A Robust Optimization Method for Laminar Wings Based on Manifold Learning

Abstract

The present invention provides a robust optimization method for laminar wings based on manifold learning, belonging to the field of aircraft design. The method includes: giving an initial geometry and design variables as the starting point for the robust optimization of laminar wings; filtering the design space based on the coupling method of geometric smoothing and Grassmann manifold, and deforming the design object; using the inverse distance interpolation method to dynamically update the volume mesh to ensure that the volume mesh is consistent with the deformed geometry; calculating the flow field information according to the updated volume mesh; obtaining the derivative information about the design variables through a laminar discrete adjoint solver; obtaining the change direction of the design variables according to the derivative information, generating new design variables, and obtaining the robust optimization result of the laminar wing if the iteration result meets the feasibility tolerance and optimality tolerance. The present invention can improve the robustness of the laminar wing optimization, reduce the influence of the geometric parameterization form on the drag reduction effect of the optimization method, and obtain a more stable optimization result.

Description

Technical Field

[0001] The present invention belongs to the field of aircraft design, and particularly relates to a robust optimization method for laminar wings based on manifold learning. Background Technique

[0002] To improve the aerodynamic performance and economy of aircraft, laminar drag reduction technology has become a research focus in the aviation field. This technology significantly reduces aerodynamic drag by maintaining a wide laminar region on the aircraft surface, which is of great significance for achieving the goal of green aviation. However, the complexity of laminar wing optimization design is much higher than that of turbulent optimization design. Especially the existence of the laminar-turbulent transition phenomenon significantly increases the optimization difficulty of this problem. Non-gradient optimization methods are widely used in laminar wing design due to their strong versatility and easy implementation. However, as the problem dimension increases, their optimization efficiency rapidly decreases. In contrast, the discrete adjoint method (Gradient Optimization Method, GOM), as a gradient-based optimization method, can effectively decouple the computational complexity and the dimension of design variables, and has become an efficient tool for solving large-scale design problems.

[0003] Geometric parameterization methods, such as Free Form Deformation (FFD) and Class function / shape function transformation (CST) methods, have been widely used in aerodynamic optimization design. However, these methods often lead to large deformations in the design space, thus exacerbating the multi-extremum characteristics and increasing the solution difficulty of gradient optimization algorithms. Research shows that when there are multiple saddle points or flat regions in the design space, the gradient information may not be sufficient to guide the optimization algorithm to jump out of the local optimum, resulting in convergence to a suboptimal solution. Research points out that the multi-extremum characteristics of the design space with different geometric parameterization forms have a significant impact on the optimization results, and appropriate design constraints can alleviate this problem and improve the optimization effect. For the laminar wing aerodynamic optimization problem, the laminar-turbulent transition and the sensitivity of aerodynamic characteristics to flow field changes significantly enhance the influence of geometric distortion on the design space, resulting in a stronger nonlinear and multi-extremum characteristics of the design space. Further, complex constraint conditions may lead to the distortion or disconnection of the feasible region, making it difficult for the gradient optimization algorithm to jump out of the local optimal solution. Therefore, how to solve the non-convexity, strong nonlinearity and multi-extremum characteristics in the laminar wing optimization problem remains a major challenge in current research.

[0004] As a non-linear dimensionality reduction method, manifold learning can extract essential features from high-dimensional data and establish a non-linear mapping from high-dimensional space to low-dimensional space. Therefore, manifold learning provides a new idea for improving the efficiency of laminar wing gradient optimization design. During the aircraft design process, manifold learning is widely used in fields such as reducing the scope of the optimization search space and quickly predicting flow field characteristics through feature extraction.

[0005] At present, the research on optimization design based on manifold learning includes: The existing technology uses ISOMAP (Isometric Mapping Algorithm) and interpolation method to study the prediction of airfoil flow field in transonic state, and compares with the results of the reduced-order model based on Principal Component Analysis (PCA). The research shows that the non-linear reduced-order model based on manifold learning can significantly improve the prediction accuracy when dealing with the high-dimensional supersonic airfoil flow field with complex characteristics compared with the traditional linear dimensionality reduction method. In the optimization design, manifold learning effectively improves the performance of the optimization algorithm by extracting the low-dimensional manifold space. The L-ISOMAP (Benchmark Point Isometric Mapping Algorithm) proposed in the existing technology enhances the optimization efficiency of the optimization algorithm by using the low-dimensional manifold space and shows good convergence in the global optimization problem. Zhan Wei extracted the low-dimensional manifold structure of the decision space based on the Locally Linear Embedding (LLE) algorithm, thus accelerating the solution of the multi-objective optimization problem. At present, the research based on manifold learning method mainly focuses on turbulent aerodynamic optimization, and there is no research on aerodynamic optimization of laminar wings. Summary of the Invention

[0006] Aiming at the above deficiencies in the existing technology, a robust optimization method for laminar wings based on manifold learning provided by the present invention solves the problems of high non-linearity and multi-extremum characteristics existing in the optimization process of laminar wings.

[0007] In order to achieve the above object, the technical solution adopted by the present invention is: A robust optimization method for laminar wings based on manifold learning, comprising the following steps:

[0008] S1. Given the initial geometry and design variables as the starting point of the robust optimization of laminar wings;

[0009] S2. Generate an airfoil database that satisfies geometric constraints through Latin hypercube sampling, wherein all airfoils in the database satisfy the geometric constraints of the optimization design;

[0010] S3. Use the principle of the smoothing algorithm to eliminate geometrically distorted airfoils in the airfoil database;

[0011] S4. Conduct manifold learning based on the airfoil database after eliminating geometrically distorted airfoils, and establish an airfoil geometric characterization model based on Grassmann manifold;

[0012] S5. In the aerodynamic optimization design, use the constructed airfoil geometric characterization model to replace the Free Form Deformation (FFD) as the geometric parameterization method to generate perturbed airfoils;

[0013] S6. Based on the perturbed airfoils, use the geometric characterization model of Grassmann manifold to reconstruct the design space originally described by the Free Form Deformation (FFD) geometric parameterization method, and complete the deformation processing of the design object;

[0014] S7. Based on the manifold learning processing results of S2 - S6, perform deformation processing on the grid and calculate the flow field, and through iterative optimization processing, obtain a robust optimization result for the laminar wing.

[0015] The beneficial effects of the present invention are as follows: Through geometric fairing and design space reconstruction, the present invention improves the multi - extreme value characteristics of the feasible region, reduces the sensitivity of the optimization effect to the change of the FFD geometric parameterization form of the original optimization problem, and improves the robustness of the optimization efficiency of the optimization method. Through the geometric representation method of the Grassmann manifold, the present invention maps the design space of the original optimization problem to a compact low - dimensional manifold space, eliminates geometric constraints, improves the non - convexity, strong non - linearity and multi - extreme value characteristics of the design problem, and provides a basis for efficient optimization.

[0016] Furthermore, the S5 includes the following steps:

[0017] A1. In aerodynamic optimization design, use the Laplacian smoothing algorithm to perform geometric fairing on the perturbed airfoil generated by the free - form deformation (FFD) geometric parameterization method. Among them, for the displacement value i of the th free - form deformation FFD point, adopt the Laplacian fairing algorithm to weight the displacement value with the displacement values of its adjacent points and update it to : ; where represents the updated displacement value, represents the weight coefficient, represents the number of adjacent points for geometric fairing, which is used to control the degree of geometric fairing, j represents the adjacent point number;

[0018] A2. Based on the airfoil samples processed in A1, by establishing the mapping between the real space and the Grassmann manifold space , represent the discrete airfoil shape as a Grassmann manifold element. Among them, n represents the real - space dimension of the Grassmann manifold, represents the manifold space, represents n the 2 - dimensional Grassmann manifold space of the

[0019] A3. Based on the Grassmann manifold element, obtain the mapping relationship between the physical coordinates of the airfoil and the Grassmann manifold element ;

[0020] A4. Construct the mapping relationship between the known manifold space and the tangent space of the data - center element, including logarithmic mapping and exponential mapping;

[0021] A5. Determine the Riemannian center of the data points by performing iterative calculations in the tangent space of the manifold;

[0022] A6. Based on the mapping relationship in A3, map all manifold elements to the tangent space of the Riemannian center through logarithmic mapping;

[0023] A7. Based on the mapping result in A6, select the first eigenvectors as the low-dimensional orthogonal basis according to the degree of eigenvalue decay in the tangent space;

[0024] A8. For , use the following formula to define a dimensional Grassmann submanifold with the linear combination of all principal components and construct the mapping relationship from the parameter space to the airfoil coordinates in the real domain space: ; ; where represents a normal coordinate in the parameter space, represents dimensional Grassmann submanifold, represents a Grassmann manifold element, represents the exponential mapping operation, represents the distance from a point to the center of the manifold space, represents the operator that converts vector splicing into a matrix;

[0025] A9. Based on the mapping relationship in A8, generate a perturbed airfoil by changing the parameter space coordinate .

[0026] The beneficial effect of the above further solution is that the present invention uses the constructed airfoil geometric characterization model to replace the free-form deformation FFD as the geometric parameterization method, ensuring that there will be no airfoils with serious geometric distortion during sample generation.

[0027] Furthermore, the expression of the mapping relationship in A3 is as follows: ; ; ; ; where represents the coordinates of the airfoil in the real space, represents the matrix formed by the centers of the airfoil data points, represents the linear transformation matrix, represents dimensional identity matrix, represents the diagonal matrix, represents the center of the airfoil data points, Indicates the number of airfoil data points, Indicates the transpose of the airfoil coordinate matrix, Indicates The identity matrix of dimension Indicates the transpose of the orthogonal matrix obtained by singular value decomposition.

[0028] Furthermore, the expression for the Riemannian center of the data points in A5 is as follows: ; where Indicates determining the Riemannian center of the data points, Indicates a data point, Indicates a data point on the Riemannian manifold, Indicates the distance from all data points on the Riemannian manifold to the Riemannian center of, Indicates the total number of data points on the Riemannian manifold, Indicates the manifold space.

[0029] Furthermore, the expression for the mapping relationship in A8 is as follows: ; where Indicates the airfoil data point matrix, Indicates the mean of the linear transformation matrices corresponding to all sample points, Indicates the mean of the translation amounts of all sample points, Indicates the Grassmann manifold element obtained by perturbation.

[0030] Furthermore, S7 includes the following steps:

[0031] S701, Mesh deformation: Based on the manifold learning processing results of S2 - S6, use the inverse distance interpolation method to dynamically update the volume mesh to ensure that the volume mesh is consistent with the deformed geometry;

[0032] S702, Flow field calculation: Calculate the flow field information according to the updated volume mesh;

[0033] S703, Gradient information acquisition: Based on the calculation result of the flow field information, obtain the derivative information about the design variables through the laminar discrete adjoint solver;

[0034] S704, Optimization iteration: Obtain the change direction of the design variables according to the derivative information, generate new design variables, and return to S2 for iterative processing;

[0035] S705, Optimization result output: If the iteration result meets the feasibility tolerance and optimality tolerance, obtain the robust optimization result of the laminar airfoil.

[0036] Furthermore, S701 is specifically:

[0037] B1. Based on the manifold learning processing results of S2 - S6, use the inverse distance interpolation method to update the volume grid coordinates according to the changes in the surface grid;

[0038] B2. Based on the updated results, use automatic differentiation to calculate the derivatives of the volume grid nodes with respect to the surface grid pattern, ensuring that the volume grid is consistent with the deformed geometry.

[0039] The beneficial effects of the above - mentioned further solution are as follows: The present invention uses the inverse distance interpolation method to dynamically update the volume grid, realizing automatic high - precision spatial grid deformation based on the surface grid, providing a basis for automatic optimization calculation. BRIEF DESCRIPTION OF THE DRAWINGS

[0040] Figure 1 It is a flowchart of the method of the present invention.

[0041] Figure 2 It is a schematic diagram of airfoil geometry fairing based on the Laplace smoothing algorithm.

[0042] Figure 3 It is a schematic diagram of the FFD control box.

[0043] Figure 4 It is a schematic diagram for comparing optimized airfoils.

[0044] Figure 5 It is a schematic diagram for comparing the pressure distributions of optimized airfoils.

[0045] Figure 6 It is a schematic diagram of the FFD box F1 with non - uniform distribution of control points.

[0046] Figure 7 It is a schematic diagram of the FFD box F2 with uniform distribution of control points.

[0047] Figure 8 It is a schematic diagram for comparing the results of optimized airfoils.

[0048] Figure 9 It is a schematic diagram for comparing the results of optimized pressure distributions.

[0049] Figure 10 For the optimized airfoil N TS It is a schematic diagram for comparing the magnification factor envelopes. DETAILED DESCRIPTION OF THE INVENTION

[0050] The following describes the specific embodiments of the present invention to facilitate those skilled in the art of the present technology to understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific embodiments. For those of ordinary skill in the art of the present technology, as long as various changes are within the spirit and scope of the present invention defined and determined by the appended claims, these changes are obvious, and all inventions made using the concept of the present invention are within the scope of protection.

[0051] Embodiment

[0052] The object of the present invention is to solve the problems of high non - linearity and multi - extreme value characteristics existing in the optimization process of laminar wings, construct a robust optimization method for laminar wings, and provide a robust gradient optimization method for the design work of laminar wings. As Figure 1 shown, the present invention provides a robust optimization method for laminar wings based on manifold learning, which is characterized by including the following steps:

[0053] S1. Input the initial geometry: Given the initial geometry shape and design variables as the starting point of the robust optimization of laminar wings;

[0054] Design space reconstruction: Filter the design space based on the coupling method of geometric fairing and Grassmann manifold, and deform the design object. The implementation method is as follows:

[0055] S2. Generate an airfoil database that satisfies geometric constraints through Latin - hypercube sampling, where all airfoils in the database satisfy the geometric constraints of the optimized design;

[0056] S3. Use the smoothing algorithm principle to eliminate geometrically distorted airfoils in the airfoil database;

[0057] S4. Conduct manifold learning based on the airfoil database after eliminating geometrically distorted airfoils, and establish an airfoil geometric characterization model based on Grassmann manifold;

[0058] S5. In the aerodynamic optimization design, use the constructed airfoil geometric characterization model to replace the free - form deformation FFD as the geometric parameterization method to generate perturbed airfoils. The implementation method is as follows:

[0059] A1. In the aerodynamic optimization design, use the Laplace smoothing algorithm to perform geometric fairing on the perturbed airfoils generated by the free - form deformation FFD geometric parameterization method. Among them, the displacement value i of the th free - form deformation FFD point adopts the Laplace fairing algorithm, weights the displacement value with the displacement values of adjacent points of this point, and updates it to : ; where represents the updated displacement value, represents the weight coefficient, j represents the number of adjacent points for geometric fairing, which is used to control the degree of geometric fairing,

[0060] A2. Based on the airfoil samples processed in A1, by establishing the real - space and the Grassmann manifold space The mapping represents a discrete airfoil shape as an element of a Grassmann manifold, where n represents the real space dimension of the Grassmann manifold, represents the manifold space, represents n the 2-dimensional Grassmann manifold space of an n-dimensional real space;

[0061] A3. Based on the Grassmann manifold element, obtain the physical coordinates of the airfoil and the mapping relationship between the Grassmann manifold element : ; ; ; ; where represents the coordinates of the airfoil in the real space, represents the matrix formed by the centers of the airfoil data points, represents the linear transformation matrix, represents the n×n identity matrix, represents the diagonal matrix, represents the center of the airfoil data points, represents the number of airfoil data points, represents the transpose of the airfoil coordinate matrix, represents the m×m identity matrix, represents the transpose of the orthogonal matrix obtained by singular value decomposition;

[0062] A4. Construct the mapping relationship between the known manifold space and the tangent space of the data center element, including the logarithmic mapping and the exponential mapping;

[0063] A5. Determine the Riemannian center of the data points by performing iterative calculations in the tangent space of the manifold: ; where represents determining the Riemannian center of the data points, represents a data point, represents the data point on the Riemannian manifold, represents the distance from all data points on the Riemannian manifold to the Riemannian center , represents the total number of data points on the Riemannian manifold, represents the manifold space;

[0064] A6. Based on the mapping relationship in A3, map all manifold elements to the tangent space of the Riemannian center through the logarithmic mapping;

[0065] A7. Based on the mapping result in A6, select the first Eigenvector As a low-dimensional orthogonal basis;

[0066] A8. For , using the following formula, the linear combination of all principal components Defines a -dimensional Grassmann submanifold, and constructs the mapping relationship from the parameter space to the airfoil coordinates in the real domain space: ; ; where, Represents a normal coordinate in the parameter space, Represents -dimensional Grassmann submanifold, Represents the Grassmann manifold element, Represents the exponential mapping operation, Represents a point to the distance from the center of the manifold space, Represents the operator that converts vector splicing into a matrix;

[0067] The expression of the mapping relationship is as follows: ; where, Represents the airfoil data point matrix, Represents the mean of the linear transformation matrices corresponding to all sample points, Represents the mean of the translation amounts of all sample points, Represents the Grassmann manifold element obtained by perturbation

[0068] A9. Based on the mapping relationship in A8, by changing the parameter space coordinates Generate a perturbed airfoil;

[0069] S6. Based on the perturbed airfoil, using the geometric characterization model of the Grassmann manifold, reconstruct the design space described by the original free-form deformation FFD geometric parameterization method to complete the deformation processing of the design object;

[0070] In this embodiment, when generating airfoil samples, geometric smoothing is first performed, and then Grassmann manifold deformation is performed on the basis of smoothing to couple geometric smoothing with the Grassmann manifold.

[0071] S7. Mesh deformation: Based on the manifold learning processing results of S2 - S6, perform mesh deformation processing and flow field calculation, and through iterative optimization processing, obtain a robust optimization result of the laminar airfoil. The implementation method is as follows:

[0072] S701. Mesh deformation: Based on the manifold learning processing results of S2 - S6, use the inverse distance interpolation method to dynamically update the volume mesh to ensure that the volume mesh is consistent with the deformed geometric shape. The implementation method is as follows:

[0073] B1. Based on the manifold learning processing results of S2 - S6, using the inverse distance interpolation method, update the volume grid coordinates according to the changes in the surface grid;

[0074] B2. Based on the updated results, use automatic differentiation to calculate the derivatives of the volume grid nodes with respect to the surface grid pattern to ensure that the volume grid is consistent with the deformed geometry;

[0075] S702. Flow field calculation: Calculate the flow field information according to the updated volume grid;

[0076] S703. Gradient information acquisition: Based on the calculation results of the flow field information, obtain the derivative information about the design variables through a laminar discrete adjoint solver;

[0077] S704. Optimization iteration: Obtain the change direction of the design variables according to the derivative information, generate new design variables, and return to S2 for iterative processing;

[0078] S705. Optimization result output: If the iteration results meet the feasibility tolerance and optimality tolerance, obtain the robust optimization results of the laminar airfoil.

[0079] In this embodiment, the airfoil geometry fairing method is introduced as follows:

[0080] Use the Laplace smoothing algorithm to perform geometric fairing on the perturbed airfoil generated by the free - form deformation FFD geometric parameterization method. As Figure 2 shown in the figure, in the figure, y represents y the x - coordinate, c represents the chord length, x represents x the y - coordinate. The displacement value i of the nth free - form deformation FFD point is weighted with the displacement values of the adjacent points of this point and updated to using the Laplace fairing algorithm. The calculation formula is: ; where, represents the updated displacement value, represents the weight coefficient, represents the number of adjacent points for geometric fairing, which is used to control the degree of geometric fairing, j represents the adjacent point number, which is used to iterate through all adjacent points of the i nth free - form deformation FFD point to accumulate the displacement values.

[0081] By smoothing the deformation displacement of the free-form deformation (FFD) control points, the configurations with severe geometric distortion generated by arbitrary deformation of the free-form deformation (FFD) in the design space can be eliminated, and the airfoil smoothness of the sample set for manifold learning can be improved.

[0082] In this embodiment, the geometric representation method based on the Grassmann manifold is introduced as follows:

[0083] Grassmann manifold learning separates the linear transformation and high-order perturbation deformation in the geometric deformation of aerodynamic design, extracts features in the design space containing only high-order surface perturbations, and can achieve dimensionality reduction. At the same time, this method can effectively control geometric constraints, construct a low-dimensional and compact design space that meets geometric constraints, thereby reducing the nonlinearity of the design space and improving the optimization performance of the gradient optimization algorithm.

[0084] The linear transformation deformation of the airfoil can be expressed as , is invertible square matrix, which represents scaling, affine, and rotational linear deformations, is invertible matrix set, briefly denoted as . represents the matrix assembled by the airfoil data point coordinates along the arc length direction. The spatial position is translated by the vector . The expression of the linear transformation matrix is as follows: ; where controls the scaling deformation, , controls the scaling deformation, controls the twisting deformation, , , and respectively represent the coefficients of the deformation matrix. For example, is the equal ratio scaling ratio, are the horizontal and vertical scaling ratios, represents the twisting angle.

[0085] To separate the linear transformation, the linear transformation represented by can be removed from the real space , and a set of parameters can be constructed to describe the high-order surface deformation. In the prior art, a method for mapping a discrete data set to the Grassmann space is proposed. This space is dimensional real space of a d - dimensional subspace, so that when sampling on the Grassmann space, the profile deformation is independent of the linear transformation. By representing the discrete airfoil shape as an element of the Grassmann manifold, the profile deformation in this space is separated from the main linear transformations in aerodynamics (changing thickness, chord length, camber, etc.).

[0086] To represent the airfoil geometry as an element of the Grassmann manifold, the LandmarkAffine Standardization (LAS) method is used to establish the mapping between the real space and the Grassmann manifold space of dimension n representing the dimension of the real space of the Grassmann manifold.

[0087] The coordinates of the airfoil in the real space are , let represent the center of the airfoil data points. The airfoil data is centered and singular value decomposition is performed: ; where . Define the linear transformation matrix as: .

[0088] The mapping relationship between the physical coordinates of the airfoil and the element of the Grassmann manifold (this mapping relationship is applied in A6 - A8 for the mapping from the parameter space to the airfoil coordinates in the real domain space) is: ; where represents the coordinates of the airfoil in the real space, represents the matrix formed by the centers of the airfoil data points, represents the element of the Grassmann manifold, represents the linear transformation matrix, represents an identity matrix of dimension used to process the center of the airfoil data points to make the dimension correspond to X, represents the center of the airfoil data points, represents the number of airfoil data points, represents the transpose of the airfoil coordinate matrix, represents an identity matrix of dimension represents the transpose of the orthogonal matrix obtained by singular value decomposition.

[0089] To the element Perform parametric description, utilize the extension of principal component analysis (PCA) on Riemannian manifolds, and establish a geometric feature representation method based on manifold learning.

[0090] To perform the extension of PCA on Riemannian manifolds, a mapping relationship between the known manifold space and the tangent space of the data center element needs to be established. Using the logarithmic mapping , map the points on the manifold space to the tangent space . The logarithmic mapping maps the point to the point in the tangent space , representing the tangent vector of the shortest path from the point to the point . Using the inverse mapping, i.e., the exponential mapping , map the vectors in the tangent space back to the manifold space. The exponential mapping transforms the velocity vector in the tangent space into a trajectory along the geodesic on the manifold. Starting from a point in the tangent space and moving along the geodesic in the direction of the tangent vector, a new point on the manifold is obtained.

[0091] To implement PCA on Riemannian manifolds, it is necessary to centralize the data on the manifold and extract the orthogonal principal directions on the manifold. In the present invention, the Riemannian center of the data points is determined by iterative calculation in the tangent space of the manifold. Given the data points , , the Karcher mean is the point that minimizes the sum of the distances from all points to the Riemannian center , i.e.: ; where represents determining the Riemannian center of the data points, represents a data point, represents the data points on the Riemannian manifold, represents the distances from all data points on the Riemannian manifold to the Riemannian center , represents the total number of data points on the Riemannian manifold, represents the manifold space, i represents the i th data point.

[0092] Map all manifold elements to the tangent space of the Riemannian center through the logarithmic mapping, and then perform PCA in the tangent space. According to the degree of eigenvalue decay in the tangent space, select the first eigenvectors as the low-dimensional orthogonal basis. In the normal coordinates of the tangent space Represents the projection corresponding to the main direction, i.e., the normal coordinate Defines the parameter space .

[0093] For , the linear combination of all principal components Defines a -dimensional Grassmann submanifold: ; where , Represents a normal coordinate in the parameter space, Represents -dimensional Grassmann submanifold, Represents an element of the Grassmann manifold, Represents the exponential mapping operation, which maps to an element of the Grassmann manifold , Represents a point to the distance from the center of the manifold space, Represents the operator that converts vector concatenation into a matrix.

[0094] Finally, a mapping relationship from the parameter space to the airfoil coordinates in the real domain space is established: ; where Represents the airfoil data point matrix, Represents the mean of the linear transformation matrices corresponding to all sample points, Represents the mean of the translation amounts of all sample points. By changing the parameter space coordinates a perturbed airfoil is generated.

[0095] In this embodiment, the design space filtering method based on the coupling method of geometric fairing and Grassmann manifold is as follows:

[0096] Through Latin hypercube sampling, an airfoil database that satisfies geometric constraints is generated using the free-form deformation FFD geometric parameterization method coupled with the Laplace smoothing algorithm. The airfoils in the database all satisfy the geometric constraints of the optimal design, and the geometrically distorted airfoils are eliminated according to the principles of the smoothing algorithm. Manifold learning is performed using the database to establish an airfoil geometric characterization model based on the Grassmann manifold: ; where Represents the airfoil data point matrix, Represents an element of the Grassmann manifold obtained through perturbation, Represents the mean of the linear transformation matrices corresponding to all sample points, Represents the mean of the translation amounts of all sample points. This airfoil geometric characterization model represents the mapping from the parameter space The mapping relationship to the airfoil coordinates in the real domain space, and the airfoil geometric representation varies with different airfoil points.

[0097] In aerodynamic optimization design, the established geometric representation model is used to replace the free-form deformation (FFD) as the geometric parameterization method to generate perturbed airfoils. Finally, the geometric representation model based on the Grassmann manifold realizes the reconstruction of the design space originally described by the FFD geometric parameterization method, and reconstructs the design space that satisfies geometric constraints and has smooth airfoils in the low-dimensional compact manifold space. The geometric design variables are transformed from the perturbation displacements of the FFD control points in the original optimization problem to the parameter space coordinates. .

[0098] In this embodiment, for the robust optimization method of laminar wings, a manifold learning enhanced gradient optimization for laminar wings (MLE-GOM) framework is adopted, which includes a design space reconstruction method, mesh deformation, a flow field solver, a gradient optimization algorithm, and a laminar discrete adjoint solution:

[0099] First, parametric deformation is performed based on the design space reconstruction method. This method filters the design space and deforms the design object based on the coupling method of geometric smoothing and the Grassmann manifold.

[0100] The next step is to modify the volume mesh to be consistent with the updated geometric shape, that is, IDW mesh deformation. When modifying, the inverse distance weighting method is used to update the volume mesh coordinates according to the changes in the surface mesh, and automatic differentiation is used to calculate the derivatives of the volume mesh nodes with respect to the surface mesh pattern.

[0101] After that, a flow field solver based on the RANS equation is used to calculate the flow field.

[0102] Finally, a laminar discrete adjoint solver and an optimizer based on the gradient method are used to handle a large number of design variables and constraints in the optimization problem.

[0103] Using the above method, the present invention describes a manifold learning enhanced gradient optimization framework for laminar wings. To start this process, an initial geometry with given design variables is given, then the design space reconstruction method is applied to obtain geometric deformation variables, then the IDW mesh deformation method is applied to modify the volume mesh, and subsequently the RANS solver will calculate the flow field information based on the volume mesh, while the transition solver will obtain information such as the growth factor and transition position. The derivative information about the design variables is obtained through the laminar discrete adjoint solver. Up to this step, the derivatives required by the optimizer are ready. Using the optimization algorithm, the optimizer will provide new design variable values, and the optimization will iterate until the feasibility tolerance and optimality tolerance are met.

[0104] In this embodiment, the aerodynamic optimization of turbulent airfoils is as follows:

[0105] The RAE 2822 airfoil is selected for aerodynamic optimization under fully turbulent conditions. The optimization conditions are as follows: , , and the optimization goal is to minimize the drag coefficient C d under the optimized conditions. Thickness constraints are imposed on the airfoil, with the constrained thickness not less than 90% of the initial thickness, and a total of 10 thickness constraints are applied. At the same time, volume constraints are imposed on the airfoil, with the constrained volume not less than 95% and not greater than 102% of the initial volume. represents the Mach number, represents the lift coefficient.

[0106] As Figure 3 shown, the free-form deformation FFD geometric parameterization method is used to perturb the RAE2822 airfoil to generate a perturbed airfoil sample library that meets the volume and thickness constraints, with a sample size of 7000. There are 7 perturbation variables on each of the upper and lower surfaces of the free-form deformation FFD control box, for a total of 14 geometric perturbation variables, and the perturbation range of each control point is ±0.02. Based on the generated perturbed airfoil sample library, a manifold learning method is used for design space reconstruction. The 14 airfoil free-form deformation FFD geometric perturbation variables are reduced to 10-dimensional geometric design variables. After the design space reconstruction, the original optimization design problem is transformed into an aerodynamic optimization problem without geometric constraints.

[0107] The classical discrete adjoint gradient optimization method (GOM) with 14-dimensional free-form deformation FFD geometric design variables and the manifold learning enhanced laminar airfoil gradient optimization method (MLE-GOM) with 10-dimensional geometric design variables are used for aerodynamic optimization design respectively. Among them, the airfoil angle of attack is also used as an optimization design variable.

[0108] The optimization results are shown in Table 1. Compared with the initial airfoil, the drag coefficient of the airfoil optimized by the classical discrete adjoint gradient optimization method (GOM) is reduced by 38.7 counts, a decrease of 25.27%, and the drag coefficient of the airfoil optimized by MLE-GOM is reduced by 37.1 counts, a decrease of 24.25%. The optimization drag reduction benefit of the laminar airfoil gradient optimization method (MLE-GOM) is slightly less than that of the traditional classical discrete adjoint gradient optimization method (GOM).

[0109] Table 1

[0110]

[0111] There are significant differences in the geometric and flow field characteristics of the airfoils optimized by the laminar airfoil gradient optimization method (MLE-GOM) and the classical discrete adjoint gradient optimization method (GOM). As Figure 4 and Figure 5 shown, Figure 4 in x / cdenote x the ratio of the chordwise coordinate to the chord length y / c denote y the ratio of the chordwise coordinate to the chord length Figure 5 in x / c is x the ratio of the chordwise coordinate to the chord length, and Cp is the pressure coefficient. Compared with the initial RAE2822 airfoil, both optimization methods result in shock - free and low - drag configurations, and the geometric and pressure distribution patterns on the upper airfoil surface have a high degree of coincidence. However, the characteristics of the lower airfoil surface are significantly different. For the laminar - wing gradient optimization method (MLE - GOM), the front - loading of the optimized airfoil decreases, while the rear - loading increases significantly. For the classical discrete adjoint gradient optimization method (GOM), the front - loading of the optimized airfoil slightly increases, and the rear - loading slightly decreases. The geometric fairing and design - space reconstruction of the laminar - wing gradient optimization method (MLE - GOM) cause changes in the feasible region and multi - extremum characteristics of the original optimization problem, which is one of the main reasons for the difference in the optimization results between it and the classical discrete adjoint gradient optimization method (GOM).

[0112] Based on the robust optimization method for laminar wings constructed above, the application verification of turbulent airfoil optimization is completed. The verification shows that there are slight differences in the drag - reduction effects between the laminar - wing gradient optimization method (MLE - GOM) and the traditional discrete - adjoint - based gradient optimization method (GOM), but both can obtain shock - free and low - drag configurations.

[0113] In this embodiment, the aerodynamic optimization of the natural laminar airfoil is as follows:

[0114] To study the influence of the perturbation form of the geometric parameterization method on the optimization efficiency of the natural laminar - wing gradient optimization design, and to compare the differences in the optimization efficiency between the laminar - wing gradient optimization method (MLE - GOM) and the traditional discrete - adjoint - based gradient optimization method (GOM), the RAE 2822 airfoil is selected for aerodynamic optimization under laminar flow conditions. The airfoil is parameterized using the FFD control box "F1" with non - uniform distribution of control points and the FFD control box "F2" with uniform distribution of control points along the chord. The number of geometric perturbation variables of the control boxes "F1" and "F2" is 14, as Figure 6 and Figure 7 shown. The optimization conditions are: , , and the optimization goal is to minimize the drag coefficient C d under the optimized state. Thickness constraints are imposed on the airfoil, with the constraint that the thickness is not less than 90% of the initial thickness, and a total of 10 thickness constraints are applied. At the same time, volume constraints are imposed on the airfoil, with the constraint that the volume is not less than 95% and not greater than 102% of the initial volume, denotes the Mach number, denotes the lift coefficient.

[0115] The natural laminar airfoil aerodynamic optimization design is carried out by using the contrast laminar wing gradient optimization method (MLE-GOM). The airfoil geometry is perturbed respectively by using the "F1" and "F2" free-form deformation FFD parameterization boxes to generate a perturbed airfoil sample library that satisfies the volume and thickness constraints. The capacity of each sample library is 7000. Manifold learning is used to reconstruct the design space, reducing the 14 airfoil free-form deformation FFD geometric perturbation variables to 10-dimensional geometric design variables. And it is compared with the classical discrete adjoint gradient optimization method GOM. In the laminar wing aerodynamic optimization, N TS = 9.0 is selected as the transition criterion.

[0116] Table 2 compares the optimization results of different methods. Based on the non-uniform "F1" free-form deformation FFD parameterization box, the aerodynamic drag reduction of the optimized configuration (GOM-F1) by the gradient optimization method (GOM) based on discrete adjoint is 41.9 counts, and the laminar flow region range on the upper airfoil surface increases by 0.159 c , and the laminar flow region range on the lower airfoil surface remains basically unchanged. While based on the uniform "F2" free-form deformation FFD parameterization box, the aerodynamic drag reduction of the optimized configuration (GOM-F2) by the gradient optimization method (GOM) based on discrete adjoint is 45.1 counts, and the laminar flow region range on the upper airfoil surface increases by 0.192 c , and the laminar flow region range on the lower airfoil surface increases by 0.032 c . For the gradient optimization method (GOM) based on discrete adjoint, the aerodynamic drag reduction benefit based on the uniform free-form deformation FFD box is greater than that of the non-uniform FFD box.

[0117] Table 2

[0118]

[0119] For the contrast laminar wing gradient optimization method (MLE-GOM), the reconstruction of the design space based on different free-form deformation FFD parameterization forms also has an impact on the optimization effect. Based on the non-uniform "F1" free-form deformation FFD parameterization box, the aerodynamic drag reduction of the optimized configuration (MLE-GOM-F1) by the contrast laminar wing gradient optimization method (MLE-GOM) is 38.6 counts, and the laminar flow region range on the upper airfoil surface increases by 0.106 c , and the laminar flow region range on the lower airfoil surface increases by 0.017 c . While based on the uniform "F2" free-form deformation FFD parameterization box, the aerodynamic drag reduction of the optimized configuration (MLE-GOM-F2) by the contrast laminar wing gradient optimization method (MLE-GOM) is 56.4 counts, and the laminar flow region range on the upper airfoil surface increases by 0.375 c , and the laminar flow region range on the lower airfoil surface increases by 0.137 cSimilar to the gradient optimization method (GOM) based on discrete adjoint, the aerodynamic drag reduction benefit of the MLE-GOM optimization based on the uniform free-form deformation FFD box is greater than that of the non-uniform free-form deformation FFD box. The main reason for this phenomenon is the difference in the design space caused by different geometric parameterizations of the free-form deformation FFD.

[0120] Comparing the optimization results, based on the non-uniform free-form deformation FFD box "F1", the aerodynamic drag reduction benefit of the optimized configuration by the laminar wing gradient optimization method (MLE-GOM) is slightly smaller than that of the optimized configuration by the gradient optimization method (GOM) based on discrete adjoint, reducing by about 3.3 counts. However, based on the uniform free-form deformation FFD box "F2", the drag reduction benefit of the MLE-GOM optimized configuration increases by 11.4 counts (25.1%), which is much greater than that of the optimized configuration by the gradient optimization method (GOM) based on discrete adjoint. At the same time, the ranges of the laminar flow regions on the upper and lower wing surfaces increase by 32.4% and 21.4% respectively. As Figure 8 and Figure 9 shown by the airfoil and pressure distribution, Figure 8 in, x / c is the ratio of the x -direction coordinate to the chord length, y / c is the ratio of the y -direction coordinate to the chord length, Figure 9 in, x / c is the ratio of the x -direction coordinate to the chord length, Cp is the pressure coefficient, where Figure 9 the circular symbol in indicates the transition position, and the airfoil shapes and pressure distribution morphological characteristics of the MLE-GOM-F1 and GOM-F1 optimized configurations are similar. Compared with the RAE2822 airfoil, the principle of drag reduction in the optimized design is to reduce the suction peak at the head, increase the favorable pressure gradient in the first 30%c region of the upper wing surface to suppress the development of T-S waves, and then perform pressure recovery in the form of a weak adverse pressure gradient and increase the camber of the airfoil trailing edge, while controlling the growth rate of Tollmien-Schlichting (T-S) and weakening the shock wave intensity. However, the airfoil shapes and pressure distribution morphological characteristics of the MLE-GOM-F2 and GOM-F2 optimized configurations are significantly different. The optimization result of MLE-GOM-F2 is a shock-free configuration, and a favorable pressure gradient is maintained throughout the first 75%c region of the upper wing surface, effectively suppressing the growth of T-S waves as Figure 10 shown in the figure, where x / c is the ratio of the x -direction coordinate to the chord length, N TS is the TS wave amplification factor, N TS= 9.0 represents the transition threshold, maintaining a large laminar flow region. At the same time, it extends the favorable pressure gradient range on the lower wing surface and delays transition. The overall morphological characteristics of the pressure distribution on the lower wing surface in the GOM-F2 optimization result are similar to those of MLE-GOM-F2, but there are problems with insufficient smoothness in the surface shape and pressure distribution. On the upper wing surface, the favorable pressure gradient range of the GOM-F2 optimized configuration only maintains up to the first 25% region, and there is a weak shock wave, causing shock-induced forced transition.

[0121] Based on the robust optimization method for laminar wings constructed above, the application verification of laminar airfoil optimization was completed. The verification shows that the classical discrete adjoint gradient optimization method for laminar wings falls into local optimal solutions, which is the main reason for the significant differences in drag reduction benefits and pressure distribution morphology between the MLE-GOM-F2 and GOM-F2 optimized configurations. The sensitivity of laminar-turbulent transition to geometric perturbation changes, as well as the shock wave boundary layer interference phenomenon, result in strong nonlinearity and prominent multi-extremum characteristics in the optimization space. This makes it impossible for the gradient optimization method (GOM) based on the uniformly distributed FFD control box "F2" to obtain a large laminar flow region while eliminating shock wave drag and cannot obtain an optimized configuration with high smoothness of the surface shape and pressure distribution. In contrast, compared with the laminar wing gradient optimization method (MLE-GOM), through geometric smoothing and design space reconstruction, it improves the multi-extremum characteristics of the feasible region, reduces the sensitivity of the optimization effect to changes in the free-form deformation FFD geometric parameterization form of the original optimization problem, and improves the robustness of the optimization method's search efficiency.

[0122] The present invention verifies the practicability and feasibility of the constructed robust optimization method for laminar wings through the RAE2822 airfoil optimization example. The example comparison results show that the robust optimization design for laminar wings can significantly reduce the influence of the geometric parameterization form on the drag reduction effect of the optimization method and improve the search efficiency and robustness of the algorithm.

Claims

1. A robust optimization method for laminar wings based on manifold learning, characterized in that, It includes the following steps: S1. Given the initial geometry and design variables as the starting point for the robust optimization of the laminar wing; S2. Generate an airfoil database that satisfies geometric constraints through Latin hypercube sampling, where all airfoils in the database satisfy the geometric constraints of the optimal design; S3. Use the principle of the smoothing algorithm to eliminate geometrically distorted airfoils from the airfoil database; S4. Perform manifold learning based on the airfoil database after eliminating geometrically distorted airfoils, and establish an airfoil geometric characterization model based on the Grassmann manifold; S5. In the aerodynamic optimal design, use the constructed airfoil geometric characterization model to replace the free-form deformation (FFD) as the geometric parameterization method to generate perturbed airfoils; S6. Based on the perturbed airfoils, use the geometric characterization model of the Grassmann manifold to reconstruct the design space originally described by the free-form deformation (FFD) geometric parameterization method, and complete the deformation process of the design object; S7. Based on the results of the manifold learning process in S2 - S6, perform deformation processing on the grid and flow field calculation, and through iterative optimization processing, obtain the robust optimization result of the laminar wing; The S5 includes the following steps: A1. In the aerodynamic optimization design, the Laplace smoothing algorithm is used to geometrically smooth the perturbed airfoil generated by the free-form deformation (FFD) geometric parameterization method. Among them, the displacement value of the i th free-form deformation FFD point adopts the Laplace smoothing algorithm to weight the displacement value with the displacement values of the adjacent points of this point and updates it to : Among them, represents the updated displacement value, represents the weight coefficient, represents the number of adjacent points for geometric fairing, which is used to control the degree of geometric fairing, j represents the adjacent point number; A2. Based on the airfoil samples processed by A1, by establishing the mapping between the real space and the Grassmann manifold space , the discrete airfoil shapes are represented as Grassmann manifold elements, where n represents the real space dimension of the Grassmann manifold represents the manifold space represents n the 2-dimensional Grassmann manifold space of the n-dimensional real space; A3. Obtain the physical coordinates of the airfoil based on the elements of the Grassmann manifold and the elements of the Grassmann manifold the mapping relationship between them; A4. Construct the mapping relationship between the known manifold space and the tangent space of the data center element, including logarithmic mapping and exponential mapping; A5. Determine the Riemannian center of the data points by performing iterative calculations in the tangent space of the manifold; A6. Based on the mapping relationship of A3, all manifold elements are mapped to the tangent space of the Riemannian center through logarithmic mapping of ; The mapping result of base A6. According to the degree of decay of the eigenvalues in the tangent space, the first eigenvectors are selected as the low-dimensional orthogonal basis; A8. For , using the following formula, the linear combination of all principal components defines a -dimensional Grassmann submanifold, and constructs the mapping relationship from the parameter space to the airfoil coordinates in the real domain space: Among them, represents a normal coordinate in the parameter space, represents a Grassmann submanifold of dimension represents an element of the Grassmann manifold, represents the exponential mapping operation, represents a point the distance to the center of the manifold space, represents the operator that converts vector concatenation into a matrix; A9. Based on the mapping relationship of A8, by changing the parameter space coordinates generate a perturbed airfoil.

2. The robust optimization method for a laminar airfoil based on manifold learning according to claim 1, wherein The expression of the mapping relationship in A3 is as follows: Among them, represents the coordinates of the airfoil in the real space, represents the matrix formed by the centers of the airfoil data points, represents the linear transformation matrix, represents the identity matrix of dimension represents the diagonal matrix, represents the center of the airfoil data points, represents the number of airfoil data points, represents the transpose of the airfoil coordinate matrix, represents the identity matrix of dimension represents the transpose of the orthogonal matrix obtained by singular value decomposition.

3. The robust optimization method for laminar airfoil based on manifold learning according to claim 1, characterized in that The expression of the Riemannian center of the data points in A5 is as follows: Among them, represents the Riemannian center for determining data points, represents a data point, represents data points on the Riemannian manifold, represents the distance from all data points on the Riemannian manifold to the Riemannian center of, represents the total number of data points on the Riemannian manifold, represents the manifold space.

4. The robust optimization method for laminar airfoil based on manifold learning according to claim 1, characterized in that The expression of the mapping relationship in A8 is as follows: Among them, represents the airfoil data point matrix, represents the mean of the linear transformation matrices corresponding to all sample points, represents the mean of the translation amounts of all sample points, represents the Grassmann manifold element obtained by perturbation.

5. The robust optimization method for laminar airfoil based on manifold learning according to claim 1, wherein The S7 includes the following steps: S701. Grid deformation: Based on the results of the manifold learning process in S2 - S6, use the inverse distance interpolation method to dynamically update the volume grid to ensure that the volume grid is consistent with the deformed geometry; S702. Flow field calculation: Calculate the flow field information according to the updated volume grid; S703. Gradient information acquisition: Based on the calculation results of the flow field information, obtain the derivative information about the design variables through the laminar discrete adjoint solver; S704. Optimization iteration: Obtain the change direction of the design variables according to the derivative information, generate new design variables, and return to S2 for iterative processing; S705. Optimization result output: If the iteration result meets the feasibility tolerance and optimality tolerance, obtain the robust optimization result of the laminar wing.

6. The robust optimization method for a laminar airfoil based on manifold learning according to claim 5, wherein The S701 is specifically: B1. Based on the results of the manifold learning process in S2 - S6, use the inverse distance interpolation method to update the volume grid coordinates according to the changes in the surface grid; B2. Based on the updated results, use automatic differentiation to calculate the derivatives of the volume grid nodes with respect to the surface grid pattern to ensure that the volume grid is consistent with the deformed geometry.

Images