Multi-Objective Reliability-Based Partial Topology Optimization of a Composite Aircraft Wing

Abstract

Reliability-based partial topology optimization (RBPTO) is a new approach for aircraft structural design; however, it brings computational complexity and makes aeroelasticity analysis quite challenging. Therefore, the present study proposed the multi-objective reliability-based partial topology optimization of a composite aircraft wing using a fuzzy-based metaheuristic (MRBPTOFBMH) approach. The objective is to obtain an optimal layout including partial topology and sizing of the aircraft wing structure. Here, an optimal aeroelastic structure is designed by taking into account the uncertain nature of material properties and the permitted transverse displacement. To increase computational efficiency in the design process, a non-probabilistic approach called a possibilistic safety index-based design optimization (PSIBDO) with fuzzy uncertainties is proposed to quantify the uncertainties due to aeroelastic and structural constraints. Various optimum partial topological shapes and sizing of aircraft wing structures with various PSI values in the range of [0.001, 1.0] have been obtained in a single optimization run. These outcomes, including deterministic and reliable optimal aircraft wing structures, demonstrate the high effectiveness of the proposed MRBPTOFBMH technique to alleviate the complexity of unconventional aircraft wing structure design. The findings also reveal the ease in cooperation of the suggested technique with a high-performance multi-objective evolutionary algorithm (MOEA) and its application in real-world multi-objective design optimization (MODO) problems with the least computational requirements against the traditional method’s multiple runs. Furthermore, the proposed methodology can generate potential aircraft wing structures in a range of m = [89.38–127.84] kg, and flutter speed = [285.61–632.78] m/s, that adhere to all the constraints requirements.

1. Introduction

In the past, deterministic topology optimization (DTO) has been used to modify the aerostructure components of commercial airplanes in an effort to make them safer and lighter. The first example of a commercial aircraft component that has been studied is the airframe structures of the Airbus A380 [1,2]. DTO was performed for the wing leading edge and trailing edge, door stops and fuselage door, and main wing box ribs and spars. The leading-edge ribs of the Boeing 787 have been studied for stiffest (minimize compliance) with a given volume constraint [3]. It achieved a reduced weight when compared with the Boeing 777. Bombardier [4] also chose DTO for the synthesis of the wing box ribs using similar objectives and constraints from [3]. These examples illustrate the high rate of DTO technique applications in aircraft industries and its advantages for redesigning aircraft structural components.

The recent development of aero-structure optimization depends on three stages of design, which can be separated into topology, sizing, and shape optimization [5]. Combining topological and size stages is the most common and popular approach to redesigning aerostructures instead of pairing them with shape optimization [6,7,8]. This is due to (i) it is not advised to alter the wing shape established during the conceptual design stage and (ii) a complicated internal structural layout is difficult to manufacture. The reason brings two alternative choices for the present design technique. Such designs included the ground element approach or discrete method generating structural layouts and dimensions at the same time, which is called partial topology optimization [7,9]. By eliminating unrealistic design outcomes, the ground element strategy has strengthened the conventional continuous approach [10]. The ground element technique discretized the design domain as a frame, truss, and panels. In this approach, the presence of material is designed by varying the pseudo-density of each member between zero and one where zero means void, while one means material presence. The possibility of defining component sizes and structural layout of the design domain at the same time in a single optimization run is its advantage. Indeed, the adaptive tailing edge structure has been synthesized with this method [11]. Later, the work was extended by many researchers to study the adaptive or morphing wing concept [12,13]. This kind of wing can improve aircraft performance in flight by shape-changing due to unconventional aircraft wing structures. The unconventional structural aircraft wing has been studied to alleviate aeroelastic phenomena, which categorizes as static and dynamic aeroelastic. The dangerous critical speed is a lower speed of the divergence speed and flutter speed. The divergence and flutter speeds are static and dynamic phenomena, respectively. The DTO techniques have been used to reveal a true structure of aircraft wing structure against violent phenomena, which may be totally different from the common structure. The common aircraft components that have been synthesized using DTO-based material density are the ribs [1,2,3,14,15], spars [15,16], stiffened panels [17], and the whole of the aircraft wing [18,19,20,21,22,23]. The structural layout and sizing of the aircraft wing can be performed using the ground structure method in one optimization run, which is called an unconventional aircraft wing structure [12,13,18]. Therefore, the design results are easier to manufacture, which can be used in practice rather than the use of continuous design variables [24].

Most of the earlier studies ignored the uncertainties in the design process of all aircraft wings. However, it is well known that there are inherent uncertainties that might cause real aircraft performance to deviate and lead to the impracticality of a design based on conventional approaches [4,25]. To alleviate the effect of uncertainty in the process of aircraft design, reliability-based design optimization (RBDO) has been proposed [26]. The uncertainty quantification is integrated into the optimization process to analyze the reliability or failure probability of the design structure. In general, the reliability index can be analyzed using the probabilistic technique. However, it has a limitation of more precise initial data requirements, which is often infeasible. Consequently, it made inefficient computation. As an alternative, non-probabilistic techniques can alleviate the computational burden. The first uncertainty quantification technique in the group of probabilistic techniques is called Monte Carlo simulation (MCS). Its main drawback is time consumption as aforementioned. This is the reason for the development of various techniques, which expect to alleviate this drawback. The first-order and second-moment (FOSM), the first-order reliability method (FORM), and the second-order reliability methods (SORM) are adaptive forms [27]. A successful method in a group of probabilistic techniques has been used to quantify the uncertainties in the design of a Goland wing, which was the polynomial chaos expansion (PCE) [28,29]. The extension of this technique has been used in design uncertainty due to the ply orientations of a composite plate wing [30]. RBDO is a combination of uncertainty quantification and the design optimization process through the iterative optimization process. The use of MCS for RBDO, however, demands a great number of function evaluations often leading to inefficient or even impossible computation [31]. Alternative methods such as Latin hypercube sampling (LHS) use fewer random variable numbers in comparison to MCS but offer less precision [32]. Optimum Latin hypercube sampling (OLHS) is an adaptation of LHS, which is expected to increase the quality of space-filling and increasing precision [33]. Later, it was combined with the surrogate model to improve its performance [34]. Nevertheless, the challenge of large computation still persists in this technique [34]. Therefore, there is a need for efficient methodologies that can overcome these limitations

A non-probabilistic approach is an alternative group of methods that do not need precise data. This root starts with a convex set [27], an interval method [35], and a fuzzy set theory [36]. The general RBDO with a probabilistic approach is a double-looped nested problem of uncertainty quantification and an optimization process. A non-probabilistic approach, on the other hand, transforms this double-loop issue into a triple-looped nested problem due to the further addition of the possibility safety index (PSI) calculation. This issue can be solved using the target-performance-based design approach (TPBDA) [37], the interval perturbation method (IMP) [38], and the multi-objective reliability-based design optimization (MORBDO) using metaheuristics (MHs) [39,40,41]. This MORBDO root begins by solving a conventional aircraft aeroelastic problem using a worst-case scenario approach [39] followed by its extension technique investigation in a reliability-based design of steering linkage [41]. Its prolongation [37] to the topology optimization problem is called multi-objective reliability-based topology optimization (MORBTO), which can reduce the computational time of triple-looped nested problems [40]. Alternatively, the work in [27] and the sequential optimization and reliability assessment (SORA) method have been used to solve the nested problem in the RBTO problem [42,43]. As previously discussed, RBTO has been used to design aero-structures to reduce their weight [4]. As presented in [4,25], the aero-structure shape layouts are significantly affected by the reliability index [4,25]. Thus, designing aircraft wing structures using the RBTO technique necessitates the inclusion of aeroelastic phenomena. Earlier, very few works investigated the non-probabilistic method for designing the aeroelastic topology of aircraft wings [44]. Recently [45], the PSI technique combined with an MH technique (PSIBDO) has been used for solving a composite aircraft wing design. The research [45] aims to reduce the mass of the composite aircraft wing to meet aeroelastic and strength constraints. Another investigation by [46,47], which is typically an extension of [45], used multi-objective reliability-based design optimization with the non-probabilistic technique for solving aeroelastic structure, which shows the potential to generate deterministic and conservative optimal aircraft structure designs in one optimization run. Both techniques are single-run techniques [46,47], but they can perform in two-step as presented in [47]. The second part of the two-step method for synthesizing airplane wing structures can be completed using a probabilistic strategy as shown in [48].

The research objective of this study is to perform topological optimization of an aircraft wing. It is based on the potential safety index-based design optimization approach with MHs. The main contributions are as follows:

• A novel fuzzy-based metaheuristic (MRBPTOFBMH) approach is suggested to optimize composite aircraft wing structure.
• A non-probabilistic approach called possibilistic safety index-based design optimization (PSIBDO) with fuzzy uncertainties is proposed to improve the computational efficiency of the aircraft wing design.
• An efficient framework that facilitates easy integration with a multi-objective evolutionary algorithm (MOEA) is suggested.
• The effectiveness of the suggested methodology is evaluated using a multi-objective aircraft wing topology optimization case study.

The remainder of this paper is divided into five sections. Section 2 introduces the aeroelastic partial topology model used in this study. Section 3 illustrates the adopted MRBPTOFMH approach. The numerical experimental setup is presented in Section 4, while the results and discussion are depicted in Section. Finally, the conclusions of this investigation and developed future scopes are illustrated in Section 6.

2. Aeroelastic Partial Topology

2.1. Aircraft Wing Model

A Goland wing has been utilized as a model of aerodynamic/ aeroelastic/ aero structure due to its simpler model than an actual complex aircraft wing [49]. A special technique to synthesize the airplane structure can be presented more effectively using a simple structure of the Goland wing. Moreover, it allows easier integration of special materials into the model. Normally, it comprises a front spar, rear spar, middle spar, tip, upper skin, lower skin, and many ribs [50] as shown in the following Figure 1.

2.2. Partial Topology Design

The partial topology and sizing optimization procedure for the present work is subjected to the ground structure approach [12,13]. Typically, the process is divided into two steps. Firstly, it is started with defining the discrete domain, which is formed by truss, frame, and panels as shown in Figure 2a. Lastly, optimization is performed with the decision to retain/or remove each element from the design domain. The design variables are the width/thickness of each element. If the element thickness is close to zero, the element was removed, as illustrated in Figure 2b, otherwise, the element was retained. With this concept, partial topology and sizing can be performed simultaneously. The decision to remove or retain can be represented as:

$$T{h}_i=\{\begin{array}{cc}0& t_i=0\\ 1& t_i>0\end{array}\to \mathrm{for} i=1\dots n$$

(1)

where $T{h}_i$ and $t_i$ are the removal parameter and element thickness of element i, respectively, and n is the total elements of the wing structure.

To find the partial topology and sizing of the internal wing structure, a discretized design domain of the wing structure was constructed, as presented in Figure 3a. The thickness of the skin, spar, and rib were selected as design variables in the structural optimization, while the removal elements considered only the last two type variables (spar and rib) to construct a realizable spar and rib layout, as shown in Figure 3b. As mentioned previously, the ground structure-based method can perform a more realizable structure than the approach based on material density [12,13].

2.3. Aerodynamic Model

In this research, aerodynamics are solved using the quasi-steady vortex ring method [45], and its reduced order is used for aeroelastic analysis. This model is considerably faster and easier to use, but less accurate compared with unsteady aerodynamics such as the doublet lattice method. The formula for aerodynamic forces exerted on the structure can be expressed as:

$$\mathit{f}\left(\mathrm{t}\right)=q{\left[G\right]}^T\left[S\right]\left[AIC\left(k\right)\right]{\left[G\right]}^T\left\{u\right\}$$

(2)

where $q$ is dynamic pressure ($q=\frac{\rho V^2}{2}$), $\left[G\right]$ is the matrix transformation, $\left[S\right]$ is the diagonal panel areas matrix, [AIC] is the matrix of aerodynamic influence coefficients, $k=\frac{L\omega }{V}$ is the reduced frequency, L is a semi-chord length, ρ is the mass density of air (depending on altitude), V is free stream velocity, ω is a circulation frequency, and {u} is a structural displacements vector of the wing (carried out using finite element analysis, FEA).

2.4. Aeroelastic Analysis

An aeroelastic analysis of the wing is needed to interface the structural and aerodynamic analysis with a surface spline interpolation technique. FEA is a tool for static and dynamic analysis, which can be coded in MATLAB to interface with the quasi-steady vortex ring code. The FEA in this research uses the quadrilateral Mindlin shell elements with a drilling degree of freedom. With the help of reduced order modeling and the interpolation technique, a discrete-time aeroelastic model can be performed, which has been proven efficient in both static and dynamic aeroelastic analysis and to use less computational time compensation [9]. The discrete-time aeroelastic equation can written as:

$$\left[M\right]\left\{\stackrel{..}{u}\right\}+\left[D\right]\left\{\stackrel{.}{u}\right\}+\left[K\right]\left\{u\right\}=\left[A_d\right(V\left)\right\{\stackrel{·}{u}\}+[A_k\left(V\right)\left]\right\{u\}$$

(3)

where [M] is the mass matrix, [D] is the damping matrix, and [K] is the stiffness matrix of the wing structure. [A_d] and [A_k] are aerodynamic damping and stiffness matrices, respectively. The last two terms are from fluid/structure interactions, as mentioned previously, which are dependent on velocity. The flutter speed is a dynamic aeroelastic phenomenon when the dynamic system becomes unstable, which occurs when the real parts of the eigenvalues are zero. It is called eigenvalues analysis.

2.5. Lift Effectiveness

The static aeroelastic phenomenon studied in this research is the lifting effectiveness, which is the ratio of lift force on the flexible structure and its rigid counterpart. The lift effectiveness is used to reflect wing aerodynamic effectiveness when the flexibility of the wing is considered. It can be expressed as [34,50]:

$$\eta =\frac{{q\mathit{S}}^T{\left[AIC\right]}_F\mathit{\alpha }}{{q\mathit{S}}^T{\left[AIC\right]}_R\mathit{\alpha }}$$

(4)

where $\mathit{\alpha }$ is the panels’ angles of attack vector and ${\left[AIC\right]}_F$ is the matrix of flexible surface aerodynamic influence coefficients.

3. Multi-Objective Reliability-Based Partial Topology Optimization Using Fuzzy-Based Metaheuristic (MRBPTOFMH)

Most real-life issues belong to multi-objective optimization problems rather than a single objective due to having a mutual conflict in the objectives, for example, a minimization of mass may cause conflict with the strength of the structure, a maximization of strength causes a high price, etc. [51,52,53]. A typical multi-objective optimization problem can be formulated as follows:

Min f_i(x)    i = 1, …, M

(5)

Subject to → g_j(x) ≤ 0, j = 1, …, N

x_l ≤ x ≤ x_u

where f_i is the objective function, g_j is the optimization constraints, x is a vector of design variables, N is the number of inequality constraints, x_l is a vector of lower bound limits, M is the total number of objective functions, and x_u is a vector of upper bound limits. If i = 1, it is called single-objective optimization, otherwise, it is called multi-objective optimization.

The solution to the multi-objective optimization problem is called the Pareto solution set. MHs are popular methods that have been widely investigated for finding the approximate Pareto solution. The solution of (5) is sometimes unrealizable due to the presence of uncertainties or random variables (y) in the design problem. Uncertainty can be divided into two groups [45]. In the first group, aleatory occurs due to random physical variation in nature. In the second group, epistemic uncertainty occurs due to the lack of knowledge. Aleatory can be quantified using the probabilistic model, while the epistemic uncertainty is handled using techniques including a convex set, a fuzzy set method, and anti-optimization. Composite aircraft wing structures are the design expectation of the present study. The wing model is composed of composite laminate skin layup in upper and lower sections, laminate skin ply thickness, ply orientation, and partial topology and sizing of the ribs and spars. The objectives are the minimization of structural weight and the maximization of flutter speed. Epistemic uncertainty is considered in this study due to the lack of knowledge of material properties and allowable transverse displacement. Furthermore, the challenge of aeroelastic topology analysis and the flaws in the MH algorithm is what prompted the authors to suggest a novel approach to overcome these challenges.

In general, uncertainty can be quantified using MCS, but it is known that this method needs a large number of calculations per set of random variables. As a result, an excessively large number of function evaluations are needed in one optimization run. This approach is impractical for the design of an aeroelastic design, although it can be exploited using a very accurate surrogate model. The process was started by quantifying uncertainties with MCS and a surrogate model. An actual physical system can be modeled using a surrogate technique, which is expected to reduce computational time consumption [34]. However, the method is still time-consuming with a great number of computations [28]. Latin Hypercube sampling (LHS) and its variant have been proven to reduce the computational burden when used to replace the MCS stage [32]. A general deterministic multi-objective design problem can be changed to a MORBDO problem as follows:

$$\mathrm{Min} \left\{f_i\right(\mathit{x}),-\beta \}$$

(6)

Subject to → Prob(g_j(x, y) ≥ 0) ≤ p_j   j = 1,2,..., N

x_l ≤ x ≤ x_u

where β is a system reliability index, Prob is a probability function, y is a vector of uncertainties or random variables, and p_j is a predefined probability of failure.

In this case, the uncertainties (y) are modeled using a fuzzy (a) function with a membership function μ_j rather than being performed with a precise technique as detailed previously. A triangular fuzzy shape has been used to represent uncertainty. Then, RBDO is used to solve the deterministic design problem by incorporating the fuzzy set theory, under a possibility safety index (PSI) constraint [40,45] or the adaptation of the fuzzy possibilistic programming (FPP) form [54,55,56], as follows.

$$\mathrm{Pos}(z\ge 0)=\underset{z\ge 0}{\sup }{\mu }_j(z)$$

(7)

where “sup” is the supremum, Pos is a possibility, and z is g_j(x, y).

From (2), (6–7), the deterministic design optimization problem can be rearranged to be the MORBDO problem based on the PSI and a fuzzy set model as:

Min {f_i(x), π_f^max}

(8)

Subject to → Pos(g_j(x,a) ≥ 0) ≤ π_fj^max   j = 1,2,..., N

max(π_fj^max) = π_f^max → j = 1,2,..., N

x_i ≤ x ≤ x_u

where a is a vector of fuzzy uncertainty variables, π_f^max is the maximum PSI, and π_fj is the PSI of each constraint.

It is known that to evaluate a PSI value, double-loop nested searching is needed to find π_fj from (9), which is derived from Figure 4 and Equation (7).

$$\mathrm{Pos} (g_j(x,a)\ge 0)=\{\begin{array}{l}0,\\ \alpha ,\\ 1,\end{array}\mathrm{where}\begin{array}{l}{g}_j^{+0}<0\\ g_j^{-1}<0<g_j^{+0}\\ g_j^{-1}\ge 0\end{array}$$

(9)

The PSI value of each constraint can be calculated if g_j⁺⁰ and g_j⁻¹ are known. If g_j⁺⁰ ≤ 0 or g_j⁻¹ ≥ 0, we can calculate that Pos(g_j(ρ,a) ≥ 0) = 0 or 1, and the solution procedure can be terminated. Conversely, if g_j⁻¹ < 0 < g_j⁺⁰, the equation g_j^−α = 0 should be solved, and its solution, α, will be the value of Pos(g_j(x,a) ≥ 0). Theoretically, the bi-section method can be used to compute the value of Pos(g_j(x,a) ≥ 0), whose procedure can be summarized as the following pseudo-code:

(1)

Start with the initial interval: let α₁ = 0, α₂ = 1; it should be noted that the specified tolerance is ε = 1 × 10⁻⁸.

(2)

Find α₃ = (α₁ + α₂)/2.

(3)

If g_j^α1 × g_j^α3 < 0, set α₂ = α₃, and go to step 2.

(4)

Otherwise, set α₁ = α₃, and go to step 2.

(5)

If (α₁ − α₂) < ε is sufficiently small, then the procedure is stopped.

(6)

The approximation of the last root is α₃ = (α₁ + α₂)/2.

When solving the optimization problem in (8) and the partial topology technique in (1), even though it becomes a triple-loop nested problem, the interesting part is a combination of the partial topology optimization of the aeroelastic structure with fuzzy uncertainties and the robustness of the MH. Its performance in solving the triple-loop problem needs to be investigated. This combined technique is called the multi-objective reliability-based partial topology optimization using fuzzy metaheuristic (MRBPTOFMH).

4. Numerical Experiment

The partial topology of a composite aircraft wing with fuzzy uncertainties in material property and allowable transverse displacement on the wing is used as a design demonstration. Unconventional aircraft wing structures can be synthesized using the traditional Goland wing model [30], which is demonstrated in Figure 3a. The traditional Goland wing geometry has chord length (c), semi-span wing (L), and wing thickness (t) of 1.216 m, 6.096 m, and ±0.0508 m, respectively. The unconventional aircraft wing structure is made of aluminum ribs and spars, as presented in Figure 3b. The unconventional structure optimization can be performed with the MRBPTOFMH technique. The skin thicknesses (3 layers) and ply orientation of the laminated carbon fiber are the design variables. The properties of both materials are presented in

Table 1. Material properties of aluminum and carbon fiber.

ALUMINUM
Properties	Value	Unit
Young’s modulus (E)	70 × 109	Pa
Poisson’s ratio (ν)	0.3	-
Density (ρ)	2700	kg/m3
CARBON FIBER
Young’s modulus (E11)	207.7 × 109	Pa
Young’s modulus (E22)	7.6 × 109	Pa
Shear modulus (G12)	5.0 × 109	Pa
Shear modulus (G13)	5.0 × 109	Pa
Shear modulus (G23)	5.0 × 109	Pa
Poisson’s ratio (ν12)	0.3	-
Density (ρ)	1800	kg/m3

. The objective functions are structural mass and flutter speed. The first objective function is aimed to save energy for the modern aircraft, while the second objective function is to protect the aircraft from wing failure caused by unstable speed. The constraints include lift effectiveness and transverse deflection, which are assigned as a constraint function to handle the aircraft structural performance. The optimization problem is expressed as:

Min f(x) = {mass, −V_f}

(10)

     Subject to u_max − u_al ≤ 0
 η_L,al − η_L ≤ 0
x_l ≤ x ≤ x_u

where x is a vector of design variables having lower and upper bounds as x_l and x_u, respectively, u_max is the maximum transverse displacement on the wing, while u_al is the allowable transverse displacement. η_L is the wing lift effectiveness, while η_L,al is the allowable lift effectiveness. V_f is a flutter speed. Air density and stream velocity are 1.2 kg/m³ and 40 m/s, respectively. Aeroelastic and structural constraints are set as η_L,al = 0.9, u_al = 0.1 m, respectively.

The design variables of the aircraft wing structure can be divided into two groups including the shape and sizing of ribs and spars and skin thicknesses and composite ply orientations. A total of 64 design variables are detailed here:

x_1–20 = front and rear spar thickness

x_21–23 = upper skin thickness

x_24–26 = lower skin thickness

x_27–56 = middle spar and rib thickness

x_57–58 = tip wing thickness

x_59–61 = lower skin ply orientations

x_62–64 = upper skin ply orientations

To make our model realizable in practice, the design variables are assigned as discrete, where the bound constraints are set as follows:

x_i∈ {0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005}m for x_1–58 and

x_i∈ {0 15 30 45 60 75 90 105 120 135 150 165} degree for x_59–64.

The combination of topology and sizing variables of the wing is carried out by assigning values to the segment thickness in the form of a discrete variable. For the middle spar and ribs, x_i ∈ [0.001, 0.005] m are the design variables for segment thickness where the lower and upper bounds are based on manufacturing tolerance. In case that x_i falls from the lower bound of 0.001, the ith ground segment will be removed from the main structure. Other thickness values that are higher than the lower bounds are used for structural sizing following the discrete variable. It is important to note that the ideal lower bounds of thickness for the topological design variables are zero, which causes a singularity in the global stiffness matrix of an airplane wing. An alternative is to use a small positive value as a lower bound. The optimizer utilized for solving the MRBPTOFMH is the adaptation of hybrid population-based incremental learning and differential evolution (aPBIL-DE), which is proven efficient for solving multi-objective optimization problems [45].

In the design, a partial topology and sizing optimization of the composite aircraft wing structure is considered along with uncertainties in the material property and allowable transverse displacement. The material property is Young’s modulus, and the allowable transverse displacement is u_al. Fuzzy uncertainty, called normal membership functions, is used to model the uncertainties where $E=exp\left\{-1/2{(\frac{r - Em}{Em \times 0.1})}^2\right\}$ and $u=exp\left\{-1/2{(\frac{r - Ualm}{Ualm \times 0.1})}^2\right\}$. The core function of Young’s modulus is Em = 70 GPa, while the allowable transverse displacement is $Ualm=0.1 \mathrm{m}$. With the proposed MRBPTOFMH technique, Equation (10) can be changed to Equation (8). In this experiment, the problem is solved with a personal computer: AMD Ryzen 5,4600 H with Radeon Graphics 3.00 GHz, 8.00 GB, 64-bit Window 10 operating system. The optimizer setting in this research considered a population size of 68 and 100 total iterations, while the Pareto archive size was 20. With the new technique, the PSI values are in the range of 0.001–1.

5. Results and Discussion

The optimum Pareto reliability-based solution set is presented in Figure 5a, while its response surface graph with surface plot and contour plot are illustrated in Figure 5b. Some selected optimum solutions are shown in

Table 2. Some selected optimum reliability-based design results of the aircraft wing design.

Sol. No.	πfmax	Mass (kg)	Vf (m/s)	η	umax (m)
1	0.4637	96.5727	446.6991	1.0636	0.0945
2	0.4363	95.9038	85.5474	1.0598	0.0949
3	0.2656	121.5729	581.5058	0.9614	0.0948
6	0.1732	101.7567	478.5590	1.0566	0.0965
8	1	126.0044	632.7823	1.0059	0.0435
10	0.0010	127.8439	419.6791	1.0458	0.0997
15	0.0390	89.3820	285.6084	1.0628	0.0885
18	0.4907	99.0811	578.8716	1.0032	0.0906
19	0.9715	106.3554	584.9849	1.0375	0.0808

and Figure 6. The result of the deterministic solution is represented with PSI π_f^max = 1. The results of some selected allowable PSIs of the composite aircraft wing design are shown as the remaining data in the same Table using a clustering technique [12]. From Table 2, it can be seen that the aeroelastic results and structural constraints can be designed to meet the proposed constraints in (10). The PSIs vary from 0.001 to 1. The contour plot (Figure 7) shows that the allowable PSIs increase with the decrease in the total mass of the aircraft structure, as illustrated with a red arrow, but a contrary case is observed at the corner zone of the Figure (depicted with a dotted circle line). The graph shows that the reliability-based design group (π_f^max < 1) obtains a higher mass when compared with the group with the deterministic design. This means the group with lower allowable PSIs generates more safe structures. Another observation from the data (Figure 7) is that the PSI values for flutter speed are not in the phase of mass. This means that it is not always true that the group with the highest mass is better equipped to endure the unstable speed (flutter speed) than the group with lower mass. According to previous studies, flutter speed is not affected by mass [57], but it is affected by ply orientation. In addition, the present findings are in accordance with our earlier research [47], which demonstrated the relationship between PSI values and reliability values (R). The decreasing π_f^max and higher R indicate a safer structure. The present study establishes the relationship between these two.

Table 3. Selected optimum design variables from Table 2.

Design Variables and Objective Function	πfmax = 0.9715 (sol.19)	πfmax = 0.4637 (sol.1)	πfmax = 0.001 (sol.10)	Deterministic (sol.8)
Upper skin layer 1 (mm.)	0.0045	0.0005	0.0040	0.0045
Upper skin layer 2 (mm.)	0.0025	0.0030	0.0030	0.0025
Upper skin layer 3 (mm.)	0.0010	0.0015	0.0005	0.0010
Lower skin layer 1 (mm.)	0.0005	0.0010	0.0050	0.0015
Lower skin layer 2 (mm.)	0.0010	0.0005	0.0005	0.0005
Lower skin layer 3 (mm.)	0.0035	0.0040	0.0005	0.0020
Lower skin layer 1 (deg.)	135	165	135	60
Lower skin layer 2 (deg.)	15	15	0	15
Lower skin layer 3 (deg.)	105	120	105	105
Upper skin layer 1 (deg.)	105	105	120	105
Upper skin layer 2 (deg.)	105	105	165	105
Upper skin layer 3 (deg.)	165	165	165	165

presents a few chosen design elements from the results listed in Table 2. This Table focuses on the effect of the composite skin of the aircraft wing on flutter speed. The outcomes illustrate that all configuration layups are anti-symmetric layups. The cross-ply orientations at 45° (grey highlight) are obtained in a group with the reliability design, while the deterministic design obtains orientations at 60° (green highlight) and 90° (yellow highlight). In some cases, a group of deterministic designs randomly obtains orientations at 45°, 60°, and 90°. These outcomes are in accordance with the previous study [47]. Related to the work [57], can be concluded that the less reliable layups are those with 60° and 90°, while layups with 30° and 45° have high reliability. In agreement with the results in Figure 6, the minimum variation in flutter speed occurs in the case with the highest reliability, which is the layup with 45°. In general, the 90° cross-ply orientations obtain medium bending and torsional strength, while the 45° cross-ply orientations obtain flexible and high torsional strength. Previous knowledge is in accordance with the results that the more reliable obtain higher transverse displacement than the less reliable. Figure 8 and Figure 9 show a few of the chosen aircraft wing structures from Figure 6 with varying π_f^max values that lead to varied aircraft geometries; the lower the π_f^max, the safer the structure. Table 3 and Figure 9 illustrate that some of the skin thicknesses of the high-reliability group realize an approach to their upper limits (dark grey highlight) in comparison to the deterministic design.

The conclusions are presented in

Table 4. The number of ribs and piece number of spars for some selected aircraft wing structures from Table 3 and Figure 8.

Sol. No.	Number of Ribs		Total	Piece Number of Spars		Total
	Patial	Full		Partial	Full
1	5	5	10	-	2	2
2	4	5	9	1	2	3
3	3	5	8	1	2	3
6	6	4	10	-	2	2
8	5	3	8	1	2	3
10	5	5	10	-	3	3
15	7	1	8	-	2	2
18	3	5	8	1	2	3
19	5	4	9	-	3	3

, which are based on the thorough analysis of the partial topology and size optimization of the undertaken aircraft wing structure.

According to the assessment, the topology optimization of aircraft wing structure using the material density technique has found the optimum structure to be composed of incomplete partial ribs and spars, leading to a nonmachinable structure. The proposed technique demonstrated the potential to solve the incomplete rib and spar, as illustrated in Figure 8. Especially for the spar case, the exceeding spar can be accomplished using the conservative design case (sol. 1, 6, 10, 15) (grey highlight), except for the group closely associated with the deterministic design (sol. 2, 3, 8 18). This means that the partial topology and sizing combined with the proposed reliability technique can accomplish the objective. In addition, the optimum structures show the middle spar can be divided into several pieces rather than a single piece. In the case of rib, the conservative design group (sol. 1, 6, 10) can produce several partial and full ribs higher than the closely deterministic group. This means a larger number of ribs and a lower PSI value cause a safer structure. Furthermore, the proposed technique can generate ribs, which are branched spars presenting a better and safer structure. The new technique can generate almost all possible reliability-based optimum solutions of the composite aircraft wing design in one optimization run. All solutions are reasonable for the designer to select later with personal experience. Instead of comparing only an objective function and the design criteria as applied in the case of deterministic design, the selection can take into account the allowable possibility of the safety index value, mass, and design criteria.

In this study, the Goland wing structural model was used for a design demonstration of the proposed technique. The wing does not have control surfaces and high lift devices. During flight, the wing is subject to mutual interaction between aerodynamic, elastic, and inertial forces, which causes aeroelastic phenomena. The practical operation of the wing structure must take into consideration of the static and dynamic aeroelastic design criteria. Lift effectiveness is one of the important static aeroelastic phenomena as a wing with too low or high lift effectiveness will cause uncertainty in a flight control model to some extent. Flutter speed, on the other hand, represents dynamic aeroelastic instability that must be avoided. One optimal design solution is chosen to analyze the wing capability. The comparison between the deterministic design (sol. 8) and reliability-based design (sol. 10) of the wing with their performances are given in Table 2. It can be seen that both design solutions meet all design constraints in such a way that the latter is a safer structure. However, the mass of sol. 8 is slightly higher than the deterministic design. The highest reliability solution can generate lift effectiveness and transverse displacement higher than the deterministic design, but the flutter speed prediction is lower. It means the former is more conservative than the latter. In conclusion, the proposed technique can be applied in the redesign of the aircraft wing, but more investigation is required before the technique can be used for a practical aircraft wing.

6. Conclusions

This research presents a new reliability-based topology optimization, which is called multi-objective reliability-based partial topology optimization using a fuzzy-based metaheuristic (MRBPTOFBMH). The new technique embedded the multi-objective reliability-based design optimization and partial topology and sizing for the design demonstration of the composite aircraft wing structure. To quantify uncertainties, a fuzzy method from the non-probabilistic approach was applied.

The obtained results prove that the proposed MRBPTOFBMH can generate the possible optimum topological shape and size of aircraft wing structures in one optimization. The solutions obtained included deterministic and conservative optimal aircraft wing structures. The findings reveal that the method can cooperate with a high-performance MOEA and easily apply to real-world MODO problems. Various optimum partial topological shapes and sizes of aircraft wing structures with various PSI values in the range of [0.001, 1.0] are realized in a single run. This proves the high effectiveness of the suggested methodology to alleviate the complexity of unconventional aircraft wing structure design. Furthermore, the proposed technique can generate possible aircraft wing structures in a range of mass m = [89.38–127.84] kg, and flutter speed V_f = [85.55–632.78] m/s, in which all constraints are met. The present study revealed that the group with lower allowable PSIs generates more safe structures with higher mass, but it is not better equipped to endure the flutter speed than the group with lower mass. The main that withstands flutter speed is equipped with composite skin in both configuration layups and cross-ply orientation. The design results found that all the configuration layups are anti-symmetric layups, while the cross-ply orientations at 45° can obtain a flexible structure and a higher reliable flutter speed. The partial topology and sizing combined with the proposed reliability technique can accomplish the incomplete rib and spar and nonmachinable structure. In addition, the optimum reliable wing structures are composed of several pieces of the middle spar and several partial and full ribs, which are branches of the spars.

Nevertheless, the technique still needs to improve its processing to assign the desired PSI at an initial stage, which is a future improvement. The PSI value can be assigned by changing the MORBDO into a two-step approach, but the second step is performed with a multiple-run technique. The desired PSI value can assign at the last step.