Airfoil design method based on risk-averse bayesian optimism optimization

Abstract

The application relates to an aviation wing airfoil design method based on risk-averse Bayesian optimism optimization, and relates to the field of intelligent optimization design of aviation equipment. The application is used to solve the problem that it is difficult to balance excellent aerodynamic performance and stable reliability under a limited simulation budget during aviation wing airfoil design. According to design requirements, the application selects to-be-designed variables of an aviation wing airfoil and a feasible space of the to-be-designed variables, iteratively optimizes the to-be-designed variables in the feasible space until a preset iteration termination condition is met, and outputs optimal variables of the aviation wing airfoil. The iteration optimization process comprises the following steps: modeling an expected response and a noise variance at all nodes in a current variable value candidate set; selecting a leaf node with the minimum evaluation value in a current search tree as an expansion node, performing space tree division, obtaining an expansion node set, and updating the search tree; constructing a risk-averse confidence boundary according to a proxy model, calculating evaluation values of the expansion nodes, and updating the variable value candidate set.

Description

Technical Field

[0001] This application belongs to the field of intelligent optimization design of aviation equipment. Background Technology

[0002] Aviation equipment is a crucial component of the modern aviation industry. Airfoils, as key geometric structures determining the aerodynamic performance of wings, directly impact an aircraft's lift, drag, flight stability, and fuel economy. In engineering practice, airfoil design typically employs parametric modeling methods to represent the airfoil's geometry parametrically. This involves designing a set of geometric parameters (such as the coordinates of control points on the upper and lower surfaces, maximum thickness and its location, maximum camber and its location, leading edge radius, and trailing edge thickness) to optimize the airfoil's aerodynamic performance. Among these parameters, the lift-to-drag ratio, as a key indicator of an airfoil's overall aerodynamic performance, is the most frequently used optimization objective in airfoil aerodynamic optimization.

[0003] However, the relationship between airfoil aerodynamic performance and design variables is highly complex, making it difficult to establish explicit analytical models. Performance evaluation primarily relies on computational fluid dynamics (CFD) numerical simulations. In practical aviation applications, aircraft operate in complex and constantly changing environments. Factors such as flight conditions, incoming flow conditions, and manufacturing errors are all random, causing random fluctuations in the lift-to-drag ratio of the same airfoil under different operating conditions. Therefore, to accurately evaluate the aerodynamic performance of an airfoil under random environments, it is usually necessary to repeatedly perform numerical simulations under multiple random samples to estimate the statistical characteristics of airfoil performance. Since single-pass numerical simulations are inherently time-consuming, and random simulations require repeated calculations for a large number of random samples, the performance evaluation cost of a single design significantly increases. Furthermore, different airfoil designs exhibit varying sensitivities to random disturbances, and the noise fluctuations in the simulation output vary with changes in design parameters.

[0004] For expensive black-box stochastic optimization problems like airfoil design, Bayesian optimization approximates the unknown objective function using a surrogate model and guides sequential sampling with a sampling function, searching for the optimal design with fewer function evaluations, thus theoretically reducing experimental costs and time. However, when existing Bayesian optimization methods are directly applied to airfoil design, there is a significant disconnect between their technical concepts and actual aerospace engineering needs, specifically in the following three aspects:

[0005] First, under expensive CFD simulation conditions, existing noise variance modeling methods struggle to balance physical consistency and numerical stability. Airfoil aerodynamic performance evaluation typically involves geometric reconstruction, computational mesh generation, flow field iteration, and convergence determination. A single high-fidelity CFD simulation is computationally expensive, and the noise variance observation information obtainable within a given computational budget is relatively limited, making it difficult to improve variance estimation accuracy through large-scale repeated simulations. Furthermore, the output space of Gaussian processes does not impose structural constraints on non-negativity, and its posterior prediction results and corresponding confidence boundaries do not naturally maintain the non-negativity of noise variance. Simultaneously, when there are significant scale differences in performance variance across different airfoil design regions, directly modeling the original variance exacerbates the numerical sensitivity of model parameter estimation. These problems further propagate to the risk avoidance objective function and sampling criteria, affecting the assessment and ranking of candidate airfoil risk levels, and consequently reducing the reliability of sequential optimization decisions. Therefore, aerospace airfoil optimization requires a heteroscedasticity modeling method suitable for limited CFD simulation budgets, capable of maintaining the non-negativity of noise variance from a structural perspective, and balancing numerical stability and prediction reliability.

[0006] Secondly, existing methods typically construct confidence boundaries for the expected response and noise variance separately, and then combine them based on risk aversion coefficients. This makes it difficult to form a unified uncertainty representation at the level of the comprehensive risk aversion objective. The final decision-making basis for airfoil design is not the independent expected aerodynamic performance and performance variance, but rather a comprehensive performance-risk index composed of both according to given risk aversion parameters. In regions where design points are unevenly distributed or where noise levels vary significantly with airfoil parameters, confidence boundaries constructed item by item and then combined cannot directly reflect the overall predictive uncertainty of the final risk aversion objective. This is not conducive to evaluating the reliability of the comprehensive performance advantage of candidate airfoils using a unified confidence scale, and may affect the consistency of node selection and candidate scheme ranking. Therefore, it is necessary to uniformly transmit the posterior uncertainty of the expected response surrogate model and the noise variance surrogate model towards the final comprehensive performance-risk objective, and construct upper and lower confidence bounds with a unified confidence radius at the comprehensive objective level, providing a consistent quantitative basis for candidate node selection and sequential sampling decision-making.

[0007] Third, existing methods require resolving the global optimization problem of the risk-avoidance acquisition function after each surrogate model update. Airfoil aerodynamic shapes are typically characterized by multiple geometric parameters, and the risk-avoidance acquisition function, composed of the desired response, noise variance, and prediction uncertainty, often exhibits non-convex and multi-modal characteristics. To determine the next sequential sampling point, traditional methods usually require resolving the problem using multi-startpoint search, random search, or other global numerical optimization methods after the acquisition function update. However, the design space search information generated in previous iterations is difficult to effectively inherit and utilize, resulting in repetitive inner-layer optimization overhead. Therefore, it is necessary to construct a hierarchical spatial partitioning structure that can be retained and reused during continuous iterations. This structure should progressively refine the design regions with optimization potential from coarse to fine, and screen candidate nodes based on risk-avoidance confidence boundaries. This allows limited computational resources to be prioritized for airfoil design regions that are more likely to achieve excellent aerodynamic performance and lower fluctuation risks.

[0008] In summary, existing technologies suffer from several shortcomings when dealing with the airfoil design problem under expensive heteroscedastic stochastic simulations. These include a lack of structural non-negativity guarantees in noise variance characterization, a lack of unified confidence evaluation methods for composite risk indicators, and excessively high optimization costs of acquisition functions during sequential search. Consequently, it is difficult to achieve both excellent aerodynamic performance and robust reliability within a limited simulation budget. Summary of the Invention

[0009] This application aims to address the challenge of balancing excellent aerodynamic performance and robust reliability within a limited simulation budget when designing aircraft wing airfoils. It presents an airfoil design method based on risk-avoidance Bayesian optimistic optimization, which fully considers the heteroscedasticity of stochastic simulation output and the risk of airfoil performance fluctuations. The optimization process is completed efficiently within a limited simulation budget, resulting in an airfoil design scheme that combines a high lift-to-drag ratio, good stability, and robustness.

[0010] The first aspect of this application provides a method for airfoil design based on risk-avoidance Bayesian optimistic optimization, including:

[0011] Based on the design requirements, select the design variables and their feasible space for the airfoil of the aircraft. Within the feasible space, iterate and optimize the design variables until the preset iteration termination condition is met, and output the optimal variables for the airfoil of the aircraft.

[0012] The iterative optimization process includes:

[0013] Stochastic Kriging modeling is performed on the expected response and noise variance at all nodes in the current variable value candidate set;

[0014] Select the leaf node with the smallest evaluation value in the current search tree as the expansion node, perform spatial tree partitioning on the expansion node to obtain the set of expansion child nodes, and update the search tree.

[0015] Construct risk avoidance confidence boundaries for each extended sub-node based on the surrogate model of the expected response and the surrogate model of the noise variance.

[0016] The evaluation value of each extended sub-node is calculated based on the risk avoidance confidence boundary, and the candidate set of variable values ​​is updated.

[0017] In one possible design, the calculation of the evaluation value of each extended sub-node based on the risk aversion confidence boundary and the updating of the candidate set of variable values ​​include:

[0018] For extended sub-nodes whose risk aversion confidence lower bound is less than or equal to the optimal simulation evaluation value of the global variable, perform repeated simulation evaluations and add them to the current variable value candidate set. Use the risk aversion confidence upper bound of the remaining extended sub-nodes as their evaluation values.

[0019] In one possible design, the evaluation value is calculated using a risk aversion objective function during the repeated simulation evaluation;

[0020] The expression for the risk aversion objective function is:

[0021] ,

[0022] in, This represents the risk aversion objective function. For the expected response, For noise variance, For risk aversion parameters, Represents a node.

[0023] In one possible design, the stochastic Kriging modeling of the expected response and noise variance at all nodes in the current variable value candidate set includes:

[0024] The average value of the simulation results obtained from repeated simulations at each node is used as the observation value to establish a Kriging surrogate model for the desired response.

[0025] The logarithm of the noise variance from repeated simulations at each node is used as the observed value, and a Kriging surrogate model for the noise variance is established with the logarithmic noise variance as the latent function.

[0026] In one possible design, constructing the risk aversion confidence boundary for each extended sub-node based on the surrogate model of the desired response and the surrogate model of the noise variance includes:

[0027] The Kriging surrogate model based on the expected response calculates the posterior prediction mean and posterior prediction variance at the representative points of each extended sub-node.

[0028] The Kriging surrogate model based on noise variance calculates the posterior expectation and posterior variance of the log-noise variance at the representative points of each extended child node.

[0029] Risk avoidance confidence boundaries are constructed based on the posterior prediction mean, posterior prediction variance, posterior expectation of log noise variance, and posterior variance of log noise variance.

[0030] In one possible design, the risk aversion confidence boundary includes a lower bound and an upper bound for risk aversion confidence, expressed as follows:

[0031] ,

[0032] ,

[0033] in, and These are the lower and upper bounds of the confidence level for risk aversion, respectively. and These are the posterior prediction mean and the posterior prediction variance, respectively. and These are the posterior expected value of the log-noise variance and the posterior variance of the log-noise variance, respectively. For risk aversion parameters, This is a composite confidence parameter.

[0034] In one possible design, the spatial tree partitioning of the extended nodes to obtain a set of extended child nodes includes:

[0035] The sub-region corresponding to the extended node is evenly divided into k sub-regions of equal size and non-overlapping. Each sub-region corresponds to an extended sub-node, and the geometric center of each sub-region is used as the representative point of the corresponding extended sub-node. k is a preset number of branches.

[0036] In one possible design, during the spatial tree partitioning process, the minimum value among all leaf node evaluation values ​​is used as a dynamic threshold, and leaf nodes with evaluation values ​​greater than the dynamic threshold are marked as nodes that are not prioritized for expansion.

[0037] The design incorporates a depth threshold that increases with the number of leaf node expansions. When the expansion depth of a branch containing a leaf node reaches the depth threshold, the search returns to the shallowest unexpanded level in the current search tree.

[0038] In one possible design, the preset iteration termination condition includes:

[0039] The number of nodes in the current candidate set of variable values ​​has reached the preset maximum simulation budget.

[0040] In one possible design, the center of the feasible space is taken as the root node, and the space tree is divided layer by layer according to the preset number of branches until the target number of child nodes is obtained, and the geometric center of the region corresponding to each child node is taken as its representative point.

[0041] Multiple repeated simulation evaluations were performed at each representative point, and the sample mean and sample variance at each representative point were calculated based on the simulation evaluation results.

[0042] All representative points are used as the initial candidate set of variable values, and the tree structure containing the root node and all child nodes is used as the initial search tree.

[0043] The second aspect of this application provides an airfoil design device based on risk-avoidance Bayesian optimistic optimization. The airfoil design device based on risk-avoidance Bayesian optimistic optimization includes a processor and a memory. The memory stores at least one instruction, which is loaded and executed by the processor to implement the airfoil design method based on risk-avoidance Bayesian optimistic optimization as described above.

[0044] A third aspect of this application provides a computer storage medium storing at least one instruction, which is loaded and executed by a processor to implement the airfoil design method for aircraft wings based on risk-avoidance Bayesian optimistic optimization as described above.

[0045] The beneficial effects of this application are:

[0046] To address the challenges of high computational costs, random performance fluctuations, and heteroscedastic noise in airfoil aerodynamic shape optimization for aviation equipment, this application integrates the desired response and noise variance to construct a risk avoidance objective function. Furthermore, it combines risk avoidance confidence boundaries with sequential sampling, enabling the sampling process to prioritize design regions with excellent desired performance and minimal performance fluctuations. This approach yields airfoil design schemes that combine superior aerodynamic performance with robustness.

[0047] Meanwhile, this application adopts a tree-based multi-scale optimistic search strategy, which adaptively determines whether candidate nodes should perform real simulation through a selective node evaluation mechanism. This eliminates the need for repeated global optimization of continuous, non-convex, and multi-peak acquisition functions. Instead, it concentrates the limited simulation budget on regions with greater optimization potential. While taking into account both global exploration and local development, it effectively reduces the computational overhead of the acquisition function optimization process and improves the overall optimization efficiency.

[0048] Furthermore, this application establishes a stochastic Kriging surrogate model for heteroscedastic stochastic simulation with objective function and noise variance, which can more accurately characterize the uncertainty of stochastic simulation output and improve the prediction accuracy and sampling decision quality of the surrogate model.

[0049] Theoretically, this application has a cumulative regret sublinear convergence guarantee under the mild regularity assumption; in the actual optimization of the aerodynamic shape of airfoils for aircraft equipment, it can achieve efficient optimization of high-dimensional parametric airfoil design variables under finite simulation budget, providing an efficient and reliable optimization method for the robust aerodynamic shape design of aircraft equipment under complex and uncertain environments. Attached Figure Description

[0050] Figure 1 This diagram illustrates the process of branch selection and expansion in the sample space using a one-dimensional test function.

[0051] Figure 2 Flowchart for extending tree-based risk perception nodes;

[0052] Figure 3 This is a flowchart of the method described in Specific Implementation Method 1. Detailed Implementation

[0053] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of this application, and not all of them. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application. It should be noted that, unless otherwise specified, the embodiments and features in the embodiments of this application can be combined with each other.

[0054] Specific Implementation Method 1: This implementation method addresses the problems in the optimization of the aerodynamic shape of airfoils for aviation equipment, such as high computational cost of stochastic simulation, heteroscedastic noise in simulation output, and shortcomings of existing Bayesian optimization methods in terms of physical consistency of noise variance, unified confidence evaluation of composite risk indicators, and sequential search efficiency.

[0055] First, a risk aversion objective function is constructed based on the expected response and noise variance of the objective function. Then, stochastic Kriging surrogate models are established for the expected response of the objective function and the logarithmically transformed noise variance, respectively. Based on these surrogate models, lower and upper bounds of risk aversion confidence are constructed. Next, the input space is hierarchically partitioned into a multi-scale tree structure. A tree-based optimistic search strategy is adopted, combined with the risk aversion confidence boundary. A selective node evaluation mechanism adaptively determines whether to perform real simulation on candidate child nodes. The global optimization of the risk aversion acquisition function is approximated by expanding the tree structure layer by layer, avoiding expensive explicit optimization in the continuous search space. Finally, the optimal design scheme under risk aversion is output based on the sequential sampling results. This results in a complete Risk-Averse Bayesian Optimistic Optimization (RABOO) method for heteroscedastic noise stochastic simulation. Details are as follows:

[0056] Step 1: Analyze the stochastic simulation system of the airfoil to identify design variables and responses.

[0057] Establish a stochastic simulation optimization model for the airfoil's aerodynamic shape.

[0058] First, the parametric geometric model of the airfoil is analyzed to determine the design variables affecting its aerodynamic performance, and these variables are used as optimization inputs. Then, combined with the airfoil simulation model, the aerodynamic performance of the airfoil is repeatedly evaluated under multiple stochastic operating conditions, using the lift-to-drag ratio as the performance response.

[0059] This step clarifies the optimization input, performance response, and sources of randomness in the stochastic optimization problem of airfoils. It integrates the parametric design variables of the airfoil, the stochastic simulation model, and the performance statistical characteristics into the optimization framework, providing a unified problem description basis for subsequent risk avoidance objective function construction, heteroscedasticity surrogate modeling, and sequential sampling optimization.

[0060] Step 2: Construct the risk aversion objective function

[0061] Since the goal of airfoil design is to maximize the lift-to-drag ratio, and the risk-avoidance Bayesian optimistic optimization framework proposed in this embodiment uses a minimization form for modeling and optimization, in order to unify the problem statement and facilitate subsequent algorithm derivation, the lift-to-drag ratio is taken as a negative value, and the original maximization problem is equivalently transformed into a minimization problem.

[0062] Given input space And the unknown objective function that can only be obtained through simulation of noisy observations. Simulation at point noise variance at Follow change.

[0063] Expected response With noise variance Through risk aversion parameters Explicit combination of risk aversion objectives embeds "robustness" into the target to be optimized, rather than relying on ex-post selection. Risk aversion objective function The expression is:

[0064] ,

[0065] in, These are risk aversion parameters used to control the desired response. With noise variance The trade-off between them The larger the value, the more sensitive the decision-maker is to noise.

[0066] This step moves the robustness requirement from the post-processing stage to the objective function definition stage, enabling all subsequent sampling decisions to automatically consider both expected performance and noise level, avoiding the selection of excessively noisy and unreliable solutions. It overcomes the limitations of existing Bayesian optimization methods that are risk-neutral and only focus on expected performance, ensuring optimal expected performance while also considering noise robustness, thus avoiding the selection of highly noisy and unreliable solutions.

[0067] Step 3: Initialize the input space tree partitioning

[0068] For input space A tree-based partitioning strategy is adopted. The tree is based on the input space. The center is the root node, initialized at the level. Given a fixed branching factor Each expanded node is divided into This generates three equal-sized, non-overlapping sub-regions. Child nodes. Hierarchy ,index The node at that point is denoted as Its child node index is Corresponding sub-region Each node Associate a representative point, defined as the geometric center of its corresponding sub-region, using... Represents a node The center point. (This is in contrast to the previously used time index notation.) Different, markings This specifies the depth and position of the corresponding node in the tree. The tree is expanded from the geometric center of the input space as the root node until the desired depth and position are obtained. At each of the representative points, execution is carried out... The simulations were repeated independently to estimate the sample mean and noise variance, and the risk aversion objective function at each node was used as its evaluation value.

[0069] This step completes the initial tree partitioning of the input space and generates an initial experimental design covering the entire design space, providing initial sample data and tree structure foundation for subsequent construction of stochastic Kriging surrogate models, calculation of risk avoidance confidence boundaries, and tree-based optimistic search.

[0070] Step 4: Perform stochastic Kriging modeling on the expected response

[0071] Given a candidate set of values ​​for the current variable Simulation results Each candidate point All places have been carried out If the observation is repeated, then... , .make This represents the estimated mean of the sample at the evaluated design points, where Considering the limited prior knowledge of the mean structure, let's assume a mean function. .

[0072] With candidate points Using the average of the simulation results from multiple repeated simulations as the observed value, a Kriging surrogate model for the desired response is established:

[0073] ,

[0074] in, To observe the response, For the expected response, This is heteroscedastic noise. It is a mean function. It is a second-order stationary Gaussian process function.

[0075] In the above model, the expected response Decomposed into mean function With a second-order stationary Gaussian process The sum. Second-order stationary Gaussian process function. Characterized by the covariance function:

[0076] ,

[0077] in, Represents the covariance function. Indicates process variance. This represents the relevant kernel function.

[0078] Heteroscedastic noise The noise variance can be estimated as:

[0079] .

[0080] For strictly second-Gaussian noise , It is a variance proxy An unbiased estimate.

[0081] For any point The posterior predicted mean of the expected response posterior distribution posterior prediction variance of the expected response posterior distribution The following can be derived:

[0082] ,

[0083] ,

[0084] in,

[0085] ,

[0086] ,

[0087] ,

[0088] ,

[0089] for The unit vector.

[0090] Step 5: Logarithmic Transformation of Noise Variance and Kriging Modeling

[0091] Take the logarithm of the noise variance As an observation, the logarithmic noise variance Establish a stochastic Kriging surrogate model for the latent function:

[0092] ,

[0093] , ,

[0094] in, The constant mean of the logarithmic noise variance latent function; It is a zero-mean second-order stationary Gaussian process; Homoscedastic noise

[0095] ,

[0096] in, The process variance of the logarithmic noise variance latent Gaussian process. Its related kernel function.

[0097] set up Represents the observed value. At any test point Place, The posterior distribution is:

[0098] .

[0099] posterior mean With prediction variance for:

[0100] ,

[0101] ,

[0102] in,

[0103] ,

[0104] ,

[0105] ,

[0106] ,

[0107] ,

[0108] for identity matrix This is the nugget effect.

[0109] because The posterior distribution is Gaussian. It follows a log-normal distribution. Therefore, the posterior expectation of the noise variance is:

[0110] .

[0111] The posterior variance is:

[0112] .

[0113] The logarithmic transformation in this step naturally ensures the non-negativity of the variance estimate and improves numerical stability. Separating the mean and (logarithmic) variance for individual proxy modeling allows their uncertainties to be characterized separately and used together in subsequent steps, improving numerical stability and modeling robustness, and avoiding the negative values ​​and numerical ill-conditioned problems that may occur when directly modeling the variance.

[0114] Step Six: Construct the lower and upper bounds of confidence for risk aversion

[0115] Based on the expected response posterior distribution obtained in step four and the log-noise variance posterior distribution obtained in step five, the composite confidence parameter is derived. And construct a risk avoidance confidence lower bound. With the upper realm :

[0116] ,

[0117] ,

[0118] in, Let be the posterior predicted mean of the posterior distribution of the expected response. Let be the posterior expected mean of the posterior distribution of the logarithmic noise variance. Let Variance be the predicted variance of the posterior distribution of the expected response. Let V be the posterior expected variance of the posterior distribution of the logarithmic noise variance.

[0119] , , This serves as a unified upper bound for the noise variance.

[0120] ,

[0121] ,

[0122] As the confidence level control parameter, the probability that the mean model and the log-variance model are jointly valid is at least 1 / 2. ; The scaling parameter is the desired response to the Gaussian process. Expected Response In the nuclear The corresponding norm in RKHS (Reproducing Kernel Hilbert Space); The scaling parameter for a log-noise variance Gaussian process; Logarithmic noise variance In the nuclear The corresponding norm in RKHS.

[0123] The confidence lower and upper bounds in this step provide a high-probability theoretical coverage of the risk aversion objective function. They serve as the criteria for determining whether candidate points are worthy of realistic simulation evaluation in subsequent tree search, enabling sequential sampling to automatically achieve a balance between exploration, utilization, and risk aversion. This is also the theoretical basis for the cumulative regret sublinear convergence guarantee of this implementation method.

[0124] Step 7: Input Space Tree Partition Update

[0125] Based on the evaluation values ​​of each leaf node in the current search tree, the leaf node with the smallest evaluation value is selected as the expanded node. Let the expanded node be... The corresponding sub-region is Following the same tree-like partitioning strategy as in step three, this sub-region is further divided into... Generate corresponding sub-regions of equal size that do not overlap. child nodes Their corresponding sub-regions are denoted as follows: The representative point of each child node is defined as the geometric center of the corresponding sub-region, denoted as . The newly generated child nodes are added to the current search tree and serve as candidate nodes for the next round of risk aversion confidence boundary assessment and real random simulation.

[0126] This step, while maintaining the consistency of the tree-like partitioning structure, adaptively refines the region with the highest optimization potential, gradually focusing the search space towards better regions. This provides new candidate design points for subsequent risk avoidance confidence boundary assessment and selective node assessment, thus balancing global exploration and local development within a limited simulation budget and improving overall optimization efficiency. It avoids costly global explicit optimization of non-convex, multi-peak acquisition functions, transforming the continuous optimization problem into a structured tree search problem, naturally supporting coarse-to-fine exploration.

[0127] Step 8: Selective Child Node Evaluation

[0128] Leaf nodes Using the risk-averse confidence boundary at its representative point An assessment will be conducted at the site. (Set up) ,set up This represents the optimal risk aversion estimate among the currently evaluated points. If Then in Execution Repeated simulations are performed to estimate the target response. and noise variance Then calculate The corresponding mean is used to update the Kriging model. Conversely, if... If a node is deemed unoptimal at the current stage, simulation resources are not allocated to it because its representative point is considered unlikely to be optimal. Instead, the node is assigned its risk aversion UCB value. However, even if a node appears unpromising at the current stage, its corresponding region may still contain promising solutions, so the node is reserved for future exploration.

[0129] To handle both sampled and unsampled nodes uniformly, a node evaluation function is introduced. Defined as:

[0130] ,

[0131] This step concentrates the limited simulation budget on truly promising candidate regions, avoiding costly simulation evaluations of obviously uncompetitive candidate points, and significantly improves computational efficiency without discarding any region search opportunities.

[0132] Step Nine: Risk Perception Tree Expansion and Pruning

[0133] The tree expands level by level. Among the leaf nodes of the current level, the node with the lowest evaluation value is selected for further expansion. Accordingly, the level... The node to be expanded is selected as The selected nodes are then divided according to step five. Each node has a child node. Subsequently, node evaluation and simulation assignment are performed according to the process described in step six. A dynamic threshold is introduced during the expansion process. (The node with the smallest evaluation value among all expanded nodes) has an evaluation value that is worse than Nodes that are no longer prioritized for expansion; at the same time, the number of expansions per node is introduced. Depth threshold of growth This limits the maximum depth that a single region can be searched. Once the current maximum depth is reached, the search returns to the shallowest, unexpanded level to continue.

[0134] This step involves dynamic threshold pruning to prevent the continued allocation of computational resources to regions with clearly uncompetitive environments; depth constraints prevent the algorithm from prematurely getting trapped in local optima and from over-exploring a single region. Together, these two measures ensure a balance between global exploration capability and local utilization efficiency. Within a limited simulation budget, computational resources are concentrated on regions that are both promising and stable, avoiding premature convergence to local optima and balancing the efficiency of global exploration with local utilization.

[0135] Step 10: Termination Judgment and Solution Output

[0136] Repeat steps seven through nine until the number of sampled design points reaches the preset maximum simulation budget; output the optimal solution of the risk avoidance objective function in the set of sampled design points as the final optimization scheme.

[0137] This step ensures that the algorithm terminates under a limited computational budget and provides a feasible and reliable final solution.

[0138] Compared with the prior art, this embodiment has the following significant advantages:

[0139] Firstly, this implementation method equivalently transforms the objective of maximizing the lift-to-drag ratio in the airfoil design problem into minimizing the negative lift-to-drag ratio. The construction of the risk avoidance objective function ensures that all subsequent sampling decisions are no longer based solely on desired performance, but rather directly weigh noise variance and desired performance under the same optimization scale. This effectively avoids selecting high-risk airfoil designs that are extremely sensitive to random operating condition fluctuations, providing fundamental optimization guarantees for the aerodynamic stability and reliability of aviation equipment in complex service environments.

[0140] Secondly, existing homoscedasticity assumptions ignore the physical heteroscedasticity of airfoil simulation outputs, leading to systematic biases in the surrogate model's predictions in variance-sensitive regions. This implementation addresses this mechanistic deficiency by using the logarithm of the sample variance as the observed variable to establish a stochastic Kriging model for the potential logarithmic noise variance. The introduction of the logarithmic transformation mathematically guarantees the non-negativity constraint of the variance estimation, avoiding negative predictions and numerical ill-conditioning problems that may arise from directly modeling variance. Simultaneously, the posterior distribution of the Gaussian process logarithm can directly derive the log-normal distribution of the noise variance, thus obtaining an accurate analytical expression of the posterior expectation of the noise variance. This technique enables the surrogate model to accurately characterize the heteroscedastic fluctuations of the stochastic simulation output as airfoil design variables change, significantly improving the prediction accuracy of the surrogate model in the global input space, and thus providing a physically consistent statistical basis for the subsequent construction of risk avoidance confidence boundaries.

[0141] Third, this implementation method, based on the posterior distributions of the expected response surrogate model and the log-noise-variance surrogate model, fuses the uncertainties of both through composite confidence parameters to construct upper and lower confidence bounds. The width of the upper and lower bounds adaptively changes with the uncertainty of the surrogate model at the sampling points. In regions with sparse samples or drastic changes in noise variance, the confidence bounds automatically widen to encourage exploration; in regions with dense samples or well-understood areas, the confidence bounds automatically narrow to focus utilization. Simultaneously, the risk aversion parameter is implicitly embedded in the confidence bounds through the composite construction of the composite confidence parameters, ensuring that high-variance regions are raised by the lower confidence bound even when expected values ​​are similar, thus theoretically covering the true risk aversion objective function with a high probability. This not only provides a unified quantitative criterion for node evaluation in tree search but also theoretically supports the guarantee of cumulative regret sublinear convergence in this implementation method, making it provable that the algorithm converges to the global optimum within a finite number of iterations.

[0142] Fourth, this implementation uses the center of the search domain as the root node and divides the input space into equal-sized, non-overlapping sub-regions layer by layer with a fixed number of branches. Each node is associated with its geometric center as a representative point, forming a multi-scale search tree from coarse to fine. This transforms the complex problem of repeatedly optimizing non-convex, multi-peak acquisition functions using multiple starting points or genetic algorithms in a continuous input space into a hierarchical decision-making problem involving ordered representative point evaluation and node expansion on discrete tree nodes. As the tree depth increases layer by layer, the density of representative points grows exponentially, gradually covering the entire continuous input space and theoretically guaranteeing near-global optimization with arbitrary precision. This completely avoids the computational bottleneck of "repeatedly performing expensive numerical optimization on the same non-convex function" in traditional acquisition function optimization, allowing the algorithm to allocate computational resources primarily to airfoil simulation evaluation rather than inner-layer optimization within the limited budget of aerospace engineering, fundamentally improving overall optimization efficiency.

[0143] Fifth, in this implementation, for newly generated candidate child nodes, the risk aversion confidence lower bound is first calculated using only the surrogate model and compared with the optimal mean-variance value among the currently sampled points; only when... A true CFD simulation evaluation is only triggered when a node is deemed suitable; otherwise, the node is retained only by assigning its upper confidence bound as an evaluation value. Nodes whose lower confidence bound is worse than the current optimal value cannot contain the global optimum even under optimistic estimations of surrogate model uncertainty, making true simulation a waste of computational resources. Retaining them and assigning an upper confidence bound ensures that the sub-region containing the node is not permanently deleted due to unsampled behavior. In subsequent iterations, if the upper confidence bound of the node remains competitive after other regions have been fully explored, it may still be reconsidered. This allows the simulation budget to be adaptively allocated to truly promising regions. Compared to the traditional strategy of simulating all candidate points indiscriminately, this approach can evaluate more potential design points with the same simulation budget, significantly improving the optimization benefits per unit simulation cost.

[0144] Sixth, this implementation introduces a dynamic threshold during tree expansion. Any node whose evaluation value is worse than this threshold will be pruned and will no longer be prioritized for expansion. Simultaneously, a depth threshold that increases with the number of expansions is introduced to limit the maximum depth that a single region can be searched. When a branch reaches its maximum depth, it automatically returns to the shallowest unexpanded level to continue searching. Dynamic threshold pruning is essentially an adaptive sparsity operation. As the number of sampled points increases, the surrogate model's understanding of the global objective function becomes clearer, and the threshold dynamically increases accordingly, continuously filtering out regions that clearly lack competitive potential based on existing information. The depth threshold limitation is a forced correction to the depth-first tendency in tree search. It forces the algorithm to periodically return to other regions globally for breadth exploration while deeply exploring a promising region, effectively preventing the algorithm from missing the global optimum by focusing too early on a local optimum. The two work together to maximize the probability of discovering a globally optimal and robust solution within a limited simulation budget.

[0145] Furthermore, from a theoretical perspective, this implementation method provides a theoretical guarantee of sublinear convergence with cumulative regret under the assumption of mild regularity. At the algorithmic level, the construction of risk-avoidance confidence boundaries ensures that the objective function is covered by the confidence bounds with a high probability. At the search strategy level, hierarchical tree partitioning ensures that any continuous region can be approximated to arbitrary precision by representative points within a finite depth. At the sampling decision level, a selective evaluation mechanism ensures that simulation resources are always allocated to potentially optimal nodes and that no region is permanently excluded. The organic combination of these three aspects guarantees that the cumulative regret of the algorithm grows at a sublinear rate, thus providing a provable theoretical basis for convergence in expensive simulation optimization problems such as airfoil design. This allows engineers to obtain an interpretable optimization quality guarantee for the final solution within a limited budget.

[0146] In summary, this implementation method achieves systematic optimization in airfoil design through the synergistic effect of six major technical means: risk avoidance objective function, logarithmic transformation heteroscedasticity surrogate modeling, composite confidence boundary construction, hierarchical tree partitioning, selective node evaluation, and dynamic pruning and depth constraint. Under limited simulation budget, it can efficiently obtain airfoil design schemes with high lift-to-drag ratio, good stability, and reliable robustness.

[0147] Specific Implementation Method Two: The airfoil design device based on risk-avoidance Bayesian optimistic optimization described in this implementation method includes a processor and a memory. The memory stores at least one instruction, which is loaded and executed by the processor to implement the airfoil design method based on risk-avoidance Bayesian optimistic optimization as described in Specific Implementation Method One.

[0148] Specific Implementation Method 3: A computer storage medium as described in this implementation method stores at least one instruction, which is loaded and executed by a processor to implement the airfoil design method based on risk-avoidance Bayesian optimistic optimization as described in Specific Implementation Method 1.

[0149] While specific embodiments of this application have been described herein with reference to them, it should be understood that these embodiments are merely examples of the principles and applications of this application. Therefore, it should be understood that many modifications can be made to the exemplary embodiments, and other arrangements can be designed without departing from the spirit and scope of this application as defined by the appended claims. It should be understood that different dependent claims and features described herein can be combined in ways different from those described in the original claims. It is also understood that features described in conjunction with individual embodiments can be used in other described embodiments.

Claims

1. An airfoil design method for aircraft wings based on risk-avoidance Bayesian optimistic optimization, characterized in that, include: Based on the design requirements, select the design variables and their feasible space for the airfoil of the aircraft. Within the feasible space, iterate and optimize the design variables until the preset iteration termination condition is met, and output the optimal variables for the airfoil of the aircraft. The iterative optimization process includes: Stochastic Kriging modeling is performed on the expected response and noise variance at all nodes in the current variable value candidate set; Select the leaf node with the smallest evaluation value in the current search tree as the expansion node, perform spatial tree partitioning on the expansion node to obtain the set of expansion child nodes, and update the search tree. Construct risk avoidance confidence boundaries for each extended sub-node based on the surrogate model of the expected response and the surrogate model of the noise variance. The evaluation value of each extended sub-node is calculated based on the risk avoidance confidence boundary, and the candidate set of variable values ​​is updated.

2. The airfoil design method for aircraft wings based on risk-avoidance Bayesian optimistic optimization according to claim 1, characterized in that, The calculation of the evaluation value of each extended sub-node based on the risk aversion confidence boundary, and the updating of the variable value candidate set, includes: For extended sub-nodes whose risk aversion confidence lower bound is less than or equal to the optimal simulation evaluation value of the global variable, perform repeated simulation evaluations and add them to the current variable value candidate set. Use the risk aversion confidence upper bound of the remaining extended sub-nodes as their evaluation values.

3. The airfoil design method for aircraft wings based on risk-avoidance Bayesian optimistic optimization according to claim 1, characterized in that, In the repeated simulation evaluation, the evaluation value is calculated using a risk aversion objective function; The expression for the risk aversion objective function is: ， in, This represents the risk aversion objective function. For the expected response, For noise variance, For risk aversion parameters, Represents a node.

4. The airfoil design method for aircraft wings based on risk-avoidance Bayesian optimistic optimization according to claim 1, 2, or 3, characterized in that, The step of performing stochastic Kriging modeling on the expected response and noise variance at all nodes in the current variable value candidate set includes: The average value of the simulation results obtained from repeated simulations at each node is used as the observation value to establish a Kriging surrogate model for the desired response. The logarithm of the noise variance from repeated simulations at each node is used as the observed value, and a Kriging surrogate model for the noise variance is established with the logarithmic noise variance as the latent function.

5. The airfoil design method for aircraft wings based on risk-avoidance Bayesian optimistic optimization according to claim 1, 2, or 3, characterized in that, The construction of risk aversion confidence boundaries for each extended sub-node based on the surrogate model of the expected response and the surrogate model of the noise variance includes: The Kriging surrogate model based on the expected response calculates the posterior prediction mean and posterior prediction variance at the representative points of each extended sub-node. The Kriging surrogate model based on noise variance calculates the posterior expectation and posterior variance of the log-noise variance at the representative points of each extended child node. Risk avoidance confidence boundaries are constructed based on the posterior prediction mean, posterior prediction variance, posterior expectation of log noise variance, and posterior variance of log noise variance.

6. The airfoil design method for aircraft wings based on risk-avoidance Bayesian optimistic optimization according to claim 5, characterized in that, The risk aversion confidence boundary includes a lower bound and an upper bound, expressed as follows: ， ， in, and These are the lower and upper bounds of the confidence level for risk aversion, respectively. and These are the posterior prediction mean and the posterior prediction variance, respectively. and These are the posterior expected value of the log-noise variance and the posterior variance of the log-noise variance, respectively. For risk aversion parameters, This is a composite confidence parameter.

7. The airfoil design method for aircraft wings based on risk-avoidance Bayesian optimistic optimization according to claim 1, 2, or 3, characterized in that, The step of partitioning the extended nodes into a spatial tree to obtain a set of extended child nodes includes: The sub-region corresponding to the extended node is evenly divided into k sub-regions of equal size and non-overlapping. Each sub-region corresponds to an extended sub-node, and the geometric center of each sub-region is used as the representative point of the corresponding extended sub-node. k is a preset number of branches.

8. The airfoil design method for aircraft wings based on risk-avoidance Bayesian optimistic optimization according to claim 7, characterized in that, During the spatial tree partitioning process, the minimum value among all leaf node evaluation values ​​is used as a dynamic threshold, and leaf nodes with evaluation values ​​greater than the dynamic threshold are marked as nodes that are not prioritized for expansion. The design incorporates a depth threshold that increases with the number of leaf node expansions. When the expansion depth of a branch containing a leaf node reaches the depth threshold, the search returns to the shallowest unexpanded level in the current search tree.

9. The airfoil design method for aircraft wings based on risk-avoidance Bayesian optimistic optimization according to claim 1, 2, or 3, characterized in that, The preset iteration termination condition includes: The number of nodes in the current candidate set of variable values ​​has reached the preset maximum simulation budget.

10. The airfoil design method for aircraft wings based on risk-avoidance Bayesian optimistic optimization according to claim 1, characterized in that, With the center of the feasible space as the root node, the space tree is divided layer by layer according to the preset number of branches until the target number of child nodes is obtained, and the geometric center of the region corresponding to each child node is taken as its representative point. Multiple repeated simulation evaluations were performed at each representative point, and the sample mean and sample variance at each representative point were calculated based on the simulation evaluation results. All representative points are used as the initial candidate set of variable values, and the tree structure containing the root node and all child nodes is used as the initial search tree.

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