AIAA 2004–1758Design of a Low-Boom SupersonicBusiness Jet Using EvolutionaryAlgorithms and an AdaptiveUnstructured Mesh MethodSeongim Choi and Juan J. AlonsoStanford University, Stanford, CA 94305-3030

Hyoung S. ChungKorean Air Force Academy, Korea

45th AIAA/ASME/ASCE/AHS/ASC Structures,Structural Dynamics and Materials Conference

April 19–22, 2004/Palm Springs, CA

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics1801 Alexander Bell Drive, Suite 500, Reston, VA 20191–4344

AIAA 2004–1758

Design of a Low-Boom Supersonic Business Jet

Using Evolutionary Algorithms and an

Adaptive Unstructured Mesh Method

Seongim Choi∗ and Juan J. Alonso†

Stanford University, Stanford, CA 94305-3030

Hyoung S. Chung‡

Korean Air Force Academy, Korea

The purpose of this study is to establish the feasibility of using constrained, multi-objective, evolutionary algorithms for the design of supersonic, low-boom aircraft incombination with high-fidelity modeling of the external aerodynamic flow. The objec-tive functions that we attempt to minimize simultaneously are the aircraft coefficient ofdrag at a fixed lift coefficient, and two different measures of the noise level generatedby the sonic boom signature at the ground. As part of this work we have developed ananalysis tool, BOOM-UA, which fully automates the process of CAD geometry and meshgeneration, solution adaptation, and signature extraction and propagation. BOOM-UAis based on the AirplanePlus unstructured tetrahedral solver for the Euler and Navier-Stokes equations. In previous work we have conducted a careful study regarding thenumber of solution adaptive refinement levels and mesh density that are required toaccurately predict the sonic boom signature at the ground. Since the cost of drivingBOOM-UA directly for reasonable design problems (> 20 design variables) with even themost efficient GAs is beyond the capabilities of modern day supercomputers, Krigingapproximation models were constructed from a database of CFD analyses and used inconjunction with the GAs in order to minimize the cost of the computation. This pro-cedure is able to generate a set of Pareto-optimal solutions for the sonic boom and dragobjectives that are also forced to satisfy the specified constraints. Optimized shapes forboth minimum initial pressure jump, ∆p, and preceived noise level, PNdB, are discussedand their boom signatures are compared. The result of this effort shows that the currentgeneration of computer clusters is able to treat significant design problems (between 20and 50 design variables) with reasonable accuracy, cost, and turnaround time.

INTRODUCTION

SONIC booms have been the main reason prevent-ing supersonic flight over inhabited areas. The

minimization of the environmental impact from thiskind of aircraft is one of the fundamental issues tobe resolved since studies have shown that such air-craft would have great market potential should theybe allowed to fly supersonically over land.1, 2 For thesereasons, research efforts have been recently focused onvarious techniques for sonic boom mitigation4, 6–8, 18

and non-linear CFD has emerged as an essential toolowing to the increasing availability of large computingresources.

The nature of this problem is more complicatedthan the traditional Aerodynamic Shape Optimizationproblem (ASO): not only must we deal with multi-ple objectives simultaneously (boom and performance)

∗Doctoral Candidate, AIAA Member†Assistant Professor, AIAA Member‡Assistant Professor, AIAA MemberCopyright c© 2004 by the authors. Published by the American

Institute of Aeronautics and Astronautics, Inc. with permission.

but the design space for the sonic boom objectivehas been shown to exhibit multiple local minima, andto be rather noisy and even discontinuous.5 Thesecharacteristics of the design problem rule out the pos-sibility of using gradient-based optimization (and thepowerful adjoint method9, 11–13) for the boom portionof the problem. Moreover, the computational cost isexacerbated by the fact that an accurate pressure dis-tribution must be computed a certain distance beneaththe aircraft (not on the aircraft surface as it is nor-mally the case for ASO) and therefore an extremelyfine mesh is required for the computations. Finally, thedetails of the geometry and the relative arrangement(distributions of volume and lift) of the componentsof the configuration are very important for accuratesonic boom predictions and, therefore, if the proce-dure is to be automated, it becomes important to haveautomatic, high-quality, geometry generation, and theability to create complex meshes without user inter-vention. The combination of all of these requirementshas lead us to the use of GAs combined with unstruc-tured adaptive meshing technologies.

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However, before sonic boom minimization work canbe credibly carried out, the accurate prediction of thefundamental sonic boom propagation problem has tobe validated. In our previous work,3 numerical andmesh resolution requirements for accurate sonic boomcomputations were established for both the computa-tion of near-field pressures and ground boom signa-tures. Results were verified with wind-tunnel experi-ments using the inviscid version of our solver and goodagreement was found for relatively fine (on the orderof 5-7 million nodes) solution-adaptive meshes.

Once the solution requirements for accurate sonicboom computation have been determined, the designof supersonic configurations can proceed. In our earlydesign efforts,4, 6 we had only considered two main ob-jective functions in an unconstrained multi-objectivedesign formulation: the drag coefficient of the aircraftat a fixed lift coefficient, and the magnitude of the ini-tial peak of its ground boom signature. In this work,additional constraints that guarantee a feasible aircraftwith the required range and suitable trim and stabilitybehavior are specified. In order to highlight the dif-ferences in the results when using two different noisemeasures, we include a comparison between the shapesthat result from using the initial pressure rise and theperceived loudness as part of our multi-objective opti-mization.

Our design procedure, as will be explained later,relies on the ability to compute a large number ofgeometrically-complex solutions for different values ofthe design variables in a short amount of time. Ouranalyses are carried out using a fully nonlinear, three-dimensional environment built from a number of inde-pendent modules including an unstructured adaptiveflow solver,19 the Centaur unstructured mesh genera-tion/adaption system,32 the AEROSURF CAD-basedgeometry kernel,17 and the ground boom predictionsoftware PCBoom3.31 The execution of all these mod-ules has been fully automated into a single analysistool which we call BOOM-UA.

But MDO methods, particularly those based onhigh-fidelity analyses, greatly increase the compu-tational burden and complexity of the design pro-cess33, 34, 36, 37 even with fully parallelized modulesbased on domain decomposition schemes. For thisreason, high-fidelity analyses typically used in singlediscipline designs may not be suitable for direct use inMDO.33, 35 In fact, even in single discipline designs, thecost of high-fidelity models may become prohibitive,particularly in combination with Genetic Algorithmsfor the optimization. Faced with these problems, thealternative of using approximation models of the ac-tual analyses has received increasing attention in re-cent years.

The Kriging technique, developed in the field of geo-statistics, has been recognized as an interesting choicefor approximation models of computationally expen-

sive CFD analyses. In theory, it is able to interpolatesample data and to model functions with multiple lo-cal extrema.27, 38 Using variations of the method ofdesign of experiments (DOE), the Kriging model canefficiently represent global trends in the design space.The basis functions chosen for the underlying Krigingapproximation determine, to a large extent, the qual-ity of the approximation.

Once an accurate approximation model is generated(at a computational cost that increases with the num-ber of design variables in the problem and the breadthof the design space), it can be used to search for opti-mum combinations of the design variables within thedesign space. Since the Kriging model can representmulti-modal functions with relatively low computa-tional cost it is a good candidate for use with globaloptimizers such as genetic algorithms for the solutionof the low-boom problem.

Our design procedure is then based on the followingkey ingredients:

• An automatic tool for the computation of flowfields and sonic boom signatures, BOOM-UA.

• The construction of Kriging approximation mod-els of the responses (aerodynamic performanceand sonic boom loudness.)

• A non-dominated sorting genetic algorithm.

• The application of constraints to limit the infea-sibility of the resulting designs.

It is worth mentioning that, when it comes to multi-objective optimization problems, there is not a singlebest solution but rather a non-dominated solution setmust be obtained (the so-called Pareto set.) In ourwork, the Non-dominated Sorting Genetic AlgorithmII (NSGA II) of Srinivas and Deb39 is used.

The procedure is straightforward. First, a largenumber of sample points in the design space (at whichCFD solutions will be computed using BOOM-UA)are generated using the Latin Hypercube Samplingtechnique. Then, Kriging approximation models areconstructed (based on this initial population) for thedrag coefficient, the initial shock strength, and the per-ceived noise level. Using these approximation modelsand two simple constraints related to trim, longitudi-nal stability and mission performance, a GA-based op-timization is performed to produce an expected Paretoset. The points on the approximation-model-basedPareto set are evaluated using BOOM-UA and the dif-ferences between the predicted and actual values canbe used to shrink or enlarge the size of a trust region.This procedure can be repeated iteratively until thepredicted and computed Pareto sets converge to eachother (the procedure can be stopped earlier if the dif-ferences are sufficiently small.) We have carried out

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such a procedure and the results are detailed in thefollowing sections.

The results show that the minimization of the loud-ness of the ground sonic boom with constraints is moredifficult to achieve compared to unconstrained opti-mizations where configuration parameters such as thesweep angle are allowed to attain unreasonable values.The amount of reduction in the boom figure of meritis not as large as for unconstrained optimization.

AERODYNAMIC ANALYSIS :

BOOM-UA

All of the necessary modules to carry out aerody-namic analyses and ground boom signature computa-tions are integrated into our analysis tool, BOOM-UA.The complete procedure is fully automated for effi-ciency and for the ability to include BOOM-UA in adesign loop. Firstly, a parameterized geometry is rep-resented directly with a CAD package (ProEngineerin this study) using a collection of surface patches. Abaseline unstructured tetrahedral mesh is generatedby the user with the Centaur software. Our geom-etry kernel, AEROSURF generates variations of thebaseline configuration. If the changes in the geometryare small enough we can perturb the baseline mesh toconform to the deformed shape without problems withdecreased mesh quality and/or edge crossings. Other-wise, an automatic mesh regeneration procedure hasbeen implemented to accommodate large changes inthe geometry. Our inviscid Euler solver, AirplanePluscalculates the surface pressure distributions and pre-dicts both CL and CD and the near-field pressureswhich can then be propagated to the ground to ob-tain ground boom signatures. A solution-adaptivemesh refinement procedure is interfaced with the flowsolver to generate refined meshes adapted according toa criterion based on the pressure gradient informationobtained by the flow solver. The boom prediction soft-ware PCBoom3 is used for the propagation portion ofthe solution procedure. Three-dimensional near-fieldpressure distributions are extracted on a cylindricalsurface several body lengths beneath the configurationand are provided to the boom propagation tool. Fig-ure 1 shows a brief schematic of all the processes thathave been integrated into BOOM-UA. In this Figure,n refers to the number of design points. Each indi-vidual module is explained in detail in the followingsubsections.

CAD Geometry Representation

High-fidelity MDO requires a consistent high-fidelitygeometry representation. In general, the geometricshape of an aircraft can be defined by an appropri-ate parameterization of the geometry. This parametricgeometry kernel is available to all of the participatingdisciplines in the design so that both cost functionsand constraints can be computed using the same ge-

n = n + 1

n = 1

Satisfied ?

Yes

AEROSURFParametrizationConfiguration

Level of accuracy

No

or

large geometric change??n > 1 and n = 1

Yes

No

Movegrid

Parallel Flow Solution(CFD)

AirplanePlus

Warping / MovementTetrahedral Mesh

Generation / RegenerationMakegrid

Tetrahedral Mesh

Adaptgrid

Boom ExtrapolationPCBOOM

Near−filed Pressure Extract

Solution AdaptiveMesh Refinement

Fig. 1 Schematic of aerodynamic analysis tool,BOOM-UA

ometry representation.

In our work, a CAD-based geometry kernel is usedto provide this underlying geometry representation.Baseline shapes are developed in a CAD-package(ProE, in our case) and constitute our parametric mas-ter model; they uniquely define the parameterizationof a configuration given a particular design intent.

Our geometry kernel (AEROSURF) is coupledwith the parametric CAD description through theCAPRI interface of Haimes,14, 17 such that it automat-ically generates watertight surface geometry patches.AEROSURF can be executed in parallel and uses adistributed geometry server to expedite the generationof a large number of different design alternatives, thusreducing the cost of running geometry regenerationsin the CAD package. Using a master/slave approachand PVM for distributed computing, arbitrary num-bers of slaves can be started simultaneously while amaster program maintains a queue of geometry re-generation requests and keeps slaves busy doing CADre-generations.

Furthermore, AEROSURF has the capability of gen-erating automatic surface meshes (both triangular andpatched-quadrilateral) that can be linked directly withthe volume mesh generation portion of Centaur. Thissurface geometry generation tool can effectively beincorporated into our unstructured mesh generationenvironment for the automatic treatment of complexconfigurations, including the presence of nacelles, di-verters, etc. Figure 2 shows a baseline geometry with

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46 surface patches and 108 design variables, which isgenerated directly in ProEngineer.

Fig. 2 Geometry representation by parametric,CAD-interfaced AEROSURF

Tetrahedral Unstructured Mesh and SolutionApproach

In this work we focus on the use of unstructuredtetrahedral meshes for the solution of the Euler equa-tions around complete aircraft configurations. Themost popular methods of generating unstructuredmeshes20–22 are the advancing-front and Delaunay tri-angulation methods. In this work, the advancing-frontmethod is chosen and the Centaur grid generation soft-ware32 is used. The advancing-front method23 involvesthe simultaneous generation of mesh points and theirconnectivity. The main process reduces to building themesh iteratively element by element, adding new ele-ments to previously generated elements, thus sweepingout a front across the entire domain. This method usu-ally relies on an explicitly defined element-size distri-bution function, which is most often constructed usinga background octree grid23, 24 for size and stretchinginformation and plays a major role in the success ofthis method.

One of the advantages of the advancing frontmethod, like any other marching method, is its restart-ing capability. Since the only data required to restartthe generation process are given by the current front,information regarding the already generated grid be-hind the current front is not necessary. Also boundaryintegrity is guaranteed since this process starts from adiscretization of the solid boundaries.

The Centaur software is directly linked with the sur-face representation obtained from AEROSURF, andis used to construct meshes for aircraft configurationsand to enhance grid quality through automatic post-processing. Only fine meshes need to be explicitlyconstructed since our multigrid algorithm is based onthe concept of agglomeration and, therefore, coarsermeshes are obtained automatically. The following Fig-ure 3 shows a triangular mesh on the body surface and

the symmetry plane of our configuration. For visual-ization purposes only, a coarsened grid is shown.

Fig. 3 Unstructured tetrahedral surface mesharound a low-boom aircraft

Fig. 4 Mesh before and after two adaption cycles

The three-dimensional, unstructured, tetrahedralAirplanePlus inviscid flow solver is used in this work.AirplanePlus is a C++ implementation of the origi-nal AIRPLANE flow solver of Jameson25 by Van derWeide with substantial enhancements to the baselinealgorithm. An agglomeration multigrid strategy isused to speed up convergence. A modified Runge-Kutta time stepping procedure with appropriately tai-lored coefficients is used to allow for high CFL num-bers. Several options for artificial dissipation and theblock-Jacobi preconditioning method are all availablein the solver. Figure 5 shows a pressure plot on the sur-face of our baseline configuration flying at M∞ = 1.5and at CL = 0.1. Figure 6 shows the pressure on thesymmetry plane for the same configuration showing

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the propagation of all shock waves at the same flowconditions.

AirplanePlus Pressure from 0.8700 to 1.0800

Fig. 5 Pressure plot around aircraft surface,M=1.5

AirplanePlus Pressure from 0.8700 to 1.0800

Fig. 6 Pressure plot on the symmetry plane,M=1.5

Solution Adaptive Refinement

Once an initial solution has been computed on thetetrahedral mesh, the grid needs to be locally refinedto better capture specific features with higher accuracyat lower cost. For the cases of steady-state flows thatwe are investigating, coarsening has only a minor per-formance impact and is omitted in this work, althoughthe mesh generation system is capable of automaticallycoarsening meshes in areas that do not require the res-olution provided.

An element subdivision (h-refinement) method isemployed. This is implemented with strict hierarchi-cal subdivision rules15 to prevent degradation of gridquality at each subsequent adaption phase. For eachedge that is flagged for refinement by the solution-based error estimation criteria, a new node is insertedat its midpoint and reconnected to form new tetrahe-dral children. Highly anisotropic children may not berefined further to ensure high grid quality.

For cells with hanging nodes resulting from the

interface of refined and non-refined regions and foranisotropic cells requiring further refinement, the chil-dren cells are removed and their parent cells areisotropically refined to ensure a compatible refinementpattern.

Finally, for boundary edges, these newly insertednodes are repositioned onto the splined surface whichdefines the original surface from the CAD package.Also, undesirable shape measures are investigated andnew local tetrahedral configurations with more desir-able shape measures are selected.

All of these refinement procedures are included inthe Centaur mesh generation package that we use forthis work.

Another important issue in solution adaptive re-finement for complex problems is the choice of anappropriate error indicator. Most widely-used refine-ment options to estimate errors or detect flow featuresare gradient-based criteria driven by the physical flowvariables. Gradients of pressure can be used to iden-tify inviscid flow features. But in sonic boom predic-tion problems, the pressure gradient in the near-field(which has a much lower magnitude) is just as impor-tant as in the neighborhood of the aircraft.

In this case, the refinement criterion is not sim-ply based on the gradient of the pressure because thestrongest shocks in the neighborhood of the aircraftwill then be the only target of refinement, leavingweaker shocks in near field unresolved.

A solution-adaptive refinement procedure with aconstraint on the cell size inside a specific region ofinterest, which can be predicted from a straightfor-ward shock angle calculation, is best suited to the sonicboom prediction problem.

A reasonable predictor of the shock location can bebased on the magnitude of the local velocity vectorprojected onto the direction of the local pressure gra-dient. A mathematical expression for this refinementcriterion is presented below. In the expression for theadaption criterion, V is the velocity vector, c is thespeed of sound and 4x is the required minimum edgelength which is often problem dependent.

ε′

=V

c·∇p

|∇p|(1)

Numerical experiments16 indicate that a modifica-tion of the previous equation to include a mesh length-scale such as:

ε =V

c·∇p

|∇p|4x (2)

produces a more effective refinement criterion forshock/expansion flows by increasing the control overthe size of triangles. The adaption procedure is re-peated until a specified accuracy is achieved or a max-imum number of refinement levels has been reached.

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Figure 4 shows two meshes. The first one is the origi-nal mesh (number of nodes = 562,057) before adaptionand the second one shows the solution-adapted mesh(number of nodes = 1,028,577) after two adaptionsteps. Typically three or four consecutive adaptioncycles are performed automatically starting from theinitial mesh to reach the necessary mesh resolution.The necessary mesh resolution for all designs in thiswork is determined by the results of a previous study.3

Ground Boom Propagation

The basic strategy for the computation of groundboom signatures can be seen in Figure 7 below. A so-lution adaptive mesh using the criteria described aboveis constructed around the aircraft. It extends a certaindistance away from it, but not to the ground plane asit would be computationally prohibitive to do so withappropriate resolution from the cruise altitude. At thenear-field plane location, the pressure signature cre-ated by the aircraft is extracted and it is propagateddown to the ground using extrapolation methods basedon geometric acoustics.

The far-field boundaries of the CFD mesh must belocated close enough so that the resulting mesh size iscomputationally manageable, but, at the same time,they must be located far enough so that the near-fieldflow field is axisymmetric and there are no remainingdiffraction effects which cannot be handled by the ex-trapolation scheme.

In this work, the pressure field on the symmetryplane 1.2 body lengths below the aircraft is obtainedand provided as an initial condition to the boom ex-trapolation program. The pressure extraction algo-rithm (for the unstructured tetrahedral mesh) is basedon advanced octree data structures and therefore in-curs very little computational cost.

In this work, we are using both the Sboom30 andPCBoom331 extrapolation methods to propagate near-field signatures into ground booms. Although the PC-Boom3 software is far more capable than Sboom, wehave only computed ground booms created by the air-craft in a steady-state cruise condition and, therefore,both codes are nearly equivalent. If ground boomscaused by maneuvering aircraft were to be computed,the capabilities of the PCBoom3 software would haveto be used.

The two sonic boom extrapolation methods accountfor vertical gradients of atmospheric properties and forstratified winds (the winds have been set to zero inthis work.) Both methods essentially rely on resultsfrom geometric acoustics for the evolution of the waveamplitude, and both utilize isentropic wave theory toaccount for nonlinear waveform distortion due to at-mospheric density gradients and stratified winds.

There are additional extrapolation/propagationmethods that are based on the concept of an F-function29 but these have not been used in this work

as they assume no variation of the near-field signaturein the azimuthal direction, which is normally not thecase in our computations.

In three-dimensional wave propagation cases, thepressure data on a cylindrical surface centered alongthe aircraft longitudinal axis is extracted instead andpropagated along rays in directions that are not neces-sarily perpendicular to the ground. The propagationscheme marches these rays down to the ground fromall azimuthal directions that may eventually reach theground. Depending on the atmospheric conditions andflight altitude, a cutoff angle will exist beyond whichno disturbance will reach the ground: refraction effectsdivert the noise propagation back toward the upper at-mosphere.

Our earlier research on low-boom aircraft designwas mainly focused on the reduction of the magnitudeof only the initial peak of the ground boom signa-ture.18 This requirement, which had been suggestedas the goal of the DARPA-sponsored Quiet SupersonicPlatform (QSP) program (∆p0 < 0.3 psf), hides theimportance of the rest of the signature, which oftenarises from the more geometrically complex aft portionof the aircraft where empennage and engine nacellesand diverters create more complicated flow patterns.Moreover, such designs often have two shock wavesvery closely following each other in the front portionof the signature,5, 7 a behavior that is not robust andis therefore undesirable.

For this reason, we have chosen to base our designson the perceived loudness of the complete signature aswell as the initial peak of ground boom. Frequencyweighting methods are used due to the unique prop-erty of the human hearing system which doesn’t havean equal response to sounds of different frequencies.In these calculations, less weighting is given to the fre-quencies to which the ear is less sensitive.

VALIDATION OF BOOM-UA

The accuracy of our aerodynamic analysis tool,BOOM-UA, has been validated in previous work3 toguarantee that the results of our optimizations can betrusted. Experimental near-field pressure data wereavailable from a NASA Langley wind-tunnel test.10

The model consists of a wing with a large outboard di-hedral/winglet, fuselage, vertical tail, and aft-fuselagemounted nacelles with diverters attached to the fuse-lage. To determine which changes correspond toincreasing geometric complexity, two additional con-figurations were also tested. To obtain these twoadditional configurations, the nacelles and divertersare first removed and, then, both the tail and na-celles/diverters are omitted.

BOOM-UA was applied to all three configurationsto extract the near-field overpressures and to predictthe ground boom signatures. The near-field overpres-sures were extracted at three distances from the air-

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Wo

F(~b)

Near Field Plane

τ

Global CFD Mesh

Atmospheric Properties

ρ(z)T (z)p(z)

Wg Ground Plane

h

Fig. 7 Schematic of sonic boom propagation pro-cedure.

craft model (9.5 in, 12 in, and 16 in) corresponding tothe locations where experimental data was taken. Thisallows us to compare the CFD predictions directlywith experiment and to compute ground signaturesextrapolated from these distances.

Our mesh resolution study required up to 4 adaptioncycles (the finest meshes had about 8 million nodes ob-tained by solution-adaptively refining an initial meshwith only 0.5 million nodes) to predict accurately thenear-field pressure distributions given in the experi-ments. The results from the finest mesh show verygood agreement with the experiment both in the near-field pressures and ground boom signatures. Figure 8shows the comparison of the near-field pressure at 9.5in below the complete configuration with nacelles andtails, and figure 9 corresponds to 12 in below the sameconfiguration. Ground boom signatures propagatedfrom those locations were also computed using BOOM-UA and compared with experimental data. Figures 10and 11 show good agreement between the computa-tions and experiments for the ground boom signatures.

Now that the accuracy of our aerodynamic analysistool, BOOM-UA, has been validated, the optimizationprocedure can be reliably combined with the approxi-mation model which is generated by BOOM-UA.

MULTI-OBJECTIVE

OPTIMIZATION

The minimization of the loudness of the groundboom normally comes at the expense of the perfor-mance of other disciplines participating in the design.For example, most shape changes that will improvethe sonic boom of an aircraft will have a detrimen-tal effect on the cruise drag coefficient. The converse

10 15 20 25 30 35 40 45 50 55 60

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

x

dp/p

computationexperiment

Fig. 8 Comparison of near-field pressure 9.5 inbelow the configuration

10 15 20 25 30 35 40 45 50 55 60

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

x

dp/p

computationexperiment

Fig. 9 Comparison of near-field pressure 12 inbelow the configuration

also tends to be true: changes to the shape that resultin improvements in aerodynamic performance tend toworsen the sonic boom characteristics.

For the low-boom problem, obtaining a solutionwhich simultaneously optimizes all the objectives isnot possible and the concept of the optimal Pareto setbecomes important, as it suggests a set of solutionswhich are superior to the rest with respect to all ob-jective criteria, but are inferior to other solutions inone or more of the objectives. Once the set of optimalsolutions has been identified, it is up to the designerto choose the one solution, out of many possible ones,that best meets the overall purpose of the design. Agenetic algorithm can use this dominance criteria ina straightforward fashion to drive the search processtoward the Pareto front.

Due to the unique features of GAs, which workwith a population of solutions, multiple Pareto opti-mal solutions can be captured in a single optimizationrun. This is the primary reason that makes GAs ide-

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0 50 100 150−1

−0.5

0

0.5

1

1.5

time (msec)

boom

ove

rpre

ssur

e (p

sf)

after 1 adptafter 4 adptexperiment

Fig. 10 Comparison of ground boom extrapolatedfrom 9.5 in below the configuration

0 50 100 150−1

−0.5

0

0.5

1

1.5

time (msec)

boom

ove

rpre

ssur

e (p

sf)

after 1 adptafter 4 adptexperiment

Fig. 11 Comparison of ground boom extrapolatedfrom 12 in below the configuration

ally suited for multi-objective optimization. Howevermany of the existing MOEAs have been criticized fortheir computational complexity (which is related tothe ranking/sorting of the individuals in a genera-tion by comparison of their fitness) and for the needto specify a sharing parameters. The Non-dominatedSorting Genetic Algorithm (NSGA-II) was used to al-leviate these problems and has shown to have fastconvergence rates.

However, the direct combination of CFD analysiswith GAs can be very expensive due to the high com-putational cost for the evaluation of the fitness of eachmember of the population and the large number of gen-erations required to reach the Pareto front: often GAsrequire large populations and many generations to lo-cate the global extremum of complex design spaces. Ifa fairly accurate global approximation model for theCFD analysis can be constructed, its combination withGAs can be tractable. A Kriging approximation modelhas been used successfully in combination with GAs.4

In this study, Kriging approximation models areconstructed from a large number of members of aninitial population using the results from repeated anal-yses carried out with BOOM-UA. The approximationmodel is combined with the NSGA-II algorithm tominimize ground boom intensity while maintaining ac-ceptable levels of aerodynamic cruise performance.

Kriging Approximation Model

The Kriging approximation technique, developed inthe field of spatial statistics, has been drawing at-tention as an alternative to expensive CFD simula-tions. This interpolation technique creates a surface fitfrom measured sample data using advanced statisticalmethods. While searching for data trends and globaland local outliers, the ability of the Kriging methodto capture multiple local extrema is fundamental inglobal optimization.

The Kriging technique uses a two-component modelthat can be expressed mathematically as

y(x) = f(x) + Z(x) (3)

where f(x) represents a global model and Z(x) isthe realization of a stationary Gaussian random func-tion that creates a localized deviation from the globalmodel.26 If f(x) is taken to be an underlying con-stant,27 β , Equation (1) becomes

y(x) = β + Z(x), (4)

which is used in this paper. The estimated model ofEquation (2) is given as

y = β + rT (x)R−1(y − f β), (5)

where y is the column vector of response data and f isa column vector of length ns which is filled with ones.R in Equation (3) is the correlation matrix which canbe obtained by computing R(xi, xj), the correlationfunction between any two sampled data points. Thiscorrelation function is specified by the user. In thiswork, we use a Gaussian exponential correlation func-tion of the form provided by Giunta, et al.28

R(xi, xj) = exp

[

−

n∑

k=1

θk|xik − x

jk|

2

]

. (6)

The correlation vector between x and the sampleddata points is expressed as

rT (x) = [R(x, x1), R(x, x2), ..., R(x, xn)]T . (7)

The value for β is estimated using the generalized leastsquares method as

β = (fT R−1f )−1fT R−1y. (8)

Since R is a function of the unknown variable θ, β

is also a function of θ. Once θ is obtained, Equation

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(3) is completely defined. The value of θ is obtainedby maximizing the following function over the intervalθ > 0

−[ns ln(σ2) + ln |R|]

2, (9)

where

σ2 =(y − f β)T R−1(y − f β)

ns

. (10)

In order to construct a Kriging approximation theonly data required are the function values at a numberof pre-specified sample locations. For many computa-tional methods, secondary information such as gradi-ent values may be available as a result of the analysisprocedure and can be used to enhance the accuracyand to lower the cost of the Kriging approximation.4

Non-dominated Sorting Genetic Algorithms

When several criteria are considered in an optimiza-tion problem, the optimal solution is generally notunique since no point can be considered best withrespect to all criteria simultaneously. Instead, opti-mal solutions on the Pareto set, which contains allnon-dominated solutions, can be identified. GeneticAlgorithms (GAs) simultaneously operate on a popu-lation of potential solutions and build a database ofsolutions which can be thought of as a cloud of pointsin solution space. At convergence, the cloud ceasesto evolve and its convex hull determines the optimalPareto set. Our results in multi-objective optimizationare based on the algorithm of Srinivas and Deb: theNon-dominated Sorting Genetic Algorithm (NSGA).39

First, this algorithm applies non-dominance defini-tions to each individual in a population according totheir fitness value. But, in a multi-objective optimiza-tion problem, different fitness values are assigned toeach candidate design corresponding to the various de-sign criteria. Therefore, to evaluate the individuals, aranking is necessary and the candidate designs are clas-sified by fronts. The non-dominated sorting procedurerequires a ranking selection method which emphasizesthe optimal points. Therefore, the sharing technique,or niche method, is used to stabilize the subpopula-tions of the ‘good’ points.

The main concern related to the use of GAs in op-timum designs is the computational effort needed. Indifficult problems, GAs require bigger populations andthis translates directly into higher computational ef-fort for the optimization algorithm itself. In the caseof our multi-objective problem, two objectives alongwith several constraints should be evaluated for eachindividual, which means that the computational effortis rather large. One of the main advantages of GAsis that they can be easily be parallelized.40 At eachgeneration, the fitness values associated with each in-dividual can be evaluated in parallel. In this study, wehave only two fitness functions at a time along with asmall number of constraints and the parallelization of

NSGA-II is not necessary. However for future prob-lems involving a large number of objective functionswith the consideration of several constraints, a paral-lelized NSGA-II can be effectively used.

TEST PROBLEMS AND RESULTS

Design Problems

A baseline configuration for the first design cycle ischosen from the result of an aero-structural design car-ried out in our previous work.17 This configuration hasbeen optimized for minimum drag with no attentionpaid to sonic boom. By choosing an initial baselinewhich was already optimized for drag we can investi-gate the effect of adding a second objective (such asreducing sonic boom.)

First, seventeen design variables are chosen to allowreasonable geometric changes from the initial baselineconfiguration. The values of these design variablesare chosen using a random Latin Hypercube samplingtechnique in order to generate parametric CAD defini-tions of the geometry for each of the initial members ofa population. In this study 232 initial sample designpoints were selected to accommodate possibly largevariations in the design space corresponding to ratherlarge geometric changes. Variations as large as 30%for each of the seventeen design variables are used inthe first cycle of this study.

Most of the geometric changes are restricted to thewing and fuselage. Fuselage camber and radii at fivesections which are equally spaced along the fuselagelength can be changed, while the location and shapeof the wing are also varied. The wing reference areais also allowed to vary, although the reference area forthe initial baseline is used to calculate aerodynamiccoefficients such as CL and CD .

The seventeen design variables which define the pa-rameterization of the configuration are

x1 = wing position along fuselagex2 = wing dihedral anglex3 = wing reference areax4 = wing sweep anglex5 = wing aspect ratiox6 = wing leading edge extensionx7 = wing twist anglex8 = fuselage camber at 16.67% of fus. lengthx9 = fuselage camber at 33.33% of fus. lengthx10 = fuselage camber at 50.00% of fus. lengthx11 = fuselage camber at 66.67% of fus. lengthx12 = fuselage camber at 83.33% of fus. lengthx13 = fuselage radius at 16.67% of fus. lengthx14 = fuselage radius at 33.33% of fus. lengthx15 = fuselage radius at 50.00% of fus. lengthx16 = fuselage radius at 66.67% of fus. lengthx17 = fuselage radius at 83.33% of fus. length

BOOM-UA is run to analyze the aerodynamic per-

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formance of each design point and to compute threeobjective functions: the aircraft drag coefficient andthe strength of the ground boom signature (using boththe initial pressure rise and the perceived noise level)for each design point. The flight altitude is set at55, 000ft at the cruise condition and CL is fixed at 0.1by allowing the angle of attack to vary during the pro-cess of a flow solution. The near-field pressure at 1.2body lengths below the configuration was extracted onthe symmetry plane. Using this signature, the groundboom signature is then computed by propagating it tothe ground plane. In addition to the initial pressurerise of the signature, the overall perceived noise level isalso computed for more practical designs of low-boomconfigurations. This measure is used because of thefact that the reduction of the initial pressure rise oftenleads to stronger shocks elsewhere in the wave whichwould result on a louder ground boom and that wouldnot be captured by this performance measure. Basedon the computed values of these objectives, Krigingapproximation models are constructed and used incombination with the NSGA-II for optimization. Fig-ure 12 shows the baseline point and the initial 232sample points. After running NSGA-II to convergence,the CFD validation of the predicted optimal Pareto setshows that some of the points in the initial databaseare very close to the optimal Pareto front itself. With-out loss of accuracy, the best sample point in the firstdesign cycle can then be used as the baseline configu-ration for the second design cycle. However with theaddition of simplified constraints for trim, longitudi-nal stability, and fuel volume (based on wing area,since the wing thickness is not changed), this pointmay turn out to be infeasible. The drag coefficientis reduced by 17.5% from 0.0112978 to 0.0093134 andthe ground boom initial pressure rise is reduced by asmuch as 13.7% from the baseline (from 0.633 to 0.546psf). The overall perceived noise level is reduced byabout 2%. This implies that substantial reductionsin the initial pressure rise do not necessarily guaran-tee that that perceived noise level will decrease in thesame proportion. Comparing the results in figures 14and 13, one can see that the disturbances in the near-field pressure distribution are much larger in the caseof the baseline configuration. Detailed near-field pres-sure distributions are compared in figure 15. Due tothe smaller disturbances in the near-field, this designpoint shows a better ground boom signature than thebaseline as can be seen in figure 16.

Realistic Constraints

Previous studies on the minimization of the groundsignature often led to unrealistic shapes due to the lackof the consideration of reasonable problem constraints.In previous work and without the ability to changethe wing reference area, the optimized shape can havea highly swept wing4 which may cause structural in-

0.005 0.01 0.015 0.020.4

0.5

0.6

0.7

0.8

0.9

1

1.1

CD

Boo

m o

verp

ress

ure

( dp

p )

173 infeasibl points59 feasible pointsdesign point 1design point 2baselinePareto front from NSGA−IICFD validation

Fig. 12 1st Design cycle results using Kriging-based NSGA-II.

AirplanePlus Pressure from 0.8700 to 1.0800

Fig. 13 Pressure plot on the symmetry plane forthe baseline configuration, M=1.5

stabilities and the inability to trim the configuration.The best sample point from the first cycle in this workshows similar lack of feasibility as can be seen in fig-ure 22. The wing area has decreased significantly andthe location of the wing has moved backwards. Thedecrease in the wing area will cause a reduction in theavailable fuel volume thus making the aircraft fail itsrange requirements. Furthermore, the increase in stallspeed will cause the field length requirements to beexceeded. The change in wing position will affect thelongitudinal stability since the center of pressure hasnow moved backwards as well. These unrealistic de-signs require the introduction of, at least, some basicconstraints. In this study, and starting from the re-sults from the first design cycle, two constraints areimposed: a simplified version of trim and longitudi-nal stability that prohibits the center of pressure frommoving further aft than a specified location, and alower limit on the wing surface area so that stall speedand fuel volume may not change significantly.

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AirplanePlus Pressure from 0.8700 to 1.0800

Fig. 14 Pressure plot on the symmetry plane forthe best sampled configuration, M=1.5

100 150 200 250 300 350 400 450 500−0.03

−0.02

−0.01

0

0.01

0.02

0.03

time (msec)

boom

ove

rpre

ssur

e (d

pp)

best sample pointbaseline

Fig. 15 Comparison of near-field pressure distri-butions of the best sample point (design point 2)and the baseline configuration.

Assuming that the center of gravity is located at65% of the fuselage length (a reasonable assumptiongiven that the aircraft has rear-mounted nacelles), therearmost location of the center of pressure of the wingcan be obtained from a simplified trim condition:

Mcg = (Xcpmax−Xcg)×W −LTmax

× (XT −Xcg) = 0,

(11)where Mcg is the total pitching moment about the cen-ter of gravity, Xcpmax

is the distance from the noseof the fuselage to the center of pressure and Xcg isthe distance to the center of gravity. W is the to-tal gross weight of the configuration and LTmax

is themaximum lift that can be generated by the horizontaltail. The location of the center of pressure is obtainedwith the assumption that the wing has an ellipticalload distribution in the spanwise direction and the lo-cal lift vector acts at the 40% chord location. Forthe initial baseline configuration, Xcpmax

was locatedat about 100 ft away from the nose of the fuselage

0 50 100 150 200 250 300−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

time (msec)

over

pres

sure

(dp

p)

with constraintwithout constraint

Fig. 16 Comparison of ground boom signaturesbetween the best sample point (design point 2) andthe baseline configuration.

(the fuselage is 150 ft in length.) The allowable min-imum wing area is set to 79% of the wing area of thebaseline configuration. With these two simplified con-straints, it turns out that the best design point inthe first design cycle was infeasible, and therefore itis not an appropriate choice for the baseline for thesecond design cycle. Instead, we select a different de-sign point that satisfies the constraints and that alsohas reasonable aerodynamic performance and boomcharacteristics. Of the 232 initial sample points, thefeasible designs are highlighted in figure 12. Amongthe 59 feasible points, design point 1 shows ratherreasonable values of both boom loudness and dragcoefficient and therefore it is chosen as the baselineconfiguration for the second design cycle. By choosingthis point as the baseline for the second design cy-cle, more realistic configurations can be selected withbetter ground boom and aerodynamic properties. Fig-ure 21 shows the two baseline configurations for eachdesign cycle. For the second design cycle, the wingof the new baseline has moved forward and the wingsweep has decreased by a small amount. In the seconddesign cycle and centered around the new baseline,another 167 sample points are chosen with a similarprocedure to that used in the first design cycle. Whenconsidering the two constraints described above, about75% of the total samples are chosen to satisfy the con-straints, and another 25% of the sample points arechosen to violate the constraints slightly. Using theseselections, the Kriging approximation model can havemore information about the location of the constraintswithin the design space. Figure 17 shows the second167 design points and the Pareto fronts for the initialpressure rise and the drag coefficient. Figure 18 showsthe Pareto front for the perceived noise level and dragcoefficient optimization. For both cases, Pareto frontswith constraints and without constraints are shown. Infigure 17, there is not much difference between the con-

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strained and unconstrained Pareto fronts, but figure 18shows larger differences between the same two Paretofronts. As one can expect, the Pareto front computedwith the constrained version of the GA has slightlyhigher values of the drag coefficient, initial pressurerise and perceived noise level. For the boom initialpressure rise optimization, the reduction in the ini-tial peak was 10.08% while the reduction in the dragcoefficient was 17.38%. For the perceived noise leveloptimization, the reduction in noise level was muchless: only about 2%. The reason for this smaller re-duction in perceived noise level is that perceived noiselevel is usually governed by the overall shapes of thewave and not only by the initial peak of the wave, andin our work the design variables are mainly involvedin changes in the wing and fuselage and they are notsufficient to reduce the 2nd peak of the wave whichis affected by the rear portions of the configurationincluding the nacelles and empennage. Additional de-sign changes in this portion of the configuration (whichis notably the most complicated to design) can resultin larger reductions in the perceived noise level and itis left for future work. A comparison of the groundboom signatures between the baseline and the boomoptimized configuration is shown in figure 19. A com-parison between the ground boom signatures of thebaseline and the perceived noise level optimized con-figuration is shown in figure 20. For both of the figures,there is not much difference in the ground boom sig-natures The corresponding shapes that generate thesesignatures are shown in figures 23 and 24.

0.009 0.01 0.011 0.012 0.013 0.014 0.015 0.016 0.0170.4

0.5

0.6

0.7

0.8

0.9

1

1.1

CD

Boo

m o

verp

ress

ure

( dp

p )

167 initial samplesconstrained pareto frontunconstrained pareto frontCFD validationbaseline

Fig. 17 2nd Design Cycle Results using Kriging-Based NSGA-II for Drag Coefficient and InitialPressure Rise.

Conclusions and Future Work

In this work, we have described a procedure basedon high-fidelity analysis for the simultaneous optimiza-tion of the aerodynamic performance (coefficient ofdrag at fixed lift) and the loudness of complete con-figuration supersonic aircraft. The procedure uses

0.009 0.01 0.011 0.012 0.013 0.014 0.015 0.016 0.01790

92

94

96

98

100

102

104

106

CD

Per

ceiv

ed n

oise

leve

l ( p

ldB

)

167 initial samplesconstrained pareto frontunconstrained pareto frontCFD validationbaseline

Fig. 18 2nd Design Cycle Results using Kriging-Based NSGA-II for Drag Coefficient and PerceivedNoise Level.

0 50 100 150 200 250 300−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

time (msec)

over

pres

sure

(dp

p)

initial baselinebm−optimized one

Fig. 19 Comparison of the Ground Boom Sig-nature Between Baseline and Initial Pressure RiseOptimized Configuration During the Second De-sign Cycle.

the BOOM-UA environment, Kriging approximations,and a non-dominated sorting genetic algorithm to ob-tain optimum designs.

A configuration that had been previously optimizedfor aerodynamic performance was chosen as the base-line for the 1st design cycle of a multi-objective opti-mization. The result of this design cycle is a configura-tion with significantly less drag and lower sonic boomloudness. However, this unconstrained design did notmeet the stability, range, and take-off and landingrequirements we had setup. For this reasons, a sub-optimal design which satisfied all constraints and hadreasonable aerodynamic performance was chosen asthe baseline for the second design cycle. The Kriging-based genetic algorithm procedure gives shapes whichare optimized for both the initial peak and perceivednoised level of the signature along with minimum drag.

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0 50 100 150 200 250 300−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

time (msec)

over

pres

sure

(dp

p)

initial baselinepnl−optimized one

Fig. 20 Comparison of the Ground Boom Sig-nature Between Baseline and boom initial peakoptimized configuration in the second design cycle

Fig. 21 Baseline configurations for the first de-sign cycle (bottom) and for the second design cycle(top)

Fig. 22 Configuration for the best design pointand the baseline

The differences between these shapes produced aresmall because of the fact that the parameterization ofthe design does not include changes in the rear part ofthe configuration which may reduce the pressure dis-turbance in the near-field and result in smaller secondpeaks in the ground boom signatures. The appropri-ate selection of the design variables to reduce the 2ndpeak of the ground boom signature will be considered

Fig. 23 Configuration optimized for minimumboom

Fig. 24 Configuration optimized for minimum pnl

in future work.

This study, however, demonstrates the feasibilityof using high-fidelity modeling for design in multi-objective situations. Even though the cost of a singleanalysis is relatively large, solution-adaptive methodswith meshes with over 5 million nodes (to appropri-ately resolve the physical phenomena) can be of prac-tical use today.

Future work will concentrate in further refinementsto the approximation techniques in order to improvethe quality of the fits for highly discontinuous func-tions such as the boom loudness. In addition, weintend to include mixtures of high- and low-fidelitymodels in order to improve our ability to achieve op-timal designs in high-dimensional spaces.

Acknowledgments

This work has been carried out under the supportof the NASA Langley Research Center, under contractNAG-1-03046. We gratefully acknowledge the supportof our point of contact, Dr. Peter Coen, in supplyingus with the geometry and experimental data neces-sary to complete the validation portion of this study.We also appreciate the invaluable help of Dr. Edwinvan der Weide for allowing us to use the AirplanePlusflow solver for this study and of Mr. Ki-Hwan Lee forcreating a parallel version of the NSGA-II optimizer.

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