High-Efficiency Data Fusion Aerodynamic Performance Modeling Method for High-Altitude Propellers

: During the overall design phase of solar-powered unmanned aerial vehicles (UAVs), a large amount of high-fidelity (HF) propeller aerodynamic performance data is required to enhance design performance, but the acquisition cost is prohibitively expensive. To improve model accuracy and reduce modeling costs, this paper constructs a multi-fidelity aerodynamic data fusion model by associating data with different fidelity. This model utilizes a low-fidelity computational method to quickly determine the design space. The constrained Latin hypercube sampling based on the successive local enumeration (SLE-CLHS) method and the expected improvement (EI) criterion were adopted to achieve the efficient initialization and fastest convergence of the Co-Kriging surrogate model within the design space. This modeling framework was applied to acquire the aerodynamic performance of high-altitude propellers, and the model was evaluated using various performance indicators. The results demonstrate that the proposed model has excellent predictive performance. Specifically, when the surrogate model was constructed using 350 high-fidelity samples, there were improvements of 13.727%, 12.241%, and 5.484% for thrust, torque, and efficiency compared with the surrogate model constructed from low-fidelity samples.


Introduction
Considering the long-endurance and solar energy utilization of solar-powered unmanned aerial vehicles (UAVs) and the low-density atmospheric conditions in near-space, the propeller is still the primary propulsion method for solar-powered UAVs [1,2].The aerodynamic performance of high-altitude propellers is an essential input parameter for the comprehensive design, control system design, and simulation of solar-powered unmanned aerial vehicles (UAVs).It directly influences the ability of solar-powered UAVs to achieve long-endurance flight [3,4].Therefore, it is crucial to quickly and accurately obtain the aerodynamic performance of high-altitude propellers across all flight conditions.The primary methods for obtaining the aerodynamic performance of high-altitude propellers include engineering algorithms, numerical simulations, and wind tunnel experiments.Engineering algorithms are flexible and cost-effective, providing abundant data, however, due to the imperfect physical models, the calculation accuracy for complex states is relatively low.Numerical simulation methods are flexible and have high computational accuracy, but acquiring a large amount of data with this method entails significant time and economic cost.Wind tunnel experiments can simulate flight states and environments, but they are subject to limitations such as interference from supports, Reynolds number effects, and wall interference.These constraints prevent a complete simulation of real flight states, Drones 2024, 8, 229 2 of 15 and obtaining a substantial amount of precise data through wind tunnel experiments also incurs significant costs [5].Therefore, to enhance the consistency of aerodynamic data, data fusion methods are gradually being applied to the construction of surrogate models using data from multiple sources [6].
Forrester et al. [7] applied the co-Kriging model for the first time in aerospace engineering design.Mohammadi-Amin et al. [8] conducted research on the fusion modeling of numerical simulation data and wind tunnel experimental data for the shape of the return capsule based on Co-Kriging.The results indicate that, for the construction of hypersonic aerodynamic databases, the data fusion model can effectively enhance the efficiency of database establishment.Under similar shapes, the efficiency of utilizing aerodynamic data can be increased by more than 80%.Alexandrov et al. [9] utilized iterative processes with low-fidelity data and monitored the progress of design optimization using high-fidelity data.Ultimately, not only did the accuracy of results improve in three-dimensional problems, but costs were also reduced by three times.Keane et al. [10] utilized the Kriging method to establish a fusion model that combines a predictive model based on wing optimization experiments and a response surface model based on computational fluid dynamics (CFD) data.This fusion led to a significant improvement in the prediction accuracy of aerodynamic data.Ghoreyshi et al. [11,12] fused wind tunnel experimental data and CFD numerical simulation results to construct a high-dimensional steady-state aerodynamic model for the aircraft, thereby establishing a multi-source database for steadystate aerodynamic data.This approach reduced the number of experimental samples required for database construction by over 30%.Mourousias et al. [13] used different surrogate models to create a 3D RANS database of propeller aerodynamic performance under high-altitude conditions and verified the effectiveness and accuracy of these models.Li et al. [14] attempted multi-fidelity aerodynamic fusion modeling of wing-distributed load by re-weighting the loss of different fidelity samples.He et al. [15] achieved the prediction of airfoil aerodynamic coefficients under variable flow conditions using a composite neural network structure.Ning et al. [16] use a novel heterogeneous aerodynamic data fusion method embedding reduced-dimension features (MHA-Net) to build an airfoil aerodynamic model.The results demonstrate that this method significantly decreases both the average error and dispersion of aerodynamic models.Wang et al. [17] fused flight test data with wind tunnel test data through a random forest model, successfully rectifying the difference between atmospheric and ground-based aerodynamic data.Nagawkar et al. [18] employed a multi-fidelity deep neural network model for the aerodynamic optimization design of airfoils and wings and obtained multi-fidelity data through calculations with grids of different resolutions.Wu et al. [19] describe a Multi-Fidelity Neural Network (MFNN)-based optimization framework for the optimization design of electric aircraft propellers and improve the accuracy of the propeller's aerodynamic force model through multi-fidelity data fusion.Many scholars employ different methods to construct surrogate models for obtaining various aerodynamic data.However, this paper not only considers the construction methods of surrogate models but also further considers the determination of sample design space and the selection of high-and low-fidelity data, thereby further saving computational resources and time.Yondo et al. [20] provided an overview of the development of aerodynamic data surrogate model methods and the impact of sampling methods on the construction of databases in the aerospace field.Simultaneously, he offered a perspective on the development of data fusion models.In the context of limited computational resources, the quality of surrogate models is closely related to the sampling methods used.Commonly utilized sampling methods include Latin hypercube sampling (LHS) [21], nearest-neighbor sampling (NNS) [22], uniform sampling methods [23], and Monte Carlo methods [24].LHS is currently one of the most popular sampling methods due to its favorable projection properties and space filling [25,26].However, high-altitude propellers need to undergo a wide range of air conditions throughout the entire operating process.Under varying wind speeds, it is necessary to adjust the rotational speed to ensure that high-altitude propellers consistently maintain positive thrust force throughout the entire operational process.Consequently, employing conventional sampling methods may result in some invalid sample points, significantly affecting both modeling accuracy and efficiency.This study initially employs a low-fidelity computation method to rapidly determine the operational conditions under which a high-altitude propeller generates positive thrust, thereby establishing a constrained design space.Within this constrained design space, a constrained LHS based on successive local enumeration (SLE-CLHS) [27] method is employed to obtain both high-fidelity and low-fidelity sample data, ensuring spatial coverage of sample points.Additionally, the expected improvement (EI) criterion is applied for sample point infilling optimization of the Co-Kriging surrogate model.This paper is organized as follows: Section 2 introduces various sampling methods, emphasizing the advantages of the sampling method used in this study.Section 3 briefly describes the Co-Kriging method.Section 4 outlines the process of constructing the surrogate model.Section 5 presents the construction results of the surrogate model and verifies its accuracy.Finally, Section 6 presents the overall conclusions of the study.

LHS Method
Latin hypercube sampling is a widely employed experimental design method for approximately random sampling from multivariate parameter distributions.Experimental designs with any number of sample points can be easily generated through LHS.Furthermore, as a stratified sampling method, LHS exhibits the capability to reduce the variance of sample point errors compared to Monte Carlo methods [28].In statistical sampling, a Latin square refers to a matrix that contains only one sample in each row and column.A Latin hypercube is an extension of the Latin square in multidimensional spaces, where each hyperplane perpendicular to an axis contains at most one sample.Assuming there are M variables, each variable can be divided into N intervals with equal probabilities, that is, In this case, it is possible to select N samples that satisfy the conditions of the LHS.The cumulative distribution of each variable is divided into equal N intervals.A random value is selected from each interval, and these N values for each variable are combined randomly with the values of other variables.

OLHS Method
The sampling points generated by the standard LHS only ensure projective characteristics.The optimal Latin hypercube sampling (OLHS) exhibits excellent spatial filling characteristics, ensuring that sampling points are evenly distributed within the design space.In order to further achieve optimal space-filling properties, Jin et al. proposed an OLHS method based on the maximum-minimum (Maximin) criterion [28].Figure 1 shows the OLHS distribution and the poor distribution of the LHS.
The Maximin criterion can be understood as the maximization of the minimum distance between sample points.As shown in the following formula max : min 1≤i,j≤n,i̸ =j d x i , x j . (2) The d(x i , x j ) is the feature distance between two sample points x i and x j , Drones 2024, 8, 229 4 of 15 ( )

SLE-CLHS Method
The sampling points generated by the SLE algorithm are evenly distributed in the design space, and the projected points in the lower dimensions are almost uniform [29].Wang et al. [27] combined the SLE algorithm with the OLHS algorithm and proposed a SLE-CLHS sampling method.The SLE-CLHS algorithm considers constraints between sample variables and can be directly applied for sampling within the constrained design space.The sampling process for SLE-CLSH is as follows: Assume that m sample points are generated in the two-dimensional design space.The OLHS algorithm is initially employed to generate m sample points in an unconstrained design space.Among the m points, there are t points satisfying the constraint conditions, where t < m.Then, these t points are arranged in ascending order based on their x-coordinates, and the distances in the x-direction between all adjacent points are calculated.The arranged set of samples is represented as Xnew = {X1, X2…, Xt}, and the set of distance values is D = {d1, d2, …, dt+1}.Select the two adjacent points where the maximum distance value is located, and place the Xt+1 point halfway along the x-direction between these two points.Simultaneously, rearrange the distances in D, generating a new distance set D = {d1, d2, …, dt+1, dt+2}.Repeat this process until all points are fixed in the x direction.Since the x-coordinate of the Xt+1 point has been determined, a search with a step size of λ is conducted in the y-direction to find the optimal position, ensuring that the feature distance of this point is maximized.Repeat this process until all points are fixed, thereby achieving the selection of sample points in the constrained space.Figure 2 shows the SLE-CLHS distribution in a two-dimensional with two constraints.

SLE-CLHS Method
The sampling points generated by the SLE algorithm are evenly distributed in the design space, and the projected points in the lower dimensions are almost uniform [29].Wang et al. [27] combined the SLE algorithm with the OLHS algorithm and proposed a SLE-CLHS sampling method.The SLE-CLHS algorithm considers constraints between sample variables and can be directly applied for sampling within the constrained design space.The sampling process for SLE-CLSH is as follows: Assume that m sample points are generated in the two-dimensional design space.The OLHS algorithm is initially employed to generate m sample points in an unconstrained design space.Among the m points, there are t points satisfying the constraint conditions, where t < m.Then, these t points are arranged in ascending order based on their x-coordinates, and the distances in the x-direction between all adjacent points are calculated.The arranged set of samples is represented as X new = {X 1 , X 2 . .., X t }, and the set of distance values is D = {d 1 , d 2 , . .., d t+1 }.Select the two adjacent points where the maximum distance value is located, and place the X t+1 point halfway along the x-direction between these two points.Simultaneously, rearrange the distances in D, generating a new distance set D = {d 1 , d 2 , . .., d t+1 , d t+2 }.Repeat this process until all points are fixed in the x direction.Since the x-coordinate of the X t+1 point has been determined, a search with a step size of λ is conducted in the y-direction to find the optimal position, ensuring that the feature distance of this point is maximized.Repeat this process until all points are fixed, thereby achieving the selection of sample points in the constrained space.Figure 2 shows the SLE-CLHS distribution in a two-dimensional with two constraints.

Co-Kriging Method
Co-Kriging represents a specific instance of multi-task or multi-output Gaussian processes, utilizing the correlation between high-fidelity and low-fidelity model data to im-

Co-Kriging Method
Co-Kriging represents a specific instance of multi-task or multi-output Gaussian processes, utilizing the correlation between high-fidelity and low-fidelity model data to improve prediction accuracy.A co-Kriging metamodel can be interpreted as a combination of two Kriging models built sequentially [30].Simultaneously, the Co-Kriging model inherits the advantages of Kriging and enables uncertainty assessment for prediction points.This method demonstrates significant advantages in scenarios involving small samples and high nonlinearity.
The Co-Kriging model considers two sets of samples, X c = {x c 1 , . .., x c nc } and X e = {x e 1 , . .., x e ne }, with dimension d obtained from the low-fidelity and high-fidelity simulators, respectively.The corresponding function values are represented by y c = {y c 1 , . .., y c nc } and y e = {y e 1 , . .., y e ne } [31].The predicted values of the Co-Kriging model are defined as In Equation ( 4), λ 1 and λ 2 represent the weighted coefficients for the responses to highand low-fidelity, respectively.Assuming the existence of two static stochastic processes corresponding to y c and y e , The covariance and cross-covariance between random variables are defined as .
In Equation ( 6), σ 1 2 and σ 2 2 represent the process variance of the stochastic processes Y 1 (x) and Y 2 (x), respectively.The estimated values of the Co-Kriging model are given by ŷ1

Infill Sampling Criterion
The EI infill-sampling criterion is also known as the Efficient Global Optimization (EGO) method [31].Let the current optimal objective function value be y min , and the predicted results of the Kriging model follow a normal distribution with a mean of ŷ(x) and a standard deviation of s(x).The probability density of Y(x) ∈ N[ ŷ(x), s 2 ] is denoted as For a minimization problem, the improvement quantity I(x) is defined as The expected value of I(x) [32] is By solving the sub-optimization problem of maximizing the Expected Improvement (EI) value: Max Thus, a new sample point x* can be obtained.

Modeling Process
This paper uses the operational states of high-altitude propellers as the design space.Altitude, wind speed, and rotational speed are considered independent variables, while thrust and torque generated by the propeller are treated as dependent variables for constructing a surrogate model.Low-fidelity sample data is obtained using the blade element momentum theory (BEMT), while high-fidelity sample data is acquired through the CFD method.A Co-Kriging model is then developed by combining both high-and low-fidelity sample data.The surrogate model adds high-fidelity sample data according to the EI criterion until the model accuracy meets the requirements or reaches the maximum number of enhancements.The flowchart for this method is as follows, along with detailed steps: Firstly, using a low-fidelity computation method to rapidly determine the operational conditions under which a high-altitude propeller generates positive thrust, thereby establishing a constrained design space.Within this design space, a uniformly constrained design method is used to obtain low-fidelity sample points, which are then computed using the BEMT.The SLE-CLHS design method is employed to obtain high-fidelity sample points and test sets, which are then computed using the CFD method.The test set consisted of fifty high-fidelity samples randomly selected from the design space, and these samples were different from the high-fidelity samples used to construct the surrogate model.Secondly, a Kriging model was constructed using low-fidelity data, and the accuracy of the Kriging model was validated using a test set.If the accuracy of the model did not meet the requirements, the number of low-fidelity samples increased.If the model's accuracy was satisfactory, the current number of low-fidelity samples was retained as the low-fidelity sample set.Both high-fidelity and low-fidelity sample data are utilized in the Co-Kriging method.The Co-Kriging method is applied to construct a surrogate model for the aerodynamic performance of high-altitude propellers.If the surrogate model reaches the convergence criterion, the final surrogate model is output.If the surrogate model does not reach the convergence criterion, the number of high-fidelity samples is increased according to the EI criterion until the model meets the convergence determination condition.Figure 3 illustrates the process of constructing surrogate models based on Co-Kriging.
Co-Kriging method.The Co-Kriging method is applied to construct a surrogate model for the aerodynamic performance of high-altitude propellers.If the surrogate model reaches the convergence criterion, the final surrogate model is output.If the surrogate model does not reach the convergence criterion, the number of high-fidelity samples is increased according to the EI criterion until the model meets the convergence determination condition.Figure 3 illustrates the process of constructing surrogate models based on Co-Kriging.

The Constrained Sampling Method
This section utilizes the aerodynamic data of the 2.8-m diameter high-altitude propeller from a specific solar-powered UAV to construct a surrogate model.The chord length, twist angle distribution, and geometry of the propeller are depicted in Figure 4.

The Constrained Sampling Method
This section utilizes the aerodynamic data of the 2.8-m diameter high-altitude propeller from a specific solar-powered UAV to construct a surrogate model.The chord length, twist angle distribution, and geometry of the propeller are depicted in Figure 4.  High-altitude propellers experience a wide range of aerodynamic conditions due to the large altitude range and the significant variation in speeds of solar-powered UAVs.In Figure 5, at the design point condition, the pressure difference occurs between the upper and lower surfaces of the propeller.This pressure difference results in positive thrust and is accompanied by stable flow.However, when the rotational speed is low and the wind speed is high, the propeller assumes a windmill state and the pressure coefficient on the upper surface of the propeller is significantly higher than the pressure coefficient on the lower surface.This pressure difference leads to the generation of negative thrust.In this state, the propeller experiences an upward shift in the stagnation point, accompanied by flow separation.Therefore, it is necessary to avoid this condition during propeller operation.High-altitude propellers experience a wide range of aerodynamic conditions due to the large altitude range and the significant variation in speeds of solar-powered UAVs.In Figure 5, at the design point condition, the pressure difference occurs between the upper and lower surfaces of the propeller.This pressure difference results in positive thrust and is accompanied by stable flow.However, when the rotational speed is low and the wind speed is high, the propeller assumes a windmill state and the pressure coefficient on the upper surface of the propeller is significantly higher than the pressure coefficient on the lower surface.This pressure difference leads to the generation of negative thrust.In this state, the propeller experiences an upward shift in the stagnation point, accompanied by flow separation.Therefore, it is necessary to avoid this condition during propeller operation.
Figure 5, at the design point condition, the pressure difference occurs between the upper and lower surfaces of the propeller.This pressure difference results in positive thrust and is accompanied by stable flow.However, when the rotational speed is low and the wind speed is high, the propeller assumes a windmill state and the pressure coefficient on the upper surface of the propeller is significantly higher than the pressure coefficient on the lower surface.This pressure difference leads to the generation of negative thrust.In this state, the propeller experiences an upward shift in the stagnation point, accompanied by flow separation.Therefore, it is necessary to avoid this condition during propeller operation.This study employs altitude, wind speed, and rotational speed as independent variables and thrust and torque as dependent variables to construct a surrogate model for the aerodynamic performance of high-altitude propellers.For the specific flight conditions of a solar-powered UAV [33], the propeller's operating parameters are set within the ranges of 0-25 km for altitude, 0-60 m/s for wind speed, and 0-1200 r/min for rotational speed.The low-fidelity calculation method uses the BEMT method, and the high-fidelity calculation method uses the CFD method.Due to the varied operational states of high-altitude propellers across the entire range of variables, a low-fidelity computational method is initially employed to calculate the propeller's aerodynamic performance.This step is used to determine the sample design space, thereby improving the accuracy of the surrogate model and reducing computational costs.Figure 6 shows the constraint plane for lowfidelity data construction.Within the design space, a uniform design method is utilized to select low-fidelity sample data, ensuring uniform distribution and coverage of samples.High-fidelity samples are selected using the SLE-CLHS method to ensure uniformity and spatial filling of samples within the design space.This study employs altitude, wind speed, and rotational speed as independent variables and thrust and torque as dependent variables to construct a surrogate model for the aerodynamic performance of high-altitude propellers.For the specific flight conditions of a solar-powered UAV [33], the propeller's operating parameters are set within the ranges of 0-25 km for altitude, 0-60 m/s for wind speed, and 0-1200 r/min for rotational speed.The low-fidelity calculation method uses the BEMT method, and the high-fidelity calculation method uses the CFD method.Due to the varied operational states of high-altitude propellers across the entire range of variables, a low-fidelity computational method is initially employed to calculate the propeller's aerodynamic performance.This step is used to determine the sample design space, thereby improving the accuracy of the surrogate model and reducing computational costs.Figure 6 shows the constraint plane for lowfidelity data construction.Within the design space, a uniform design method is utilized to select low-fidelity sample data, ensuring uniform distribution and coverage of samples.High-fidelity samples are selected using the SLE-CLHS method to ensure uniformity and spatial filling of samples within the design space.
lation method uses the CFD method.Due to the varied operational states of high-altitude propellers across the entire range of variables, a low-fidelity computational method is initially employed to calculate the propeller's aerodynamic performance.This step is used to determine the sample design space, thereby improving the accuracy of the surrogate model and reducing computational costs.Figure 6 shows the constraint plane for lowfidelity data construction.Within the design space, a uniform design method is utilized to select low-fidelity sample data, ensuring uniform distribution and coverage of samples.High-fidelity samples are selected using the SLE-CLHS method to ensure uniformity and spatial filling of samples within the design space.

Surrogate Model Construction and Convergence Assessment Criteria
This section presents the verification of the proposed method using a solar-powered UAV propeller model.The independent variables in this study are altitude, wind speed, and rotational speed, while the dependent variables are thrust and power, and the efficiency is calculated from the thrust and power.Based on the convergence assessment of the Kriging model constructed with low-fidelity data, 900 low-fidelity sample data points were selected to build the surrogate model.Initially, 150 high-fidelity sample data are combined with low-fidelity data to construct the surrogate model.Subsequently, additional highfidelity sample points are incrementally added based on the EI criterion until the surrogate model satisfies the convergence conditions.To assess the accuracy of the surrogate model, 50 high-fidelity samples are selected as the test set.
In order to comprehensively assess the prediction accuracy of the surrogate model and quantify the deviation between the predicted values and the true values, three indicators are used as measures of measurement accuracy.These indicators include the root mean square error (RMSE), the mean relative error (MRE) and the mean absolute error (MAE).The expressions for RMSE, MRE and MAE are as follows: Among them, y e (m) and ŷ(m) are the actual value and predicted value under the first m verification point, and N represents the number of samples in the test set, which is set to 50.
In this study, the convergence criteria for the surrogate model are defined as the number of increasing sample sizes exceeding 300.

Analysis of Modeling Accuracy
In this section, a surrogate model for the aerodynamic performance of high-altitude propellers is constructed using 900 low-fidelity sample data and 150 high-fidelity sample data.Subsequently, additional high-fidelity sample points are incrementally added based on the EI criterion.The surrogate model for high-altitude propellers is divided into three components: the thrust surrogate model (TSM), the torque surrogate model (QSM), and the efficiency surrogate model (ESM).The efficiency surrogate model is obtained through calculations involving thrust and torque.First, the model's accuracy is assessed based on the RMSE values of the surrogate model.The following figure shows the RMSE values of the surrogate model on the test set after each addition of high-fidelity samples.The horizontal axis represents the number of newly added high-fidelity sample points, while the left vertical axis represents the RMSE values of thrust and torque, and the right vertical axis represents the RMSE values of efficiency.
From Figure 7, it can be observed that as the number of high-fidelity samples increases, the RMSE values of the TSM and QSM exhibit fluctuations but show an overall decreasing trend.The fluctuations are mainly due to the extensive design space of the surrogate model.When relatively few high-fidelity sample data points are used, the accuracy of the surrogate model is easily affected by the high-fidelity samples.The ESM also follows the aforementioned trend.However, when the number of added high-fidelity samples reaches 180-200, there is a noticeable deviation: while the RMSE values for TSM and QSM show a decreasing trend, the RMSE value for ESM suddenly increases.The reason for this discrepancy is that the propeller efficiency is calculated based on the propeller thrust and torque.The RMSE value is related to the square of the difference between the predicted values and the actual values.When the predicted thrust is relatively higher and the predicted torque is relatively lower than the actual values, this situation will occur.Therefore, in order to accurately construct a surrogate model for the aerodynamic performance of high-altitude propellers, it is important to ensure high accuracy in the surrogate models for thrust, torque, and efficiency.
Drones 2024, 8, x FOR PEER REVIEW 11 of 15 decreasing trend.The fluctuations are mainly due to the extensive design space of the surrogate model.When relatively few high-fidelity sample data points are used, the accuracy of the surrogate model is easily affected by the high-fidelity samples.The ESM also follows the aforementioned trend.However, when the number of added high-fidelity samples reaches 180-200, there is a noticeable deviation: while the RMSE values for TSM and QSM show a decreasing trend, the RMSE value for ESM suddenly increases.The reason for this discrepancy is that the propeller efficiency is calculated based on the propeller thrust and torque.The RMSE value is related to the square of the difference between the predicted values and the actual values.When the predicted thrust is relatively higher and the predicted torque is relatively lower than the actual values, this situation will occur.Therefore, in order to accurately construct a surrogate model for the aerodynamic performance of high-altitude propellers, it is important to ensure high accuracy in the surrogate models for thrust, torque, and efficiency.In order to further validate the accuracy of the surrogate model, surrogate models are selected with 150, 250, 350, and 450 high-fidelity samples.Subsequently, the MRE and MAE values are verified on the test set for each of these surrogate models.Figure 8 displays the MRE and MAE values of the surrogate models constructed with varying numbers of high-fidelity sample points.In order to further validate the accuracy of the surrogate model, surrogate models are selected with 150, 250, 350, and 450 high-fidelity samples.Subsequently, the MRE and MAE values are verified on the test set for each of these surrogate models.Figure 8 displays the MRE and MAE values of the surrogate models constructed with varying numbers of high-fidelity sample points.
In order to further validate the accuracy of the surrogate model, surrogate models are selected with 150, 250, 350, and 450 high-fidelity samples.Subsequently, the MRE and MAE values are verified on the test set for each of these surrogate models.Figure 8  Combining Figure 8 and Table 1, it is evident that the accuracy of the surrogate model is positively correlated with the number of high-fidelity sample points.When using 150 high-fidelity samples to construct the surrogate model, compared to the low-fidelity Kriging surrogate model, the prediction accuracy of thrust, torque, and efficiency has only increased by 3.829%, 5.569%, and 0.911% in terms of MRE values.As for the MAE values, the prediction errors for thrust, torque, and efficiency are greater than 20 N, 9 N•m, and 2.5%, respectively.This result suggests that the improvement in predictive accuracy of the surrogate model is not worth the computational cost of using 150 high-fidelity samples.When the number of high-fidelity sample points used for constructing the surrogate model exceeds 350, the MRE values for thrust, torque, and efficiency are all below 3%, meeting the accuracy requirements of the surrogate model.However, it is worth noting that when the number of high-fidelity sample points is increased from 350 to 450, the improvement in model accuracy becomes less significant; the improvement in predictive accuracy for thrust, torque, and efficiency is only 0.42%, 0.082%, and 0.723%, respectively.The benefits brought by the improvement in model accuracy are lower than the increased modeling costs.Therefore, constructing a surrogate model for the aerodynamic performance of high-altitude propellers using 350 high-fidelity sample points is more cost-effective.In terms of MRE values, compared to the low-fidelity Kriging surrogate model, when the number of high-fidelity sample points is 350, the predictive accuracy for thrust, torque, and efficiency improves by 13.727%, 12.241%, and 5.484%, respectively.Regarding MAE values, when the number of high-fidelity sample points is 350, the prediction errors for thrust, torque, and efficiency are less than 4 N, 2 N•m, and 1%, respectively.When constructing the surrogate model using 350 high-fidelity samples, the aerodynamic parameters of the propeller at an altitude of 25 km are extracted, as shown in Figure 9.It is evident that when wind speed is held constant, an increase in rotational speed results in a gradual increment in the propeller's generated thrust.However, at high wind speeds and low rotational speeds, the propeller produces negative thrust.These characteristics are completely consistent with the physical laws governing high-altitude propeller aerodynamic parameters.When constructing the surrogate model using 350 high-fidelity samples, the aerodynamic parameters of the propeller at an altitude of 25 km are extracted, as shown in Figure 9.The white region in the bottom right corner of the figure represents the area where negative thrust is generated, while the white region in the top left corner represents the area where the propeller torque is excessive.These regions are excluded when selecting the sample design space, thus enabling efficient modeling.It can be observed that the operating conditions of the propeller vary significantly at different altitudes.This is mainly due to the significant impact of changes in air density between high and low altitudes, which also highlights the importance of selecting the sample space.It is evident that when wind speed is held constant, an increase in rotational speed results in a gradual increment in the propeller's generated thrust.When the wind speed is low, the efficiency of the propeller is relatively low as well.These characteristics are completely consistent with the physical laws governing high-altitude propeller aerodynamic parameters.
From the above figures and table, it can be observed that, due to the extensive design space of high-altitude propellers, constructing a surrogate model for propeller aerodynamic performance with a small number of high-fidelity samples does not meet the usage requirements.Moreover, using a small number of high-fidelity samples not only signifi- When constructing the surrogate model using 350 high-fidelity samples, the aerodynamic parameters of the propeller at an altitude of 25 km are extracted, as shown in Figure 9.The white region in the bottom right corner of the figure represents the area where negative thrust is generated, while the white region in the top left corner represents the area where the propeller torque is excessive.These regions are excluded when selecting the sample design space, thus enabling efficient modeling.It can be observed that the operating conditions of the propeller vary significantly at different altitudes.This is mainly due to the significant impact of changes in air density between high and low altitudes, which also highlights the importance of selecting the sample space.It is evident that when wind speed is held constant, an increase in rotational speed results in a gradual increment in the propeller's generated thrust.When the wind speed is low, the efficiency of the propeller is relatively low as well.These characteristics are completely consistent with the physical laws governing high-altitude propeller aerodynamic parameters.
From the above figures and table, it can be observed that, due to the extensive design space of high-altitude propellers, constructing a surrogate model for propeller aerodynamic performance with a small number of high-fidelity samples does not meet the usage requirements.Moreover, using a small number of high-fidelity samples not only significantly improves the model accuracy but also increases the modeling cost.However, when a larger number of high-fidelity sample data are utilized to construct the surrogate model, there is a significant improvement in the accuracy of propeller aerodynamic performance prediction.Of course, this also means an increase in the cost required to build the surrogate model.
Overall, in view of the importance of high-altitude propeller aerodynamic performance for the overall design of solar-powered UAVs and flight control simulation, the benefits brought by the improvement in propeller aerodynamic performance prediction accuracy outweigh the time and computational resource costs consumed in modeling.Therefore, using the Co-Kriging method with combined high-and low-fidelity sample data to construct a surrogate model for the aerodynamic performance of high-altitude propellers is entirely feasible.

Conclusions
This paper proposes a method for constructing a surrogate model for the aerodynamic performance of high-altitude propellers.This method can also be applied to obtain various aerodynamic data, enabling the acquisition of high-precision aerodynamic data with relatively low time costs and computing resources.This paper utilizes low-fidelity sample data to quickly determine the de-sign space, which can eliminate the nonlinear operating states of high-altitude propellers and avoid the waste of computational resources.The SLE-CLHS method is employed to select low-and high-fidelity sample data, greatly improving the construction efficiency of the surrogate model due to its excellent spatial filling characteristics.The following conclusions can be drawn from the surrogate model results: When calculating the efficiency surrogate model using thrust and torque surrogate models, even if the accuracy of the thrust and torque surrogate models is relatively high, it does not necessarily mean that the accuracy of the efficiency surrogate model is equally accurate.Further accuracy validation of the efficiency surrogate model is required.
When constructing the surrogate model using 350 high-fidelity samples, compared to the Kriging model based on low-fidelity samples, the proposed model in this paper improves by 13.727%, 12.241%, and 5.484% for thrust, torque, and efficiency, respectively.This fully demonstrates the effectiveness and superiority of the sampling method, as well as the feasibility of utilizing surrogate models.

Figure 1 .
Figure 1.Comparison of LHS and OLHS results in a two-dimensional space with ten sample points.(a) Sampling results of OLHS method.(b) Poor distribution of LHS.

Figure 1 .
Figure 1.Comparison of LHS and OLHS results in a two-dimensional space with ten sample points.(a) Sampling results of OLHS method.(b) Poor distribution of LHS.

Drones 2024, 8 , 15 Figure 2 .
Figure 2. SLE-CLHS results in a two-dimensional space with ten sample points and two constraints.

Figure 2 .
Figure 2. SLE-CLHS results in a two-dimensional space with ten sample points and two constraints.

Figure 3 .
Figure 3.The process of constructing surrogate models based on Co-Kriging.

Figure 3 .
Figure 3.The process of constructing surrogate models based on Co-Kriging.

Figure 4 .
Figure 4. Geometry of the propeller.(a) Airfoil geometries of several sections across the blade.(b) Chord, pitch angle, and thick distribution.(c) Propeller model.

Figure 4 .
Figure 4. Geometry of the propeller.(a) Airfoil geometries of several sections across the blade.(b) Chord, pitch angle, and thick distribution.(c) Propeller model.

Figure 5 .
Figure 5. Analysis of propeller aerodynamic performance.(a) Pressure coefficient distribution map on the blade surface under the design point conditions.(b) Airfoil streamline diagram of blade section at 75% radius under the design point conditions.(c) Pressure coefficient distribution map on the blade surface under windmill state.(d) Airfoil streamline diagram of blade section at 75% radius windmill state.

Figure 5 .
Figure 5. Analysis of propeller aerodynamic performance.(a) Pressure coefficient distribution map on the blade surface under the design point conditions.(b) Airfoil streamline diagram of blade section at 75% radius under the design point conditions.(c) Pressure coefficient distribution map on the blade surface under windmill state.(d) Airfoil streamline diagram of blade section at 75% radius windmill state.

Figure 6 .
Figure 6.(a) A certain type of solar-powered UAV [33].(b) High-altitude propeller sample design space and constraint plane.

Figure 7 .
Figure 7.The variation of RMSE values of the surrogate model with the number of high-fidelity samples.

Figure 7 .
Figure 7.The variation of RMSE values of the surrogate model with the number of high-fidelity samples.

Figure 8 .
Figure 8.Comparison of surrogate model accuracy constructed with different numbers of high-fidelity samples.(a) MRE.(b) MAE.

Table 1 .
The MRE and MAE values of surrogate models constructed using different numbers of high-fidelity sample points.