1 Background

Airship is a lighter-than-air air vehicle guided by its own power system [1] flows in stratosphere layer which has approximately constant air properties [2, 3]. Most airships are unmanned aerial vehicles (UAVs) with vertical take off and landing (VTOL) and solar power propulsion system. They are utilized in various missions like weather forecasting, communications, aerial navigation, earth exploration, remote sensing, traffic monitoring and control, military applications, etc. [4,5,6,7,8]. Therefore, improving the airship performance is demanding. This article discusses a methodology to enhance airship aerodynamic performance by optimizing its shape using MATLAB genetic algorithm toolbox. Genetic algorithm is an optimization technique that simulates the nature selection across the population to produce better populations according to a cost function with selected order of crossover and random occurrence of mutations to avoid local extrema [9,10,11]. This iterative numerical optimization method is used in the process of the airship shape optimization by searching for the optimal geometry that satisfies the cost function. The aerodynamic analysis of the airship hull was developed by Munk [12] based on slender body assumption and potential flow theory, then modified by Allen and Perkins [13] with empirical part to simulate the effect of viscosity on the normal force per unit hull length considering that each local cross section as an infinite-length circular cylinder placed normal to the flow. Also, Hopkins [14] developed a semi-empirical equation to compute the normal force per unit hull length. The solution over the hull body splitted into two parts. The first one is computed by Munk’s model [12], whereas the second one is obtained by Hopkins’ model [14] based on body revolution at low angles of attack. The effect of fins is added by Jones and DeLaurier [15] considering the analysis is divided into two regimes. The first one starts from airship’s nose to the position of fins leading edge and the second is the extended body shape. Although these models were developed for uniform flow, they can achieve stability for large side-slip angle of finned airships [16]. Jones and DeLaurier model [15] was verified to be the best one [17]. The equations of this model depend on two categories of free parameters. The first category is related to the airship hull which are the minor axes (b), the average of the major axes (a) and the location of the leading edge of airship fin \((l_h)\). The second category is related to the airship fin which are fin chord (c), fin span \((b_f)\) and the maximum thickness to chord ratio of fin \((t/c)_{\max}\). Yuwen Li and Meyer Nahon [18] use Hopkins and Finck [14, 19] semi-empirical model to deduce the aerodynamic governing equation in space and verify it with CFD. In this study, the kinetics of aerodynamics in space is developed by Jones and DeLaurier model [15]. Genetic algorithm is utilized to establish the optimal airship shape with minimum drag coefficient [20,21,22,23,24,25,26], whereas the effect of the side wind is considered in the optimization process of this work.

Figure 1 shows a flowchart of the work throughout the article. In the next Sect. 2, the aerodynamic model is derived with considering the effect of side-slip angle \(\beta\). The optimization problem with its parameters, and cost function are also formulated. MATLAB genetic algorithm toolbox is utilized to get the results of Sect. 3. Result comparison and the method of choosing the optimal airship shape are discussed in Sect. 4. The article is concluded in Sect. 5.

Fig. 1
figure 1

Airship optimal shape design flowchart

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2 Methods

2.1 Aerodynamic kinetics

The semi-empirical aerodynamic equations for uniform flow over airship hull and fins were developed by Jones and DeLaurier [15] as shown in Fig. 2. The current model is extended to consider the side flow effect by introducing the side-slip angle \((\beta )\) combined with the angle of attack \((\alpha )\) as presented in Fig. 3. The two flow angles can be obtained from

$$\begin{aligned} \alpha= & {} \tan ^{-1}\left( \dfrac{w}{u}\right) \end{aligned}$$
(1)
$$\begin{aligned} \beta= & {} \sin ^{-1}\left( \dfrac{v}{V_t}\right) \end{aligned}$$
(2)
Fig. 2
figure 2

Schematic of an Airship [15]

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The aerodynamic equations developed by Jones and DeLaurier [15] were introduced at airship nose for uniform flow, so the heading velocity projection is taken in sr-plane, see Fig. 2, to be

$$\begin{aligned} \begin{bmatrix} F_{s,\alpha }\\ F_{r,\alpha }\\ L_{s,\alpha }\\ M_{q,\alpha } \end{bmatrix}=\dfrac{1}{2}\rho V_{t,\alpha }^2 \begin{bmatrix} C_{X,\alpha }\\ C_{Z,\alpha }\\ C_{L,\alpha }\\ C_{M,\alpha } \end{bmatrix}\quad \quad (3) \end{aligned}$$

where

$$\begin{aligned} C_{X,\alpha }= & {} C_{X1}\cos ^2(\alpha ) +C_{X2}\sin (2\alpha )\sin \left( \dfrac{\alpha }{2}\right) \nonumber \\ \end{aligned}$$
(3a)
$$\begin{aligned} C_{Z,\alpha }= & {} C_{Z1}\cos \left( \dfrac{\alpha }{2}\right) \sin (2\alpha )+C_{Z2}\sin (2\alpha )\nonumber \\{} & {} +C_{Z3}\sin (\alpha )\sin (\vert \alpha \vert )\nonumber \\{} & {} +C_{Z4}\left( \delta _{eL}+\delta _{eR}\right) \end{aligned}$$
(3b)
$$\begin{aligned} C_{L,\alpha }= & {} C_{L1}\left( \delta _{eR}-\delta _{eL}\right) \end{aligned}$$
(3c)
$$\begin{aligned} C_{M,\alpha }= & {} C_{M1}\cos \left( \dfrac{\alpha }{2}\right) \sin (2\alpha )+C_{M2}\sin (2\alpha )\nonumber \\{} & {} +C_{M3}\sin (\alpha )\sin (\vert \alpha \vert )\nonumber \\{} & {} +C_{M4}\left( \delta _{eL}+\delta _{eR}\right) \end{aligned}$$
(3d)
Fig. 3
figure 3

Airship aerodynamic axes

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The aerodynamic constants \(C_{X1},\) \(C_{X2},\) \(C_{Z1},\) \(C_{Z2},\) \(C_{Z3},\) \(C_{Z4},\) \(C_{M1},\) \(C_{M2},\) \(C_{M3}\) and \(C_{M4}\) are given by

$$\begin{aligned} C_{X1}= & {} C_{Dh0}S_h+C_{Df0}S_f \end{aligned}$$
(3e)
$$\begin{aligned} C_{X2}= & {} -\left( k_2-k_1\right) \eta _kI_1 \end{aligned}$$
(3f)
$$\begin{aligned} C_{Z1}= & {} -C_{X2} \end{aligned}$$
(3g)
$$\begin{aligned} C_{Z2}= & {} \dfrac{1}{2}\left( \dfrac{\partial C_l}{\partial \alpha }\right) _fS_f\eta _f \end{aligned}$$
(3h)
$$\begin{aligned} C_{Z3}= & {} C_{Dch}J_1+C_{Dcf}S_f \end{aligned}$$
(3i)
$$\begin{aligned} C_{Z4}= & {} \left( \dfrac{\partial C_l}{\partial \delta }\right) _fS_f\eta _f \end{aligned}$$
(3j)
$$\begin{aligned} C_{L1}= & {} \left( \dfrac{\partial C_l}{\partial \delta }\right) _fS_f\eta _fl_{f3} \end{aligned}$$
(3k)
$$\begin{aligned} C_{M1}= & {} -\left( k_2-k_1\right) \eta _kI_3 \end{aligned}$$
(3l)
$$\begin{aligned} C_{M2}= & {} -\dfrac{1}{2}\left( \dfrac{\partial C_l}{\partial \alpha }\right) _fS_f\eta _fl_{f1} \end{aligned}$$
(3m)
$$\begin{aligned} C_{M3}= & {} -\left( C_{Dch}J_2+C_{Dcf}S_fl_{f2}\right) \end{aligned}$$
(3n)
$$\begin{aligned} C_{M4}= & {} -\left( \dfrac{\partial C_l}{\partial \delta }\right) _fS_f\eta _fl_{f1} \end{aligned}$$
(3o)

In the current study, these equations will be introduced in sq-plane by taking the projection of the heading velocity in this plane, but the sign convention of the side-slip angle \((\beta )\) and moment about r-axis \(\left( N_{r,\beta }\right)\) will violate the equations axes configuration. This violation will be considered in the aerodynamic constants, so the analysis can be written as

$$\begin{aligned} \begin{bmatrix} F_{s,\beta }\\ F_{q,\beta }\\ L_{s,\beta }\\ N_{r,\beta } \end{bmatrix}=\dfrac{1}{2}\rho V_{t,\beta }^2 \begin{bmatrix} C_{X,\beta }\\ C_{Y,\beta }\\ C_{L,\beta }\\ C_{N,\beta } \end{bmatrix} \quad \quad (4) \end{aligned}$$

where

$$\begin{aligned} C_{X,\beta }= & {} C_{X1}\cos ^2(\beta )\nonumber \\{} & {} +C_{X2}\sin (2\beta )\sin \left( \dfrac{\beta }{2}\right) \end{aligned}$$
(4a)
$$\begin{aligned} C_{Y,\beta }= & {} C_{Y1}\cos \left( \dfrac{\beta }{2}\right) \sin (2\beta )+C_{Y2}\sin (2\beta )\nonumber \\{} & {} +C_{Y3}\sin (\beta )\sin (\vert \beta \vert )\nonumber \\{} & {} +C_{Y4}\left( \delta _{rT}+\delta _{rB}\right) \end{aligned}$$
(4b)
$$\begin{aligned} C_{L,\beta }= & {} C_{L1}\left( \delta _{rB}-\delta _{rT}\right) \end{aligned}$$
(4c)
$$\begin{aligned} C_{N,\beta }= & {} C_{N1}\cos \left( \dfrac{\beta }{2}\right) \sin (2\beta )+C_{N2}\sin (2\beta )\nonumber \\{} & {} +C_{N3}\sin (\beta )\sin (\vert \beta \vert )\nonumber \\{} & {} +C_{N4}\left( \delta _{rT}+\delta _{rB}\right) \end{aligned}$$
(4d)

The aerodynamic constants \(C_{Y1},\) \(C_{Y2},\) \(C_{Y3},\) \(C_{Y4},\) \(C_{L1},\) \(C_{N1},\) \(C_{N2},\) \(C_{N3}\) and \(C_{N4}\) are given by

$$\begin{aligned} C_{Y1}= & {} -C_{Z1} \end{aligned}$$
(4e)
$$\begin{aligned} C_{Y2}= & {} -C_{Z2} \end{aligned}$$
(4f)
$$\begin{aligned} C_{Y3}= & {} -C_{Z3} \end{aligned}$$
(4g)
$$\begin{aligned} C_{Y4}= & {} -C_{Z4} \end{aligned}$$
(4h)
$$\begin{aligned} C_{N1}= & {} C_{M1} \end{aligned}$$
(4i)
$$\begin{aligned} C_{N2}= & {} C_{M2} \end{aligned}$$
(4j)
$$\begin{aligned} C_{N3}= & {} C_{M3} \end{aligned}$$
(4k)
$$\begin{aligned} C_{N4}= & {} C_{M4} \end{aligned}$$
(4l)

So the full aerodynamic forces and moments can be expressed as a sum of Eqs. 3 and 4 as

$$\begin{aligned} \begin{bmatrix} F_{s}\\ F_{q}\\ F_{r}\\ L_s\\ M_q\\ N_r \end{bmatrix}= & {} \dfrac{1}{2}\rho V_t^2 \begin{bmatrix} C_X\\ C_Y\\ C_Z\\ C_L\\ C_M\\ C_N \end{bmatrix}\nonumber \\= & {} \dfrac{1}{2}\rho V_t^2 \begin{bmatrix} C_{X,\alpha }\cos ^2(\beta )+C_{X,\beta }\cos ^2(\alpha )\\ C_{Y,\beta }\cos ^2(\alpha )\\ C_{Z,\alpha }\cos ^2(\beta )\\ C_{L,\alpha }\cos ^2(\beta )+C_{L,\beta }\cos ^2(\alpha )\\ C_{M,\alpha }\cos ^2(\beta )\\ C_{N,\beta }\cos ^2(\alpha ) \end{bmatrix} \end{aligned}$$
(5)

The aerodynamic constants in Eqs. 3e–3o and 4e–4l and the geometric variables shown in Fig. 2 are given by

$$\begin{aligned} S_{fh}= & {} a_2b\left[ \dfrac{(x_2-a_1)\sqrt{a_2^2-(x_2-a_1)^2}}{a_2^2}\right. \nonumber \\{} & {} -\dfrac{(x_1-a_1)\sqrt{a_2^2-(x_1-a_1)^2}}{a_2^2}\nonumber \\{} & {} +\sin ^{-1}\left( \dfrac{x_2-a_1}{a_2}\right) - \left. \sin ^{-1}\left( \dfrac{x_1-a_1}{a_2}\right) \right] \end{aligned}$$
(6)
$$\begin{aligned} I_1= & {} \int _0^{l_h}\dfrac{{\text{d}}A(s)}{{\text{d}}s}{\text{d}}s\nonumber \\= & {} \pi b^2\left[ 1-\left( \dfrac{l_h-a_1}{a_2}\right) ^2\right] \end{aligned}$$
(7)
$$\begin{aligned} I_3= & {} \int _0^{l_h}s\dfrac{{\text{d}}A(s)}{{\text{d}}s}{\text{d}}s\nonumber \\= & {} \dfrac{\pi b^2}{3}\left[ a_1-\dfrac{1}{a_2^2}\left( 2l_h^3-3a_1l_h^2+a_1^3\right) \right] \end{aligned}$$
(8)
$$\begin{aligned} J_1= & {} \int _0^{l_h}2r(s){\text{d}}s\nonumber \\= & {} \dfrac{1}{2}a_1b\pi +a_2b\left[ \dfrac{(l_h-a_1)\sqrt{a_2^2-(l_h-a_1)^2}}{a_2^2}\right. \nonumber \\{} & {} +\left. \sin ^{-1}\left( \dfrac{l_h-a_1}{a_2}\right) \right] \end{aligned}$$
(9)
$$\begin{aligned} J_2= & {} \int _0^{l_h}2r(s)s{\text{d}}s\nonumber \\= & {} 2a_1^2b\left( \dfrac{\pi }{4}-\dfrac{1}{3}\right) \nonumber \\{} & {} +2a_2b\left[ \dfrac{a_2}{3}\left( 1-\left( 1-\left( \dfrac{l_h-a_1}{a_2}\right) ^2\right) ^{3/2}\right) \right. \nonumber \\{} & {} -\dfrac{a_1}{2}\left( \left( \dfrac{l_h-a_1}{a_2}\right) \sqrt{1-\left( \dfrac{l_h-a_1}{a_2}\right) ^2}\right. \nonumber \\{} & {} +\left. \left. \sin ^{-1}\left( \dfrac{l_h-a_1}{a_2}\right) \right) \right] \end{aligned}$$
(10)

The drag coefficients \(C_{Dh0}\) and \(C_{Df0}\) can be obtained from Hoerner [27] and Sadraey [28], respectively,

$$\begin{aligned} C_{Dh0}= & {} \dfrac{1}{Re_{l_h}^{1/6}}\left[ 0.172\left( \dfrac{l_h}{2b}\right) ^{1/3}\right. +0.252\left( \dfrac{l_h}{2b}\right) ^{-1.2} \nonumber \\{} & {} +\left. 1.032\left( \dfrac{l_h}{2b}\right) ^{-2.7}\right] \end{aligned}$$
(11)
$$\begin{aligned} C_{Df0}= & {} C_f\left( \dfrac{S_{wet}}{S_{f}}\right) \left( \dfrac{C_{d_{\min}}}{0.004}\right) ^{0.4} \left( 1-0.08M_{no}^{1.45}\right) \nonumber \\{} & {} \quad \left[ 1+2.7\left( \dfrac{t}{c}\right) _{\max}+100\left( \dfrac{t}{c}\right) _{\max}^4\right] \end{aligned}$$
(12)
$$\begin{aligned} C_f= & {} {\left\{ \begin{array}{lcl} \displaystyle \dfrac{1.327}{\sqrt{Re_f}} &{},&{} \text {Laminar flow} \\ \displaystyle \dfrac{0.455}{\left[ \log _{10}Re_f\right] ^{2.58}} &{},&{} \text {Turbulent flow} \end{array}\right. }\end{aligned}$$
(13)
$$\begin{aligned} S_{wet}= 2\left[ 1+0.5\left( \dfrac{t}{c}\right) _{\max}\right] S_{f} \quad \text {(For Rectangular fin)} \end{aligned}$$
(14)
$$\begin{aligned} S_{f}= & {} b_fc\ \ \text {(For Rectangular fin)} \end{aligned}$$
(15)

The cross-flow drag coefficient, \(C_{Dch},\) obtained by Robinson [29] and \(C_{Dcf}\) computed by a regression formula developed by Wardlaw [30], see Fig. 4, are as follows,

$$\begin{aligned} C_{Dch}= & {} {\left\{ \begin{array}{lcl} 1&{},&{}\text {Laminar flow} \\ 0.5&{},&{}\text {Turbulent flow} \end{array}\right. } \end{aligned}$$
(16)
(17)

The fin-efficiency factor \(\eta _f\) and hull-efficiency factor \(\eta _k\) were computed by a regression formula developed by Jones and DeLaurier [15] and are shown in Figs. 5 and 6, respectively, and the axial and lateral apparent-mass coefficients \(k_1,k_2\) were computed by a regression formula developed by Munk [31] shown in Fig. 7,

$$\begin{aligned} \eta _f= & {} 5.649\left( \dfrac{S_{fh}}{S_f}\right) ^7 - 17.28\left( \dfrac{S_{fh}}{S_f}\right) ^6 \nonumber \\{} & {} + 18.18\left( \dfrac{S_{fh}}{S_f}\right) ^5 - 5.409\left( \dfrac{S_{fh}}{S_f}\right) ^4\nonumber \\{} & {} - 3.558\left( \dfrac{S_{fh}}{S_f}\right) ^3 + 3.538\left( \dfrac{S_{fh}}{S_f}\right) ^2 \nonumber \\{} & {} - 2.115\left( \dfrac{S_{fh}}{S_f}\right) + 1.006 \end{aligned}$$
(18)
$$\begin{aligned} \eta _k= & {} 1.638\left( \dfrac{S_f\cos ^2(\Gamma )}{J_1}\right) ^4 \nonumber \\{} & {} - 2.444\left( \dfrac{S_f\cos ^2(\Gamma )}{J_1}\right) ^3 \nonumber \\{} & {} + 2.398\left( \dfrac{S_f\cos ^2(\Gamma )}{J_1}\right) ^2\nonumber \\{} & {} + 0.02692\left( \dfrac{S_f\cos ^2(\Gamma )}{J_1}\right) \nonumber \\{} & {} + 1.003 \end{aligned}$$
(19)
$$\begin{aligned} k_1= & {} 0.4964\left( \dfrac{a}{b}\right) ^{-1.16} - 0.02019 \end{aligned}$$
(20)
$$\begin{aligned} k_2= & {} -0.000004864\left( \dfrac{a}{b}\right) ^6 + 0.0002078\left( \dfrac{a}{b}\right) ^5 \nonumber \\{} & {} - 0.003632\left( \dfrac{a}{b}\right) ^4 + 0.03358\left( \dfrac{a}{b}\right) ^3 \nonumber \\{} & {} - 0.1781\left( \dfrac{a}{b}\right) ^2 + 0.5506\left( \dfrac{a}{b}\right) + 0.09385 \end{aligned}$$
(21)

However, the kinetic analysis of the airship is usually derived at the center of volume (C.V.). Eq. 5 can be expressed in xyz axes as follows,

$$\begin{aligned} \begin{bmatrix} F_{x,a}\\ F_{y,a}\\ F_{z,a}\\ L_a\\ M_a\\ N_a \end{bmatrix}= \begin{bmatrix} -F_s\\ F_q\\ -F_r\\ -L_s\\ M_q+x_nF_r\\ -N_r+x_nF_q \end{bmatrix}= \dfrac{1}{2}\rho V_t^2 \begin{bmatrix} -C_X\\ C_Y\\ -C_Z\\ -C_L\\ C_M+x_nC_Z\\ -C_N+x_nC_Y \end{bmatrix} \end{aligned}$$
(22)
Fig. 4
figure 4

Fin cross-flow drag coefficient [30]

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Fig. 5
figure 5

Fin-efficiency factor \(\eta _f\) [15]

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Fig. 6
figure 6

Hull-efficiency factor \(\eta _k\) [15]

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Fig. 7
figure 7

Axial and lateral apparent-mass coefficients \(k_1,k_2\) [31]

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where \(x_n\) is the nose position in x-direction with respect to xyz axes,

$$\begin{aligned} x_n=a_1+\dfrac{4}{3\pi }(a_2-a_1) \end{aligned}$$
(23)

2.2 Problem formulation

The airship can be considered as a merge of two ellipsoids with the same minor axes. The parameters of airship shape will be selected according to a certain cost function which improves the overall performance partially using genetic algorithm optimization technique with some assumptions to simplify the problem and reduces the selected parameters as follows:

  1. 1

    The range of change of angle of attack \((\alpha )\) and side-slip angle \((\beta )\) is \(\left[ -20^\circ ,20^\circ \right]\),

  2. 2

    Neglect the effect of the fin deflection \(\left( \delta _{rT}=\delta _{rB}=\delta _{eR}=\delta _{eL}=0\ \Rightarrow \ L_a=0\right)\),

  3. 3

    Symmetric airfoil of airship’s fin \(\left( \Rightarrow l_{f1}=l_{f2}=l_h+\dfrac{c}{4}\right)\),

  4. 4

    No taper ratio

  5. 5

    No dihedral angle \((\Gamma =0^\circ )\).

  6. 6

    low speed \(\left( M_{no}<0.3\right)\),

  7. 7

    Airship length is between two and three meters \((2m\le l\le 3m)\),

  8. 8

    Airship’s fin is NACA-0006 \(\left( (t/c)_{\max}=0.06\right)\) with chord \(c=0.15l\) and span \(b_f=2.1b\) and the location of airship fin leading edge is \(l_h=0.79l\), where b is the airship minor axis.

Genetic algorithm optimization technique is used to solve this problem under some constraints to get the regular shape of the airship as shown in Fig. 8, which are

  1. 1

    Rear major axis is greater than front major axis \(\left( a_2>a_1\right)\),

  2. 2

    Front major axis is greater than minor axis \(\left( a_1>b\right)\) and

Fig. 8
figure 8

Regular Airship Shape

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The optimization problem is to maximize the following cost function:

$$\begin{aligned} J=\dfrac{\Vert f_{z,a} \Vert + \Vert M_{a} \Vert }{\Vert f_{x,a} \Vert + \Vert f_{y,a} \Vert + \Vert N_{a} \Vert } \end{aligned}$$
(24)

This cost function is constructed to reduce the effect of wind load \(\left( f_{y,a}, N_a\right)\) and drag force \(\left( f_{x,a}\right)\), to improve energy consummation efficiency, and to increase lift force \(\left( f_{z,a}\right)\) and pitching moment \(\left( M_a\right)\), to increase weight capacity and endurance range in case of engine failure. Table 1 shows the sign change in aerodynamic forces and moments according to the sign change in angle of attack \(\alpha\) and side-slip angle \(\beta\) with the same magnitude. If \(\alpha\) has a fixed positive or negative value and the sign of \(\beta\) changes with the same magnitude, the sign of \(f_{y,a}\) and \(N_{a}\) changes also with the same magnitude. Same conclusion is valid for \(F_{z,a}\) and \(M_a\) if \(\beta\) is fixed. So, the operating domain can be reduced to a quarter as the value of the cost function J depends on the absolute values of the aerodynamic forces and moments. The aerodynamic forces and moments have the same magnitude for a fixed absolute values of angle of attack and side-slip angle. This is due to the nature of the airship shape as it has two planes of symmetry.

The airship shape can be developed by various ways. In this case, three configurations are used to build airship shape,

Table 1 Aerodynamic forces and moments signs according to the sign of angle of attack and side-slip angle
Full size table
  1. 1

    NPL shape suggested by the National Physics Laboratory [32]: NPL shape can be considered as an intersection of two ellipses with different major axes and same minor axes. The equations of shape construction are clarified in Fig. 9.

  2. 2

    GNVR shape developed by National Aerospace Laboratories [33]: GNVR shape consists of three parts: semi-ellipse, sector of a circle and sector of parabola. The equations of shape construction are clarified in Fig. 10.

  3. 3

    Wang shape [34]: Wang shape is developed by a parametric equation clarified in Fig. 11.

Fig. 9
figure 9

NPL Shape

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Fig. 10
figure 10

GNVR Shape

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Fig. 11
figure 11

Wang Shape

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3 Results

Genetic algorithm optimization solutions are carried out using MATLAB toolbox with population size equal to 50 and the number of generations equal to 50 of NPL, GNVR and Wang shapes at each corresponding angle of attack \((\alpha )\) and side-slip angle \((\beta )\) are presented as

  1. 1.

    NPL optimal solutions: NPL shape is constructed by two parameters, the optimal solution of “\(a_N\)” parameter at every \(\alpha\) and \(\beta\) is shown in Fig. 12 and “\(b_N\)” parameter in Fig. 13. The value of the cost function at every possibility is presented in Fig. 14. The solution results show that the optimal solutions of “\(a_N\)” and “\(b_N\)” have the same behavior for all values of \(\alpha\) and \(\beta\). As shown, “\(a_N\)” starts with extreme maximum value for all values of \(\alpha\), then has a step drop, and then increases as \(\beta\) increases. Also, “\(b_N\)” begins with extreme minimum value for all values of \(\alpha\) then increases as \(\beta\) increases. And the cost function J decreases as \(\alpha\) increases for all values of \(\beta .\)

  2. 2.

    GNVR optimal solutions: Figure 15 shows the optimal solution of “\(d_G\)”, the single variable of GNVR shap, at each \(\alpha\) and \(\beta\). The value of cost function is shown in Fig. 16. The solution results show that the optimal solutions of “\(d_G\)” are approximately constant for all values of \(\alpha\) and \(\beta\) and the cost function J increases as \(\beta\) increases for all values of \(\alpha .\)

  3. 3.

    Wang optimal solutions: Wang shape is built by a parametric equation. Figures 17, 18, 19, 20 and 21 show the optimal solution of “\(a_W\)”, “\(b_W\)”, “\(c_W\)”, “\(d_W\)” and “\(l_W\)” at each \(\alpha\) and \(\beta\), respectively. The value of the cost function is presented in Fig. 22. The solution results show that the optimal solutions of “\(a_W\)” and “\(b_W\)” have the same pattern for all values of \(\alpha\) and \(\beta\). As shown, their values decrease as \(\alpha\) increases for all values of \(\beta\). Also, “\(c_W\)” and “\(d_W\)” have the same behavior for all values of \(\alpha\) and \(\beta\). “\(c_W\)” and “\(d_W\)” optimal values are approximately constant except for \(10^\circ \le \alpha \le 20^\circ\) and \(0^\circ \le \beta \le 10^\circ\) where the values have rapid changes. In addition, “\(l_W\)” optimal values are approximately constant except for the upper values of \(\alpha\) and \(\beta\). And the cost function J decreases as \(\alpha\) increases for all values of \(\beta .\)

Fig. 12
figure 12

NPL “\(a_N\)” parameter

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Fig. 13
figure 13

NPL “\(b_N\)” parameter

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Fig. 14
figure 14

Cost function values of NPL shape

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Fig. 15
figure 15

GNVR “\(d_G\)” parameter

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Fig. 16
figure 16

Cost function values of GNVR shape

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Fig. 17
figure 17

Wang “\(a_W\)” parameter

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Fig. 18
figure 18

Wang “\(b_W\)” parameter

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Fig. 19
figure 19

Wang “\(c_W\)” parameter

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Fig. 20
figure 20

Wang “\(d_W\)” parameter

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Fig. 21
figure 21

Wang “\(l_W\)” parameter

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Fig. 22
figure 22

Cost function values of wang shape

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4 Discussion

The optimized solution is performed in two steps. First one is to choose the shape type which has the best performance for various values of \(\alpha\) and \(\beta\). Figure 23 shows the values of the cost functions of the three types, and the difference between these cost functions is shown in Fig. 24. Figure 24 visualizes the difference sign regardless the magnitude, since the yellow color indicates that the difference is positive and negative otherwise. It is clear that GNVR and NPL shapes are better than Wang shape. In addition, the statistics in Table 2 clarify that there is no major difference between these two types. However, Fig. 24 shows that GNVR is better at high side-slip angle \(\left( \beta >10^\circ \right)\). So, GNVR shape is the optimized solution in this study. Reddy Desham and Rajkumar S. Pant [35] CFD work shows that GNVR has the smallest volumetric drag coefficient among the other shapes NPL and Wang in a certain case, which give us an intuition to the solution.

The second step is to find the best GNVR shape parameter which fits the optimal solutions at every \(\alpha\) and \(\beta\). The optimized parameter can be determined using a weighted function for averaging as follows:

$$\begin{aligned} \left( \begin{array}{c} \text {Optimized}\\ \text {shape}\\ \text {parameter} \end{array} \right) =\dfrac{\sum \text {Shape parameter} \times \text {Weight}}{\sum \text {Weight}}, \end{aligned}$$
Fig. 23
figure 23

Cost Function Values of NPL, GNVR and Wang Shapes

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Fig. 24
figure 24

Cost Function Difference between NPL, GNVR and Wang Shapes

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Table 2 Statistical Analysis of \(\left| J_{GNVR}-J_{NPL} \right|\)
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Different optimal values of the parameter “\(d_G\)” corresponding to different weights are shown in Table 3. So, it is clear that there is no major difference of the values of the parameter “\(d_G\)” for different weights and this leads to choosing the optimal parameter as \(d_{G,opt.}=0.655748\) and Fig. 25 shows a diagram of the optimized airship.

Table 3 Optimal Shape Parameter with Different Weights
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Fig. 25
figure 25

Schematic diagram of optimized airship

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5 Conclusion

This article presents a further analytical development of the semi-empirical aerodynamic model in Eq. 3 of airship to consider the effect of side wind in Eq. 4 depending on the symmetrical shape of the airship. This model depends only on airship main geometric parameters. The shape optimization problem is formulated by NPL, GNVR and Wang shapes to choose the one which achieves the best performance index, minimizes drag and wind load and maximizes lift and pitching moment, using the genetic algorithm optimization technique. GNVR shape exceeded their counterparts and consequently was adopted.