Control method for adaptive morphing wing based on hessian vector product
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- NORTHWESTERN POLYTECHNICAL UNIV
- Filing Date
- 2026-06-10
- Publication Date
- 2026-07-10
Smart Images
Figure CN122365729A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of aircraft design technology, specifically relating to a control method for adaptive deformable wings based on Hessian vector product. Background Technology
[0002] With the aviation industry's ever-increasing demands for aircraft efficiency, performance, and environmental friendliness, adaptive morphing wing technology has become a cutting-edge research hotspot. This technology changes the wing shape in real time to maintain optimal aerodynamic performance under different flight conditions, thereby significantly reducing fuel consumption and emissions. Among them, morphing technology based on trailing edge camber variation shows great potential in improving cruise efficiency because it can effectively control lift distribution, delay flow separation, and reduce shock wave intensity.
[0003] Despite the promising future of morphing technology, its practical application in commercial aircraft still faces significant challenges. According to industry feedback, without mechanical locking mechanisms, current technology makes it difficult for control surfaces to maintain the ideal morphing angle precisely throughout flight, with control errors typically reaching 1°. Even after optimization, the error range can still reach 0.5°. This control uncertainty directly impacts the wing's aerodynamic performance, potentially causing designs optimized under ideal conditions to perform poorly in actual flight, or even negating the benefits of morphing. Therefore, this uncertainty must be considered during the wing design phase, and RDO (Robust Design Optimization) is a key method for addressing this issue because it incorporates UQ (Uncertainty Quantification) into the optimization framework.
[0004] An efficient RDO framework relies on two main pillars: optimization algorithms and UQ methods. Regarding optimization algorithms, gradient optimization algorithms based on discrete adjoint methods have become the mainstream for high-fidelity aerodynamic optimization. The cost of calculating the gradient of the objective function with respect to design variables is almost independent of the number of design variables, offering a significant efficiency advantage over gradient-free optimizers when dealing with high-dimensional problems. In terms of UQ, the generalized multinomial chaos (gPC) method is favored for its high accuracy and efficiency. When the QoI (quantity of interest, i.e., the target parameter of the perturbed random variable of interest) is available, the gradient-enhanced GEgPC method can be further employed. This method, compared to gPC, can improve accuracy and further reduce computational cost with a limited number of sampling points, which is crucial for computing expensive high-fidelity CFD simulations.
[0005] For deformable wings, the number of random variables considered is generally greater than three, making the GEgPC method well-suited for this problem. However, in existing optimization frameworks, when using gradient-based optimizers, it is necessary to obtain the accurate gradient of the objective function with respect to the design variables. This requires calculating the second derivative of the sample function. Traditional GEgPC methods require at least [a certain number of steps] at each sample point when solving for the gradient. Only after several additional calculations can the accurate gradient of the design variables be obtained, where The number of random variables. The large number of design variables presents a significant computational bottleneck. For the optimization problem of adaptive deformable wings with multi-source uncertainties, existing methods struggle to efficiently and accurately solve for the gradient of the objective function. Therefore, a gradient calculation method that eliminates the need to construct a second-order derivative matrix while maintaining both efficiency and accuracy is urgently needed. Summary of the Invention
[0006] To address the aforementioned shortcomings in existing technologies, the control method for adaptive deformable wings based on Hessian vector product provided by this invention solves the computational efficiency bottleneck problem in high-fidelity robust design optimization of adaptive deformable wings.
[0007] To achieve the aforementioned objectives, the technical solution adopted by this invention is: a control method for an adaptive deformable wing based on Hessian vector product, comprising the following steps: S1. Define the design variables for the wing and generate normalized sample points based on the random variable distribution. S2. Update the design variables based on gradient information using the optimizer; S3. Update the wing geometry using the FFD geometric parameterization method based on the design variables, and adjust the fluid computation mesh based on the updated geometry. S4. Calculate the aerodynamic performance data and first-order gradient of the sample points based on the normalized sample points and the adjusted fluid computational grid. S5. Input the aerodynamic performance data and first-order gradient of the sample points into the UQ module and the Hessian vector product module. Calculate the robust objective function and perturbation vector through the UQ module. Calculate the total gradient of the robust objective function relative to the design variables using the perturbation vector through the Hessian vector product module. S6. Input the robust objective function and total gradient into the optimizer. The optimizer determines whether the wing configuration determined by the design variables has converged based on the robust objective function and total gradient. If it has, the wing configuration is taken as the final robust optimal wing design. If not, the total gradient is taken as the gradient information and returned to S2.
[0008] Furthermore: In S1, the design variables of the wing include the angle of attack design variable and the local geometry, twist angle and control surface deflection angle design variables obtained by the FFD geometric parameterization method.
[0009] Furthermore, S4 includes the following sub-steps: S41. Normalized sample points generated by sampling based on the distribution of random variables are used to obtain actual sample points through the current design variables, where each sample point represents a possible realization of an uncertainty; S42. Based on the normalized sample points and the actual sample points, call the CFD solver to calculate the aerodynamic performance data under the sample point conditions, obtain the aerodynamic performance data of the sample points, and call the discrete adjoint solver to calculate the first-order gradient of the aerodynamic performance relative to the design variables.
[0010] Furthermore, in S5, based on the gradient-enhanced generalized polynomial chaos method, the UQ module calculates the robust objective function. I The specific expression is: In the formula, For the measurement matrix containing gradient terms, This is the sample result vector obtained by the high-fidelity solver. for 3D real vector space, The number of sample points. For design variables, for 3D real vector space, To determine the number of design variables, For random variables, for 3D real vector space, The number of random variables. To transform operator notation, representing It is by and Sure.
[0011] Furthermore: In S5, the Hessian vector product module calculates the second derivative of the robust objective function using the finite difference method or the complex step-adjoint method, and then solves for the total gradient of the robust objective function with respect to the design variables; Specifically, the Hessian vector product is calculated using the finite difference method or the complex step-adjoint method, thereby obtaining the second derivative of the robust objective function. The Hessian vector product is then calculated using the finite difference method. The expression is: In the formula, Let be the perturbation vector. For random variables, Step size, For design variables; Calculate the Hessian vector product using the complex variable method. The expression is: In the formula, To separate the imaginary part from a complex number expression, i It is the imaginary unit.
[0012] Compared with the prior art, the present invention has the following beneficial effects: (1) Improved computational efficiency: This invention accurately calculates gradients using the HVP (Hessian-Vector Product) method, reducing the cost of gradient computation from that of the traditional GEgPC method. The magnitude of the problem has been reduced to the same level as that of single uncertainty quantification analysis, which basically solves the problem of finding the accurate gradient in gradient-based robust optimization using GEGPC, and greatly improves the computational availability of robust optimization methods based on GEGPC when there are many random variables.
[0013] (2) High-precision gradient calculation is possible: While ensuring efficiency, the gradient provided by this invention has a relative error that can be controlled within 0.002% and an absolute error of approximately 10 compared with the high-precision central finite difference result. -8 The magnitude ensures the reliability and convergence of the optimization process.
[0014] (3) Excellent overall performance: The optimized design obtained by applying this invention can significantly improve the robustness of the wing while maintaining the aerodynamic benefits brought by the adaptive deformation technology. In the presence of control errors and fluctuations in flight conditions, this design can not only maintain a lower average drag (expected) but also greatly reduce the range of drag fluctuations (standard deviation), thereby ensuring the reliability and economy of the aircraft in actual operation.
[0015] In summary, this invention successfully overcomes the core bottleneck of high-fidelity robust optimization through a key HVP algorithm, achieving a balance between efficiency and accuracy. It also delivers an engineering design solution with excellent robustness in an easily integrated manner, providing a powerful tool for solving the uncertainty problem in advanced aircraft design. Attached Figure Description
[0016] Figure 1 This is a flowchart of the control method for an adaptive deformable wing based on Hessian vector product according to the present invention.
[0017] Figure 2 This is a method for calculating the Hessian vector product based on the complex step-adjoint method (CSAD).
[0018] Figure 3 For function 1 with different numbers of random variables With the number of design variables The number of function evaluations is increased with the increase in robust design optimization (RDO).
[0019] Figure 4 For function 1 with different numbers of random variables With the number of design variables The number of function evaluations increases with the number of robust design optimizations (RDO).
[0020] Figure 5 For function 1 with different numbers of random variables With the number of design variables The number of function evaluations increases with the number of robust design optimizations (RDO).
[0021] Figure 6 For function 1 with different numbers of random variables With the number of design variables The number of function evaluations is increased with the increase in robust design optimization (RDO).
[0022] Figure 7 It is an L2 multi-block structured mesh.
[0023] Figure 8 It is a standard flap-aileron configuration.
[0024] Figure 9 This is the FFD frame used for wing deformation.
[0025] Figure 10 This is the FFD frame used for flaperon deformation.
[0026] Figure 11 A violin plot of the optimization results for the initial stage of cruise is used to show the overall performance distribution of the optimization results under uncertain conditions.
[0027] Figure 12 A violin plot of the optimization results for the midpoint of the cruise phase is used to show the overall performance distribution of the optimization results under uncertain conditions.
[0028] Figure 13 For the initial phase of cruise Optimized results: Standardized lift and elliptical distribution difference diagram.
[0029] Figure 14 Midpoint of cruise phase Optimized results: Standardized lift and elliptical distribution difference diagram.
[0030] Figure 15The image shows a comparison of the four airfoil cases at the first cross-section.
[0031] Figure 16 This is a comparison diagram of the four airfoil cases at the second section.
[0032] Figure 17 This is a comparison diagram of the four airfoil cases at the third section.
[0033] Figure 18 This is a comparison diagram of the four airfoil cases at the fourth section.
[0034] Figure 19 Comparison of cloud maps for Case 1 during the cruise start phase.
[0035] Figure 20 Comparison of cloud maps for Case 2 during the mid-cruise phase.
[0036] Figure 21 Comparison of pressure distribution at the first section during the cruise start phase.
[0037] Figure 22 Comparison of pressure distribution at the second section during the cruise start phase.
[0038] Figure 23 Comparison of pressure distribution at the third section during the cruise start phase.
[0039] Figure 24 Comparison of pressure distribution at the fourth section during the cruise start phase.
[0040] Figure 25 Comparison of pressure distribution at the first section during the mid-cruise phase.
[0041] Figure 26 Comparison of pressure distribution at the second section during the mid-cruise phase.
[0042] Figure 27 Comparison of pressure distribution at the third section during the mid-cruise phase.
[0043] Figure 28 Comparison of pressure distribution at the fourth section during the mid-cruise phase. Detailed Implementation
[0044] 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.
[0045] Example 1: like Figure 1 As shown, in one embodiment of the present invention, the control method for an adaptive deformable wing based on Hessian vector product includes the following steps: S1. Define the design variables for the wing and generate normalized sample points based on the random variable distribution. S2. Update the design variables based on gradient information using the optimizer; S3. Update the wing geometry using the FFD geometric parameterization method based on the design variables, and adjust the fluid computation mesh based on the updated geometry. S4. Calculate the aerodynamic performance data and first-order gradient of the sample points based on the normalized sample points and the adjusted fluid computational grid. S5. Input the aerodynamic performance data and first-order gradient of the sample points into the UQ module and the Hessian vector product module. Calculate the robust objective function and perturbation vector through the UQ module. Calculate the total gradient of the robust objective function relative to the design variables using the perturbation vector through the Hessian vector product module. S6. Input the robust objective function and total gradient into the optimizer. The optimizer determines whether the wing configuration determined by the design variables has converged based on the robust objective function and total gradient. If it has, the wing configuration is taken as the final robust optimal wing design; otherwise, the total gradient is used as the gradient information, and the process returns to S2. In each iteration, the optimizer updates the design variables based on the total gradient feedback from the previous iteration, generating a new set of design variables, thereby initiating a new round of wing configuration updates and convergence determination.
[0046] The purpose of this invention is to overcome the bottleneck of large computational cost of second derivative terms in gradient optimization of quantization methods based on gradient-enhanced polynomial chaos (GEgPC) by using Hessian vector product. It can obtain high-precision gradients of the objective function with respect to design variables without explicitly constructing the second derivative matrix, and the cost of calculating the gradient of a single objective function is only equivalent to one GEgPC evaluation. It basically solves the problem of finding accurate gradients in gradient-based robust optimization using GEgPC, greatly improves the computational availability of the GEgPC method in robust optimization when there are many random variables, and achieves a balance between efficiency and accuracy, ensuring the reliability and economy of the aircraft in actual operation.
[0047] In S1, the wing's design variables include the angle of attack design variable and the local geometry, twist angle, and control surface deflection angle design variables obtained by the FFD geometric parameterization method. This embodiment generates sample points using Latin hypercube sampling.
[0048] In S3, mesh deformation is performed using methods such as inverse distance weighting, and the basic structural mesh of the fluid computation mesh can be obtained using ICEM software.
[0049] S4 includes the following steps: S41. Normalized sample points generated by sampling based on the distribution of random variables are used to obtain actual sample points through the current design variables, where each sample point represents a possible realization of an uncertainty; In this embodiment, each sample point is a specific set of values drawn from all random variables (in this embodiment 2, the random variables include 4 control surface angle errors and 1 Mach number), which together constitute a specific and possible flight condition to simulate a concrete implementation of uncertainty. S42. Based on the normalized sample points and the actual sample points, call the CFD solver to calculate the aerodynamic performance data under the sample point conditions, obtain the aerodynamic performance data of the sample points, and call the discrete adjoint solver to calculate the first-order gradient of the aerodynamic performance relative to the design variables.
[0050] In S5, the uncertainty quantification method used in the UQ module is GEgPC. The derivation process will first be based on the basic gPC, and then extended to the GEgPC method that includes gradient information. It can be considered that the quantity of interest (QoI) is simultaneously influenced by design variables. With random variables The effects take the following form: In the formula, For truncation error, No. orthogonal basis functions for The corresponding coefficients, The last term in the polynomial expansion is numbered, and the total number of terms selected is... The number of terms is determined by the polynomial order. With the number of random variables The calculation formula is jointly determined as follows: In the formula, The upper limit of the polynomial order; We will first consider solving for a single metric of interest (QoI) and assume that the coefficients are obtained through regression methods. Therefore, the coefficients... It can be solved by the following formula: In the formula, For the measurement matrix, for A real vector space of dimensionless vectors, whose elements are... , For the matrix of the first The row is an orthogonal polynomial basis function In the sample points The value at that location, For the first 1 sample point. Let be the polynomial chaotic coefficient vector to be determined. for 3D real vector space, This is the sample result vector obtained by the high-fidelity solver.
[0051] The solution can always be expressed in the following form: in The transformation operator symbol is derived from... and A specific matrix. When for After the sampling is determined, That is, the known quantity. For a specific regression method (such as least squares LSQ), from The generated matrix and It's irrelevant; at this time... , It is a pseudo-inverse matrix.
[0052] Specifically, for the gradient-enhanced generalized polynomial chaos (GEgPC) method, the UQ module computes the robust objective function. I The specific expression is: In the formula, For the measurement matrix containing gradient terms, This is the sample result vector obtained by the high-fidelity solver. for 3D real vector space, The number of sample points. For design variables, for 3D real vector space, To determine the number of design variables, For random variables, for 3D real vector space, The number of random variables. To transform operator notation, representing It is by and Sure.
[0053] In S5, the Hessian vector product module calculates the second derivative of the robust objective function using the finite difference method (FDAD) or the complex step-adjoint method (CSAD), and then solves for the total gradient of the robust objective function with respect to the design variables. Assuming any two positive input variables, regardless of whether they come from design variables Or random variable They are independent of each other, and the total gradient of the robust objective function with respect to the design variables is calculated. The specific expression is: In the formula, It is the transpose symbol. The sign of the partial derivative. The differential symbol, This is the sample result vector obtained by the high-fidelity solver. For the first i Function values for each sample point For design variables, For random variables, n This represents the number of sample points. In the total gradient In the expression, and It can be obtained by uncertainty quantification regression algorithm. The second term can be efficiently calculated using the standard discrete adjoint method, and it contains a second-order differential term. Traditional methods for calculating this term are extremely costly. Therefore, this invention focuses on the second term, considering only one sample point within the second term. : In the formula, For the first A random variable, m The number of random variables; Represents a vector of fixed random variables Except for the first All components other than the one component (i.e.) In the case of ), the first Each component is replaced with its perturbation value. Then proceed Calculate the value.
[0054] If the present invention includes This vector is denoted as Therefore, the above summation is essentially a Hessian matrix. With disturbance vector The core computational challenge has been simplified from calculating the entire Hessian matrix to calculating a single Hessian vector product. This invention employs two methods to calculate the Hessian vector product of a single sample: the finite difference method or the complex step-adjoint method. This yields the second derivative of the robust objective function, cleverly solving for the second term in the total gradient, and thus obtaining the total gradient of the robust objective function relative to the design variables. The Hessian vector product is calculated using the finite difference method (FDAD). The expression is: In the formula, Let be the perturbation vector. For random variables, Step size, For design variables; Calculate the Hessian vector product using the method of complex variables (CSAD) The expression is: In the formula, This is an operation to separate the imaginary part from a complex number expression.
[0055] For each sample and a single objective function Its computational cost is one additional solution to the nonlinear equation and one solution to the adjoint equation. Due to the uncertainty quantization (UQ), a total of There are samples, therefore the total computational cost of calculating the derivative is . Solving the sub-nonlinear equations and Solving the adjoint equation. The algorithm for calculating the Hessian vector product using the complex step-adjoint method (CSAD) is as follows: Figure 2 As shown. Among them. Step size, For the state variable vector, For the adjoint solution vector, For the objective function For design variables The total derivative. Although the CSAD method can theoretically achieve higher accuracy, current mainstream nonlinear solvers primarily rely on real number operations. Therefore, unless otherwise specified, the Hessian vector product calculation defaults to the FDAD method.
[0056] Similar to the adjoint method, when there are multiple objective functions or constraint functions based on uncertainty, the Hessian vector product needs to be calculated multiple times accordingly. Furthermore, limited by current real-valued nonlinear solvers, the method described in this invention can only achieve first-order accuracy. Although the central difference scheme theoretically provides higher accuracy, it doubles the number of nonlinear solver calls required, thus necessitating a trade-off between computational efficiency and accuracy.
[0057] Example 2: This embodiment, in connection with Embodiment 1, provides a gradient-based high-fidelity aerodynamic multidisciplinary robust design optimization scenario. Specifically, it discloses a performance evaluation method for the RDO process, relying on an improved Rosenbrock function to complete the test, accurately verifying the optimization performance of the GEgPC method combined with different differential methods, and providing data support for the engineering application of the RDO framework.
[0058] This embodiment first introduces design variables and random variables, denoted as function 1, and the expected value of function 1. Standard deviation The minimum points are all [1, 1, ... 1]. Let the design variable be... The random variable is ,in and These represent the number of design variables and random variables, respectively. Subsequently, based on the classic Rosenbrock function, a family of RDO test functions containing both design variables and random variables is constructed. Function 1... As shown below: In the formula, and For the Kronecker function, For the first One design variable, For the first One design variable, For the first A random variable, For the first A random variable, To determine the number of design variables, The number of random variables. For the first A random variable, For the first One random variable; In the UQ settings, GEgPC uses an order. The sparsity coefficient is 0.9, and the oversampling rate is... The expansion coefficients are then solved using regression methods. When calculating the derivative, the first-order differential within GEGPC is implemented using complex-step differentiation. When using HVP and FD... and FD When calculating the second derivative, the step size of the finite difference method is set to 1 × 10⁻⁶. -6 Among them, FD FD represents the finite difference approximation for the design variables. This indicates a finite difference approximation for the random variable, where the computational cost of the remaining first derivatives is comparable to solving the adjoint equation. The objective function is set as follows: , Let be the mean of function 1. Let be the standard deviation of function 1. For function 1, the optimization process employs the commonly used Sequential Quadratic Programming (SQP) optimizer, SLSQP. The number of function evaluations required for the optimizer to converge is as follows: Figures 3-6 As shown.
[0059] exist Under the same conditions, the number of iterations required for the solver to converge varies. It exhibits approximately linear growth. Under the same single-objective function and other conditions, HVP requires the fewest evaluations. Under the same circumstances, with Increase, with FD In comparison, the number of iterations required for HVP convergence decreases more significantly. When At that time, FD Compared to FD It is more efficient. Furthermore, HVP is more efficient than FD. It can reduce the number of iterations by about an order of magnitude, and the speedup increases with... Increase and improve.
[0060] In summary, this embodiment demonstrates that the RDO framework using GEgPC combined with the HVP method significantly reduces the number of function evaluations required for optimization convergence compared to traditional finite difference methods for design variables and random variables. It improves the computational efficiency of robust optimization in high-dimensional multivariate systems, balancing optimization accuracy and engineering practicality. It can be directly applied to high-fidelity aerodynamic and multidisciplinary robust design optimization scenarios.
[0061] Example 3: This embodiment, based on Embodiment 1, provides a control method for an adaptive deformable wing of a simple transonic wing model for passenger aircraft.
[0062] This embodiment adopts Figures 7 to 10The model shown establishes a robust design optimization problem for an adaptive deformable trailing-edge wing. The main objective of this section is to investigate the comprehensive benefits of implementing trailing-edge deformation technology on an optimized STW model configuration, considering the uncertainty of control surface deflection angles under operating conditions. Four control surfaces are arranged along the spanwise, and the deflection schemes of the control surfaces are as follows: Figures 7 to 10 As shown.
[0063] In setting the design variables, since only aerodynamic shape is considered and structural factors are ignored, the design variables include angle of attack, control surface deflection angle, airfoil, and twist angle. The robust objective function for the final optimization target under ideal conditions is... The expression is as follows: In this formula, all input variables are design variables, including deterministic design variables and variables whose probability distributions can be designed. For the initial angle of attack distribution, To correspond to the angle of attack distribution of the optimal control surface deflection strategy, For any desired control surface deflection combination under fixed angle of attack and free-flow Mach number conditions (which can be regarded as a fixed state in the random space of the incoming flow), a distribution is then formed through probability integration. Let represent the distribution of the optimal expected control surface deflection strategy under random free flow conditions. Here, the input variable is the control surface deflection strategy, which represents the expected distribution. As constraints, for The subproblem represents the objective function that considers the uncertainty of the control surface deflection only under the condition of fixed free flow. For the local geometric design variables and twist angle of the wing, To achieve the final, uncertainty-based robust objective function The best performance is achieved by returning the minimum value. This refers to finding a specific angle of attack (AoA) and a specific control surface deflection angle. This improves performance when considering the uncertainty of the control surface deflection angle in the current state. Optimal performance (e.g., minimum resistance) returns the point of minimization; however, in actual calculations... In this case, the uncertainty of the control surface still needs to be considered, and the probability integration needs to be performed again to obtain the actual distribution. The uncertainty of the control surface is assumed to follow a normal distribution, and its 3 The value is approximately 0.5°. These represent the local geometric design variables and twist angle of the wing; these are all deterministic design variables.
[0064] The final design variables include 210 wing local shape variables, 8 section twist angle variables, 4 control surface deflection angle variables for the initial cruise phase, and 2 angle of attack variables (corresponding to the initial cruise and midpoint cruise states, respectively), for a total of 224 design variables.
[0065] Step 1: Update the wing design variables based on gradient information using the optimizer; Step 2: Update the wing geometry using FFD geometric parameterization based on the design variable information. Mesh deformation can be performed using methods such as inverse distance weighting. Adjust the fluid dynamics calculation mesh based on the updated geometry. Step 3: Calculate the aerodynamic performance data and first-order gradient of the sample points based on the sample points generated by the random variable distribution and the adjusted fluid computational grid; In step three, the random variable is defined as follows: During the initial cruise phase, five random variables are defined: the incoming Mach number (which follows a normal distribution N(0.785, 0.02)). 2 The standard deviation of the deflection angles of the four control surfaces (all following a normal distribution) is 0.16667°.
[0066] During the midpoint cruise phase, define one random variable: the incoming Mach number (distribution as before), and assume that the control surface is mechanically locked with no deflection uncertainty.
[0067] Step 4: Input the aerodynamic performance data and its first-order gradient of the sample points into the UQ (Uncertainty Quantization) module and the Hessian vector product module. The UQ module calculates the robust objective function and perturbation vector, and the Hessian vector product module uses the perturbation vector to calculate the total gradient of the robust objective function with respect to the design variables. The ideal optimization scheme has too high a computational cost, and the two-layer optimization is difficult to guarantee reliability. Mechanical locking in the non-deformable state makes the control surface deviation negligible, while the uncertainty of the control surface becomes significant during FFD deformation. In reality, maintaining a quasi-steady state and achieving real-time optimal control during flight is required, which is difficult to achieve in actual flight. Therefore, in step four, an optimized robust objective function is used to approximate the upper limit of the deformation benefit. The specific expression of the optimized robust objective function is as follows: In the formula, Let be the objective function for the initial stage of cruise. The objective function for the midpoint of the cruise phase; This embodiment sets the optimization target as two flight state points: Mid Cruise and Cruise Start. Constraints include leading edge radius constraints, thickness constraints, and wing box volume constraints, as well as aerodynamic design constraints including lift coefficient constraints, pitch moment coefficient constraints, and angle of attack constraints.
[0068] Step 5: Input the robust objective function and total gradient into the optimizer. The optimizer uses the robust objective function and total gradient to determine whether the wing configuration determined by the design variables has converged. If it has, the wing configuration is taken as the final robust optimal wing design; if not, the total gradient is taken as the gradient information and the process returns to Step 1.
[0069] As comparative cases, in addition to robust optimization considering deformation techniques (Case 1), the present invention also carried out the following comparative studies: deformation-free robust optimization based on uncertainty quantification (Case 2), deterministic optimization considering deformation techniques (Case 3), and deterministic optimization without deformation techniques (Case 4), as shown in Table 1.
[0070] Table 1. Comparative studies of Cases 1 to 4 In Table 1, Formula 1 is specifically as follows: in, The drag coefficient, This represents the drag coefficient during the initial stage of cruise. This represents the drag coefficient during the midpoint of the cruise phase. For the expectation, Standard deviation To robustly optimize the objective function during the initial cruise phase. To robustly optimize the objective function for the cruise midpoint phase, AOA(CruiseStart) represents the angle of attack during the initial cruise phase, and AOA((Mid Cruise)) represents the angle of attack during the cruise midpoint phase. The lift coefficient, The longitudinal moment coefficient is given by the formula, and the fuel volume is given by the formula. Let be the airfoil thickness in the z-direction. Let be the initial airfoil thickness in the z-direction of the outer wing box. Where is the leading edge radius, Let the initial leading edge radius be . For the wing spars height, The initial wing sparsity height, These are the actual deflection angles of the four control surfaces during the initial phase of cruise. It follows a normal distribution. This refers to the Mach number at the start of cruise. This refers to the Mach number during the midpoint of the cruise phase. The optimization results are shown in Table 2. Figure 11 , Figure 12 As shown, The results are shown in Table 3. Figure 11 and Figure 12 In this context, expectation represents the expected value, ±1 STD 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, as well as between optimizations that consider deformation and those that do not.
[0071] Table 2 Optimization Results Table 3 The result The robust optimization considering deformation, Case 1, performs best under uncertain conditions. Compared to Case 3, it significantly reduces the standard deviation of drag, demonstrating extremely strong robustness. However, at the defined baseline, the drag is slightly higher than Case 3, successfully achieving high performance robustness while retaining most of the deformation benefits. The robust optimization without considering deformation, Case 2, has lower expected drag and standard deviation compared to Case 4, indicating that robust optimization is also effective for fixed configurations. However, since Case 2 does not utilize deformation, its performance is among the worst in all cases, inferior to any design considering deformation. The deterministic optimization considering deformation, Case 3, has the lowest drag and best performance at the defined baseline design point. However, it is extremely sensitive to uncertainty. Once actual control errors and Mach number fluctuations are considered, its expected drag and standard deviation rise sharply, resulting in poor robustness and a significant drop in actual performance. Case 4, as a traditional optimization result that does not consider deformation and ignores uncertainty, and as the baseline design, has the worst performance among all cases, exhibiting the highest drag both at the baseline and under uncertain conditions. When considering uncertainties, the optimized morphing wing maintains its performance advantage under high-lift cruise conditions, regardless of whether robust or deterministic optimization methods are used. This advantage primarily stems from reduced induced drag. It is particularly noteworthy that robust design optimization that simultaneously considers morphing techniques, geometry, and uncertainties can further reduce the expected drag value and standard deviation while preserving morphing benefits, thereby enhancing robustness. Figure 13 and Figure 14 The distribution of spanwise normalized lift differences at the baseline flight point under four operating conditions is shown. The horizontal axis represents the spanwise position, and the vertical axis represents the difference between the normalized lift and the elliptical lift distribution under each operating condition. This aims to illustrate the impact of deformation techniques on the variation of spanwise lift distribution. Under the considered flight conditions, deterministic optimization consistently yields a lift distribution closer to the ideal elliptical distribution than robust optimization. Airfoil profiles were extracted at four stations: y = 0.13, 5.6, 10.5, and 13.3 m. Figures 15 to 18This paper presents a comparison of optimized airfoils for four operating conditions under no control surface deflection. The optimized airfoils are generally similar, but their specific shapes differ. At y = 0.14m, the lower surface of the root airfoil is mainly affected by volume and moment constraints. The robustly optimized airfoil at this location has a flatter upper surface compared to the deterministic optimization result: Case 1 achieves this feature by increasing the thickness of the middle chord segment of the lower surface, while Case 2 achieves this by increasing the overall thickness of the entire airfoil. Furthermore, a more significant feature is that the negative twist angle of the outer airfoil segment is relatively larger in the robustly optimized result. Figure 19 and Figure 20 The study presents a comparison of pressure cloud maps for Case 1 under conditions at the start and middle stages of cruise, while... Figures 21 to 24 and Figures 25 to 28 The pressure distribution curves for the corresponding stages are presented in comparison. During the mid-cruise phase, the robustly optimized pressure distribution exhibits a more significant trailing-edge load distribution compared to the deterministic optimization. For the morphing wing, the optimization results without considering variable camber (Case 2 and Case 4) show a more pronounced local supersonic region at approximately x / c=0.6 in the middle and outer wing sections (y=5.6m, y=10.5m), which aims to reduce shock wave intensity under high lift coefficients. Furthermore, except for the wing root region, the cases employing the morphing wing design (Case 1 and Case 3) exhibit a larger negative pressure peak at the leading edge, which helps reduce pressure drag. During the initial cruise phase, the robustly optimized results show trailing-edge load characteristics similar to those in the mid-cruise phase, but their significance is more pronounced than that of the deterministic optimization, and the morphing configuration (Case 1 and Case 3) produces a stronger trailing-edge load compared to the fixed camber design. These phenomena are expected, as all control surfaces are set to positive deflection angles. At this point, the magnitudes of the leading-edge suction peaks converge, but the shock intensity in the robust optimization results is significantly lower than that in the deterministic optimization. Specifically, Case 1 and Case 3 exhibit weaker shock intensity than Case 2 and Case 4, respectively. Although theoretically a weaker shock would lead to lower drag, the higher drag observed at the baseline in the robust optimization indicates that the induced drag effect dominates the shock-induced drag variation in this configuration.
[0072] In the description of this invention, the above are merely preferred embodiments and are not intended to limit the scope of protection of this invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this invention should be included within the scope of protection of this invention.
Claims
1. A control method for an adaptive deformable airfoil based on Hessian vector product, characterized in that, Includes the following steps: S1. Define the design variables for the wing and generate normalized sample points based on the random variable distribution. S2. Update the design variables based on gradient information using the optimizer; S3. Update the wing geometry using the FFD geometric parameterization method based on the design variables, and adjust the fluid computation mesh based on the updated geometry. S4. Calculate the aerodynamic performance data and first-order gradient of the sample points based on the normalized sample points and the adjusted fluid computational grid. S5. Input the aerodynamic performance data and first-order gradient of the sample points into the UQ module and the Hessian vector product module. Calculate the robust objective function and perturbation vector through the UQ module. Calculate the total gradient of the robust objective function relative to the design variables using the perturbation vector through the Hessian vector product module. S6. Input the robust objective function and total gradient into the optimizer. The optimizer determines whether the wing configuration determined by the design variables has converged based on the robust objective function and total gradient. If it has, the wing configuration is taken as the final robust optimal wing design. If not, the total gradient is taken as the gradient information and returned to S2. In S5, the UQ module computes the robust objective function based on the gradient-enhanced generalized polynomial chaos method. I The specific expression is: In the formula, For the measurement matrix containing gradient terms, This is the sample result vector obtained by the high-fidelity solver. for 3D real vector space, The number of sample points. For design variables, for 3D real vector space, To design the number of variables, For random variables, for 3D real vector space, The number of random variables. To transform operator notation, representing It is by and Sure.
2. The control method for an adaptive deformable wing based on Hessian vector product according to claim 1, characterized in that, In S1, the design variables of the wing include the angle of attack design variable and the local geometry, twist angle and control surface deflection angle design variables obtained by the FFD geometric parameterization method.
3. The control method for an adaptive deformable wing based on Hessian vector product according to claim 1, characterized in that, S4 includes the following steps: S41. Normalized sample points generated by sampling based on the distribution of random variables are used to obtain actual sample points through the current design variables, where each sample point represents a possible realization of an uncertainty; S42. Based on the normalized sample points and the actual sample points, call the CFD solver to calculate the aerodynamic performance data of the adjusted fluid computational grid under the sample point conditions, obtain the aerodynamic performance data of the sample points, and call the discrete adjoint solver to calculate the first-order gradient of the aerodynamic performance relative to the design variables.
4. The control method for an adaptive deformable wing based on Hessian vector product according to claim 1, characterized in that, In S5, the Hessian vector product module calculates the second derivative of the robust objective function using the finite difference method or the complex step-adjoint method, and then solves for the total gradient of the robust objective function with respect to the design variables. Specifically, the Hessian vector product is calculated using the finite difference method or the complex step-adjoint method, thereby obtaining the second derivative of the robust objective function. The Hessian vector product is then calculated using the finite difference method. The expression is: In the formula, Let be the perturbation vector. For random variables, Step size, For design variables; Calculate the Hessian vector product using the complex variable method. The expression is: In the formula, To separate the imaginary part from a complex number expression, i It is the imaginary unit.