Aircraft robust multidisciplinary design optimization methodology based on fuzzy preference function

Ali Reza BABAEI,Mohammad Reza SETAYANDEH,Hamid FARROKHFAL

Department of Mechanical and Aerospace Engineering,Malek-Ashtar University of Technology,Isfahan,Shahinshahr 115/83145,Iran

KEYWORDS Fuzzy logic;Multidisciplinary design optimization;Preference function;Robust design;Unmanned Aerial Vehicle(UAV)

AbstractThis paper presents a Fuzzy Preference Function-based Robust Multidisciplinary Design Optimization(FPF-RMDO)methodology.This method is an effective approach to multidisciplinary systems,which can be used to designer experiences during the design optimization process by fuzzy preference functions.In this study,two optimizations are done for Predator MQ-1 Unmanned Aerial Vehicle(UAV):(A)deterministic optimization and(B)robust optimization.In both problems,minimization of takeoff weight and drag is considered as objective functions,which have been optimized using Non-dominated Sorting Genetic Algorithm(NSGA).In the robust design optimization,cruise altitude and velocity are considered as uncertainties that are modeled by the Monte Carlo Simulation(MCS)method.Aerodynamics,stability and control,mass properties,performance,and center of gravity are used for multidisciplinary analysis.Robust design optimization results show 46%and 42%robustness improvement for takeoff weight and cruise drag relative to optimal design respectively.

1.Introduction

The design of aerospace systems is a complex,time-consuming and expensive problem,which consists of three phases:conceptual design,preliminary design,and detailed design.1The design of these systems is a multidisciplinary phenomenon so that disciplines are strongly coupled to each other(input of one subsystem is an output of another subsystem and vice versa).2The classical design methods usually include many design loops,and therefore,these methods are timeconsuming due to high iterations and the results from these methods are not optimum almost.3,4The mentioned problems of classical design methods,multidisciplinary nature of complex systems and the recent trends in development,and implementation of accurate and rapid analysis tools for seeking an optimal solution led to the emergence of the Multidisciplinary Design Optimization(MDO)method.5,6Furthermore,increasing computational capabilities and development of optimization techniquesforsolvingcomplex problemshad an important influence upon the appearance of this concept.MDO is a design approach to engineering systems,which uses multidisciplinary analysis to identify more appropriate solutions during optimization and design process of complex systems.Monolithicand multilevelformulationsaretwo categories of this approach.The main advantages of this approach which has been proposed as a useful method of design in the aerospace industry are as follows3,4,7:

(1)High speed in the analysis and design(reducing design time and cost).

(2)Achieving optimal solutions.

(3)Deleting different connections among involved disciplines in design(considering all disciplines at once).

(4)Increasing fiexibility versus changes in each subsystem.

Hu and Yu8proposed an optimization strategy for multidisciplinary design optimization of the unmanned combat air vehicle.Simultaneous use of surrogate modeling and multilevel optimization are the advantages of the presented strategy.The aims of optimization are minimizing aerodynamic drag coefficient under the constraint of stealth and minimizing the structural weight.Leifsson et al.9discussed multidisciplinary design optimization of Blended-Wing-Body(BWB)transport aircraft with distributed propulsion.The main ideas of this study are distributed propulsion(with the aim of reducing aircraft noise)and using the advantages of MDO.The conceptual design of aircraft is obtained through MDO.Nguyen et al.10studied the multidisciplinary design optimization of the unmanned air vehicle.The most important advantage of their study is the use of Multi-Fidelity Model(MFM)to improve the accuracy of the design.Low-fidelity codes are developed for conceptual design and then high-fidelity codes are used to improve the accuracy of the analysis.

Complex engineering systems have an uncertain nature(in analysis,design,production and operational phases)as well as multidisciplinary essence.Studies show that about 40%of failures are due to lack of attention to uncertainties in the design phase and rest of failures depend upon other factors such as production phase,operational condition,etc.6,11So considering that uncertainties are unavoidable in the analysis and design of these systems,classical design methods use a safety factor to consider the uncertainty.This method has the following problems12:(A)determination of the safety factor is difficult for new systems and material because there is not any past experience,(B)reliability(robustness)measurement is difficult for the design process,and(C)using this method may limit the feasible design space in the optimization problems.Using optimization methods for the design process(such as MDO)had been limited to deterministic problems because these methods do not consider the uncertainty.3Robust Design Optimization(RDO)is one of the main algorithms that is developed for considering uncertainties in the design optimization problems.The aim of this method is achieving an optimal design with lower sensitivity to uncertainties(minimizing the variance of objective functions).1

Daskilewicz et al.13discussed the effects of uncertainty in the multi-objective design optimization.The decision maker can evaluate system performance and robustness by analyzing the Pareto frontier variations due to uncertainties.And the designer can distinguish the favorable or high-risk regions of design space.The robust design of structures is formulated as a multi-criterion optimization problem wherein both mean value and the standard deviation of the objective function are minimized.14In this reference,the two-criterion optimization problem is converted into single optimization problem and is solved by a gradient-based optimization algorithm.Jaeger et al.2proposed a procedure for robust optimization of an aircraft at the conceptual design phase.The main advantage of this approach is that it permits designers to update uncertainties from the historical database at each step of optimization.To reduce the computational cost,response surface approximations are constructed by Monte Carlo Simulation(MCS).Messac and Ismail-Yahaya15developed a Physical Programming-based Robust Design Optimization(PP-RDO)method.This technique is based on physical programming and robust design optimization methods.The main advantage of their method is that it allows the designer to say robustness wishes in physical meaningful terms.Nguyen et al.5applied Possibility-Based Design Optimization(PBDO)foran electric-powered Unmanned Aerial Vehicle(UAV)to obtain reliable design.For this purpose,an in-house integrated UAV analyzer is developed at first,and then,PBDO solver is used for uncertainty modeling.Design speed,density correction,design altitude,payload,battery weight,battery capacity,battery amperage and propeller efficiency are considered as uncertainties.Nguyen et al.16discussed a multidisciplinary robust optimization framework for UAV conceptual design.A new objective function which consists of adjusted mean and variance function is generated in the robust design process.The fiight altitude and speed are considered as uncertainties in this study.Zaman and Mahadevan17presented a methodology for reliability-based design optimization under both aleatory and epistemic uncertainty.Four-parameter fiexible Johnson family of distributions is used for uncertainty modeling.The main advantage of the proposed method is that it does not use a separate expression for aleatory and epistemic uncertainties and both of uncertainties are treated with a unifi ed probabilistic format.So this topic reduces the computational effort and simplifies the optimization.

Fuzzy logic is a methodology based on the experience of humanity and is 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 that are created from the experience of experts.Some of the advantages of 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.18

Azizi et al.19proposed a method based on artificial intelligence(fuzzy logic and neural network)which can effectively be used to select the suitable combination of engine thrust,wing area,and aircraft weight.Reducing the aircraft design cycle time is the main benefit from this method.On the other hand,the main design outputs(such as engine thrust,wing area,and aircraft weight)can be achieved without long calculations.Huang et al.20developed a fuzzy interactive multiobjective optimization model based on Pareto solutions.The proper feature of this method is that it can create the Pareto optimal set with the maximum satisfaction degree and the minimum distance from the ideal solution.The final optimal solution can be selected by analyzing the trade-off matrix and collaborative sensitivity.Huang et al.21proposed the use of the fuzzy models for collaborative optimization in order to construct the sufficiency degree for constraints and the satisfaction degree for objectives in each discipline.Achieving the optimal solution and decision making is complex because some design variables or constraints contain vague(fuzzy)information.To avoid such problems the fuzzy satisfaction degree and fuzzy sufficiency degree models have been proposed.These concepts are rational and practical approaches for decision making in multidisciplinary design optimization.

In this paper,an efficient robust design methodology is presented in the title Fuzzy Preference Function-based Robust Multidisciplinary Design Optimization(FPF-RMDO).Using the designer experiences during optimization by the fuzzy preference functions and performing robust optimization with variable degrees of robustness are the differences of this approach relative to previous works.In this method,the designer must determine the desirable and undesirable ranges of mean and standard deviation values of the objective functions and constraints in the physical meaningful terms.The determination of these ranges is the first use of designer experience during design optimization.Then the fuzzy preference function must be formed.Preference function is a function which shows the satisfaction degree of the system response.These functions are used for the optimization as new objective functions and constraints and the aim of the optimization algorithm is maximizing these functions(achieve the maximum satisfaction degree).Creating fuzzy rules is the second use of designer experience.Using the experienced people during multidisciplinary design optimization is one of the great advantages of this method.The second major advantage of this method is that the designer can easily change the degree of robustness by changing the fuzzy rules.In other words,the designer can do highly robust design(more focus on the standard deviation of the objective function),robust design(same focus on the mean values and the standard deviation of the objective function)and/or semi-robust design(more focus on the mean values of objective functions).Customer’s need,design requirements and/or designer experience determine the type and the degree of robust design,which can be easily implemented by changing the fuzzy rules of preference functions.Finally,the proposed method is applied to robust multidisciplinary design optimization of an unmanned aerial vehicle with 33 design variables and 23 practical constraints.

The organization of the paper is as follows.The FPFRMDO method has been introduced in Section 2.In Section 3,the implementation of this methodology for Predator MQ-1 UAV is described.In Section 4,design optimization results are expressed and a probability analysis has been done for robustness investigation.Finally,conclusions are presented in Section 5.

2.Fuzzy preference function-based robust multidisciplinary design optimization methodology

The proposed methodology for robust design optimization is based on the fuzzy logic concept and preference function creation.The fiowchart of this methodology has been shown in Fig.1,in which lJi,rJiand lCi,rCiare the mean and variance values of Jiand Ci.The details of this method are described in the following manner.

2.1.Optimization problem definition

An optimization problem definition involves the following four steps:(A)Design variables definition,(B)Design parameters calculation,(C)Cost functions definition,and(D)Constraints definition.Design variables are parameters that explain the optimal design and are the interface between optimizer and multidisciplinary design analyzer.Design parameters depend on design variables.For example,by determining design variables,some geometric properties are calculated.These parameters(along with the design variables)exchange information among different disciplines for multidisciplinary design analysis.The cost functions are performance indicators that depend on design variables and/or design parameters and with regard to the optimization problem must be minimum or maximum.Also,constraints depend on design variables and/or design parameters and determine the design limitations.These factors should be determined in the first step of optimization problems.

Fig.1Flowchart of proposed FPF-RMDO method.

2.2.Mission definition

In this step,the aircraft fiight profile(mission)is determined.In addition to specifying the different parts of the profile,altitude of each section is also determined.Other parameters such as takeoff and/or landing distances,flight endurance,range,etc.can be determined in this step.Often,every aerial vehicle is designed for a specific mission with certain cruise altitude and speed.It is very economical which one aircraft can be optimum or near-optimum for a range of cruise altitude and speed.In other words,instead of designing an aircraft for a certain altitude and speed,it is designed for a range of cruise altitude and speed(several cruise altitudes and speeds)and the aircraft has optimal performance in those ranges.To achieve such a goal,the concept of robust design must be used in aerospace vehicle design.

2.3.Uncertainty modeling

In this step,uncertainties are created.Uncertainty modeling is an important task because this process plays an important role in the quality of the answers.There are many different techniques to uncertainty modeling.To learn these techniques,see Ref.11written by Yao et al.MCS method is one of the most widely used methods for uncertainty modeling.Although this method is the most basic and simplest approach among all probabilistic methods,it is a time-consuming technique.Unlike many probability methods,this technique requires little information about the statistic and probability.The results from the MCS method are completely accurate if enough simulations to be used.6,11Fig.2 shows the fiowchart of uncertainty modeling.In Fig.2,uiand fuiare sample points and uncertainty output respectively.

2.4.Multidisciplinary design analysis

Various disciplines in the design of an aircraft are modeled in this step.Modeling of disciplines can be done by using lowfidelity models,high fidelity models or a combination of both models.According to various applications,the number of disciplines can be different.With various analyses in this step,the numerical values of the cost functions and the constraints are calculated and finally,by uncertainties modeling through MCS,their mean and variance values are calculated and sent to the next step.

2.5.Fuzzy preference function

Fig.2Flowchart of uncertainty modeling.

In this step,the preference functions corresponding to objective functions and constraints are generated by fuzzy logic.These functions allow the use of the designer experiences during system optimization.In this method,some satisfaction degrees are defined for the objective functions(constraints)by the designer(in the horizontal axis).In other words,the designer classifies horizontal axis to different regions in terms of satisfaction of the objective function(and constraint),and each region is categorized using verbal variables from a qualitative point of view.Then the vertical axis,which indicates that the preference function is divided into several regions same as the horizontal axis and each region,like the horizontal axis,is categorized using verbal variables from a qualitative point of view.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.Fig.3 shows an example of preference function.Then the optimizer uses preference function instead of objective functions during optimizations.22,23Since the aim of the robust design is the minimization of objective function variance due to noise parameters,the logic of this method is as follows.The variance and the mean values of the objective functions(the constraints)should initially be calculated.Then,like the above explanation,different satisfaction degrees are determined for the variance and the mean values and the preference functions of the objective functions(the constraints)are produced with the fuzzy logic.This process is shown in Fig.4.Now,maximizing these preference functions is the aim of optimization algorithms(the highest satisfaction degree).In other words,objective functions of the optimization problem are the set of preference functions all of which must be maximized.

2.6.Multi-objective optimization

During the last four decades,many algorithms have been developed for solving different engineering optimization problems.Most of these algorithms are based on linear and nonlinear programming methods that require gradient information.These numerical optimization algorithms have created a useful strategy for finding the local minimum in simple problems.But many actual engineering optimization problems are very complex and completely difficult to solve.In contrast,emersion of evolutionary algorithms creates a new source for an optimization problem.Evolutionary algorithms present a more efficient and robust approach to solve complex problems.Because these algorithms are stochastic,they have a less probability to be caught in the local minimum.Among evolutionary algorithms,Genetic Algorithm(GA)is most popular.This algorithm is a global optimization algorithm based on the principle of survival of the fittest,natural selection mechanism and reproducing.One kind of GA is Non-dominated Sorting Genetic Algorithm(NSGA)that finds set of optimal solutions(Pareto frontier)by adding an essential operator to general singleobjective GA.This operator determines a preference criterion(rank)based on the non-dominated sorting of the population.24To learn more about this optimization algorithm,see Ref.25written by Kalyanmon et al.

2.7.Final selection

Fig.4Process of fuzzy preference function generation.

In the multi-objective optimization problems,the optimizer creates a set of optimal solutions named Pareto frontier.Each of these solutions has no absolute superiority to each other,and each of them can be selected as the optimal solution.But this selection is not easy.There are various ways to select afinal solution such as design requirements,designer experience,and fuzzy logic.In this research,another concept is presented for the final selection.This concept is a distance between the utopian point and Pareto points.In this method,the Pareto frontiers are classified based on this criterion and finally,a Pareto point that has the lower distance(the nearest point to the ideal points)is selected.Using this criterion,the best compromise is created among multiple objective functions.This criterion is calculated for each Pareto point as

where i is the number of objective function,fiis the value of the ith objective function,futis an ideal optimum value that is obtained from a single objective optimization process for each objective function,and dutis utopian distance.The purpose of this criterion is to find a Pareto point and the distance between it and utopian values is minimal.

3.Aircraft design by using FPF-RMDO methodology

3.1.Aircraft optimization problem definition

In this study,two design optimizations are done for Predator MQ-1 UAV.The first case is a deterministic optimization.This case is a multidisciplinary design optimization and MultiDisciplinary Feasible(MDF)approach is used to implement.In this optimization process,the problem can be formulated as

where WTO,Dcr,and G are take-off weight,drag of cruise phase,and constraints respectively.

The second case is a robust multidisciplinary design optimization so that the proposed method has been applied to solve it.In this optimization,new objective functions and constraints are created by fuzzy logic.These new functions are called preference functions.So in this optimization,the problem can be formulated as follows:

In the above equation,FWTO,FDcrand FGiare preference functions of take-off weight,drag of cruise phase,and constraints respectively.

Because the optimization problems are constrained,the penalty function method has been used to apply constraints in both problems.It is worth noting that the uncertainties in the second design optimization are cruise altitude and velocity.For both design optimizations,33 design variables and 23 constraints are considered.Furthermore,other required parameters are considered as design parameters for aircraft design optimization.Design variables and design constraints with their numerical ranges are shown in Tables 1 and 2 respectively.For the better understanding of some design variables,see Fig.5.abA;ayoutA;abfand ayinfare the constants which are multiplied in the wingspan and aileron span,outer distance of aileron,flap span and inner distance of fiap are obtained respectively.l42is considered to control fuselage length to diameter ratio in a proper range.It is worth nothing that,it is presented that ranges have been determined based on special limitations,requirements,and similar aircraft database.

3.2.Aircraft mission definition

The considered mission has been shown in Fig.6 for both design optimizations.In both cases,the aim is the design of UAV so that its endurance is 24 h,cruise altitude is 4500 m,cruise velocity is 45 m/s,payload weight is 204 kg,and takeoff distance is 801 m.

3.3.Flight uncertainty modeling

As already mentioned,MCS method is used for uncertainty modeling in this study and cruise altitude and its speed have been considered as uncertainties.A normal distribution is usedto generate uncertainties(X=Xm+ (DX=3)?randn(1;N)).In this study,the values of hm;Vm;Dh;DV are 4500 m,45 m/s,3000 m,and 15 m/s respectively.

Table 1Design variables.

3.4.Aircraft multidisciplinary design analysis

In this study,the multidisciplinary analysis section consists of the following modules:input,geometry,performance,weight,aerodynamics,center of gravity,trim,and dynamic stability.Fig.7 shows the relationship among these disciplines.

It is worth nothing that in Fig.7,Wpayis payload weight,qmatis density of material,SFC is specific fuel consumption,STOis take-off distance,SLAis landing distance,ROC is rate of climb,R is range,E is endurance,Vstallis stall velocity,WFuelis fuel weight,Wiis aircraft weight in each fiight phase,WEis empty weight,Wwis wing weight,WEMPis tails wight,CD0is zero-lift drag coefficient,CLis lift coefficient,CDis drag coefficient,CLqand Clrare stability and control derivatives,CLmaxis maximum lift coefficient,(X;Y;Z)CGis center of gravity position,IXX;IYY;IZZare moment of inetia.

Table 2Design constraints.

3.4.1.Input

In this module,the parameters that are fixed during design optimization such as payload weight,type of engine(propeller or jet),airfoil parameters,etc.are determined.In other words,all parameters that need to be considered as input parameters for the analysis of each discipline are determined in this section.

3.4.2.Geometry

In this section with attention to considered design variables and the outputs of the input module,all geometrical parameters of the wing,fuselage,vertical and horizontal tails and landing gears are determined by using available relationships.With considered design variables and calculated geometrical parameters in this module,full configuration of design is achieved.

3.4.3.Performance

In this module,the performance of each fiight phase is calculated by using available equations.Some of the input parameters for this section are desirable take-off distance,the maximum rate of climb or maximum climb angle,cruise velocity,range or endurance.Maximum required power or thrust,fuel weight,velocity profile and landing distance are some of the important outputs of this module.The obtained results indicate the acceptable accuracy of this module.

3.4.4.Weight

Fig.5Definition of some variables of UAV configuration.

Fig.6Intended fiight profile.

In this module,the UAV weight is divided into the following main parts:wing,fuselage,vertical and horizontal tails,landing gears,engine,fuel system,fuel tank,and subsystem.Experimental and quasi-experimental equations are used for weight calculation.Good accuracy of weight estimation has been shown in Fig.8.

3.4.5.Aerodynamics

To develop aircraft design algorithms,aerodynamic specifications should reasonably be predicted with sufficient accuracy and computation time.Since the outputs of this module are sent to most other modules,this section is important.26–28In this study,the aerodynamic module is composed of three parts:(A)lift estimation,(B)drag estimation,(C)stability and control derivative estimation.To estimate the aerodynamic characteristics,this module has been prepared using empirical relationships.It is worth noting that the evaluation of the static stability of the aircraft is also done in this module.Boeing 747,Beach 100 king air and Navion aircraft are used for validation of this module.Fig.9 and Table 3 show the fine precision of this section.

3.4.6.Center of gravity

Determining the center of gravity is a critical step in the aircraft design because the stability,control,and trim calculations depend on this step.Experimental equations are used for estimation of this center in this study.

3.4.7.Trim and dynamic stability

Finding the angle of attack,sideslip angle,and control surface deflection angles are the purposes of the trim module.Dynamic stability characteristics are calculated in the last module.The outputs of this module show the dynamic stability of UAV in the fiight profile.The outputs of these two modules,as mentioned before,generate the constraints of the optimization problem.

Fig.7Aircraft multidisciplinary analysis fiowchart.

Fig.8Some results of weight and performance modules for Predator MQ-1.

Fig.9Validation of lift and drag coefficients.

3.5.Fuzzy preference function

In this study,preference functions are generated by using the product inference engine,singleton fuzzifier,and center average defuzzifier.Membership functions of input and output parameters are shown in Figs.10 and 11 respectively.Nine fuzzy rules are used for the preference function generation,which are expressed in Table 4.In this module,FWTO;FDcrand FGiare created and sent to optimizer as new objective functions and constraints.

4.Design optimization results

As mentioned,two design optimizations are done in this study.The first optimization is a deterministic optimization,and the second is a robust optimization.The considered objective functions and constraints are the same for both design optimizations.

For deterministic design optimization,the multi-objective genetic algorithm yielded four optimal designs(Pareto frontier).Fig.12 shows the Pareto frontier set.Utopian distance concept is used for the final selection.Tables 5 and 6 show the deterministic optimization results and the utopian distances for the Pareto frontier points respectively.It is obvious that Pareto frontier 2 is final optimal design because this point has a lower utopian distance.

Mentioned uncertainties are modeled by MCS for robust multidisciplinary design optimization.For this optimization,new objective functions(preference functions)are made by fuzzy logic.With the implementation of FPF-RMDO method,three Pareto frontiers have been obtained(see Fig.13).The values of the objective functions of these points are given in Table 7.Similar to deterministic optimization,utopian distance is calculated for final selection(see Table 8).With attention to the utopian distances,it is clear that Pareto frontier 1 is the final selection.Specification of obtained designs is given in Tables 9–11.

Table 3Validation of stability and control derivative(Navion aircraft).

Fig.10Membership function of input parameters(mean or variance values).

Fig.11Membership function of output parameter(preference function).

Table 4Fuzzy rule set.

Fig.12Pareto points for deterministic optimization.

According to the obtained results,it is obvious that configurations of the base,optimal,and robust designs are different to each other.A better comparison can be done using the span,root chord and tip chord of the wing,horizontal tail,and vertical tail.The results in Table 12 show that there are not significant differences in the wing root chord of base,optimal,and robust designs.The robust design has the largest wingspan and then wingspan of the optimal design is the largest.With attention to the values of root and mean chords and wing span,it can be said that optimization algorithm has suggested the longer wing for robust design.The horizontal tail of robust design has the greatest span among three designs,although there are not drastic differences among base,optimal,and robust designs from the horizontal tail root chord viewpoint.So the results show that the optimization algorithm has suggested the narrower horizontal tail for the robust design and a chubby configuration for the optimal design.From the vertical tail viewpoint,the optimal design has the greatest root chord and the robust design is secondary.The robust and base designs have larger vertical tail span respectively.So the vertical tail configuration of the robust design is the largest.Finally,although the robust design has larger length and diameter than the optimal design,the fuselage length to diameter is close together.All in all,we can say that the configuration of robust design is greater than two other designs.

With attention to the optimal design results,it can be understood that good suggestions have been offered.At first,it is discussed on drag force.Drag consists of two parts:(A)configuration drag(zero-lift drag)and(B)drag due to lift.All parts of the aircraft(such as wing,tails,fuselage,etc.)are effective in configuration drag production.As already noted,the fuselage length of optimal design is larger than base design.This leads to an increase in the Reynolds number of the fuselage and ultimately reduces the friction coefficient of the fuselage.Another parameter that affects the fuselage zero-lift drag coefficient is the length to diameter ratio.The greater value of this ratio reduces the fuselage zero-lift drag coefficient.The optimal design has the larger value relative to the basedesign.In the case of the wing and horizontal tail,it should be said that their friction coefficients of the optimal design increase because the mean chords of this design are smaller than base design,and this topic reduces the Reynolds numbers of optimal design.But since there is no significant difference between the mean chords of two designs,no drastic effect is created.A parameter that has a greater effect on the wing and horizontal tail zero-lift drag coefficient is their wetted area.Because the horizontal tail area of optimal design is smaller than base design,its wetted area reduces,and this issue reduces the horizontal tail zero-lift drag coefficient.Although the wing area of the optimal design is larger than that of the base design,there is no significant difference between them.Thevertical tail portion in the zero-lift drag coefficient is exactly the opposite of the cases expressed in the wing and horizontal tail.About the drag due to lift,the first issue is the aspect ratio.Larger aspect ratio reduces the drag due to liftis Oswald efficiency factor).The second issue is the lift coefficient.About the lift coefficient,we should pay attention to the significant impact of less weight of the optimal design.Because the optimal design has lower weight,this design will require a lower lift coefficient in the cruise phase.

Table 5Deterministic optimization results.

Table 6Utopian distance of Pareto frontiers for deterministic optimization.

Fig.13Pareto points for robust optimization.

Table 7Robust optimization results.

Table 8Utopian distance of Pareto frontiers for robust design.

Table 9Objective function minimum values.

In the case of weight,the results show that the wing weight of the optimal design is similar to that of the base design,but the horizontal tail weight of the optimal design has been reduced relative to the base design but the vertical tail weight of the optimal design has been increased.Although the fuselage length of optimal design has increased,its diameter has decreased and in total,the fuselage weight has decreased for the optimal design.Another reason for weight reduction of the optimal design is that the decrease in the drag reduces the required fuel weight relative to the base design.Therefore,it is understood that the optimization algorithm suggestions are suitable and have been able to well reduce the objective functions.

About the robust design,it can be stated that this design has greater dimensions,and this leads to greater take-off weight and drag relative to the optimal and base designs.In the case of the fuselage length of the robust design,it should be said that because take-off weight and fuel weight of this design are more than two other designs,the length of the fuel cabin and the length of other cabins have increased.This causes that fuselage length to diameter ratio of the robust design increases too(although the diameter of the robust design has increased relative to that of the optimal design).Although larger dimensions and more weight of the robust design increase the drag of this design,in other words,the cost of achieving a robust design is increasing the objective functions,it guarantees optimality in the range of altitude and speed.A probabilistic analysis is done for robustness analysis.The target take-off weight and cruise drag are considered as 1020 kg and 490 N respectively.2000 points as sampling points are considered for fiight velocity and altitude.Figs.14,15 and Table 13 show the results from this analysis.

As the results show,the robust design has been able to improve robustness by 46%and 27%for take-off weight and by 42%and 40%for cruise drag compared with the optimal and base designs respectively.After the robust design,the base design has more robustness than the optimal design.In other words,the optimal design could optimize objective functions and satisfy the constraints in desired fiight condition(specified cruise height and velocity),but this design does not have enough robustness.Instead,the robust design does not have a good situation in terms of optimality relative to the base and optimal designs,but guarantees the optimality of design in

the range of cruise velocity and altitude.According to the operational condition of UAV and the purpose of the mission,both optimal and robust designs are worthwhile from the optimality and the robustness viewpoint and selection of each design(optimal design or robust design)is related to the mission and customer’s need.

Table 10Design variables of base,optimal and robust designs.

Table 11Optimal and robust design constraints.

Table 12Some specifications of base,optimal,and robust designs.

Fig.14Probability Density Function(PDF)of take-off weight.

Fig.15Probability density function of cruise drag.

Table 13Results of probabilistic analysis.

5.Conclusions

In this paper,a robust multidisciplinary design optimization methodology is presented based on fuzzy preference function definition.Some advantages of this method are:using designer experience during design,performing design optimization with different degrees of robustness,reducing design time,simplicity and no need for complex calculations.In this research,two optimizations(robust and deterministic)are done for Predator MQ-1 UAV.Intended objective functions are takeoff weight and cruise drag and these functions are converted into new objective functions(preference functions)using fuzzy logic.The optimization algorithm is the Non-dominated Sorting Genetic Algorithm(NSGA)that generates the set of optimal solutions(Pareto frontier).Utopian distance is used forfinal selection of the optimal design.Cruise altitude and velocity are considered as noise parameters(uncertainties)and the MCS method is used for uncertainty modeling.The results of deterministic and robust optimizations show drastic differences between the two designs,but each of them is worthwhile.A probabilistic analysis is done for robustness validation.The obtained results show that the robust design has a good resistance relative to other designs.