ORIGINAL PAPER
doi: 10.5028/jatm.v10.934
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A New Approach for Robust Design Optimization Based on the Concepts of Fuzzy Logic and Preference Function Ali Reza Babaei1, Mohammad Reza Setayandeh1
Babaei AR Setayandeh M
http://orcid.org/0000-0002-8092-2931 http://orcid.org/0000-0002-8541-1495
How to cite Babaei AR; Setayandeh MR (2018) A New Approach for Robust Design Optimization Based on the Concepts of Fuzzy Logic and Preference Function. J Aerosp Technol Manag, 10: e3618. doi: 10.5028/jatm.v10.934.
ABSTRACT: In this study, an efficient methodology is proposed for robust design optimization by using preference function and fuzzy logic concepts. In this method, the experience of experts is used as an important source of information during the design optimization process. The case study in this research is wing design optimization of Boeing 747. Optimization problem has two objective functions (wing weight and wing drag) so that they are transformed into new forms of objective functions based on fuzzy preference functions. Design constraints include transformation of fuel tank volume and lift coefficient into new constraints based on fuzzy preference function. The considered uncertainties are cruise velocity and altitude, which Monte Carlo simulation method is used for modeling them. The non-dominated sorting genetic algorithm is used as the optimization algorithm that can generate set of solutions as Pareto frontier. Ultimate distance concept is used for selecting the best solution among Pareto frontier. The results of the probabilistic analysis show that the obtained configuration is less sensitive to uncertainties. KEYWORDS: Robust design, Optimization, Fuzzy logic, Preference functions.
INTRODUCTION Engineering design cycle is a process of the subsystems formulation to meet human needs. To increase design performance, it is necessary to use the design optimization methodologies in the systems design (Hao et al. 2015). Complex systems usually have multiple conflicting disciplines, which difficult its compromise. Moreover, sequential or single objective optimization will result in low-efficiency solutions or even non-optimal solutions (Huang et al. 2008). Multi-objective optimization is a method that optimizes a group of objective functions simultaneously (Huang et al. 2006). In today’s world, systems should also be robust in the face of uncertainties. In other words, the performance of the optimal solutions must have fewer changes because of the variations of design variables and operational conditions. So a measure of robustness should be introduced during optimization and design (Gaspar-Cunha and Covas 2008). Taguchi introduces robust design at first to enhance product quality so that the quality is not sensitive to variations (Wan et al. 2011). The purpose of this method is to minimize the deviation of system response due to uncertainties. In classical design methods for considering uncertainty in a system design, the designers apply drastic tolerances and large safety factors (Jun et al. 2011). Assigning the values of these parameters is based on past experience of designer and has these drawbacks: a) specifying the values of safety factor for new systems and materials without any experience is difficult; b) in the design process, it is not easy to measure robustness and 1.Malek-Ashtar University of Technology – Department of Mechanical and Aerospace Engineering – Shahinshahr/Isfahan – Iran Correspondence author: Ali Babaei | Malek-Ashtar University of Technology – Department of Mechanical and Aerospace engineering | Ferdowsi Street – Malek-Ashtar University of Technology | CEP: 83.145-115 – Shahinshahr/Isfahan – Iran | E-mail:
[email protected] Received: Jun. 4, 2017 | Accepted: Sept. 19, 2017 Section Editor: Paulo Greco
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reliability, so the designer cannot compare the system from standpoint of resistant and optimality; c) using this method in the optimization problem limits design space and therefore the new design is very conservative. So with attention to these drawbacks, it is necessary to use robust design optimization for considering uncertainty in system design (Roshanian et al. 2012). Fuzzy logic is a methodology based on the experience of human and has been developed to deal with vague and uncertain systems. Fuzzy theory is a systematic process to convert the experience of human into nonlinear mapping and it is an important aspect of this methodology. This technique is used as a modeling method for complex systems. The main core of a fuzzy system is a set of “If-then” rules obtained from the experience of experts. Some of the advantages that make this theory as an efficient technique in engineering applications are proper simplicity and speed, no need for any complex calculations, finding acceptable answers in a short period of time, and using the experience of experts (Wang 1997). Jaeger et al. (2013) proposed a new method for considering uncertainties in the aircraft conceptual design phase. This method is based on the uncertainties distribution so that their standard deviations are variable. Uncertainties have been considered in design and model variables as probabilistic setting and the uncertainties update from the historical database at each step of the optimization process. Shah et al. (2015) presented a robust optimization algorithm for airfoil design under mixed uncertainties (inherent and epistemic) using by the multi-fidelity approach. This method uses stochastic expansions to generate surrogate models. To reduce the computational cost, the high-fidelity CFD model is used. Probabilistic modeling and interval analysis methods are combined to uncertainty modeling. Dashilewicz et al. (2011) studied the disciplinary uncertainty effects in multi-objective optimization for aircraft conceptual design. By analyzing the Pareto frontier, the decision makers can judge between the expected performance and resistance so they can determine regions of design space that are proper or dangerous. Zhang et al. (2012) studied an effective computational approach for aerodynamic robust optimization under aleatory and epistemic uncertainties using by stochastic expansions that are based on non-intrusive polynomial method. The stochastic surfaces are used as surrogates in the optimization process. The stochastic surface is a function of the design variables and uncertain variables. In this paper, two stochastic optimization formulations have been investigated: a) optimization under pure aleatory uncertainty; and b) optimization under mixed uncertainty. Mach number is the aleatory uncertain variable, and a k factor multiplied by the turbulent eddy viscosity coefficient is the epistemic uncertain variable. Messac and Ismail-Yahaya (2002) developed a physical programming-based robust design optimization (PP-RDO) method. This technique is based on physical programming and robust design optimization methods that allow the designer to express robustness wishes in physical meaningful terms. Du and Wang (2016) discussed a method with dynamic surrogates models based on fuzzy clustering analysis to improve the efficiency of uncertainty analysis and reduce the effect of error propagation on uncertainty models. Fuzzy clustering analysis applied to the sample points before the generation of surrogate models. The results show that the amount of uncertainty analysis can be decreased effectively, while the optimized performance can satisfy the reliability and robustness. Bashiri and Hosseininezhad (2009) proposed a methodology for optimizing multi-response surface in robust design that optimizes mean and variance by applying fuzzy set theory. A fuzzy programming method is expressed to solve the problem by applying the degree of satisfaction from each objective. Park et al. (2011) proposed the reliability-based design optimization (RBDO) and possibility-based design optimization (PBDO) methods to handle uncertainties in multidisciplinary design optimization when low fidelity analysis tools are used. The RBDO method uses probability density function and PBDO method uses fuzzy input for uncertain parameters. Zhao and Xue (2010) studied a new parametric design approach with non-deterministic neural network and fuzzy relationships for considering uncertainties. In this paper, probabilistic uncertainties are presented by neural network relationships and possibility uncertainties are presented by fuzzy relationships. This new parametric design method has been used in the design of a solid oxide fuel cell system. In this study, a robust design optimization methodology based on the concepts of fuzzy logic and preference function is proposed. Preference function is a function that shows the satisfactory of system response. In this paper, the preference functions of objective functions and constraints are formed using fuzzy logic. Using the experience and knowledge of experts (designers) J. Aerosp. Technol. Manag., São José dos Campos, v10, e3618, 2018
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with preference functions definition during design optimization is one of the great advantages of this method, used for wing robust optimization of Boeing 747. The optimization algorithm is the non-dominated sorting genetic algorithm (NSGA). Objective functions are wing weight and wing drag force, and lift coefficient and the volume of fuel tank are constraints of the optimization problem. Cruise altitude and velocity are uncertainties modeled by Monte Carlo Simulation (MCS) method. Finally, a comparison is done between optimum, robust, and baseline configurations from the optimally and robustness viewpoint.
PROPOSED METHODOLOGY FOR ROBUST DESIGN OPTIMIZATION The proposed method is presented in Fig. 1. This method consists of six blocks that include: definition of objective functions, constraints and parameters (block 1); uncertainty simulation (block 2); modeling of objective functions and constraints and calculation of the mean and standard deviation (block 3); fuzzy performance function (block 4); optimization (block 5); and final solution selection (block 6). Block 1 Design variables definition Design parameters calcultion Objective function definition Constraints definition
Block 2 Uncertainty simulation
Block 3 Objective functions (J)
Constraints (C)
Calculation of the mean and standard deviation
Fuzzy preference function of objective functions
Fuzzy preference function of constraints Block 5
Optimization Yes Block 6
Pareto frontier and final selection
No
Figure 1. Flowchart of the proposed methodology.
PARAMETER DEFINITION For any optimization objective functions, design variables, design parameters, and the constraints of optimization problem must be determined at first. Design variables are parameters that describe the obtained design from optimization algorithm. Design parameters are parameters which are dependent on design variables and these parameters must be calculated to compute the objective functions and constraints. The objective functions and constraints are criteria dependent on design variables and/ or design parameters.
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UNCERTAINTY SIMULATION AND MODELING OF OBJECTIVE FUNCTIONS AND CONSTRAINTS There are many approaches to modeling uncertainty, from which one of the most famous is MCS. This method was developed in 1940 by Stan Ulam and John Von Neumann and is a computational algorithm that performs repetitive sampling and simulation. Unlike the many probability methods, this technique needs to a little information about the statistic and probability. If the sufficient number of sample points to be used, the results of MCS are completely accurate. So this method is a standard reference for evaluating other methods (Yao et al. 2011). The flowchart of MCS is shown in Fig. 2.
Sampling points
Sampling outputs
Figure 2. Monte Carlo Simulation.
To do simulations for different sample points, the modeling of objective functions and constraints should be performed at first. After simulation, the mean and standard deviation of objective functions and constraints are produced. These values are inputs to the fuzzy system to form preference functions (see Fig. 1). FUZZY PREFERENCE FUNCTION Brief Description of Fuzzy Logic Fuzzy systems are knowledge-based systems. The core of a fuzzy system is a knowledge base that has been consisted of fuzzy If-Then rules. A fuzzy If-Then rule is an If-Then statement in which some words are characterized by continuous membership functions. For example, the following statement is a fuzzy If-Then rule: If the speed of a car is high, Then apply less force to the accelerator where the words “high” and “less” are characterized by the membership functions (Wang 1997). The general configuration of a fuzzy system is shown in Fig. 3.
Fuzzy rule base
Crisp inputs
Fuzzifier
Fuzzy inference engine
Defuzzifier
Crisp outputs
Figure 3. Basic configuration of a fuzzy system.
It is composed of four main parts: fuzzifier, fuzzy rule base set, fuzzy inference engine, and defuzzifier. Fuzzy sets consist of a number of so-called membership functions to describe the fuzziness of variables. The fuzzifier component converts crisp input values into fuzzy sets. Fuzzy sets enter the inference engine as inputs. The decisions are made upon the fuzzy If-Then linguistic rule base set. The inference engine as a mathematical mechanism is applied to manipulate fuzzy sets and aggregate the rules to make the decisions. The outputs of the inference engine are fuzzy, but the system output must be crisp. Therefore, the defuzzifier part maps the fuzzy sets to crisp outputs (Oroumieh et al. 2013). J. Aerosp. Technol. Manag., São José dos Campos, v10, e3618, 2018
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Preference Function Creation In this block, the mean value (μ) and standard deviation (σ) of objective functions and constraints are transformed into the new objective functions and new constraints as preference functions. The reason for using this function is that experience of the designer (or experts) can be used during design optimization (by expressing his or her preferences relative to objective functions and constraints) and increases the ability to achieve more practical solutions. To create the preference function, the designer defines satisfaction degrees for any objective function (and constraint) in the horizontal axis. In other words, the designer classifies horizontal axis to different regions in terms of satisfaction of objective function (and constraint) (see Fig. 4). Then the vertical axis, which indicates the preference function, is divided into several regions same as the horizontal axis. It is worth noting that the preference function value is between zero and one, and maximization of preference function (the greatest satisfaction degree) is the purpose of optimization. In this study, the relationship between horizontal and vertical axis is created using fuzzy logic. On the other hand, the preference functions are generated using fuzzy logic. Fuzzy logic is used because we can also benefit from the experience of experts in the creation of fuzzy rules (in addition to determining the satisfaction degrees for any objective function and constraint). In other words, we can use designer experiences in design optimization using the fuzzy preference function definition by two methods: • Expressing the satisfaction degrees for objective functions and constraints. Genetic algorithms are stochastic optimization algorithms suitable for optimization of • Creating of fuzzy rules.
complex problems with unknown search space. These algorithms are programming techniques Preference function
1 that use genetic evolution as a pattern of problems solution. This is a global optimization
algorithm basedVeryongood the principle of survival of fittest, natural selection mechanism, and reproduction. Because of the genetic algorithm’s ability in solving complex problems, it is Good
selected as the optimization algorithm. Since we are dealing with more than one objective function, an
Medium Low improved0version of GA called non-dominated sorting genetic algorithm Objective function J1 J2 J3 Low Medium High Very high
is used in
this study. This algorithm finds set of optimal solutions (Pareto frontiers) by adding an essential Figure 4. An example of preference function for minimization problem
operator to general single objective GA. Figure 5 shows the flowchart of this algorithm (Babaei
OPTIMIZATION ALGORITHM AND THE FINAL SELECTION METHOD Genetic et algorithms al. 2015).are stochastic optimization algorithms suitable for optimization of complex problems with unknown search space. These algorithms are programming techniques that use genetic evolution as a pattern of problems solution. This is a global optimization algorithm the principle survival of fittest, naturaldon’t selection mechanism, and reproduction. Because of Sincebased theonsolutions inofthe Pareto frontier have the preference to each other the genetic algorithm’s ability in solving complex problems, it is selected as the optimization algorithm. Since we are dealing with more thancompletely, one objectivethe function, an improved version of GA is called genetic is used in this concept of utopian distance usednon-dominated in this studysorting to select thealgorithm final solution. This study. This algorithm finds set of optimal solutions (Pareto frontiers) by adding an essential operator to general single objective distance defines as follows (Eq. 1):(Babaei et al. 2015). GA. Figure 5 shows the flowchart of this algorithm Since the solutions in the Pareto frontier don’t have the preference to each other completely, the concept of utopian distance is used in this study to select the final solution. This distance defines as follows (Eq. 1):
𝑑𝑑$% = ∑1234 ((
*+ ,*-.+ *-.+
)0
(1)
(1)
where: dut is utopian distance; i is the number of objective function; and fut is the utopian value obtained from a single objective
where: dut is utopian distance; i is the number of objective function; and fut is the utopian value
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obtained from a single objective optimization for each objective function. Among the Pareto
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optimization for each objective function. Among the Pareto frontier, the solution that has the lower value of dut is the best solution because the standpoint of objective functions is closest to the utopian values. cost1 = f1 (population) cost2 = f2 (population)
[cost1, ind1] = sort (cost1)
Reorder based on sorted cost1 Cost2 = cost2 (indx1)
Population = Population (indx1)
Rank = 1
Assign chromosome on pareto front Cost = rank
Remover all chromosome assigned the value of rank from population
Rank = Rank + 1
For chromosome n Cost (n) = Rank
Tournament selection
Population
No
Converge Yes Figure 5. The flowchart of NSGA algorithm.
APPLICATION OF THE PROPOSED METHOD FOR WING ROBUST DESIGN OPTIMIZATION Definition of Objective Functions and Design Variables In this study, two design optimizations have been done for wing optimization, which includes: deterministic optimization (DO), and robust optimization (RO). In the first case, the problem can be formulated as follows (Eq. 2): J. Aerosp. Technol. Manag., São José dos Campos, v10, e3618, 2018
A New Approach for Robust Design Optimization Based on the Concepts of Fuzzy Logic and Preference Function
Minimize Ww & Drcr
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(2)
subject to V & CL where: WW is wing weihgt, Drcr is wing cruise drag, V is fuel tank volume, and CL is lift coefficient. In the second case, the problem can be formulated as follows (Eq. 3):
Maximize FWw & FDr
cr
subject to FV & FC
(3)
L
where: FWw , FDr , and FC are the preference functions of wing weight, wing cruise drag, fuel tank volume, and lift coefficient, cr L respectively. Because the main purpose of this paper is to present a new method for robust design and for simpler presentation of the application of the proposed method, only three variables are considered. Wing root chord (Cr), wingspan (b), and sweepback angle (Λ0.5) are considered as design variables for both optimizations. Numerical ranges of these variables have been shown in Table 1. Table 1. Limits for design parameters. Design arameters
Cr
b
Λ0.5
Upper limits
20
75
45
Lower limits
10
55
25
UNCERTAINTY SIMULATION Aircraft design usually is accomplished based on a certain mission. Often every aerial vehicle is designed for a specific mission with certain cruise altitude and speed. It is very economical that one aircraft can be optimum or near-optimum for other missions. In other words, it is suitable that one aircraft is designed for a range of cruise altitude or speed. This topic could also be the case for transport jet. Therefore, cruise altitude and flight speed are considered as uncertainties in this research. A normal distribution is used to generate sampling points (X = Xm + (∆x/3) randn (1,N)). In this study, the values of hm, Vm, Δhm and ΔV are considered 25000 ft, 762 ft/s, 10000 ft and 182 ft/s, respectively. The values of hm and Vm are selected based on cruise condition of Boeing 747, and the values of Δh and ΔV are selected based on designer experiences. MODELING OF OBJECTIVE FUNCTIONS AND CONSTRAINT As already noted, wing weight and drag force of Boeing 747 are considered as objective functions while fuel tank volume and lift coefficient are as constraints. The considered condition for wing design optimization is cruise phase. It is necessary to extract equations of wing weight, drag force, fuel tank volume and lift coefficient. Like all modeling, modeling of this research has been done with consideration of assumptions. It is worth noting that some parameters are considered constant (such as taper ratio, Mach number, dynamic pressure, wing incidence angle, cruise angle of attack, fuselage diameter, etc.) are related to Boeing 747. In robust optimization, the parameters like Mach number and dynamic pressure are considered variable in modeling process (unlike the deterministic optimization). Modeling of Wing Weight The wing weight of an aircraft can be calculated from Eq. 4 (Sadraey 2012):
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i 1 process (unlike the deterministic f utf i foptimization). n i W uti &2 Dr Minimize w ) cr d ut i 1 ( Minimize W & f w ut to Modeling Vi & C L Drofcr Wing Weight Minimize Wsubject w & Drcr Minimize Minimize Ww to &W Dr subject V w& CDr cr& L cr subject to V & C L Minimize W Dr subject subject to to CcrV & C wV&& The wing weight of an aircraft can be L L calculated from Eq. 4 (Sadraey 2012): Babaei AR; Setayandeh MR subject toWV & CL Minimize Maximize w & Drcr FW & FDr w cr Maximize FWw & FDrcr subject to V & C L to FV & FC L Maximize Fsubject Ww & FDrcr Maximize FWwto&F &FFDrcr subject FFWVDrw& % Maximize cr ij . 1 C L -k. q.r q.qt 𝑊𝑊= = 𝑆𝑆subject . 𝑀𝑀𝑀𝑀𝑀𝑀. ( ) . 𝜌𝜌 . 𝐾𝐾 . ( ) 𝜆𝜆 𝑔𝑔 (4) (4) to F & F = Zd% Maximize &to AV Zde Dr subject subject toFCWL F &FF Fhcr &Alm (n F o.p ) w V
CVL
CL
subject to FFWV & FCDrL Maximize w cr AR . nult 0.6 0.04 t ) ratio,gρmat is density of Ww (t/c)max S w .MAC .( maximum )max . mat .K .( where: SW is wing area, MAC is mean aerodynamic chord, is the of thickness to. n chord AR t subject to FV & FC L .04 ult cos( ) c )0.g6 is0gravity g constant. W S . MAC .( ) . . K .( 0 5 . AR n t w n is w ultimate load maxfactor, mat .6 taper .04 ratio and ult λ 0is construction material, Kρ is density factor, AR is aspect ratio, %0cos( AR . n AR . n ult . t .K .(tc ) g W S . MAC .( ) ) 0 6 0 04 0 6 0 04 . . . . ult ult max mat 0 5 . where: SW is wingwarea,w W MAC is mean aerodynamic chord, ( ) is the maximum of thickness area ) ) g g W S wwc.MAC S w ..( MAC )max.(.cos( )max.K. .(5 .)K (Eq. .( A5):Zde mat w wing mat If root chord, taper ratio, and span are known, then 0.AR .nultcos( tis calculated cos( c c as follows 0.5 0).6 00..504) ) g Ww S w .MAC .( )max . mat .K .( cos( ) c 0 5 . to chord ratio, ρmat is density of construction material, K is density factor, AR is aspect ratio, nult AR . n t ρ ult b )0.6 0.04 g Ww S wS.wMAC Cr.(( 1 )max ). matb.K .( (5) cos( 0.5 ) Sbw Ccr ( 1 2 ) is ultimate load factor, b g isb2gravity constant. ) ratio and S w Cλr (is 1 taper S w CS2rw( 1Cr ()1 ) b 2 2 placement of this value in Eq. 5, obtaining (Eq. 6): Taper ratio for Boeing 747 is 0.28, thereforeSthe wing and is)obtained Cr ( 1area w ratio, If root chord, taper span arewith known, then wing area is calculated as follows (Eq. 2 S w 0.64 b Cr b S w Cr ( 1S ) 0.64C b (6) r 5): S w 0.64Cr b w 2 S w 0S.64 C0r b.64Cr b w Cr b7): The mean chord of the wing is calculated asSfollows (Eq. w 0.64 2 2 0.28 C C2r ( 1 C 0.707Cr 2) S w 0v.64C b 0.28 2r 3 1 C Cr(021.28 2 ) C 0.707Cr 2 𝑆𝑆= = 𝐶𝐶 (5) (7) CB (1 +C𝜆𝜆) )23 C0.280.707 Cr 1 0.28 r ( 10 2 C 0C .707 C 0.r707Cr 3 C 1C Cr( 1 C2r ( 1 ) ) 2 3 31 1 0. 28 C Cr ( 1 ) C 0.707Cr 2 3 1 t 0.12 2 0.28 It is assumed that the airfoil of the wing isCNACA2412, 8): C( r ( 1)max ) C 0.707Cr tso (Eq. c 3 1 ( ) 0. 12 t for Boeing 747max Taper ratio is 0.28, therefore the wing area is obtained with placement of ( )max 0.t 12 tc (8) ( ( ) ) 0. 12 0. 12 c t c max c max ( )max (Eq. 0.126): this value in Eq. 5, obtaining c kg t The density of construction material and (wing density factor for the,aerospace aluminum alloy are (Eq. 9): 2711 k 0.004 )max mat 0.12 m 3 kg , k 0.004 c kg mat 2711 mat 2711 3 , k kg 0.004kg m3 (9) matm2711 , k 3 0, k.004 mat 2711 0.004 3 𝑆𝑆= = 0.64𝐶𝐶B𝑏𝑏 2711 kgm, k m0.004 (6) mat 3 max 3 nult kg 1m.5 is nmax n nult 4.5 of ultimate load factor is considered It is assumed that the maximum load factor of Boeing 747 3 and therefore thevalue nmax 3 mat 2711 , k 0 . 004 n 3 n 1 . 5 n 3 ult max ult 4.5 max (Eq. 10) (Roskam 1997): m nult 1.5 nmax n nult nmax 34.n5 max 3 nult 1n.ult 5 nmax 1.5n n 4n.ult5 4.5 max ult The mean chord isn calculated as 3 max nultofthe 1.5 wing nmax n 4 .5follows (Eq. 7): (10) b2 b ult b AR nbmax 2 3 AR 1.56 b nult2 1.5 nAR .5 AR 1.56 S 0. 64nCultbr 4b Cr b bmax 2 w 2 Note that 1.5 is a safety factor and it isAR considered The bequation bS wAR b 0requirements. b b of aspect ratio according to the 1 . 56 basedbon structural .64 C r C AR0.264 ARC r .56 1.56 r AR 1AR S C w r Eq. 6 is expressed as follows (Eq. 11): b S w 0Sb.w64 C0r .64 C r b Cr Cr AR AR 1.56 0.64 C r equation of bwing C r weight is obtained with placement above e Finally, bS2 w the bmain (11) AR AR 1.56 S w 0.64 C r Cr Eq. 4, obtaining (Eq. 12): Finally, the main equation of wing weight is obtained with placement above equations in Eq. 4, obtaining (Eq. 12):
𝑊𝑊= =
4ê.ê4VTë.ívë.ì
(12)
(îïñ óo.p )o.ì
Modeling of Wing Drag The main equation of the drag is expressed as follows (Eq. 13):
Modeling of Wing Drag
The main equation of the drag is expressed as follows (Eq. 13):
J. Aerosp. Technol. Manag., São José dos Campos, v10, e3618, 2018
(1
The main 𝐷𝐷 =equation 𝑞𝑞ô𝑆𝑆𝐶𝐶R of the drag is expressed as follows (Eq. 13): Modeling of Wing Drag
main equation of theand dragCDis isexpressed as follows 𝐷𝐷The ô𝑆𝑆𝐶𝐶 where: 𝑞𝑞= ô is𝑞𝑞 dynamic drag coefficient that(Eq. it is13): obtained from R 1.4 pressure, 17.71 Cr b1.6 09/22 xx/xx Ww of this research Since the purpose is wing optimization, therefore it is evident that wetted (cos 0.5 )0.6
A New Approach for Robust Design Optimization Based on the Concepts of Fuzzy Logic and Preference Function
area changes in addition to wing area in the optimization process and finally equivalent parasit 0 (13) 𝐷𝐷𝐶𝐶= 𝑞𝑞ô𝑆𝑆𝐶𝐶Rq where: 𝑞𝑞ô is dynamic from Eq R + 𝑘𝑘𝐶𝐶L and CD is drag coefficient that it is obtained R = 𝐶𝐶pressure, area and zero-lift drag will change. So, to calculate zero-lift drag coefficient, an D coefficient q SCD
– where: q is dynamic pressure, and CD is drag coefficient that it is obtained from Eq. 14:
equation for the wetted area must be determined at first and then according to Eqs. 15 and 16 Since 𝑞𝑞ôthe purpose ofpressure, isD wing optimization, therefore it is evident th 0this research where: is and C is drag coefficient it is(14) obtained 𝐶𝐶R In = dynamic 𝐶𝐶Rq14, + 𝑘𝑘𝐶𝐶 Eq. CD0 is zero-lift coefficient that can bethat calculated as (Eq.from 15) ( 2L CD can CDbe kCL 0 calculated. zero-lift drag coefficient The wetted area is shown in Fig. 6. area changes in addition to wing area in the optimization process and finally equivalen
In Eq. 14, CD0 is zero-lift coefficient that can be calculated as (Eq. 15) (Roskam 1997):
area and zero-lift drag* coefficient will change. So, to calculate zero-lift drag coeffi 0 𝐶𝐶R𝐶𝐶14, = 𝐶𝐶 In Eq. CRq is 𝑘𝑘𝐶𝐶 zero-lift coefficient that can be calculated as (Eq. 15) (Ros L (15) D0 + Rqf = é CD 0 equation for the wetted area must be determined at first and then according to Eqs. 15 S
where: f is equivalent parasite area. This parameter is obtained as follows (Eq. 16):
zero-lift drag coefficient can be calculated. The wetted area is shown in Fig. 6. * In Eq. zero-lift coefficient that can be calculated as (Eq. 15) (R 𝐶𝐶Rq = 14, CD0 is where: parasite area. This parameter is obtained (16) as follows (Eq. éa b log S wet logf10fisequivalent 10
where: a and b are constant, and Swet is the wetted area. Figure 6. Definition of wing wetted area. 6 * it is evident that wetted area changes in addition to wing area Since the purpose of this research is wing17 optimization, therefore .71Cr1.4 b𝐶𝐶1.Rq = éQõ.This parameter is obtained as follows (Eq. 16): * 2( parasite C C r )area. f is equivalent W where:parasite 𝑙𝑙𝑙𝑙𝑙𝑙 𝑎𝑎t + 𝑏𝑏𝑙𝑙𝑙𝑙𝑙𝑙 C [é = x 4q 4q in the optimization process and finally wequivalent and zero-lift drag]coefficient will change. So, to calculate zero-lift r (cos 0.5 y)0.6area b drag coefficient, an equation for the wetted area must be determined at first and then according to Eqs. 15 and 16, zero-lift drag Toarea find the equation coefficient can be calculated. The wetted is shown in Fig. 6. for the wetted area in terms of design variables, it is necessary to
éQõ. area. This parameter is obtained as follows (Eq. 16 * where: f is parasite = 𝑎𝑎 + 𝑏𝑏𝑙𝑙𝑙𝑙𝑙𝑙 (4q Ct C a4qequivalent and bchord are constant, and Sfwet is the the wingspan. wetted area.This equation is obtaine find an to 𝑙𝑙𝑙𝑙𝑙𝑙 express distribution along D equation q SCDwhere: r )d S wet S wet [ Cr Ct Figure w ][ b of d fwing ] wetted area. 6. Definition b using a simple geometric relationship as follows (Eq. 17):
2 éQõ. * CD where: CD 0 kC =constant, 𝑎𝑎 + 𝑏𝑏𝑙𝑙𝑙𝑙𝑙𝑙 L 𝑙𝑙𝑙𝑙𝑙𝑙 a and b are and is the wetted area. 4q 4q6.S wet C C ) area in terms of design variables, it is nec To find the( equation for the 5(wetted 2.5229 log [ C C ][ b 6.5 ]) b f0( V 10 . , VT ) 10
r
t
t
r
𝑦𝑦 = an 𝐶𝐶Bequation +[ ]𝑥𝑥 (17) find v to express chord distribution along the wingspan. This equation is
Figure 6. Definition of wing wetted area. f C Dusing where: a and b are constant, andasSwet is the(Eq. wetted 0 relationship 17):area. 6.5( Cfollows S a simple geometric t Cr ) ][ b . ]) 2 5229 6 5 ( . log [ C C t To find the equation for the wetted area in terms of design variables, 10it isr necessary b to find an equation to express chord 10 C distribution along the wingspan. This equation is obtained simple relationship as follows (Eq.the 17):wetted area, the chord i D0 using aand where: y is chord distribution, x isgeometric along wing span. To find 0.64Cthe rb S wet f 0( V , V ) ülog (17) † 10 a b 𝑦𝑦log = 10𝐶𝐶 + [ . T ]𝑥𝑥 (17)
𝑥𝑥 =
0
is the only unknown parameter that is obtained from Eq. 17. Fuselage diameter is 6.5 B v
2area, AR the chord in x = d /2 is the only unknown where: y is chord distribution,meter and x for is along the wing To find wetted Cspan. Boeing 747. thethe main equation is obtained forf the wetted area (Eq. 18): LFinally, 2 2 2 parameter that is obtained from Eq. 17. Fuselage diameter is 6.5 meter for Boeing 747. Finally, main equation is obtained for tan the AR 2( Ct Cr ) 0.5 2 4 ( 1 ) y C [ ] x 2 2 r the wetted area (Eq. 18): where: y is chord k x is along the wing span. To find the wetted area, the b distribution, and ü
(V
)ü
. ,VT † 𝑥𝑥𝑆𝑆=°% = †=is¢𝐶𝐶the only unknown parameter Fuselage £ [𝑏𝑏 − 𝑑𝑑* ] that is obtained from Eq. 17. (18) (18) diame B + 𝐶𝐶% + 0 v
( Ct Cr )d f S wetmeter [ Cfor Ct 747. Finally,][the b main d f ] equation is obtained for the wetted area (Eq. 18) r Boeing b where: Ct is wing tip chord, and df is fuselage diameter. The equivalent parasite area (Eq. 19) and zero-lift drag coefficient (Eq. 20) can be calculated using Eqs. 6, 15, 16 and 18. 6.5( Ct C r (V ) . ,VT )ü† ][ b 6.5 ])
𝑆𝑆=°% = ¢𝐶𝐶B + 𝐶𝐶%b +
( 2.5229 log10 [ C r Ct
f 10
( 2.5229 log10 [ Cr Ct
CD0
10
v
£ [𝑏𝑏 − 𝑑𝑑* ]
(19)
J. Aerosp. Technol. Manag., São José dos Campos, v10, e3618, 2018 6.5( Ct Cr ) ][ b 6.5 ]) b
0.64Cr b
(18)
To (©|.p||´¨k≠Æ calculate the dragì.p(® coefficient according to Eq. 14, an equation for lift . ©®T )∞[™©ì.p]) Ø® ¨® ¨ ëo
4q
10/22 xx/xx
Babaei AR; Setayandeh MR
T
.
™
𝐶𝐶Rq =To calculate theq.rtV drag according Eq. 14, an equation for(20) lift c T v coefficient be achieved. The lift coefficient is assumed as Cto L0 + CLα, where CL0 is lift co (,0.¶00ß}àlÖ
}V }
ì.p(®. ©®T )
£[v,r.¶]) ëo ¢VT . ™ 𝑓𝑓 = 10 be (19) achieved. The lift assumed + Cangle CL0 is The lift coe Lα, where angles of attack, CLα coefficient is lift curveis slope andasα CisL0the of attack. lif
angles of attack, CLαcoefficient is lift slope1997). and α is14,the of attack. The lift To calculate drag according to Eq. an angle equation for lift coefficie calculated asthe follows (Eq. 21)curve (Roskam )
ì.p(®. ©®T Ø®coefficient ∞[™©ì.p]) (©|.p||´¨k≠Æ ¨®. ¨ (Eq. calculated 21) (Roskam be achieved. The Cas lift is assumed CL0 + CLα, where CL0 is(20) lift coefficient Tairfoil ëofollows where: M is Mach number and is lift slope. as1997). lα ™ curve 4q 𝐶𝐶Rq = (20) q.rtVT v 0≤ij anglesnumber of attack, CLα is lift slope condition and α is the angle therefore, of attack. 𝐶𝐶The lift iscurve The Mach of 747curve in cruise is 0.85, à± ª 𝐶𝐶 Boeing =
L±
0}(t}
d% Ω
.ã∑|∏o.p ≥¥|µ| (4} ) ∂| µ|
0≤ij To calculate the drag coefficientcalculated according toasEq. 14, an=equation lift coefficient must be achieved. The lift coefficient is follows (Eq. 21)for(Roskam 1997). 𝐶𝐶 L±The value ≥¥ |µ calculated as follows (Eq. 24). of 𝐶𝐶|à± ªd% Ω3q is assumed to be equal to the value of the .ã∑|∏o.p ( assumed as CL0 + CLα, where CL0 is To lift coefficient zerodrag angles of attack, is lift curve angle of attack. lift 0} t} C (4} )slope | Lα | to calculateatthe coefficient according Eq.and 14,α is anthe equation for liftThe coefficient must ∂
µ
curve slope is calculated as follows (Eq. 1997). and Clα is airfoil lift curve slope. where: M21) is (Roskam Mach number
NACA 2412. be achieved. The lift coefficient isk and assumed CL0 + from CLα, where lift23. coefficient at zero In Eq. 21, β are as obtained the Eqs.CL0 22isand 0≤ij 𝐶𝐶 = The Mach number 𝐶𝐶à± ª (21) L± (21) |of | Boeing .ã∑|∏o.p747 in cruise condition is 0.85, therefore, d% Ω (t}≥¥ |µ (4} 0} ) | angles of attack, CLα isInlift andobtained α is thefrom angletheofEqs. attack. The23.lift curve slope is Eq.curve 21, k slope andµβ are 22 and ∂
calculated 𝐶𝐶as follows (Eq. 24). The value of 𝐶𝐶à± ª is assumed to be equal to the value of ª d% Ω3q 𝐶𝐶à± = √1 d% Ω √1 − 𝑀𝑀0 NACA 2412.In Eq. 21,𝛽𝛽k=and β−are 0 from the Eqs. 22 and 23. √1 𝑀𝑀obtained (22) ©ë 2 Vk± |º¿o 3r.q0 Bdü ø≥®≥ |íë| 1 M ,4 0≤ij ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯Å 𝐶𝐶à± ª = 11.43 𝑟𝑟𝑟𝑟𝑟𝑟 (24) d% Ω3q.~¶ 𝐶𝐶L± = (21) V ∏ª .ã∑|k± ≥¥|µ| ã. º o.p (23) 0}(t} | (4} ) 𝑘𝑘 = | ∂ µ 0≤ à± ª− 𝛽𝛽 = 𝐶𝐶√1 𝑀𝑀V0k± ªã. º (22) d% Ω3q 𝐶𝐶à± ª C= 𝑘𝑘 = 0 d% Ω l 0≤ − 𝑀𝑀 at√1 M where: M is Mach number and Clα is airfoilk lift slope. curve With placement of Eqs. 22, 23, 24, and aspect ratio in Eq. 21, the final equation (Eq. 25) 2 The Mach number of Boeing 747 in cruise condition is 0.85, therefore, Clα |atM is calculated as follows (Eq. 24). The value of ©ë Vk± |º¿o 3r.q0 Bdü In Eq. 21,of ktheand β are obtained from the Eqs. 22 and ,4 23. ø≥®≥ |íë| Clα |atM = 0 is assumedistoobtained be equal tofor thelift value NACA 2412. ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯Å 𝐶𝐶à± ª = 11.43 𝑟𝑟𝑟𝑟𝑟𝑟 (24) curve slope. Vk± ª d% Ω3q.~¶ ã. º 𝑘𝑘 = 0≤ (23) Cl at M 0 Cl 0 at M 𝛽𝛽 = √1 − 𝑀𝑀 ™1| M 2 (22) 0≤ With placement of Eqs. 22, 23, 24, and aspect ratio in Eq. 21, the final equation (Eq. 2 (24) ¬ M 0 1 𝐶𝐶L± = (25) Cl NACA 2412 1 í 6.02 rad ™ | Cl ) 11.43 rad 0}( t}q.q~t | (4}^.r %d1 óo.p is obtained for lift ¬curve slope. at M 0.85 à± d% Ω3q as follows 21) (Roskam0 1997). In Eq. 21, k and β arecalculated obtained 22(Eq. and𝛽𝛽23. ª from the Eqs. = − 𝑀𝑀
Vk± ª
𝑘𝑘 = With placement of Eqs. 22, 23, 24, and aspect ratio in Eq. 21, the final equation (Eq. 25) is obtained for lift curve slope. 0≤ ã. º
(23)
b 2of attack is not constant, similar to zero-lift drag | 2 The lift coefficient for zero ™angles 0≤ S ¬ 𝐶𝐶LC±L= (25) (25) ™í 4 | ó process. (t}q.q~t b coefficient and changes 0} during the ¬optimization This parameter is calculated as follows o.p ) 2 | (4}^.r %d1 2 4 0.084 2 ( 1 3.6 tan 0.5 ) S (Eq. 26):
The lift coefficient for zero angles of attack is not constant, similar to zero-lift drag coefficient and changes during the optimization process. This parameter is calculated as follows (Eq. 26):
The lift coefficient for zero S angles of attack is not S constant, similar to zero-lift dr C L 0 C L 0 wf C L h h ( h )( ih 0 h ) C L c c ( c )( ic 0 c ) S S é√ é coefficient and+changes This calculated as follo (26) 𝐶𝐶Lo = 𝐶𝐶Lo Q† 𝐶𝐶L±√ 𝜂𝜂≈ ∆during « (𝑖𝑖≈ the − 𝜀𝜀optimization ∆ S « (𝑖𝑖A − 𝜀𝜀qA )parameter is (26) q≈ ) + 𝐶𝐶L±S 𝜂𝜂A process. é
(Eq. 26):
é
C L wf ( of iw wing 0body, where: CL0 is lift coefficient at zero angles C ofLattack combination 0 Cfor L 0 wf LW ) is horizontal tail lift curve slope, ηh is horizontal wf tail efficiency, Sh is horizontal tail area, ih is horizontal tail incidence angle, ε0h is downwash angle at the horizontal tail, is canard lift curve slope, ηc is canard efficiency, Sc is canard area, ic is canard incidence angle, and ε0c is up-wash angle at the canard. é éS The last term in the Eq. 26 is related to𝐶𝐶the canard that+has this𝜀𝜀q≈ study. second ) +The 𝐶𝐶Lo Q† 𝐶𝐶Lbeen 𝜂𝜂 neglected ∆ √ « (𝑖𝑖≈in− 𝐶𝐶L±S 𝜂𝜂A ∆ term « (𝑖𝑖isA related − 𝜀𝜀qA )to horizontal Lo = ±√ ≈ é é lift 0coefficient for0 l zero at M angles of attack is calculated from Eq. 27: tail, which is negligible compared to the first term. Therefore { ( ) }{ } 0 LW
0l
J. Aerosp. Technol. Manag., São José dos Campos, v10, e3618, 2018
C L 0 wf k wf C L wf
t
t
0l
at M 0.3
(26)
last term in the Eq. 26 is related to the canard that has been neglected in thi C LThe 4 b2 b 2 4 0.084 b 2 ( 1 3.6 tan 22 0.5 ) 2 The second2term is related to horizontal Stail, which is negligible compared to the fir S C L S (25) C L b4 2 b 4 coefficient2for 4 0angles .084 2of( 1attack 3.6 tan 0.5 ) from Eq. 27: 2 zero Therefore lift is calculated 2 A New 4 Approach 0.084for Robust ( 1 Design 3.6Optimization tan 0.5Based ) on the Concepts of Fuzzy Logic and Preference Function S xx/xx 11/22 S2 Sh Sc C L 0 C L 0 wf C L h h ( )( ih 0 h ) C L c c ( )( ic 0 c ) S S Sh S 𝐶𝐶Lo = (27) S 𝐶𝐶LCo Q† =C(𝑖𝑖=wf + 𝛼𝛼 S(L± Q† CqLP)𝐶𝐶 )( ih 0 h ) C L c c ( c )( ic (27) 0c ) C L 0 C L 0 wf C L h h ( h )( ihL 0 0 h L)0 C L cL ch ( h c )( i ) (26) S c S 0c S S C L 0 C L 0 wf C L wf ( iw 0 LW )
where: iw is wing incidence angle, α0LW is wing angle of attack for zero lift and is wing body lift curve slope. The incidence angle is 2° for Boeing 747iand is C calculated using incidence angle, α(0LW where: Eq. C L28: iw iswing w isαwing 0LW L0 CL0 wf 0 LW ) angle of attack
C L 0 C L 0 wf C L wf ( iw 0 LW )
wf
body lift curve slope. 0 ) }{ t 0 LW { 0 l (
0l
at M
}
for zero lift and 𝐶𝐶L± Q† (27)
(28)
t 0 l at M 0.3 0 l atand 0Boeing 747 The incidence angle is 2° M α0LW is calculated using Eq. 28: 0 l at M { ( for 0 ) }{ } LW l t 0 0 where: α0l is airfoil zerolift angle, Δα /εt is( the variation angle, and εt is wing twist angle. ) t }{ of wing zero lift } angle of tattack to wing (28) 0 l twist 0 LW { 00l at M 0.3 t in Eq. 28, 0the l at value With the placement of relevant parameters of this parameter is obtained –1.92. Wing body lift curve slope M 0.3 C L 0 wf k wf C L wf
is obtained as follows (Eq. 29):
C L 0 wf k wf C L wf
ÀÃ
à |
𝛼𝛼qLP = 𝛼𝛼qà + ∆ Õ o « 𝜀𝜀% Œ {à |ok ã. º } C L 0 wf k wf.C L wf ok ã. º¿o.Ü
df
df
(29)
(28) (29)
0.16 10.56 1 is0calculated )( 2 ) k wfwhich .025( from ) 0Eq. .2530: ( )2 k wf 1 ( where: kwf is wing-body interference factor, b b b b ∆Ão d d angle 10 of .56 attack where: α0l is airfoil zero lift angle,f Õ is the variation 0.16 d f 6of .5 wing zero lift f df . 0.250 .56 k 1 ( (30)) ( 2 ) kd f 1 d0f .025 ) (.16 )2 10 6.5 ( kbwf 1 ( b ) ( 2 ) wf k wf 1 0.025( ) 0.25( wf )2 b (30) b b b b b twist angle, and εt is wing twist angle. d f 6.5
The final equation for lift curve slope is (Eq. 31):
With the placement of relevant parameters in Eq. 28, the value of this param 9.82 b 0.16b 10.56 2 (31) )as follows (Eq. 29): C L wf ( obtained –1.92. bWing )( 0.body 16b lift 10.curve 56 2 slope is obtained 9.82 bC ) (31) C Lr wf ( 0.205b)( 2 0b.5 2) tan 0.205 56bC4r C 2 ( 1 39..682 b 2 0.16b 102. 2 2 4 ( 1 3.6 tan ) 0.5 ) C L wf ( )( r 2 2 Cr 0.205b 82 b 2 0.16b 10.56 bCr 2 of9.attack 4 is calculated 0.532): ( 1 as3follows .)6 tan2 (Eq. ) (31) C L wf toEqs. ( 27 and 29, lift coefficient )( for zero angles 2 Finally, according 2 C bCr r 0.205b 2 2 4 ( 1 3.6 tan 0.5 ) CL 0 ( 2 1.92 )CL wf 0.08CL wf C 2 (32) r (32) CL 0 ( 2 1.92 )CL wf 0.08CL wf 2
CL 0 ( 2 1.92 )CL wf 0.08CL wf Cr drag force (Eq. 33):2 2 Now using Eqs. final)Cequation obtained for CL 0 13 D(and 2 q14, 1 0is.08 C0L.25 (32) (33) wf ( 0.92 .64CLr b wf){ C ][ 0.08k w D0 [ 0. 25]CrC L 2 2 D q ( 0.64bCr b ){ CD0 [ ][ 0.08k w ] C L b 0.25Cr (33) D q ( 0.64Cr b ){ CD0 [ ][ 0.08k w ] 2 C L2 b 0.25C 4k .68C D q ( 0.64Cr b ){ CD0( 2.5229 [ log10 [ Cr r ][ 0 . 08 b ]62.5C]) L2 (33) 0.28C r w r ][ 0.162 10.56 b 4.68C[( b r 1 ( 72attack 8 916 ( 10 for the cruise condition ) . ) ( )) 8706 Dynamic pressureDand angle .of of log Boeing 747 are 8706.72 kg/ms and 2.5°, respectively. The 0 28 6 5 ( 2.5229 [ C . C ][ b . ]) 0.16 10.56 10 r r b) 8.916[( b 21 ( ( 1020, 25 and 30 in Eq. 33: b ) ( 2 )) 8706.72 main drag equation (Eq. 34) is obtained byDsubstituting Eqs. 4 68 . C r b log210 [2C r 0.28C r ][ b 6.5 ]) 0.16 10.b56 2.5 D2 8706.72( 10( 2.5229 9.81 b (34) b ) ( )) 2 ]2 ) 8.916[( 1 ( ( )] [ 2 2.5 2 2 4.68Cr 9.81 b b b 180 0.28[ )] ( 2.5229 (log10 [0 C.r 205 0.16 10].56 bCr b ][ b 6.5 ])2 2 2) 2 2 ) 8 916 1 D 8706.72(210 . [( ( ) ( )) 2 4 1 3 6 ( ( . tan ) 180 . 0 5 0 205 . b 2 2 .5 2 b Cr ( 2 9.481 ( 1 3.6 tan]2b0.5 ) )2 b ( )] [ 2 2 2 2 C (34) 180 r 0.205b 2 2 2.5 2 (34) 4 b ( 2 9.81 ( 1 3.6 tan ( )] [ ] 0.5 ) ) 2 2 C 180 r 0.205b (2 4 ( 1 3.6 tan2 0.5 ) )2 2 b Cr b V 2 ( A0 A1 4 Am ) (35) 6 b V 2 ( A0 A1 4 Am ) J. Aerosp. Technol. Manag., São José dos Campos, v10, e3618, 2018 6 V 2 ( A0 A1 4 Am ) b 6 V 2Frontspar ( A0 A1 40A (35) (36) .15 , Re arspar 0.7 m )
12/22 xx/xx
Babaei AR; Setayandeh MR
Modeling of the Fuel Tank Volume One of the wing tasks is to provide an enclosure for the fuel tank in addition to producing the lift force. Since the design variables change in the optimization process the fuel tank volume will also change, and because the aircraft mission depends on the volume of fuel so this volume should not change relative to the initial volume. The cross-section fuel tank and its parameters are shown in Fig. 7. It is assumed that the fuel tank is between wing spars in this study. z Front spar
Rear spar
9.82 b 2 0.16b 10.56 ) C ( )( x 2d L wf In the actual condition, the wing his the bC taper and the maximum of 0.205b 2 airfoil thickness r 2 4 ( 1 3.6 tan2 0.5 ) 2 9 82 b 2 0.16b 10.56 . C r (modeled ) )( variable thickness. 2w L wf is changes from root to tip, so fuel Ctank as taper with Fig. 8 shows bC 2 r 0 205 . b2 2 0 16 10 56 9 82 b . b . . 2 )(4 ( 1 3.6 tan 0.5 ) Figure 7. Wing cross-section 2 ) C L wf (and fuel tank parameters. Cr 0.205b 2 the considered fuel tank. bCr 2 ( 1 3.6 tan 0.5 ) CL 0 ( 2 1.92 )CL wf 0.08CL wf 2 4 In the actual condition, the wing is the taper and the maximum of airfoil thickness changes from rootCtor 2tip, so fuel tank is
modeled as taper with variable thickness. Fig. 8 shows the considered fuel tank.
CL 0 ( 2 1.92 )CL wf 0.08CL wf 0.25C D qC ( 0L 0.64 ][C0L.08 k ] 2 C L2 C ( 2r b){ 1.C 92D0)CL[ wf 0.r08 wf w b 0.25Cr D q ( 0.64Cr b ){ C D0 [ ][ 0.08k w ] 2 C L2 b 0.254C .68C log .5 ])k ] 2 C 2 0.16 10.56 10 [ C r 0 D q ( 0.64( C2.5229 b ){ C [.28Cr br r][][ b0.608 r D0 D 8706.72( 10 )w 8.916[(L1 ( )( 2 ) b b b 2 42.68C r 28C r b 0..81 ( 2.5229 log10 [ C r 9 ][ b 6.5 ]) 2.5 2 0.16 10.56 b )].72[( 10 ] [(1 ( D( 8706 ) 8.916 ) ( 2 )) 2 180 8.8.Fuel b b blog10 [ Cr 0.28Cr 4.268Cr ][ b 62.5 ]) Figure tanktank model. Figure Fuel model. .2205 (0 .5229 0.16 10 2 4 1 3 6 ( ( . tan b0.5 ) ) 2 2 2 ) 8.916[(1 ( )( 2.5 D2 8706.72( 10 C9r .81 b b ( )] [ ] In Fig. 8, 2w is the length of the fuel tank, 2d is180 the width of the fuel tank andbh2 is the fuel 2tank 2 shell thickness. 0 205 . 2.5(2 2 4 b 2 ) )2 6 tan ( 1 93..81 0.5 The fuel tank volume is obtained from the following(equation 2 )](Eq. [ 35): ] C 2 r 180 0.205of b the fuel tank In Fig. 8, 2w is the length of the fuel tank, 2d is the width 2 and h 2 is the fuel (2 4 ( 1 3.6 tan 0.5 ) ) b 2 Cr (35) V 2 ( A0 A1 4 Am ) tank shell thickness. b6
TheFig. fuel tankto volume is obtained from thevalues following 35): It is clear from 6 that, calculate the fuel the of 2d, 2wequation and h must(Eq. be determined in the root, mean V tank 2 volume, (A 0 bA1 4 Am ) 6 and tip chords at first. Another important point is the position of spars along the wing because 2w and 2d are dependent on this Frontspar 0A.15 , Rearspar V 2as(follows 436) Am(Roskam ) 0.7 1997): position. The position of the front and rear spars are assumed (Eq. 0 A 1 6
𝑉𝑉 =
™ |
Frontspar 0.15, Re arspar 0.7 (𝐴𝐴q + 𝐴𝐴4 + 4𝐴𝐴Z ) 2d r 0.15,h 0.02m 2w Frontspar 0.15, Re arspar 0.7
(35)
(36)
From Eq. 36, the distance of two spars is 0.55 C, so the value of 2w is 0.55 C in different parts of the wing. The values of 2d and h for Boeing 747 are assumed (Eq. 37) (Setayandeh 2011): 2d
0.15,h 0.02m 2w It is clear from Fig. 6 that, Ato1 calculate tank of 2d, and ( 22dwr02.15 hthe )(,h2fuel 2hm) volume, ( 0.55Cr the 0.values 04 )( 0.082 Cr 2w 0.04 ) h d r0 .02 (37) 2w A 0.0451Cr2 0.025Cr 0.0016 must be determined in the root, mean1 and tip chords at first. Another important point is the A1 ( 2wr 2h )( 2d r 2h ) ( 0.55Cr 0.04 )( 0.082Cr 0.04 )
J. Aerosp. Technol. Manag., São José dos Campos, v10, e3618, 2018
2
A1 A02w .0451 Cr 2d 0are .025dependent Cr 0.0016on this position. position of spars along the wing because and 0.04 )( 0The .082position Cr 0.04 ) 1 ( 2 wr 2h )( 2d r 2h ) ( 0.55C C r C & 0.28 A0 ( 0.55Ct 0.04 )( 20.082Ct 0.04 ) A1 02.0451Cr 0.025Cr 0.0016 of the front and rear spars are assumed follows 36) A as 0.0035 C (Eq. 0.007 C (Roskam 0.0016 1997): t
r
0.16 10.56 10 r r D 8706.72( 10( 2.5229log10 [ Cr 0.28Cr 4.68bb Cr ][ b 6.5 ]) ) 8.916[(1 ( 0.16 ) ( 10.256 )) 4 68 . C D 8706.72( 10 ( 2.5229log10 [ Cr 0.28Cr ) ( b.256 )) r ][ b 6.5 ]) ) 8.916[( 1 ( 0b b b.16 ) ( 10 b )) 8 916 1 D2.58706.72( 10 ) . [( ( 2 2 4.68C r 2 2b 0.28 ( 2.5229 log10 [9 C.r81 C2r ][ b 6.5 ]) b b.256 0 16 10 . (D2.58706 )] 2 .[72( 10 ].916[(1 ( 9.81 b b 8 ) ) ( )) 2 ( 180 ] 0.2059b.2812 b 2 2.5 )] 2[ b b2 2 2 180 2 4 1 3 6 ( ( . tan ) ) 0 205 . b A )] New [ Approach for Robust Design Based on2 the Concepts ( ] Fuzzy Logic and Preference Function 13/22 2 of 2 Optimization 2 23.6 tan 0.5 ) ) xx/xx 2 (1 0.5 2180 .5 2 ( 2 4 0.C r 29b.81 b 205 2 2 ( )] [ ( 2 4 Cr 2 ( 1 3.6 tan 0.5 ) ) ] 180 Cr b 2 0.205 (2 4 ( 1 3.6 tan2 0.5 ) )2 2 Finally, the values of 2w and 2d are listed in Table 2. Cr b b Table 2. Final values of fuel tank. b ( A0 A1 4 Am ) V2 2 V 6 ( A0 A1 2w 4 Am ) Parameter 2d V 6b2 ( A0 A1 4 Am ) Wing root 0.55 Cr 0.15 (2w) = 0.15 (0.55 Cr) = 0.082 Cr 6 V 2 ( A0 A1 4–Am ) – Mean chord 0.55 C 0.082 C 6 Frontspar 0.15, Re arspar 0.7 Frontspar 0.0.55 15, Re Wing tip Ct arspar 0.7 0.082 Ct Frontspar 0.15, Re arspar 0.7 Frontspar 0.15, Re arspar 0.7 2d for the fuel tank volume is calculated as follows (Eqs. 38, 39, 40, and 41): Therefore, the final equation ,h 0.02m 2d 0.15 3 The fuel tank volume is2148.8 for 0.02mdesign using Eq. 41. This value is considered as the first constraint in the ,h baseline wd 0m.15 2 2 w 0.02m 0 15 . , h optimization design (ΔV = –148.8 = 0). 2w d 0.15,h 0.02m 2w A1 ( 2wr 2h )( 2d r 2h ) ( 0.55Cr 0.04 )( 0.082Cr 0.04 ) A1 ( 2wr 2h )( 2d r 2h ) ( 0.55Cr 0.04 )( 0.082Cr 0.04 ) 2 (38) 0.0451 AA (2w 2hC)( 2d0.025 2hC) (00..0016 55C 0.04 )( 0.082Cr 0.04 ) 1 A11 0.r0451Crr2 0r.025Crr 0.0016r A1 A1( 2w0r.0451 2hC)(r2 2d0r .025 2hC)r( 0.0016 55Cr 0.04 )( 0.082Cr 0.04 ) A1 0.0451Cr2 0.025Cr 0.0016 C C & 0.28 t A0 ( 0.55Ct 0.04 )( 0.082Ct 0.04 ) C r t C r & 0.28 A0 ( 0.55Ct 0.04 )( 0.082Ct 0.04 ) 2 Ct C r & 0.28 AA00 0( .00035 0 . 007 C 0 . 0016 .55CCt r 0 . 04 )( 0 . 082 C 0 . 04 ) t A0 0.0035Cr2 0.007Crr 0.0016 C C & .28 0 t 2 0.04 )( 0.082C 0.04 ) .55CC r A00 (00.0035 t r 0.007Cr 0.t0016
(39)
A0 0.0035Cr2 0.007Cr 0.0016 2 2 0.28 2 2 0.28 C CCr0(.1707 C ) C 0.707 C r C C r ( 1 2 ) 3 1 r 2 0.28 3 1 C C r ( 1 ) C 0.707 C r 3 1
(34 (34 (3
(3
(35 (35 (3
(3 (3 (3 (3
(3 (37 (37 (3
(3
(3 (3 (3
(3
(39 (39 (3
(3
()(00.55 CC 0.04 )( 0) . 082 0.04 ) Am ( 0.55C A0m.04 .082 0.04 C (40) Am ( 0.55C 2 0.04 )( 0.0822C 0.04 ) A 0 . 022 C 0 . 018 C 0 . 0016 A0 0.022Cr2 00 .018Cr 0r .0016 r A0 0.022Cr 0.018Cr 0.0016
(40) (40)
b 2 b ( 0.137 0.104 V b ( 0.137CVr22 0.104 Cr C0r .0096 ) Cr 0.0096 ) 12 V 12 ( 0.137Cr 0.104Cr 0.0096 ) 12
(41) (41)
(41)
Modeling of Lift Coefficient n m n Using the described equations form the llift coefficient, the yi )l ()lift coefficient l ( xiwas ) ) calculated 0.28 for baseline design, so this value x n l( m yl ( 1 l 1 Ai i l 1 1 Ai ( x ) ) i f ( x ) y ( l f ( x ) is considered as the second constraint. This constraint the n obtained lift coefficient from optimization algorithm (as an mi A makes f ( x ) l mm1 ( nn i 1 l (i xil )1() i 1 Al ( xi ) ) important parameter in cruise phase) not Ai compared to the ibaseline design (ΔCL = CL –0.28 = 0). l 1 to change, 1 i ( ( x ))
l 1
i 1
Ail
i
MAKING PREFERENCE FUNCTION USING FUZZY LOGIC It has been mentioned that an important advantage of this method is using the experience of experts in the design optimization in defining preference function. It was also stated that preference functions are created using fuzzy logic. For this purpose, fuzzy rules are defined by the designers (experts) and based on their experiences. In addition, simplicity and reducing the computational time due to the use of fuzzy logic are other advantages of this function. To use these advantages, preference functions are formed for all objective functions and constraints.
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(42) (42)
computational time due to the use of fuzzy logic are other advantages of this functio
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The fuzzy preference function is achieved using product inference engine, fuzzifier, and center average defuzzifier, as follows (Eq. 42):
The fuzzy preference function is achieved using product inference engine, singleton fuzzifier, and center average defuzzifier, as follows (Eq. 42):
𝑓𝑓 (𝑥𝑥 ) =
∑ä ô k(fl∑ k¿ë fi +¿ë ‡≥k (e+ )) +
(42)
∑ ∑ä k¿ë(fl+¿ë ‡≥k (e+ )) +
– In Eq. 42, y is the center of the fuzzy set, l is the number of fuzzy rules and i is the number of inputs. To form preference functions of drag and lift coefficient, because both of them change with uncertainties, two inputs (μ, σ) must be considered. Mean values and standard deviation are divided into four and three sections, respectively (the membership functions of them are shown in Fig. 9), so 12 fuzzy rules must be created to form preference function. To form preference functions of weight and fuel tank volume only one input (μ) must be considered (these parameters don’t change with uncertainties). The mean values of them are divided into four sections so four fuzzy rules must be created to form preference function. The preference function (as the output of fuzzy logic) is divided into four sections. The membership function of preference function is shown in Fig. 9. Membership function
Low
Medium
0.35
0.7
High
Very low
0
1
Preference function
Figure 9. Membership function of preference functions.
Preference Function of Drag Since the uncertainties affect the drag, standard deviation and the mean value of drag are as inputs of the fuzzy system for making preference function of drag. This process is shown in Fig. 10. μD Fuzzy system
σD
F (μ, σ)
Figure 10. Fuzzy preference function of drag.
The membership functions of the mean and standard deviation values are shown in Figs. 11 and 12, respectively. The fuzzy rule set for drag is: 1. If μD is very high and σD is high then drag preference function is very low (0). 2. If μD is very high and σD is medium then drag preference function is very Low (0). 3. If μD is very high and σD is low then drag preference function is low (0.35). 4. If μD is high and σD is high then drag preference function is very low (0). 5. If μD is high and σD is medium then drag preference function is low (0.35). J. Aerosp. Technol. Manag., São José dos Campos, v10, e3618, 2018
(42
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6. If μD is high and σD is low then drag preference function is medium (0.7). 7. If μD is medium and σD is high then drag preference function is Low (0.35).
8. If μD is medium and σD is medium then drag preference function is medium (0.7). 9. If μD is medium and σD is low then drag preference function is high (1). 10. If μD is low and σD is high then drag preference function is low (0.35). 11. If μD is low and σD is medium then drag preference function is high (1). 12. If μD is low and σD is low then drag preference function is high (1).
Membership function Low
0
Medium
High
Very high
0.5
0.7
1.5
μ (Dr)
Figure 11. Membership functions of the mean value of normalized drag.
Membership function Low
1
0
Medium
High
1.5
3
σ (Dr)
Figure 12. Membership function of the standard deviation of normalized drag.
Preference Function of Wing Weight Unlike the drag, uncertainties don’t affect the wing weight, so there is no need to calculate mean and standard deviation in this case. The values of normalized wing weight are as input to the fuzzy system (Fig. 13).
w
Fuzzy system
F (w)
Figure 13. Fuzzy preference function of weight force.
The membership function of normalized wing weight is shown in Fig. 14. The weight preference function is similar to Eq. 42 with a difference in the number of fuzzy rules. J. Aerosp. Technol. Manag., São José dos Campos, v10, e3618, 2018
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The fuzzy rule set for wing weight is: 1. If μW is very high then wing weight preference function is very Low (0). 2. If μW is high then wing weight preference function is low (0.35). 3. If μW is medium then wing weight preference function is medium (0.7). 4. If μW is low then wing weight preference function is high (1). Membership function Low
1
0
Medium
High
Very high
0.05
0.5
0.7
μ (WT0)
Figure 14. Membership function of normalized wing weight.
Preference Function of Lift Coefficient Similar to drag force, uncertainties also affect the lift coefficient. Standard deviation and the mean value of ΔCL are inputs to the fuzzy system. Membership functions of the mean and standard deviation, and the rule set are shown in Figs. 15 and 16, respectively. The lift coefficient preference function is similar to Eq. 42. Membership function Low
1
0
Medium
High
Very high
0.35
0.7
1.05
μ (ΔCL)
Figure 15. Membership function of the mean value of normalized lift coefficient.
The lift coefficient preference function is similar to Eq. 42 and the fuzzy rule set for lift coefficient is: 1. If μΔCl is very high and σΔCl is high then lift coefficient preference function is very low (0). 2. If μΔCl is very high and σΔCl is medium then lift coefficient preference function is very low (0). 3. If μΔCl is very high and σΔCl is low then lift coefficient preference function is low (0.35). 4. If μΔCl is high and σΔCl is high then lift coefficient preference function is very low (0). 5. If μΔCl is high and σΔClv is medium then lift coefficient preference function is very low (0). 6. If μΔCl is high and is low then lift coefficient preference function is low (0.35). 7. If μΔCl is medium and is high then lift coefficient preference function is low (0.35). J. Aerosp. Technol. Manag., São José dos Campos, v10, e3618, 2018
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8. If μΔCl is medium and σΔCl is medium then lift coefficient preference function is medium (0.7). 9. If μΔCl is medium and σΔCl is low then lift coefficient preference function is medium (0.7). 10. If μΔCl is low and σΔCl is high then lift coefficient preference function is high (1).
11. If μΔCl is low and σΔCl is medium then lift coefficient preference function is high (1). 12. If μΔCl is low and σΔCl is low then lift coefficient preference function is high (1).
Membership function Medium
Low
1
0
High
0.75
1.5
σ (ΔCL)
Figure 16. Membership function of the standard deviation of normalized lift coefficient.
Preference Function of Fuel Tank Volume Like the wing weight, uncertainties don’t affect the fuel tank volume. The process of making preference function for this constraint is similar to Fig. 13 and normalized ΔV is the only input to the fuzzy system. The membership function of normalized fuel tank volume is shown in Fig. 17. Membership function Low
1
Medium
High
Very high
0.15
0.3
0.45
μ (ΔV) 0
Figure 17. Membership function of normalized fuel tank volume.
The fuzzy rule set for fuel tank volume is: 1. If μΔV is very high then fuel tank volume preference function is very low (0). 2. If μΔV is high then fuel tank volume preference function is low (0.35). 3. If μΔV is medium then fuel tank volume preference function is medium (0.7). 4. If μΔV is low then fuel tank volume preference function is high (1) J. Aerosp. Technol. Manag., São José dos Campos, v10, e3618, 2018
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OPTIMIZATION ALGORITHM As mentioned before, the NSGA algorithm is considered as the optimizer. Since the optimization problem is a constrained problem, penalty function method is used for implementation. The specifications of NSGA for optimization in this study are in Table 3. Table 3. Specifications of NSGA. Parameter
Value
Generation
30
Population size
100
Mutation rate
0.1
Selection rate
0.5
DESIGN OPTIMIZATION RESULTS As already mentioned, two design optimizations have been done in this paper that includes: deterministic optimization, and robust optimization. It was also stated that the utopian distance concept is used for final solution selection so the utopian values of wing drag and weight must be calculated at first. From a single objective optimization, utopian values of drag and weight and their specifications were obtained as follows (Tables 4 and 5): Table 4. Utopian values for drag. Cr (m)
b (m)
Λ0.5 (deg)
Dut (N)
15.29
57.17
41.83
49910
Cr (m)
b (m)
Λ0.5 (deg)
Wut (N)
15.42
56.16
40.68
6.0631 × 105
Table 5. Utopian values for weight.
Four optimal solutions (Pareto frontier) were achieved for deterministic optimization, presented in Table 6. It is worth noting that the details of deterministic optimization are available in Babaei et al. (2015). Table 6. Results of deterministic optimization. Pareto point
Cr(m)
b (m)
Λ0.5(deg)
Dut (N)
1
14.92
58.6
39.2
0.83
2
15.77
55.29
38.7
0.76
3
14.8
59.2
38
0.86
4
15.4
57.5
41.3
0.73
As already noted, each point that has lower distance is the best solution among Pareto frontier. So for deterministic optimization, number four has lower distance and is the final solution. The full specifications of this point are given in Table 7. Table 7. The full specification of final solution. Cr (m)
b (m)
Λ0.5 (deg)
S (m2)
AR
CL
V (m3)
W (N)
D (N)
15.4
57.5
41.3
566.72
5.83
0.28
147.5
6.32 × 105
86597
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To perform robust optimization (with the proposed method), the considered uncertainties were simulated, and then the mean and standard deviation of the objective functions and constraints were calculated. Finally, preference functions were formed as new objective functions and new constraints. The goal of this optimization is maximizing the considered preference function. The results of robust optimization are presented in Table 8. Table 8. Results of robust optimization. Pareto point
Cr (m)
b (m)
Λ0.5 (deg)
Dut(N)
1
13.65
72.09
42.4
4.76
2
14.73
61.66
36.1
5.5
3
15.15
57.89
33.8
5.6
4
15.18
57.8
28.65
6.21
The best solution for robust optimization is number 1. The full specification of this point is in Table 9. Table 9. Full specification of final solution. Cr (m)
b (m)
Λ0.5 (deg)
S (m2)
AR
CL
V (m3)
W (N)
D (N)
13.65
72.09
42.4
629.78
8.25
0.27
144.33
7.744 × 105
94871
The properties of optimal and robust designs are listed in Table 10 for a better comparison. Table 10. Comparison of deterministic and robust configuration. D (N)
Cr (m)
b (m)
Λ0.5 (deg)
CL
V (m3)
6.32 × 105
86597
15.4
57.5
41.3
0.28
147.5
7.744 × 10
94781
13.65
72.09
42.4
0.27
144.3
Design
W (N)
Optimal Robust
5
From the results of Table 10, it is clear that Optimal Design (OD) and Robust Design (RD) configurations are completely different, and that robust design has failed to provide good results in terms of objective functions. But this plan is more resistant in the face of uncertainties. On the other hand, the cost that must be paid to achieve a robust design is increasing the values of objective functions. The performance of robust design will be optimized for a wider range of flight altitude and velocity while the optimum plan is optimized only at a certain flight altitude and velocity. In terms of dimensions, the optimum plan suggests a chubby configuration, whereas robust plan suggests a slender configuration. From the standpoint of the root chord and the wingspan, there are significant differences between optimum plan and robust plan configurations. The robust plan has smaller root chord and larger wing span than optimum plan. The root chord and wing span of robust plan configuration lead to wing area and aspect ratio of this design have a larger value than optimum plan configuration. For better comparison, the geometry of three designs is shown in Fig. 18. To analyze the robustness of different designs, the probabilistic analysis is done for three configurations. For this purpose, 2000 points are considered as the sampling point for flight velocity and altitude as uncertainties and MCS method is used for simulation. Probability distribution function (PDF) of three configurations is shown in Fig. 19. Figure 19 shows that robust design has more robustness than the other two configurations. The results in Table 11 show that the variance of RD is improved 51.1% and 20.2% compared to OD and baseline configurations, respectively. The results show that RD configuration is less sensitive to uncertainties and increases the reliability of wing performance. Therefore,
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it can be argued that the proposed method is a simple and efficient method for robust design optimization, which can be used from the experience of experts as a valuable source of information during the optimization.
RD OD BD
PDF
Figure 18. Wing configurations of robust (RD), optimum (OD), and baseline designs (BD). 0.3
RO
0.25
DO Baseline
0.2 0.15 0.1 0.05 0
-4
-2
0
2
4
6
8
Normalized Drag Force Figure 19. Probability distribution function of normalized drag force for robust design (RD), optimum design (OD) and baseline design. Table 11. Variance comparison of deterministic, robust and baseline configuration.
Variance (N)
Baseline Configuration
OD Configuration
RD Configuration
8.99 × 105
11.3 × 105
7.476 × 105
The importance of design cycle should be noted. In aircraft design, different disciplines and the communications between them are considered. The final design will be obtained from the compromise between these disciplines. The main purpose of this paper is to present an efficient method for robust design optimization in which wing design optimization is considered for implementation. In this paper, the wing performance is optimized alone, and more precise considerations with more analysis must be considered to replace these designs (optimal and robust) with the base design.
CONCLUSION In this study, a robust optimization methodology was proposed and was applied for the wing of Boeing 747. As already mentioned, the main advantage of this method is the preference function definition using fuzzy logic. The designer can help optimization algorithm to find more practical design using the preparation of fuzzy rules and also determining desirable and undesirable ranges of objective functions and constraints. The preference function definition has advantages such as using of designer experience, good accuracy, and simplicity. Cruise altitude and velocity were considered as uncertainties, and MCS method was used for uncertainty modeling. Deterministic optimization and robust optimization were done in this study and their results J. Aerosp. Technol. Manag., São José dos Campos, v10, e3618, 2018
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were compared to each other. The results show significant differences between RD and OD configurations and a probabilistic analysis showed that robust design has more robustness than optimal and baseline designs.
AUTHOR’S CONTRIBUTION Conceptualization, Babaei A; Methodology, Babaei A and Setayandeh M; Investigation, Babaei A and Setayandeh M; Writing – Original Draft, Setayandeh M; Writing – Review and Editing, Babaei A and Setayandeh M; Funding Acquisition, Babaei A; Resources, Babaei A and Setayandeh M; Supervision, Babaei A.
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