Two-Stage Global–Local Aerodynamic/Stealth Optimization Method Based on Space Decomposition

Abstract

The design of the flying wing airfoil must consider aerodynamic stealth and trim requirements, with their coupling exacerbating the complexity of the design problem by introducing multiple local minimum points in design space. In addition, it would need a broad space with high dimensionality to obtain the ideal result, and expansion of design space could lead to more local minimum points, causing significant challenges to traditional optimization design methods. A Two-Stage Global–Local Constrained Optimization Method (TGLCOM) was proposed to address these issues. The parametric space was divided into a large-scale global space and a high-dimensional local space. A surrogate-based global constrained optimization method was applied in the large-scale global space, followed by a gradient-based algorithm in the high-dimensional local space to refine the design and obtain the global optima. The efficiency and robustness of the proposed TGLCOM were verified through the airfoil and flying wing layout aero/stealth design. The results indicated a minor conflict between the RCS drag and pitch moment performance. Moreover, the stealth design of the airfoil improved the stealth performance of the flying wing layout in both the yaw and pitch directions.

1. Introduction

The design of the flying wing (FW) has long been a prominent topic in aircraft design [1,2,3,4,5,6]. It involves multiple disciplines, including aerodynamics, stealth, and stability. The sectional airfoil’s characteristics significantly influenced the flying wing, especially as the tail was removed and the fuselage and wing were highly integrated. It must possess strong aerodynamic trim and scattering performance to fully leverage the layout’s advantages, driving aero/stealth coupled optimization design [7,8,9,10,11,12,13].

Numerous significant studies have been conducted on aero/stealth optimization design. D.S. Lee et al. [14] employed a robust multi-objective evolutionary algorithm for the design and optimization of airfoil sections and wing planform shapes. Jiang et al. [8] introduced an integrated optimization method based on the Radial Basis Function (RBF) to enhance the aerodynamic and stealth performance of helicopter rotors and rotor airfoils. Wu Di et al. [7] applied an aero-structure-stealth coupled optimization method to enhance the high aspect ratio wing, utilizing an Enhanced Adaptive Response Surface (EARS) method to reduce computational costs. Zhou et al. [10] proposed a discrete adjoint equation of the integral Maxwell equation, which was employed for aerodynamic/stealth optimization of the X-47B configuration. Although the gradient-based algorithm significantly improves optimization efficiency, it generally cannot guarantee global optima across the entire design space. Liu et al. [15] applied a portfolio-based Bayesian optimization framework to aero and aero/stealth design optimizations of the RAE2822 airfoil, integrating several single-infill criteria and assimilating a sample based on a meta-criterion. Taj et al. [12] proposed an aero/stealth multidisciplinary design exploration framework based on a multi-objective genetic algorithm (GA) and surrogate model to develop fighter aircraft, using a shared parameterized geometry for high-fidelity aero-stealth analysis. Liu et al. [16] combined a convolved multiple-output Gaussian process model with a correlated Pareto-frontier entropy search infill sampling strategy to capture the complex relationships between airfoil aerodynamic and stealth characteristics.

It was demonstrated that both the surrogate-based optimization method (SBO) and gradient-based optimization method (GBO) were utilized in these studies for aerodynamic/stealth optimization. The design of a flying wing airfoil should generally consider both aerodynamic stealth and trim characteristics, which necessitates a broad parametric space to achieve optimal results [11,13]. However, multidisciplinary design requirements often lead to multiple local minima, presenting significant challenges for SBO [17]. Additionally, extending the size interval of design variables further intensifies multimodality in the design space. SBO has the potential to address large-scale design problems with high multimodality, but it suffers from the “Curse of Dimensionality” in high-dimensional spaces. GBO can efficiently search for ideal results in high-dimensional space; however, there is a high possibility that it may become trapped in local minima when applied to large-scale design problems with multiple local minima. Additionally, aerodynamic constraints not only significantly reduce feasible regions but also position the global optima near the boundary of these regions, making it difficult to search for the global optima.

Various studies have developed methods that combine SBO and GBO to address these challenges. Chernukhin et al. [18] proposed a hybrid algorithm that combines a genetic algorithm with a gradient-based algorithm to solve aerodynamic design problems characterized by high multimodality. Li et al. [19] introduced a surrogate-assisted gradient-based optimization method that combines dimension reduction through surrogate modeling with a gradient-based algorithm to solve multipoint aerodynamic design problems. Zhang et al. [20] utilized the adjoint method to locate sensitive regions and applied Efficient Global Optimization to search for the global optima within those regions. Li et al. [21] introduced a sensitivity-based Bezier surface FFD that utilized the adjoint method to scale the design space, improving the performance of surrogate-based optimization methods in high-dimensional problems. However, in these studies, the gradient-based algorithm was primarily used to compensate for the shortcomings of local convergence in surrogate-based algorithms or to enhance the performance of surrogate-based algorithms in high-dimensional problems by introducing gradient information. The applicability of algorithms to different design spaces was often overlooked, resulting in an imbalance between global searching and design efficiency in multidisciplinary design problems.

Based on the above assessment, a new Two-Stage Global–Local Constrained Optimization Method (TGLCOM) was proposed in this study. First, the broad parametric space was divided into large-scale and high-dimensional spaces, and the Niching-based Adaptive Space Reconstruction method (NASR) was applied to search for the most promising local minima located near the global optima [13]. During this process, a Non-Parametric Adaptive Penalty Function method (NPAPF) was used to handle multiple constraints, ensuring the optimal results near the boundary of the feasible region. A high-dimensional local space was then constructed based on the promising local minima, and the GBO was adopted to obtain the global optima. By decomposing the design space, the complex multidisciplinary design problem was simplified into a highly multimodality, large-scale problem and a high-dimensional problem. The proposed Two-Stage Global–Local Constrained Optimization Method leveraged the strengths of both SBO and GBO, preventing the optimization from becoming trapped in local minima and obtaining the global optima more efficiently.

The remainder of this paper is organized as follows. Section 2 explores the multimodality of the aerodynamic and stealth design space, while Section 3 focuses on the TGLCOM procedure. Section 4 presents two applications of TGLCOM: the optimization of two-dimensional airfoils and three-dimensional layout aero/stealth shapes. Finally, Section 5 concludes the paper with a summary.

2. Analysis of Aerodynamic and Stealth Objective Function

In this section, two cases based on the NACA653-018 airfoil are examined to study the multimodality and feasible regions of the aerodynamic and stealth objective functions. A significant amount of research has been conducted on the characteristics of aerodynamic design spaces [18,22,23], while less research has focused on stealth design spaces, according to the author’s knowledge. The first case examines the multimodality and variation trends of the stealth objective function, as well as the influence of geometric constraints, such as maximum thickness, on the feasible region. The second case examines the multimodality and variation trends of aerodynamic performance, as well as the influence of aerodynamic constraints, such as the pitch moment coefficient.

Due to the difficulty of visually displaying high-dimensional design spaces, the Generative Topology Mapping (GTM) model [24] was applied to transform the original high-dimensional space into a two-dimensional space. GTM defines a mapping relationship from the low-dimensional manifold to high-dimensional data. Bayes’ theorem was then applied to perform posterior classification in the low-dimensional space, obtaining the probability distribution function of the low-dimensional variable in the high-dimensional space, represented by the coefficient matrix and the inverse of the GTM model’s variance. The maximum likelihood estimation was used to solve the model hyperparameters.

2.1. Problem Formulation

The baseline geometry had a relative thickness of 0.18. The mesh is shown in Figure 1. The design variables consisted of the z-coordinates from the B-spline parameterization [25], and all optimizations shared the same design space, i.e., the boundaries for the design variables. There were six design variables on each surface of the airfoil, with a size interval for each variable of [−0.30, 0.30].

In the following calculations, the aerodynamic design point was $Ma=0.70,C_l=0.25$ [26], and the Reynolds number for the unit chord length was $2\times {10}^7$. The incident frequency for the stealth design point was f = 9 GHz, and the frontal sector was defined as $\varphi { = 150}^{\circ }~210^{\circ }$, as shown in Figure 2. The aerodynamic and electromagnetic computational states are summarized in

Table 1. Computational parameters.

Parameter	Value
Cruise Mach number	0.70
Cruise lift coefficient	0.25
Reynolds number (unit chord length)	2 × 107
Incident wave frequency (GHz)	9
Incident angle	150°~210°

.

The parallel CFD solver PMB3D [27,28], based on a multiblock structural grid and finite-volume method, was used for aerodynamic computation, and the two-equation Shear-Stress Transport (SST) turbulence model was adopted [29]. The reliability of PMB3D has been verified in numerous cases [10,27,30]. The stealth performance of the airfoils was evaluated using the two-dimensional Method of Moments (MoM) [31], which solves the integral form of Maxwell’s equations. For two-dimensional airfoil stealth performance evaluations, the two-dimensional MoM had a clear efficiency advantage over a three-dimensional solver [11,13].

2.2. Shape Variation by GTM

Using the GTM method, the original 12-D design space, defined by the B-spline method, was transformed into a 2-D manifold space, enabling visualization of the design space and objective functions.

2.3. Analysis of Stealth Objective Function

In this section, only the stealth characteristics are considered. The optimization objective was to reduce the frontal RCS while maintaining maximum thickness. The optimization model is shown in Equation (1). The upper surface of the airfoil was perturbed using the B-spline parameterization method, and the lower surface was obtained symmetrically from the upper surface.

$$\begin{array}{l}Min RCS\\ s.t. thic{k}_{max}\ge thic{k}_{initial}\end{array}$$

(1)

Figure 3 and Figure 4 show the characteristics of frontal RCS and the objective function, respectively, where the objective was $Obj=RCS+1\ast \max (0.18-thic{k}_{max})$. Figure 3 demonstrates that, when only considering the frontal RCS, there were three local minima regions in the current design space. When the thickness constraint is included in the objective function, as shown in Figure 4, the red area on the left side of the design space is penalized for not meeting the thickness constraint, resulting in a sharp increase in the objective function value. This reduces the feasible regions in the design space and creates a discontinuous state. Furthermore, the intense changes in objective values at the local minima would create difficulties in fitting the Kriging model [32].

2.4. Analysis of Aerodynamic Objective Function

In this section, the aerodynamic characteristics are studied. The aerodynamic design status is shown in Table 1. The design objective was the drag reduction, and the aerodynamic constraint were that the pitch moment coefficient must be larger than 0.03, the lift coefficient must equal 0.25, and the geometric constraint was that the maximum thickness of the airfoil could not be reduced. The design model was shown as follows:

$$\begin{array}{l}Min C_d\\ s.t. C_l=0.25\\ C_m\ge 0.03\\ thic{k}_{max}\ge thic{k}_{initial}\end{array}$$

(2)

Using the penalty function method to handle constraints, the ratio of the penalty term to the objective term was set to 1. Thus, the design objective was set as $Obj=C_d+1\ast \max (0.03-C_m)+1\ast \max (0.18-thic{k}_{max})$. Figure 5 presents the characteristics of $C_d$ in the current design space. It can be observed that, under the current design state, there are few local minima regions in the design space, and the changes in $C_d$ are relatively gentle and smooth. Figure 6 shows the contour distribution of the objective function. Due to the presence of constraint terms, the objective value increased sharply, resulting in a significant difference in the position of the global optimal solution in the design space compared to Figure 5.

Figure 7 showed the distribution of sample points under pitch moment constraints, where the white rectangle represented the sample points that did not meet the constraint. Compared to Figure 6, the feasible region of the design space under the constraint was greatly reduced, and the remained regions were not adjacent to each other. The optimal solution was located at the boundary of the feasible region, making it difficult for the algorithm to search for the global optima.

There were multiple local minima in the aerodynamic and stealth objectives, and their coupled design would further aggravate the complexity of the objective function, leading to difficulties in fitting the surrogate model and searching for the global optima. In addition, expanding the size interval would not only increase the number of local minima but also lead to a sparse distribution of samples. Thus, a large number of samples would be required to construct a highly accurate surrogate model, which would further complicate aerodynamic/stealth optimization design.

3. Two-Stage Global–Local Constrained Optimization Method

The flying wing airfoil must be optimized in the high-dimensional and large-scale design space to meet aerodynamic stealth and trim requirements, which presents challenges for traditional optimization design methods. To address these issues, a Two-Stage Global–Local Constrained Optimization Method (TGLCOM) based on space decomposition is proposed in this paper. This method achieves high efficiency and convergence in optimization and leverages the advantages of numerical optimization design.

3.1. Non-Parametric Adaptive Penalty Method

The results in Section 2 demonstrated that aerodynamic and geometric constraints would cause feasible regions to become discontinuous, with the global optima generally located at the boundary of feasible regions. This creates challenges for traditional constraint handling methods, such as the constrained EI function [33]. Thus, the Non-Parametric Adaptive Penalty (NPAP) method was adopted in this study. For constrained optimization problems, the original objective function can be transformed into an unconstrained problem by introducing a penalty term:

$$Min\hspace{0.33em}F(x)=f(x)+{\displaystyle \sum _{j=1}^k{\lambda }_j\max (g_j(x)-0, 0)}, j=1,2\dots ,k$$

(3)

where $F(x)$ was the objective function and ${\lambda }_j$ was the penalty factor corresponding to each constraint.

According to the principle of minimum punishment, the ideal penalty term or penalty factor should be as small as possible and meet the requirements to exclude all unfeasible solutions [34]. Assume that the current design space $D\subseteq {ℝ}^d$, $f:D\to {ℝ}^d$ was an arbitrary function, and ${\mathrm{g}}_{j,j=1\dots k}:D\to {ℝ}^d$ were the constraint functions. Then, $f$ had a unique optimal solution $x_{feasible}^{*}$ in the design space, satisfying $\left\{x_{feasible}^{*}\right|f(x_{feasible}^{*})<f(x_{feasible}):\hspace{0.33em}{x}_{feasible}\in D_{feasible}\}$, where $D_{feasible}\subseteq {ℝ}^d$ was the feasible domain space. In addition, $x_{infeasible}^{*}$ existed in the infeasible regions $D_{infeasible}\subseteq {ℝ}^d$, and it would satisfy $\left\{x_{infeasible}^{*}\right|f(x_{infeasible}^{*})<f(x_{infeasible}):x_{infeasible}\in D_{infeasible}\}$ and $\left\{x_{infeasible}^{*}\right|{\mathrm{g}}_i(x_{infeasible}^{*})\le {\mathrm{g}}_i(x_{infeasible}):x_{infeasible}\in D_{infeasible}\}$. Suppose that there existed a very small positive real number $\epsilon$, when the penalty coefficient of each penalty term satisfied Equation (4), then it would satisfy Equation (5). Therefore, the algorithm could exclude the infeasible solution and efficiently obtain the optimal solution.

$$\begin{array}{l}{\lambda }_j>\hspace{0.33em}{\displaystyle \frac{f(x_{feasible}^{*})+\epsilon -f(x_{infeasible})}{g_j(x_{infeasible})}}>{\displaystyle \frac{f(x_{feasible}^{*})+\epsilon -f(x_{infeasible}^{*})}{g_j(x_{infeasible}^{*})}}\\ x_{feasible}^{*}\in D_{feasible}\hspace{0.33em}\hspace{0.33em},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{x}_{infeasible}\in D_{infeasible}\end{array}$$

(4)

$$F(x_{infeasible})>F(x_{infeasible}^{*})>f(x_{feasible}^{*})$$

(5)

Generally speaking, there were no feasible samples, or the number was small in the early stage of optimization. In order to make the algorithm quickly find the feasible solution, a larger penalty factor should be set to improve the influence of the penalty term on the optimization objective function. As the optimization progressed, the optimizer would gradually approach the global optima across the feasible regions, and the influence of the sample objective function should be increased to improve the optimization convergence and efficiency, so the penalty coefficient should be reduced.

Therefore, a Non-Parametric Adaptive Penalty method (NPAP) was adopted, which adaptively adjusted the penalty factor based on the quantitative variation of feasible solutions during the optimization process. Firstly, the mean of the objective function and constraint condition in the sample set was used as the normalization factor to normalize the objective function and constraint respectively to avoid artificially setting the penalty factor. Then the optimization problem was converted to:

$$Min\hspace{0.33em}\hspace{0.33em}F(x)=-{\displaystyle \frac{EI(x)}{f_{avg}}}+{\displaystyle \sum _{j=1}^k\frac{\max (g_j(x)-0,\hspace{0.33em}0)}{c_{j,avg}}}$$

(6)

where $f_{avg}$ was the mean value of the sample points in the current sample set, and $c_{j,avg}={\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^m\max (g_{i,j}(x_i)-0,\hspace{0.33em}0)}$ was the mean of the jth constraint.

Secondly, from the above discussion, it could be seen that as the algorithm approached the feasible regions, the penalty term should be gradually reduced to improve the optimization efficiency, so the Equation (7) was introduced:

$$v(x)={\displaystyle \frac{number\hspace{0.33em}of\hspace{0.33em}{x}_{infeasible}}{number\hspace{0.33em}of\hspace{0.33em}all\hspace{0.33em}samples}}{\displaystyle \sum _{j=1}^k\frac{\max (g_j(x)-0,\hspace{0.33em}0)}{c_{j,avg}}}$$

(7)

In the early stage of optimization, the number of infeasible solutions usually occupied a large proportion, at this time, the larger $v(x)$ could make the penalty term work at full capacity. As the number of feasible solutions gradually increased, $v(x)$ would be decreased to making the role of the objective term more prominent.

The final form of the objective function was:

$$\begin{array}{l}Min\hspace{0.33em}\hspace{0.33em}F(x)=-{\displaystyle \frac{EI(x)}{f_{avg}}}+{\displaystyle \sum _{j=1}^k\frac{\max (g_j(x)-0,\hspace{0.33em}0)}{c_{j,avg}}}+v(x)\\ s.t.\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{x}_i^l\le x_i\le x_i^u i=1,2,\dots ,D\end{array}$$

(8)

3.1.1. Illustrative Example

The constrained Branin function [28] is employed to indicate how the NPAP acted. The function expression is shown in Equation (9). Figure 8 presents the original Branin function, and it is shown that this $2\text{-}D$ function was usually evaluated on the ${[0,1]}^2$, with three global optima. The constraint function is defined by Gomez#3 function, as shown in Figure 9. Compared to the Branin function, the feasible solution region of the constrained Branin function is extremely reduced, exhibiting discontinuous characteristics, as shown in Figure 10. It is shown that the constrained Branin function had three discontinuous feasible regions, and the optimal solution located at the boundary of the feasible region in the lower right corner, where the red dots (0.9406, 0.3171) represented the location of global optima, with an optimal value of 12.0050.

$$\begin{array}{l}\min \hspace{0.33em}f(x)=(X_2-{\displaystyle \frac{5.1}{4{\pi }^2}}{X}_1^2+{\displaystyle \frac{5}{\pi }}{X}_1-6{)}^2\hspace{0.33em}+(10-{\displaystyle \frac{10}{8\pi }})cos(X_1)+{\displaystyle \frac{X_1+35}{3}}\hspace{0.33em}\\ where\hspace{0.33em}\hspace{0.33em}{X}_1=15x_1-5\hspace{0.33em},\hspace{0.33em}{X}_2=15x_2\\ s.t.\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}g(x)=-((4-2.1X_1^2+{\displaystyle \frac{X_1^4}{3}})X_1^2+X_1{X}_2+6(4X_2^2-4)X_2^2+3\sin (6(1-X_1))+3\sin (6(1-X_2)))\le 0\\ where\hspace{0.33em}\hspace{0.33em}{X}_1=2x_1-1\hspace{0.33em},\hspace{0.33em}{X}_2=2x_2-1 \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\end{array}$$

(9)

The Optimized Latin Hypercube Sampling method (OLHS) [35] is adopted to generate initial samples with the size of $2(D+1)$ to construct Kriging model. The termination condition is that the error between the objective function value and the optimal value is less than 1%.

Figure 11 presents the convergence history of the objective function, from which it is shown that the NPAP finds the optimal solution in the 14th generation. Figure 12 shows that during the optimization process, the algorithm is not trapped in the discontinuous feasible domain, and it could efficiently converge to the boundary of the feasible domain where the optimal solution exists. For the problems where the optimal solution is located at the boundary of the feasible domain, when the algorithm approaches from the feasible domain to the boundary, the penalty function terms are all 0, and the objective term is completely dominated. When the algorithm approaches the boundary of the feasible domain, the penalty term gradually decreases, avoiding the algorithm from directly crossing the boundary and concentrating optimization within the feasible domain. The dynamic penalty function strategy based on sample analysis avoids the algorithm from falling into local optima due to discontinuous feasible domains, enabling it to quickly converge to the feasible domain near the global optima. Especially for problems where the optimal solution is located at the boundary of the feasible domain, the optimization efficiency is significantly improved.

3.1.2. Numerical Experiments and Discussion

Five classical nonlinear constraint functions (G1, G3, G5, G6, G24) are selected in order to furtherly test and verify the efficiency and robustness of proposed method. More details are shown in

Table 2. Benchmark problems for global optimization.

Function	Dim	Noc	Domain	Target Value
G1	13	9	${[0,1]}^9\times {[0,100]}^3\times [0,1]$	−15
G3	10	1	[0,10]10	−1.0005
G5	4	5	${[0,1200]}^2\times {[-0.55,0.55]}^2$	5126.49
G6	2	2	[0,10]2	−6961.8
G24	2	2	$[0,3]\times [0,4]$	−5.508

, where the Dim denotes the number of dimensions and the Noc is the abbreviation of “Number of Constraints”. In addition, the Target value are from the published papers on constrained problems [32]. These functions have different dimensions and constraint characteristics, which together could simulate many problems.

The traditional constrained EGO Infilling method (CEI) [36] and the Static Penalty Function (SPF) are used to make comparisons with the proposed method. The LHS is used to generate initial samples with size $D+1$ are the same for all optimization algorithms on test problems [32]. Thirty runs on each of these test functions have been made to eliminate randomness, and 200 sample points are updated in each run.

These methods are compared in terms of the average number of function evaluations (over 30 trials) required to attain a solution with $relative\hspace{0.33em}error<1\%$. If $f_{target}$ is the global minima value over the domain $D$ and $f_{min}$ is the best value obtained by an algorithm, then the relative error of the algorithm is given by:

$$relative\hspace{0.33em}error\hspace{0.33em}=\left\{\begin{array}{l}{\displaystyle \frac{\left|f_{min}-f_{target}\right|}{\left|f_{target}\right|}},\hspace{0.33em}\hspace{0.33em}{f}_{target}\ne 0\\ 0.001\hspace{0.33em},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em} f_{target}=\hspace{0.33em}0\end{array}\right.\hspace{0.33em}$$

(10)

If all 30 runs get the target value, then the number inside the parenthesis is the standard error of the mean. The symbol “$>$” indicates that at least one trial failed to obtain the target value, and N_e is the trial number that does not reach the target value within 200 function evaluations.

Table 3. Average number of function evaluations (over 30 trials).

Function	CEI		SPF		NPAP
	Result	Ne	Result	Ne	Result	Ne
G1	200 (-)	0	>124.9 (-)	28	115.1 (17.48)	30
G3	200 (-)	0	>191.7 (-)	21	175.3 (9.6)	30
G5	200 (-)	0	53.6 (1.75)	30	44.8 (2.71)	30
G6	74.6 (3.55)	30	32.6 (1.21)	30	14.3 (0.99)	30
G24	22.4 (3.08)	30	13.5 (1.11)	30	8.6 (1.35)	30

The CEI is constrained EGO Infilling method, and the SPF is the Static Penalty Function, the NPAP is the Non-Parametric Adaptive Penalty Method.

shows that for G1, G3, and G5 problems with higher dimensions or more constraints, CEI method could not find the target value within 200 times. The optimization results of SPF are much better than that of CEI, but the robustness is not that good since the optimization convergence could not be guaranteed every time. Compared to CEI and SPF, the results demonstrate that the NPAP could find the optimal solution in all 30 runs of all test problems at the price of fewer sample points, showing higher optimization efficiency and convergence. For G6 and G24 problems with low dimensions and fewer constraints, these three methods could all find the global optimal solution, but the NPAP has the best optimization efficiency compared to alternatives, indicating that it is a robust method for constrained problems.

3.2. Two-Stage Global–Local Constrained Optimization Method

In this method, the surrogate global optimization method named Niching-based Adaptive Space Reconstruction method (NASR) [13] is adopted to search for the most promising sample point in large-scale design space first. Niching-based multiple points sampling method is adopted to simultaneously search for the locations of multiple local minima of the EI function. Different expansion strategies are applied to adjust the size of design intervals in different iterations. More details of NASR can be found in reference [13]. Then, the aerodynamic/stealth optimization design method based on discrete adjoint [10] is applied to conduct fine design in the high-dimensional design space to obtain the global optima.

The TGLCOM included the following steps:

Step 1: Construct large-scale design space with low-dimensionality;

Step 2: Use surrogate-based global optimization method to search for the most promising sample point;

Step 3: The global optimization terminates when the number of iterations reaches the threshold;

Step 4: Construct high-dimensional design space with small scale;

Step 5: The gradient algorithm was adopted to optimize the design in the high-dimensional design space.

The flowchart of TGLCOM is shown in Figure 13.

4. Aerodynamic/Stealth Optimizations and Discussion

In this section, two benchmark problems are carried out. The first case is the optimization of flying wing airfoil based on NACA65,3-018, as shown in Section 2, and the second is for the flying wing layout based on X-47B [37]. Both of them consider the aerodynamic, stealth and trim requirements. The Niching-based Adaptive Space Reconstruction method (NASR) [13] combined with Non-Parametric Adaptive Penalty method (NPAP) is adopted for global optimization in both cases, while gradient-based optimization method with adjoint approach [10] is applied in local optimization.

The PMB3D [27,28] is employed for aerodynamic computation, and the two-dimensional Method of Moment (MOM) is used for the stealth computational of two-dimensional airfoil. As for the three-dimensional layouts, the computational cost of MOM could be too high for electrically large objects, making it unaffordable for engineering design. Thus, a parallel CEM solver named multilevel fast multipole method (MLFMA) [10], where the Bi-CGSTAB [38] algorithm is adopted in the iterative solution. The MLFMA is the high precision and fast solution method of MOM, it could reduce the computational complexity of MOM by accelerating the matrix-vector multiplications. The efficiency and reliability have been proven in [10].

4.1. Airfoil Optimization

4.1.1. Problem Formulation

The aerodynamic computational state is Ma = 0.71, C_l = 0.25 and Re = $2\times {10}^7$, and in stealth computations, the incident frequency of stealth design point is f = 9 GHz, and the frontal sector is defined as $\varphi { = 150}^{\circ }\sim 210^{\circ }$. More details could be referred to Section 2.

The design objective is to reduce the drag coefficient and improve the frontal stealth performance. The constraints are that the pitch moment coefficient should be no less than 0.03, and the maximum thickness is not reduced. The drag coefficient and the frontal average RCS are normalized by the values of the baseline, as described in (11), where $w_1=1$ and $w_2=1$ are used.

$$\begin{array}{l}Min\hspace{0.33em}\hspace{0.33em}\hspace{0.33em} F=w_1{C}_d/C_{d\_baseline}+w_{2}RCS/RC{S}_{baseline}\\ s.t.\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{C}_l=0.25\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{C}_m\ge 0.03\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{t}_{max\_opt}\ge t_{max\_initial}\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{x}_{i,low}\le x_i\le x_{i,up}\end{array}$$

(11)

where $C_d$ is the drag coefficient, C_l is the lift coefficient, $C_m$ is the pitch moment coefficient, $t_{max}$ is the maximum thickness of airfoils, and $x_i$ is the design variable. The incident angle of RCS is shown in Figure 2.

4.1.2. Global Optimization

The large-scale design space for global optimization is constructed by 12 design variables, and the size interval of each design variable is [−0.06, 0.06]. There could be multiple local optimum points in aero/stealth design space, and the size interval of design variable should be large enough to contain the region where the global optimum point is most likely to exist. Since it is the first stage of the optimization, there is no need to choose too many design variables.

Table 4. Comparison of optimization results.

	Alpha	Cl	Cd	Cm	RCS (m)
NACA 65,3-016	1.98°	0.25	0.0114	−0.0100	0.0594
Global_FW_foil	2.94°	0.25	0.0082	0.0305	0.0335

tablets the computational results of airfoils. It is shown that the aerodynamic and stealth performance of the global optimized airfoil named Global_FW_foil has significantly improved. The drag coefficient of Global_FW_foil has a reduction of 32counts (1count = 0.0001), and the frontal RCS is reduced by 0.0259 m. Figure 14 and Figure 15 present the shape and pressure coefficient distribution, respectively. it is shown that the upper surface of the Global_FW_foil tends to be flatter, which is beneficial to reduce the shock wave. It is shown that the strong shock on the upper surface of the NACA 65,3-016 is weakened into two weak shocks, leading to better drag divergence performance, as shown in Figure 16, and the shock wave on the lower surface is completely eliminated.

It is shown that the positive camber on the leading edge and negative camber on the trailing edge of the Global_FW_foil both increased, and it would expand the positive loading on the leading edge as well as the negative loading on the trailing edge of the airfoil, which is conducive to the improvement of the positive pitch moment performance, as shown in Figure 17. Moreover, the increment of the positive camber on the leading edge could help decrease the leading-edge radius, resulting in good frontal stealth performance, as shown in Figure 18. Thus, the RCS $\varphi { = 180}^{\circ }~210^{\circ }$ was reduced a lot. However, the decrement of RCS at $\varphi { = 150}^{\circ }~180^{\circ }$ is not that notable due to the shape variation on the upper surface, indicating the conflict between the aerodynamic and stealth performance.

4.1.3. Local Optimization

Based on the Global_FW_foil, a high-dimensional design space is constructed with 24 design variables, whose value range is [−0.03, 0.03]²⁴, as shown in Figure 19. Then, the optimization would be started with the point searched by the NASR method in the section above, and there would be no significant change in shape, such as the thickness and camber distributions. Thus, it needs more design variables to depict the local shapes in detail. The Sequential Quadratic Programming (SQP) algorithm is used to optimize the design in a high-dimensional design space. The design state is the same as the global optimization.

The computational results of optimized airfoils are tabulated in

Table 5. Comparison of optimization results.

	Alpha	Cl	Cd	Cm	RCS (m)
NACA 65,3-016	1.98°	0.25	0.0114	−0.0111	0.0453
Global_FW_foil	2.94°	0.25	0.0082	0.0305	0.0335
Global/Local_FW_foil	2.51°	0.25	0.0081	0.0356	0.0090

Alpha is the angle of attack, C_l is the lift coefficient, C_d is the drag coefficient and C_m is the pitch moment coefficient.

. It is shown that the aerodynamic performance of the optimized airfoil in this section, named Global/Local_FW_foil, is similar to that of the Global_FW_foil, but the stealth performance has significantly improved. The drag coefficient of the Global/Local_FW_foil has a reduction of 1 count compared to the Global_FW_foil, and the pitch moment coefficient reached 0.0356, meeting the constraint. Furthermore, the average value of the frontal RCS was reduced by 0.0245 m, verifying the effect of the local optimization.

In Figure 20, it is shown that the leading-edge radius of the Global/Local_FW_foil is further reduced and the thickness becomes smaller,. However, it would also decrease the positive loading on the leading edge, and the negative loading should be increased to satisfy the pitch moment constraint, leading to the expansion of the negative-lift regions, as shown in Figure 21. The upper surface of the Global/Local_FW_foil becomes much flatter compared to the Global_FW_foil, and the two weak shock waves are reduced to a single weak shock wave, the position moves a little afterward. However, this phenomenon is not good for the drag divergence characteristics of the airfoil, as shown in Figure 22. The pitch moment characteristics of the Global/Local_FW_foil are similar to those of the Global_FW_foil. resulting in better stealth performance at all angles, as shown in Figure 23 and Figure 24.

4.2. Flying Wing Optimization

4.2.1. Problem Formulation

The flying wing optimization consists of the two-dimensional (2D) airfoil optimizations based on surrogate-based global optimization and three-dimensional (3D) sectional shapes optimization by gradient-based local optimization. In addition, 2D airfoil optimizations could provide a good start for 3D sectional shapes optimization. In all optimizations, the aerodynamic and stealth performances in the frontal sector are considered together by the weighted sum method, and the weighted factors were 1:1 [9].

In 2D airfoil optimization, symmetry airfoil NACA 65,3-018 would be adopted as the baseline of various zones, which is shown as Figure 25, and the corresponding relative thickness at root, kink, and tip are 0.16, 0.13, and 0.11, respectively. The incident frequency of RCS is f = 9 GHz for all airfoil optimizations, and the incident angle is the same as in Figure 2. Since the discipline characteristics requirements of airfoils are distinct in various zones of the flying wing, the design conditions of airfoils are different, which can be referred to in [39].

In 3D sectional shapes optimization, the baseline aircraft is similar to the X-47B layout [10] in size and configuration, and the tip is revised to simplify the optimization problem. Parameters of the baseline aircraft are summarized in

Table 6. Layout parameters of the baseline aircraft.

Geometric Parameter	Value
Span (m)	18.93
Reference wing area (m2)	87.42
Reference center (m)	(6.17,0,0)
Mean aerodynamic chord (m)	6.57

, where the coordinate of the nose is (0, 0, 0), and the FFD control volumes are shown in Figure 26. The design variables are the z coordinate of the control points distributed on the FFD volume. There are 8 points stream-wise and 5 points span-wise, resulting in 80 design variables in total.

4.2.2. Global Optimization

Root Airfoil Aerodynamic/Stealth Optimization

The design point is Ma = 0.65, Re = 2.0e7, C_l = 0.25, and the baseline airfoil is NACA 65,3-016 airfoil. To improve the pitch moment trimming characteristics of 3D layout, the pitch moment coefficient of the optimized airfoil should be greater than 0.05. The objective of root airfoil is to reduce drag and average RCS, with fixed lift and moment constraints. The problem could be expressed as follows:

$$\begin{array}{l}Min:\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}F=C_d/C_{d\_baseline}+RCS/RC{S}_{baseline}\\ \hspace{0.33em}s.t.\hspace{0.33em}:\hspace{0.33em}\hspace{0.33em}{C}_l=0.25\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{C}_m\ge 0.05\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}thic{k}_{max}\ge thic{k}_{max\_initial}\end{array}$$

(12)

The aerodynamic and stealth results of airfoils are compared in

Table 7. Comparison of design results.

Configuration	Alpha	Cl	Cd	Cm	AOA	RSC (m)
NACA 65,3-016	1.51°	0.250	0.0067	0.0058	1.51°	0.0341
Aero/RCS_RootOpt	2.65°	0.250	0.0066	0.0519	2.65°	0.0041

Alpha is the angle of attack, C_l is the lift coefficient, C_d is the drag coefficient and C_m is the pitch moment coefficient.

. The aerodynamic performance of optimized airfoil, named Aero/RCS_RootOpt, is similar to that of baseline airfoil, NACA 65,3-018. However, the average value of frontal RCS decreased by 87.9%, and the pitch moment coefficient reached to 0.0519, satisfying the constraint. Figure 27 and Figure 28 show the comparison of shape and pressure coefficient distribution, respectively. It is shown that the maximal thickness position of Aero/RCS_RootOpt move forward, and the camber increases at leading edge and decreases at trailing edge, due to the pitch moment constraint. In addition, the leading-edge radius of Aero/RCS_RootOpt reduces a lot, tending to be “olecranon”, which is obviously good for frontal RCS performance, as shown in Figure 29. Since there is no obvious shock occurred during the optimization, the lift–drag characteristics has little changes.

Kink Airfoil Aerodynamic/Stealth Optimization

In this section, the NACA 65013 airfoil is adopted as the baseline airfoil, and the design point is Ma = 0.73, Re = 2e7, C_l = 0.55. The objective of airfoil on Kink zone is to improve the lift–drag, drag divergence and the frontal RCS characteristics. On general, the Kink zone is close to Center of Gravity (C.G.), so it would need to keep positive moment no less than 0.01. The problem could be expressed as follows:

$$\begin{array}{l}Min: F=C_d/C_{d\_baseline}+RCS/RC{S}_{baseline}\\ s.t.: C_l=0.55\\ C_m\ge 0.01\\ C_{d\_0.75}-C_{d\_0.73}\le 0.001\\ t/c\ge t/c_{initial}\end{array}$$

(13)

Aerodynamic and stealth results of airfoils are summarized in

Table 8. Comparison of design results.

Configuration	Alpha	Cl	Cd	Cm	AOA	RSC (m)
NACA 65,3-013	1.74°	0.550	0.0079	−0.0113	1.74°	0.0299
Aero/RCS_KinkOpt	2.25°	0.550	0.0068	0.0105	2.25°	0.0077

Alpha is the angle of attack, C_l is the lift coefficient, C_d is the drag coefficient and C_m is the pitch moment coefficient.

\Figure 30 presented that the leading edge of Aero/RCS_KinkOpt got sharper and the upper surface has become much flatter. The strong shock wave of baseline airfoil was eliminated into two weak shock waves, and the suction peak increased, as shown in Figure 31. It is shown that the drag coefficient of the Aero/RCS_KinkOpt was reduced by 13.9% compared to the NACA 65,3-013, with the pitch moment constraint satisfied. The frontal RCS was reduced to 0.0077 m, resulting in better RCS performance, as shown in Figure 32.

Tip Airfoil Aerodynamic/Stealth Optimization

The design requirements of the tip airfoil are to improve lift–drag performance and stealth performance, with fixed lift and moment constraints. The baseline airfoil is the NACA 65,3-011, whose design states are Ma = 0.73, Re = 2e7, C_l= 0.35.

$$\begin{array}{l}Min: F=C_d/C_{d\_baseline}+RCS/RC{S}_{baseline}\\ s.t.: C_l=0.35\\ C_m\ge 0.00\\ C_{d\_0.75}-C_{d\_0.73}\le 0.001\\ t/c\ge t/c_{initial}\end{array}$$

(14)

From

Table 9. Comparison of design results.

Configuration	Alpha	Cl	Cd	Cm	AOA	RSC (m)
NACA 65,3-011	2.80°	0.350	0.0077	−0.0072	2.80°	0.0219
Aero/RCS_TipOpt	2.96°	0.350	0.0069	0.0000	2.95°	0.0070

Alpha is the angle of attack, C_l is the lift coefficient, C_d is the drag coefficient and C_m is the pitch moment coefficient.

, it demonstrates that the drag coefficient of the optimized airfoil, Aero/RCS_TipOpt, is reduced to 0.0069 compared to the baseline airfoil NACA 65,3-011, and the pitch moment coefficient reached to 0.000. In addition, the frontal RCS performance of Aero/RCS_TipOpt significantly improved, whose average value decreased by 68%. Figure 33 shows that the leading edge of optimized airfoil also becomes sharper, and the upper surface tended to be flat, which was beneficial to pressure recovery, as shown in Figure 34; thus, the strong shock of baseline is eliminated entirely, resulting in good RCS performance, as shown in Figure 35.

To test and verify the effect of airfoil optimizations, the optimized airfoils are assembled onto the various zones of the 3D layout named GlobalOpt, shown in Figure 25, and compared to the BaseModel based on the corresponding baseline airfoils. The computational results are shown in

Table 10. Computational results of all configurations.

Configuration	Alpha	CL	CD	CM	Yaw (−60–60°) dBSm	Pitch (−45–45°) dBSm
BaseModel	1.38°	0.30	0.02795	−0.042	0.3755	−3.4616
GlobalOpt	1.97°	0.30	0.02071	−0.017	−2.2782	−22.3780

. It is presented that the drag characteristics of GlobalOpt with three optimized airfoils have significantly improved, the drag coefficient was reduced from 0.02795 to 0.02071, and the pitch moment coefficient reaches −0.017, leading to better pitch moment trimming performance. The RCS characteristics have also improved in both yaw and pitch directions, especially for the pitch direction.

Figure 36 shows that the shock wave on the inboard wing of the GlobalOpt is completely eliminated, and on the outboard wing is also largely weakened, compared to the BaseModel. The low-pressure zones on the leading edge of the GlobalOpt extend to the wingtip, and the pressure recovery on the trailing edge tends to be much gentler, resulting in good aerodynamic performance. Figure 37 and Figure 38 present the distribution of RCS. It is shown that the RCS in pitch direction is mainly focused on the frontal sector, which is attributed to shape variations on the leading edges of three optimized airfoils. What is more, the decrement of the airfoils’ leading edge radius would also make the leading edge of the whole 3D layout shaper, which is also good for the RCS performance in the yaw direction, where the RCS wi was evidently reduced, especially for the RCS peak at around $\varphi { = 58}^{\circ }$ corresponding to the leading edge of the inboard wing, and $\varphi { = 26}^{\circ }$ corresponding to the leading edge of the outboard wing.

4.2.3. Local Optimization

Take the GlobalOpt as the baseline model, and the gradient-based discrete adjoint optimizer [10] is adopted to find the ideal sectional shapes. The main design objective is to improve the drag and stealth performances in the frontal sector. Thickness constrains are imposed on five wing cross-sections in order to meet realistic engineering requirements. The moment coefficient is constrained to be larger than the baseline model moment coefficient. Moreover, the Mach number of design point was 0.8 with a fixed lift coefficient of 0.30; unit chord length Reynolds number is 6.8e6 and the Reynold number is $4.47\times {10}^7$ based on the mean aerodynamic chord. The incident frequency is f = 500 MHz, the wave length for f = 500 MHz is $\lambda =0.6m$ ($c_{mean}/\lambda =11$), which belonged to the high-frequency optics region [10]. Then, the optimization problem formulation would be described as:

$$\begin{array}{l}Min: F=C_D/C_{D\_baseline}+RCS/RC{S}_{baseline}\\ s.t.: C_L=0.3\\ C_M\ge C_{M\_ori}\\ t_n\ge t_{n\_low},n=1,2,\dots ,5\end{array}$$

(15)

Table 11. Computational results of all configurations.

Configuration	Alpha	CL	CD	CM	Yaw (−60–60°) dBSm	Pitch (−45–45°) dBSm
BaseModel	1.38°	0.30	0.02795	−0.042	0.3755	−3.4616
GlobalOpt	1.97°	0.30	0.02071	−0.017	−11.2782	−22.3780
Global/LocalOpt	2.41°	0.30	0.01720	−0.017	−13.7520	−23.8039

tablets the computational results of all design models, and it is shown that the aerodynamic and stealth performance of the GlobalOpt layout has furtherly improved by the local optimization. The drag coefficient of the GlobalOpt was reduced by 35.1 counts, with the pitch moment coefficient maintained. The average value of RCS in pitch direction decreased from −22.378 dBSm to −23.8039 dBSm, with a decrement of 1.4259 dBSm, and in the yaw direction it reaches −13.7520 dBSm with a decrement of 2.4738 dBSm.

From Figure 39, it was presented that the low-pressure regions on the outboard wing surface extended to the inboard wing, and the shock wave intensity on the outboard wing are significantly reduced. In addition, the shock waves on the Kink and Tip zones are almost eliminated completely. This is because that the upper surface of the sectional shapes of the Global/LocalOpt at Y = 4.72 m (the Kink zone) and Y = 8.27 m (the Tip zone) tends to be flatter, and the camber on the trailing edge increases, leading it similar to the supercritical airfoils, which is beneficial for weakening the shock waves, as shown in Figure 40. Moreover, the leading-edge radius of the sectional shapes at Y = 0.00 m and Y = 2.33 m became shaper, which is beneficial for improving the stealth performance on the inboard wing and reducing the RCS at the peak angle around $\varphi { = 26}^{\circ }$ in the yaw direction and at the frontal sector in pitch direction, as shown in Figure 41 and Figure 42. The leading-edge radius of the sectional shape at the Y = 9.46 m is also decreased, resulting in a little diminution at the peak angle around $\varphi { = 58}^{\circ }$ in the yaw direction.

Figure 41, Figure 42, Figure 43 and Figure 44 present the aerodynamic performance comparisons of the optimized layouts. It is shown that the lift characteristics of the GlobalOpt and Global/LocalOpt have been degraded, with a noticeable kink in the lift curve at 5 degrees, compared to the BaseModel. This is because, in order to improve stealth and drag characteristics, the leading-edge radius has been reduced across various cross-sections of the GlobalOpt and Global/LocalOpt, resulting in flatter upper surfaces and decreased camber. The drag characteristics have been significantly improved, as shown in Figure 42. Within the range of 2° to 8°, the GlobalOpt exhibits better drag characteristics compared to the Global/LocalOpt. As illustrated in Figure 43, the longitudinal moment trim characteristics of the GlobalOpt and Global/LocalOpt are notably enhanced compared to the BaseModel; however, there is a slight loss in terms of the moment breakpoint angle of attack. From the lift-to-drag ratio comparison, it can be seen that the GlobalOpt and Global/LocalOpt achieved a significant improvement in lift-to-drag ratio between 2° and 6°. However, the lift-to-drag characteristics within the range of 0° to 2° were slightly degraded. This is because, after optimization, the cruising angle of attack increased from 1.38° to 2.41°, resulting in a substantial improvement in lift-to-drag ratio near the cruising point, as shown in Figure 45 and Figure 46.

By comparing the above calculation results with those in the literature [40], it could be seen that improvements in stealth characteristics were mainly attributed to the reduction of the leading-edge radius in the optimized layout, leading to a loss of maximum lift coefficient and stall characteristics. On the other hand, reducing the bend of each section in the design layout will enhance drag characteristics; however, this also results in a loss of lift curve slope, causing an increase in cruising angle of attack and a reduction in lift-to-drag ratio at small angles of attack.

From the above discussions, it can be observed that improvements in drag and stealth characteristics for the optimized shape result in a loss of lift characteristics, while also causing a forward shift in the moment breakpoint. This demonstrates the inherent conflicts between different design objectives.

5. Conclusions

The flying wing design should take aerodynamic, stealth, and trimming requirements into consideration. It needs a broad design space to obtain the ideal results, which would bring challenges for traditional optimization methods. In addition, the coupling of multiple disciplines would lead to high multimodality, furtherly exacerbating the complexity of the problem. To address these issues, a global/local coupling aerodynamic/stealth optimization design method is proposed in this study, which could achieve high efficiency and convergence optimization, and give full play to the advantages of numerical optimization design. Then, it is verified through the airfoil and flying wing aero/stealth design optimizations. The conclusions of this article are summarized as follows:

(1)

There are multiple local minima in aerodynamic and stealth objectives, respectively, and the coupled design would furtherly aggravate the complexity of the design optimization. In addition, the geometric constraints could reduce the feasible region, and the aerodynamic constraint, like pitch moment, would not only lead to discontinuous feasible regions, but also make the global optima exist on the boundary of the feasible region, bringing difficulty for traditional optimization method in searching for the global optima.

(2)

To address these issues, a Two-Stage Global–Local Constrained Optimization Method (TGLCOM) based on space decomposition is proposed in this study. It divides the board design space into a large-scale global space constructed by few design variables with larger size interval, where the surrogate-based global optimization method is adopted to search for the promising local minima, and a high-dimensional local space based on large number of design variables with small range, where the gradient-based local optimization method would be employed to obtain the global optima. In addition, a Non-Parametric Adaptive Penalty method (NPAP) is proposed to deal with multiple constraints in aerodynamic/stealth optimizations, and the effectiveness of the proposed method is verified by a flying wing airfoil and configuration aero/stealth optimizations.

(3)

The airfoil aerodynamic/stealth optimization result demonstrates that the positive moment of the airfoil relies on the positive loading on the leading edge and negative loading on the trailing edge. However, the RCS objective requires a smaller leading-edge radius, decreasing the positive loading. To fulfill the moment constraint, the negative loading should be extended, which would damage the drag divergence performance. Thus, the conflict between the aerodynamic, stealth, and trimming requirements exists in airfoil design.

(4)

The aerodynamic/stealth shape optimization result demonstrates that the airfoil design could significantly improve the aerodynamic, pitch moment trimming, and stealth performance of the three-dimensional configuration. The shock wave on the inboard wing was completely eliminated, and on the outboard is largely weakened. Especially, the stealth design of the airfoil could decrease the RCS of the configuration both in pitch and yaw direction efficiently. The sectional shape optimization could furtherly reduce the shock waves on the Kink and Tip zones, improving the lift–drag characteristics. In addition, the RCS is evidently reduced in both yaw and pitch directions, especially for the RCS peak at around $\varphi { = 58}^{\circ }$ corresponding to the leading edge of the inboard wing.

(5)

It is shown that the improvements in stealth characteristics are mainly attributed to the reduction of the leading-edge radius in the optimized layout, which leads to a loss of maximum lift coefficient and stall characteristics. On the other hand, reducing the camber of sectional shapes in the design layout enhanced drag characteristics; however, this also results in a loss of lift curve slope, causing an increase in cruising angle of attack and a reduction in lift-to-drag ratio at small angles of attack. This demonstrated the inherent conflicts between different design objectives.

In addition, the literature [41] indicates that the ground effect has a significant impact on an aircraft’s aerodynamic characteristics and static stability. Vehicles flying close to the ground experience higher lift and lower induced drag, thus demonstrating better aerodynamic efficiency and static stability. In future work, we will continue to conduct comprehensive aerodynamic and stealth designs considering ground effects in order to achieve even better design outcomes.