A physical-enhanced graph neural network-based aerodynamic optimization method

Abstract

This invention discloses an aerodynamic optimization method based on a physically augmented graph neural network (PBR), comprising: acquiring airfoil design requirements and multiple operating condition constraints; parameterizing the airfoil using a function-like / shape-function transformation method to generate candidate airfoil geometry; constructing a physically augmented graph structure based on discrete points on the candidate airfoil surface, and using node coordinates, local curvature, and unit normal vectors as node features; training the PBR using low-fidelity and high-fidelity aerodynamic data to obtain a surrogate model for predicting airfoil aerodynamic performance; using a differential evolution algorithm to iteratively optimize the surrogate model under multiple operating condition constraints; archiving historical trajectories of candidate airfoil parameters, predicted aerodynamic performance values, and constraint states during the optimization process, and extracting explicit design rules between airfoil parameters and aerodynamic performance through symbolic regression. This method can reduce the cost of high-fidelity simulation calls and improve the efficiency of multi-condition airfoil optimization.

Description

Technical Field

[0001] This invention relates to the field of aircraft aerodynamic shape optimization technology, specifically an aerodynamic optimization method based on physical augmented graph neural networks. Background Technology

[0002] Low-altitude long-endurance UAVs have high requirements for lift-to-drag ratio, cruise drag, stall margin, and structural volume. As the fundamental unit of wing aerodynamic performance, the airfoil's geometry directly affects cruise drag, climb efficiency, maneuverability, and overall endurance. In actual missions, UAVs often experience multiple flight states simultaneously, including cruise, climb, and maneuvering. Therefore, low-drag airfoils obtained under a single operating condition are prone to performance degradation after deviating from the design point, making it difficult to meet the multi-point robust design requirements of engineering applications.

[0003] Traditional airfoil optimization typically couples high-fidelity computational fluid dynamics solvers with global search algorithms such as genetic algorithms and differential evolution algorithms. While this approach can yield relatively reliable aerodynamic evaluation results, each candidate airfoil evaluation requires solving the governing equations. If the optimization process requires thousands of iterations, the computational cost is extremely high, making it difficult to quickly explore a large design space during the scheme design phase.

[0004] To reduce computational load, existing technologies often use surrogate models to replace high-fidelity solvers, such as Kriging, radial basis functions, support vector machines, multilayer perceptrons, or convolutional neural networks. While these methods can accelerate evaluation, they still have significant drawbacks: First, traditional statistical surrogate models are susceptible to the curse of dimensionality in high-dimensional parameter spaces; second, multilayer perceptrons treat airfoil parameters as ordinary vectors, making it difficult to express the spatial topological relationships between airfoil boundary nodes; and third, convolutional neural networks rely on regular Cartesian grids, requiring the continuous airfoil boundaries to be rasterized, which can easily lead to the loss of sub-pixel geometric details such as leading-edge radius and local curvature, resulting in increased drag prediction errors.

[0005] Graph neural networks can transmit information on non-Euclidean structures, making them suitable for representing airfoil boundary nodes and their neighborhood relationships. However, existing graph models often directly use coordinates or grid nodes as input, failing to explicitly embed physical and geometric priors such as curvature and unit normals, which directly influence boundary layer development, suction peaks, pressure recovery, and separation locations. Therefore, in predicting aerodynamic parameters like drag, which are highly sensitive to local geometric disturbances, the model may introduce significant uncertainties, which can be exploited by evolutionary algorithms during optimization, resulting in pseudo-optimal solutions with unreasonable shapes or weak physical meaning.

[0006] Furthermore, existing deep learning-based aerodynamic optimization methods can only output optimized discrete coordinates or aerodynamic coefficients, lacking transparency in the internal decision-making mechanisms and failing to provide designers with a clear analytical relationship between geometric parameters and aerodynamic performance. Engineering design not only requires finding a low-drag airfoil shape but also understanding why that shape is effective and transforming optimization experience into reusable design principles. Therefore, it is necessary to propose an airfoil optimization method that combines high efficiency, physical reliability, and interpretable knowledge extraction capabilities.

[0007] Existing airfoil optimization methods typically rely on extensive computational fluid dynamics simulations, resulting in high optimization costs. Traditional machine learning surrogate models often use airfoil parameters or images as input, making it difficult to fully utilize prior physical information such as curvature and normal vectors in airfoil geometry. Furthermore, deep learning surrogate models often exhibit black-box characteristics, making it difficult to reveal the explicit relationship between airfoil geometric parameters and aerodynamic performance. Therefore, there is an urgent need for an airfoil optimization method that can balance optimization efficiency, the ability to express physical constraints, and the interpretability of design principles. Summary of the Invention

[0008] The technical problem to be solved by this invention is that, in the current prediction of aerodynamic parameters, purely data-driven models may lack physical priors and are prone to producing pseudo-optimal or non-physical geometric problems in evolutionary optimization.

[0009] To solve the above-mentioned technical problems, the present invention provides the following technical solution:

[0010] An aerodynamic optimization method based on a physically enhanced graph neural network includes: Obtain airfoil design requirements and multiple operating condition constraints, parameterize the airfoil based on the function-like / shape function transformation method, and obtain the geometric parameters of candidate airfoils; Based on the candidate airfoil geometric parameters, discrete points on the airfoil surface are generated, and a physical augmentation graph structure is constructed based on the discrete points on the airfoil surface. The node features of the physical augmentation graph structure include at least node coordinates, local curvature, and unit normal vector. A surrogate model for predicting airfoil aerodynamic performance is obtained by multi-fidelity training of a physical augmented graph neural network based on low-fidelity and high-fidelity aerodynamic data. A fitness function is constructed using aerodynamic performance indicators and geometric constraint indicators under multiple operating conditions, and a differential evolution algorithm is used to call the surrogate model to iteratively optimize the candidate airfoil. Historical trajectory archiving is performed on candidate airfoil parameters, predicted aerodynamic performance, constraint satisfaction status, and fitness values ​​during the iterative optimization process; Symbolic regression is performed based on the historical trajectory archive results to extract the explicit design rules between airfoil parameters and aerodynamic performance, and the optimized airfoil and the explicit design rules are output.

[0011] Preferably, the physical enhancement graph structure is constructed, including: The airfoil boundary is discretized into N nodes, and an undirected spatial graph G=(V,E) is constructed as a physical augmentation graph; where each node Let E represent a sampling point on the airfoil boundary, where boundary E describes the local spatial interactions between nodes; and the feature vector of each node is explicitly embedded with local geometric-physical priors; where... Represented as airfoil boundary node index, For airfoil boundary nodes.

[0012] Preferably, the initial feature vector of each airfoil boundary node is represented in the following way: In the formula, Represented as the first The initial feature vectors of the nodes, Represented as the first The coordinates of each node, Represented as the first The curvature of each node, Represented as the first The unit normal vector components of each node. This is the matrix transpose.

[0013] Preferably, the surrogate model uses the Phys-GNN surrogate model, which includes a graph convolutional feature extraction layer, a hybrid readout layer, and a multi-task prediction layer; The graph convolutional feature extraction layer uses a normalized adjacency matrix with self-loops for neighborhood aggregation and feature propagation; the hybrid readout layer performs global average pooling and global max pooling on the feature vectors corresponding to the last layer nodes, and then concatenates the two to form the airfoil global latent vector; finally, the multi-task prediction layer outputs the drag coefficient and lift coefficient simultaneously through a multilayer perceptron.

[0014] Preferably, the training of the Phys-GNN surrogate model is divided into two stages. In the first stage, the complete Phys-GNN surrogate model is trained on low-fidelity aerodynamic data to obtain the initial parameters of the graph convolutional feature extraction layer and the hybrid readout layer, so that the Phys-GNN surrogate model can learn the basic mapping from a wide range of airfoil geometry to aerodynamic response. In the second stage, the graph convolutional feature extraction layer is frozen, and only the hybrid readout layer is updated on high-fidelity aerodynamic data.

[0015] Preferably, the candidate airfoil geometry parameters are obtained through the following methods: The CST method is used to parameterize the upper and lower surfaces of the airfoil, generating airfoil parameters based on the airfoil surface coordinates; For airfoil parameters, Latin hypercube sampling is used to generate initial airfoil samples within the given parameter boundaries, which serve as low-fidelity samples; Then, a clustering algorithm is used to select representative airfoil samples from the low-fidelity samples as high-fidelity samples.

[0016] Preferably, low-fidelity and high-fidelity aerodynamic data are obtained through the following methods: For low-fidelity samples, a global topological mapping between airfoil geometry and aerodynamic coefficients is learned based on a potential flow solver, a fast airfoil analysis tool, or a NeuralFoil-like model, and used as low-fidelity aerodynamic data. The Reynolds-averaged Navier-Stokes equations were used to solve the high-fidelity samples to obtain the lift coefficient and drag coefficient, which were then used as high-fidelity aerodynamic data.

[0017] Preferably, the curvature and unit normal vector are obtained as follows: based on the continuous airfoil profile defined by CST parameterization, the unit normal vector and curvature of the local element are analytically calculated; wherein, the unit normal vector is expressed as: ; The curvature is calculated using the standard differential geometric expression for a plane curve: ; In the formula, For unit normal vector, for The first derivative, Represented as airfoil surface coordinates, For chord coordinates, For matrix transpose, For curvature, for The first derivative.

[0018] Preferably, during the process of using the differential evolution algorithm to call the surrogate model to iteratively optimize the candidate airfoil, the airfoil samples that meet the constraints and have a fitness better than the corresponding parent individuals, along with their geometric features, aerodynamic prediction values, constraint satisfaction status, and fitness values, are recorded in the historical archive for subsequent symbolic pattern extraction.

[0019] Preferably, subsequent symbol pattern extraction includes: selecting a Pareto optimal subset H from historical archives based on aerodynamic performance and geometric diversity. The maximum curvature, maximum curvature location, CST leading edge radius, cross-sectional area, and maximum thickness location are used as inputs for symbolic regression. Symbolic regression uses the maximum curvature, maximum curvature location, CST leading edge radius, cross-sectional area, and maximum thickness location as candidate variables. It fits the implicit resistance variation law in the optimized trajectory of the Phys-GNN surrogate model through algebraic operations, trigonometric functions, exponential functions, and / or combinations.

[0020] Compared with the prior art, the beneficial effects of the present invention are: First, this invention constructs discrete points on the airfoil surface into a physical augmentation graph structure and introduces node coordinates, local curvature, and unit normal vectors into the node features, enabling the surrogate model to simultaneously characterize the airfoil boundary topological relationship and local physical geometric features, thereby improving the stability and physical reliability of aerodynamic performance prediction.

[0021] Second, this invention uses low-fidelity aerodynamic data and high-fidelity aerodynamic data for multi-fidelity training, enabling the surrogate model to obtain higher prediction accuracy with fewer high-fidelity samples, thereby reducing the number of high-fidelity CFD simulation calls and lowering the cost of airfoil optimization.

[0022] Third, this invention uses the differential evolution algorithm to call the trained Phys-GNN surrogate model for multi-condition constraint optimization, and uses the penalty function to handle lift constraints and thickness constraints, so that the optimization process can take into account multiple conditions such as cruise, climb and maneuver.

[0023] Fourth, this invention archives the historical trajectories of candidate airfoil parameters, predicted aerodynamic performance values, and constraint states during the optimization process, and uses symbolic regression to extract explicit design rules from the historical trajectories, so that the optimization results include not only the optimized airfoil, but also interpretable and reusable geometric-aerodynamic relationships. Attached Figure Description

[0024] Figure 1 This is a schematic diagram of the overall process of an airfoil method based on physical augmented graph neural network and symbolic regression according to an embodiment of the present invention.

[0025] Figure 2 This is a schematic diagram illustrating the mapping from the CST parametric design space to the physical augmentation graph representation in an embodiment of the present invention.

[0026] Figure 3 This is a schematic diagram of the parameterized airfoil geometry sample cluster generated by Latin hypercube sampling in an embodiment of the present invention.

[0027] Figure 4 This is a framework diagram of airfoil optimization based on a physical augmented graph neural network according to an embodiment of the present invention.

[0028] Figure 5 This is a schematic diagram of multi-fidelity transfer learning training according to an embodiment of the present invention.

[0029] Figure 6 This is a schematic diagram of differential evolution multi-condition constraint optimization and multi-fidelity historical trajectory convergence in an embodiment of the present invention.

[0030] Figure 7 This is a comparison chart of the prediction results of the proxy model under different node feature configurations in this embodiment of the invention. Detailed Implementation

[0031] To facilitate understanding of the technical solution of the present invention by those skilled in the art, the technical solution of the present invention will now be further described in conjunction with the accompanying drawings.

[0032] The terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Therefore, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature. In the description of this application, "multiple" means two or more, unless otherwise explicitly specified.

[0033] like Figure 1 As shown, this invention provides an aerodynamic optimization method based on a physical augmented graph neural network, comprising: design requirements / operating condition constraints 101 input to an airfoil parameterization module 102, which generates candidate airfoil geometry; a physical augmented graph construction module 103 constructing a physical augmented graph structure containing coordinates, curvature, and normal vectors based on the candidate airfoil geometry; a Phys-GNN multi-fidelity training module 104 training a surrogate model based on low-fidelity and high-fidelity aerodynamic data; a differential evolution multi-operating condition constraint optimization module 105 calling the trained surrogate model to perform multi-operating condition constraint optimization; a historical trajectory archiving module 106 recording candidate airfoils, aerodynamic performance, and constraint states during the optimization process; and a symbolic regression knowledge extraction module 107 extracting interpretable design patterns based on historical trajectories and outputting the optimized airfoil and design patterns 108. Specifically, the method includes: S1. Obtain airfoil design requirements and multiple operating condition constraints, and parameterize the airfoil based on the function-like / shape function transformation method to obtain the geometric parameters of the candidate airfoil.

[0034] S2. Generate discrete points on the airfoil surface based on the candidate airfoil geometric parameters, and construct a physical enhancement graph structure based on the discrete points on the airfoil surface, wherein the node features of the physical enhancement graph structure include at least node coordinates, local curvature, and unit normal vector.

[0035] S3. Based on low-fidelity and high-fidelity aerodynamic data, a physical augmentation graph neural network is trained in multiple fidelity modes to obtain a surrogate model for predicting airfoil aerodynamic performance.

[0036] S4. Construct a fitness function using aerodynamic performance indicators and geometric constraint indicators under multiple operating conditions, and use the differential evolution algorithm to call the surrogate model to iteratively optimize the candidate airfoil.

[0037] S5. Archive the historical trajectory of candidate airfoil parameters, predicted aerodynamic performance, constraint satisfaction status and fitness values ​​during the iterative optimization process.

[0038] S6. Based on the historical trajectory archive results, perform symbolic regression to extract the explicit design rules between airfoil parameters and aerodynamic performance, and output the optimized airfoil and the explicit design rules.

[0039] It should be noted that this embodiment uses the Eppler 638 airfoil as the initial airfoil and focuses on the cruise, climb, and maneuvering states of a low-altitude long-endurance UAV. This invention is not limited to this airfoil and can be extended to other low-speed airfoils, wing sections, or three-dimensional wing shape optimizations.

[0040] Please see Figure 2 The diagram illustrates the mapping from the CST parametric design space to the physical augmentation map representation. In one embodiment of the invention, airfoil parametricization and design space construction involve using the CST (Class Shape Transformation) method to parametricate the upper and lower surfaces of the airfoil. For the normalized chord coordinates... The airfoil surface coordinates can be expressed as: y ; ; ; In the formula, y For the airfoil surface coordinates, For chord coordinates, For class functions, The shape functions are composed of Bernstein basis functions. For trailing edge thickness, , These are the characteristics of a blunt leading edge and a sharp trailing edge, respectively. In this embodiment, their values ​​are set to 0.5 and 1, respectively. For the first One geometric weight parameter to be optimized. The coefficients are binomial coefficients. The order of the Bernstein polynomial is given. By adjusting the Bernstein coefficients on the upper and lower surfaces, the leading edge radius, maximum curvature, maximum thickness location, and morphology near the trailing edge can be continuously controlled.

[0041] In one embodiment of the present invention, the unit normal vector and curvature of the local element are analytically calculated based on the continuous airfoil profile defined by CST parameterization. The unit normal vector is expressed as: ; The curvature is calculated using the standard differential geometric expression for a plane curve: ; In the formula, For unit normal vector, for The first derivative, Represented as airfoil surface coordinates, For chord coordinates, For matrix transpose, For curvature, for The first derivative.

[0042] In one embodiment of the present invention, for airfoil parameters, Latin hypercube sampling is used to generate initial airfoil samples within a given parameter boundary as low-fidelity samples. Then, a clustering algorithm is used to select representative airfoil samples from the low-fidelity samples as high-fidelity samples. Please refer to [link to relevant documentation]. Figure 3 As shown, this is a schematic diagram of the parameterized airfoil geometry sample cluster generated by Latin hypercube sampling.

[0043] Specifically, 5,000 low-fidelity samples are first generated to cover the design space, and then 1,000 representative geometric samples are selected from this sample pool using K-means clustering for high-fidelity RANS (Reynolds-Averaged Navier-Stokes) calculation, thus balancing data coverage and physical accuracy.

[0044] In one embodiment of the present invention, multi-fidelity aerodynamic data is generated: the multi-fidelity aerodynamic data includes low-fidelity aerodynamic data (DLF) and high-fidelity aerodynamic data (DHF), wherein the low-fidelity aerodynamic data (DLF) provides a broad-coverage, low-cost training foundation, and the high-fidelity aerodynamic data (DHF) provides sparse but accurate physical correction information.

[0045] In this embodiment, for low-fidelity samples, a global topological mapping between airfoil geometry and aerodynamic coefficients is learned based on a potential flow solver, a fast airfoil analysis tool, or a NeuralFoil-like model, serving as low-fidelity aerodynamic data. For high-fidelity samples, the Reynolds-averaged Navier-Stokes equations are used to obtain lift and drag coefficients, serving as high-fidelity aerodynamic data. This is particularly useful for correcting nonlinear flow phenomena such as adverse pressure gradients, laminar separation bubbles, viscous drag, and high angle-of-attack stall, which are difficult to capture when training Phys-GNN surrogate models using low-fidelity aerodynamic data.

[0046] Please see Figure 4 As shown, in one embodiment of the present invention, the airfoil boundary is discretized into N nodes, and an undirected spatial graph G=(V,E) is constructed as a physical augmentation graph. Each node... This represents a sampling point on the airfoil boundary. The boundary E is used to describe the local spatial interactions between nodes, specifically established using the K-nearest neighbor method. Represented as airfoil boundary node index, These are the airfoil boundary nodes. The G structure in this graph directly acts on the sampling points of the continuous airfoil boundary, without needing to map the airfoil contour to a regular pixel grid. Therefore, it can reduce the leading edge curvature loss and boundary aliasing error caused by rasterization.

[0047] In this embodiment, the initial feature vector of each node not only includes normalized coordinates, but also explicitly embeds local geometric-physical priors, specifically: ; In the formula, Represented as the first The initial feature vectors of the nodes, Represented as the first The coordinates of each node, Represented as the first The curvature of each node, Represented as the first The unit normal vector components of each node. This is the matrix transpose. Curvature is used to highlight the leading edge, suction peak, and regions sensitive to adverse pressure gradients, while the unit normal vector is used to characterize the local inflow direction and the relative attitude of the boundary layer. Through these feature embeddings, the network can acquire a stronger physical perception ability when learning aerodynamic responses than with simple coordinate input.

[0048] In one embodiment of the present invention, the Phys-GNN surrogate model is constructed by building a Phys-GNN surrogate model based on a physically augmented graph representation. The Phys-GNN surrogate model includes a graph convolutional feature extraction layer, a hybrid readout layer, and a multi-task prediction layer.

[0049] The graph convolutional feature extraction layer uses a normalized adjacency matrix with self-loops for neighborhood aggregation, and its propagation form can be expressed as: ; In the formula, For the first Layer node feature matrix For the first The layer node feature matrix consists of the feature representations of all nodes. It is a non-linear activation function. To augment the adjacency matrix, for The corresponding angle matrix, , It is an adjacency matrix. It is the identity matrix. , The first The layer weights and biases are specifically trainable parameters. To establish topological connectivity, the edge set E defines the interaction paths between adjacent boundary elements along the discretized airfoil profile. Each boundary node... Each node is connected to its adjacent nodes in the ordered contour sequence, and bidirectional edges are used to achieve bidirectional information propagation. Accordingly, a sparse unweighted adjacency matrix is ​​defined. Multi-layer graph convolution progressively expands the receptive field of nodes, enabling shallow layers to capture high-frequency geometric details such as leading edge radius and local curvature, while deeper layers capture global topological features such as maximum curvature distribution and thickness distribution.

[0050] The hybrid readout layer performs global average pooling and global max pooling on the feature vectors corresponding to the last layer nodes, and then concatenates the two to form the airfoil's global latent vector. Among them, global average pooling is used to characterize the overall load trend, and global max pooling is used to highlight local separation, suction peaks, and geometric singularities.

[0051] Finally, the multi-task prediction layer synchronously outputs the drag and lift coefficients through a multilayer perceptron. The training loss uses weighted mean squared error with an L2 regularization term to balance the differences in lift and drag scales and suppress overfitting.

[0052] Please see Figure 5 As shown, in one embodiment of the present invention, multi-fidelity transfer learning training is performed in two stages when training the Phys-GNN surrogate model. In the first stage, a complete Phys-GNN surrogate model is trained on low-fidelity aerodynamic data to obtain the initial parameters of the graph convolutional feature extraction layer and the hybrid readout layer, enabling the Phys-GNN surrogate model to learn the basic mapping from a wide range of airfoil geometry to aerodynamic response. In the second stage, physical fine-tuning is performed on high-fidelity aerodynamic data (DHF). To retain the geometric topology awareness acquired in the first stage, the graph convolutional feature extraction layer is frozen, and only the readout layer parameters are updated, allowing the Phys-GNN surrogate model to learn the viscosity effect, separation penalty, and high angle-of-attack nonlinear correction contained in the high-fidelity data. This training method reduces the risk of overfitting caused by relying entirely on a small number of high-fidelity samples for training, while significantly reducing the number of high-fidelity samples required to build a high-precision surrogate model.

[0053] In one embodiment of the present invention, the multi-point aerodynamic optimization objective and constraints are as follows: Three representative operating conditions are set under the conditions of Ma=0.3 and Re=3×10^6: cruise angle of attack 3°, weight 0.6; climb angle of attack 6°, weight 0.3; and maneuver angle of attack 9°, weight 0.1. The optimization objective is to minimize the weighted average drag coefficient. ; In the formula, To optimize the objective, To optimize the design variable vector, different values ​​of which correspond to different airfoil geometries, , , These correspond to cruise, climb, and maneuvering conditions, respectively. , , These represent the drag coefficients at different cruise angles of attack, climb angles of attack, and maneuver angles of attack. , , These are the weighting coefficients for each working condition.

[0054] Simultaneously set lift and structural thickness constraints: weighted lift coefficient Maximum relative thickness And ensure that the design variables are within the upper and lower limits. To facilitate the differential evolution algorithm in handling constraints, an external penalty function method is introduced to transform the constrained problem into an augmented objective function: + ; In the formula, To broaden the objective function, , These are two penalty factors. This objective function prompts the evolutionary search to meet lift and structural volume requirements while reducing multi-condition weighted drag.

[0055] In one embodiment of the present invention, differential evolution search and trajectory archiving are based on a Phys-GNN surrogate model. The present invention does not limit the optimization method of the surrogate model; this application provides a method based on differential optimization: A differential evolution population P0={x1,…,xNp} is randomly initialized. For each target individual, different parent vectors are selected to perform mutation and crossover, generating candidate offspring u. k For each candidate offspring, airfoil boundaries are generated based on its CST parameters, curvature and unit normal vector are calculated, and a physical enhancement map G is constructed. k And input the trained Phys-GNN surrogate model to directly predict the C of the candidate airfoil under various operating conditions. d and C l Where k represents the index of the candidate airfoil.

[0056] If the candidate airfoil does not meet the lift or thickness constraints, a penalty is applied to the predicted drag, i.e., a penalty is imposed. If the constraints are met, the predicted weighted resistance is used as the fitness, i.e., execution... .

[0057] The parent is replaced only when the candidate offspring has a better fitness than the parent, and the successfully replaced airfoil sample, its geometric features, and aerodynamic predictions are recorded in the historical archive H. Since the archived samples all come from effective improvement steps in the optimization process, this dataset has a higher physical signal-to-noise ratio than randomly sampled data, making it suitable for subsequent symbolic pattern extraction.

[0058] In one embodiment of the present invention, knowledge extraction is based on symbolic regression of the optimized trajectory: after optimization, a Pareto optimal subset H is selected from the historical archive according to aerodynamic performance and geometric diversity. , as the input for symbolic regression; symbolic regression uses the maximum curvature c max Location of maximum curvature x c CST leading edge radius R CST Cross-sectional area A, location of maximum thickness x t Using geometric features as candidate variables, the implicit resistance variation patterns in the optimized trajectory of the Phys-GNN surrogate model are fitted by searching for combinations of algebraic operations, trigonometric functions, and exponential functions. Specifically, in this embodiment, the concise dominant law obtained by symbolic regression can be expressed as: C d ×100≈c max / x c +RCST+0.5436; This expression indicates that excessive camber gradient and an overly blunt leading-edge radius near the leading edge will increase drag on low-speed airfoils. This expression is merely an exemplary design principle derived from the optimized trajectory in this embodiment and does not constitute a limitation on the scope of protection of this invention. Further high-precision expressions can introduce variables such as cross-sectional area and maximum thickness location to describe the nonlinear coupling relationship between camber, thickness, and leading-edge radius. Therefore, this invention not only outputs optimized airfoils but also provides interpretable and reusable design criteria.

[0059] In one embodiment of the present invention, the optimization results and three-dimensional verification are as follows: The Eppler 638 baseline airfoil is optimized at multiple points using the method described above. High-fidelity verification results show that the optimized airfoil achieves drag reduction at cruise angles of attack of 3°, climb angles of attack of 6°, and maneuver angles of attack of 9°; the weighted average drag coefficient decreases from 0.0117 to 0.0111, a reduction of 5.13%; the weighted average lift-to-drag ratio increases by 3.37%, and the weighted lift meets the requirements. Constraints. As shown in Table 1.

[0060] Table 1. Airfoil optimization results

[0061] After extending the two-dimensional optimized airfoil to a three-dimensional finite wing and integrating it into the UAV model, high-fidelity three-dimensional RANS verification was performed under low-speed cruise conditions. The results show that the overall lift coefficient increased from 0.4512 to 0.4699, the drag coefficient decreased from 0.0285 to 0.0256, and the lift-to-drag ratio increased from 15.83 to 18.37, an improvement of 16.05%. See Table 2 for details.

[0062] Table 2 Comparison of airfoil configurations

[0063] Please see Figure 6 As shown, Figure 6 The diagrams show a comparison of loss convergence for different surrogate models during training, illustrating the training stability of the physically augmented graph neural network used in this invention for predicting airfoil aerodynamic performance. For comparison, this embodiment includes a multilayer perceptron model based on airfoil parameters or coordinate input, a convolutional neural network model based on a regular grid representation, a graph neural network model using only coordinate node features, and a graph neural network model incorporating local curvature and / or unit normal vectors into the node features. Figure 6 It can be seen that, under the same training data and conditions, the graph neural network incorporating the airfoil boundary topology exhibits a more stable convergence process compared to the ordinary vector model and the regular grid model. Furthermore, by adding local curvature and unit normal vectors to the node features, the model training loss is further reduced. These results demonstrate that constructing the discrete points of the airfoil boundary as a physically enhanced graph structure and explicitly embedding local geometric-physical priors helps improve the surrogate model's ability to represent the airfoil's aerodynamic response. Figure 6 This is only used to illustrate the training effect of the proxy model in this embodiment and does not constitute a limitation on the scope of protection of this invention.

[0064] Please see Figure 7 As shown, Figure 7 A comparison chart of surrogate model prediction results with different node feature configurations is provided to illustrate the impact of physically enhanced node features on the prediction accuracy of airfoil aerodynamic performance in this invention. Figure 7 The drag coefficient prediction plot and lift coefficient prediction plot are used to compare the consistency between the surrogate model predictions and the reference aerodynamic data; the mean square error index and the coefficient of determination index are used to quantitatively evaluate the prediction error and fit of different model configurations.

[0065] Depend on Figure 7It can be seen that when the graph neural network uses only the coordinates of the airfoil boundary nodes as input features, the model can obtain a certain predictive ability by utilizing the topological relationship of the airfoil boundary. When local curvature and unit normal vectors are further introduced as node features, the consistency between the prediction results and the reference aerodynamic data is improved, and the prediction errors of the drag coefficient and lift coefficient are reduced. This result shows that local curvature can reflect the geometric changes in regions such as the leading edge, suction surface, and pressure recovery section, and unit normal vectors can reflect the local attitude of the airfoil boundary. Using these two as node features can enhance the surrogate model's ability to identify aerodynamically sensitive areas. Figure 7 The results shown are used to verify the effectiveness of the physical augmentation map representation in this embodiment, and do not limit the invention to the use of this method. Figure 7 All listed comparison models or evaluation indicators.

[0066] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above, and that the invention can be implemented in other specific forms without departing from its spirit or essential characteristics. Therefore, the embodiments should be considered illustrative and non-limiting in all respects, and the scope of the invention is defined by the appended claims rather than the foregoing description. Thus, all variations falling within the meaning and scope of equivalents of the claims are intended to be included within the present invention, and no reference numerals in the claims should be construed as limiting the scope of the claims.

[0067] The above embodiments are merely examples of implementation methods of the invention. The scope of protection of the present invention is not limited to the above embodiments. For those skilled in the art, several modifications and improvements can be made without departing from the concept of the present invention, and these all fall within the scope of protection of the present invention.

Claims

1. A physical-enhanced graph neural network based aerodynamic optimization method, characterized in that, include: Obtain airfoil design requirements and multiple operating condition constraints, parameterize the airfoil based on the function-like / shape function transformation method, and obtain the geometric parameters of candidate airfoils; Based on the candidate airfoil geometric parameters, discrete points on the airfoil surface are generated, and a physical augmentation graph structure is constructed based on the discrete points on the airfoil surface. The node features of the physical augmentation graph structure include at least node coordinates, local curvature, and unit normal vector. A surrogate model for predicting airfoil aerodynamic performance is obtained by multi-fidelity training of a physical augmented graph neural network based on low-fidelity and high-fidelity aerodynamic data. A fitness function is constructed using aerodynamic performance indicators and geometric constraint indicators under multiple operating conditions, and a differential evolution algorithm is used to call the surrogate model to iteratively optimize the candidate airfoil. Historical trajectory archiving is performed on candidate airfoil parameters, predicted aerodynamic performance, constraint satisfaction status, and fitness values ​​during the iterative optimization process; Symbolic regression is performed based on the historical trajectory archive results to extract the explicit design rules between airfoil parameters and aerodynamic performance, and the optimized airfoil and the explicit design rules are output.

2. The physical-enhanced graph neural network based aerodynamic optimization method of claim 1, wherein, Constructing the physical augmentation graph structure includes: The airfoil boundary is discretized into N nodes, and an undirected spatial graph G=(V, E) is constructed as a physical enhancement graph; wherein each node represents a sampling point on the airfoil boundary, the boundary E is used to describe the local spatial interaction between nodes; and the feature vector of each node explicitly embeds the local geometric physical prior; wherein, wherein, represents the airfoil boundary node index, is the airfoil boundary node.

3. The aerodynamic optimization method based on a physically augmented graph neural network according to claim 2, characterized in that, The initial eigenvector of each airfoil boundary node is represented as follows: In the formula, Represented as the first The initial feature vectors of the nodes, Represented as the first The coordinates of each node, Represented as the first The curvature of each node, Represented as the first The unit normal vector components of each node. This is the matrix transpose.

4. The aerodynamic optimization method based on a physically augmented graph neural network according to claim 1, characterized in that, The surrogate model uses the Phys-GNN surrogate model, which includes a graph convolutional feature extraction layer, a hybrid readout layer, and a multi-task prediction layer; The graph convolutional feature extraction layer uses a normalized adjacency matrix with self-loops for neighborhood aggregation and feature propagation; the hybrid readout layer performs global average pooling and global max pooling on the feature vectors corresponding to the last layer nodes, and then concatenates the two to form the airfoil global latent vector; finally, the multi-task prediction layer outputs the drag coefficient and lift coefficient simultaneously through a multilayer perceptron.

5. The aerodynamic optimization method based on a physically augmented graph neural network according to claim 4, characterized in that, The training of the Phys-GNN surrogate model is divided into two stages. In the first stage, the complete Phys-GNN surrogate model is trained on low-fidelity aerodynamic data to obtain the initial parameters of the graph convolutional feature extraction layer and the hybrid readout layer, so that the Phys-GNN surrogate model can learn the basic mapping from a wide range of airfoil geometry to aerodynamic response. In the second stage, the graph convolutional feature extraction layer is frozen, and only the hybrid readout layer is updated on high-fidelity aerodynamic data.

6. The aerodynamic optimization method based on a physically augmented graph neural network according to claim 1, characterized in that, The candidate airfoil geometry parameters are obtained through the following method: The CST method is used to parameterize the upper and lower surfaces of the airfoil, generating airfoil parameters based on the airfoil surface coordinates; For airfoil parameters, Latin hypercube sampling is used to generate initial airfoil samples within the given parameter boundaries, which serve as low-fidelity samples; Then, a clustering algorithm is used to select representative airfoil samples from the low-fidelity samples as high-fidelity samples.

7. The aerodynamic optimization method based on a physically enhanced graph neural network according to claim 6, characterized in that, Low-fidelity and high-fidelity aerodynamic data are obtained through the following methods: For low-fidelity samples, a global topological mapping between airfoil geometry and aerodynamic coefficients is learned based on a potential flow solver, a fast airfoil analysis tool, or a NeuralFoil-like model, and used as low-fidelity aerodynamic data. The Reynolds-averaged Navier-Stokes equations were used to solve the high-fidelity samples to obtain the lift coefficient and drag coefficient, which were then used as high-fidelity aerodynamic data.

8. The aerodynamic optimization method based on a physically enhanced graph neural network according to claim 1, characterized in that, The curvature and unit normal vector are obtained as follows: based on the continuous airfoil profile defined by CST parametric methods, the unit normal vector and curvature of the local elements are analytically calculated; where the unit normal vector is expressed as: ； The curvature is calculated using the standard differential geometric expression for a plane curve: ； In the formula, For unit normal vector, for The first derivative, Represented as airfoil surface coordinates, For chord coordinates, For matrix transpose, For curvature, for The first derivative.

9. The aerodynamic optimization method based on a physically enhanced graph neural network according to claim 1, characterized in that, During the process of using the differential evolution algorithm to iteratively optimize the candidate airfoil by calling the surrogate model, the airfoil samples that meet the constraints and have a fitness better than the corresponding parent individuals, along with their geometric features, aerodynamic prediction values, constraint satisfaction status, and fitness values, are recorded in the historical archive for subsequent symbolic pattern extraction.

10. The aerodynamic optimization method based on a physically enhanced graph neural network according to claim 1, characterized in that, Subsequent symbol pattern extraction includes: selecting a Pareto optimal subset H from historical archives based on aerodynamic performance and geometric diversity. The maximum curvature, maximum curvature location, CST leading edge radius, cross-sectional area, and maximum thickness location are used as inputs for symbolic regression. Symbolic regression uses the maximum curvature, maximum curvature location, CST leading edge radius, cross-sectional area, and maximum thickness location as candidate variables. It fits the implicit resistance variation law in the optimized trajectory of the Phys-GNN surrogate model through algebraic operations, trigonometric functions, exponential functions, and / or combinations.

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