Aerodynamic Design Optimization for Flying Wing Gliders Based on the Combination of Artificial Neural Networks and Genetic Algorithms

Abstract

Gliders are engineless aircraft capable of maintaining altitude for extended periods and achieving long ranges. This paper presents an optimal aerodynamic design method for flying wing gliders, leveraging a combination of artificial neural networks (ANNs) as a surrogate model and genetic algorithms (GAs) for optimization. Data for training the ANN is generated using the vortex-lattice method (VLM). The study identifies optimal aerodynamic shapes for two objectives: maximum flight endurance and maximum range. A key finding is the inherent conflict between aerodynamic performance and static stability in tailless designs. By introducing a stability constraint via a penalty function, we successfully generate stable and high-performance configurations. For instance, the stabilized RG15 airfoil design achieves a maximum glide ratio of 24.1 with a robust 5.1% static margin. This represents a calculated 11.5% performance reduction compared to its unstable theoretical optimum, quantitatively demonstrating the crucial trade-off between stability and performance. The methodology provides a computationally efficient path to designing practical, high-performance, and inherently stable flying wing gliders.

1. Introduction

One of the significant advantages of gliders is that they do not use engines, thereby considerably reducing takeoff weight compared to other aircraft. This advantage has made gliders become popular with many applications. However, to achieve maximum operational efficiency, gliders must be designed to suit the specific characteristics of each mission. A crucial problem in the design phase is finding the optimal aerodynamic shape for the glider. The optimal glider shape must satisfy certain mission criteria, such as achieving maximum range or maximum flight endurance under given flight conditions.

In practice, aerodynamic optimization problems have been addressed for various types of aircraft. Lyu and Martins [1] used CFD (computational fluid dynamics) simulations and gradient-based optimization methods for blended wing body aircraft. The CFD method used in Lyu and Martins’ research offers high accuracy as it directly solves the Navier–Stokes equations, but it requires significant computation time and resources.

To overcome this computational bottleneck, the field has increasingly relied on surrogate-based optimization (SBO), which replaces the expensive high-fidelity model with a cheap-to-evaluate approximation. A comprehensive overview of these techniques can be found in several key review articles. Among the most popular surrogate models is Kriging (or Gaussian Process Regression) [2,3,4], which is highly valued for its ability to provide not only a prediction but also a measure of prediction uncertainty, enabling intelligent and efficient global optimization strategies. Other prominent methods include Radial Basis Functions (RBFs) [5,6,7], valued for their conceptual simplicity and effectiveness, and Polynomial Response Surfaces [8,9,10]. More recently, machine learning techniques have gained significant traction. Artificial neural networks (ANNs) [11,12,13], particularly deep neural networks, excel at capturing complex, highly non-linear relationships in high-dimensional design spaces. The choice of optimization algorithm is equally crucial. Gradient-based algorithms are highly efficient for problems with smooth, convex design spaces, but are susceptible to converging to the nearest local minimum in the multi-modal landscapes typical of aerodynamics. In contrast, global, population-based methods such as genetic algorithms (GAs) [14,15,16] and Particle Swarm Optimization (PSO) [16,17,18] are exceptionally robust at exploring the entire design space. They are particularly adept at handling multi-objective problems.

Despite the maturity of these individual components, a significant research gap exists in their integrated application to complex problems like flying wing design, where aerodynamic efficiency and longitudinal static stability are fundamentally conflicting objectives. The necessity of this study, therefore, arises from the need for an efficient computational framework that can navigate this trade-off effectively. The novelty of our work lies in the specific combination of a fast-evaluating ANN surrogate, trained on VLM data, with a robust GA optimizer to efficiently solve a multi-objective design problem for flying wings, explicitly considering this crucial stability–performance conflict. Within the scope of this paper, we investigate the aerodynamic shape optimization problem for flying wing gliders using GAs and ANN. Training data for the ANNs are obtained from simulation results based on the VLM. The use of ANN can significantly reduce the program’s computation time. In this paper, we assume that the glider is released from a predetermined altitude and gradually descends to the ground. The glider’s aerodynamic shape is optimized for two scenarios: requiring maximum flight endurance and requiring maximum range.

2. Methodology

2.1. Equations of Motion for a Flying Wing Glider

A flying wing glider is an aircraft without tails; thus, all aerodynamic forces are generated by the main wing. Under trimmed flight conditions, the sum of external forces acting on the glider, including weight W, lift L, and drag D, must be zero (Figure 1). By projecting these force components onto the horizontal and vertical axes, we obtain the following system of equations of motion:

$$\left\{\begin{array}{l}D\cos \left(\alpha -\theta \right)=L\sin \left(\alpha -\theta \right)\\ W=L\cos \left(\alpha -\theta \right)+D\sin \left(\alpha -\theta \right)\end{array}\right.$$

(1)

If we assume that the angle of attack α and the pitch angle θ are small, Equation (1) can be simplified to the following:

$$\left\{\begin{array}{l}D=L\left(\alpha -\theta \right)\\ W=L\end{array}\right.$$

(2)

2.2. Method for Determining Aerodynamic Forces Based on ANNs

2.2.1. Building the ANN

As presented in Section 1, there are currently many methods for calculating aerodynamic forces acting on aircraft with varying degrees of accuracy. However, for optimization problems, many calculations must be performed to find the most suitable solution. Therefore, the method for determining aerodynamic forces must have short computation time while still ensuring accuracy. The primary motivation for using an ANN is the significant reduction in computational time during the optimization loop.

Table 1. Comparison of computational costs.

Task	Method	Time
Generation of a 3000-point training dataset	VLM	30 min
ANN Training	Levenberg–Marquardt	5 min
Single Optimization Run (10,000 individuals)	ANN + GA	2 min
Estimated Single Optimization Run (same parameters)	VLM + GA	3 h

provides a quantitative comparison of the computational costs associated with the direct VLM approach versus the proposed ANN-based surrogate approach. All timings were performed on a standard desktop computer (Intel Core i7, 16 GB RAM).

As the Table 1 illustrates, the surrogate-based optimization is orders of magnitude faster than a direct-coupling approach, making the design exploration feasible.

In this paper, an ANN is built to predict the lift coefficient C_L and the induced drag coefficient C_Di acting on the glider with given input geometric parameters. The component of viscous drag is determined later based on an empirical formula. The diagram of the artificial neural network is depicted in Figure 2 with three layers: the input layer (containing input parameters i), the hidden layer (containing neurons z), and the output layer (containing output parameters o). In this study, the hidden layer consists of 70 neurons.

Additionally, the diagram includes bias terms b, which act as supplementary neurons and are not connected to the preceding layers. The use of bias terms allows us to flexibly shift the position of the activation function in the neurons to the right or left, thereby potentially increasing the efficiency of the network training process. Sigmoid and linear activation functions are used in the hidden and output layers. The connections between the input layer and the hidden layer have weights W(i), while the set of weights W(o) is used to connect the hidden layer and the output layer. Detailed descriptions of artificial neural networks are presented in reference [19,20].

Based on the characteristic shape of a flying wing glider, six input parameters for the neural network are introduced: root chord cᵣ, half wing length b, taper ratio t (t = c_t/c_r), sweep angle Λ, angle of attack α, and dihedral angle γ (Figure 3). The output parameters include the lift coefficient C_L and the induced drag coefficient C_Di.

2.2.2. Training the ANN

Before the ANN can be used for prediction, it must be trained on a pre-existing dataset. The total dataset, comprising 3000 unique wing geometries and corresponding aerodynamic coefficients, was carefully partitioned into two distinct subsets: a training set (90% of the data) and a testing set (10%). The training set is used to adjust the network’s weights and biases via the Levenberg–Marquardt algorithm [19,20]. The validation set is used during the training process to monitor performance and prevent overfitting; training is stopped early if the error on the validation set begins to increase.

The hyperparameters of the network were determined through a systematic process. A single hidden layer was chosen for simplicity and efficiency. A sensitivity analysis, the results of which are not detailed here for brevity, was performed to select the number of hidden neurons; architectures with 50, 70, and 90 neurons were tested, and 70 neurons were found to provide the optimal balance between accuracy on the validation set and model complexity. Furthermore, a sigmoid activation function was used for the hidden layer to capture the non-linear aerodynamic relationships, which is standard practice for regression tasks where the output is a continuous value. The final performance of the trained model was then assessed using the unseen testing set to provide an unbiased evaluation of its generalization capability.

The linear VLM is employed to obtain the ANN training data [21,22]. The VLM was configured with a mesh resolution of 15 chordwise and 20 spanwise panels. A uniform spacing distribution was used in both directions. To establish the validity of the VLM code used to generate the training data, a validation case was performed against well-established experimental data conducted by Selvanose et. al. [23].

The numerical results were compared against the experimental data for the wing with NACA2412 airfoil at a Reynolds number of 1.5 × 10⁵ [23]. Figure 4 presents this comparison, showing the lift coefficient C_L as a function of the angle of attack α and the drag polar C_D vs. C_L.

As shown in the left panel of Figure 4, the lift curve predicted by our model shows excellent agreement with the experimental data within the linear, pre-stall region. This confirms that the VLM accurately captures the potential-flow lift characteristics. The right panel shows that the drag polar, which combines the VLM’s induced drag and the empirical skin friction drag as presented in Equation (4) in Section 2.3.2, also agrees well with the experimental measurements. The deviations at higher positive and negative lift coefficients are expected, as our simplified model does not account for complex viscous phenomena like the onset of flow separation, which become more prominent away from the zero-lift condition. Overall, this strong agreement with experimental data validates our aerodynamic modeling approach and provides a high degree of confidence in the quality of the dataset used for training the ANN.

The training data for the ANN are obtained from the VLM with 2700 different calculation cases corresponding to 3000 sets of input and output parameters. After being trained, the ANN is used to predict the lift and induced drag coefficients for the next 300 cases to verify the reliability of the method (Figure 5). The calculations here correspond to a glider with an RG15 airfoil. The comparison results between the predicted data by the ANN and the numerical data determined by the VLM have affirmed the reliability of the ANN used in this study.

In addition, key error metrics were calculated on the unseen test set to provide a more quantitative assessment of the ANN’s accuracy. The results are summarized in

Table 2. ANN performance metrics on the test set.

Output Coefficient	Mean Absolute Error (MAE)	Mean Squared Error (MSE)	R-Squared (R2)
Lift Coeff. (CL)	0.0021	6.5 × 10−6	0.993
Induced Drag Coeff. (CDi)	0.00015	3.1 × 10−8	0.990

.

The high R² values and low error metrics confirm that the ANN surrogate model predicts the aerodynamic coefficients with a high degree of accuracy.

2.3. Optimization Using Genetic Algorithm

2.3.1. Genetic Algorithm Flowchart

A genetic algorithm (GA) was chosen for the optimization task for several key reasons. Firstly, the aerodynamic design space is often highly complex and non-convex, with many local optima. As a global, population-based search method, GA is less susceptible to getting trapped in these local minima compared to traditional gradient-based optimizers [24]. Secondly, GA is a derivative-free method, which adds robustness to the process as it does not rely on the potentially noisy gradients of the ANN surrogate. Finally, this framework offers flexibility for future extensions, such as the inclusion of discrete design variables. The flowchart describing the genetic algorithm is shown in Figure 6. First, a set of individuals of the initial generation is created, where each individual has a genetic code, and within this genetic code, there are chromosomes corresponding to the optimization parameters. In this paper, the set of selected optimization parameters includes root chord c_r, half wing length b, taper ratio t, sweep angle Λ, angle of attack α, dihedral angle γ, and velocity V. Next, the value of the objective function corresponding to each individual is determined. The objective function is constructed based on the optimization goal of the problem. From the second iteration, the program checks the stopping condition. The program stops when the change in the objective function between two consecutive iterations is sufficiently small. In other words, the objective function has converged to an optimal value. The GA was configured with the following parameters: a population size of 10,000 individuals, a crossover rate of 0.8, and a mutation rate of 0.05. The creation of new individuals is performed through crossover and mutation operations. For crossover, offspring individuals inherit chromosomes from one of the two parent individuals. Mutation creates offspring individuals with chromosomes that do not belong to the parent individuals. Mutated offspring are created to avoid local optima. In addition to crossover and mutation, some of the best individuals (5% of the population) pass on their genes intact to the next generation. The optimization was run for a maximum of 200 generations, with an additional convergence criterion that terminated the process if the average change in the objective function over 20 consecutive generations was less than 10⁻⁶.

2.3.2. The Objective Function

From Figure 1, we see that for the range optimization problem (finding the maximum range) of the glider, we need to ensure that the difference between the angle of attack and the pitch angle α − θ is minimized. For the flight endurance optimization problem (finding the maximum flight time), we then minimize (α − θ) × V. From Equation (1), we have α − θ = D/L = C_D/C_L. In addition, the optimal result must satisfy the condition L = W. Thus, the objective function can be written as follows:

$$\begin{array}{l}{J}_1={\displaystyle \frac{C_D}{C_L}}+K\times \left|L-W\right|\\ J_2={\displaystyle \frac{C_D}{C_L}}\times V+K\times \left|L-W\right|\end{array}$$

(3)

where J₁ is the objective function for the range optimization problem, and J₂ is the objective function for the flight endurance optimization. $K\times \left|L-W\right|$ is the penalty function with the coefficient K chosen as 0.01 and 0.1 for the flight range and flight endurance optimization problems, respectively. Using a penalty function can transform the constrained optimization problem into an unconstrained optimization problem. In the case of constrained optimization with static stability, the objective function now includes a penalty term, which utilizes the Heaviside step function H to activate only when the static margin falls below a desired positive threshold of 5% MAC, and the penalty coefficient K₂ was set to 1 in this study.

$$\begin{array}{l}{J}_1={\displaystyle \frac{C_D}{C_L}}+K_1\times \left|L-W\right|+K_{2}H\left(\left(X_{np}-X_{cp}\right)+0.05\right)\\ J_2={\displaystyle \frac{C_D}{C_L}}\times V+K_1\times \left|L-W\right|+K_{2}H\left(\left(X_{np}-X_{cp}\right)+0.05\right)\end{array}$$

(4)

For the flight endurance optimization problem, the goal is to maximize the time the glider can stay in the air, which is equivalent to minimizing its sink rate. The sink rate is given by $V_s=V\times \sin \left(\gamma \right)$, where $\gamma$ is the flight path angle. For small angles, $\sin \left(\gamma \right)\approx \tan \left(\gamma \right)={\displaystyle \raisebox{1ex}{$D$}\!\left/ \!\raisebox{-1ex}{$L$}\right.}$. Therefore, minimizing the sink rate is equivalent to minimizing the product $V\times {\displaystyle \frac{D}{L}}$. Since L = W must be satisfied, this simplifies to minimizing $V\times {\displaystyle \frac{C_D}{C_L}}$. This directly leads to the objective function J₂.

The lift coefficient C_L is determined directly through the ANN, while the drag coefficient C_D is the sum of the induced drag coefficient C_Di and the viscous drag coefficient C_D0. C_Di is obtained from the ANN, C_D0 is calculated according to the Prandtl–von Karman Formula [25]:

$$C_{D_0}={\displaystyle \frac{2\times 0.074}{{\mathrm{Re}}^{0.2}}}$$

(5)

where Re is the Reynolds number determined according to the mean aerodynamic chord of the glider wing. Lift L is determined by L = 0.5ρV²SCʟ, where ρ is the air density, and S is the wing area.

3. Results

3.1. Optimization for Maximum Range

As presented above, optimizing for maximum range involves determining the parameters such that the objective function J₁ reaches its minimum value. In this case, we use an RG15 airfoil, which is widely used in glider design [26]. When using the genetic algorithm, the program creates 10,000 individuals for each generation, corresponding to 10,000 calculation cases. The parameter limits are shown in

Table 3. Parameters and their limits.

Parameters	Meaning	Lower Bound	Upper Bound
c	Root chord length (m)	0.15	0.5
b	Half wing length (m)	0.3	0.75
t	Taper ratio	0.01	1.0
Λ	Sweep angle (°)	0	45.0
α	Angle of attack (°)	1.0	5.0
γ	Dihedral angle (°)	0	20.0
V	Velocity (m/s)	5.0	50.0

.

Table 4. Optimization results for the maximum range at different altitudes.

Parameters	h = 0 km	h = 2 km	h = 4 km	h = 6 km	h = 8 km	h = 10 km
c (m)	0.15	0.15	0.15	0.15	0.15	0.15
b (m)	0.75	0.75	0.75	0.75	0.75	0.75
t	0.14	0.13	0.12	0.13	0.10	0.13
Λ (°)	11.0	9.8	12.0	10.5	11.1	11.2
α (°)	5.0	5.0	5.0	5.0	5.0	5.0
γ (°)	0.1	0	0	0.1	0.3	0.1
V (m/s)	13.1	14.1	16.1	17.9	20.3	22.6
Δl/Δh	35.4	35.0	34.7	34.4	34.0	33.6

presents the optimization results for the maximum range objective when the glider mass is fixed at 1.0 kg and the altitude varies from 0 to 10 km. The data in

Table 5. Optimization results for the maximum range with different masses.

Parameters	m = 0.5 kg	m = 1.0 kg	m = 2.0 kg	m = 3.0 kg	m = 4.0 kg
c (m)	0.15	0.15	0.15	0.15	0.15
b (m)	0.75	0.75	0.75	0.75	0.75
t	0.12	0.12	0.14	0.10	0.13
Λ (°)	11.1	12.0	11.3	12.6	12.4
α (°)	5.0	5.0	5.0	5.0	5.0
γ (°)	0	0	0.2	0.1	0.1
V (m/s)	11.4	16.1	22.5	28.2	32.2
Δl/Δh	32.9	34.7	36.5	37.7	38.6

corresponds to an altitude of 4 km and glider mass varying from 0.5 to 4.0 kg. To evaluate the results, the ratio between flight range and altitude drop, Δl/Δh, is used. For a glider in steady, unpowered flight, this glide ratio (Δl/Δh) is equivalent to the lift-to-drag ratio (L/D) of the aircraft. The geometric parameters do not change with altitude or mass. To achieve the optimal shape for maximum range, the glider must have a root chord equal to the lower limit and a wingspan equal to the upper limit. In addition, the glider with the optimal shape has no dihedral angle; the angle of attack reaches the maximum allowable value (5 degrees). The angle of attack is kept no greater than 5 degrees to ensure that flow separation does not occur during flight. The glider’s velocity increases with altitude as well as mass. This is because at high altitudes, the air density is lower, so the glider is forced to increase its flight speed to ensure sufficient lift. Similarly, for larger masses, the glider must also fly at a higher speed to generate lift equal to its weight. As altitude increases, the Δl/Δh ratio decreases, while as mass increases, this ratio increases.

3.2. Optimization for Maximum Flight Endurance

For the glider to have maximum flight endurance, the objective function J₂ must reach its minimum value. Similar to the previous case, the optimization results for the maximum flight endurance objective are also presented when changing the altitude and mass of the glider (

Table 6. Optimization results for the maximum flight endurance at different altitudes.

Parameters	h = 0 km	h = 2 km	h = 4 km	h = 6 km	h = 8 km	h = 10 km
c (m)	0.33	0.33	0.31	0.32	0.33	0.33
b (m)	0.75	0.75	0.75	0.75	0.75	0.75
t	0.34	0.35	0.36	0.36	0.37	0.36
Λ (°)	13.5	13.0	12.3	12.0	12.0	12.6
α (°)	5.0	5.0	5.0	5.0	5.0	5.0
γ (°)	0	0.1	0.2	0.1	0.1	0.2
V (m/s)	8.9	9.8	11.0	12.1	13.4	15.2
ΔT/Δh (h/km)	0.85	0.77	0.69	0.61	0.54	0.47

and

Table 7. Optimization results for the maximum flight endurance with different masses.

Parameters	m = 0.5 kg	m = 1.0 kg	m = 2.0 kg	m = 3.0 kg	m = 4.0 kg
c (m)	0.33	0.31	0.32	0.32	0.31
b (m)	0.75	0.75	0.75	0.75	0.75
t	0.36	0.36	0.33	0.33	0.35
Λ (°)	12.4	12.3	13.0	13.2	12.9
α (°)	5.0	5.0	5.0	5.0	5.0
γ (°)	0.1	0.2	0	0	0.3
V (m/s)	7.6	11.0	15.6	19.0	21.9
ΔT/Δh (h/km)	0.93	0.69	0.51	0.42	0.37

). To evaluate the results, we calculate the ratio of flight time to the corresponding altitude drop, ΔΤ/Δh. This endurance parameter (ΔΤ/Δh) is inversely proportional to the glider’s sink rate. Specifically, ΔΤ/Δh can be expressed as (L/D)/(V×k), where L/D is the lift-to-drag ratio as mentioned previously, V is the flight velocity, and k = 3.6 is a conversion factor (where 1 s/m is equivalent to 1/3.6 h/km). Similar to the previous case, the optimal geometric parameters for the maximum flight endurance objective of the glider do not depend on the altitude or mass of the glider. The glider’s velocity increases with altitude and mass. The ΔΤ/Δh ratio decreases with increasing altitude and glider mass.

3.3. Evaluation of Results

First, to confirm that the results presented in Section 3.1 and Section 3.2, which were based on the RG15 airfoil, are indeed optimal, we compare the calculated ratios Δl/Δh and ΔΤ/Δh for 30 random cases generated by geometric parameters within the limits specified in Table 1 with the optimal results as illustrated in Figure 7. These calculations were performed for a glider with a mass of 1.0 kg at an altitude of 4 km. It can be observed that the range and flight endurance of these 30 random cases do not exceed the optimal results. This indicates that the obtained optimal results are reliable.

3.4. Comparison of Optimal Results for Different Airfoils at 5000 m Altitude

3.4.1. Optimization for Maximum Range for Different Airfoils

Table 8. Optimization results for the maximum range for different airfoils.

Parameters	NACA0012	NACA2412	RG15	MH78	GOE765
c (m)	0.15	0.15	0.15	0.15	0.15
b (m)	0.75	0.75	0.75	0.75	0.75
t	0.03	0.07	0.08	0.03	0.04
Λ (°)	7.8	15.6	14.7	7.5	7.6
α (°)	5.0	5.0	5.0	5.0	5.0
γ (°)	1.0	2.9	1.7	0.3	1.0
V (m/s)	22.2	18.1	17.5	22.4	23.5
Δl/Δh	23.7	26.9	27.3	23.2	23.5
Xnp-Xcp (%MAC)	0	−8.4	−8.2	9.9	11.0

presents the optimization results for maximizing the flight range of gliders designed with five different airfoils: NACA0012, NACA2412, RG15, MH78, and GOE765. The primary objective is to maximize the glide ratio Δl/Δh. Regarding range performance, the RG15 airfoil stands out as the top performer, achieving the highest glide ratio of 27.3, closely followed by the NACA2412 with a ratio of 26.9. This demonstrates the effectiveness of cambered airfoils such as RG15 and NACA2412 for high-efficiency gliding applications. Conversely, reflexed airfoils MH78 and GOE765 show the lowest performance with glide ratios of just 23.2 and 23.5, respectively. Symmetrical airfoil NACA0012 yields a slightly higher glide ratio of 23.7. These outcomes reinforce the principle that camber generally enhances lift production, leading to a superior lift-to-drag (L/D) ratio.

This analysis also provides the evaluation of longitudinal static stability, represented by the static margin (Xnp-Xcp) and expressed as a percentage of the mean aerodynamic chord (%MAC). A positive static margin indicates a stable design, while a negative value signifies static instability. Table 8 shows the trade-off between aerodynamic performance and static stability. The highest-performing configurations (RG15 and NACA2412) are statically unstable, with significant negative static margins of −8.2% and −8.4%, respectively. This means that while they are optimized for maximum range, they would lack the natural tendency to return to a stable flight path after a disturbance, thus requiring an active control system or constant pilot input to maintain equilibrium.

In contrast, the most stable configurations are the lowest performers in range. The gliders with reflexed airfoils MH78 and GOE765 have large positive static margins of 9.9% and 11.0%, indicating robust self-stabilizing tendencies. This inherent stability comes at the cost of aerodynamic efficiency. The NACA0012 airfoil presents a unique intermediate case, achieving a perfectly neutral static stability (static margin of 0) with a moderate glide ratio.

The other geometric and operational parameters were adjusted to achieve these optimal points for each airfoil. While the root chord (c = 0.15 m) and the half wing length (b = 0.75 m) remain largely consistent, parameters such as taper ratio (t), sweep angle (Λ), and dihedral angle (γ) show some differences. The optimization favors significant sweep angles for the unstable configurations; the RG15 has a sweep of 14.7°, and the NACA2412 has a sweep of 15.6°. On the other hand, the neutrally stable NACA0012 has a sweep of 7.8°, and stable configurations with airfoils MH78 and GOE 765 have sweeps of 7.5° and 7.6°, respectively. The optimization results showed a low sensitivity to the dihedral angle. The optimal configurations consistently converged to dihedral values approaching zero, indicating it is not a primary parameter in this case. The taper ratio is highest for the highly swept RG15 and NACA2412 configurations, with values of 0.08 and 0.07, respectively, whereas the MH78 and GOE765 have much lower values of 0.03 and 0.04. The angle of attack is consistently set to its upper limit of 5.0 degrees for all configurations, suggesting this is the optimal angle for achieving the maximum L/D ratio under the given constraints. The optimal flight velocity also differs, ranging from 17.5 m/s for the RG15 to 23.5 m/s for the GOE765, reflecting the distinct aerodynamic characteristics of each airfoil.

The results in Table 8 illustrate the core design conflict between maximizing performance and static stability. The choice of an airfoil and its corresponding design would depend entirely on the design objective: prioritize pure aerodynamic efficiency, which may need complex control systems, or enhance stability at the expense of flight range. Figure 8 illustrates the optimal configurations with various airfoil selections.

The consistent convergence to the upper bound of the angle of attack is noteworthy. It is hypothesized that within this pre-stall flight regime, increasing the angle of attack is the most efficient way for the optimizer to achieve the required lift (to satisfy the L = W constraint) while maximizing the lift-to-drag ratio. To rigorously verify this hypothesis, the optimization for maximum range (unconstrained by stability, as in Table 8) was re-run for all five airfoils with the angle of attack expanded to [1.0°, 7.0°]. This wider range allows the optimizer to find a potentially unconstrained optimum. The results, summarized in

Table 9. Comparison of optimization results with expanded angle of attack bound.

Airfoil	Case 1: αmax = 5°	Case 2: αmax = 7°
NACA0012	5°	6.1°
NACA2412	5°	6.5°
RG15	5°	6.4°
MH78	5°	5.7°
GOE765	5°	5.8°

, confirm the hypothesis.

As the table clearly shows, by expanding the search space, the optimizer was no longer bounded by the 5.0° limit. Instead, it converged to new optimal angles in the range of 5.7° to 6.5°. This indicates that the true aerodynamic optimum for these configurations lies in this higher angle of attack. Therefore, this analysis confirms that the original 5-degree limit was an active and restrictive constraint that prevented the optimizer from reaching the true maximum. The 5-degree limit was nonetheless retained for the main body of this study to conservatively ensure that all analysis remains well within a flight regime where the attached-flow assumption of the VLM is highly reliable. This limitation defines the scope of our design’s validity. The resulting configurations are optimized for efficient cruising flight but are not intended for mission scenarios requiring high-agility, high-angle-of-attack maneuvers or flight in highly unsteady conditions (e.g., severe atmospheric turbulence). Applying this method to such scenarios would necessitate the integration of higher-fidelity, unsteady aerodynamic models (e.g., RANS or DES) and a reformulation of the optimization objectives to include metrics for dynamic stability and control.

3.4.2. Optimization for Maximum Flight Endurance for Different Airfoils

The results in

Table 10. Optimization results for the maximum flight endurance for different airfoils.

Parameters	NACA0012	NACA2412	RG15	MH78	GOE765
c (m)	0.26	0.29	0.24	0.28	0.38
b (m)	0.75	0.75	0.75	0.75	0.75
t	0.29	0.26	0.28	0.25	0.24
Λ (°)	4.9	13.1	11.1	9.2	7.5
α (°)	5.0	5.0	5.0	5.0	5.0
γ (°)	0	0	0	0	0
V (m/s)	12.6	12.8	13.6	12.0	11.2
ΔT/Δh (h/km)	0.34	0.44	0.45	0.33	0.28
Xnp-Xcp (%MAC)	0	−8.5	−10.1	11.7	14.3

clearly illustrate the trade-off between flight endurance and static stability. The highest-performing configurations, which use the RG15 and NACA2412 airfoils, are statically unstable, with significant negative static margins of −10.1% and −8.5%, respectively. In contrast, the stable configurations with airfoils MH78 and GOE765 are the lowest performers in endurance. The gliders with reflexed airfoils MH78 and GOE765 have large positive static margins of 11.7% and 14.3%, indicating self-stabilizing tendencies. This static stability comes at the cost of aerodynamic efficiency, resulting in shorter flight endurance. The NACA0012 airfoil again presents an intermediate case, achieving neutral static stability with moderate endurance performance.

The other geometric and operational parameters were adjusted to achieve these optimal points for each airfoil. The optimal sweep angle varies significantly across the different airfoil configurations, the NACA2412 has a sweep of 13.1°, and the RG15 has a sweep of 11.1°. On the other hand, the MH78 has a sweep of 9.2°, the GOE765 has a sweep of 7.5°, and the NACA0012 was minimized to just 4.9°. The taper ratios are generally higher, and the angle of attack is consistently set to its 5.0-degree upper limit to maximize the lift coefficient. The optimal flight velocities are also significantly lower, ranging from 11.2 m/s for the GOE765 to 13.6 m/s for the RG15, reflecting the shift in focus from minimum drag to minimum power required. Similar to the flight range case, all optimal configurations converged to dihedral values approaching zero.

The results in Table 10 illustrate the core design conflict between maximizing flight endurance and static stability. The choice of an airfoil and its corresponding design would depend entirely on the mission objective: enhance pure flight endurance, which may need complex control systems, or enhance stability at the expense of flight time. Figure 9 illustrates the optimal endurance configurations with various airfoil selections. This results in wings with a lower aspect ratio than their range-optimized counterparts. The optimization for maximum flight endurance reshapes the glider’s planform, favoring slower speeds and lower aspect ratio wings. However, it also fails to resolve the core conflict between performance and static stability. The designs that can stay in the air the longest are unstable, while the most stable designs have the worst performance.

3.4.3. Optimization for Maximum Range and Endurance Using Wing Twist

The practical solution to this classic trade-off between performance and stability lies in aerodynamic design. To make high-range and high-endurance designs such as the RG15 configuration, an engineer would need to utilize features that restore stability without severely sacrificing performance. The most effective method would be to apply wing twist, which modifies the lift distribution to move the center of pressure rearward, thereby creating a positive static margin.

The results presented in Table 8 and Figure 10 mark a significant change from the previous configurations to more practical designs. By introducing wing twist as a key design variable and refining the optimization’s objective function with a penalty to enforce static stability, we have successfully reconciled the conflict between performance and stability.

The results presented in

Table 11. Optimization results using wing twist for the maximum range for different airfoils.

Parameters	NACA0012	NACA2412	RG15
c (m)	0.15	0.15	0.15
b (m)	0.75	0.75	0.75
t	0.14	0.07	0.08
Λ (°)	27.8	39.1	40.1
α (°)	5.0	5.0	5.0
γ (°)	3.1	6.6	5.5
V (m/s)	22.9	21.1	20.3
Δl/Δh	21.1	23.8	24.1
Twist angle (°)	−1.34	−2.61	−3.0
Xnp-Xcp (%MAC)	5.49	5.70	5.10

and Figure 10 mark a significant change from the previous configurations to more practical designs. By introducing wing twist as a key design variable and refining the optimization’s objective function with a penalty to enforce static stability, we have successfully reconciled the conflict between performance and stability. The primary achievement is the successful stabilization of the configuration with RG15 and NACA2412, which previously exhibited negative static margins, now have positive values of 5.10% and 5.70% MAC, respectively. The neutrally stable NACA0012 is more stable with a 5.49% margin. This change was primarily enabled by utilizing negative wing twist. By reducing the angle of attack at the wingtips relative to the root, the washout effectively shifts the wing’s overall center of pressure rearward, creating the restoring moment necessary for stable flight. The required twist angles, ranging from −1.34° for the NACA0012 to −3.0° for the RG15 airfoil, represent a subtle yet powerful form of aerodynamic design.

It is remarkable to note that this method of achieving stability is functionally identical to the mechanism of reflexed airfoils, such as the MH78 and GOE765 airfoils analyzed previously. In essence, by geometrically twisting a high-lift, conventional airfoil such as the RG15, we are artificially creating a ‘reflexed wing’. The downward-deflected wingtips provide a stabilizing nose-down pitching moment, which precisely mimics the effect of the upward-curved trailing edge of reflexed airfoils. This demonstrates that stability can be engineered either at the 2D reflexed airfoil or the 3D wing level.

However, this gain in stability is associated with an aerodynamic cost; a direct comparison reveals this trade-off quantitatively. The NACA2412 configuration, for instance, saw its glide ratio decrease from a theoretical peak of 26.9 in its unstable state to a stable value of 23.8, a performance reduction of approximately 11.5%. Similarly, the RG15’s glide ratio dropped from 27.3 to 24.1.

Perhaps the most interesting outcome is that achieving this new, stable configuration required a complete geometric re-configuration of the wing planform, not just the addition of twist. The sweep angles for all three designs increased dramatically; for example, the NACA2412’s sweep jumped from 15.6° to 39.1°, while the RG15’s sweep increased from 14.7° to 40.1°, and the NACA0012’s sweep surged from 7.8° to 27.8°.

The results presented in

Table 12. Optimization results using wing twist for the maximum endurance for different airfoils.

Parameters	NACA0012	NACA2412	RG15
c (m)	0.48	0.17	0.15
b (m)	0.75	0.75	0.75
t	0.21	0.25	0.31
Λ (°)	44	37.5	35.7
α (°)	5.0	5.0	5.0
γ (°)	1.8	4.6	3.5
V (m/s)	15.7	18.9	18.7
ΔT/Δh	0.252	0.33	0.344
Twist angle (°)	−2	−3.2	−3.7
Xnp-Xcp (%MAC)	5.10	5.41	5.17

and Figure 11 demonstrate the designs optimized for maximum flight endurance. By again employing wing twist and the objective function’s penalty term to enforce a positive static margin, the conflict between endurance performance and stability has been effectively resolved.

The primary achievement is the successful stabilization of all three configurations. The previously unstable RG15 and NACA2412 designs now exhibit robust positive static margins of 5.17% and 5.41% MAC, respectively. The neutrally stable NACA0012 configuration was also enhanced to a 5.10% margin. This stability was obtained through the application of significant negative wing twist, as mentioned previously in the case of flight range. The required twist angles are even more pronounced in this low-speed regime, ranging from −2.0° for the NACA0012 to −3.7° for the RG15 airfoil.

However, this enforced stability comes at a performance cost. A comparison with the results in Table 12 reveals this trade-off. The NACA2412 configuration, for instance, saw its endurance performance metric decrease from 0.44 to 0.33, a performance reduction of approximately 25%. Similarly, the RG15’s performance dropped from 0.45 to 0.344, and the NACA2412 configuration exhibited a decrease in performance from 0.44 to 0.33.

However, a radical re-configuration of the wing planform was observed. Unlike the range-optimized cases, the root chord varies dramatically, with the NACA0012 adopting a very large chord of 0.48 m to create a low aspect ratio wing, while the RG15 uses a much smaller chord of 0.15 m. The sweep angles remain very high, ranging from 35.7° to 44°. Furthermore, all configurations have significant dihedral angles, which were previously zero. The angle of attack remains fixed at its 5.0° limit, while the optimal flight velocities are very low, confirming the focus on the minimum power required flight condition.

To investigate the influence of the stability penalty term on the optimization outcome, a sensitivity study was conducted on the penalty coefficient K₂. The optimization for the NACA2412 airfoil for maximum range with a stability requirement (static margin ≥ 5% MAC) was repeated for several values of K₂, ranging from a very low penalty K₂ = 0.1 to a very high penalty K₂ = 10. The results are summarized in

Table 13. Sensitivity of NACA2412 optimal design to stability penalty coefficient K₂.

Penalty Coeff. (K2)	Static Margin (%MAC)	Glide Ratio (Δl/Δh)	Sweep Angle (Λ°)	Twist Angle (°)	Performance Change
0 (unconstrained)	−8.4% (unstable)	26.9	15.6°	0° (N/A)	Baseline
0.1 (low penalty)	5.05%	24.5	32.1°	−2.1°	−8.90%
1.0 (this study)	5.70%	23.8	39.1°	−2.61°	−11.50%
10 (high penalty)	6.52%	22.9	44.5°	−3.1°	−14.90%

.

This analysis demonstrates that K₂ is a critical tuning parameter that allows a designer to explicitly control the desired level of stability robustness. A low K₂ will yield a higher-performance design that just meets the minimum stability requirement, while a high K₂ will produce a more robustly stable aircraft at the expense of aerodynamic efficiency. The value of K₂ = 1.0 used in this study was chosen as it represents a good balance, enforcing the stability constraint effectively without imposing an excessive performance penalty.

4. Discussion and Conclusions

This paper has presented a comprehensive aerodynamic shape optimization methodology for flying wing gliders, leveraging a combination of genetic algorithms and artificial neural networks. The study systematically pursued two distinct objectives—maximum flight range and maximum flight endurance; it uncovered and ultimately resolved a fundamental conflict in tailless aircraft design.

The initial investigation confirmed that the optimal glider shape is intrinsically linked to its mission. Configurations optimized for maximum range are characterized by higher aspect ratios and flight speeds to achieve the best lift-to-drag ratio. Conversely, designs optimized for maximum endurance feature larger wing areas and lower flight speeds to minimize the power required for flight. A comparison of various airfoils, including the NACA0012, NACA2412, RG15, MH78, and GOE765, revealed that while cambered airfoils consistently outperformed symmetric ones, a critical issue of static stability was observed.

The core finding of the unconstrained optimization was the conflict between aerodynamic performance and longitudinal static stability. The analysis of the center of pressure and neutral point locations revealed that the airfoils delivering the highest performance, such as the RG15 and NACA2412, consistently resulted in statically unstable configurations. Conversely, airfoils with reflex, such as the MH78 and GOE765, were the only ones to produce stable designs, but this stability came at a significant cost to their aerodynamic efficiency. This presented an engineering problem: the best-performing designs were unstable, and the most flyable designs were underperforming.

To bridge this gap, the research progressed to a more sophisticated optimization method. A stability constraint, implemented as a Heaviside penalty function, was integrated into the objective function to enforce a positive static margin. The primary mechanism for achieving this was the introduction of wing twist as a design variable. This refined approach proved successful, changing the previously unstable, high-performance designs into robustly stable configurations. A key insight from this result is that applying a geometric twist to a conventional airfoil functionally mimics the effect of a reflexed airfoil, effectively creating a “stable wing” from an “unstable wing section.”

However, this stability was achieved at an aerodynamic cost. The stable designs exhibited a performance decrease of approximately 11.5% in maximum range and a more substantial 25% in maximum endurance compared to their unstable theoretical counterparts. Furthermore, the optimizer found that stability was most efficiently achieved through a combination of high wing sweep and twist, leading to dramatic changes in the final wing planforms.

While the aerodynamic modeling approach has been validated against experimental data for a fundamental case, it is important to acknowledge that the final, optimized designs presented in this paper represent theoretical optima based on this model. Consequently, the performance of these specific configurations must be verified using higher-fidelity methods before any practical application. Future work should therefore focus on validating these promising designs through Reynolds-Averaged Navier–Stokes (RANS) computational fluid dynamics simulations and, ultimately, wind tunnel experiments. Such steps are essential to confirm the predicted performance gains and ensure that the flight characteristics are accurately captured before proceeding to a physical prototype.

Beyond the aerodynamic focus of this study, it is vital to recognize that the optimized geometric parameters have significant structural implications. Designs that converge toward a high aspect ratio and large sweep angle, while aerodynamically efficient, will experience higher bending moments at the wing root and are more susceptible to aeroelastic issues such as divergence and flutter. For instance, the designs in Table 8, which feature high sweep angles to achieve static stability, would require significant structural reinforcement to maintain stiffness, thereby increasing the overall weight and potentially offsetting some of the aerodynamic gains.

Our design philosophy follows a sequential optimization approach, which is common in conceptual aircraft design. The core of this approach is the observation that for a given flight condition (altitude and velocity), the optimal aerodynamic shape for metrics like the lift-to-drag ratio is largely independent of the aircraft’s mass. Therefore, the structural problem can be effectively decoupled and addressed after the aerodynamic shape has been optimized. In this framework, the optimal aerodynamic shapes found in this study serve as a high-quality baseline. The subsequent structural design phase would then focus on realizing this shape with the minimum possible mass while satisfying all strength, stiffness, and aeroelastic constraints.

This highlights a key limitation of single-discipline optimization. A logical and necessary extension of this research is the development of a Multidisciplinary Design Optimization (MDO) framework. Such a framework would integrate a structural model (e.g., a finite element model) into the optimization loop, allowing for the simultaneous consideration of aerodynamic efficiency and structural constraints like weight, stress, and aeroelastic stability. This would lead to designs that are not only high-performing but also structurally robust and feasible.

In conclusion, this research went from identifying a fundamental design conflict to implementing and analyzing a more practical engineering solution. It has been demonstrated that, through advanced optimization techniques that incorporate aerodynamic mechanisms, such as wing twist and well-posed stability constraints, it is possible to develop high-performance flying wing designs that are both aerodynamically efficient and inherently stable. Future work should build upon these stable baselines to explore aero-structural optimization, the integration of control surfaces for maneuverability, and experimental validation of these promising designs.