test point to obtain the target function value after iteration. The iterative trial points are merged with the particles of the previous generation of optimal clusters, and the optimal clusters are re-clustered and iterated according to the objective function value. The offset between the centroid of the iterative optimal cluster and the centroid of the previous generation is calculated. Based on the relationship between the offset and the current exploration step size, the exploration step size of the next generation is adaptively adjusted. Repeat the above iterative process, determine the first convergence condition based on the size of the exploration step, and determine the second convergence condition based on the dispersion of the design optimization parameters of the optimal cluster. When the first and second convergence conditions are satisfied at the same time, select the particle with the best performance from the optimal cluster as the final design optimization parameters and output it.

2. The method according to claim 1, characterized in that, Each particle represents a set of design optimization parameters for the shape; The step of generating an initial particle swarm based on the value range of multiple design optimization parameters of the aerodynamic shape to be optimized is as follows: Based on multiple design optimization parameters of the aerodynamic shape to be optimized, the value range of all design optimization parameter types is determined, and the initial number of particles to be generated is set. Then, the value range of each design optimization parameter type is evenly divided into small intervals with the same number of initial particles. Next, each design optimization parameter type is randomly sampled without replacement. From all the small intervals of each design optimization parameter type, one interval is selected, and any value within that interval is taken as the value of the design optimization parameter type on the corresponding particle. After all design optimization parameters have been independently sampled, the values ​​of each design optimization parameter type on each particle are combined to form a complete set of design optimization parameter vectors. Finally, the generated particles are boundary checked to ensure that all values ​​do not exceed the preset range. If the values ​​exceed the limit due to numerical error, they are corrected to the nearest boundary value, resulting in an initial particle swarm that is evenly distributed in the design space.

3. The method according to claim 1, characterized in that, The steps for performing clustering are as follows: Using the objective function value corresponding to each particle as the basis for clustering, the number of clusters is set, and an initial number of cluster centers is randomly preset, the same as the number of clusters. For each cluster feature, the absolute difference between it and each cluster center is calculated, and it is assigned to the cluster represented by the center with the smallest difference. After all values ​​are assigned, the arithmetic mean of all cluster features in each cluster is recalculated and used as the new center of that cluster. The above assignment and update steps are repeated until the cluster centers no longer change or the preset maximum number of iterations is reached. The clusters are sorted according to the size of the center value of each cluster, and the cluster with the smallest center value is selected as the cluster with the optimal objective function value.

4. The method according to claim 1, characterized in that, The steps for obtaining the variance contribution rate of the main search direction are as follows: A matrix is ​​formed with the particle types in the optimal cluster as rows and the design optimization parameter types of each particle as columns. Take the average value of the design optimization parameter types for each column of the matrix; Subtract the mean of the design optimization parameter type from the value of each design optimization parameter type for each particle, so that the data is distributed around the origin, and a centered matrix is ​​obtained to eliminate the effect of translation. Using a centered matrix, the covariance between different types of design optimization parameters is calculated to obtain a covariance matrix. The diagonal of the matrix represents the variance of each variable, while the off-diagonal represents the covariance between the variables. Eigenvalue decomposition of the covariance matrix yields a set of eigenvalues ​​and corresponding eigenvectors. The magnitude of the eigenvalues ​​reflects the degree of dispersion of the data along the direction of the eigenvector. The top few eigenvectors with the largest eigenvalues ​​are selected as principal directions, and the variance contribution rate of each principal direction is calculated to measure the importance of each principal direction.

5. The method according to claim 1, characterized in that, The method for determining the initial exploration step size is as follows: Set a baseline step size value, and perform a linear mapping between the variance contribution rate of each principal direction and the baseline step size value to obtain the exploration step size corresponding to each principal direction.

6. The method according to claim 1, characterized in that, The strategy for generating test points along both the positive and negative sides of each main direction is as follows: Obtain the average distance between adjacent particles, determine the number of test points based on the average distance and the exploration step size, and based on the number of test points, move the center point along the positive and negative directions of the main direction according to the exploration step size to generate the corresponding number of test points.

7. The method according to claim 1, characterized in that, The method for calculating the offset of the centroid of the optimal cluster after iteration from the centroid of the previous generation is as follows: Take all particles from the previous generation's optimal cluster, calculate the average value of each design optimization parameter, and obtain the previous generation's centroid; Take all particles from the new optimal cluster obtained from the current iteration, calculate the average value of each design optimization parameter, and obtain the new centroid; Subtract the corresponding components of the new centroid from the old centroid, square them, sum them, and then take the square root to obtain the offset.

8. The method according to claim 1, characterized in that, The method for adaptively adjusting the exploration step size for the next generation is as follows: Based on the average step size of all main directions, calculate the ratio of the centroid displacement to the average value, and set an adjustment threshold. When the ratio is less than the set adjustment threshold, it is determined that the population has gathered in a stable region. Therefore, the exploration step size in each main direction is multiplied by a convergence factor less than 1 to achieve a more refined search. When the ratio is not less than the set adjustment threshold, the exploration step size remains unchanged, and the exploration continues at the same scale.

9. The method according to claim 1, characterized in that, The method for determining whether the first convergence condition is met is as follows: Obtain the maximum value of the exploration step size in all main directions, and set the step size convergence threshold. When this maximum value is less than the step size convergence threshold, it is determined that the current iteration number satisfies the first convergence condition.

10. The method according to claim 1, characterized in that, The method for determining whether the second convergence condition is met is as follows: For each design optimization parameter, calculate the difference between the maximum and minimum values ​​of the particles in the current optimal cluster for that design optimization parameter, and divide this difference by the corresponding range of the initial design space; When the relative range of all design optimization parameters is less than the preset ratio, the current iteration number is judged to meet the second convergence condition.

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