(c)2002 American Institute of Aeronautics & Astronautics ...aero-comlab.stanford.edu/Papers/aiaa2002-260.pdf · technology in aircraft design process.2"5 The cou-pling between the

(c)2002 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

VIIA VI

AIAA 2002-0260Adaptive Aerodynamic Optimizationof Regional Jet AircraftB.I. SoemarwotoNational Aerospace Laboratory NLR, The NetherlandsM.LabanNational Aerospace Laboratory NLR, The NetherlandsA. JamesonStanford University, U.S.A.A.L. MartinsEmpresa Brasileira de Aeronautica S.A., BrasilB. OskamNational Aerospace Laboratory NLR, The Netherlands

40th AIAA Aerospace SciencesMeeting and Exhibit

January 14-17, 2002/Reno, NVFor permission to copy or republish, contact the American Institute of Aeronautics and Astronautics1801 Alexander Bell Drive, Suite 500, Reston, VA 20191-4344

Dow

nloa

ded

by S

TA

NFO

RD

UN

IVE

RSI

TY

on

Dec

embe

r 1,

201

5 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

002-

260


Adaptive Aerodynamic Optimization ofRegional Jet Aircraft

B.I. Soemarwoto *National Aerospace Laboratory NLR, The Netherlands

M. Laban *National Aerospace Laboratory NLR, The Netherlands

A. Jameson tStanford University, U.S.A.

A.L. Martins *Empresa Brasileira de Aerondutica S.A., Brasil

B. Oskam §

National Aerospace Laboratory NLR, The Netherlands

Nomenclature

A cross-sectional areaA aerodynamic constraint functionalC curvature

CD total drag coefficientCi sectional lift coefficient

Cm sectional pitching moment coefficientCp normalized pressure coefficient

Cdw normalized wave drag coefficientCdv normalized viscous drag coefficient

F flux vectorQ geometric constraint

H 3D boundary layer shape factorM Mach numberP aerodynamic objective functionalQ conservative flow variables

RLE leading edge radiusRe Reynolds numberW flow condition parameters

b wing spanna number of aerodynamic constraints

* Research Scientist, Computational Fluid Dynamics andAeroelasticity Department, Fluid Dynamics Division, Na-tional Aerospace Laboratory NLR.

tProfessor, Department of Aeronautics and Astronau-tics, Stanford University.

* Research Engineer, Aeronautical Engineering, EmpresaBrasileira de Aeronautica S.A.

§Head, Fluid Dynamics Division, National AerospaceLaboratory NLR.

Copyright © 2001 by National Aerospace LaboratoryNLR. Published by the American Institute of Aeronautics andAstronautics, Inc. with permission.

np number of aerodynamic objectivesnf number of design pointng number of geometric constraintst/c thickness ratio

x,y,z Cartesian axesSTE trailing edge included angle

IJL fuzzy membership functionV y/bF circulationA sweep angle0 geometry control variable

AbstractThe paper addresses an aerodynamic optimiza-

tion methodology for obtaining an optimal winggeometry of wing/body configuration of regionaljet aircraft. An adaptive optimization schemeis presented, in which optimization problems aresolved sequentially to obtain the best design thatmeets given design criteria, by means of a combi-nation of CFD analysis and design tools. Each op-timization problem represents a subset of the de-sign criteria, defining a subspace of physically fea-sible designs. The optimization results are evalu-ated, and if further improvements are deemed nec-essary, the optimization problem is adapted, suchthat a relatively well-posed optimization problemcan be formulated in the final design cycles. Toaccount for different levels of flow modeling thatunderly the CFD tools, a defect-correction proce-dure is applied. Results of three-dimensional andsuccessive two-dimensional wing design optimiza-tions are presented.

1 OF 12

AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS PAPER 2002-0260

Dow

nloa

ded

by S

TA

NFO

RD

UN

IVE

RSI

TY

on

Dec

embe

r 1,

201

5 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

002-

260


IntroductionThe starting point of any detailed aircraft design

is the specification of the design criteria in relationto the mission that the aircraft has to fulfill. Thedesigner's task is not only to meet the specifica-tion with a minimum or at least acceptable risk,but also to provide an adequate level of compet-itiveness in his design product quality relative toany potential rival in the aircraft industry.

For the specific case of the regional aircraftmarket, the issue of competitiveness has grownvery importantly during the past decade. Theinsertion of specifically designed regional jets hasbrought operational and comfort standards to amuch higher level, compared to the typical re-gional turboprop aircraft from a previous genera-tion. On the other hand, the level of performancerequirements has been elevated, with time avail-able for development considerably shortened dueto an extremely demanding competition environ-ment. As a result, the application of robust aero-dynamic design techniques has become a factor ofprime importance in the development of new gen-eration regional jet aircraft, in order to attain thechallenging mission requirements imposed by themarket.

The mission is generally specified as a flight pro-file along which the aircraft has to perform, withthe flight condition changing continuously. Asdealing directly with the whole continuous missionprofile is a formidable task, it is common practiceto divide the profile into segments which are de-cisive for the overall quality of the design. Foreach individual segment, a design point and de-sign criteria are specified, noting that propertieson a design point cannot be isolated from those ofthe other design points.

In general, an aircraft has quite a complex ge-ometry. It is practically impossible to handle theaircraft as a single entity during the detailed de-sign phase. Hence, it is also common practiceto decompose the aircraft into components havingdistinct functions and properties. To each com-ponent, a subset of the design criteria is assigned,taking into consideration that there are interac-tions between different components.

Another dimension of aircraft design problemis due to traditional classification into such disci-plines as aerodynamics, structures, performance,stability and control. In this respect, the prob-lem can be divided into a number of interrelatedmono-disciplinary problems.

Although in the above approach, i.e. a priorisubdividing a design problem into smaller sub-problems, there is an inevitable risk of obtaining adesign solution which is optimal only locally insidethe domain of a subproblem,1 significant payoffscan be gained from advances in analysis and syn-

thesis methods within disciplines. Indeed, thisline of thought has been followed in exploiting ad-vances in Computational Fluid Dynamics (CFD)technology in aircraft design process.2"5 The cou-pling between the subproblems can then be doneby means of an iterative method.6

This paper addresses a CFD-based optimizationmethodology for wing design of a wing/body con-figuration of regional jet aircraft at design pointsassociated with transonic flow conditions. Thecross-sectional geometries of the wing are to be de-signed. The design criteria represent such goals astarget fuel-tank volume, aerodynamic forces andmoments coefficients, flow conditions, buffet-onsetcharacteristics, and constraints which also incor-porate multi-disciplinary aspects. Bearing in mindthat design criteria specification is also a crucialstep in a design process, this paper concentrates onthe approach of solving optimization problems se-quentially to obtain a best design that meets givendesign criteria, by means of a combination of CFDanalysis and design tools.

In the optimization methodology, the design cri-teria must be formulated in a mathematical ex-pression stating the optimization problem. Theproblem statement defines the objectives to beminimized and the constraints to be satisfied asfunctions of a vector of control (design) variables.Formulating the design criteria into a suitable setof objectives, constraints and control variables,and selecting the starting point of optimization,are not trivial tasks. The optimization problemformulation should lead to a design space, repre-senting a subspace of physically feasible designs,that includes an optimum point corresponding tothe best design. The selection of the startingpoint should imply a favorable path towards theoptimum point to be found by the optimizationalgorithm.

In general, the human designer's knowledge onthe relation between the design criteria, the opti-mization problem statement and the implied de-sign space is limited. A new project generallyimplies a unique set of design criteria. Hence, ini-tially an optimization problem statement is at bestan educated guess on a mathematical expression ofthe design criteria. After examining the solutionof an initial optimization problem, there shouldbe a provision to modify the optimization problemfor the next design cycle, based on the designer'sexperience and insight. The design process maythen be considered as a learning process for thedesigner, to know the nature of the design criteriaand to understand the relations between variousdesign requirements, such that a relatively well-posed optimization problem can be formulated atthe final design stages.

The optimization problem formulation is in-

2 OF 12


Dow

nloa

ded

by S

TA

NFO

RD

UN

IVE

RSI

TY

on

Dec

embe

r 1,

201

5 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

002-

260


evitably dependent on the CFD analysis and de-sign optimization tools used, computational re-sources available, budget and time allocated. Oneshould also be aware that the CFD analysis anddesign tools may have different levels of represen-tation of the flow physics, and that there maybe restrictions on the type of objectives and con-straints, and flow conditions, that can be dealtwith by the CFD tools. On the other hand, for ameaningful numerical optimization, the CFD anal-ysis and design tools must also be of sufficientsophistication to facilitate timely and accurate ex-amination by the designer.

In the present design exercise, a 3D analysiscode MATRICS-V,7 a 3D design code SYNSS8'10

and a 2D design code AEROPT11'12 are appliedduring the design process. These codes are con-sidered to represent the best combination, selectedfrom the available CFD tools to ensure adequatedesign results obtained with short turn-aroundtime between design cycles, as required by theindustrial environment. Yet, this combination isconsidered to have a large potential to produceimproved designs.

This paper is organized as follows. First, thedesign criteria are presented, followed by an expla-nation of the adaptive optimization scheme. Theelements of the scheme are described in terms ofthe CFD analysis code, a generic form of opti-mization problem statement, and the CFD designcodes. Computational results are then presented,demonstrating the effects of adapting optimizationproblem for multi-point transonic wing design, andfinally conclusions are drawn in the last section.

Design Criteria SpecificationGiven a fixed wing planform, with the sweep an-

gle and the area not to be altered, the geometriesof the wing cross-sections are to be optimized:

• to achieve a minimum increase of 5.4% in thenet fuel-tank volume,

• to maintain manufacturability and structuralstrength,

while at the same time:

• to improve the cruise performance (in termsof lift-to-drag ratio),

• to improve the off-design characteristics withrespect to buffet-onset boundary,

• to satisfy a maximum nose-down pitching mo-ment constraint in relation with maximumlimits on the tail-load and the trim drag,

• to provide sufficient margins from trailingedge flow separation at low Reynolds numbersto allow wind tunnel testing,

Designer-in-the-loop,visualization & geometry

processing tools

CFDanalysis code

Optimizationproblem definition

and starting geometry

CFDdesign code 'L

Fig. 1 Adaptive optimization scheme.

• to maintain adequate margins from super-critical velocities on the lower wing surfacenear the wing kink to take into account en-gine/airframe integration.

Adaptive Optimization SchemeTo formulate all requirements simultaneously

into an optimization problem is a formidable task.The adaptive optimization strategy consists of de-sign cycles, where in each cycle a subset of thedesign criteria is dealt with in order to understandthe relations between various design requirements,while approaching the best design solution thatfulfills the complete set of the design criteria.

Figure 1 depicts the process of adaptive opti-mization scheme. The scheme begins with thespecified design criteria and an initial wing geom-etry of the wing/body configuration as an initialdesign iterate. The flow around this configurationis then calculated by means of a CFD analysiscode. The designer, as man-in-the-loop, assessesthe geometric properties and the flow character-istics with respect to the design criteria, usingvisualization and geometry processing tools. Iffurther improvements are desired, a subset of thedesign criteria is identified, based on which the de-signer defines an initial optimization problem (oradapts a previous one).

The new optimization problem should be for-mulated as such that its solution would lead tothe desired improvements. The designer decideswhether the current design iterate or a variationthereof should be used as a starting point for thenew optimization. The CFD design code performsthe optimization, and the resulting optimal solu-tion is treated as a new design iterate. The processis then repeated until the best design is obtained.It should be noted that in a larger design-loopthe best design can be further assessed, whichmay lead to an adjustment of the design criteria.

3 OF 12


Dow

nloa

ded

by S

TA

NFO

RD

UN

IVE

RSI

TY

on

Dec

embe

r 1,

201

5 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

002-

260


For example, some variations of the wing plan-form may become allowable. However, this largerdesign-loop is beyond of the scope of this paper.

CFD Analysis CodeThe CFD analysis code performs flow analyses

of wing design iterates of the wing/body configu-ration to provide the flow solutions. A validatedcode should be used with sufficient capabilitiesto capture important flow characteristics. At thesame time, as it will be used in an iterative de-sign scheme, the analysis code should also providea quick turn- around.

The MATRICS-V7 code is employed to performthe flow analyses. The flow model is based on thefull-potential flow in quasi-simultaneous interac-tion with boundary-layers on the wing, allowingconsiderable extent of flow separation to be mod-eled. To support the identification of possiblewing design improvements, MATRICS-V gives abreakdown of the total drag into separate physicalcomponents:

• induced drag, produced by wing trailing edgevortex flow,

• viscous drag, originating from viscous stressesin boundary-layers and wakes,

• wave drag, resulting from losses associatedwith shock waves.

Spanwise distributions of these components of thetotal drag are presented such that wing sectionsthat need to be improved can be identified conve-niently. Also, other important aerodynamic char-acteristics, such as drag-divergence Mach number,buffet-onset boundary and boundary layer proper-ties, are available from the flow analyses. Condi-tions that can lead to critical shock-induced andtrailing-edge boundary layer separations are indi-cated by the growth of a three-dimensional bound-ary layer shape factor H.Optimization Problem Statement

In order to apply the optimization methodology,the design criteria must be formulated as an opti-mization problem. A generic form of optimizationproblem statement can be expressed as

Minimize (Qj ,W j , 0) ,

Subject to:•Akj(Qj,0) < 0, fc = l,..,n

Gi(0) < 0, J = l,..,np

(1)

where 6 is the control variables defining the wingcross-sectional geometries, W^ represents the pa-rameters specifying the flow condition at a design

point j, Pij and Akj refer to aerodynamic objec-tive and constraint functionals, while Gi denotesgeometric constraints. Multi-objective optimiza-tion problem is implied if np > 1. The totalnumber of constraints consists of na aerodynamicconstraints and ng geometric constraints. Multi-point optimization is allowed by specifying n/ > 1for the number of design points. In the adaptiveoptimization scheme, the dimensions of the opti-mization problem (na, ng, np and n/), the typeof objectives and constraints (P, A and £), andthe flow models may vary from one design cycle toanother.

For given 9 and Wj, the values of the flow vari-ables Qj are obtained by solving the governingflow equations expressed in a conservative form as

V-F,-(Q;,W,-)=0, j = i,..,n/, (2)

with F the fluxes, subject to the boundary condi-tions

BJ-(QJ-,WJ-,fl)=0, j = l,..,n/. (3)CFD Design Code

The CFD design code performs optimization tofind a new design iterate as a solution to the cur-rent optimization problem. While the algorithmof the CFD design code should be efficient, itshould also incorporate the relevant flow physics,such that the geometric modifications producedwill correspond to a new design that is closerto meeting the design criteria. An inviscid 3Ddesign code SYN888~10 and a viscous 2D designcode AEROPT11'12 are used in parallel at variousstages of the design process.

SYN88 provides an aerodynamic optimizationtool for wing design of wing/body configurationsin an inviscid flow governed by the Euler equa-tions. A cell-centered finite-volume scheme is em-ployed for space discretization of the Euler equa-tions. Second-order dissipation terms are addedto the fluxes of the flow equations for capturingthe shock wave. The flow solutions are obtainedthrough time integration towards a steady-stateusing an explicit Runge-Kutta scheme. For thestability of the time integration, fourth-order dis-sipation terms are added to the fluxes. As theoptimization is performed in a three-dimensionalinviscid flow domain, this code facilitates three-dimensional minimization of drag consisting ofwave and vortex drag, subject to aerodynamic andgeometric constraints.

AEROPT provides an aerodynamic optimiza-tion tool for airfoil design in a viscous flow gov-erned by the Reynolds-Averaged Navier-Stokesequations with the Baldwin-Lomax algebraic tur-bulence model.13'14 A cell-vertex finite-volumescheme is used for space discretization. Second-order dissipation terms are added to the fluxes of

4 OF 12


Dow

nloa

ded

by S

TA

NFO

RD

UN

IVE

RSI

TY

on

Dec

embe

r 1,

201

5 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

002-

260


the flow equations for capturing the shock wave.The flow solutions are obtained through time inte-gration towards a steady-state using an explicitRunge-Kutta scheme. For the stability of thetime integration, fourth-order dissipation termsare added to the fluxes. As the optimization is per-formed in a two-dimensional viscous flow domain,this code facilitates two-dimensional minimizationof sectional drag consisting of the viscous and wavedrag components, subject to aerodynamic and ge-ometric constraints.

Both design codes employ gradient-based opti-mization algorithms. The variational method isapplied for efficient computations of the gradientsthrough solutions of continuous adjoint equations,obtained with the same numerical scheme used forsolving the flow equations. A general form of aero-dynamic functionals is considered as follows,

•-/. (4)

where the notation T applies for both P and v4,while Sw is the surface of the aerodynamic shapeto be optimized. -0 is a local functional represent-ing a large class of aerodynamic goals, includingthose expressed in terms of lift, drag, and pitchingmoment coefficients. For each pair of an aerody-namic functional and a design point (specified bya constant value of W), a Lagrangian is defined as

•LfJs

T-B(Q,W,0)dS, (5)

where A is the so-called vector of adjoint variables,T is the vector of Lagrange multipliers associatedwith the boundary conditions, and fJ is the flowdomain. The variation of the Lagrangian is evalu-ated as

SC = 8CX -f 8Cr + 6£Q (6)

where 8C\ refers to the variation of C due to thevariation of A while the other variables are keptfixed, and similarly for the other terms. The vari-ational method consists of a procedure for settingthe variations 8C\, SC? and S£Q equal to zero.Setting 8C\ = 0 and 8Cr = 0 is equivalent withsatisfying the flow equations and the boundaryconditions, providing the values of the flow vari-ables Q. Setting 8£,Q — 0 leads to a set of adjointequations and boundary conditions, the solution ofwhich provide the values of the adjoint variablesA and the Lagrange multipliers Y. The variation8C becomes

6£ = 6£0, (7)

from which an expression for gradient of the aero-dynamic functional T with respect to 9 can beobtained. The gradient information is passed onto the optimization algorithm. The computationaleffort of solving the adjoint equations is compara-ble to that of solving the flow equations. Thus,the computational effort of obtaining the gradientis about twice that of solving the flow equations,irrespective of the dimension of the control vari-ables. This feature is important because it givesthe designer a large flexibility in formulating theoptimization problem.

It should now be noted that, from the 3D vis-cous flow model of MATRICS-V, the viscous ef-fects are missing in the 3D inviscid flow modelof SYN88, while the three-dimensional effects aremissing in the 2D viscous flow model of AEROPT.Because the flow modeling underlying the CFD de-sign codes are less complete than that underlyingthe CFD analysis code, the optimization must beperformed in a defect-correction fashion as follows.

An evaluation of a current design iterate givesthe pressure distribution over the wing surface.This process is symbolized as

C, = N6, (8)

where Cp represents the pressure coefficient dis-tribution, N is an operator representing the flowanalysis using the more complete flow model, and9 refers^o the current wing geometry. Now, an op-erator N is introduced for the flow analysis usingthe less complete flow model underlying the designcode. An inverse procedure, defined as

§ = N-ICP, (9)is performed to obtain a geometry 9. In the lesscomplete flow model, this geometry produces apressure distribution Cp,

Cp = N0. (10)

where Cp approximates Cp following a conditionthat \\CP — Cp\\2 is minimum. In this approximatesense, the geometry 9 can be interpreted as

0 = 9 + 89, (11)where 89 is a geometric correction that incorpo-rates the missing flow physics. An operator Mis introduced for optimization using the less com-plete flow model, taking 9 as the starting geometrywhile assuming 89 constant. The optimization re-sults in a new geometry 0*:

0* = MB. (12)

Because 89 is assumed constant, the geometry 6*can also be interpreted as

r = 0* + so, (13)5 OF 12


Dow

nloa

ded

by S

TA

NFO

RD

UN

IVE

RSI

TY

on

Dec

embe

r 1,

201

5 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

002-

260


where 9* represents an actual geometry in themore complete flow model. The new design iterate9* can be determined from

with cj a relaxation factor. A subsequent eval-uation of the new design iterate provides a newpressure distribution C*,

r<* — i\Tft* f-\ z\^p ~ MV . (Lb)

If the examination of C* and 9* by the designerstill suggests further improvement, then the prob-lem statement is updated accordingly, and theprocess is repeated for the next design cycle withthe new optimization problem.

N in the above scheme represents MATRICS-V that solves the full-potential equations withthe boundary layer equations on the wing in athree-dimensional domain around the wing/bodyconfiguration. N represents AEROPT (SYN88)that solves the two-dimensional RANS (three-dimensional Euler) equations, while the symbols9 and 69 refer to two-dimensional airfoil shapes(three-dimensional wing shapes), p and p are lo-cal cross-sectional (surface wing) pressure distri-butions, and M, is an operator for constrainedmulti-point aerodynamic optimization of airfoils(wings of the wing/body configuration).

Computational ResultsMATRICS-V and AEROPT were executed on

the National Aerospace Laboratory's NEC SX-5supercomputer in the Netherlands, while SYN88was executed on a Pentium-based computer atStanford University in the United States. 3D and2D optimizations were performed in parallel at thetwo sites. Internet technology has been exploitedfor communication and data exchange. It was ex-perienced that the 10-hour time difference betweenthe two geographical locations could shorten theelapsed time of consecutive design cycles. For ex-ample, computational results of an 8-hour longdesign run that finished around midnight at onesite could be assessed immediately by the designerat the other site, which in turn.could adapt theproblem and submit a new design run.

Starting from an infeasible fuel tank volume, itbecame soon obvious during the design processthat the minimum 5.4% increase of fuel volumein combination with the fixed wing planform wasdemanding. This combination seems to imply anarrow feasible design space for aerodynamic op-timization aimed at cruise performance and off-design characteristics with adequate margins totrailing edge flow separation and supercritical ve-locities on the lower surface.

The design exercise produced 17 different wings,resulting from 2D and 3D design cycles with dif-ferent sets of objectives and constraints. Quan-titatively, in terms of the fuel tank volume andaerodynamic coefficients, the results of 2D and 3Doptimizations are comparable. The best design hasbeen determined based also on the qualitative as-pects, such as the characteristics of the pressuredistribution, expected buffet-onset properties, andthe risks related to wind tunnel testing and en-gine/airframe integration.

The best design, resulting from the two-dimensional optimizations, represents 6.7% in-crease in the fuel volume (larger than the mini-mum target). The cruise performance has beenenhanced, with 57% reduction of wave drag, 1.4%increase in viscous drag and practically unchangedvortex drag, giving a net reduction of 2 dragcounts. The margin to buffet-onset boundary isenlarged by 1.5% in terms of CL> Other con-straints considering the tail-load and trim drag(maximum nose-down pitching moment), windtunnel testing (attached flow at low Reynoldsnumber), and engine/airframe integration (max-imum velocities on the lower surface) are satisfied.Wind tunnel testing has been performed, and theresults verified the outcome of the design exercise.

Some design cycles are less successful than theother ones, but these cycles provide valuable in-formation to adapt the optimization. Only a fewdesign cycles will be discussed here, bearing inmind that the other cycles were also importantsteps in the adaptive optimization strategy. Theresults to be presented are limited to those of con-secutive design cycles demonstrating the effect ofadapting an optimization problem.

The defect-correction approach explained in theprevious section is followed. Both design codeshave the capability of performing an inverse shapedesign through minimizing a functional of the type

J~ in -(p-p}2dS. (16)

The operator N~l in equation (9) represents thealgorithm to minimize the functional (16) em-ployed by the design codes to yield a geometry0.3D Optimization

A wing geometry 9 of a current design iter-ate is considered. A viscous flow analysis us-ing MATRICS-V is performed for this geometryand, after an evaluation, further improvements aredeemed necessary through global modification ofthe wing surface. Figure 2 shows distributions ofthe pressure coefficient at an inner and a near-kink wing station. Cp(9) (indicated by circles)is the viscous pressure distribution computed by

6 OF 12


Dow

nloa

ded

by S

TA

NFO

RD

UN

IVE

RSI

TY

on

Dec

embe

r 1,

201

5 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

002-

260


C at an inner wing station

.... Cp(e) (3D inviscid flow)

— Cp@) (3D inviscid flow)o Cp(0) (3D viscous flow)

-L.cr

buffet boundary

....4 \3 \

M

Fig. 3 Design points in 3D optimization.

C at a near-kink wing station

—- CjO) (3D inviscid flow)

— Cfl) (3D inviscid flow)o C(0) (3D viscous flow)

Fig. 2 3D inverse design.

MATRICS-V for a current wing design iterate atthe cruise design point. For the same geometryand the same flow condition (also the same CL),the solution of the Euler equations gives Cp(0)(dashed lines), obtained by the analysis mode ofSYN88. The difference between Cp(0) and Cp(&)gives an idea about the viscous effects due to theboundary layer displacement, which in this caseare clearly significant.

Applying the inverse mode of SYN88 with Cp(0)as the target yields a geometry 0. The figure showsthat some discrepancies are still visible betweenthe obtained Cp(§) and the target Cp(0), in partic-ular near the shock wave. This may be attributedto the fact that the inviscid flow model producessharp shock waves and therefore cannot match thepressure of the shock wave boundary layer inter-action with the lower gradient. Nevertheless, thepressure distributions shown represent a minimumdeviation in the sense of (16), such that the ge-ometry 0 may be considered as incorporating theviscous effects in the inviscid flow domain aroundthe wing/body configuration.

A 3-point constrained drag optimization prob-lem is stated as follows:

minimizesubject to:

^ * /C-max, initial 5 (17)

where the subscripts 1,2, and 3 indicate the threedesign points specified by the lift coefficients and

n = 0.80

initialoptimizedoptimized w/spanload constraint

Fig. 4 Results of 3D optimizations.

Mach numbers depicted in Figure 3. It should benoted that for all design points the fixed CL valuesare implicitly satisfied together with the solutionsof the Euler equations. The objective to be min-imized is a weighted sum of drag coefficients atthese design points. It is expected that, while try-ing to minimize the drag at the cruise condition,minimizing the drag at the second and third designpoints would weaken the shock waves at condi-tions near the buffet-onset boundary, such that theshock-induced boundary layer separation would bedelayed. The purpose of the maximum thicknessconstraint is to maintain the fuel volume currentlyrepresenting 5% increase from that of the baseline.

The geometry 0 obtained from the above in-verse procedure is taken as the starting geometry.The optimization using SYN88 yields a geometry0*, for which the thickness constraint is activelysatisfied. The drag coefficients at the three de-sign points in the inviscid flow have been reducedsignificantly, while the new inviscid pressure dis-tributions indicate weaker shock waves.

The actual geometry 6* is obtained throughequation (14), which is subsequently evaluated us-ing MATRICS-V. Figure 4_shows the upper sur-face pressure distributions (Cp) before (black) andafter (green) the optimization. The new pressuredistribution indicates reduction of the shock wave

7 OF 12


Dow

nloa

ded

by S

TA

NFO

RD

UN

IVE

RSI

TY

on

Dec

embe

r 1,

201

5 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

002-

260


7.5

1.0

0.5

0.0

0.4

0.2

1200

1000

800

600

400

2000 0.5 0.75

Fig. 5 Results of 3D optimizations.

strength, leading to a lower wave drag. Figure 5shows spanwise distributions of the sectional wavedrag (Cdw), viscous drag (<?<*„) and circulation (F),normalized with respect to the respective initialintegrated values. The sectional wave drag is re-duced along the span, and more significantly onthe outer wing stations, giving 40% reduction ofthe wave drag. The new geometry produces highersectional viscous drag. It is observed that down-stream of the shock wave there is a rapid growth ofthe boundary layer, which explains the higher vis-cous drag (0.6% increase in the viscous drag). Thecirculation distribution is slightly pushed towardsan elliptical distribution, giving 1% reduction ofthe vortex drag.

A control over the spanwise load distributionis considered necessary to obtain a more trian-gular distribution, which is expected to yield:(i) reduced nose-down pitching moment implyingsmaller trim drag and tail-load, (ii) delayed buffet-onset and less sectional viscous drag on the outerwing, and also (iii) less root bending moment po-tential for weight reduction.

The optimization problem is adapted by intro-ducing an equality constraint of the form:

(18)

where C\t is the desired distribution. It shouldbe noted that this constraint is applied in the in-viscid flow, as the optimization is still performedby SYN88, meaning that an equality constraint

on the spanwise load in the inviscid flow does notnecessarily imply the same spanwise load in theviscous flow.

The new results are indicated in red in theabove figures, where the effect of the constraintis evident. A significant amount of twist (in thewash-out orientation) has been introduced by theoptimization algorithm. This provides a lower sec-tional angle of attack on the outer wing, givinga lower sectional lift implying lower wave dragand viscous drag. On the other hand, a signif-icant increase in the velocity is observed on thelower surface, which may not be favorable for en-gine/airframe integration. The more triangulardistribution results in a slight increase of the vor-tex drag, but this is compensated by significantgains from the reductions of both wave and vis-cous drag. Relative to the initial values (black),46% wave drag and 0.2% viscous drag reductionshave been achieved, with no noticeable change inthe vortex drag.

2D optimizationA wing geometry 0 of a current design iter-

ate is considered. A viscous flow analysis us-ing MATRICS-V is performed for this geometryand, after an evaluation, further improvements aredeemed necessary through local modifications ofthe wing sections.

In order to perform a meaningful two-dimensional optimization, three-dimensional ef-fects must be incorporated in the two-dimensionaldomain. As a first approximation, the simple-sweep theory is applied. The flow condition (Machand Reynolds numbers), the local sectional viscouspressure distribution given by MATRICS-V, andthe geometry are scaled as follows,

Ms =

t/c(es) =Cp(0)cos-2A,*/c(0)cosA,

(19)(20)(21)(22)

where A is the sweep-angle taken along the wingquarter-chord. Cp(0) is the sectional distributiongiven by MATRICS-V for the current wing designiterate. The last expression is the scaling of thethickness of the sectional geometry 9. Figure 6shows CP(6S) (dashed-line) obtained using the two-dimensional viscous flow model at the scaled ge-ometry and scaled flow condition. There is still asignificant deviation between Cps(0) and CP(0S),which means that the simple-sweep theory doesnot sufficiently incorporate the three-dimensionaleffects. The finite wing span, and the spanwisevariations of the chord and the airfoil shape, musthave contributed to the three-dimensional effectsnot modeled by the simple-sweep theory.

8 OF 12


Dow

nloa

ded

by S

TA

NFO

RD

UN

IVE

RSI

TY

on

Dec

embe

r 1,

201

5 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

002-

260


Cp at a mid-span wing statio

—- Cp(0s) (2D viscous flow)

— Cp(8s) (2D viscous flow)

0 C (9) (3D viscous flow)

Fig. 6 2D inverse design.

An inverse procedure using AEROPT is thenperformed, with Cpa(0) as the target and 6S asthe starting geometry. The results is a geometry6S which produces CP(0S) in the two-dimensionalviscous flow domain. Figure 6 shows that CP(0S)closely recovers the target Cpa(9). In this sense,the geometry Os may be considered as incorporat-ing the three-dimensional effects.

Three defining wing sections are considered, lo-cated at an inner, an outer, and a kink area of thewing. The wing surface is defined by interpolationbetween these sections. Three successive designcycles are performed. The first cycle is aimed atoptimizing the inner section. In the second cycle,after evaluation of the new viscous flow solutionobtained from the first cycle, the outer section isoptimized. Finally, based on the outcome of thesecond cycle, an optimization problem for the kinksection is formulated and solved.

In these design cycles, the aerodynamic designproblems are formulated as multi-objective fuzzyoptimization problems,15 which can be expressedin a general form as follows,

Maximize

Subject to:

> ^ fuel,

RLESTE

initial")

TE, initial,

(23)

(24)(25)(26)(27)(28)

The subscript i refers to four design points illus-trated in Figure 7. The flow solution procedure isperformed in the fixed G\ mode. Design point 1corresponds to the cruise condition. Design points2 and 3 are associated with conditions near thebuffet-onset boundary. Design point 4 is includedto suppress the tendency of a singular behavior ofoptimal solution at the cruise design point. Thisrefers to a situation in which a very low drag value

is typically achieved precisely at the cruise con-dition, but high drag values occur at conditionsslightly different from that of the cruise condition.

~L.cr(2D)

2t

buffet boundary

-rr—!\M,cr(2D) M(2D)

Fig. 7 Design points in 2D optimization.

In fuzzy optimization, an objective is expressedin terms of a monotone function /x, 0 < /z < 1,referred to as the membership function. An objec-tive value giving \JL — 1 is considered most satisfac-tory, while }JL = 0 indicates a very undesirable valueof the objective. Thus, for a (bad) high drag sit-uation, a value close to zero is assigned to fi.(Cd),while a (good) low drag situation corresponds toa value of p,(Cd) close to one. This formulationprovides a natural way of adapting the optimiza-tion problem, as the designer is expected to havea good idea about bad or good values that shouldbe achieved or avoided after evaluating a currentdesign iterate.

Constraint (24) is an aerodynamic constraintthat limits the nose-down pitching moment atcruise. Constraint (25) maintains the fuel tankvolume currently representing a 6.7% increase rel-ative to that of the baseline wing. Constraint (26)limits the leading edge radius for low-speed stallconsideration. Constraint (27) in combinationwith constraint (28) take into account aspectsof manufacturability and structural strength, byavoiding geometries with excessively thin trailingedge region.

Figures 8 and 9 present the pressure distribu-tions (Cp) and spanwise distributions^ of the sec-tional wave drag (Cdw), viscous drag (Cdv) and cir-culation (F), before (black) and after optimizationof the inner-section (blue), outer-section (green)and kink-section (red). From the pressure distri-butions, it can be observed that optimization ofthe inner-section has quite a local effect, i.e. inthe outer stations (rj = 0.60 and 77 = 0.80) theblue curves overwrite almost completely the black(initial) curves. The same applies to optimizationof the outer-section, where the green curves almosttotally overwrite the blue curves in the inner sta-tions (q ~ 0.225 and T? = 0.35).

The outcome of the first design cycle (inner-section optimization) can be identified as: highersuction peak near the leading edge, slightly more

9 OF 12


Dow

nloa

ded

by S

TA

NFO

RD

UN

IVE

RSI

TY

on

Dec

embe

r 1,

201

5 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

002-

260

initialinner-wing optimizedouter-wing optimizedmid-span optimized

Fig. 8 Pressure distributions of 2D optimiza-tion results.

7.5

1.0

0.5


crease of the sectional viscous drag at the outerstations, which can be related with a strongergrowth of the boundary layer towards the trail-ing edge. The spanwise load distribution remainsthe same.

The third design cycle (kink-section optimiza-tion) is targeted at reduction of the sectional wavedrag in the mid-span area. This optimization af-fects both the inner and outer wing areas. Alongthe span, higher suction peak near the leading edgeand more sloping roof-top type of pressure distri-bution are observed. The strong re-accelerationflow in the outer station is alleviated, while thepressure gradient clearly indicates weaker shockwaves. The sectional wave drag in the mid-spanarea has successfully been reduced, at the expenseof a slight increase of the wave drag in the in-ner part of the wing. The wave drag becomesalmost uniformly distributed along the span. Thespanwise load distribution remains practically thesame. In the inner stations, there is a remarkablereduction of the viscous drag.

Higher sectional viscous drag can be attributedto a stronger growth of the boundary layer towardsthe trailing edge. This type of flow is susceptible totrailing edge separation, especially in low Reynoldsnumbers like those occur in the wind tunnel. Thisstronger boundary layer growth is due to a re-gion with high surface curvatures. These tran-sonic bumps are known to be physical mechanisms,which are apparently recognized by the optimiza-tion algorithm, to weaken shock waves. To copewith this situation, a constraint of the form

C(x/c) < Cmax, 0.50 < x/c < 0.98 (29)

is introduced to limit the upper surface curva-tures behind the shock wave. The effect of theconstraint at r? = 0.60 can be observed in Fig-ure 10. The lower picture shows that, after thecurvature constraint is applied, less growth of theboundary layer shape factor H is achieved, givinga significantly lower value of H at the trailing edge,thus providing sufficient margin to trailing edgeseparation. Remarkably, the curvature constraintdid not contradict the reduction of wave drag, aseven a lower wave drag was achieved. The pres-sure distribution appears to have adapted itself,by transferring a slight amount of the aft-load to-wards the middle part by slightly moving the shockwave aft, to allow a more gentle deceleration of theflow towards the trailing edge.

Another example from 2D optimization con-cerns a constraint on the lower wing surface of awing section at the kink area. This constraint canbe introduced to control the velocity on the lowersurface indirectly. Allowing the the lower surfaceto move downward would shift the fuel-tank vol-ume to the lower part, and therefore would provide

'J0.3

0.2

1200

WOO

800

600

400

2000.25 0.5 0.75

Fig. 9 Spanwise characteristics of 2D opti-mization results.

aft-loading, and lower sectional wave drag in theinner stations which is also indicated by more gen-tle gradient near the shock wave. There are nosignificant effects on the sectional viscous drag andspanwise load distributions.

The outcome of the second design cycle (outer-section optimization) can be identified as: highersuction peak near the leading edge, more gen-tle gradient near the shock wave, sloping roof-top pressure distribution, and a rather strong re-acceleration of the flow after the shock wave. Thesectional wave drag in the outer stations have beenreduced significantly. There is a noticeable in-

10 OF 12


Dow

nloa

ded

by S

TA

NFO

RD

UN

IVE

RSI

TY

on

Dec

embe

r 1,

201

5 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

002-

260

0.8

3.2

2.4

1.6

before optimizationafter optimization

0.5

Fig. 10 Upper surface Cp and H distributionsat r? = 0.60.


ConclusionsAn adaptive optimization strategy applied to

wing design of a wing/body configuration of re-gional jet aircraft has been described. The designcriteria have been met by successive optimizationstages, where on each stage a subset of the de-sign criteria is dealt with. The adaptive strategyallows one to understand the relations betweenvarious design requirements to approach the bestdesign solution. The optimization results shownemphasize that obtaining an optimal design space,through a well-posed optimization problem, is aprerequisite to achieving the best design solutionmeeting all the design criteria.

A combination of CFD tools, consisting of a3D viscous analysis code based on full-potentialand boundary layer equations, a 3D inviscid de-sign code based on the Euler equations, and a2D viscous design code based on the RANS equa-tions, has been used in the optimization process. Adefect-correction approach has been applied, suchthat this combination with different levels of flowmodeling leads to an actual design improvement inthe flow model with the highest representation ofthe flow physics involved (three-dimensional vis-cous flow). However, it should be noted that abetter perspective for the future may be given by acommon high fidelity modeling, such as that pro-vided by three-dimensional RANS equations, forthe analysis and design (adjoint) codes.

The results of 3D and 2D optimizations arecomparable quantitatively in terms of global ge-ometric properties and aerodynamic coefficients.In the end, the best design is determined basedalso on qualitative aspects such as characteristicsof the pressure distribution, expected buffet-onsetproperties and risks related to engine/airframe in-tegration and wind tunnel testing.

Transonic wind tunnel tests have been per-formed at the DNW-HST in Amsterdam, con-forming that the results from the present designexercise have led to the full attainment of cruiseperformance requirements for a new regional jetaircraft. From an industrial standpoint, the designmethodology has proven to be sufficiently robustfor solving real-life industrial design problems.

References1Liddel, P., "Technical Evaluation Report," Aerody-

namic design and optimisation of flight vehicles in aconcurrent multi-disciplinary environment, RTO-MP-35,2000.

2Jameson, A. and Vassberg, J., "Computational FluidDynamics for Aerodynamic Design: Its Current and FutureImpact," AIAA Paper 2001-0538, 2001.

3Nielsen, E. and Anderson, W., "Recent improve-ments in aerodynamic design optimization on unstructuredmeshes," AIAA Paper 2001-0596, 2001.

4Soemarwoto, B., Labrujere, T., Laban, M., and Yan-syah, H., "Inverse Aerodynamic Shape Design for Improved

1.10

| 1.08I>l 1.06

1.04

1.02

1.00

0.98

0.96

•• current+ optimized (lower surface constraint strictly applied)•• optimized (lower surface constraint fully released)

W/B at cruise CL, Pe

0.97 0.98 0.99 1.00

M/M .

Fig. 11 Effect of lower surface constraint todrag rise.

some flexibility for streamlining the upper surface.As a consequence, bringing the volume towardsthe lower part may imply higher velocity on thelower surface, thus may give additional risks withrespect to airframe/engine integration. Figure 11gives an idea of the possibilities that can be ob-tained by using such a control within optimization.While the improvements of the cruise performanceare practically the same, the properties below andabove the cruise Mach number are significantlydifferent, representing conflicting characteristics:potentially higher drag-divergence Mach numberbut higher drag at lower Mach numbers and largerrisks on airframe/engine integration (the lowersurface velocity has become supersonic), againstlower drag at lower Mach numbers and less riskson airframe/engine integration but possibly lowerdrag-divergence Mach.

11 OF 12


Dow

nloa

ded

by S

TA

NFO

RD

UN

IVE

RSI

TY

on

Dec

embe

r 1,

201

5 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

002-

260


Wing Buffet-Onset Performance," Inverse Problems in En-gineering Mechanics II, edited by M. Tanaka and G. Du-likravich, Elsevier, 2000.

5Soemarwoto, B., "Airfoil optimization using theNavier-Stokes equations by means of the variationalmethod," AIAA Paper 98-2401, 1998.

6Arian, E., Convergence Estimates for Multidisci-plinary Analysis and Optimization, ICASE Rep. 97-57,1997.

7van der Wees, A., van Muijden, J., and van derVooren, J., A fast and robust viscous-inviscid interactionsolver for transonic flow about wing/body configurations onthe bases of full potential theory, NLR-TP-93214 U, 1993.

8Jameson, A., "Optimum Aerodynamic Design UsingControl Theory," CFD Review, 1995, pp. 495-528.

9Reuther, J. and Jameson, A., "Aerodynamic ShapeOptimization of Wing and Wing-Body Configurations Us-ing Control Theory," AIAA Paper 95-0123, 1995.

10Jameson, A., Alonso, J., Reuther, J., Martinelli, L.,and Vassberg, J., "Aerodynamic shape optimization tech-niques based on control theory," AIAA Paper 98-2538,1998.

11Soemarwoto, B., Multi-Point Aerodynamic Design byOptimization, Ph.D. thesis, Delft University of Technology,also NLR TR 96534 L, 1996.

12Soemarwoto, B., The Variational Method for Aero-dynamic Optimization using the Navier-Stokes Equations,ICASE Rep. 97-71/NASA CR-97-206277, 1997.

13Brandsma, F., "Description of the Method Used byNLR," EUROVAL-A European Initiative on Validation ofCFD Codes, edited by H. W. et.al, Vol. 42 of Notes onNumerical Fluid Mechanics, Vieweg, Brunswick, 1993.

14Hagmeijer, R., "Grid Adaption Based on ModifiedAnisotropic Diffusion Equations Formulated in the Para-metric Domain," J. of Computational Physics, Vol. 115,No. 1, Nov 1994, pp. 169-183.

15Soemarwoto, B. and Labrujere, T. E., "Airfoil designand optimization methods: recent progress at NLR," Intl.J. Numer. Meth. Fluids, Vol. 30, 1999, pp. 217-228.

12 OF 12


Dow

nloa

ded

by S

TA

NFO

RD

UN

IVE

RSI

TY

on

Dec

embe

r 1,

201

5 | h

ttp://

arc.

aiaa

.org

| D

OI:

10.

2514

/6.2

002-

260

Documents

(c)2002 American Institute of Aeronautics & Astronautics ...aero-comlab.stanford.edu/Papers/aiaa2002-260.pdf · technology in aircraft design process.2"5 The cou-pling between the