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Accepted Manuscript Fast reconstruction of aerodynamic shapes using evolutionary algorithms and virtual nash strategies in a CFD design environment J. Periaux, D.S. Lee, L.F. Gonzalez, K. Srinivas PII: S0377-0427(08)00545-1 DOI: 10.1016/j.cam.2008.10.037 Reference: CAM 7011 To appear in: Journal of Computational and Applied Mathe- matics Received date: 19 July 2007 Revised date: 7 January 2008 Please cite this article as: J. Periaux, D.S. Lee, L.F. Gonzalez, K. Srinivas, Fast reconstruction of aerodynamic shapes using evolutionary algorithms and virtual nash strategies in a CFD design environment, Journal of Computational and Applied Mathematics (2008), doi:10.1016/j.cam.2008.10.037 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Accepted Manuscript

Fast reconstruction of aerodynamic shapes using evolutionary

algorithms and virtual nash strategies in a CFD design environment

J. Periaux, D.S. Lee, L.F. Gonzalez, K. Srinivas

PII: S0377-0427(08)00545-1

DOI: 10.1016/j.cam.2008.10.037

Reference: CAM 7011

To appear in: Journal of Computational and Applied Mathe-

matics

Received date: 19 July 2007

Revised date: 7 January 2008

Please cite this article as: J. Periaux, D.S. Lee, L.F. Gonzalez, K. Srinivas, Fast reconstruction

of aerodynamic shapes using evolutionary algorithms and virtual nash strategies in a CFD

design environment, Journal of Computational and Applied Mathematics (2008),

doi:10.1016/j.cam.2008.10.037

This is a PDF file of an unedited manuscript that has been accepted for publication. As a

service to our customers we are providing this early version of the manuscript. The manuscript

will undergo copyediting, typesetting, and review of the resulting proof before it is published in

its final form. Please note that during the production process errors may be discovered which

could affect the content, and all legal disclaimers that apply to the journal pertain.

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Fast Reconstruction of Aerodynamic Shapes

using Evolutionary Algorithms and Virtual

Nash Strategies in a CFD Design

Environment

J. Periaux a,∗, D.S Lee b, L.F Gonzalez c, K. Srinivas d

aSenior Visiting Professor, CIMNE/UPC, Barcelona, Spain.bPostgrad student, AMME, University of Sydney, NSW 2006, Australia.

cLecturer, Queenslan University of Technology, QLD 4001, Australia.dSenior Lecturer, AMME, University of Sydney, NSW 2006, Australia.

Abstract

This paper compares the performances of two different optimisation techniques forsolving inverse problems; the first one deals with the Hierarchical Asynchronous Par-allel Evolutionary Algorithms software (HAPEA) and the second is implementedwith a game strategy named Nash-EA. The HAPEA software is based on a hierar-chical topology and asynchronous parallel computation. The Nash-EA methodologyis introduced as a distributed virtual game and consists of splitting the wing designvariables - aerofoil sections - supervised by players optimising their own strategy.The HAPEA and Nash-EA software methodologies are applied to a single objec-tive aerodynamic ONERA M6 wing reconstruction. Numerical results from the twoapproaches are compared in terms of the quality of model and computational ex-pense and demonstrate the superiority of the distributed Nash-EA methodology ina parallel environment for a similar design quality.

Key words: Nash equilibrium, Shape Optimisation, Evolutionary Optimisation

∗ Corresponding author.Email addresses: [email protected] (J. Periaux),

[email protected] (D.S Lee),[email protected] (L.F Gonzalez), [email protected](K. Srinivas).

Preprint submitted to Elsevier January 7, 2008

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Nomenclature

α = angle of attack Γ = dihedral

AR = aspect ratio L/D = lift to drag ratio

b = span length M∞ = free stream Mach number

BP = break point or crank Re = Reynolds number

c = aerofoil chord length S = wing wetted area

CD = drag coefficient λ = taper ratio

CD0 = drag coefficient at zero lift Λ = sweep angle

CL = lift coefficient ψ = yaw angle

CR = root chord length

1 Introduction

Aerodynamic shape optimisation using EAs has been explored by several re-searchers (1; 2; 3; 4; 5). The aim in EA research for aerodynamic shape op-timisation is to develop efficient optimisation techniques with high qualityof solutions. The paper investigates two different optimisation techniques theHierarchical Asynchronous Parallel Evolutionary Algorithms (HAPEA) andNash game with EAs - Nash-EAs- for inverse problems. HAPEA relies ontwo major ingredients including a hierarchical topology for exploration andrefinement and asynchronous parallel computation for continuous evaluationof candidate solutions. The Nash-EAs methodology consists of several playersfocused on local surfacic pressure distribution of airfoils using their strategy tooptimise their local criteria but coupled with other players via the flow envi-ronnement modeled by non linear PDEs. The optimisation methods HAPEAand Nash-EA are applied to solve global single-objective aerodynamic designinverse problem. In this study, Nash games play the role of pre-conditionnersto speed up the capture of global single objective optimisation. Numericalresults from two approaches are compared in terms of both quality of modeland computation time expense. The approach is implemented in a CFD de-sign environment for the minimisation of the pressure difference between apre-defined pressure and candidate pressure distribution over an aircraft wingoperating at transonic flight conditions in Euler or Potential flows. In thisresearch, we use a framework for multi-objective and multidisciplinary designoptimisation. This framework has a graphical user interface (GUI) and has dif-ferent modules for aerofoil, wing, aircraft, Unmanned Aerial Vehicles (UAVs)and configuration design. Details on framework can be found in reference (6).

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2 Methodology

The evolutionary algorithm used in this paper is based on Evolution Strate-gies (ES)(7). The first method HAPEA couples EA with several aerodynamicanalysis tools and incorporates the concepts of Covariance Matrix Adapta-tion (CMA)(8; 9), a hierarchical topology(10), asynchronous evaluation anda Pareto tournament selection(11; 12). The hierarchical topology can providedifferent models including precise, intermediate and approximate models. Eachnode belonging to the different hierarchical layer can be handled by a differ-ent EA code. The second method couples Nash-EA with several aerodynamicanalysis tools. The Nash game players choose their own strategy to improvetheir own objective.

2.1 Hierarchical Asynchronous Parallel Evolutionary Algorithms (HAPEA)

In this study, we use a robust multi-criteria optimisation software tool; a Hier-archical Asynchronous Parallel Evolutionary Algorithm (HAPEA) full detailscan be found in references (13; 14; 15).

Hierarchical TopologyThe optimiser has capabilities to handle multiple fidelity models for the solu-tion. The bottom layer can be entirely devoted to exploration, the intermedi-ate layer is a compromise between exploitation and exploration and the toplayer concentrates on refining solutions. To take full benefit of a hierarchicalstructure, the top layer uses a very precise model meaning a time consumingsolution. But at the same time, the subpopulations of the bottom layer neednot yield a very precise result, as their main goal is to explore the search space.That means that they can make good use of simple models, with fast solvers.Individual migrates up and down during the optimisation. Figure 1a shows arepresentation of this formulation.

Parallel Computing and Asynchronous EvaluationAnother feature of HAPEA is the use of parallel computing. EAs are wellsuited to parallel computing; individuals can be sent to remote machines,evaluated and incorporated back into the optimisation process. In this study,the optimiser was parallelised on a cluster of computers. The system has tenmachines with performances varying between 2.0 and 2.8 GHz. The mastercomputer carries on the optimisation process while the remote machines com-pute the solver code. The message-passing model used is the Parallel Vir-tual Machine (PVM)(16). The parallel implementation requires modificationsto the canonical ES, which ordinarily evaluates entire populations simultane-ously. The distinctive method of an asynchronous approach is that it generates

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only one candidate solution at a time and only re-incorporates one individualat a time, rather than an entire population at every generation as is usual withtraditional EAs(7). Consequently solutions can be generated and returned outof order. This allows the implementation of an asynchronous fitness evaluationgiving the method its name. Figure 1b shows a schematic representation ofthis approach.

Figure 1. a) Hierarchical topology b) Asynchronous parallel computation.

2.2 Nash Game Strategies

Nash equilibrium is a result of a game based on symmetric information ex-changed between different players. Each player is in charge of one objective,has its own strategy set and its own criterion. During the game, each playerlooks for the best strategy in its search space in order to improve its own crite-rion while criteria of other players are fixed. The Nash equilibrium is reachedafter a series of strategies tried by players in a rational set until no player canimprove its score by changing its own strategy. For instance, f = xy be thestring representing the potential solution for a dual objective optimisation,where x corresponds to the first criterion and y to the second one. The firstplayer Player1 is assigned for the optimisation of x and the optimisation of yto Player2. Player1 optimises f with respect to the first criterion by modify-ing x, while y is fixed by Player2. Symmetrically, Player2 optimises f withrespect to the second criterion by modifying y while x is fixed by Player1 asillustrated in figure 2. Details of Nash and game strategies can be found inreferences (4; 5).In the sequel Nash games are used as a preconditioner to speed up the captureof a single objective and not to solve a conflictual multi-criteria optimisationproblem. In the former particular situation, the Nash games are classified asvirtual games whereas in the later case the real game denomination associatedwith the conflictual physics of the problem prevails.

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Figure 2. Nash strategy with Player1 and 2.

3 Aerodynamic Analysis Tools

There are several complexities involved in transonic shocked flows due to com-pressible effects. By increasing the flight regime from subsonic to transonic,velocity reaches the critical Mach number, a name given to the lowest (sub-sonic) free-stream Mach number for which the maximum value of the localvelocity first becomes sonic. This flow field contains regions of locally sub-sonic and locally supersonic velocities with shocks, hence a proper selectionand validation of aerodynamic analysis tools is required before using results ofthe flow solver during an optimisation process to evaluate candidate solutions.The flow solver should meet some essential conditions such as: result accuracy,time consumption and robustness. It is always desirable to use a high fidelityNavier-Stokes solver that accounts for the flow complexities such as boundarylayers, flow separation and shocks. The major problem with this approach isthe high computational expense for evaluating a solution since the CPU costof one computation might take several hours on a parallel environment (clus-ter, grids, supercomputer).Therefore it is convenient to define and introduce low/middle fidelity solverssuch as a full potential flow solver with viscous effects which contains com-pressible error correction; the results obtained from a potential flow solverwith viscous effects can be then compared to wind tunnel experimental datafor validation.

3.1 Potential Flow Solver (FLO22) and Friction

In this paper, two analysis tools are utilised; the potential flow solver FLO22(17)written by Jameson and Caughey and the FRICTION program developed byHendrickson(18). FLO22 is designed for analysing inviscid, isentropic, tran-sonic flow past 3D swept wing configurations. The free stream Mach numberis restricted only by the isentropic assumption and weak shock waves areautomatically located wherever they occur in the flow. The finite-differenceform of the full equation for the velocity potential is solved by the method ofrelaxation, after the flow exterior to the airfoil is mapped to the upper half

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plane. The input data includes wing geometric configurations and aerofoil sec-tions information at each section and flow conditions input data such as Machnumber, angle of attack and friction drag(CD0). Friction drag is computedexternally using the FRICTION program.This program was developed by Hendrickson and provides an estimate of lam-inar and turbulent the skin friction suitable for use in aircraft preliminarydesign. Details of the FLO22 code validation can be found in reference (19)and it is shown that the results obtained by FLO22 are in good agreementwith experimental data(20). FLO22 has capabilities to provide accurate re-sults and to solve the aerodynamic characteristics for 3D wings operating attransonic speeds. FLO22 provides some advantages: The first benefit is goodaccuracy even considering the inviscid flow assumption. The other advantagewhen compared full Navier-Stokes solver is the computational time; a singlecomputation takes only 50 to 70 seconds on a computational grid of 96×12×16with 200 iterations. Therefore, the authors have confidence on the capabilitiesof the solver and its accuracy for its coupling with the evolutionay optimiser.

4 Reconstruction of Pressure Distribution on ONERA M6 Wing

In this study, the aerodynamic design reconstruction of a ONERA M6 wing isinvestigated. Two approaches are considered; first approach uses the HAPEAsoftware with three layers hierarchical topology. The second approach uses theNash-EA software implemented with three players placed at root, crank andtip aerofoil sections.

4.1 Design Variables

The external geometry of the wing planform is fixed and illustrated in Fig. 3and table 1. The control points that define the aerofoil sections at 3 spanwisestations represent the design variables. The aerofoil geometry is representedusing Bezier splines with the combination of a mean line and thickness dis-tribution, which is a very common concept in classical aerodynamics(21). Avariable number of intermediate control points whose x -positions are fixed inadvance and whose y-heights form the problem unknowns as illustrated in Fig.4.

Table 1. ONERA M6 wing configuration.

V ariables AR b ΛIn ΛOut λIn λout Γ

V alues 3.806 2.39ft 30o 30o 0.781 0.562 0o

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Figure 3. ONERA M6 wing geometry.

Figure 4. Aerofoil section mean and thickness control points design envelope.

4.2 Fitness Function and Termination Criteria of HAPEA and Nash-EA

This reconstruction problem deals with a single-objective and consists of min-imisation the difference between pre-computed ONERA M6 wing surface pres-sures and computed wing pressure distributions. The flow conditions are pro-vided in table 2. The fitness function and termination criteria are as follows;

f = min{

1n+m

[abs

[∑ni=1

[∑mj=1 (CpTarget − CpCandidate)

]]]}

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where i and j indicate chord-wise and span-wise number of wing sections.

Termination Criteriaif (fitness ≤ 0.0167 || Evaluation T ime ≥ 150 hours)

The optimisation will be terminated when the value of fitness is less than0.0167 or when the evaluation time goes over 150 hours.

Table 2. Flight Condition.

V ariables M∞ α ψ Re

V alues 0.84 3.06o 0o 11.72× 106

4.2.1 ONERA M6 Wing Reconstruction using HAPEA

Problem DefinitionThe first approach considers the use of HAPEA as a methodology and theapplication of single-objective inverse aerodynamic design of ONERA M6 wingoperating at transonic speeds.

Design VariablesThe wing geometry is fixed and illustrated in figure 3 and table 1. Threeaerofoil sections are considered and FLO22 will interpolate the aerofoil shapebetween sections. The computed pressure distributions obtained from 21 span-wise and 107 chordwise sections are compared to pre-defined pressure distri-bution.

ImplementationThe FLO22 solver is utilised and the following specific parameters are consid-ered for the evolutionary optimiser using a hierarchical topology.First Layer : Population size of 20 with a computational grid of 96× 12× 16cells (fine grid).Second Layer : Population size of 40 with a computational grid of 82× 12× 16cells (intermediate grid).Third Layer : Population size of 60 with a computational grid of 68× 12× 16cells (coarse grid).

ResultThis problem was run for 2010 function evaluations of the head node on two2.4GHz processors and was stopped after thirty hours. Figure 5 shows the op-timisation convergence history for this approach. The geometries of optimisedaerofoil section at root, break and tip are designed very close to baseline aero-foil sections shape and compared to target aerofoil baseline in figure 6 where

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it can be seen there is slightly difference at 30 to 60% of root and tip aerofoilsection. Figure 7 compares between target and optimised pressure distribu-tions at 21 spanwise sections where the cross (+) presents target pressure andthe circle is for optimised pressure. Spanwise pressure distributions at 0, 20,40, 60, 80 and 90% of the span are illustrated in figures 8 to 10.

Figure 5. Optimisation convergence history for inverse aerodynamic design usingHAPEA.

Figure 6. Comparison between target and optimised aerofoil geometries.

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Figure 7. Comparison between target and optimised pressure distribution at allspanwise sections.

Figure 8. Comparison between target and optimised pressure distribution at 0 and20% of span.

Figure 9. Comparison between target and optimised pressure distribution at 40 and60% of span.

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Figure 10. Comparison between target and optimised pressure distribution at 80and 90% of span.

4.2.2 ONERA M6 Wing Reconstruction using Nash-EA

Problem DefinitionThis test case considers the Nash-EA approach denoted virtual Nash game andits application to single-objective inverse aerodynamic design of the ONERAM6 wing operating at transonic speeds. Players 1, 2 and 3 are placed at root,crank and tip aerofoil section respectively (aerofoil1, aerofoil2 and aerofoil3).Player 1 will minimise the difference of candidate surface pressure and targetedsurface pressure for Root section aerofoil while sending best root aerofoil toPlayer 2 and 3. A similar task process is assigned to Player 2 and Player 3 asillustrated in table 3 and figure 11.

Table 3. Nash-EA Player 1, 2 and 3.

Players Root Crank T ip

P layer1 NewCandidate Elite(Player2) Elite(Player3)

Player2 Elite(Player1) NewCandidate Elite(Player3)

Player3 Elite(Player1) Elite(Player2) NewCandidate

Figure 11. Nash-EA Players for reconstruction of ONERA M6 wing.

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Design VariablesThe wing geometry is fixed as illustrated in figure 3 and table 1. Three aerofoilsections are considered and FLO22 will interpolate the aerofoil shape betweensections. The computed pressure distributions obtained from 21 spanwise and107 chordwise sections are compared to pre-defined pressure.

ImplementationThe FLO22 solver is utilised and the following specific parameters are consid-ered for the evolutionary optimiser using three players in terms of fine grid.Player1: Population size of 10 with a computational grid of 96× 12× 16 cells.Player2: Population size of 10 with a computational grid of 96× 12× 16 cells.Player3: Population size of 10 with a computational grid of 96× 12× 16 cells.

ResultThe problem was run for 315 function evaluations of each Player 1, 2 and 3and took approximately five hours on two 2.4 GHz processors. Figure 12 showsthe optimisation convergence history for this test case. The fitness reached topre-defined value after five hours that is only 16% of HAPEA time expense.The geometries of reconstructed aerofoil section at root, break and tip com-pare quite well with baseline aerofoil sections shape and compared to targetaerofoil in figure 13. As illustrated, there is a good match between aerofoilsgeometries. Figure 14 shows the comparison between target and optimisedpressure distributions at 21 spanwise sections where the cross (+) presentstarget pressure and the circle is for optimised pressure. Spanwise pressure dis-tributions at 0, 20, 40, 60, 80 and 90% of the span are illustrated in figures 15to 17.

Figure 12. Optimisation convergence history for inverse aerodynamic design usingNash-EA.

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Figure 13. Comparison between target and optimised aerofoil geometries.

Figure 14. Comparison between target and optimised pressure distribution at allspanwise sections.

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Figure 15. Comparison between target and optimised pressure distribution at 0 and20% of span.

Figure 16. Comparison between target and optimised pressure distribution at 40and 60% of span.

Figure 17. Comparison between target and optimised pressure distribution at 80and 90% of span.

Concluding this case, it can be observed from the numerical experiments thatNash-EA approach has a significant potential to save CPU cost for any single

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criteria inverse problem. This might be due to the distributing design variablesto Nash-Players instead of dealing whole design variables. However further re-search is needed for multi-criteria inverse problems and optimisation problems.The methodologies including HAPEA and Nash-EA software are verified andeasily coupled to aerodynamic analysis tools. These approaches have both flex-ible capabilities to find optimal shapes for inverse aerofoil sections and shapeoptimisation problems. Without any problem specific knowledge of the flowanalyzer, the methodologies HAPEA and Nash-EA have captured the correctgeometries and pressure distribution over different aerofoil sections operatingat transonic shocked flow regimes.

5 Discussion

This paper explored the use of Nash-EA and HAPEA for inverse aerodynamicdesign optimisation. Results from the test cases arise two distinctive discus-sion points;

1. CPU time costNash-EA seems to be more efficient than HAPEA for an inverse design prob-lem. This is mainly because the Nash-EA distributes design variables to Nash-Players while HAPEA manages entire design variables. Another reason is thatHAPEA is based on a Pareto-EAs approach which is effective in finding thewide range solutions but leads to expensive CPU time cost.As a further investigation, there are two possible ways to reduce the computa-tional expense of the Nash-EA approach; one is by using a cluster of computers.The second is implementing hierarchical strategy into the Nash-EA approachwhich can be called Nash-HAPEA.

2. Pareto front with Nash-EquilibriumIt is necessary to improve the Nash-EA methodology before considering multi-objective or multidisciplinary design problem since Nash-EA is only capableof producing a single solution. In addition, it is important in many cases toproduce Pareto front that can show the trend of non-dominated solutions foreach objective. To produce Pareto non-dominated solution using Nash-EA, avirtual player which based on Pareto-EA needs to be implemented. Currenttests are being conducted using Nash-HAPEA with virtual Player.

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6 Conclusion

Numerical results obtained from Nash-EA and HAPEA optimisation approachesare compared in terms of efficiency and model quality. The two approachesoffer alternative choices to the designer when solving single inverse design prob-lems. Current research focus on direct design and multi-objective optimisationproblems using the above methodology and other conflicting game strategieslike Nash-HAPEA or hierarchical game like Stackelberg for distributed virtualor real games are presently under investigation.

7 Acknowledgements

The authors gratefully acknowledge E. J. Whitney and M. Sefrioui for fruitfuldiscussions on Hierarchical and Asynchronous Parallel EA. We are also grate-ful to A. Jameson and S. Obayashi for accessing the FLO22 software, and toHendrickson the FRICTION software used in optimisation procedures.

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