Nomenclature

Definitions/Abbreviations

UAV

unmanned aerial vehicle

ANFIS

adaptive neuro-fuzzy inference system

CFD

computational fluid dynamics

PSO

particle swarm optimisation

ACO

ant colony optimisation

GA

genetic algorithm

m

UAV mass

pitch

propeller pitch

S

wing area

rpm

propeller revolution per minute

∧

wing swept angle

λ

taper ratio of wing

U ₀

UAV velocity

Q

dynamic pressure

C _L0

aircraft reference lift coefficient

AR

wing aspect ratio

b

span

$\bar c$

wing mean aerodynamic chord

C _D0

UAV reference drag coefficient

P _req

power require

T _req

thrust require

D

drag force

L

lift force

V

velocity

C _L

lift coefficient

C _D

drag coefficient

W

UAV weight

ρ

air density

e

Oswald number

2.0 Theoretical background for cfd and optimisation methods

In this section, the design criteria for a UAV with a variable swept angle are briefly described. In Fig. 1, the dimensions of the designed UAV are given, and the morphing mechanism used to change the swept angle of the wing is schematically illustrated. The morphing mechanism consists of one servo motor and two gears placed on the leading edge of each wing. In addition, the wing is supported by bearings mounted on the fuselage, and the morphing is constrained on the horizontal plane.

Figure 1. The concept design of swept wing UAV.

The technical specifications of the designed swept UAV are presented in Table 1.

Table 1. Technical specification of the swept wing UAV

2.1 Numerical method

2.1.1 The grid independence study

Grid independence is a very important parameter, especially in the numerical analysis of computational fluid dynamics (CFD) applications. The network structure parameters, namely the number of cells, or the number of nodes in the analysis, greatly affect the accuracy of the solution. An increased number of nodes or number of cells used lead to more accurate results but at the expense of increased usage of computational resources in the analysis. For this reason, studies should be independent of the network structure. A grid independence study was carried out to determine the optimum mesh number. With the grid independence study, the optimum number of meshes can be solved in the shortest time [Reference Douvi, Tsavalos and Margaris15].

Since time and central processing unit (CPU) power are significant parameters in the analysis process, an optimisation should be performed to find the balance between the accuracy of, and the computational cost of finding, the solution. Therefore, the optimum mesh element number is determined and given in Fig. 2.

Figure 2. The effect of mesh element number on the lift coefficient.

2.1.2 Boundary conditions

While determining the flow area, a distance of 15 $\bar c$ above and below the nose of the aircraft was determined to be the limit. On the trailing edges, a distance of 20 $\bar c$ to the outward was determined as the limit. No-slip boundary conditions have been used on solid surfaces. Figure 3 presents the flow field boundary conditions.

2.1.3 Mesh properties

As seen in Fig. 4, the mesh structure created in the calculation area is intense in the regions close to the aerofoil, while being less intense in regions that are more distal. The mesh number is optimised using the 1.2 growing mesh size from the wing surface to the outer flow region. The reason for the application of such a network structure is to take up less space on the computer on which the numerical analysis is performed, and to aim to conduct the analyses in a shorter time. To monitor the flow events (flow separation bubble, reverse flow, reattachment, turbulent flow, etc.) on the wing surface more clearly, a boundary layer was created with the inflection parameter. Mesh properties and skewness and orthogonal quality mesh metrics spectra are given in Table 2.

Table 2. Mesh properties and skewness and orthogonal quality mesh metrics spectra

Figure 3. The dimensions and boundary conditions of the computational domain.

Figure 4. Mesh inflation image.

2.3 Turbulence models used in aircraft analysis

2.3.1 SST k-ω turbulence model

The shear stress transport (SST) k-ω turbulence model, which is accepted as the turbulence model that gives the best results in external flow aerodynamic applications or in simulations where flow separations are important, is used together with k-ω and k-ε. Among these models, the k-ω model gives better results in regions close to the wall, but it is inadequate in free-flow regions that are more distal. On the other hand, the k-ε turbulence model gives good results in free-flow regions.

(1) \begin{equation}{{\partial \!\left( {{\rm{\rho \omega }}} \right)} \over {\partial t}} + \nabla \!\left( {{\rm{\rho }}V{\rm{\omega }}} \right) = \nabla \!\left( {\left( {{\rm{\mu }} + {{{{\rm{\mu }}_t}} \over {{{\rm{\sigma }}_{\rm{\omega }}}}}} \right){\nabla _{\rm{\omega }}}} \right) + {{\rm{\gamma }} \over {v_t}}{P_k} - {\rm{\beta \rho }}{{\rm{\omega }}^2} + \underbrace {2\!\left( {1 - {F_1}} \right){{{\rm{\rho \sigma }}{{\rm{\omega }}_2}} \over {\rm{\omega }}}{\nabla _k}:{\nabla _{\rm{\omega }}}}_1\end{equation}

The part indicated by “1” in Equation (1) is the additional term. The expression at the end of the additional term is the inner product tensor. The expansion of the inner product tensor is given in Equation (2).

(2) \begin{equation}{\nabla _k}:{\nabla _{\rm{\omega }}} = \frac{{{\partial _k}}}{{{\partial _{xj}}}}\frac{{{\partial _{\rm{\omega }}}}}{{{\partial _{xj}}}} = \frac{{{\partial _k}}}{{{\partial _x}}}\frac{{{\partial _{\rm{\omega }}}}}{{{\partial _x}}} + \frac{{{\partial _k}}}{{{\partial _y}}}\frac{{{\partial _{\rm{\omega }}}}}{{{\partial _y}}} + \frac{{{\partial _k}}}{{{\partial _z}}}\frac{{{\partial _{\rm{\omega }}}}}{{{\partial _z}}}\end{equation}

The SST turbulence model was preferred in the solution for the aerodynamic analysis in this study because it aims to overcome the shortcomings of the standard k-ω model, the dependence of k and ω on the free flow values, and can provide effective solutions for flow separations [Reference Menter19–Reference Menter21].

2.4 Prediction methods

In the literature, there are various modeling approaches used in the modeling and estimation of the dynamical system responses [Reference Sevim, Bilgic, Cansiz, Ozturk and Atis22–Reference Morgado, Vizinho, Silvestre and Páscoa24]. While approaches such as regression and multilinear regression are frequently used to model and estimate linear systems, linear approaches do not give effective results in modeling and estimating nonlinear systems. For this reason, artificial intelligence algorithms have recently been successfully used in the modeling of nonlinear systems [Reference Oktay, Arik, Turkmen, Uzun and Celik25, Reference Bagis and Konar26]. In this section, the modeling approaches used in this study are briefly described.

2.4.1 Multilinear regression method

The Multilinear regression model is used in modeling of linear systems; it is used to estimate a dependent variable with more than one independent variable. It is based on optimising the coefficients of the dependent variable to estimate an independent variable, and can be expressed according to Equation (3)

(3) \begin{equation}y = \beta + {w_1}{x_1} + {w_2}{x_2} + {w_3}{x_3} + \ldots + {w_n}{x_n}\end{equation}

where y, x _i , and w _i express the independent variable, dependent variable, bias variable and the weight of the independent variable, respectively. Although the multilinear regression model generally gives successful results in modeling linear systems, this is less the case in modeling nonlinear systems. To address this issue, many artificial intelligence-based modeling approaches have been proposed in the literature [Reference Bilgic, Guvenc, Çakır and Mistikoglu27–Reference Mert, Bilgic, Yağlı and Koç29].

2.4.2 Adaptive neuro-fuzzy inference systems (ANFIS)

ANFIS, proposed by Jang in 1993, is a modeling approach that combines the parallel computing capabilities of artificial neural networks and the heuristic computing capabilities of fuzzy logic systems. ANFIS can often be trained without specialist knowledge to design a classical fuzzy logic system. An input-output-based dataset is generally needed to apply the ANFIS method. The typical architecture of ANFIS is presented in Figs 5 and 6 [Reference Jang30].

Figure 5. ANFIS structure [Reference Jang30].

Figure 6. The ANFIS premise process [Reference Jang30].

For simplicity, the architecture of ANFIS is presented in Fig. 5 for a two-input one-output system. Figure 6 presents a visual representation of the theoretical model of ANFIS.

Although ANFIS is easier to design than a classical fuzzy logic system, tuning its parameters significantly increases modeling success [Reference Isermann31]. Various algorithms have been proposed in the literature to determine optimum parameters. Differing from classical optimisation approaches, various metaheuristic search algorithms have been proposed in the literature [Reference Shehabeldeen, Abd Elaziz, Elsheikh and Zhou32, Reference Han, Chen, Shao and Wu33].

In this section, the genetic algorithm, artificial colony algorithm, and ant colony algorithms used in the training of ANFIS are briefly discussed. A genetic algorithm, inspired by the science of genetics, is a heuristic algorithm for solving complex problems that cannot be answered with classical optimisation techniques [Reference Holland John34]. Genetic algorithms are frequently used in optimisation processes and their success has been proven in the literature [Reference Mirjalili35]. Particle swarm optimisation (PSO) is a heuristic algorithm inspired by the swarm behaviour of animals when they meet basic needs such as finding food [Reference Eberhart and Kennedy36]. Ant colony optimization (ACO), a heuristic optimisation algorithm, was introduced in 1991 by Dorigo et al. [Reference Dorigo, Maniezzo and Colorni37]. Since the algorithm was introduced, various versions have been developed and applied to different optimisation problems [Reference Song and Zhao38, Reference Chen, Bai, Zhan, Hu and Liu39].

The optimisation parameters are determined by testing traditional functions such as Rastrigin, Ackley, and Goldstein–Price functions. In this study, ACO, Genetic Algorithm (GA) and PSO parameters were chosen by examining the literature [Reference Sevim, Bilgic, Cansiz, Ozturk and Atis22, Reference Rao, Savsani and Vakharia40–Reference Tam, Ong, Ismail, Ang and Khoo43]. Optimisation parameters for metaheuristic algorithms discussed in the study are presented in Table 3.

Table 3. Optimisation parameters of the GA, ACO and PSO algorithms

Figure 7. Schematic diagram of the proposed approach.

3.0 Decision support system design

The decision support system consists of two stages, as presented in Fig. 7. In the first stage, the lift and drag coefficients were numerically computed via CFD methods provided by ANSYS. In CFD analysis, 350 datasets, including the independent variables (velocity, swept angle and angle-of-attack) and the dependent variables (drag and lift forces), were created using 5 velocities, 14 swept angles, and 5 angles of attack to cover all possible flight conditions. The accuracy of the numerical results has been confirmed by the experimental results presented in Refs (Reference Homa, Wróblewski, Majkut and Strozik13, Reference Kheir-aldeen and Hamid14). In the second stage, three nonlinear estimation models were created using the PSO, GA and ACO algorithms in the ANFIS structure, where the datasets obtained in the first stage were utilised as training datasets. In addition, a multilinear regression model was also designed to allow for a comparative analysis to investigate the effectiveness of nonlinear models.

Figure 8 shows that lift force and drag force gradually increase with angle-of-attack at different velocities. Although the lift coefficient tends to decrease with the change in the swept angle, the increase in the swept angle eliminates the possibility of an abrupt stall. Although the increase in the swept angle reduces the total lift force (F) and the lift coefficient, extra lift control is needed for high velocity flight. Drag force (D) also increases with increasing angle-of-attack, but a sudden increase occurs after the stall angle. When the graphs are examined, the increase in the swept angle significantly reduces the variation of the drag force with angle-of-attack. Although the increase in the swept angle causes an increase in the drag coefficient, the use of a variable swept angle will result in a lower drag force at high velocity flight. This means that there will be fuel savings and increased aircraft flight performance as variable swept designs at higher velocities are exposed to reduced levels of drag. Similarly, increasing the swept angle reduces the lift and drag force due to the reduced wing area. However, this reduces the lift force more than the drag force, which has a negative effect on aerodynamic efficiency. The variation of aerodynamic fines (E _max) with different velocities is given in Fig. 9.

Figure 8. Variation of aerodynamic force coefficients according to different velocities and swept angles.

Figure 9. Variation of aerodynamic fines (E _max) according to different velocities.

The Clark-Y aerofoil used in the study was compared with an experimental study in the literature. It was compared with Silvestren’s design, with Re = 365,000 and AR = 7.2, and a design with Re = 300,000 and AR = 10 [Reference Silverstein44]. The results are presented in Fig. 10. When the figure is examined, the aerodynamic analysis shows certain similarities.

Figure 10. Comparison of lift coefficient variation with angle-of-attack.

In this study, the aerofoil of the variable swept aircraft design is Clark-Y. Fuselage and wings were analysed together to determine aerodynamic parameters. Therefore, when the angle-of-attack increases, the hull’s contribution to the carrying force is substantial. The lift and drag forces given in the aerodynamic parameters are for the fuselage and wing form. However, to be able to compare with the literature, the lift and drag coefficients of the plain wing for a velocity of 30m/s were also calculated. The Clark-Y aerofoil achieved its maximum lift coefficient at an approximately 22º angle-of-attack [Reference Kheir-aldeen and Hamid14].

The 350 datasets obtained from the aerodynamic analysis were used as model input to estimate drag and lift force. First, 250 variations of swept angle, angle-of-attack, and velocity values were selected as independent variables to estimate the drag force. The drag force corresponding to each dataset was defined as the dependent variable. First, a multilinear regression approach analysis, which is one of the conventional regression approaches frequently used in the literature, was performed. The multilinear regression equation for drag force is presented in Equation (4).

(4) \begin{equation}{\rm{Drag\ Force}} = - 4.881 + 0.316v - 0.105\Lambda + 0.608\alpha\end{equation}

To demonstrate the model’s success, 100 previously unused datasets were applied to multiple linear regression analysis; model success was evaluated via statistical parameters. In the statistical analysis, the R ², RMSE, and MAE values for the 250 training sets were respectively calculated as 0.694, 3.85, and 2.794. For 100 test data, the R ², RMSE, and MAE values were respectively calculated as 0.659, 4.416, and 3.096.

To increase the model success obtained with the multilinear regression approach, ANFIS models, based on metaheuristic algorithms that have been applied in different areas, have been proposed in the literature. Unlike conventional ANFIS structures, the ANFIS training parameters are optimised using the ACO, GA and PSO algorithms. Training and test responses from ACO-, GA- and PSO-based ANFIS trainings are presented in Fig. 11.

While performing metaheuristic-based ANFIS training, 250 different variations of swept angle, angle-of-attack and velocity values were defined as independent variables to estimate drag force. The drag force corresponding to each dataset was defined as the dependent variable. The purpose of using 100 datasets, which were never used in the training process, for testing is to avoid memorisation of the model and to provide a proper learning process. The statistical parameters calculated for the model responses based on metaheuristic algorithms are presented in Table 4. When Table 4 is examined, the PSO-based ANFIS model apparently shows the best performance with an R ² of 0.983.

Table 4. Statistical results for drag

Figure 11. Drag prediction using the MLRM model and the ACO-/GA-/PSO-based ANFIS models.

Scatter plots are presented in Fig. 12 a, b, c and d that reveal the success of the models obtained within the scope of the study. In the scatter graphs, the x-axis defines the model responses, whilst the y-axis defines the target to be estimated. In the scatter plot, the linear line expresses the change of target data according to target data. The markers represent the change in the target data according to the model output. For this reason, the model’s success is measured according to the linearity of the markers. As can be clearly seen from the scatterplots, the PSO-based ANFIS model predicted the drag force more successfully than all other models.

Figure 12. Scatter graphs for drag predictions (a) MLRM, (b) ACO-based ANFIS, (c) GA-based ANFIS and (d) PSO-based ANFIS.

In the estimation of the lift force, a process similar to the one followed in the estimation of the drag force was followed. First, 150 of the 180 datasets were used to determine the parameters of the multilinear regression model. The model equation for the lift force is presented in Equation (5).

(5) \begin{equation}{\rm{Lift\ Force}} = - 54.805 + 4.367v - 1.691\Lambda + 6.999\alpha \end{equation}

The success of the multilinear regression model in predicting the lift force was demonstrated by evaluating 100 previously unused datasets with Equation (5). In the statistical analysis, the R ², RMSE and MAE values for the 250 training datasets were respectively calculated as 0.723, 48.287 and 35.347. For 100 test data, R ², RMSE and MAE values were respectively calculated as 0.706, 55.364 and 40.937. The results of the multilinear regression model for the training and test data are presented in Fig. 13.

Estimation of lift force was calculated using metaheuristic-based ANFIS models. The model results in which the ANFIS training parameters were optimised with the ACO, GA and PSO algorithms are presented in Fig. 13.

While training the metaheuristic-based ANFIS models in the estimation of lift force, 250 different variations of swept angle, angle-of-attack and velocity values were defined as independent variables, in a similar manner to the training process for the drag force. The lift force corresponding to each data was defined as the dependent variable. One hundred datasets that had never been used in the training process were used for the test. The statistical parameters calculated for ANFIS based on the ACO, GA and PSO models are presented in Table 5. When the table is examined, it is clear that the GA-based ANFIS model has the highest performance with an R ² of 0.995.

Table 5. Statistical results for lift

Figure 13. Lift prediction using the MLRM model and the ACO-/GA-/PSO-based ANFIS models.

In order to demonstrate the success of the models obtained within the scope of the study, scatter plots of the lift force are presented in Fig. 14 a, b, c and d. When the scatter plots are examined, the results of the PSO-based ANFIS model are clustered on the linear line. This indicates that the PSO-based ANFIS model predicts lift force more successfully than all other models.

4.0 Swept angle estimation approach for minimum lift force

The decision support system enables us to estimate the maximum swept angle producing the minimum lift required for the steady-level flight of the morphing UAV. In addition, it also generates the minimum drag force. The application of the decision support system is presented in Fig. 15.

Figure 14. Scatter graphs for lift predictions (a) MLRM, (b) ACO-based ANFIS, (c) GA-based ANFIS and (d) PSO-based ANFIS.

Figure 15. The application of the decision support system.

It is a well-known fact that lower required power leads to higher endurance. The endurance is inversely proportional to the reduction in the required power, and the required power is directly related to the minimum drag force. Thus, the minimum power (maximum endurance) condition occurs when the drag force is at a minimum. Because the morphing mechanism’s adjustment of the swept angle of the wing enables a decrease in drag force, endurance can be maximised.

Therefore, the endurance efficiency can be computed as follows:

The drag force coefficient is given by

(6) \begin{equation}{C_D} = {C_{{D_0}}} + \frac{{{C_L}^2}}{{\pi eAR}}\,{\rm{where}}\,L = \frac{1}{2}\rho {V^2}S{C_L}\;{\rm{and}}\,D = \frac{1}{2}\rho {V^2}S{C_D}\end{equation}

By using Equation (6), the drag force can be computed as follows:

(7) \begin{equation}D = \frac{1}{2}\rho {V^2}S{C_{{D_0}}} + \frac{{{W^2}}}{{\frac{1}{2}\rho {V^2}S}}\left( {\frac{1}{{\pi eAR}}} \right)\end{equation}

For the steady level flight, the required thrust must be equal to the drag force (Equation (7)), i.e. ${T_{req}} = D$ . Therefore, the required power becomes

(8) \begin{equation}{P_{req}} = {T_{req}}V = DV\end{equation}

Finally, substituting Equation (7) into Equation (8) yields:

(9) \begin{equation}{P_{req}} = \frac{1}{2}\rho {V^3}S{C_{{D_0}}} + \frac{{{W^2}}}{{\frac{1}{2}\rho VS}}\left( {\frac{1}{{\pi eAR}}} \right)\end{equation}

which is used to compute the required power.

In the simulation analysis, two different aircraft configurations are considered. In the first configuration, the swept angle was set to zero, whereas the second configuration was built using a swept wing. To assess the effectiveness of the proposed method, 11 datasets composed of various velocities ranging from 15 to 33m/s and angles of attack ranging from 1º to 12º were randomly selected. The swept angles that produce the optimal lift forces for these selected velocities and angles of attack are estimated using the decision support system introduced in Section 3. The estimated lift and drag forces are then compared with the numerical results obtained from the CFD analyses to determine the accuracy of the estimation approach. In addition, the drag force for the aircraft configuration with non-morphing wings was also computed in a CFD analysis. The percentage reduction in drag force was then computed when a swept wing configuration was utilised. The comparative results are presented in Table 6.

Table 6. Comparison of DSS and CFD results

According to the results in Table 4, the lift and drag forces obtained with the estimated swept angle are consistent with those obtained from CDF analysis. The proposed method is able to reduce the drag force significantly for all flight conditions. The maximum reduction is approximately 73%, which is achieved for the flight condition given by a velocity of 29m/s and @@an angle-of-attack of 15.1º. Moreover, the proposed DSS also improves the endurance of the morphing UAV. When a standard lipo battery is used for the designed, electrically powered UAV, the instantaneously required current can be calculated by considering the drag force. Endurance (min) can be calculated for battery capacity and instantaneous current values. The results show that it can increase endurance by varying between 1.23 and 6.26 min.