Aerodynamic Optimization Method for Propeller Airfoil Based on DBO-BP and NSWOA

Abstract

To address the issues of tedious optimization processes, insufficient fitting accuracy of surrogate models, and low optimization efficiency in drone propeller airfoil design, this paper proposes an aerodynamic optimization method for propeller airfoils based on DBO-BP (Dum Beetle Optimizer-Back-Propagation) and NSWOA (Non-Dominated Sorting Whale Optimization Algorithm). The NACA4412 airfoil is selected as the research subject, optimizing the original airfoil at three angles of attack (2°, 5° and 10°). The CST (Class Function/Shape Function Transformation) airfoil parametrization method is used to parameterize the original airfoil, and Latin hypercube sampling is employed to perturb the original airfoil within a certain range to generate a sample space. CFD (Computational Fluid Dynamics) software (2024.1) is used to perform aerodynamic analysis on the airfoil shapes within the sample space to construct a sample dataset. Subsequently, the DBO algorithm optimizes the initial weights and thresholds of the BP neural network surrogate model to establish the DBO-BP neural network surrogate model. Finally, the NSWOA algorithm is utilized for multi-objective optimization, and CFD software verifies and analyzes the optimization results. The results show that at the angles of attack of 2°, 5° and 10°, the test accuracy of the lift coefficient is increased by 45.35%, 13.4% and 49.3%, and the test accuracy of the drag coefficient is increased by 12.5%, 39.1% and 13.7%. This significantly enhances the prediction accuracy of the BP neural network surrogate model for aerodynamic analysis results, making the optimization outcomes more reliable. The lift coefficient of the airfoil is increased by 0.04342, 0.01156 and 0.03603, the drag coefficient is reduced by 0.00018, 0.00038 and 0.00027, respectively, and the lift-to-drag ratio is improved by 2.95892, 2.96548 and 2.55199, enhancing the convenience of airfoil aerodynamic optimization and improving the aerodynamic performance of the original airfoil.

1. Introduction

Propeller aircraft are characterized by their good economy and wide adaptability, making them widely used in both military and civilian aviation, where they play an irreplaceable role in low-altitude domains. With the continuous development of the low-altitude economy, the performance requirements for propeller drones are also steadily increasing. The optimal design of propeller blades is one of the key factors for enhancing their flight performance. In the optimization design of propellers, the airfoil serves as the foundation of the blades and fundamentally determines their performance. Only by selecting the appropriate airfoil can the efficiency of the propeller be maximized [1].

Wang et al. investigated the influence of the maximum thickness, the position of maximum thickness, the trailing edge camber angle, and the trailing edge camber position on the aerodynamic performance and aerodynamic noise of airfoils. Subsequently, they conducted a multi-objective optimization design of the airfoil with the objectives of high lift-to-drag ratio and low aerodynamic noise, effectively improving the aerodynamic characteristics and noise [2]. He et al. combined the MOGA (Multi-Objective Genetic Algorithm) with numerical computation methods to optimize the airfoil. The results showed that, compared with the original NACA4412 airfoil, the optimized airfoil exhibited significant improvements in both lift coefficient and lift-to-drag ratio [3]. Zhang et al. took the traditional wind turbine NACA63418 airfoil as the research object and achieved the optimization of the airfoil’s aerodynamic performance by combining the adaptive genetic algorithm with the XFOIL software [4]. Luo et al. enhanced the performance of the NACA4412 airfoil by employing the NACA parameter method, integrating a response surface surrogate model with quadratic programming. This approach, utilizing a limited number of design variables, resulted in superior optimization outcomes, as referenced in [5]. Kaya conducted aerodynamic optimization of two-dimensional airfoils based on the discrete adjoint method, effectively improving the lift-to-drag characteristics of the airfoils [6]. Wang et al. proposed a constant/steady or unsteady aerodynamic optimization design method for airfoils based on the discrete adjoint method, significantly reducing the time-averaged drag [7]. Sun et al. integrated a fully turbulent continuous adjoint solver with the Shear Stress Transport (SST) turbulence model within the Reynolds Averaged Navier–Stokes (RANS) equations to optimize the NACA4412 airfoil. The implementation of a free deformation parameterization method and dynamic mesh deformation techniques further enhanced the aerodynamic performance of the airfoil, surpassing that of the original design [8]. Wang et al. and others introduced an optimization method for high-speed ground-effect airfoil lift-to-drag characteristics based on the Kriging model. By optimizing the NACA4512 airfoil under specified conditions with the lift-to-drag ratio as the optimization target, they effectively increased the airfoil’s lift-to-drag ratio [9]. Ju et al. and colleagues utilized artificial neural networks to establish a surrogate model and combined it with genetic algorithms to optimize the FX63-167 airfoil. This approach can shorten the optimization time and improve efficiency. However, the prediction accuracy of the surrogate model can affect the final optimization results [10].

From the aforementioned literature, it is evident that the aerodynamic optimization methods for propeller airfoils can generally be categorized into the following three types: The first category is the gradient-based optimization algorithm. This algorithm’s computational load is independent of the dimensionality of the design variables, making it highly efficient for solving multidimensional problems. However, when multiple extrema are present, this algorithm is prone to getting trapped in local optima [11,12,13,14,15,16,17]. The second category comprises heuristic optimization algorithms, which exhibit superior global search capabilities. However, the optimization process requires numerous numerical analysis calculations, leading to significant computational costs. The computational load increases with the addition of variable dimensions [18,19,20]. The third category involves surrogate optimization methods, which optimize by constructing surrogate models of the objective and constraint functions [21,22,23,24,25].

The method of constructing surrogate models to replace Computational Fluid Dynamics (CFD) simulations is gaining widespread application. However, in the process of optimizing propeller airfoils using surrogate models, the fitting accuracy and efficiency of the surrogate models significantly influence the reliability of the entire optimization [26]. When optimizing airfoils by combining Back Propagation (BP) neural network surrogate models with optimization algorithms, Wang et al. found that the prediction accuracy of the BP neural network model for aerodynamic performance is not high when the number of retraining samples is insufficient, but when the number of training samples is too large, the computational load becomes excessively high [27]. Therefore, enhancing the accuracy of surrogate models is key to such aerodynamic optimization methods.

Currently, the mainstream approach is to optimize the prediction accuracy of surrogate models through optimization algorithms. Based on the aerodynamic parameters of computational airfoil samples, Wang et al. established a surrogate model based on Radial Basis Function (RBF) neural networks. By adopting a coarse-to-fine outer iteration of the surrogate model, they significantly improved the accuracy and efficiency of the original RBF neural network surrogate model, ensuring the reliability of the optimization results [28]. In order to enhance the accuracy and efficiency of evaluating the production capacity of oil and gas reservoirs, thereby improving the efficiency of oil field development, Tian et al. introduced the Dung Beetle Optimization (DBO) algorithm into the existing BP neural network prediction model. They established a DBO-BP production capacity prediction model, and experiments showed that both the training and testing accuracies of the DBO-BP model were effectively improved, enhancing the production capacity prediction performance of karst reservoirs [29].

To address the issue mentioned in this paper regarding the low accuracy of aerodynamic performance prediction by BP neural network surrogate models when the sample data are limited, this study proposes a scheme that combines the Dung Beetle Optimization (DBO) algorithm with the BP neural network model. The DBO optimization algorithm is characterized by its strong optimization capability, high solution accuracy and fast convergence speed [30]. By using the error of the BP neural network model as a reference for the fitness function, it seeks to find the optimal initial weight factors of the BP neural network, with the aim of optimizing the accuracy of the BP neural network model.

Numerous scholars have employed genetic algorithms (GAs) and particle swarm optimization (PSO) for airfoil optimization using surrogate models [31,32]. However, the classical swarm intelligence algorithms, represented by GA and PSO, still exhibit certain deficiencies in optimization performance, which may lead to suboptimal airfoil optimization outcomes. Compared to GA and PSO, the multi-objective whale optimization algorithm (NSWOA) not only offers higher search efficiency but also demonstrates superior convergence and global search capabilities [33]. In light of these advantages, this study will utilize the NSWOA algorithm for subsequent aerodynamic performance optimization of airfoils.

This study takes the NACA4412 airfoil as the research object, employing Computational Fluid Dynamics (CFD) methods to calculate the airfoil’s characteristics and to establish a dataset. A surrogate model is constructed by integrating a Back Propagation (BP) neural network. The BP neural network surrogate model is then optimized using the Dung Beetle Optimization (DBO) algorithm to enhance the accuracy of the BP neural network surrogate model, ensuring the credibility of the surrogate model. Subsequently, a Multi-Objective Whale Optimization (NSWOA) algorithm is applied for multi-objective optimization of the airfoil, resulting in a set of airfoils with high lift-to-drag ratios. Experimental validation confirms that the DBO-BP neural network surrogate model developed in this study can effectively improve the precision of the original BP surrogate model, thereby increasing the reliability of the surrogate model and the optimization outcomes. Moreover, the aerodynamic performance of the propeller is significantly enhanced after the airfoil optimization.

2. Airfoil Model Establishment and Example Verification

2.1. CST Airfoil Parametric Model

The CST (Class function/Shape function Transformation) method is a parametric approach developed by Boeing employee Brenda Kulfan in 2006 [33]. This method, which is based on Bezier curves, comprises a simple analytical shape function and a class function. The former controls parameters such as the leading edge radius, the trailing edge angle at the trailing edge and the closure of a specific tail thickness, while the latter extends the method to various geometric shapes.

Using the CST parametric method to describe airfoils, this approach has the advantages of simplicity, intuitiveness, few parameters and high precision, as shown in Equations (1) and (2).

Upper surface:

$$({\begin{array}{c}\zeta )\end{array}}_{\mathrm{Upper}}=C_{N_2}^{N_1}(\psi )\cdot S_u(\psi )+\psi \cdot \Delta {\xi }_{\mathrm{Upper}}$$

(1)

Lower surface:

$${(\zeta )}_{\mathrm{Lower}}=C_{N_2}^{N_1}\left(\psi \right)\cdot S_l\left(\psi \right)+\psi ·\Delta {\xi }_{\mathrm{Lower}}$$

(2)

In the equations: $c$ represent the chord length of the airfoil; $\psi =x/c$, $\zeta =y/c$ represent the non-dimensional coordinates of the airfoil; $\Delta {\xi }_{\mathrm{Upper}}=z{u}_{\mathrm{TE}}/c$ is the thickness ratio at the leading edge of the upper surface of the airfoil; $\Delta {\xi }_{\mathrm{Lower}}=z{l}_{\mathrm{TE}}/c$ is the thickness ratio at the trailing edge of the lower surface of the airfoil.

The class function $C_{N_2}^{N_1}(\psi )$ is defined as shown in Equation (3):

$$C_{N_2}^{N_1}(\psi )={\psi }^{N_1}\cdot {(\begin{array}{c}1\end{array}-\psi )}^{N_2}$$

(3)

The shape function $S\left(\psi \right)$ is defined as follows:

$$S_u(\psi )={{\displaystyle \sum }}_{i=1}^{n}A{u}_i\cdot S_i(\begin{array}{c}\psi )\end{array}$$

(4)

$$S_l(\psi )={{\displaystyle \sum }}_{i=1}^{n}A{l}_i\cdot S_i(\begin{array}{c}\psi )\end{array}$$

(5)

$$S_i(\psi )={\displaystyle \frac{n!}{i!(n-i)!}}{\psi }^i{(1-\psi )}^{n-i}$$

(6)

In the equation, for a general circular leading edge and sharp trailing edge airfoil, $N_1$ is taken as 0.5, and $N_2$ as 1.0; $A{u}_i$ and $A{l}_i$ represent the coefficients of the upper and lower surface shape functions of the airfoil, respectively; the shape function $S_i(\begin{array}{c}\psi )\end{array}$ is commonly represented by Bernstein polynomials, where the parameter $n$ denotes the order of the Bernstein polynomials for the upper and lower surfaces of the airfoil.

Therefore, the upper and lower surface coordinate equations represented by the CST parameterization method are shown in Equations (7) and (8):

$$\begin{array}{cc}{(\zeta )}_{\mathrm{Upper}}& ={\psi }^{N1}\cdot {(1-\psi )}^{N2}\cdot {{\displaystyle \sum }}_{i=1}^n\Lambda u_i\cdot \hfill \\ & {\displaystyle \frac{n!}{i!\left(n-i\right)!}}{\psi }^i{(1-\psi )}^{n-i}+\psi ·\Delta {\xi }_{\mathrm{Upper}}\hfill \end{array}$$

(7)

$$\begin{array}{cc}{(\zeta )}_{\mathrm{lower}}\hfill & ={\psi }^{N1}\cdot {(1-\psi )}^{N2}\cdot {{\displaystyle \sum }}_{i=1}^{n}A{l}_i\cdot \hfill \\ & {\displaystyle \frac{n!}{i!\left(n-i\right)!}}{\psi }^i{(1-\psi )}^{n-i}+\psi ·\Delta {\xi }_{\mathrm{lower}}\hfill \end{array}$$

(8)

2.2. Comparison of Precision Under Different Polynomial Orders

According to the literature, the order of Bernstein polynomials in CST parameterization has a certain impact on the fitting accuracy [34,35,36]. Cezem conducted research on the airfoil parameterization defined by 36th-order Bernstein polynomials and found that excessively high orders can severely ill-condition the parameterization process [37]. Moreover, different airfoil shapes vary significantly. For subsonic airfoils with smaller leading edge radii and maximum thickness positions further aft, and low Reynolds number airfoils with greater leading edge thickness and more pronounced curvature, the required order of polynomials varies. Therefore, selecting an appropriate order is both reasonable and necessary.

By comparing the residuals generated in the creation of the NACA4412 airfoil to determine the appropriate order of Bernstein polynomials, Figure 1 displays the airfoil fitting residual chart. The chart provides a comparison of the errors in fitting the upper and lower surfaces of the NACA4412 airfoil with Bernstein polynomials of orders 2 to 6. It can be observed that the fitting accuracy of the airfoil increases with the increase in the order of the polynomial. From Figure 1, it is evident that the maximum fitting errors of Bernstein polynomials from the second to the sixth order are all around 0.0001. Considering the balance between the accuracy of the order description, the convergence speed of the algorithm, and the time consumption in building the surrogate model, a fourth-order Bernstein polynomial is used for the parametrization process of the airfoil. With 5 design variables each for the upper and lower surfaces, a total of 10 design variables can adequately describe the airfoil with relatively few parameters [38].

2.3. Simulation Calculation and Verification

A C-type unstructured grid was adopted, with the dimensionless airfoil chord length c set to 1.0. The outer boundary of the computational domain extended 50 times the chord length in all directions. The details of the computational grid and the refined region near the airfoil are shown in Figure 2. The flow field solutions for the airfoil are based on the SST (Shear-Stress Transport) turbulence model within Fluent’s RANS (Reynolds-Averaged Navier–Stokes) equations. This model combines the accuracy of the k-ω model near the wall region with the robustness of the k-ε model in the far wall region. The time term is discretized using a second-order implicit scheme, the convective terms are discretized using a second-order upwind scheme, and the diffusive terms are discretized using a central scheme. Relevant conditions were set based on the hotwire anemometer wind speed measurement experiments conducted by Coles and Wadcock, with the incoming flow angle of attack at 13.87° and Mach number at 0.079. The Reynolds number (Re) based on the chord length and incoming flow conditions was 1.52 × 10⁶, with the Y+ value of the airfoil wall surface ranging between 0.1 and 2.0 [39]. The pressure coefficient from the experiment is used as the benchmark for comparison, and grid independence is verified. The number of grid points is 417 × 131, 447 × 151 and 491 × 171, As shown in

Table 1. Three different grid density settings and lift–drag comparison.

Case	CL	CD
Experiment	1.66	/
Related Literature [40]	1.64	0.036
Coarse grid (417 × 131)	1.69	0.037
Moderate grid (447 × 151)	1.72	0.035
Refined grid (491 × 171)	1.71	0.035

, and the Y+ value is taken as 1. After constructing the grids, comparisons were made to determine the surface pressure coefficient (C_P), lift coefficient (C_L) and drag coefficient (C_D) for the airfoil under three different grid sets. The calculated values were compared with experimental results, as shown in Figure 3 and Table 1.

As can be seen from Figure 3, the computational results from all three grid sets are in good agreement with wind tunnel test data and related literature data, meeting the requirements of grid independence and eliminating the influence of the grid on the computational results. Some differences exist between the coarse grid and the other two grids, while no significant differences are observed between the medium-density grid and the fine grid. It can be concluded that the computational results are independent of the grid size at this point. Considering the balance between refinement level, grid quantity and reliability of experimental results, the medium-density grid is selected for aerodynamic computations in subsequent airfoil optimization.

3. Airfoil Optimization Method Based on DBO-BP Surrogate Model

To address the issue of low prediction accuracy in BP neural network surrogate models due to insufficient training data, this study integrates the Dung Beetle Optimizer (DBO) algorithm, renowned for its robust optimization capabilities, high solution precision, and rapid convergence rate, to develop a DBO-BP neural network surrogate model. The initial weights and thresholds of the BP neural network surrogate model are optimized using the DBO optimization algorithm. Subsequently, the aerodynamic characteristics of the airfoil are used as the objective function for further optimization analysis. The process framework is shown in Figure 4.

3.1. DBO-BP Model

3.1.1. BP Neural Network

The BP (Backpropagation) neural network is a common type of artificial neural network model. It is an error-backpropagation neural network where neurons are arranged in layers. The network is trained using the backpropagation algorithm to perform tasks such as pattern recognition and function approximation. Its structure is depicted in Figure 5, with input parameters being the CST parameters [$x_1,x_2,x_3\dots x_8$], and the output parameters being the lift coefficient Cl and the drag coefficient Cd.

As shown in Equation (9), the calculation principle for the neuron net is as follows:

$$ne{t}_j={{\displaystyle \sum }}_{i=1}^n{w}_{ij}{x}_i,j=1,2\cdots m$$

(9)

In Equation (8), $w_{ij}$ represents the weight of the connection between the $i$-th input neuron and the $j$-th neuron in the hidden layer; $x_i$ is the input value to the neuron, and $m$ is the number of neurons in the hidden layer.

As shown in Equations (10) and (11), the weights and biases are updated in the reverse direction according to the optimization algorithm. This iterative process continues until the loss function converges or a certain number of iterations is reached.

$$\Delta w_{ij}=\eta \left(d_j-y_j\right)f^{\prime }\left(ne{t}_j\right)x_i$$

(10)

$$w_{ij}\left(k+1\right)=w_{ij}\left(k\right)+\Delta w_{ij}\left(k\right)$$

(11)

where $\eta$ is the learning rate, $d_j$ is the desired output of the $j$-th node, $y_j$ is the actual output of the $j$-th node, and k is the number of iterations.

The final output of the neuron is as shown in Equation (12).

$$ne{t}_j^l=f\left(w_{ij}^l\cdot ne{t}_j^{l-1}+b_j^l\right)$$

(12)

In the equation, $l$ represents the neuron index, and $b$ represents the threshold value.

3.1.2. Dung Beetle Optimization (DBO) Algorithm

The Dung Beetle Optimization (DBO) algorithm is an innovative swarm intelligence optimization algorithm inspired by the biological behavior of dung beetles, proposed by Professor Shen Bo’s team at Donghua University. It mainly includes the following four parts: rolling behavior, breeding behavior, foraging behavior, and stealing behavior.

• Ball-Rolling Behavior

During the forward motion, in the absence of obstacles, the dung beetle updates its position by rolling a ball, as depicted in Equation (13). Conversely, in the presence of obstacles, the dung beetle resorts to a dancing behavior to update its position.

$$x_i^{k+1}=\{\begin{array}{ll}{x}_i^k+\alpha \cdot e\cdot x_i^{k-1}+b\cdot \left(\left|x_i^k-X^{\omega }\right|\right)& \mathrm{i}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} \mathrm{i}\mathrm{s} \mathrm{a}\mathrm{n} \mathrm{o}\mathrm{b}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{l}\mathrm{e}\\ x_i^k+\mathrm{t}\mathrm{a}\mathrm{n}\left(\theta \right)\cdot \left(\left|x_i^k-x_i^{k-1}\right|\right)& \mathrm{i}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} \mathrm{i}\mathrm{s} \mathrm{n}\mathrm{o} \mathrm{o}\mathrm{b}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{l}\mathrm{e}\end{array}$$

(13)

In the aforementioned equation, $k$ represents the current iteration coefficient; $x_i^k$ denotes the position of the $i$-th dung beetle during the $k$-th iteration; $x_i^{k-1}$ signifies the position of the $i$-th dung beetle during the ($k-1$)-th iteration; $x_i^{k+1}$ is the position of the $i$-th dung beetle at the ($k+1$)-th iteration; $\alpha$ is a natural coefficient, taking a value of either 1 or −1, where a value of 1 indicates no deviation from the original direction, and a value of −1 indicates deviation from the original direction; $e$ is the deflection coefficient, which is a constant within the interval (0, 0.2]; $b$ is a constant within the interval (0, 1); $X^{\omega }$ represents the global worst position; $\theta$ is the deflection angle, which is an angle within the range [0, π]; when $\theta$ takes values of 0, π/2 or π, the position of the dung beetle is not updated, whereas for other values of $\theta$, the position of the dung beetle is updated.

2.

Reproductive Behavior

The strategy emulates the selection process by which scarab beetles identify optimal regions for oviposition, thereby ensuring that their offspring are incubated and nurtured within an environment conducive to their survival and development. The boundary selection mechanism, which models the oviposition site determination in scarab beetles, is articulated in Equation (14) as follows:

$$\{\begin{array}{l}{L}_{*}=max\left(X^{*}\cdot \left(1-R\right),L\right)\\ U_{*}=min\left(X^{*}\cdot \left(1+R\right),U\right)\end{array}$$

(14)

where $U_{*}$ and $L_{*}$ represent the upper and lower bounds of the boundary region, respectively; $U$ and $L$ denote the upper and lower bounds of the optimization problem, respectively; $X^{*}$ signifies the current local optimal position; and $R$ is an intermediate variable defined as $R=1-k/K_{max}$, where $K_{max}$ is the maximum number of iterations.

According to the definition of boundary selection, it is evident that the dung beetle’s egg-laying area varies dynamically with the number of iterations. Consequently, the position of the egg ball also changes dynamically, as represented in Equation (15):

$$B_i^{k+1}=X^{*}+r_1\cdot \left(B_i^k-L_{*}\right)+r_2\cdot \left(B_i^k-U_{*}\right)$$

(15)

where $B_i^{k+1}$ denotes the position of the $i$-th egg ball at the k-th iteration; $r_1$ and $r_2$ are independent random vectors of size $1\times D$; and $D$ represents the dimensionality.

3.

Foraging Behavior

Upon reaching sufficient maturity, dung beetles seek out food sources. By establishing an optimal foraging area, mature dung beetles are directed to forage. The definition of the optimal foraging boundary area is presented in Equation (16):

$$\{\begin{array}{l}{L}_{\mathrm{b}}=max\left(X^{\mathrm{b}}\cdot \left(1-R\right),L\right)\\ U_{\mathrm{b}}=min\left(X^{\mathrm{b}}\cdot \left(1+R\right),U\right)\end{array}$$

(16)

where $U_{\mathrm{b}}$ and $L_{\mathrm{b}}$ represent the upper and lower bounds of the foraging area, respectively, and $X^{\mathrm{b}}$ denotes the global optimal position.

4.

Thieving Behavior

Within the population, certain dung beetles engage in the behavior of stealing dung balls from their conspecifics; these individuals are designated as thief dung beetles. The positional update of these beetles is delineated in Equation (17).

$$x_i^{k+1}=X^{\mathrm{b}}+s\cdot g\cdot \left(\left|x_i^k-X^{*}\right|+\left|x_i^k-X^b\right|\right)$$

(17)

where $s$ is a constant, and $g$ is a random vector of size $1\times D$, which is normally distributed.

3.1.3. Surrogate Model of DBO-BP Neural Network

The initial weights and thresholds of the BP neural network are optimized through the DBO algorithm. The fitness function is set as the mean square difference between the predicted output value and the actual output value; when the fitness function reaches the set precision, the network parameters of the BP neural network are obtained.

The specific implementation steps of the DBO-BP neural network model are as follows:

• Data preprocessing. Normalize the data to accelerate the network’s convergence speed.
• Algorithm initialization. Initialize the parameters of the DBO algorithm and the BP neural network.
• Update the positions of the dung beetles according to the improved dung beetle optimization algorithm and update the initial weights and thresholds of the BP neural network.
• Calculate the fitness value based on the fitness function.
• Determine whether the set precision of the network is reached; if not satisfied, return to step 3; if the set precision is achieved, output the optimal weights and thresholds to the BP neural network.

The initial weights optimization of the BP neural network through the DBO algorithm essentially solves an optimization problem. By combining the dung beetle optimization algorithm with the BP neural network and leveraging the former’s excellent global search capabilities, the global optimization performance is enhanced. This approach addresses the issues of getting stuck at saddle points or converging to local minima during the original training process, thereby improving the neural network’s fitting accuracy and generalization ability in a complementary manner.

3.2. Non-Dominated Sorting Whale Optimization Algorithm

The Whale Optimization Algorithm (WOA), proposed by Mirjalili and Andrew in 2016, is an optimization algorithm inspired by the hunting behavior of humpback whales. This algorithm emulates the foraging tactics of humpback whales, particularly their unique bubble-net feeding method. The process of the algorithm searching for the optimal solution is akin to whales locating and preying upon their targets. The hunting strategy of whales, known as the bubble-net feeding strategy, is primarily divided into the following three parts:

• Encircling Prey.

Humpback whales share information about the prey they have detected and then approach the closest whale to the prey within their group, gradually tightening the encirclement of the entire pod around the prey. During this process, the position update of the whales is illustrated by Equation (18):

$$X\left(t+1\right)=X^{*}\left(t\right)-A\cdot \left|C{X}^{*}\left(t\right)-X\left(t\right)\right|$$

(18)

where $X^{*}\left(t\right)$ represents the position information of the current optimal individual; $X\left(t\right)$ represents the position information of the current individual; $t$ represents the iteration number; and $A$ and $C$ are related coefficients, which can be expressed as Equation (19):

$$\{\begin{array}{l}A=2a\cdot r-a\\ C=2\cdot r\\ a=2\left(t_{max}-t\right)/t_{max}\end{array}$$

(19)

where $r$ is a random number between [0,1]; $a$ is the convergence factor; and $t_{max}$ is the maximum number of iterations.

2.

Spiral Approaching Prey.

When whales search for prey, they use a spiraling ascent to gradually approach the target, and the searching process is described by Equation (20):

$$\{\begin{array}{c}{D}^{*}=\left|C{X}^{*}\left(t\right)-X\left(t\right)\right|\hfill \\ X\left(t+1\right)=X^{*}\left(t\right)+D^{*}{e}^{bl}\mathrm{c}\mathrm{o}\mathrm{s}\left(2\pi l\right)\hfill \end{array}$$

(20)

where $D^{*}$ represents the distance between the current whale and the prey, which is the optimal solution found so far; b is the constant for the shape of the logarithmic spiral, with $\mathrm{b}=1$; $l$ is a random number between 0 and 1.

3.

Random Search for Prey.

In this stage, the whales no longer update their positions based on the location of the optimal solution but instead randomly select a whale’s position for updating, thereby increasing the search range of the pod. When |A| ≥ 1, it indicates that the whales are outside the encircling perimeter and adopt a random search method to enhance the search capability of the WOA algorithm; when |A| < 1, it signifies that the whales are within the encircling circle, opting for a spiral encircling search method to update the position of the current optimal solution. The random search update expression is shown in Equation (21):

$$X\left(t+1\right)=X_{rand}\left(t\right)-A\cdot \left|C{X}_{rand}\left(t\right)-X\left(t\right)\right|$$

(21)

where $X_{rand}\left(t\right)$ represents the position vector of a whale chosen at random.

Non-Dominated Sorting Whale Optimization Algorithm (NSWOA) incorporates logistic chaos mapping and non-dominated sorting into the traditional Whale Optimization Algorithm (WOA), enhancing the diversity and universality of the population and effectively preventing the phenomenon where better individuals are not selected due to the limited search space.

4. Airfoil Optimization and Results Comparison

4.1. Airfoil Optimization Process

The NACA4412 was selected as the baseline airfoil for optimization. The design conditions were set as follows: an incoming flow angle of attack of 2°, 5° and 10°, an incoming Mach number of 0.25, and a Reynolds number of $7.5\times {10}^6$. The same grid division method and medium-density grid as used in the aforementioned grid independence verification were adopted. The turbulence model used was SST k-ω, the time term used a second-order implicit scheme, the convective terms were discretized using a second-order upwind scheme, and the diffusive terms were discretized using a central scheme.

For this optimization problem, following the optimization process shown in Figure 4, Latin hypercube sampling was first used for Design of Experiments (DOE) to construct the sample space. Subsequently, numerical simulation of fluid dynamics was performed on the design points to obtain the initial dataset. After normalizing the samples, they were divided into training and testing sets in a certain proportion, and then an initial BP neural network was established. The selected 10 CST parameters were used as input signals for the input layer, with the number of neurons in this layer being 10 and the number of neurons in the hidden layer is 14; the output layer chooses the airfoil’s lift coefficient and drag coefficient as output signals, hence the number of neurons in the output layer is 2. After initializing the dung beetle population, the initial weights and thresholds of the initialized BP neural network are mapped into the Dung Beetle Optimization (DBO) algorithm for optimization. The DBO algorithm then outputs the optimal weights and thresholds.

After the establishment of the DBO-BP neural network surrogate model, the multi-objective Whale Optimization Algorithm (NSWOA) is combined to perform multi-objective optimization of the airfoil aerodynamic parameters. The optimization variables are the airfoil parameters, with the objectives being to maximize the lift coefficient and minimize the drag coefficient. Additionally, to ensure that the optimized airfoil shape is reasonable, the airfoil CST parameters should be set within a reasonable optimization range, as shown in

Table 2. Optimization bounds for CST parameters.

Lower Bound	Upper Bound
$0.15095$	$0.28033$
$0.18632$	$0.34602$
$0.18661$	$0.34655$
$0.17037$	$0.31639$
$0.20264$	$0.37633$
−0.18011	−0.09698
−0.03774	−0.02032
$-0.07271$	$-0.03915$
$-0.00616$	$-0.00331$
$-0.0376$0	$-0.02024$

. The optimal solution under this surrogate model is obtained, and a separate CFD validation is performed for this design point, ultimately outputting the optimized airfoil. This paper focuses on optimizing the aerodynamic performance of the NACA4412 airfoil at different angles of attack (2°, 5°and 10°) to validate the effectiveness of the proposed method.

4.2. Optimization Results Analysis

Under the aforementioned optimization framework, the DBO algorithm continuously trains and tests the BP surrogate model with the optimization goal of minimizing the loss function until the loss function of the training sample set no longer decreases, indicating the completion of training. The schematic diagram of the lift and drag test results for the BP surrogate model and the DBO-BP surrogate model can be obtained, as shown in Figure 6, Figure 7 and Figure 8. Subsequently, the accuracy of the BP surrogate model and the DBO-BP surrogate model before and after optimization is compared by RMSE (Root Mean Squared Error), MAE (Mean Absolute Error), and R² (Coefficient of Determination). As shown in

Table 3. Comparison of lift training and testing errors between BP surrogate model and DBO-BP surrogate model at different angles.

AOA (°)	Model	Cl	RMSE	MAE	R2
2	BP	Training Set	0.00609	0.00385	0.98328
		Test Set	0.00688	0.00594	0.96517
	DBO-BP	Training Set	0.00290	0.00217	0.99159
		Test Set	0.00376	0.00300	0.98202
5	BP	Training Set	0.00482	0.00376	0.97758
		Test Set	0.00627	0.00793	0.93618
	DBO-BP	Training Set	0.00369	0.00290	0.98276
		Test Set	0.00543	0.00452	0.95573
10	BP	Training Set	0.00746	0.00497	0.90088
		Test Set	0.00945	0.00672	0.83904
	DBO-BP	Training Set	0.00243	0.00163	0.98938
		Test Set	0.00479	0.00353	0.94474

and

Table 4. Comparison of drag training and testing errors between BP surrogate model and DBO-BP surrogate model at different angles.

AOA (°)	Model	Cd	RMSE	MAE	R2
2	BP	Training Set	0.00014	0.00488	0.96113
		Test Set	0.00016	0.00524	0.95829
	DBO-BP	Training Set	0.00012	0.00007	0.93999
		Test Set	0.00014	0.00011	0.77678
5	BP	Training Set	0.00013	0.00009	0.87789
		Test Set	0.00023	0.00018	0.50737
	DBO-BP	Training Set	0.00011	0.00008	0.94350
		Test Set	0.00014	0.00011	0.91343
10	BP	Training Set	0.00027	0.00020	0.92339
		Test Set	0.00029	0.00022	0.90937
	DBO-BP	Training Set	0.00016	0.00008	0.96765
		Test Set	0.00025	0.00019	0.92337

, at the angles of attack of 2°, 5° and 10°, the test accuracy of the lift coefficient is increased by 45.35%, 13.4% and 49.3%, and the test accuracy of the drag coefficient is increased by 12.5%, 39.1% and 13.7%. Therefore, it can be concluded that optimizing the BP surrogate model with the DBO algorithm can effectively improve the prediction accuracy of the BP surrogate model for airfoil aerodynamic performance.

Further validation of the accuracy of the DBO-BP surrogate model’s predicted data and the reliability of the optimization results is carried out using CFD methods. As shown in the data in

Table 5. Comparison of errors between predicted values by surrogate models and actual simulation experiment results at different angles.

AOA (°)	Parameter	Forecast Value (DBO-BP)	True Value (CFD Result)	Error
2	Cl	0.65410	0.66403	1.52%
	Cd	0.01115	0.01127	1.08%
	Cl/Cd	58.664	58. 920	0.44%
5	Cl	0.93025	0.94054	1.11%
	Cd	0.01235	0.01260	2.02%
	Cl/Cd	75.324	74.646	0.90%
10	Cl	1.42927	1.43818	0.62%
	Cd	0.02132	0.02107	1.17%
	Cl/Cd	67.039	68.257	1.82%

, the predicted values of the surrogate model are close to the actual values from the simulation, meeting the expected requirements.

Under the optimization framework, the comparison of the shapes and pressure coefficients of the airfoil before and after optimization is shown in Figure 9. The lift, drag and moment coefficients before and after optimization are shown in

Table 6. Comparison of aerodynamic performance of airfoil before and after optimization at different angles of attack.

AOA (°)	Parameter	Original Airfoil	Optimized Airfoil	Variation
2	Cl	0.62061	0.66403	0.04342
	Cd	0.01109	0.01127	0.00018
	Cl/Cd	55.96123	58.92014	2.95892
	Cm	0.24185	0.25577	0.01392
5	Cl	0.92898	0.94054	0.01156
	Cd	0.01296	0.01260	0.00038
	Cl/Cd	71.68056	74.64603	2.96548
	Cm	0.31458	0.31475	0.00017
10	Cl	1.40215	1.43818	0.03603
	Cd	0.02134	0.02107	0.00027
	Cl/Cd	65.70525	68.25724	2.55199
	Cm	0.42558	0.41194	0.01364

. It can be seen that all three sets of airfoils have a trend of bending and thinning after optimization while still maintaining high smoothness. The optimized airfoil arches upward in the front and middle sections and overlaps with the original airfoil at the trailing edge, slightly resembling a spindle shape. This deformation can increase the lift and reduce the drag of the airfoil, improving the propulsion efficiency of the propeller. However, due to the bending of the airfoil, local stress concentration may occur, affecting the structural strength of the propeller. In addition, it can be observed that compared to the original airfoil, the optimized airfoil has a better distribution of pressure coefficients. A strong suction area forms on the front and middle parts of the entire wing surface, and the pressure center moves forward, shortening the moment arm of the aerodynamic force relative to the elastic axis, thereby increasing the pitch moment coefficient. Therefore, when using this method for aerodynamic optimization of the airfoil, it is necessary to comprehensively consider both the aerodynamic performance and structural strength of the airfoil.

5. Conclusions

This paper investigates an aerodynamic optimization method for propeller airfoils based on the DBO-BP and NSWOA algorithms, constructing an efficient and reliable optimization framework.

• Under this optimization framework, the NACA4412 airfoil was subjected to aerodynamic optimization at attack angles of 2°, 5° and 10°, and the lift coefficients of the airfoil increased by 0.04342, 0.01156 and 0.03603. The drag coefficients decreased by 0.00018, 0.00038 and 0.00027. The lift-to-drag ratios improved by 2.95892, 2.96548 and 2.55199.
• A Dung Beetle Optimization (DBO) algorithm is proposed to optimize the BP neural network for airfoil optimization. By employing the DBO algorithm to optimize the initial weights and thresholds of the BP neural network, issues such as slow convergence speed and susceptibility to local optima are addressed. At attack angles of 2°, 5° and 10°, the lift coefficient test accuracy is improved by 45.35%, 13.4% and 49.3%, and the drag coefficient test accuracy is enhanced by 12.5%, 39.1% and 13.7%, effectively improving the prediction accuracy of the BP neural network, ensuring the reliability of the surrogate model’s predictions for airfoil aerodynamic parameters.
• The optimization framework that combines the Design of Experiments (DOEs) based on Latin hypercube sampling, the DBO-BP neural network surrogate model, and the multi-objective Whale Optimization Algorithm (NSWOA) for airfoil optimization has broad applicability and high convenience, which is of great assistance in the engineering design of airfoils.