Laminar flow wing robust optimization method based on manifold learning

Abstract

The invention provides a laminar flow wing robust optimization method based on manifold learning, and belongs to the field of aircraft design, and the method comprises the steps: giving an initial geometrical shape and a design variable as a starting point of laminar flow wing robust optimization; filtering the design space based on a geometric fairing and Grassmann manifold coupling method, and deforming a design object; dynamically updating the volume grid by using an inverse distance interpolation method to ensure that the volume grid is consistent with the deformed geometrical shape; calculating flow field information according to the updated volume grid; obtaining derivative information about a design variable through a laminar flow discrete adjoint solver; and obtaining a design variable change direction according to the derivative information, generating a new design variable, and obtaining a steady optimization result of the laminar flow wing if an iteration result meets feasibility tolerance and optimality tolerance. According to the method, the robustness of laminar flow wing optimization can be improved, and the influence of a geometric parameterization form on the drag reduction effect of the optimization method is reduced, so that a more stable optimization result is obtained.

Description

Technical Field

[0001] The invention belongs to the field of aircraft design, and in particular relates to a laminar wing robust optimization method based on manifold learning. Background Art

[0002] In order 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 area on the surface of the aircraft, which is of great significance for achieving green aviation goals. However, the complexity of laminar wing optimization design is much higher than that of turbulent optimization design, especially the existence of laminar-turbulent transition phenomenon, which significantly increases the difficulty of optimization of this problem. Non-gradient optimization methods are widely used in laminar wing design because of their strong versatility and easy implementation, but their optimization efficiency decreases rapidly with the increase of problem dimension. In comparison, the discrete adjoint method (Gradient Optimization Method, GOM), as a gradient-based optimization method, can effectively decouple computational complexity and design variable dimension, 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, which aggravates the multi-extreme characteristics and increases the difficulty of solving the gradient optimization algorithm. Studies have shown that when there are multiple saddle points or flat areas in the design space, the gradient information may not be sufficient to guide the optimization algorithm out of the local optimum, resulting in convergence to a suboptimal solution. Studies have pointed out that the multi-extreme characteristics of the design space in different geometric parameterization forms have a significant impact on the optimization results. Appropriate design constraints can alleviate this problem and improve the optimization effect. For the aerodynamic optimization problem of laminar wings, the laminar-turbulent transition and the sensitivity of aerodynamic characteristics to flow field changes significantly enhance the impact of geometric distortion on the design space, resulting in stronger nonlinear and multi-extreme characteristics in the design space. Furthermore, complex constraints may cause distortion or disconnection of the feasible domain, 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-extreme value characteristics in laminar wing optimization problems remains a major challenge in current research.

[0004] As a nonlinear dimensionality reduction method, manifold learning can extract essential features from high-dimensional data and establish a nonlinear 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. In the process of aircraft design, manifold learning is widely used in narrowing the optimization search space range and quickly predicting flow field characteristics through feature extraction.

[0005] At present, the optimization design research based on manifold learning includes: the existing technology uses ISOMAP (iso-distant mapping algorithm) and interpolation method to study the prediction of airfoil flow field under transonic state, and compares it with the results of the principal component analysis (PCA) reduction model. Studies have shown that the nonlinear reduction model based on manifold learning can significantly improve the prediction accuracy when dealing with 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 low-dimensional manifold space. The L-ISOMAP (reference point iso-distant mapping algorithm) proposed in the prior art enhances the optimization efficiency of the optimization algorithm by utilizing 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 local linear embedding (LLE) algorithm, thereby accelerating the solution of multi-objective optimization problems. At present, the research based on manifold learning methods mainly focuses on turbulent aerodynamic optimization, and there is no aerodynamic optimization research on laminar wings. Summary of the invention

[0006] In view of the above-mentioned deficiencies in the prior art, the present invention provides a laminar wing robust optimization method based on manifold learning, which solves the problems of highly nonlinear and multi-extreme value characteristics existing in the laminar wing optimization process.

[0007] In order to achieve the above objectives, the technical solution adopted by the present invention is: a laminar wing robust optimization method based on manifold learning, comprising the following steps: S1, given initial geometry and design variables as the starting point for robust optimization of laminar wing; 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 optimized design; S3, using the smoothing algorithm principle to eliminate geometrically distorted airfoils in the airfoil database; S4, performing manifold learning based on the airfoil database with geometrically distorted airfoils removed, and establishing an airfoil geometry representation model based on Grassmann manifold; S5. In aerodynamic optimization design, the constructed airfoil geometry representation model is used to replace the free deformation FFD as a geometric parameterization method to generate a perturbed airfoil; S6. Based on the perturbed airfoil and using the geometric representation model of Grassmann manifold, the design space originally described by the free deformation FFD geometric parameterization method is reconstructed to complete the deformation processing of the design object. S7. Based on the manifold learning processing results of S2-S6, the grid is deformed and the flow field is calculated, and through iterative optimization processing, a robust optimization result of the laminar wing is obtained.

[0008] The beneficial effects of the present invention are as follows: the present invention improves the multi-extreme value characteristics of the feasible domain through geometric smoothing and design space reconstruction, 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 method's optimization efficiency. The present invention maps the original optimization problem design space to a compact low-dimensional manifold space through the geometric representation method of Grassmann manifold, eliminates geometric constraints, improves the non-convexity, strong nonlinearity and multi-extreme value characteristics of the design problem, and provides a basis for efficient optimization.

[0009] Furthermore, the S5 comprises the following steps: A1. In aerodynamic optimization design, the Laplace smoothing algorithm is used to perform geometric smoothing on the disturbed airfoil generated by the free deformation FFD geometric parameterization method. i Displacement value of free deformation FFD point Using the Laplace smoothing algorithm, the displacement value The displacement values ​​of the points adjacent to this point are weighted and updated to : ;in, represents the updated displacement value, represents the weight coefficient, Indicates the number of adjacent points used for geometric smoothing, which is used to control the degree of geometric smoothing. j Indicates the adjacent point number; A2, based on the airfoil sample processed by A1, by establishing a real space With Grassmann manifold space The discrete airfoil shape is represented as a Grassmann manifold element by a mapping of n represents the real space dimension of the Grassmann manifold, represents the manifold space, express n 2D Grassmann manifold space in real space; A3. Based on the Grassmann manifold element, the physical coordinates of the airfoil are obtained With Grassmann manifold elements The mapping relationship between them; A4. Construct the mapping relationship between the known manifold space and the tangent space of the data center elements, including logarithmic mapping and exponential mapping; A5. Determine the Riemann center of the data point 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 Riemann center through logarithmic mapping The tangent space ; The mapping result of A7 and base A6 is selected according to the degree of attenuation of the eigenvalue of the tangent space. feature vector As a low-dimensional orthogonal basis; A8. For , using the following formula, the linear combination of all principal components Defines a dimensional Grassmann submanifold and construct from the parameter space The mapping relationship to the real-domain airfoil coordinates is: ; ;in, represents a normal coordinate in parameter space, express dimensional Grassmann submanifold, 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 for concatenating vectors into matrices; A9, based on the mapping relationship of A8, by changing the parameter space coordinates Generate a perturbed airfoil.

[0010] The beneficial effect of the above further solution is that the present invention uses the constructed airfoil geometry characterization model to replace the free deformation FFD as the geometry parameterization method, thereby ensuring that no airfoil with serious geometric distortion will appear when the sample is generated.

[0011] Furthermore, the expression of the mapping relationship in A3 is as follows: ; ; ; ;in, represents the coordinates of the airfoil in real space, represents the matrix composed of the centers of the airfoil data points, represents the linear transformation matrix, express dimensional unit matrix, represents a diagonal matrix, represents the center of the airfoil data point, represents the number of airfoil data points, represents the transpose of the airfoil coordinate matrix, express dimensional unit matrix, Represents the transpose of the orthogonal matrix obtained by singular value decomposition.

[0012] Furthermore, the expression of the Riemann center of the data point in A5 is as follows: ;in, represents the Riemann center of the determined data point, represents a data point, represents a data point on the Riemann manifold, Represents all data points on the Riemann manifold to the Riemann center The distance represents the total number of data points on the Riemann manifold, Represents a manifold space.

[0013] Furthermore, the expression of the mapping relationship in A8 is as follows: ;in, represents the airfoil data point matrix, Represents the mean of the linear transformation matrices corresponding to all sample points, represents the mean value of the translation of all sample points, represents the element of the Grassmann manifold obtained by perturbation.

[0014] Furthermore, the S7 comprises the following steps: S701, mesh deformation: based on the manifold learning processing results of S2-S6, the volume mesh is dynamically updated using the inverse distance interpolation method to ensure that the volume mesh is consistent with the deformed geometric shape; S702, flow field calculation: calculate flow field information according to the updated volume grid; S703, gradient information acquisition: based on the flow field information calculation results, the derivative information about the design variables is acquired through the laminar discrete adjoint solver; S704, optimization iteration: obtain the change direction of the design variable according to the derivative information, generate new design variables, and return to S2 for iterative processing; S705, output of optimization results: If the iteration results meet the feasibility tolerance and the optimality tolerance, a robust optimization result of the laminar wing is obtained.

[0015] Furthermore, the S701 is specifically as follows: B1, based on the manifold learning processing results of S2-S6, the volume grid coordinates are updated according to the changes of the surface grid using the inverse distance interpolation method; B2. Based on the updated results, automatic differentiation is used to calculate the derivatives of the volume mesh nodes relative to the surface mesh pattern to ensure that the volume mesh is consistent with the deformed geometry.

[0016] The beneficial effect of the above further scheme is that the present invention uses the inverse distance interpolation method to dynamically update the volume grid, realizes the automatic high-precision spatial grid deformation based on the surface grid, and provides a basis for automatic optimization calculation. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] Figure 1 The present invention is a flow chart of the method.

[0018] Figure 2 Schematic diagram of airfoil geometry smoothing based on the Laplace smoothing algorithm.

[0019] Figure 3 This is a schematic diagram of the FFD control box.

[0020] Figure 4 Schematic diagram for comparison of optimized airfoils.

[0021] Figure 5 Schematic diagram for comparison of pressure distribution for optimized airfoil.

[0022] Figure 6 Schematic diagram of FFD box F1 with non-uniform distribution of control points.

[0023] Figure 7 Schematic diagram of FFD box F2 with uniformly distributed control points.

[0024] Figure 8 Schematic diagram for comparison of optimized airfoil results.

[0025] Fig. 9 Schematic diagram for comparison of optimized pressure distribution results.

[0026] Fig.10 To optimize the airfoil N TS Schematic diagram of comparison of amplification factor envelopes. DETAILED DESCRIPTION

[0027] The specific implementation modes of the present invention are described below so that those skilled in the art can understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific implementation modes. For those of ordinary skill in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the attached claims, these changes are obvious, and all inventions and creations utilizing the concept of the present invention are protected.

[0028] Example The purpose of the present invention is to solve the problems of highly nonlinear and multi-extreme characteristics in the optimization process of laminar wings, to construct a robust optimization method for laminar wings, and to provide a robust gradient optimization method for the design of laminar wings. Figure 1 As shown, the present invention provides a laminar wing robust optimization method based on manifold learning, characterized in that it includes the following steps: S1. Input initial geometry: given initial geometry and design variables as the starting point for robust optimization of laminar wing; Design space reconstruction: Filter the design space based on the coupling method of geometric smoothing and Grassmann manifold, and deform the design object. The implementation method is as follows: 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 optimized design; S3, using the smoothing algorithm principle to eliminate geometrically distorted airfoils in the airfoil database; S4, performing manifold learning based on the airfoil database with geometrically distorted airfoils removed, and establishing an airfoil geometry representation model based on Grassmann manifold; S5. In aerodynamic optimization design, the constructed airfoil geometry characterization model is used to replace the free deformation FFD as a geometric parameterization method to generate a perturbed airfoil. The implementation method is as follows: A1. In aerodynamic optimization design, the Laplace smoothing algorithm is used to perform geometric smoothing on the disturbed airfoil generated by the free deformation FFD geometric parameterization method. i Displacement value of free deformation FFD point Using the Laplace smoothing algorithm, the displacement value The displacement values ​​of the points adjacent to this point are weighted and updated to : ;in, represents the updated displacement value, represents the weight coefficient, Indicates the number of adjacent points used for geometric smoothing, which is used to control the degree of geometric smoothing. j Indicates the adjacent point number; A2, based on the airfoil sample processed by A1, by establishing a real space and Grassmann manifold space The discrete airfoil shape is represented as a Grassmann manifold element by a mapping of n represents the real space dimension of the Grassmann manifold, represents the manifold space, express n 2D Grassmann manifold space in real space; A3. Based on the Grassmann manifold element, the physical coordinates of the airfoil are obtained With Grassmann manifold elements The mapping relationship between them is: ; ; ; ;in, represents the coordinates of the airfoil in real space, represents the matrix composed of the centers of the airfoil data points, represents the linear transformation matrix, express dimensional unit matrix, represents a diagonal matrix, represents the center of the airfoil data point, represents the number of airfoil data points, represents the transpose of the airfoil coordinate matrix, express dimensional unit matrix, represents the transpose of the orthogonal matrix obtained by singular value decomposition; A4. Construct the mapping relationship between the known manifold space and the tangent space of the data center elements, including logarithmic mapping and exponential mapping; A5. Determine the Riemann center of the data point by iterative calculation in the tangent space of the manifold: ;in, represents the Riemann center of the determined data point, represents a data point, represents a data point on the Riemann manifold, Represents all data points on the Riemann manifold to the Riemann center The distance represents the total number of data points on the Riemann manifold, represents a manifold space; A6. Based on the mapping relationship of A3, all manifold elements are mapped to the Riemann center through logarithmic mapping The tangent space ; The mapping result of A7 and base A6 is selected according to the degree of attenuation of the eigenvalue of the tangent space. feature vector As a low-dimensional orthogonal basis; A8. For , using the following formula, the linear combination of all principal components Defines a dimensional Grassmann submanifold and construct from the parameter space The mapping relationship to the real-domain airfoil coordinates is: ; ;in, represents a normal coordinate in parameter space, express dimensional Grassmann submanifold, 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 for concatenating vectors into matrices; The expression of the mapping relationship is as follows: ;in, represents the airfoil data point matrix, Represents the mean of the linear transformation matrices corresponding to all sample points, represents the mean value of the translation of all sample points, represents the element of the Grassmann manifold obtained by perturbation A9, based on the mapping relationship of A8, by changing the parameter space coordinates Generate a disturbance airfoil; S6. Based on the perturbed airfoil and using the geometric representation model of Grassmann manifold, the design space originally described by the free deformation FFD geometric parameterization method is reconstructed to complete the deformation processing of the design object. In this embodiment, when generating the airfoil sample, geometric smoothing is first performed, and then Grassmann manifold deformation is performed on the basis of the smoothing, so as to couple the geometric smoothing with the Grassmann manifold.

[0029] S7, mesh deformation: Based on the manifold learning processing results of S2-S6, the mesh is deformed and the flow field is calculated. Through iterative optimization processing, a robust optimization result of the laminar wing is obtained. The implementation method is as follows: S701, mesh deformation: Based on the manifold learning processing results of S2-S6, the volume mesh is dynamically updated using the inverse distance interpolation method to ensure that the volume mesh is consistent with the deformed geometric shape. The implementation method is as follows: B1, based on the manifold learning processing results of S2-S6, the volume grid coordinates are updated according to the changes of the surface grid using the inverse distance interpolation method; B2. Based on the updated results, automatic differentiation is used to calculate the derivatives of the volume mesh nodes relative to the surface mesh pattern to ensure that the volume mesh is consistent with the deformed geometry; S702, flow field calculation: calculate flow field information according to the updated volume grid; S703, gradient information acquisition: based on the flow field information calculation results, the derivative information about the design variables is acquired through the laminar discrete adjoint solver; S704, optimization iteration: obtain the change direction of the design variable according to the derivative information, generate new design variables, and return to S2 for iterative processing; S705, output of optimization results: If the iteration results meet the feasibility tolerance and the optimality tolerance, a robust optimization result of the laminar wing is obtained.

[0030] In this embodiment, the airfoil geometry smoothing method is described as follows: The Laplace smoothing algorithm is used to perform geometric smoothing on the disturbed airfoil generated by the free deformation FFD geometric parameterization method, such as Figure 2 As shown in the figure, y express y Towards coordinates, c represents the chord length, x express x Direction coordinates.i Displacement value of free deformation FFD point Using the Laplace smoothing algorithm, The displacement values ​​of the points adjacent to this point are weighted and updated to , the calculation formula is: ;in, represents the updated displacement value, represents the weight coefficient, Indicates the number of adjacent points used for geometric smoothing, which is used to control the degree of geometric smoothing. j Indicates the adjacent point number, used to traverse the i All neighboring points of a free deformation FFD point are used to accumulate displacement values.

[0031] By smoothing the deformation displacement of the free deformation FFD control points, the configurations with severe geometric distortion generated by the arbitrary deformation of the free deformation FFD in the design space can be eliminated, and the smoothness of the airfoil in the sample set used for manifold learning can be improved. In this embodiment, the geometric representation method based on Grassmann manifold is introduced as follows: Grassmann manifold learning separates the linear transformation and high-order disturbance deformation in the geometric deformation in aerodynamic design, and extracts features in the design space containing only high-order surface disturbances, which can achieve dimensionality reduction. At the same time, this method can effectively control the geometric constraints and construct a low-dimensional and compact design space that satisfies the geometric constraints, thereby reducing the nonlinearity of the design space and improving the optimization performance of the gradient optimization algorithm.

[0032] The linear transformation deformation of the airfoil can be expressed as , Reversible square matrix, representing scaling, affine and rotational linear deformations, Reversible The matrix set is abbreviated as . The matrix representing the coordinates of the airfoil data points assembled along the arc length direction. Perform translation deformation of spatial position. Linear transformation matrix The expression is as follows: ;in, Control scaling deformation, , Control scaling deformation, Control torsional deformation, , , and They represent the coefficients of the deformation matrix, for example For proportional scaling, for horizontal and vertical scaling, Indicates the twist angle.

[0033] In order to separate the linear transformation, we can start from the real space Remove A linear transformation represented by , constructs a set of parameters to describe the high-order surface deformation. In the prior art, a method for mapping a discrete data set to a Grassmann space is proposed. Real Space of dimensional subspace, so that when sampling on Grassmann space, the deformation of the profile is independent of linear transformations. By representing the discrete airfoil shape as an element of the Grassmann manifold, the deformation of the profile in this space is separated from the main linear transformations in aerodynamics (changing thickness, chord length, camber, etc.).

[0034] In order to represent the airfoil geometry as Grassmann manifold elements, the real space LandmarkAffine Standardization (LAS) is used to establish and Grassmann manifold space The mapping of n represents the real space dimension of the Grassmann manifold.

[0035] The coordinates of the airfoil in real space are ,make Indicates the center of the airfoil data point. Centralize and perform singular value decomposition: ;in, . Define the linear transformation matrix for: .

[0036] The physical coordinates of the airfoil can be obtained With Grassmann manifold elements The mapping relationship between (this mapping relationship is applied in A6-A8 to The mapping from the coordinates of the airfoil to the real space is: ;in, represents the coordinates of the airfoil in real space, represents the matrix composed of the centers of the airfoil data points, represents an element of the Grassmann manifold, represents the linear transformation matrix, express dimensional unit matrix (used to process the center of the airfoil data point so that the dimension corresponds to X), represents a diagonal matrix (used to process the centers of airfoil data points so that the dimension corresponds to X), represents the center of the airfoil data point, represents the number of airfoil data points, represents the transpose of the airfoil coordinate matrix, express dimensional unit matrix, Represents the transpose of the orthogonal matrix obtained by singular value decomposition.

[0037] In order to A parametric description is performed, and the extension of principal component analysis (PCA) on Riemann manifold is utilized to establish a geometric feature representation method based on manifold learning.

[0038] To perform principal component analysis (PCA) on a Riemannian manifold, it is necessary to establish a mapping relationship between the known manifold space and the tangent space of the data center elements. , the manifold space The points on the map are mapped to the tangent space In. Logarithmic mapping Point Map to point The tangent space In, it means from point To point The tangent vector of the shortest path. Using the inverse mapping, that is, the exponential mapping Map the vector in the tangent space back to the manifold space. The exponential mapping converts the velocity vector in the tangent space into a trajectory along the geodesic on the manifold. Starting from a point in the tangent space, move along the geodesic in the direction of the tangent vector to obtain a new point on the manifold.

[0039] In order to implement PCA on a Riemann manifold, it is necessary to perform centralization on the data on the manifold and extract the orthogonal principal directions on the manifold. The present invention determines the Riemann center of the data point by iterative calculation in the tangent space of the manifold. Given a data point on a Riemann manifold , , Karcher mean is such that all points are close to the Riemann center Distance The point where the sum is the smallest, that is: ;in, represents the Riemann center of the determined data point, represents a data point, represents a data point on the Riemann manifold, Represents all data points on the Riemann manifold to the Riemann center The distance represents the total number of data points on the Riemann manifold, represents the manifold space, i Indicates i data points.

[0040] Map all manifold elements to Riemann centers by logarithmic mapping The tangent space , and then carry out principal component analysis PCA in the tangent space. According to the degree of attenuation of the eigenvalues ​​in the tangent space, select the previous feature vector as a low-dimensional orthogonal basis. Normal coordinates in tangent space Represents the projection corresponding to the main direction, that is, the normal coordinate Defines the parameter space .

[0041] for , the linear combination of all principal components Defines a dimensional Grassmann submanifold: ;in, , represents a normal coordinate in parameter space, express dimensional Grassmann submanifold, represents an element of the Grassmann manifold, Represents the exponential mapping operation, Mapping to Grassmann manifold elements , Represents a point The distance to the center of the manifold space, Represents the operator for concatenating vectors into matrices.

[0042] Finally, we establish a parameter space The mapping relationship to the real-domain airfoil coordinates is: ;in, 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 of all sample points. By changing the parameter space coordinates Generate the perturbed airfoil.

[0043] In this embodiment, the design space filtering method based on the coupling method of geometric smoothing and Grassmann manifold is as follows: Through Latin hypercube sampling, the free deformation FFD geometric parameterization method coupled with the Laplace smoothing algorithm is used to generate an airfoil database that meets geometric constraints. The airfoils in the database all meet the geometric constraints of the optimized design, and the geometrically distorted airfoils are eliminated according to the principles of the smoothing algorithm. The database is used for manifold learning, and an airfoil geometry representation model based on Grassmann manifold is established: ;in, represents the airfoil data point matrix, represents the element of the Grassmann manifold obtained by perturbation, Represents the mean of the linear transformation matrices corresponding to all sample points, Represents the mean value of the translation of all sample points. The airfoil geometry representation model represents the displacement from the parameter space The mapping relationship from the airfoil to the real-domain space airfoil coordinates, the airfoil geometric representation varies with the airfoil points.

[0044] In aerodynamic optimization design, the established geometric representation model replaces the free deformation FFD as the geometric parameterization method to generate the perturbed airfoil. Finally, the geometric representation model based on Grassmann manifold realizes the reconstruction of the design space described by the original free deformation FFD geometric parameterization method, and reconstructs the design space that satisfies the geometric constraints and the airfoil is smooth in the low-dimensional compact manifold space. The geometric design variables are transformed from the perturbation displacement of the free deformation FFD control point of the original optimization problem to the parameter space coordinates. .

[0045] In this embodiment, a robust optimization method for laminar wings is implemented using a manifold learning enhanced laminar wing gradient optimization (MLE-GOM) framework, which includes a design space reconstruction method, grid deformation, flow field solver, gradient optimization algorithm, and laminar discrete adjoint solution: 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 Grassmann manifold; The next step is to modify the volume mesh to make it consistent with the updated geometry, i.e., IDW mesh deformation. During the modification, the volume mesh coordinates are updated according to the changes in the surface mesh using the inverse distance weighted method, and the derivatives of the volume mesh nodes with respect to the surface mesh pattern are calculated using automatic differentiation; Then the flow field is calculated using a flow field solver based on the RANS equation; Finally, a laminar discrete adjoint solver and a gradient-based optimizer are used to handle a large number of design variables and constraints in the optimization problem.

[0046] Using the above method, the present invention describes a laminar wing gradient optimization framework enhanced by manifold learning. To start the process, an initial geometry with given design variables is given, and then the design space reconstruction method is applied to obtain the geometric deformation variables, and then the IDW grid deformation method is applied to modify the volume grid. Subsequently, the RANS solver will calculate the flow field information based on the volume grid, and the transition solver will obtain information such as the growth factor and the transition position. The derivative information about the design variables is obtained through the laminar discrete adjoint solver. Until 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.

[0047] In this embodiment, the aerodynamic optimization of the turbulent airfoil is as follows: The RAE 2822 airfoil was selected for aerodynamic optimization under full turbulence conditions. The optimized conditions are: , , the optimization target is the drag coefficient under the optimized state C d Minimum. The thickness of the airfoil is constrained to be no less than 90% of the initial thickness. A total of 10 thickness constraints are applied. At the same time, the volume of the airfoil is constrained to be no less than 95% of the initial volume and no more than 102% of the initial volume. represents the Mach number, represents the lift coefficient.

[0048] like Figure 3 As shown in the figure, the free deformation FFD geometric parameterization method perturbs the RAE2822 airfoil to generate a perturbed airfoil sample library that satisfies the volume and thickness constraints, with a sample capacity of 7000. Among them, the free deformation FFD control frame has 7 perturbation variables on the upper and lower wing surfaces, 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, the manifold learning method is used to reconstruct the design space. The 14 free deformation FFD geometric perturbation variables of the airfoil are reduced to 10-dimensional geometric design variables. After the design space is reconstructed, the original optimization design problem is transformed into an aerodynamic optimization problem without geometric constraints.

[0049] The aerodynamic optimization design was performed using the classic discrete adjoint gradient optimization method (GOM) with 14-dimensional free-deformation FFD geometric design variables and the manifold learning enhanced laminar wing gradient optimization method (MLE-GOM) with 10-dimensional geometric design variables, respectively. The airfoil angle of attack was also used as an optimization design variable.

[0050] The optimization results are shown in Table 1. Compared with the initial airfoil, the drag coefficient of the airfoil after optimization 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 after optimization 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 smaller than the optimization effect of the traditional classical discrete adjoint gradient optimization method (GOM).

[0051] Table 1

[0052] There are significant differences between the geometric and flow field characteristics of the airfoil optimized by the laminar airfoil gradient optimization method (MLE-GOM) and the classical discrete adjoint gradient optimization method (GOM), such as Figure 4 and Figure 5 As shown, Figure 4 middle, x / c express x The ratio of the axial coordinate to the chord length, y / c express y The ratio of the axial coordinate to the chord length, Figure 5 middle, x / c for x The ratio of the radial coordinate to the chord length, and Cp is the pressure coefficient. Compared with the initial RAE2822 airfoil, both optimization methods obtain shock-free and low-drag configurations, and the upper wing geometry and pressure distribution morphology are highly consistent. However, the characteristics of the lower wing are significantly different. The front loading of the airfoil optimized by the laminar wing gradient optimization method (MLE-GOM) is reduced, and the rear loading is significantly increased. The front loading of the airfoil optimized by the classical discrete adjoint gradient optimization method (GOM) is slightly increased, while the rear loading is slightly reduced. The geometric smoothing and design space reconstruction of the laminar wing gradient optimization method (MLE-GOM) cause changes in the feasible domain range and multi-extreme characteristics of the original optimization problem, which is one of the main reasons for the difference in optimization results between it and the classical discrete adjoint gradient optimization method (GOM).

[0053] Based on the robust optimization method for laminar airfoils constructed above, the application verification of turbulent airfoil optimization was completed. The verification shows that the drag reduction effect of the laminar airfoil gradient optimization method (MLE-GOM) method is slightly different from that of the traditional gradient optimization method (GOM) based on discrete adjoint, but both methods can obtain a shock wave-free low-drag configuration.

[0054] In this embodiment, the aerodynamic optimization of the natural laminar airfoil is as follows: In order to study the influence of the disturbance form of the geometric parameterization method on the performance of the gradient optimization design of natural laminar wing, the difference in optimization performance between the laminar wing gradient optimization method (MLE-GOM) and the traditional gradient optimization method (GOM) based on discrete adjoint is compared. The RAE 2822 airfoil is selected for aerodynamic optimization under laminar flow conditions. The airfoil is parameterized using the FFD control frame "F1" with non-uniform distribution of control points and the FFD control frame "F2" with uniform distribution of control points along the chord direction. The number of geometric disturbance variables of the control frames "F1" and "F2" is 14, such as Figure 6 and Figure 7 The optimized working condition is: , , the optimization target is the drag coefficient under the optimized state C d Minimum. The thickness of the airfoil is constrained to be no less than 90% of the initial thickness. A total of 10 thickness constraints are applied. At the same time, the volume of the airfoil is constrained to be no less than 95% of the initial volume and no more than 102% of the initial volume. represents the Mach number, represents the lift coefficient.

[0055] The comparative laminar wing gradient optimization method (MLE-GOM) is used for aerodynamic optimization design of natural laminar airfoils. The airfoil geometry is perturbed using the "F1" and "F2" free deformation FFD parameterization boxes to generate a perturbed airfoil sample library that meets the volume and thickness constraints. The capacity of each sample library is 7000. Manifold learning is used to reconstruct the design space, and the 14 airfoil free deformation FFD geometric perturbation variables are reduced to 10-dimensional geometric design variables. It is compared with the classic discrete adjoint gradient optimization method GOM. In the laminar wing aerodynamic optimization, the selection N TS = 9.0 as the transition criterion.

[0056] Table 2 compares the optimization results of different methods. Based on the non-uniform "F1" free-deformation FFD parameterization frame, the optimized configuration (GOM-F1) based on the discrete adjoint gradient optimization method (GOM) has a 41.9 count reduction in aerodynamic drag and a 0.159 increase in the laminar flow area on the upper wing. c , the laminar flow area of ​​the lower wing surface remains basically unchanged. Based on the uniform "F2" free deformation FFD parameterization frame, the gradient optimization method (GOM) optimized configuration (GOM-F2) based on discrete adjoint has aerodynamic drag reduction of 45.1 counts, and the laminar flow area of ​​the upper wing surface increases by 0.192 c , the laminar flow area of ​​the lower wing surface increases by 0.032 c For the discrete adjoint gradient optimization method (GOM), the aerodynamic drag reduction benefit of the uniform free deformation FFD frame is greater than that of the non-uniform FFD frame.

[0057] Table 2

[0058] For the MLE-GOM, the reconstruction of the design space based on different free-deformation FFD parameterizations also has an impact on the optimization effect. Based on the non-uniform "F1" free-deformation FFD parameterization frame, the MLE-GOM optimized configuration (MLE-GOM-F1) has a 38.6 count reduction in aerodynamic drag and a 0.106 increase in the laminar flow area on the upper wing. c , the laminar flow area of ​​the lower wing surface increases by 0.017 c Based on the uniform "F2" free deformation FFD parameterization frame, compared with the laminar wing gradient optimization method (MLE-GOM) optimized configuration (MLE-GOM-F2), the aerodynamic drag reduction is 56.4 counts, and the laminar flow area on the upper wing surface is increased by 0.375 c , the laminar flow area of ​​the lower wing surface increases by 0.137 c Similar to the gradient optimization method (GOM) based on discrete adjoint, the aerodynamic drag reduction benefit of MLE-GOM based on uniform free deformation FFD frame is greater than that of non-uniform free deformation FFD frame. The difference in design space caused by different free deformation FFD geometric parameterization forms is the main reason for this phenomenon.

[0059] Comparing the optimization results, the aerodynamic drag reduction benefit of the laminar wing gradient optimization method (MLE-GOM) optimized configuration based on the non-uniform free deformation FFD frame "F1" is slightly smaller than that of the gradient optimization method (GOM) optimized configuration based on discrete adjoint, which is reduced by about 3.3 counts. However, based on the uniform free deformation FFD frame "F2", the drag reduction benefit of the MLE-GOM optimized configuration increased by 11.4 counts (25.1%), which is much larger than the gradient optimization method (GOM) optimized configuration based on discrete adjoint. At the same time, the laminar flow area of ​​the upper and lower wing surfaces increased by 32.4% and 21.4%, respectively. Figure 8 and Fig. 9 The airfoil and pressure distribution shown show that Figure 8 middle, x / c for x The ratio of the axial coordinate to the chord length, y / c for y The ratio of the axial coordinate to the chord length, Fig. 9 middle, x / c for x The ratio of the axial coordinate to the chord length, Cp is the pressure coefficient, where Fig. 9The circular symbol in the middle marks the transition position. The airfoil shapes and pressure distribution morphological characteristics of the optimized configurations of MLE-GOM-F1 and GOM-F1 are similar. Compared with the RAE2822 airfoil, the drag reduction principle of the optimized design is to reduce the suction peak of the head, increase the forward pressure gradient in the 30%c area in front of the upper wing to suppress the development of TS waves, and then perform pressure recovery in the form of a weak reverse pressure gradient and increase the curvature of the airfoil trailing edge, while controlling the Tollmien-Schlichting (TS) growth rate and weakening the shock wave intensity. However, the airfoil shapes and pressure distribution morphological characteristics of the optimized configurations of MLE-GOM-F2 and GOM-F2 are significantly different. The optimization result of MLE-GOM-F2 is a shock wave-free configuration, and the forward pressure gradient is maintained in the 75%c area in front of the entire upper wing, which effectively suppresses the growth of TS waves. Fig.10 As shown in the figure, x / c for x The ratio of the axial coordinate to the chord length, N TS is the TS wave amplification factor, N TS =9.0 indicates the transition threshold, maintaining a wide range of laminar flow. At the same time, the range of the pressure gradient along the lower wing is extended, delaying the transition. The overall pressure distribution morphology of the GOM-F2 optimization results is similar to that of MLE-GOM-F2, but the surface and pressure distribution have insufficient smoothness. On the upper wing, the pressure gradient range of the GOM-F2 optimization configuration is only maintained to the first 25% area, and there is a weak shock wave, which causes a forced transition of the shock wave.

[0060] Based on the robust optimization method for laminar airfoils constructed above, the application verification of laminar airfoil optimization was completed. The verification shows that the classic laminar airfoil discrete adjoint gradient optimization method falls into the local optimal solution, which is the main reason for the significant difference in drag reduction benefits and pressure distribution forms between the MLE-GOM-F2 and GOM-F2 optimized configurations. The sensitivity of the laminar-turbulent transition to geometric disturbance changes and the interference phenomenon of the shock wave boundary layer cause the optimization space to have strong nonlinearity and prominent multi-extreme characteristics, which makes it impossible for the discrete adjoint gradient optimization method (GOM) based on the uniformly distributed FFD control frame "F2" to obtain a large range of laminar areas while eliminating shock wave drag, and cannot obtain an optimized configuration with high smoothness of the profile and pressure distribution. Compared with the laminar airfoil gradient optimization method (MLE-GOM), the multi-extreme characteristics of the feasible domain are improved through geometric smoothing and design space reconstruction, the sensitivity of the optimization effect to the changes in the geometric parameterization form of the free deformation FFD of the original optimization problem is reduced, and the robustness of the optimization method's optimization performance is improved.

[0061] The present invention verifies the practicability and feasibility of the robust optimization method for laminar wing constructed by the RAE2822 airfoil optimization example. The example comparison results show that the robust optimization design for laminar wing can significantly reduce the influence of geometric parameterization on the drag reduction effect of the optimization method, and improve the optimization efficiency and robustness of the algorithm.

Claims

1. A robust optimization method for laminar wings based on manifold learning, characterized in that: The following steps are involved: S1, given initial geometry and design variables as the starting point for robust optimization of laminar wing; 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 optimized design; S3, using the smoothing algorithm principle to eliminate geometrically distorted airfoils in the airfoil database; S4, performing manifold learning based on the airfoil database with geometrically distorted airfoils removed, and establishing an airfoil geometry representation model based on Grassmann manifold; S5. In aerodynamic optimization design, the constructed airfoil geometry representation model is used to replace the free deformation FFD as a geometric parameterization method to generate a perturbed airfoil; S6. Based on the perturbed airfoil and using the geometric representation model of Grassmann manifold, the design space originally described by the free deformation FFD geometric parameterization method is reconstructed to complete the deformation processing of the design object. S7. Based on the manifold learning processing results of S2-S6, the grid is deformed and the flow field is calculated, and through iterative optimization processing, a robust optimization result of the laminar wing is obtained.

2. The laminar wing robust optimization method based on manifold learning according to claim 1 is characterized in that: The S5 comprises the following steps: A1. In aerodynamic optimization design, the Laplace smoothing algorithm is used to perform geometric smoothing on the disturbed airfoil generated by the free deformation FFD geometric parameterization method. i Displacement value of free deformation FFD point Using the Laplace smoothing algorithm, the displacement value The displacement values ​​of the points adjacent to this point are weighted and updated to : in, represents the updated displacement value, represents the weight coefficient, Indicates the number of adjacent points used for geometric smoothing, which is used to control the degree of geometric smoothing. j Indicates the adjacent point number; A2, based on the airfoil sample processed by A1, by establishing a real space and Grassmann manifold space The discrete airfoil shape is represented as a Grassmann manifold element by a mapping of n represents the real space dimension of the Grassmann manifold, represents the manifold space, express n 2D Grassmann manifold space in real space; A3. Based on the Grassmann manifold element, the physical coordinates of the airfoil are obtained With Grassmann manifold elements The mapping relationship between them; A4. Construct the mapping relationship between the known manifold space and the tangent space of the data center elements, including logarithmic mapping and exponential mapping; A5. Determine the Riemann center of the data point 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 Riemann center through logarithmic mapping The tangent space ; The mapping result of A7 and base A6 is selected according to the degree of attenuation of the eigenvalue of the tangent space. feature vector As a low-dimensional orthogonal basis; A8. For , using the following formula, the linear combination of all principal components Defines a dimensional Grassmann submanifold and construct from the parameter space The mapping relationship to the real-domain airfoil coordinates is: in, represents a normal coordinate in parameter space, express dimensional Grassmann submanifold, 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 for concatenating vectors into matrices; A9, based on the mapping relationship of A8, by changing the parameter space coordinates Generate a perturbed airfoil.

3. The laminar wing robust optimization method based on manifold learning according to claim 2 is characterized in that: The expression of the mapping relationship in A3 is as follows: in, represents the coordinates of the airfoil in real space, represents the matrix composed of the centers of the airfoil data points, represents the linear transformation matrix, express dimensional unit matrix, represents a diagonal matrix, represents the center of the airfoil data point, represents the number of airfoil data points, represents the transpose of the airfoil coordinate matrix, express dimensional unit matrix, represents the transpose of the orthogonal matrix obtained by singular value decomposition.

4. The laminar wing robust optimization method based on manifold learning according to claim 2 is characterized in that: The expression of the Riemann center of the data point in A5 is as follows: in, represents the Riemann center of the determined data point, represents a data point, represents a data point on the Riemann manifold, Represents all data points on the Riemann manifold to the Riemann center The distance represents the total number of data points on the Riemann manifold, Represents a manifold space.

5. The laminar wing robust optimization method based on manifold learning according to claim 2, characterized in that: The expression of the mapping relationship in A8 is as follows: in, represents the airfoil data point matrix, Represents the mean of the linear transformation matrices corresponding to all sample points, represents the mean value of the translation of all sample points, represents the element of the Grassmann manifold obtained by perturbation.

6. The laminar wing robust optimization method based on manifold learning according to claim 1, characterized in that: The S7 comprises the following steps: S701, mesh deformation: based on the manifold learning processing results of S2-S6, the volume mesh is dynamically updated using the inverse distance interpolation method to ensure that the volume mesh is consistent with the deformed geometric shape; S702, flow field calculation: calculate flow field information according to the updated volume grid; S703, gradient information acquisition: based on the flow field information calculation results, the derivative information about the design variables is acquired through the laminar discrete adjoint solver; S704, optimization iteration: obtain the change direction of the design variable according to the derivative information, generate new design variables, and return to S2 for iterative processing; S705, output of optimization results: If the iteration results meet the feasibility tolerance and the optimality tolerance, a robust optimization result of the laminar wing is obtained.

7. The laminar wing robust optimization method based on manifold learning according to claim 6 is characterized in that: The S701 is specifically as follows: B1, based on the manifold learning processing results of S2-S6, the volume grid coordinates are updated according to the changes of the surface grid using the inverse distance interpolation method; B2. Based on the updated results, automatic differentiation is used to calculate the derivatives of the volume mesh nodes relative to the surface mesh pattern to ensure that the volume mesh is consistent with the deformed geometry.

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