Received 08 April 2010, Revised 15 July 2010

Defence Science Journal, Vol. 60, No. 6, November 2010, pp. 628-638Ó 2010, DESIDOC

1. INTRODUCTIONIn the past, many engineering design problems were

tried to be solved largely by trial and error depending onengineers� experience and intuition. Thus, a lot of manpower,time or cost was inevitable to obtain an aerodynamicallyoptimal shape. Nowadays, keeping pace with the rapidgrowth of computing power, commercial software replacesengineers� efforts in the field of aircraft design, such as3-D geometric modelling, structure analysis, and flowcomputations. Computational software also contributesin reducing errors caused by human factors, such as incompleteknowledge based on poor experience, mental and/or physicalfatigue. However, unlike CAD and FEM-based structuralanalysis software, reliability and commercialisation of CFDsoftware still relatively stays at low level, neverthelessconsiderable efforts to develop advanced CFD methods.In general, it requires hundreds of flow solutions to completea reliable aerodynamic optimisation process. For that reason,flow analysis and aerodynamic design optimisation techniquesmust be carefully devised or selected by optimising thetrade-off between numerical accuracy and computationalefficiency. Assuming that the computing power steadilygrows, the paper reviews literature on some essential elementsrequired for high-fidelity aerodynamic flow analysis anddesign optimisation. Design methodology based on Euleror Navier-Stokes equations can be roughly classified into

gradient and non-gradient methods depending on the usageof sensitivity analysis process. Non-gradient optimisationmethod is mostly a kind of global optimisation method,while gradient-based method is local optimisation method.Each optimisation approach has its own merits and demeritsdepending on design problems. Generally, computationalaerodynamic shape optimisation consists of the four essentialelements: (i) flow solver, (ii) sensitivity analysis solver(in case of gradient-based optimisation), (iii) grid generator(or grid modifier), and (iv) optimisation algorithm1.

Among the four elements, an accurate and efficientflow solver is the first concern because it provides flowfield information and integral aerodynamic loads such aslift or drag, which are intrinsic ingredients of objectivefunction. To obtain robust and reliable flow field information,high-fidelity numerical schemes that are able to resolvecomplex flow phenomena, are thus the foremost. From thisperspective, high-fidelity shock-stable schemes, such asRoeM and AUSMPW+ have been examined2,3. RoeM scheme,based on the Roe�s approximate Riemann solver, is a shock-stable scheme without any tunable parameters while maintainingthe accuracy of the original Roe scheme2. AUSMPW+scheme is an improved version of AUSMPW scheme. Usingpressure-based weighting functions, AUSMPW+ can reflectboth physical properties across a cell interface adequately.As a result, oscillations and overshoots behind shocks

Computational Elements for High-fidelity AerodynamicAnalysis and Design Optimisation

Chongam KimSeoul National University, Seoul�151 744, Korea

Email: chongam@snu.ac.kr

ABSTRACT

The study reviews the role of computational fluid dynamics (CFD) in aerodynamic shape optimisation,and discusses some of the efficient design methodologies. The article in the first part, numerical schemesrequired for high-fidelity aerodynamic flow analysis are discussed. To accurately resolve high-speed flowphysics, high-fidelity shock-stable schemes as well as intelligent limiting strategy mimicking multi-dimensionalflow physics are essential. Exploiting these numerical schemes, some applications for 3-D internal/externalflow analyses were carried out with various grid systems which enable the treatment of complex geometries.In the second part, depending on the number of design variables and the way to obtain sensitivities or designpoints, several global and local optimisation methods for aerodynamic shape optimisation are discussed. Toavoid the problem that solutions of gradient-based optimisation method, (GBOM) are often trapped in localoptimum, remedy by combining GBOM with global optimum strategy, such as surrogate models and geneticalgorithm (GA) has been examined. As an efficient grid deformation tool, grid deformation technique usingNURBS function is discussed. Lastly, some 3-D examples for aerodynamic shape optimisation works basedon the proposed design methodology are presented.

Keywords: Aerodynamic shape optimisation, high-fidelity numerical methods, gradient-based optimisation method,meta modelling, genetic algorithm, adjoint variable method, aerodynamic analysis.

REVIEW PAPER

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and near a wall, which are typical symptoms of AUSM-type schemes, are successfully eliminated. Both RoeMand AUSMPW+ schemes are among the recently developedadvanced numerical fluxes for gas dynamics. In addition,to achieve high-order spatial accuracy by incorporatingmulti-dimensional flow physics, multi-dimensional limitingprocess (MLP) are discussed in 2-D and 3-D setting4-6.

The sensitivity analysis method commonly used canbe summarised by four techniques, (i) finite differencemethod (FDM)7, (ii) direct differentiation (DD)8, (iii) complexstep derivative9, and (iv) continuous/discrete adjointvariables7,10. Among them, adjoint variable approachesare most popular since computational cost is almost independentof the number of design variables and roughly the sameas the cost of flow analysis. Thus, adjoint method isparticularly useful in the problems with many design variables.In the present work, sensitivity analysis by discrete adjointapproach has been examined. Furthermore, the extensionof the adjoint approach to various grid systems, such asmulti-block for internal flow application and overset meshsystem for multiple-body external problem has been discussed.

Regarding the geometric modification required in shapeoptimisation process, various shape functions have beenused. Smoothness in shape change and a flexible degreeof freedom (DOF) in design space are the top prioritiesin choosing a shape function. For that reason, a non-uniform rational B-spline (NURBS) equation can be employedas a new shape function, and the control points of NURBSsurface are then used as design variables. NURBS canmaintain grid smoothness since NURBS equations canpreserve a certain level of higher-order derivatives at eachknot of the surface. Hence, gradient smoothing affectingthe accuracy of sensitivity can no longer be necessary.

Lastly, as an optimisation strategy to avoid the potentialdanger, that solutions of GBOM are often trapped in localoptimum in highly nonlinear design space, an efficientapproach by combining surrogate models and genetic algorithm(GA) has been examined. Exploiting all of the elements forthe high-fidelity flow analysis and ASO, some three-dimensionaldesign works including inlet and wing/body design havebeen presented.

2. NUMERICAL SCHEMES FOR FLOW ANALYSISFor compressible flow analysis, a numerical design of

inviscid fluxes, namely a numerical flux function at a cell-interface, should guarantee a high level of accuracy, efficiency,and robustness. Upwind-basing is the most physical approachto capture various linear and nonlinear waves, which canbe categorised as either flux difference splitting (FDS) orflux vector splitting (FVS) schemes. FDS schemes are basedon the idea of Godunov, and the Riemann problem is usedlocally. Many researchers have tried to simplify the stepof numerical flux calculation, which leads to the family ofGodunov-type schemes or approximate Riemann solvers,such as Roe�s FDS, HLLE(M), Osher�s FDS, and so on. FVS,such as van Leer�s and other variants, has advantages inview of robustness and efficiency. At the same time, it is

also well-known that these schemes suffer accuracy problemsin resolving shear layer region due to excessive numericaldissipation. On the other hand, AUSM-type schemes, originallyproposed by Liou, et al., turn out to be advantageous inseveral aspects.

In an effort to improve the accuracy and robustnessof previous numerical flux schemes, several high-fidelityflux schemes such as RoeM2, AUSMPW+3 are designed.As the same time, multi-dimensional limiting process (MLP)4-6

is also proposed to achieve high-order spatial accuracyand to incorporate multi-dimensional effect. Exploiting thesenumerical schemes, reliable two- and three-dimensionalinternal/external flow analyses can be carried out withvarious grid systems. However, the accuracy of high-speedaerodynamic simulation is always limited by computationalburden, because high-fidelity simulation usually meanshigh computational cost, and thus a slow design cycle.Therefore, appropriate set of additional numerical strategieshas to be chosen for high Reynolds number flow designconsidering computational efficiency and numerical accuracy:

� First, efficient time integration technique to reducecomputational cost, such as multi-grid with residualsmoothing or GMRES.

� Second, adequate turbulence modelling for separatedflows, such as S-A or k-w SST model.

� Third, grid system to capture delicate flow phenomenaespecially in boundary layer around 3-D realisticconfiguration, such as overset mesh system.

2.1 RoeMRoeM2 scheme is an improved Roe-type scheme that

is free from the shock instability and still preserves theaccuracy and efficiency of the original Roe�s FDS11. Roe�sFDS is known to possess good accuracy but suffers fromthe shock instability, such as the carbuncle phenomenon.As the first step towards a shock-stable scheme, Roe�sFDS is compared with the HLLE scheme to identify thesource of the shock instability. Through a linear perturbationanalysis of the odd-even decoupling problem, dampingcharacteristic was examined and Mach number-based functionsf and g were introduced to balance damping and feedingrates, which leads to a shock-stable Roe scheme. To satisfythe conservation of total enthalpy, which is crucial inpredicting surface heat transfer rate in high-speed steadyflows, an analysis of dissipation mechanism in the energyequation was carried out to find the error source and tomake the proposed scheme preserve the total enthalpy.Modifying the maximum-minimum wave speed, the problemof expansion shock and numerical instability in the expansionregion could also be remedied without sacrificing the exactcapturing of contact discontinuity. From these analyses,the newly formulated RoeM scheme is proposed as follows:

1 2 1 *1 2 1 21/ 2 ^

1 2 1 2 1 2

1

1

j j

j

b F b F b b b bF Q g B Q

b b b b b bM

++

´ - ´ ´ ´= + D - ´ D

- - -+

(1)

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630

*2

1 0

�,

���

x

y

u u n Uu pQ B Q f

v v n Uv c

H HH

æ ö æ öræ öç ÷ ç ÷ç ÷ D - Dr Dæ ö ç ÷ ç ÷ç ÷D = D = Dr - + rç ÷ ç ÷ ç ÷ç ÷ D - Dr è ø ç ÷ ç ÷ç ÷ ç ÷ç ÷r Dè ø è øè ø

(2)

1 1 2� �� � � �max(0, , ), min(0, , )j jb U c U c b U c U c+= + + = - - (3)

where

2 2� �1, 0

� , elsewhereh

u vf

M

ì + =ï= íïî

(4)

, (1/2) (1/2), (1/2), 1 (1/2), 11 min( , , , )i j i j i j i jh P P P P+ - - + + += - (5)

, , 1, (1/2)

1, ,

min ,i j i j

i ji j i j

p pP

p p

++

+

æ ö= ç ÷ç ÷

è ø (6)

and

1

11 min , 2

2

� �, 0

�1, 0

j j

j j

p p

p pM Mg

M

+

+

æ öç ÷- ç ÷è ø

ìï ¹= íï =î

(7)

2.2 AUSMPW+Typical symptoms appearing in the application of AUSM-

type schemes in high-speed flows, such as pressure wigglesnear a wall and overshoots across a strong shock, canbe cured by introducing weighting functions based onpressure (AUSMPW)12. The main feature of AUSMPW isthe removal of the oscillations of AUSM+ near a wall oracross a strong shock by introducing pressure-based weightfunctions. AUSMPW uses the pressure-based weight functionf to treat the oscillations near a wall and w to remove theoscillation across a strong shock. The starting point ofAUSMPW is to observe the fact that AUSM+ and AUSMDare complementary to each other. AUSM+ considers theleft cell density only while AUSMD takes both cell densities.This may be the reason for the numerical oscillations ofAUSM+ and carbuncle phenomena of AUSMD.

Thus, by incorporating the property of the right cellp

R, numerical oscillations near a wall can be eliminated.

A newly improved version of the AUSMPW scheme3, calledAUSMPW+, is developed to increase the accuracy andcomputational efficiency of AUSMPW in capturing obliqueshock without compromising robustness. With a new definitionof numerical speed of sound at a cell interface, obliqueshock can be captured accurately, and it can be provedthat AUSMPW+ completely excludes unphysical expansionshock3. The AUSMPW+ flux at a cell interface can besummarised as

1 1 1 3 3

2 2 2 16 16

| |L L R R L L R RF M C M C P P P P+ - + -

a= a=

æ öç ÷= F + F + +ç ÷è ø

(8)

( ) ( )

( )

( )

( ) ( )

1/2

1/2

if m 0,

1 1

1

elseif m 0,

1

1 1

L L R R L

R R R

L L L

R R L L L R

M M M f f

M M f

M M f

M M M f f f

+ + -

- -

+ +

- - +

³

= + × - w × + -é ùë û

= ×w× +

£

= ×w× +

= + × - w × + + -é ùë û

(9)

with ( )3

, 1 min ,L RL R

R L

p pp p

p p

æ öw = - ç ÷

è ø. ,L Rf is simplified as

( )( )

2

1, 1, 2, 2,,

,

min , , ,1 min 1, , 0

min ,

0, elsewhere

L R L RL Rs

L R s L R

p p p ppp

f p p p

ì æ öæ öï ç ÷- ¹ïç ÷= ç ÷íè ø è øïïî

(10)

where s L L R Rp P p P p+ -= + , and the Mach number and pressure

splitting functions of AUSMPW+ at a cell interface areas follows.

( )

( )

211 , 1

41

, 12

M M

M

M M M

±

æ ± ± £ç= ç

ç ± >çè

(11)

( ) ( ) ( )

( )( )

22 211 2 1 , M 1

41

1 , M 1 2

M M M M

P

sign M

±

a

æ ± ± ± a - £ç= ç

ç ± >çè

m

(12)

2.3 Multi-dimensional Limiting ProcessSince the late 1970s, numerous ways to control oscillations

have been studied and several limiting concepts havebeen proposed. Most representatives would be TVD13,14,TVB15, and ENO/WENO16,17. Most oscillation-free schemeshave been largely based on the mathematical analysis ofone-dimensional convection equation, and applied to multi-dimensional applications with dimensional splitting. Theyare successful in many cases, but quite often, it is insufficientor almost impossible to control oscillations near shockdiscontinuity in multiple dimensions. This manifests thenecessity to design oscillation control method for multi-dimensional flow physics.

By extending the one-dimensional monotonic conditionto 2-D and 3-D flows, the multi-dimensional limiting conditionis proposed, and with this limiting condition, the multi-dimensional limiting process (MLP)4-6 can be formulated.The starting point is the observation that the dimensionalsplitting extension does not possess any information onproperty distribution at cell vertex points, whose informationis essential when property gradient is not aligned withlocal grid lines. To derive the multi-dimensional limitingfunction, the vertex point value is expressed in terms of

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variations across cell-interface, and then, they determinethe variation to satisfy the multi-dimensional limiting conditionusing the limiting coefficient a. The coefficient a possessesthe information of multi-dimensionally distributed physicalproperty. With the coefficient a, the multi-dimensionallimiting function can be formulated. Finally, a new familyof limiters to control oscillations in multi-dimensional flowscan be developed. For a 3-D flow,

( )

( )

1/2, , 1/2, ,, , ,

1/2, , 3/2, ,, , ,

0.5 , ,

0.5 , ,

Li j k ijk L L i j kL i j k

Ri j k ijk R R i j kR i j k

r

r

x+ -

x+ +

F = F + f a b DF

F = F - f a b DF (13)

where a is the multi-dimensional restriction coefficientwhich determines the baseline region of MLP, and b isthe local slope evaluated by a higher-order polynomialinterpolation. The interpolated values of 1/2, ,

Li j k+F and 1/2, ,

Ri j k+F

are based on the final form of MLP. Since the calculationsof interpolated values are independent of a numerical flux,MLP can be combined with any numerical flux. a

L,R and

bL,R

in Eqn. (13) can be summarised as follows.

Along the x-direction, if 0pxDF ³ ,

max, , , ,, , ,

1/2, ,

, ,

min, , , ,, , ,

3/2, ,

1, ,

2 max(1, )( ),

1

2 max(1, )( )

1

p q r i j kL i j k

L q

i j kq p

i j k

p q r i j kR i j k

R q

i j kq p

i j k

g

g

x

gh x

+x x

x

gh x

+x x +

é ùê úê úg F - Fê úa =ê úæ öDFDFê úç ÷+ + DFç ÷ê úDF DFè øë û

é ùê úê úg F - Fê úa =ê úæ öDFDFêç ÷+ + DFç ÷ê DF DFè øë û

úú

(14)

where 1/2, ,

, , ,1/2, ,

i j k

L i j ki j k

r+x

-

DF=

DF , 1/2, ,

, , ,3/2, ,

i j k

R i j ki j k

r+x

+

DF=

DF

and ( )( )( ) max 1, min 2,g x x= .

Along the h- and x-direction, the left and right valuesat the cell interface can be calculated in the same way.With b in the form of a third-order polynomial and a fifth-order polynomial, they finally obtain MLP3 and MLP5,respectively. For detailed explanation, Fig. 1 shows thecomputed results of shock-vortex interaction4-6. Comparedto conventional limiting, MLP can control oscillations acrossthe shock discontinuity and capture local flow structurein detail.

3. AERODYNAMIC SHAPE OPTIMISATIONThough the progress of computing environment makes

the choice of optimisation methods flexible, a design strategyhas to be judiciously chosen by considering the characteristicsof design problems. Global optimisation method may providethe global optimum value within the specified design space.For example, GA originated from the theory of naturalevolution is widely used as a global optimisation tool18-

20 but it is generally costly in imitating an accurate evolutionalprocess. Particularly for 3-D aerodynamic design problemswith a lot of design variables, it requires an enormousamount of computational time in evaluating experimentaldata at each design point. Thus, some approximation, calledmeta-modelling originated from statistics, is popularly adopted,such as RSM21,22 or Kriging23,24. These modelling methodsmay still require a huge computational cost to obtain sufficientexperimental data for building up the response model, ifgeometric shape is complex or the number of design variableis large. Furthermore, if sample experimental points representingobjective function values are not appropriate, design resultscan be poorer than other optimisation tools. As an improvedmeta-modelling, optimisation based on Kriging model isapplied for the robust exploration of the global optimumvalue. Jones25, et al. firstly introduced the expected improvementmethod originally proposed by Mockus26, et al. into Krigingmodel.

Adjoint approach based gradient-based optimisationmethod is also popular among local optimisation methodsbecause computational cost is essentially independent ofthe number of design variables. In addition, it exhibits a

good convergence characteristic becauseGBOM uses the gradient vector of the objectivefunction which usually provides an optimaldirection in design space. It is particularlypowerful in case of wing surface designwith a lot of design variables. Jameson10,27,28,et al. proposed continuous adjoint approach,and applied it to aerodynamic shapeoptimisation problems of various wing/bodygeometries with wing planform and surfacedesign variables. Lee29, et al. extended thediscrete adjoint method to overset meshsystem, which can be applied to complexgeometries with a relatively simple gridtopology. Through these applications,continuous or discrete adjoint variable methodshave demonstrated the capability to produceFigure 1. Comparison of density distributions for shock wave-vortex interaction.

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a substantial improvement of aerodynamic performance.Despite the superior performance in aerodynamic designproblems, GBOM has a potential danger to be trapped inlocal optimum during design process, especially in casesof noisy nonlinear design spaces.

3.1 Genetic AlgorithmGenetic algorithm is a class of stochastic algorithm

inspired by natural evolution, and has been applied tofind optimum values in various fields. Starting with a randomlygenerated population of chromosomes, a GA goes througha process of fitness-based selection and recombinationto produce successor population or the next generation.During recombination, parent chromosomes are selectedand their genetic material is recombined to generate thechild generation. As this process is progressed iteratively,a sequence of successive generations evolves and averagefitness of the chromosomes tends to increase until stoppingcriterion is reached. In this way, a GA evolves into thebest solution to a given design problem. An advantageof GA is that, unlike other optimisation algorithms, it doesnot need gradient information. Therefore, if there are manylocal extrema and discontinuous properties in objectivefunctions and constraints, GA is more suitable in findingthe global optimisation point and design variable set thangradient-based optimisation methods. At the same time,a GA requires substantial computational cost because ofa large number of function evaluations. For that reason,it is prohibitive to directly apply it to complex aerodynamicshape optimisation with a large number of design variables.

3.2 Kriging ModelMeta-modelling techniques are commonly used to create

approximation of the mean and variation of response innoisy design space. A meta-model is adopted as a surrogateapproximation for actual experimental data or numericalanalysis during design process. Among the meta-modellingtechniques, RSM and Kriging model are the most populartechniques in aerodynamic shape design. RSM employsa simple polynomial function using the least square regressiontechnique. For that reason, RSM has a limitation if physicalphenomena are highly nonlinear or noisy wrt design variables.Kriging method is more flexible in dealing with aerodynamicdesign problems of highly nonlinear design space23-26.Kriging method was developed in the field of geo-statistics,and it is useful in predicting correlated data temporallyand spatially. A most distinguished advantage of Krigingmodel is that it can exactly interpolate sample data, andcan represent a function with multiple local extrema.

3.3 Expected ImprovementOnce a Kriging model is constructed, GA is used to

search the global optimum value within the specified designspace. Thus, there is a possibility that the global optimumvalue given by GA may not be the global optimum in thereal design space. As a way to find more accurate responsesurface and more efficient exploration for global optimum

point, expected improvement has been proposed25,26.The main idea is that the uncertainty of the predicted

value should be taken into account in sampling additionalpoint for the initial Kriging model. In this process, expectedimprovement basically provides a kind of figure of merit(or a value of expected improvement) when additionalsample point is added to the existing sample data. Forexample, during the minimising process of objective function,expected improvement can be formulated and expressedin a closed form as in Eqns. (15) and (16).

[ ]min minmin

� �( ) ,�( ) max ( ),0

0 ,otherwise

f y x if y fI x f y x

- <ì= = -í

î (15)

[ ] min minmin

� ��( ) ( )

f y f yE I x f y s

s s

- -æ ö æ ö= - F + jç ÷ ç ÷è ø è ø

(16)

where minf is the current minimum value obtained by flowsolver and y� is predicted by the Kriging model, s is theroot mean square error of the predictor representing someuncertainty at the predicted point, F and f are the normaldistribution and normal density function, respectively.From Eqn. (16), the maximum expected improvement pointcan be evaluated, and the value of the sample point obtainedby CFD solver is added. If the objective function valueof the sample point is smaller than the current minimumvalue, is newly updated. This process is iteratively performedand stopped until the expected improvement becomes lessthan some threshold criterion. Through the Kriging-expectedimprovement process, the point nearer the global optimumcan be predicted in the specified design space23-25.

3.4 Sensitivity Analysis-based on Discrete AdjointApproachDiscrete adjoint variable method is applied to get

sensitivity information by fully hand differentiating thethree-dimensional Euler and N-S equations. The symbolicformulation of the discrete residual R for the steady-stateflow equations can be written as

{ } { } { }( , , ) 0R R Q X D= =

(17)

where Q is the flow variable vector, X is the position ofcomputational grid and D is the vector of design variables.Without evaluating the vector dQ/dD, the sensitivity derivativeof the objective function, F = F(Q,X,D), can be calculateda s

T T

T

dF F dQ F dX F

dD Q dD X dD D

R dQ R dX R

Q dD X dD D

ì ü¶ ¶ ¶ì ü ì ü ì ü ì ü ì ü= + +í ý í ý í ý í ý í ý í ý¶ ¶ ¶î þ î þ î þ î þ î þî þ

æ öé ù¶ ¶ ¶ì ü é ù ì ü ì ü+L + +ç ÷í ý í ý í ýê ú ê ú¶ ¶ ¶î þ ë û î þ î þë ûè ø

(18)

if and only if the adjoint vector L satisfies the followingadjoint equation.

{ }0T

TTF R

Q Q

ì ü é ù¶ ¶+ L =í ý ê ú¶ ¶î þ ë û

(19)

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The solution vector L is then obtained by solvingEqn. (19) with the Euler implicit method in a time-iterativemanner as

1

,T

m

m m

I R R F

J t Q Q Q

+

æ öé ù é ù ì ü¶ ¶ ¶+ DL = - L -ç ÷ í ýê ú ê úD ¶ ¶ ¶ë û ë û î þè ø

L = L + DL

(20)

where I is the identity matrix, J the Jacobian matrix, andthe subscript VL means the van Leer flux Jacobian. Adjointformulation on the overset boundary can be similarly derivedby slightly modifying the conventional adjoint boundarycondition, which can be expressed as

{ }0TT TSM M

TM SFFM M M

RR F

Q Q Q

é ùé ù ì ü¶¶ ¶ï ïL + L + =ê úê ú í ý¶ ¶ ¶ï ïê úë û î þë û

(21)

{ }0TT TMS M

TS SFFS S M

RR F

Q Q Q

é ùé ù ì ü¶¶ ¶ï ïL + L + =ê úê ú í ý¶ ¶ ¶ï ïê úë û î þë û

(22)

{ }0T T T

MM MTM MF

FM M MF F F

RR F

Q Q Q

é ù é ù ì ü¶¶ ¶ï ïL + L + =ê ú ê ú í ý¶ ¶ ¶ê ú ê ú ï ïë û ë û î þ

(23)

{ }0T T T

SS STS SF

FS S SF F F

RR F

Q Q Q

é ù é ù ì ü¶¶ ¶ï ïL + L + =ê ú ê ú í ý¶ ¶ ¶ê ú ê ú ï ïë û ë û î þ

(24)

where the subscript F indicates fringe cell. The superscriptM and S represents the main-grid and sub-grid domain,respectively. By solving the four equations sequentially,overset boundary value on the main- and sub-grid canbe updated. The update procedure of the adjoint variableson the overset boundary Eqns. (21) to (24) is reverse tothe conventional overset flow analysis because of thetransposed operation in the adjoint formulation.

3.5 Geometric Modification�NURBSIn automatic shape design for 3-D geometry, a grid

generator, which guarantees a sufficient and flexible designspace, is important. NURBS surface equations can be employedas new shape functions. The benefits of NURBS havealready been mentioned and the determination of controlpoints and NURBS blending functions, grid generationprocess from the evaluated NURBS equations are discussed.The coordinate vectors of NURBS curve, X(u), are expressedb y

( )( )0 ,

ni i i i kh P N u

X uh

=S= (25)

where homogeneous coordinate h represents

( ),0

n

i i k

i

h h N u=

= å

(26)

and ( ), ,i i i iP x y z= is a position vector of the thi control

point in 3-D space. Homogeneous coordinate acts as a

weighting factor for each control point. As the value ofh increases, the corresponding NURBS curve is closer tothe control point. To impose equal weighting for eachcontrol point, all the homogeneous coordinates can beset to 1. The value of n indicates the number of controlpoint. Also, blending functions are defined as

( )( ) ( ) ( ) ( )

( )( )

, 1 1 1, 1,

1 1

1,0

,

1

0 otherwise

i i k i k i k

i k

i k i i k i

i ii

u t N u t u N uN u

t t t t

t u tN u

- + + + -

+ + + +

+

- -= +

- -

ì £ £= í

î

(27)

where ti (i = 0, 1, 2, . . .) are knot-values, and subscript

k indicates the order of NURBS blending function30.The grid points are approximated by the least-square

method with NURBS. After this, control points andhomogeneous coordinates (weighting factors) can be usedas design variables. The grid sensitivity is then finallyevaluated as in Eqns. (28) and (29). For the i th controlpoint,

( ) ( )( )

,

0 ,

i i k

ni i i i k

h N uX u

P h N u=

¶=

¶ S (28)

And, for the i th weighting factor,

( ) ( ) ( ) ( ){ }

( )

, 0 ,

2

0 ,

ni k j i j j j k

nii i i k

N u P P h N uX u

h h N u

=

=

é ùS -¶ ë û=¶ é ùSë û

(29)

In case of NURBS surface approximation, the overallprocedure is similar to NURBS curve approximation exceptit is formulated by 2-D NURBS equations.

4. DESIGN CASE STUDY4.1 Turbulent Internal Flow Design

Adjoint approach, which has been very successfulin external flow problems, does not seem to be appliedto highly viscous internal flow problems yet. This is mainlydue to the difficulty in differentiating turbulence transportequations7. In addition, awfully time-consuming work indeveloping a reliable adjoint code may discourage itsapplication to internal flow design problems. Most researcheson optimal intake design employs parametric studies, GBOMusing finite difference method, or other global optimisationmethods based on the modelling of design space. Theycarry out the design optimisation of a subsonic intakebased on the discrete adjoint approach. The sensitivityanalysis for two-equation turbulence model is performedto allow a large number of design variables and to globallymodify the intake (or duct) geometry. The computationalcost involved in two adjoint equations for turbulencetransport equations is side-stepped by the parallelisedadjoint method31. As a result, the computational cost forsensitivity analysis with the parallelised adjoint methodis almost equal to that of the flow solver.

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The objective function is to maximise the total pressurerecovery with minimising the loss of other performancemeasures. The maximisation of the volume-averaged totalpressure (VATP) is adopted as a useful objective function.The VATP is defined by

0 /V V

VATP P dv dv= ò ò

(30)

where Vò implies the volume integration from (x/D = 0)

(D: throat diameter) to the outlet boundary, and 0P is the

total pressure in the duct.Figure 2 shows the geometric change between the

baseline model and the designed model. The duct geometryafter design is somewhat beyond expectation, and it appearsthat this kind of shape change can rarely be obtained fromconventional design works. Intuitively, three noticeablefeatures can be observed from Fig. 2. as follows:

the back pressure condition. It is observed that the designedmodel shows a better performance than the baseline modelin all the off-design test cases.

Through design and off-design condition tests, thepresent optimisation approach using a discrete adjointcode and NURBS shape modification tool successfullydemonstrates that the designed model exhibits a goodperformance in various flight conditions. Even if it maynot guarantee the globally optimal shape, several testsat off-design conditions confirm that the designed modelstill yields desirable performance over a wide range offlow conditions.

4.2. Extension to Complicated Overset Mesh SystemThe overset grid technique possesses several attractive

properties which can be beneficial to large scale flowanalysis and design optimisation.

At first, the grid topology to represent the deforminggrid is relatively simple. Secondly, the movement of thesub-domain grid system or the movement of a local component,such as the movement of engine nacelle along the wingsurface, can be readily realised without re-generating grid.Thirdly, a high-quality flow solution can be obtained bya relatively small number of grid points. Finally, the fullyautomatic grid generation is possible because of the simplegrid topology. These characteristics of the overset meshtechnique can derive the overall aerodynamic designoptimisation process into the final goal, i.e., �fully automaticaerodynamic design from the CAD models�.

In the present work, the seven block overset meshsystem of DLR-F4 wing/body configuration, which wasthe test geometry in the first Drag Prediction Workshop(DPW-I), has been considered32. The total number of meshpoint is about 1.22 million. The design conditions are thefree-stream Mach number of 0.75 and the angle of attackof 0.0 degree, which corresponds to the cruising condition.

Optimisation is performed using the Broydon-Fletcher-Goldfarb-Shanno (BFGS) variable metric method, which isa kind of non-constrained optimisation technique. As a

� Firstly, curved region at the lower surface is stretchedafter design. As a result, the suction effect at thelower surface becomes weak and the size of flowseparation is remarkably reduced.

� Secondly, the exhaust region of the designed intakeis changed into an elliptical shape from a circularsection. This reduces the intensityof swirling flow over the wholeduct region.

� Lastly, a smooth bump appearsalong the lower surface near theexhaust region, which stabilisesflow into the engine face.The performance of the VATP

designed model is examined at severaloff-design conditions. The performancecoefficients of the baseline anddesigned geometry are compared bychanging the mass flow rate into theengine. For each design case, totalpressure recovery, distortion and massflow rate are compared under the samefree-stream conditions and also under

Figure 2. Streamline comparison (VATP maximisation, Up:baseline, Down: designed).

Figure 3. Comparison of flow pattern between the baseline model and the designed model(Re-design of DLR-F4 wing/body configuration): (a) baseline model (DLR-F4)and (b) designed model.

(a) (b)

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standard application of the overset GBOM tool, a dragminimisation with maintaining constant lift coefficient isfirstly performed. The objective function is defined byEqn. (33) with the constraint of Eqn. (32). To balance thevariation of the penalty function in the objective function,the weighting factor of the lift constraint is given by theratio of the lift sensitivity to the drag sensitivity wrt angleof attack.

Minimise: CD (31)

Subject to: 0 0

Lift Coefficient of the,

Baseline ModelL L LC C C

æ ö³ = ç ÷

è ø (32)

( )0

Objective Function min 0, ,D L L

D L

F C Wt C C

C CWt

é ù= + ´ -ë û¶ ¶

=¶a ¶a

(33)

Figure 4. Comparison of Cp curve and geometric change (Re-design of DLR-F4 wing/body configuration).x /cx / c

x / cx / c

x / c x / c

�C

p�

Cp

�C

p

yy

y

DLR-F4(y/Span=33.1%)DESIGNED MODEL (y/Span=33.1%)

DLR-F4(y/Span=63.6%)DESIGNED MODEL (y/Span=63.6%)

DLR-F4(y/Span=84.4%)DESIGNED MODEL (y/Span=84.4%)

DESIGNED 33.1%BASELINE 33.1%

DESIGNED 63.6%BASELINE 63.6%

DESIGNED 84.4%BASELINE 84.4%

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The total number of design variables is 200, locatedalong 10 different design sections of the wing surface.At each design section, 20 Hicks-Henne functions wereused. Three component blocks - collar block, wing block,and tipcap block - were overlapped on the wing surface.The deformation of the overlap surface meshes was carriedout by the mapping technique from the wing-surface domainto the planform domain.

After design, the drag reduction of the wing only wasabout 17 per cent, which is quite reasonable consideringthe drag portion of the fuselage. The L/D was changedfrom 32.26 per cent to 36.25 per cent. It can be observedfrom Fig. 3 that the shock strength on the wing surfacewas substantially mitigated after design. At the sectionof 33 per cent wing span as shown in Fig. 4, the frontshock on the upper surface almost disappears becausethe shelving leading-edge relieves strong expansion.

Regarding the rear shock, the relatively mild slopearound x/c = 75 per cent prevents drastic flow expansionafter the front shock. The maximum thickness of the wingsection is conserved to maintain the lift constraint. Asshown at the sections of 84.4 per cent wing span, theposition of the maximum thickness along the span-wisedirection shifts toward the trailing edge because of thetwist angle of the baseline wing. Thus, the flow regionafter the maximum thickness is not sufficient to transformthe rear shock wave into the region of gradual pressurechange. Furthermore, the geometric change of the upperwing surface after the maximum thickness region is limitedbecause the wing section gets closer to the wing-tip. Thissuggests that planform design should be introduced toobtain more refined design solutions such as a shock-freewing. Especially, leading edge sweepback angle can diminishthe shock strength on the upper wing surface. Thus planformdesign variables such as leading edge sweepback, wingspan, taper ratio and twist angle are added.

By combining surface design and planform design efficiently,the two-stage design approach is performed33. The first stagedesign is the planform design using global optimisationmethod based on surrogate model combined with GA. The

second stage is the wing surface design through the discreteadjoint approach on overset mesh system. This multi-stage,multi-fidelity design strategy incorporating the discrete adjointapproach and the overset mesh system is certainly expectedto provide optimal aerodynamic shape for complex 3-Dconfigurations as in Fig. 5.

5. CONCLUSIONSSome essential elements required for high-fidelity

aerodynamic analysis and design optimisation have beenbriefly discussed. First of all, AUSMPW+ and RoeMschemes are introduced as accurate, efficient, and robustnumerical fluxes for high-fidelity flow analysis. As a multi-dimensional limiting strategy, which ensures multi-dimensionalmonotonicity and higher-order spatial accuracy, the basicidea of MLP is presented and its performance examined.For the aerodynamic shape design optimisation, variousdesign methods have been discussed depending on theusage of sensitivity analysis process, and the sensitivityanalysis technique using discrete adjoint approach isdiscussed. For robust design optimisation without beingtrapped in local optimum, multi-stage ASO strategy isemployed by combining GBOM for surface design withsurrogate models/GA for planform design. Through various2-D and 3-D applications, the proposed strategy for high-fidelity aerodynamic analysis and design has been verified.However, some challengeable issues for high-fidelityaerodynamic shape optimisation still remain, such as robustconvergence of adjoint solver over complex configuration,efficient exploration of highly nonlinear design space,and mesh generation techniques to assure a proper resolutionof boundary layers and complex geometries.

ACKNOWLEDGEMENTSThe author appreciates the financial support by National

Space Laboratory (NSL) programme through the NationalResearch Foundation of Korea funded by the Ministry ofEducation, Science and Technology (Grant 20090091724),and by the Korea Science and Engineering Foundation(KOSEF) grant funded by the Korea Government (MEST)

Figure 5. (a) Two-stage design history, and (b) pressure contour of designed model.(a) (b)

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(No.20090084669). The author also appreciates help fromMr JinWoo Yim and Mr JunSok Yi in preparing this manuscript.

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Contributor

Dr Chongam Kim is currently a Professorin the School of Mechanical and AerospaceEngineering at Seoul National University.He did his PhD in Aerospace Engineeringfrom Princeton University, USA underthe supervision of Antony Jameson Hisresearch interests include: Numericalmethods for conservat ion laws,aerodynamic shape optimisation and flow

control, compressible and cryogenic multi-phase flows, bio-mimetic aerodynamics and multi-disciplinary flow analysis.