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    Robust design optimisation using multi-objective evolutionary algorithms.pdf

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    Robust design optimisation using multi-objective evolutionary algorithms.pdf

    , JBarcelona7hierarchicalandselectedis shown how the approach can provide robust solutions using game theory in the sense that they are less sensitive to little changes ofinput parameters. Starting from a statistical definition of stability, the method captures, simultaneously Pareto non-dominated solutionsgeometry wingspan, length, angles or to operational flight uncertainties and compared to a traditional single pointboth performance and stability.The proposed above methodology is implemented in aframework design environment for the drag minimisationof a wing operating at transonic flight conditions in Euleror potential flows with a variation in the Mach number.Then both the mean lift to drag ratio and its variance are*Corresponding author.E-mail addresses dongseop.leeaeromech.usyd.edu.au D.S. Lee,l.gonzalezqut.edu.au L.F. Gonzalez, jperiauxgmail.com J. Periaux,raghaeromech.usyd.edu.au K. Srinivas.Available online at www.sciencedirect.comComputers ...;xn;xn1; ...;xmperformance at desired flight condition however, it con-tains poor off-design characteristics. Design 2 made stableperformance over a range of operative conditions withoutfluctuations.Taguchi optimisation method minimises the variabilityof the performance under uncertain operating conditions.Therefore in order to perform an optimisation underrandom fluctuations the best way is to define two differentobjectives associated to the function to optimise the meanvalueC22f and its variance dfC22f 1KXKj1fjand df 1K C0 1XKj1jfjC0C22fjwhere K denotes the number of subintervals of variationflow conditions.2.2. Multi-criteria and robust design2.2.1. Definition of design problembers. Fig. 1 presents the fluctuation on operated conditionsand shows that traditional optimisation approach couldwhere x1,...,xnrepresent design parameters andxn1,...,xmrepresent uncertainty parameters that are infine step size.w yaw angleThe minimisation of inverse lift on drag with variabilityon the Mach number can be defined asdCM∞DesignM1Design2DesignFig. 1. Drag coefficient comparison between stable design 1 and unstabledesign 2.1LD11LD andMost real world problems involve a number of insepara-number of solutions located on a Pareto front among theevolved population.3. Hierarchical asynchronous parallel multi-objectiveevolutionary algorithms HAPMOEAIn this work we use a robust multi-criteria optimisationsoftware tool; a hierarchical asynchronous parallel evolu-tionary algorithm HAPEA developed by Whitney [16,17]with some extensions for multidisciplinary and multi-crite-ria analysis [18].3.1. Hierarchical topologyThe optimiser has capabilities to handle multiple fidelitymodels for the solution. The bottom layer can be entirelydevoted to exploration, the intermediate layer is a compro-...;N 1Subject to constraints gjx0 j 1; ...;Mhkx 6 0 k 1; ...;Kwhere fiare the objective functions, N is the number ofobjectives and x is an N-dimensional vector where its argu-ments are the decision variables. For a minimisation prob-lem, a vector x1is said partially less than vector x2iff8i fix1 6 fix2 and 9i fix1 fix22If 2 holds, then that the solution x1dominates the solutionx2.min1LD at M12MbC0e;MbeC138 subject toa aC3; tc tcC3where [MbC0 e,Mb,Mb e] is the variability of the Machnumber and Mbis the standard design point. The angleof attack and thickness ratio are fixed *. According theTaguchi concept, the above design problem can be con-structed as the following multi-criteria robust optimisationXKD.S. Lee et al. / ComputersAs EAs consider multiple points design simultaneouslyas vector valued functions, they are capable of finding amise between exploitation and exploration and the toplayer concentrates on refining solutions. To take full benefitof a hierarchical structure, the top layer uses a very precisemodel meaning a time consuming solution. But at the sametime, the subpopulations of the bottom layer need not yielda very precise result, as their main goal is to explore thesearch space. That means that they can make good use ofsimple models, with fast solvers. Fig. 2 shows a representa-tion of this formulation.3.2. Parallel computing and asynchronous evaluationAnother feature of the HAPMOEA approach is the useof parallel computing. EAs are well suited to parallel com-puting; individuals can be sent to remote machines, evalu-ated and incorporated back into the optimisation process.In this work the optimiser was parallelised at USYD ona cluster of computers. The system has ten machines withperformances varying between 2.0 and 2.8 GHz. The mas-ter computer carries on the optimisation process while theFig. 2. Hierarchical topology.the potentialflow solver FLO22 [21] written by Jameson and Caugheyand the FRICTION program developed by Hendricksonas described in Mason [22]. FLO22 is designed for analy-sing inviscid, isentropic, transonic flow past 3D swept wingconfigurations. The free-stream Mach number is restrictedonly by the isentropic assumption and weak shock wavesare automatically located wherever they occur in the flow.The finite-difference form of the full equation for the veloc-ity potential is solved by the method of relaxation, after theflow exterior to the airfoil is mapped to the upper halfplane. The input data includes wing geometric configura-tions and aerofoil sections information at each sectionand flow conditions input data such as Mach number,angle of attack and friction drag CD0. Further detailscan be found in the manual of FLO22. Friction drag iscomputed externally using the FRICTION program. Thisprogram was developed by Hendrickson and provides anestimate of laminar and turbulent the skin friction suitablefor use in aircraft preliminary design.Details of the FLO22 code validation can be found inauthor’s previous work [23] and it is shown that the resultsobtained by FLO22 are in good agreement with experimen-tal data [24]. FLO22 has capabilities to provide accurateresults and solve the aerodynamic characteristics for 3Dwings operating at transonic speeds. FLO22 provides someadvantages The first benefit is good accuracy even consid-ering the inviscid flow assumption. The other advantagewhen compared full Navier–Stokes solver is the computa-tional time; a single computation takes only 50–70 s on acomputational grid of 96 12 16 with 200 iterations.Therefore, the authors have confidence on the capabilitiesof the solver for its coupling with the optimiser.5. ONERA M6 wing aerofoil section design optimisationIn this work, the airfoil section optimisation of ONERAM6 wing has been investigated. Its aerofoil sections aredesign and optimised to improve aerodynamic efficiencyat variability of transonic flow conditions while the qualityand sensitiveness of wing is maximised. The quality andsensitiveness of a model can be represented by using statis-tic formulation mean and variance in fitness functionsthat are called Uncertainty approach. The results obtainedfrom single and multi-objective optimisation are compared.As multi-objective techniques, the Taguchi method uncer-tainty is applied to provide stable wing. In this section, thedesign variables for optimisation are presented before thepractical applications. Four test cases are considered; firsttest case focuses on the ONERA M6 wing reconstructiondesign optimisation to check the robustness and well cou-pling between aerodynamic analysis tool and the method-ology. The second and third test case considers single andFluids 37 2008 565–583multi-points design and the final test focus on mutli-objec-tive optimisation problem with uncertainties.M6 wing operating at transonic speeds. This reconstructionproblem deals with a single-objective and consists of mini-mising the difference between computed wing surface pres-sures and pre-computed ONERA M6 wing pressuredistributions. It is very important to check both the robust-ness and the coupling between aerodynamic analysis tooland inverse methodology before considering the practicaltest case. The flow conditions are provided in Table 2. Fit-ness function is as followsf min1nmabsXni1Xmj1CpTargetC0CpCandidates“ where i and j indicate chordwise and spanwise number ofwing sections.5.2.2. Design variablesThe wing geometry is fixed as illustrated in Fig. 6 andTable 3. Three aerofoil sections are considered andFLO22 will interpolate the aerofoil shape between sections.The computed pressure distributions obtained from 21spanwise and 107 chordwise sections are compared toparameters are considered for the evolutionary optimiserFig. 4. ONERA M6 wing geometry.D.S. Lee et al. / Computers we conclude that these solutionscan be called supercritical wings at two flow conditionswhile the shocks form on baseline design and single popu-lation optimum. Figs. 17a–17c show the Mach sweep on liftand drag coefficient and lift to drag ratio where it can beseen that all Pareto members produce higher lift coefficientat Mach sweep range when compared to single optimumand baseline design. These figures compared between bestsolutions for flow conditions 1 and 2 as well as compro-mised solutions Pareto members 6, 7, 8 where it can beseen that Pareto member 7 produced the highest lift curvefrom Mach 0.8 to 0.93 and the best solution 1 indicates thelowest drag curve after Mach 0.89. The solution fromsingle population optimisation produce lower drag coeffi-cient until Mach 0.87 and its lift to drag ratio was higherbefore Mach 0.85. In stability aspect, best solution 1 Par-eto member 1 made the most stable aerodynamic perfor-mance. Pressure contours for best solutions of objective 1and 2 and Pareto member 6 are illustrated in Figs. A.5–A.10.Fig. 16a. Pressure coefficient comparison between compromisedthe ONERA M6 wing aerofoil sections at variability ofMach numbers with new robust design methodology.Instead of designing at a fixed flow point or points, uncer-tainty concept considers the stability of each model at theoffset operating Mach numbers; i.e. candidate model oper-ating at M12 [MbC0 e,Mb,Mb e] where Mbis the stan-dard design point. This optimisation is set as twoobjectives problem where the fitness functions are minimi-sation of discrete mean and variance of the inverse of lifton drag.f1 min1LD 1KXKi11LDiM21iM2b– AverageM1i208195;08295;08395;08495;08595C138 anda 306C14f2 mind1LD1K C0 1XKi11LDiM21iM2bC0 1LDC18C192– Variancesolution and baseline M 0.8359.

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