Robust optimization design method for wing based on hessian eigenvector dimension reduction

Abstract

The application discloses a wing robust optimization design method based on Hessian eigenvector dimension reduction, and belongs to the technical field of aircraft design optimization, and comprises the following steps: determining initial design variables corresponding to an initial geometric shape of a wing; updating a wing geometric shape, and then adjusting a fluid calculation grid by using a grid deformation module; performing dimension reduction on an uncertainty space based on a Hessian eigenvector, and calculating aerodynamic performance data and a first-order gradient of sample points; inputting the aerodynamic performance data and the first-order gradient of the sample points into a UQ module, and calculating gradient information of the initial design variables; and judging, by an optimizer, whether a wing configuration determined by the initial design variables is convergent. The application reduces the calculation cost and improves the calculation efficiency while ensuring the accuracy, and provides an efficient technical path for engineering-level aerodynamic robust design under high-dimensional uncertainty conditions.

Description

Technical Field

[0001] This invention belongs to the technical field of aircraft design optimization, specifically relating to a robust optimization design method for wings based on Hessian eigenvector dimensionality reduction. Background Technology

[0002] As the aviation industry's demands for high efficiency, high maneuverability, and high mission adaptability of aircraft continue to increase, aerodynamic shape optimization design has gradually shifted from improving performance under single operating conditions to ensuring comprehensive performance under complex operating conditions. Against this backdrop, the sensitivity of aerodynamic design to uncertainty has significantly increased: shape errors caused by manufacturing, assembly deviations, and service deformation introduce geometric uncertainty; fluctuations in flight state parameters (such as angle of attack, Mach number, and altitude) introduce state uncertainty. The superposition of these two types of disturbances can be amplified into significant aerodynamic changes, thereby weakening performance stability across the entire flight envelope. Therefore, traditional deterministic optimization often fails to guarantee that the configuration maintains the expected aerodynamic performance under multi-source disturbances, leading to unstable benefits or even failure in engineering applications.

[0003] To address the aforementioned issues, robust optimization design has gradually become a cutting-edge research hotspot in aerodynamic optimization design. Robust optimization design typically embeds Uncertainty Quantification (UQ) into the optimization framework. By statistically characterizing the propagation laws of disturbances, it constructs objective functions and constraints using statistical moments such as mean and standard deviation, thereby obtaining designs insensitive to random disturbances. Efficient and robust optimization design frameworks rely on optimization algorithms and UQ methods. Regarding optimization algorithms, gradient optimization methods based on discrete adjoint theory have become the mainstream choice for high-dimensional aerodynamic optimization due to the weak correlation between gradient solution cost and the size of design variables, making them suitable for handling complex design problems. As for UQ methods, Polynomial Chaotic Expansion (PCE) has attracted widespread attention due to its excellent statistical moment calculation properties. In particular, non-intrusive PCE does not require rewriting the governing equations; it only needs to select sample points in random space, call the solver, and regress to solve the expansion coefficients to obtain the statistical moments.

[0004] However, directly coupling UQ methods such as PCE with robust optimization still faces a key bottleneck: to ensure the accuracy of statistical moment calculation, the number of samples required by PCE-based uncertainty quantification methods is proportional to the dimensionality of the uncertainty space. Therefore, the computational cost of uncertainty quantification increases rapidly with the increase of the uncertainty space dimensionality, making robust optimization design unsustainable in high-dimensional uncertainty scenarios. Especially in integrated machine design, uncertainties are widespread and have high dimensionality. If the uncertainty space is not effectively compressed, the engineering usability of robust optimization design will be greatly limited. Existing dimensionality reduction methods mostly rely on engineering experience and parameter selection, resulting in high computational complexity. Therefore, to reduce the sample requirements caused by high-dimensional uncertainty, it is urgent to develop an uncertainty space dimensionality reduction method adapted to the gradient optimization framework. This method would then construct the statistical moment objective function and constraints in the low-dimensional space using UQ methods, and finally solve the gradient using the adjoint method to complete robust optimization. This would significantly reduce the number of uncertain variables while ensuring the accuracy of uncertainty quantification, thereby effectively saving computational resources. Summary of the Invention

[0005] The purpose of this invention is to address the above-mentioned shortcomings in the prior art by providing a robust optimization design method for wings based on Hessian eigenvector dimensionality reduction, thereby solving the problems of large sample size and high computational cost in robust optimization design with multiple sources and high-dimensional uncertain variables.

[0006] To achieve the above objectives, the technical solution adopted by the present invention is as follows: A robust optimization design method for airfoils based on Hessian eigenvector dimensionality reduction includes the following steps: S1. Given the initial geometry of the wing, determine the initial design variables corresponding to the initial geometry; S2. Update the wing geometry using a parametric method based on the initial design variables, and then adjust the fluid calculation grid using a mesh deformation module based on the updated wing geometry. S3. Construct uncertainty variables based on design state disturbance and geometric random disturbance, reduce the dimension of uncertainty space based on Hessian eigenvectors, and generate sample points in the dimension-reduced uncertainty space. For each sample point, perform flow field calculation based on the fluid computing grid corresponding to the initial design variables to obtain the aerodynamic performance response of the sample point and its first gradient with respect to the initial design variables. S4. Input the aerodynamic performance response of the sample points and its first-order gradient with respect to the initial design variables into the UQ module, construct the statistical moment objective function, and calculate the gradient information of the statistical moment objective function with respect to the initial design variables in combination with the analytical expression of the statistical moments; at the same time, determine the constant lift constraint and pitch moment coefficient constraint based on the flow field calculation results in S3, and determine the thickness constraint based on the thickness of the current configuration, and calculate the gradient information of the constant lift constraint, pitch moment coefficient constraint and thickness constraint with respect to the initial design variables respectively. S5. Input the gradient information of the statistical moment objective function, constant lift constraint, pitch moment coefficient constraint and thickness constraint in S4 relative to the initial design variables into the optimizer. The optimizer determines whether the wing configuration corresponding to the current design variable has converged. If yes, the wing configuration is taken as the final robust optimal design. If not, the optimizer updates the design variables according to the gradient information of the objective function and constraint functions, and returns the updated design variables to S1 to continue optimization.

[0007] Furthermore, step S3 includes the following sub-steps: S31. Standardize the uncertainty variables corresponding to the design state disturbance and geometric random disturbance to obtain the uncertainty variables with independent standard normal distribution; S32. In the standardized space, the Hessian vector product module is used, and the Hessian matrix of the objective function with respect to the uncertain variables is approximately constructed using the finite difference method. S33. Perform eigenvalue decomposition on the Hessian matrix. Based on the dimension r after dimensionality reduction, select the eigenvectors corresponding to the eigenvalues ​​with the highest absolute values ​​as the principal directions and form the principal eigenvector matrix. The column vectors of the principal eigenvector matrix together span to form the principal subspace, which is the dimensionality reduction space for sampling uncertain variables. S34. Based on the reduced space, sample the uncertain variables of the independent standard normal distribution, and then map the sampled low-dimensional coordinates back to the original uncertainty space to obtain a sample of uncertain variables for the UQ module to use for calculation. S35. Based on the initial design variables, use the fluid computational grid to calculate the aerodynamic performance data and its first gradient of the sample points in the uncertainty variable sample.

[0008] Furthermore, S32 specifically includes: Define the Hessian matrix composed of the second derivatives of the objective function with respect to the uncertain variables, and establish the linear transformation relationship between the Hessian matrix and the Hessian matrix in the standardized space. Then, by applying a perturbation to the uncertain variables using the finite difference method, the Hessian vector product is calculated. This finite difference method is repeated several times to approximate the Hessian matrix of the original space. The Hessian matrix of the original space is then linearly transformed to obtain the Hessian matrix in the normalized space.

[0009] Furthermore, in S4, the construction of the statistical moment objective function includes: The mean and standard deviation of the drag coefficient of the UQ module computer wing are calculated, and the two are weighted and combined to construct a statistical moment objective function to balance aerodynamic performance and robustness.

[0010] Furthermore, in S4, the PCE method is used to calculate statistical moments and solve for the gradient of deterministic design variables, including: calculating the gradients of the mean and standard deviation of the drag coefficient with respect to the initial design variables.

[0011] Furthermore, in S4, during the process of calculating statistical moments using the PCE method and solving for the gradient of the deterministic design variables, the gradient of the PCE model coefficients with respect to the deterministic design variables is obtained by solving the PCE model. That is, by using the partial derivatives of the system response with respect to the deterministic design variables obtained by the adjoint method, combined with the multivariate orthogonal polynomial basis functions of the PCE model, the gradient of the PCE model coefficients with respect to the deterministic design variables is obtained inversely.

[0012] Furthermore, in S4, an elastic net algorithm is introduced when solving the PCE model coefficients, and L1 and L2 penalty terms are added to the minimum value solution expression to obtain the coefficient matrix of the PCE model. The expression for finding the minimum value is as follows: ; The expression for solving the coefficient matrix of the PCE model is: In the formula, The system response, representing the uncertainty variable, is the drag coefficient. This is the coefficient matrix of the PCE model. for Penalty terms are used to induce sparse solutions. for Penalty terms are used to improve the stability of numerical solutions. The coefficient matrix is ​​composed of the values ​​of the polynomial basis functions at each sampling point. for The square of the norm, , For regularization parameters, for Norm; It is an identity matrix.

[0013] Furthermore, in S5, the optimizer uses a gradient-based sequential quadratic programming algorithm to set the design variables of the wing, and the design variables are geometric variables.

[0014] The robust optimization design method for wings based on Hessian eigenvector dimensionality reduction provided by this invention has the following beneficial effects: (1) Significantly reduces dimensionality and saves computational resources: This invention introduces a spatial transformation of random variables, projecting the original uncertain variables in the standardized space onto a dimensionality-reduced space spanned by the principal eigendirections of the Hessian matrix. This significantly reduces the uncertainty dimension while maintaining consistency in key sensitive directions, thereby reducing sampling and computational costs. Simultaneously, by constructing the Hessian matrix of the objective function with respect to the uncertain variables using the Hessian vector product module and finite difference approximation, the cost of explicitly solving for the second derivative to construct the Hessian matrix is ​​avoided, thus improving computational efficiency.

[0015] (2) It can perform high-precision UQ and integrate with the gradient optimization framework: Non-intrusive PCE is used for statistical moment calculation, and elastic net regularization is introduced to improve the stability and robustness of regression solution. The total gradient of the statistical moment objective function with respect to the design variables is calculated by discrete adjoint method and analytical method, which can be effectively combined with the gradient robust optimization framework.

[0016] (3) Excellent overall performance: The optimized design obtained by applying this invention can significantly improve the robustness of the design while ensuring greater aerodynamic benefits. In the presence of flight state fluctuations and geometric errors, this design can not only maintain a lower average drag (mean) but also greatly reduce the range of drag fluctuations (standard deviation) while significantly reducing the number of uncertainty bits and sampling scale, thereby ensuring the reliability of the aircraft in actual operation.

[0017] In summary, this invention successfully overcomes the problems of large sample size and high computational cost in robust optimization under multi-source uncertain inputs. It can effectively improve the robustness of optimization results while ensuring aerodynamic performance gains, and finally forms an engineering optimization design scheme with excellent robustness, providing a practical reference for the robust optimization problem of aircraft design under multi-source uncertain conditions. Attached Figure Description

[0018] Figure 1 This is a flowchart of the robust optimization design method for wings based on Hessian eigenvector dimensionality reduction in Example 1.

[0019] Figure 2 This is a flowchart of the uncertainty space dimensionality reduction method based on Hessian eigenvectors in Example 1.

[0020] Figure 3 The RAE2822 airfoil structure mesh is shown in Example 2.

[0021] Figure 4 This is a schematic diagram of the FFD deformation control points of the RAE2822 airfoil in Example 2.

[0022] Figure 5 The curves showing the mean and standard deviation of the lift coefficient under different uncertainty variable dimensions are shown in Example 2 when the UQ of the RAE2822 airfoil is reduced.

[0023] Figure 6 The curves showing the mean and standard deviation of the drag coefficient under different uncertainty variables are shown in Example 2 when the UQ of the RAE2822 airfoil is reduced.

[0024] Figure 7 The curves showing the mean and standard deviation of the pitching moment coefficient under different uncertainty variable dimensions are shown in Example 2 when the UQ of the RAE2822 airfoil is reduced.

[0025] Figure 8 The curves showing the changes in the mean and standard deviation of the lift coefficient under different uncertainty variable dimensions are shown in Example 2 for the RAE2822 airfoil when the geometric disturbance is increased to reduce the UQ dimension.

[0026] Figure 9 The curves showing the mean and standard deviation of the drag coefficient under different uncertainty variable dimensions are shown in Example 2 for the RAE2822 airfoil when the geometric disturbance is increased to reduce the UQ.

[0027] Figure 10 The curves showing the mean and standard deviation of the pitching moment coefficient under different uncertainty variable dimensions are shown in Example 2 for the RAE2822 airfoil when the geometric disturbance is increased to reduce the UQ.

[0028] Figure 11 The structure grid of the M6 ​​wing in Example 3.

[0029] Figure 12 This is a schematic diagram of the FFD deformation control points of the M6 ​​wing in Example 3.

[0030] Figure 13 The curves showing the mean and standard deviation of the lift coefficient under different uncertainty variable dimensions are shown in Example 3 when the UQ of the M6 ​​wing is reduced.

[0031] Figure 14 The curves showing the mean and standard deviation of the drag coefficient under different uncertainty variable dimensions are shown in Example 3 when the UQ of the M6 ​​wing is reduced.

[0032] Figure 15 The curves showing the mean and standard deviation of the pitching moment coefficient under different uncertainty variable dimensions are shown in Example 3 when the UQ of the M6 ​​wing is reduced.

[0033] Figure 16 For the initial configuration, deterministic optimized configuration, and robust optimized configuration (Unopt) in Example 4, at the spanwise position Comparison of airfoil shape and pressure distribution at the location.

[0034] Figure 17 For the initial configuration, deterministic optimized configuration, and robust optimized configuration (Unopt) in Example 4, at the spanwise position Comparison of airfoil shape and pressure distribution at the location.

[0035] Figure 18 For the initial configuration, deterministic optimized configuration, and robust optimized configuration (Unopt) in Example 4, at the spanwise position Comparison of airfoil shape and pressure distribution at the location.

[0036] Figure 19 For the initial configuration, deterministic optimized configuration, and robust optimized configuration (Unopt) in Example 4, at the spanwise position Comparison of airfoil shape and pressure distribution at the location.

[0037] Figure 20 This is a comparison of the pressure distribution cloud maps at the design point for the initial configuration, deterministic optimization configuration, and robust optimization configuration (Unopt) in Example 4.

[0038] Figure 21 Example 4 shows the differences in normalized lift and elliptic distribution for the initial configuration, the deterministic optimized configuration, and the robust optimized configuration.

[0039] Figure 22 Example 4 presents violin plots of UQ for the initial configuration, deterministic optimization configuration (Deopt), and robust optimization configuration (Unopt) to illustrate the comprehensive performance distribution of the optimization results under uncertain conditions.

[0040] Figure 23 Example 4 shows the initial configuration, the deterministic optimized configuration, and the robust optimized configuration (Unopt) at the spanwise position. Pressure coefficient distribution and mean at the location deviation zone Figure 24 Example 4 shows the initial configuration, the deterministic optimized configuration, and the robust optimized configuration (Unopt) at the spanwise position. Pressure coefficient distribution and mean at the location deviation zone Figure 25 Example 4 shows the initial configuration, the deterministic optimized configuration, and the robust optimized configuration (Unopt) at the spanwise position. Pressure coefficient distribution and mean deviation zone Figure 26 Example 4 shows the initial configuration, the deterministic optimized configuration, and the robust optimized configuration (Unopt) at the spanwise position. Pressure coefficient distribution and mean at the location Quasi-error zone. Detailed Implementation

[0041] The specific embodiments of the present invention are described below to enable those skilled in the art to understand the present invention. However, it should be understood that the present invention is not limited to the scope of the specific embodiments. For those skilled in the art, various changes are obvious as long as they are within the spirit and scope of the present invention as defined and determined by the appended claims. All inventions utilizing the concept of the present invention are protected.

[0042] Example 1 This embodiment provides a robust optimization design method for wings based on Hessian eigenvector dimensionality reduction, referencing... Figure 1 Specifically, it includes the following: S1. Given the initial geometry of the wing, determine the initial design variables corresponding to the initial geometry; use the initial design variables as the starting point for robust optimization iteration, and in subsequent iterations, the optimizer regenerates the design variables based on gradient information as the starting point for robust optimization.

[0043] S2. Update the wing geometry using a parametric method based on the initial design variables, and then adjust the fluid calculation mesh using the mesh deformation module based on the updated wing geometry. S3. Construct uncertainty variables based on design state disturbances and geometric random disturbances, reduce the dimension of the uncertainty space based on Hessian eigenvectors, and generate sample points in the reduced uncertainty space. For each sample point, perform flow field calculations based on the fluid computation grid corresponding to the initial design variables to obtain the aerodynamic performance response of the sample point and its first-order gradient with respect to the initial design variables. This specifically includes the following steps: S31. Standardize the uncertainty variables corresponding to the design state disturbance and geometric random disturbance to obtain the uncertainty variables with independent standard normal distribution; The expression for the standardized transformation is: In the formula, Represents the original uncertain variable. Represents a standard normally distributed variable. express The mean, express The standard deviation.

[0044] S32. Within the standardized space, using the Hessian vector product module and the finite difference method, approximately construct the Hessian matrix of the objective function with respect to the uncertainty variables. ; In some embodiments, a target function is defined. The Hessian matrix formed by the second derivatives of the uncertain variables And establish the linear transformation relationship between the Hessian matrix and the standardized space Hessian matrix; Among them, the Hessian matrix Represented as: In the formula, For the objective function abbreviation, , It is an uncertain variable; The linear transformation relationship between the Hessian matrix and the normalized space Hessian matrix is ​​expressed as follows: In the formula, The normalized space Hessian matrix; Then, by applying a perturbation to the uncertain variables using the finite difference method, the Hessian vector product is calculated, which is expressed as: In the formula, The first element of the identity matrix List, For disturbance quantity; For gradient operators; A vector of uncertain variables; By repeating the finite difference method several times, the Hessian matrix of the complete original space can be approximately obtained. A linear transformation is performed on the Hessian matrix in the original space to obtain the Hessian matrix in the normalized space. .

[0045] S33. Perform eigenvalue decomposition on the Hessian matrix, specifically on the Hessian matrix in the normalized space. Perform eigenvalue decomposition. Based on the reduced dimensionality r, select the eigenvectors corresponding to the top r eigenvalues ​​by absolute value as the principal directions and construct the principal eigenvector matrix. , principal eigenvector matrix The column vectors together span to form a principal subspace, which is the dimensionality-reduced space for sampling uncertain variables; The expression for eigenvalue decomposition is as follows: In the formula, The eigenvector orthogonal matrix, It is the eigenvalue matrix.

[0046] S34. Based on the dimensionality reduction space, sample the uncertain variables of the independent standard normal distribution, and then map the sampled low-dimensional coordinates back to the original uncertainty space to obtain the uncertainty variable samples used by the UQ module for calculation. This process is expressed as: In the formula, Represents the coordinates of high-dimensional uncertain variables in the original space; Represents the low-dimensional coordinates of the subspace.

[0047] S35. Based on the initial design variables, use a fluid computational grid to calculate the aerodynamic performance data and its first gradient of the sample points in the uncertainty variable sample.

[0048] S4. Input the aerodynamic performance response of the sample points and its first-order gradient with respect to the initial design variables into the UQ module to construct the statistical moment objective function, and calculate the gradient of the statistical moment objective function with respect to the design variables using the analytical expression of the statistical moments; simultaneously, determine the constant lift constraint (i.e., ...) based on the flow field solution results in S3. , Constraints for the target lift coefficient and pitching moment coefficient (similar to) The inequality constraint, i.e., compared to the initial pitching moment coefficient The change amount does not exceed 0.01), and the thickness constraint is determined by the thickness of the current configuration (i.e.: The optimized thickness is no less than a times the initial thickness, and the gradient information of each constraint relative to the design variables is calculated respectively. In one specific embodiment, the construction of the statistical moment objective function includes: The UQ module performs a weighted fusion of the mean and standard deviation of the wing drag coefficient to obtain a statistical moment objective function that balances the wing's average aerodynamic performance and disturbance rejection robustness. It is represented as: In the formula, This represents the average drag coefficient. Standard deviation The weighting coefficients corresponding to the mean. The weighting coefficient is the standard deviation.

[0049] In one specific embodiment, the PCE method is used to calculate statistical moments and solve for gradients of deterministic design variables, specifically as follows: The gradients of the mean and standard deviation of the drag coefficient with respect to the initial deterministic design variables were calculated using analytical methods. The gradient solution is expressed as: In the formula, The system response, representing an uncertain variable, is here the drag coefficient. Its mean, Its standard deviation; Design variables for deterministic purposes; These are the coefficients of the PCE model, with the number of terms being... , Denotes the basis functions of a multivariate orthogonal polynomial. The sample size required for the UQ method based on non-invasive PCE is denoted as . ,in, For oversampling rate, The number of random variables. The PCE model order is... For sample index.

[0050] In one specific embodiment, the gradient of the PCE model coefficients with respect to deterministic design variables Solving the PCE model involves using the partial derivatives of the system response with respect to deterministic design variables obtained through the adjoint method, and combining this with the multivariate orthogonal polynomial basis functions of the PCE model to inversely calculate the gradients of the PCE model coefficients with respect to deterministic design variables. In the formula, For uncertain variables, For the first i Deterministic design variables; In one specific embodiment, when solving the PCE model coefficients The elastic net algorithm is introduced to avoid the ill-conditioned or even near-singular coefficient matrix problem that may exist in the least squares method in the data space. That is, the expression that needs to solve for the minimum value is transformed into ; Solving the coefficient matrix of the PCE model , is represented as: In the formula, for Penalty terms are used to induce sparse solutions. for Penalty terms are used to improve the stability of numerical solutions. The coefficient matrix is ​​composed of the values ​​of the polynomial basis functions at each sampling point. for The square of the norm, , For regularization parameters, for Norm; It is an identity matrix.

[0051] S5. Input the gradient information of the statistical moment objective function, constant lift constraint, pitch moment coefficient constraint and thickness constraint in S4 relative to the initial design variables into the optimizer. The optimizer determines whether the wing configuration determined by the initial design variables has converged. If it has, this wing configuration is taken as the final robust optimal design. If not, the optimizer regenerates the design variables based on the total gradient and returns to S1 as the new initial design variables to continue optimization. The optimizer uses a gradient-based sequential quadratic programming algorithm to set the design variables for the wing, which are geometric variables.

[0052] Example 2 This case study is based on the uncertainty space dimensionality reduction method based on Hessian eigenvectors in Example 1, specifically the content corresponding to S3 in Example 1; for the RAE2822 airfoil, according to Figure 2 The framework shown is dimensionality-reduced and quantized.

[0053] according to Figure 2 The design flow shown is for Figure 3 and Figure 4The computational grid and deformation control points shown are subjected to uncertainty reduction quantization, with a total grid size of 17,000. In the FFD (Free-Form Deformation) control point diagram, the red control points are fixed points, and the black control points are geometric uncertainty variables. The airfoil geometry is changed only by the displacement of the black control points. The design conditions are set as Mach number Ma = 0.73, angle of attack AoA = 2.79°, and Reynolds number Re = 6.5 × 10⁻⁶. 6 Target lift coefficient Considering Mach number, angle of attack, and 10 geometric uncertainties, all assumed to follow a normal distribution, with the following distributions: , Geometric uncertainty variables To address the aforementioned uncertainties, a third-order PCE model is used for approximation, namely... And select oversampling rate .

[0054] Table 1 and Figure 5 , Figure 6 and Figure 7 The influence of different numbers of uncertain variables on the mean and standard deviation of corresponding aerodynamic performance parameters within the reduced-dimensional space is presented. In the figure, mean represents the mean, and std represents the standard deviation. In the table, 12-ori (ori represents the original space) represents the sample obtained based on 12 uncertain variables in the original space, and its mean corresponds to... Figure 5 , Figure 6 and Figure 7 The red dashed line corresponds to the standard deviation, while the blue dashed line represents the standard deviation. The numbers 1 to 12 represent samples obtained from the corresponding number of uncertain variables in the reduced-dimensional space. Compared to the evaluation results based on samples with 12 uncertain variables in the original space, when the number of uncertain variables is reduced to 2... , as well as The maximum relative error of the mean is 0.4%, and the maximum relative error of the standard deviation is 4.1%. It also enables a rapid reduction in the sample size from 910 to 20.

[0055] Table 1. Comparison of the impact of dimensionality reduction on aerodynamic performance - RAE2822 airfoil Furthermore, to investigate the impact of increased geometric perturbation on aerodynamic performance, the standard deviations of the 10 geometric uncertainty variables were increased, i.e. , , We will analyze its impact on dimensionality reduction of the space of uncertain variables.

[0056] Table 2 and Figure 8 , Figure 9 , Figure 10 The effects of different numbers of uncertainties in the reduced-dimensional space on the mean and standard deviation of the corresponding aerodynamic performance parameters are presented. The results show that increasing the geometric perturbation has the greatest impact on the moment standard deviation of the RAE2822 airfoil. To ensure the accuracy of the lift coefficient standard deviation prediction, at least five uncertainties in the reduced-dimensional space are required. , When the number of uncertain variables is reduced to 3, the relative error of the standard deviation can still be guaranteed to be less than 3.4%. At the same time, it can achieve a sharp reduction in the sample size from 910 to 40.

[0057] Table 2 Comparison of the impact of dimensionality reduction on aerodynamic performance - RAE2822 airfoil (increased geometric disturbance) Example 3 In problems with high geometric dimensions, aerodynamic uncertainties mainly arise from operating condition variables such as Mach number and angle of attack, which have relatively low dimensionality. In contrast, geometric uncertainties typically have significantly higher dimensionality, and geometric deviations often dominate the source of uncertainty in aerodynamic response. Therefore, in high-dimensional geometric uncertainty problems, dimensionality reduction based on Hessian eigenvectors can be applied only to the geometric space, while Mach number and angle of attack can be directly retained as low-dimensional operating condition uncertainties.

[0058] This paper selects the M6 ​​wing as a three-dimensional transonic configuration example to verify the effectiveness of the dimensionality reduction method in Example 2 above in high-dimensional problems. The method flow is as follows: Figure 2 As shown. The mesh and deformation control point distribution used are as follows. Figure 11 and Figure 12 As shown, the total mesh size is 110W. In the FFD control point diagram, the airfoil section control points and leading and trailing edge points are fixed points; the remaining control points are treated as geometric uncertainty variables, and the airfoil geometry is changed only through the displacement of these control points. The design conditions are set as Ma = 0.8395, AoA = 3.06°, and Reynolds number Re = 11.72 × 10⁻⁶. 6 Target lift coefficient Considering Mach number, angle of attack, and 30 geometric uncertainties, all assumed to follow a normal distribution, with the following distributions: , , To address the aforementioned uncertainties, a third-order PCE model is used for approximation, namely... And select oversampling rate Sample points are generated and sampled within a reduced-dimensional subspace constructed based on Hessian eigenvectors, and the statistical properties of the aerodynamic response are further evaluated. (Selection...) , as well as As the output of interest, its mean and standard deviation are calculated as statistical moments.

[0059] Table 3 and Figure 13 , Figure 14 , Figure 15 The influence of different numbers of uncertainties in the reduced-dimensional space on the mean and standard deviation of the corresponding aerodynamic performance parameters is presented. Here, 32-ori represents the sample obtained based on 32 uncertainties in the original space, and its mean corresponds to... Figure 13 , Figure 14 and Figure 15 The red dashed line corresponds to the standard deviation, while the blue dashed line represents the sample size. The values ​​3 to 32 represent samples obtained from corresponding numbers of uncertain variables in the reduced-dimensional space. Compared to the evaluation results based on samples with 32 uncertain variables in the original space, when the number of uncertain variables is reduced to 3 (i.e., the geometric uncertainty variable is reduced to 1), , as well as The maximum relative error of the mean is 1.8%, and the maximum relative error of the standard deviation is 5.9%. It also enables a rapid reduction in the sample size from 1186 to 26.

[0060] Table 3 Comparison of the impact of dimensionality reduction on aerodynamic performance - M6 wing Example 4 This embodiment is based on embodiment 3, and further according to... Figure 1 The robust optimization framework based on Hessian eigenvector dimensionality reduction is shown, and robust optimization of the M6 ​​wing is carried out. The main purpose of this part is to investigate whether, after introducing the Hessian eigenvector-based dimensionality reduction method, the robust optimization framework can significantly save computational resources while improving flight performance robustness compared to deterministic optimization in a space of less dimensional uncertain variables. The computational grid, FFD control frame, and deformation control points are shown in the diagram. Figure 11 and Figure 12 The specific process is as follows: Step 1: Given an initial geometry, determine the initial design variables corresponding to the initial geometry. Use the initial design variables as the starting point for robust optimization iterations. Subsequent iterations use the optimizer to generate design variables based on gradient information as the starting point for robust optimization. Initial design variables include angle of attack, airfoil, and twist angle, with the airfoil and twist angle geometric design variables using global design variables. Constraints include: constant lift constraint ( Pitch moment coefficient constraint () That is, the change in moment coefficient does not exceed 0.01), and thickness constraints ( (That is, the thickness is not less than 98% of the initial thickness). Therefore, the mathematical expression of the robust optimization problem is: In the formula, ,Right now The statistical moment function is composed of the mean and standard deviation with a weight ratio of 3:1. For the local geometric design variables and twist angle of the wing, This is the design variable geometry space. The design variables ultimately include 30 global wing geometry variables and 1 angle of attack variable. The initial spanwise section thickness, To optimize the thickness of the cross-section, This is the initial torque coefficient.

[0061] Step 2: Update the geometry using a parametric method based on the initial design variables, and adjust the fluid calculation mesh by the mesh deformation module based on the updated geometry; Step 3: Construct uncertainty variables based on design state disturbances and geometric random disturbances. Reduce the dimensionality of the uncertainty space using Hessian eigenvectors and generate sample points in the reduced uncertainty space. For each sample point, perform flow field calculations based on the fluid computation grid corresponding to the current initial design variables to obtain the aerodynamic performance response of the sample point (e.g., ...). , , (etc.) and its first gradient with respect to the initial design variables.

[0062] In step three, a total of 32 random variables are defined, including: The design point has an incoming Mach number that follows a normal distribution N(0.8395, 0.02). 2 The angle of attack (follows a normal distribution N(3.06, 0.2)) is... 2 )) and 30 geometric design variables (from a normal distribution N(0, 0.001) 2 )).

[0063] Step 4: Input the aerodynamic performance response and its first-order gradient of the sample points into the UQ module, construct the statistical moment objective function, and calculate the gradient of the statistical moment objective function with respect to the design variables using the analytical expression of the statistical moments; simultaneously, determine the constant lift constraint (i.e., ...) based on the flow field solution results in S3. , Constraints for the target lift coefficient and pitching moment coefficient (similar to) The inequality constraint, i.e., compared to the initial pitching moment coefficient The change amount does not exceed 0.01), and the thickness constraint is determined by the thickness of the current configuration (i.e.: The optimized thickness is no less than a times the initial thickness, and the gradient information of each constraint relative to the design variables is calculated respectively. When performing UQ in step four, the order is used. Oversampling rate sparsity coefficient As a parameter of the PCE method, it can be guaranteed that the normalized root mean square deviation (NRMSD) of PCE during the quantization process is less than 0.01.

[0064] Step 5: Input the statistical moment objective function, each constraint information, and their gradient information relative to the design variables into the optimizer. The optimizer determines whether the configuration determined by the design variables has converged. If so, this configuration is taken as the final robust optimal design; if not, the total gradient is taken as the gradient information, and the process returns to Step 1 to regenerate the design variables based on the gradient information.

[0065] As a comparative case, deterministic optimization was also performed, with the problem settings shown below. Except for the objective function, which is to minimize the drag coefficient, all other settings are the same as for robust optimization.

[0066] The initial configuration, deterministic optimized configuration, and robust optimized configuration are compared and named Initial, Deopt, and Unopt, respectively. The optimization results are shown in Table 4. Figures 16 to 26 As shown. Figure 22 In this context, expectation represents the mean, ±1STD represents plus or minus one standard deviation, and 95% confidence represents the 95% confidence interval. These results highlight the core differences between robust and deterministic optimization.

[0067] Table 4 Comparison of Optimization Results Depend on Figure 8 The airfoil geometry and profile pressure distribution at different spanwise positions show that, compared to the Initial configuration, the drag reduction of Deopt and Unopt primarily stems from the reconstruction of the airfoil profile load distribution: both optimized configurations make the pressure recovery process in the mid-to-rear chord more continuous and gradual, thereby reducing the impact of unfavorable pressure gradients, significantly weakening shock waves, and reducing pressure drag near the trailing edge. Compared to the Initial and Deopt configurations, Unopt more significantly reduces the airfoil's rear loading and shifts the maximum thickness position forward, resulting in a forward shift of the pressure center at each wing position and a reduction in nose-down moment. Simultaneously, the outer wing section exhibits a more significant increase in leading-edge nose radius and stronger torsion, reducing and blunting the leading-edge suction peak and decreasing the overall undulation of the profile pressure curve. Furthermore, combined with... Figure 21Under design point conditions, both Unopt and Deopt reduce induced drag to some extent through load redistribution, thereby improving aerodynamic performance, but the difference in their circulation distribution is relatively small. Furthermore, due to... Figure 22 As can be seen from the violin diagram, compared to Initial and Deopt, Unopt... The distribution pattern is more "short and stout" and A shorter interval indicates a more concentrated and less volatile sample drag, resulting in a smaller mean drag. In contrast, while Deopt reduces the pressure difference in the middle and later chord segments at the mean level, its larger longitudinal span and more pronounced tail-end extension suggest that drag remains highly sensitive to disturbances. This statistical conclusion is consistent with... Figure 23 , Figure 24 , Figure 25 , Figure 26 The pressure coefficient shown The shaded areas show consistent changes: the standard deviation in the Deopt key area remains relatively large, leading to more significant drag fluctuations. In contrast, the Unopt shaded area is narrower overall, reflecting a smaller range of drag fluctuations, corresponding to the robustness advantage shown in the violin chart. This characteristic is more pronounced in the outer wing segment.

[0068] Example 5 For the robust optimized configuration Unopt in Example 4, the following is adopted: Figure 2 The dimensionality reduction method shown further performs dimensionality reduction quantization, compressing the original 32 uncertainties into 3 uncertainties used in the optimization (including only 1 geometric uncertainty variable), and then performs uncertainty quantization analysis on each. The design conditions are the same as in Example 2. Table 5 shows a comparison of the start-up feasibility between the dimensionality reduction quantization results and the full-space quantization results.

[0069] Wherein, 32 represents the samples obtained based on 32 uncertain variables, while 3 represents the samples obtained based on 3 uncertain variables in the reduced dimension space. It can be seen that the invented uncertainty space dimensionality reduction method significantly reduces the number of samples from 1186 to 26 while compressing the uncertainty dimension from 32 to 3, but the obtained statistical moments remain highly consistent with the original 32-dimensional results. Specifically, the mean... , and The relative errors were 0.2014%, -0.2641%, and 0.0491%, respectively, and the standard deviations were... , and The relative errors were -4.2639%, -2.9578%, and -3.8133%, respectively, with the overall error controlled within 5%. This indicates that the invented dimensionality reduction method can effectively capture the main direction of uncertainty propagation and, on this basis, achieve rapid evaluation of aerodynamic response statistical moments. That is, while significantly reducing the uncertainty dimension and sample cost, it can still maintain reliable predictive ability for key statistical features such as mean and standard deviation.

[0070] Table 5 Comparison of the impact of dimensionality reduction on aerodynamic performance - M6 wing Although specific embodiments of the invention have been described in detail with reference to the accompanying drawings, this should not be construed as limiting the scope of protection of this patent. Various modifications and variations that can be made by a person skilled in the art without inventive effort within the scope described in the claims still fall within the scope of protection of this patent.

Claims

1. A robust optimization design method for airfoils based on Hessian eigenvector dimensionality reduction, characterized in that, Includes the following steps: S1. Given the initial geometry of the wing, determine the initial design variables corresponding to the initial geometry; S2. Update the wing geometry using a parametric method based on the initial design variables, and then adjust the fluid calculation grid using a mesh deformation module based on the updated wing geometry. S3. Construct uncertainty variables based on design state disturbance and geometric random disturbance, reduce the dimension of uncertainty space based on Hessian eigenvectors, and generate sample points in the dimension-reduced uncertainty space. For each sample point, perform flow field calculation based on the fluid computing grid corresponding to the initial design variables to obtain the aerodynamic performance response of the sample point and its first gradient with respect to the initial design variables. S4. Input the aerodynamic performance response of the sample points and its first-order gradient with respect to the initial design variables into the UQ module, construct the statistical moment objective function, and calculate the gradient information of the statistical moment objective function with respect to the initial design variables in combination with the analytical expression of the statistical moments; at the same time, determine the constant lift constraint and pitch moment coefficient constraint based on the flow field calculation results in S3, and determine the thickness constraint based on the thickness of the current configuration, and calculate the gradient information of the constant lift constraint, pitch moment coefficient constraint and thickness constraint with respect to the initial design variables respectively. S5. Input the gradient information of the statistical moment objective function, constant lift constraint, pitch moment coefficient constraint and thickness constraint in S4 relative to the initial design variables into the optimizer. The optimizer determines whether the wing configuration corresponding to the current design variable has converged. If yes, the wing configuration is taken as the final robust optimal design. If not, the optimizer updates the design variables according to the gradient information of the objective function and constraint functions, and returns the updated design variables to S1 to continue optimization.

2. The robust optimization design method for wings based on Hessian eigenvector dimensionality reduction according to claim 1, characterized in that, S3 includes the following steps: S31. Standardize the uncertainty variables corresponding to the design state disturbance and geometric random disturbance to obtain the uncertainty variables with independent standard normal distribution; S32. In the standardized space, the Hessian vector product module is used, and the Hessian matrix of the objective function with respect to the uncertain variables is approximately constructed using the finite difference method. S33. Perform eigenvalue decomposition on the Hessian matrix. Based on the dimension r after dimensionality reduction, select the eigenvectors corresponding to the eigenvalues ​​with the highest absolute values ​​as the principal directions and form the principal eigenvector matrix. The column vectors of the principal eigenvector matrix together span to form the principal subspace, which is the dimensionality reduction space for sampling uncertain variables. S34. Based on the reduced space, sample the uncertain variables of the independent standard normal distribution, and then map the sampled low-dimensional coordinates back to the original uncertainty space to obtain a sample of uncertain variables for the UQ module to use for calculation. S35. Based on the initial design variables, use the fluid computational grid to calculate the aerodynamic performance data and its first gradient of the sample points in the uncertainty variable sample.

3. The robust optimization design method for wings based on Hessian eigenvector dimensionality reduction according to claim 2, characterized in that, Specifically, S32 includes: Define the Hessian matrix composed of the second derivatives of the objective function with respect to the uncertain variables, and establish the linear transformation relationship between the Hessian matrix and the Hessian matrix in the standardized space. Then, by applying a perturbation to the uncertain variables using the finite difference method, the Hessian vector product is calculated. This finite difference method is repeated several times to approximate the Hessian matrix of the original space. The Hessian matrix of the original space is then linearly transformed to obtain the Hessian matrix in the normalized space.

4. The robust optimization design method for wings based on Hessian eigenvector dimensionality reduction according to claim 1, characterized in that, In step S4, the construction of the statistical moment objective function includes: The mean and standard deviation of the drag coefficient of the UQ module computer wing are calculated, and the two are weighted and combined to construct a statistical moment objective function to balance aerodynamic performance and robustness.

5. The robust optimization design method for wings based on Hessian eigenvector dimensionality reduction according to claim 1, characterized in that, In step S4, the PCE method is used to calculate statistical moments and solve for the gradient of deterministic design variables, including: calculating the gradients of the mean and standard deviation of the drag coefficient with respect to the initial design variables.

6. The robust optimization design method for wings based on Hessian eigenvector dimensionality reduction according to claim 5, characterized in that, In S4, during the process of calculating statistical moments and solving the gradient of deterministic design variables using the PCE method, the gradient of the PCE model coefficients with respect to the deterministic design variables is obtained by solving the PCE model. That is, by using the partial derivatives of the system response with respect to the deterministic design variables obtained by the adjoint method, combined with the multivariate orthogonal polynomial basis functions of the PCE model, the gradient of the PCE model coefficients with respect to the deterministic design variables is obtained inversely.

7. The robust optimization design method for wings based on Hessian eigenvector dimensionality reduction according to claim 6, characterized in that, In S4, an elastic net algorithm is introduced when solving the PCE model coefficients. L1 and L2 penalty terms are added to the minimum value solution expression to obtain the coefficient matrix of the PCE model. The expression for finding the minimum value is as follows: ; The expression for solving the coefficient matrix of the PCE model is: In the formula, The system response, representing the uncertainty variable, is the drag coefficient. This is the coefficient matrix of the PCE model. for Penalty terms are used to induce sparse solutions. for Penalty terms are used to improve the stability of numerical solutions. The coefficient matrix is ​​composed of the values ​​of the polynomial basis functions at each sampling point. for The square of the norm, , For regularization parameters, for Norm; It is an identity matrix.

8. The robust optimization design method for wings based on Hessian eigenvector dimensionality reduction according to claim 1, characterized in that, In S5, the optimizer uses a gradient-based sequential quadratic programming algorithm to set the design variables of the wing, and the design variables are geometric variables.

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