Supersonic Business Jet Design using aKnowledge-Based Genetic Algorithm with anAdaptive, Unstructured Grid Methodology

Hyoung-Seog Chung∗

Seongim Choi†

Juan J. Alonso‡

Stanford University, Stanford, CA 94305

In the design of supersonic low-boom aircraft, it is important to balance the aero-dynamic performance and sonic boom requirements in a way that represents the bestcompromise for the overall design. Since the ground sonic boom is typically not a smoothfunction of the design variables and may actually contain multiple local minima, it isimportant to select an optimization algorithm that is able to cope with this kind of de-sign space. In this work, we study the use of Kriging approximation models for bothboom and performance and use them in conjunction with genetic algorithm techniquesto investigate the computational cost and characteristics of such an approach for thedesign optimization of a low-boom supersonic business jet (SBJ). Direct use of geneticalgorithms with high-fidelity CFD analysis tools has been limited by the inherently largecomputational cost of genetic algorithms (GAs). The use of computationally inexpensiveapproximation models in lieu of high-fidelity CFD greatly improves the robustness andefficiency of the design process for searches in relatively large design spaces. In order toimprove the performance of this method, a new hybridization strategy that combines aGA with gradient information is proposed and its improved convergence rate is demon-strated. Regardless, the proposed procedures still require a large number of evaluationsof the flow and boom patterns for different points in the design space. For this purposewe have built two automated Euler analysis tools that use a CAD-based geometry engine,and both multiblock-structured and unstructured, adaptive meshing techniques (they arenamed QSP107 and QSP-UA respectively). QSP-UA, has been developed to handle thegeometric detail of the complete configuration including wing, fuselage, nacelles, divert-ers, empennage, etc., and to provide accurate performance and boom data. Results ofsample test problems, and a 15-dimensional design case are presented and discussed.

1. Introduction

FOR decades, the development of economicallyand environmentally acceptable supersonic air-

craft has been identified as a key step toward the nextgeneration of aviation history that could improve manyaspects of human life and foster economic growth. Thepartial consensus view of prospective manufacturers ofsupersonic aircraft is that the supersonic business jet(SBJ) is a near-term-realizable class of vehicle havingsignificant economic potential and with an estimatedmarket of at least 200 aircraft over a 10-year period.1

The key technology barrier for this class of aircraft isthe elimination, or reduction to acceptable levels, ofthe sonic boom for flight over land while guarantee-ing the challenging performance requirements of theother major disciplines. This essentially means that allmajor disciplines including aerodynamics, structures,stability and control, mission, and propulsion shouldbe taken into account from the early stages of design

∗Doctoral Candidate, AIAA Member†Doctoral Candidate, AIAA Member‡Assistant Professor, Department of Aeronautics and Astro-

nautics , AIAA Member

process; therefore, boom reduction must be consideredas an additional aspect of the MDO problem.

MDO methods, particularly those based on high-fidelity analyses, greatly increase the computationalburden and complexity of the design process.2–5 Forthis reason, high-fidelity analysis software typicallyused in single discipline designs may not be suitablefor direct use in MDO.3,6 Faced with these problems,the alternative of using approximation models of theactual analysis software has received increased atten-tion in recent years.

The Kriging technique, developed in the field of geo-statistics, has been recognized as an alternative to thetraditional polynomial function-based Response Sur-face method in generating approximation models ofcomputationally expensive CFD analyses. In theory,it is able to interpolate sample data and to model func-tions with multiple local extrema.8,9 Using variationsof the method of design of experiments (DOE), theKriging model can efficiently represent global trendsof design space.

Once a approximation model is generated, it caneasily be used to search for optimum combinations

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of design variables within the design space. Sincethe Kriging model can represent multimodal functionswith relatively low computational cost, it would bereasonable to choose a global optimization scheme asthe optimizer.

Genetic algorithms (GAs) are search algorithmsbased on the mechanics of natural selection wherethe encoded design variables (or so-called genes) aremanipulated using genetic-like operators such as se-lection, crossover and mutation to generate new popu-lations of candidate design points. One of the biggestadvantages of GAs is their robustness. Since GAs donot require sensitivity information, they can be appliedto realistic design environments where discontinuities,multimodality and noisy responses may exist. Theyalso have considerable advantages for multiobjectivedesign problems to obtain Pareto optimal sets. A ma-jor drawback of the GA approach is that it requiresmany generations to locate the global optimum point.Because of this reason, the utilization of GAs in high-dimensional design spaces with high-fidelity analyseshas been impractical, if not impossible. Therefore,the use of accurate and efficient approximation mod-els with the ability to simulate the multimodal natureof some objective functions can help decrease the costin realistic design problems. Goldberg introduced theidea of using approximation models with GAs underthe name of “knowledge-based GAs.10” In this work,and starting from this strategy, a new hybridizationapproach that combines GAs with the available gra-dient information has been proposed and its improvedconvergence rate has been demonstrated. We have alsostudied the use of Kriging models as gradient estima-tors for CFD-based GAs in the low-boom SBJ designproblem.

There have been many difficulties in analyzing sonicboom signatures using CFD methods. These problemsare associated with issues of mesh resolution, arti-ficial dissipation formulation, two-dimensional versusthree-dimensional signature propagation methods, andthe exact formulation of these propagation procedures.The fundamental difficulty in predicting sonic boomsaccurately is that a near-field pressure signature mustbe obtained sufficiently far from the aircraft so that thepressure disturbance can be modeled using geometricalacoustics. Given the dissipative nature of CFD meth-ods, very high mesh resolution is required between thelower surface of the aircraft and the near-field loca-tion (typically 1-5 body lengths below the aircraft)leading to extremely large meshes even for inviscid cal-culations. This mesh resolution/distribution issue istightly coupled with the fact that complex geometryrepresentation (full configurations including nacelles,diverters, etc.) is typically necessary in the shaping oflow sonic boom aircraft. Complete configurations alsorequire higher mesh resolution for capturing shock andexpansion waves around the aircraft.

Unstructured meshes are ideally suited to addressthese meshing issues since they can easily conform toarbitrarily complex aircraft shapes and they can belocally adapted to capture the relevant physical phe-nomena. This adaptive mesh refinement strategy canimprove computational efficiency in the presence of fi-nite computing resources.

An automated, nonlinear integrated boom anal-ysis tool, QSP-UA, which incorporates a routinefor adaptive unstructured mesh generation, a three-dimensional Euler flow solver, and a boom propagationprocedure has been developed and is used here for thecomputation of accurate near-field and ground pres-sure distributions. This tool is used for collectingsample data for the creation of approximation mod-els.

In this paper, our intention is to explore the ap-plicability and efficiency of using genetic algorithmtechniques in conjunction with approximation mod-els representing response functions with multiple localminima and sharp discontinuities in the multidimen-sional design optimization of a low-boom supersonicbusiness jet configuration. The goal of this multiob-jective design optimization problem is to reduce thesonic boom signature at the ground by modifying theaircraft configuration parameters while preserving orimproving aerodynamic performance.

2. Overview of Kriging MethodThe Kriging technique uses a two component model

that can be expressed mathematically as

y(x) = f(x) + Z(x), (1)

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.12 If f(x) is taken to be an underlying con-stant,9 β , Equation (1) becomes

y(x) = β + Z(x), (2)

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

y = β + rT (x)R−1(y − f β), (3)

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, the authors use a Gaussian exponential correla-tion function of the form provided by Giunta, et al.7

R(xi, xj) = exp

[−

n∑

k=1

θk|xik − xj

k|2]

. (4)

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

rT (x) = [R(x, x1), R(x,x2), ..., R(x,xn)]T . (5)

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

β = (fT R−1f)−1fT R−1y. (6)

Since R is a function of the unknown variable θ, βis also a function of θ. Once θ is obtained, Equation(3) is completely defined. The value of θ is obtainedby maximizing the following function over the intervalθ > 0

− [ns ln(σ2) + ln |R|]2

, (7)

where

σ2 =(y − f β)T R−1(y − f β)

ns. (8)

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. Alternatively, the gradient vector can becomputed with very little additional cost, as is thecase in the adjoint method.4 Gradient informationis usually well cross-correlated with the function val-ues and thus contains useful additional information.The efficiency and accuracy of Kriging models canbe greatly improved by incorporating these secondaryfunction values.13 This technique, known as the Cok-riging method, has been investigated and validated inour previous work.14,15

3. Advanced Genetic Algorithms3.1 Multiobjective Genetic Algorithms

Many real-world optimization problems, especiallyin MDO situations, require the simultaneous optimiza-tion of possibly conflicting multiple objectives: thisapproach is often referred to as multiobjective opti-mization. Unlike single-objective optimization whereonly one optimal solution is pursued, a typical multi-objective optimization problem produces a set of solu-tions which are superior to the rest with respect to allobjective criteria, but are inferior to other solutions inone or more objectives. These solutions are known asPareto optimal solutions or non-dominated solutions.None of the solutions in the Pareto optimal set is ab-solutely better than any others with respect to all ofthe objectives being considered; therefore, any one ofthem is an acceptable solution. Once the set of opti-mal solutions is identified, it is left to the designer tochoose one solution out of the many possible ones.

A genetic algorithm can use this dominance criteriain a straightforward fashion to drive the search process

Selection

Crossover

PopulationInitialization

New Population

Elitism

NominalConvergence?

Yes

No

External Memory(Pareto Set Archive)

from Archive?Yes

No

(Mutation)

Fitness Evaluation/Pareto Ranking

Elitism

Fig. 1 Flow Chart for Multiobjective GA

toward the Pareto front. Due to the unique featuresof GAs, which work with a population of solutions,multiple Pareto optimal solutions can be captured ina single run. This is the primary reason that makesGAs ideally suited for multiobjective optimization.

A recent study by Coello16 proposed a micro-GA-based multiobjective optimization that uses an exter-nal file of non-dominated vectors found in previousgenerations to accelerate the multiobjective optimiza-tion process. The method implemented an additionalelitism strategy and an adaptive grid-type techniqueto accelerate the convergence and to keep the diver-sity in the Pareto front. The Micro-GA algorithm isa specialized GA that works with a very small pop-ulation size of usually 3-6 and a reinitialization step.Studies16,17 have shown that micro-GAs achieved afaster convergence rates than simple GAs. In thepresent research, some of the ideas of Coello’s workhave been adapted to a single objective micro-GA al-gorithm along with the traditional Goldberg’s Paretoranking approach in order to develop an efficient androbust design framework. The authors have modified amicro-GA algorithm originally developed by Carroll.18

The procedure is illustrated in Figure 1. First, a ran-dom population is generated and their objective valuesare calculated as in the original micro-GA. Then, toensure that all the non-dominated individuals havesame level of reproductive potential, Goldberg’s non-dominated sorting procedure is implemented. There-fore, the fitness level of each individual is determinedbased on the non-domination criterion rather than the

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GA−BasedGlobal Search

Gradient−BasedLocal Search

GA−BasedGlobal Search

Gradient−BasedLocal Search Selection

Crossover Selection

CrossoverMutation

Mutation

(Traditional Hybrid Strategy) (New Hybrid Strategy)

Fig. 2 New Hybrid GA Strategy

objective function value itself.Based on the rank of non-dominance, the popula-

tion goes through the usual operations of micro-GA,namely selection and crossover, and check to see ifthe nominal convergence among the population pointshas been reached. If the problem is not converged,the algorithm returns to the function evaluation andnon-dominance ranking steps for the new generation,otherwise it continues on to the reinitialization step.

Two types of elitism are implemented in the reini-tialization step. The first type carries on the bestsolutions from the previous nominal convergence stage.This is the same elitism strategy used in the single ob-jective micro-GA. The second type involves the storingof non-dominated vectors produced from each cycle ofmicro-GA to an external file and inserting some of thebest solutions generated so far in the reinitialized pop-ulation for the micro-GA. This process is applied atregular intervals to improve the non-dominated solu-tions by getting closer to the true Pareto front or byachieving a better distribution.

3.2 Gradient Enhanced Micro-GA

The convergence history of GAs can be character-ized as having a fast portion during the early stage of arun followed by very slow progress in the later stages.This means that GAs work reasonably well up to anear-optimal solution but they consume an unaccept-able number of generations and function evaluationsto attain the real optimum. Because of this, an ob-vious improvement to the GA is to take advantage ofthe characteristics of gradient-based optimizers aftera certain level of convergence has been achieved bythe GA. The strategy of traditional hybridization isto obtain some of the candidates for the optimal solu-tion from the GA and to then apply a gradient-basedoptimizer starting from those points to accelerate theconvergence to the optimal point. This approach canbe seen in Figure 2 below.

PopulationInitialization

New Population

Elitism

NominalConvergence?

Yes

No

Fitness & Gradients

Gradient−BasedLocal Search

Selection

Crossover

Evaluation

(Better Population)

Fig. 3 Flow Chart for Gradient-Enhanced Micro-GA

A new strategy of hybridization using gradient in-formation is proposed in the present research. Insteadof combining a gradient-based optimizer at the end ofa GA run, the proposed method implements the gra-dient information at every generation of the GA. Theschematic procedure of the method is shown in Fig-ure 3. The only difference between the new hybridmicro-GA and the original micro-GA is that all theindividuals in a population are updated or improvedusing the steepest descent method with the gradientinformation available before they go through the pro-cesses of selection and crossover at each and everygeneration. That is, the hybridization is achieved byinserting a gradient-based optimizer (steepest descentmethod) at the GA operator level. This results in moreclosely combined hybridization procedures.

3.3 Approximation Model based micro-GA

Once fairly accurate global approximation modelsare constructed with computationally efficient tech-niques such as the Kriging or Cokriging methods,combining them with GAs becomes an obvious choiceto overcome the computational burden presented bydirectly-coupled GA methods. Since the computa-tional cost of estimating the objective function throughthe approximation models is trivial, the slow conver-gence rate of GAs leading to many generations and

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function evaluations to get to the optimal solution doesnot matter any more. If the approximation model isaccurate enough, it can even provide auxiliary infor-mation such as gradients to accelerate the process evenfurther. In addition, GAs are a more suitable approachthan gradient-based methods for use with approxima-tion models that can support multimodal functions.

The approximation model itself can benefit from theuse of a direct GA by recycling the rich dataset pro-duced with each population. The mutual benefits ofcombining these two methods can provide the efficientand robust design framework necessary in MDO. Yetanother way of using approximation models for accel-erating the GA process is to use them as a gradientestimators to provide the necessary gradient informa-tion for the hybrid micro-GA proposed in the previoussection. Note that even though adjoint methods canprovide gradient information cheaply and efficiently,the derivation of the adjoint equations and boundaryconditions can not be carried out for arbitrary costfunctions. In addition, an adjoint code implementa-tion might take months of validation. Therefore, theapproximation models can be a useful and inexpensivealternative for computing the gradients.

4. Test Problem :Low-Boom Supersonic Business Jet

(SBJ) DesignThe design problem in question involves the simul-

taneous ground boom and drag minimization of asupersonic business jet wing-body-tail (and possiblynacelles with the unstructured method) configurationat a specified lift coefficient, CL = 0.07104, whichcorresponds to a cruise weight of 100, 000 lbs at acruise altitude of 50, 000 ft. The wing reference area is1, 032 ft2. The free-stream flow conditions were fixedat M∞ = 2.0. The aircraft geometry and flow condi-tions were parameterized directly in CAD using 108design variables.19 The list of geometric design vari-ables for the 15-dimensional design problem which wewill describe later is given below:

x1= wing position along fuselagex2= wing dihedral anglex3= wing sweep anglex4= wing aspect ratiox5= wing leading edge extensionx6= upper fuselage radius at 12.50% of fus. lengthx7= upper fuselage radius at 18.75% of fus. lengthx8= upper fuselage radius at 25.00% of fus. lengthx9= upper fuselage radius at 31.25% of fus. lengthx10= upper fuselage radius at 37.50% of fus.

lengthx11= lower fuselage radius at 12.50% of fus. lengthx12= lower fuselage radius at 18.75% of fus. lengthx13= lower fuselage radius at 25.00% of fus. lengthx14= lower fuselage radius at 31.25% of fus. length

x15= lower fuselage radius at 37.50% of fus. length

The wing planform for this configuration was de-signed with experience from our previous supersonicwork to have a portion with a subsonic leading edgefollowed by an outboard panel with a supersonic lead-ing edge. This helps increase the span to achieve betterlow-speed performance at a small cost in cruise perfor-mance. The airfoil sections for the outboard portionof the main wing and the horizontal tail were cho-sen to be simple biconvex airfoils of varying thickness,while an RAE2822 was used for the inboard part of themain wing with a subsonic leading edge. Fuselage aft-mounted nacelles are included in the baseline design,although their presence was only modeled in the op-timization results based on our unstructured analysisenvironment.

A CFD/boom analysis of the two objective criteriaconsidered in this study yields values for the baselinedesign of an inviscid coefficient of drag, CD = 0.010046and an initial shock pressure rise, ∆p = 0.91157psfrespectively. The performance of the baseline config-uration at this low CL condition is admittedly low sothat we can analyze the speed with which the pro-posed algorithms approach the Pareto fronts. We arecurrently repeating the optimizations presented in thispaper with a baseline design that has been thoroughlyoptimized (for aerodynamic performance alone) usingour multiblock adjoint design program.19 All improve-ments in either or both of these criteria are measuredagainst the value for this baseline configuration.

5. Design ToolsIn order to develop Kriging and Cokriging approxi-

mation models, a large number of CFD computationsfor different geometries must be carried out automat-ically. For this purpose, we have developed two sepa-rate nonlinear integrated boom analysis tools, QSP107and QSP-UA, that can provide both ground boomand aerodynamic performance information by simplychanging a small set of configuration variables that areprovided in an input file. QSP107 is our structured-mesh analysis suite, while QSP-UA relies on an un-structured adaptive tetrahedral flow solver.

5.1 QSP107

QSP107 is a nonlinear integrated tool for both sonicboom prediction and aerodynamic performance analy-sis based on fully nonlinear CFD. This tool couplesthe multiblock Euler and Navier-Stokes flow solverfor structured mesh, FLO107-MB,20 to a CAD-basedgeometry kernel and a mesh perturbation procedurefor efficient mesh regeneration, and to the PC Boomsoftware for far-field propagation developed by WyleAssociates.22 A flowchart of the automated analysisprocess can be seen in Figure 5.

The procedure starts with a CAD-based geome-

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try generation module called AEROSURF that usesa parametric CAD model to automatically generatethe necessary surface meshes to describe the configu-ration in question.19 This geometry module is basedon a parametric aircraft description with 108 designvariables of which only a small subset (either two orfifteen) are chosen for our optimization applications.AEROSURF relies on the CAPRI CAD interface ofHaimes19,23 to provide access to a number of paramet-ric CAD software programs including Pro/Engineer,CATIA V5, SolidWorks, and I-DEAS. Parametric air-craft models of arbitrary complexity can be createdand used within this environment, thus automatingsome of the most complicated geometry manipulationsthat are the heart of any high-fidelity design proce-dure.

The surface meshes generated by AEROSURF, to-gether with an initial multiblock mesh that is gen-erated using the Gridgen software24 are passed to amesh perturbation routine called Meshwarp that canhandle arbitrary configurations and generate volumemeshes corresponding different surface geometries. Asconstructed, the meshes have higher resolution in theareas where shock waves and expansions are presentbelow the aircraft, and the grid lines are slanted at theMach angle to maximize the resolution of the pressuresignature at a distance of the order of one to threefuselage lengths below the aircraft. The user mayspecify the location of an arbitrary cylindrical surfacewhere the near-field signature is extracted from themultiblock flow solution and then provided as an inputto a modified version of PC Boom which propagatesa full three-dimensional signature along all rays thatreach the ground. This allows for the calculation ofarbitrary cost functions (not only ground-track initialoverpressure) that may involve weighted integrationof the complete sonic boom footprint. In this work,however, only the ground track overpressure has beenconsidered.

The flow solver, FLO107-MB, combines advancedmultigrid procedures and a preconditioned explicitmultistage time-stepping algorithm which allows fullparallelization. Because of the advanced solution algo-rithms and parallelization, the integrated tool providesfully nonlinear simulations with very rapid turnaroundtime. Using typical meshes with over 3 × 106 meshpoints we can obtain a complete flow solution andground signature in around 7 minutes, using 16 proces-sors of a Beowulf cluster made up of AMD AthlonXP2100 processors. In this work, QSP107 has been usedrepeatedly to generate Kriging and Cokriging approx-imation models and it has also been directly coupledwith a genetic algorithm for the design optimizationprocess.

5.2 QSP-UA

QSP107 imposes some limitations on the magnitudeof the shape perturbations that can be handled since itrelies on a single multiblock mesh (generated prior tothe start of the design procedure) and a mesh pertur-bation routine. If the surface perturbations are large,the mesh perturbation procedure may fail.

In order to overcome these limitations when explor-ing very large design spaces, QSP-UA includes andunstructured adaptive mesh generation/perturbationcapability based on the Centaur mesh generation soft-ware.25 As in QSP107 the procedure is fully auto-mated from the specification of the design variablesfor a given configuration to the computation of boththe coefficient of drag and the ground boom overpres-sures.

This tool integrates the three-dimensional Eulerflow solver AirplanePlus which is a C++ implemen-tation of the original AIRPLANE solver of Jame-son28 with an agglomeration multigrid strategy, andMPI-based parallelization, and the ability to solve theReynolds-Averaged Navier-Stokes (RANS) equationson unstructured tetrahedral meshes. As in QSP107,the unstructured adaptive solution is passed to thePC Boom software to propagate the near-field pres-sure distributions to the ground.

QSP-UA functions in a manner that is similar toQSP107 except for the surface and volume mesh gen-eration and the actual flow solver. It also uses AERO-SURF to provide the surface definition of the aircraftgeometry. With the input of this CAD-defined geome-try and pre-defined far-field boundaries, Centaur usesan advancing-front method to generate both surfaceand volume meshes.

Tetrahedral Mesh GenerationThe current methods available for the generation of

tetrahedral meshes29–31 are typically based on eitheradvancing front or Delaunay triangulation ideas. Inour work, the advancing-front method26 in the form ofthe Centaur software is used for mesh generation andperturbation.

As mentioned above, the Centaur software is usedin our work to construct meshes for aircraft configura-tions and to enhance grid quality through automaticpost-processing. Only the fine meshes need to be ex-plicitly constructed since our multigrid algorithm isbased on the concept of agglomeration and, therefore,coarser meshes are obtained automatically.

Figure 4 shows a typical, unadapted, tetrahedralmesh around our baseline configuration. The figurecontains only the surface triangles of the aircraft andthe symmetry plane for visualization purposes.

Mesh Adaptation ProceduresOnce an initial solution has been computed on the

tetrahedral mesh, the grid needs to be locally adapted

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Fig. 4 Unstructured tetrahedral surface mesharound a low-boom aircraft

to better capture specific features with higher accu-racy at lower cost. This can be achieved through amore optimal distribution of grid points for each com-puted solution: unstructured tetrahedral elements arewell suited for cell adaptation. For the cases that wehave studied, coarsening has only a minor performanceimpact in steady-state calculations and is omitted inthis study.

The adaptation procedure utilizes h-refinement orsubdivision techniques. For each edge that is flaggedby the error estimation technique, new mesh nodes areinserted at the midpoint. For boundary edges, thesepoints are repositioned onto the splined surface whichdefines the original geometry from the CAD pack-age. The current post-adaptation grid-improvementscheme employs face and edge swapping. Undesir-able shape measures are investigated and new localtetrahedral configurations with more desirable shapemeasures are selected.

The adaption procedure is, of course, recursive, andit proceeds until a certain level of error has beenachieved or a maximum number of refinement levelshas been accomplished. Figure 6 (a) shows the meshafter the first adaptation. The selection of the initialmesh resolution is important to capture the underly-ing pressure gradients. Since boom prediction is thedriving feature to be captured in this study and shockangle is relatively predictable, uniform local adapta-tion (mainly in the region under the aircraft) is per-formed after the initial mesh is generated. Figure 6 (b)shows the solution-adapted mesh after two adaptationsteps. Typically, four consecutive adaption cycles areperformed automatically starting with the initial meshto reach the necessary solution quality.

The most popular refinement options for fluid-flowproblems are heuristically derived gradient-based cri-teria, which involve a single or multiple physical flowvariables. The gradient of pressure can be used to

identify inviscid flow features. But in sonic boomprediction problems the pressure gradient in the nearfield is as important as in the neighborhood of theaircraft. In addition, the direction of the gradientshould be taken into account as well. In this study,a pre-specified range of velocity gradient magnitudeswhich are projected onto the direction of the local pres-sure gradient work successfully to predict the shocklocation and to capture small pressure gradients innear-field.

In the following expression for the adaption criterionV is the velocity vector, c is speed of sound and 4x isrequired minimum edge length.

ε′=

Vc· ∇p

|∇p| (9)

Numerical experiments32 indicate that the modifica-tion of the previous equation to include a local meshlength-scale such as:

ε =Vc· ∇p

|∇p|4x (10)

produces a more effective refinement criterion forshock/expansion flows. This is partially due to thefact that while the simple gradient-based criteria de-creases in magnitude as the mesh is refined in smoothregions of flow, it remains approximately constant inthe vicinity of shock waves, since the shock wave profilesteepens as the mesh is refined, and the jumps re-main almost constant. However, even in the regions ofsmooth flow, the additional length scale weights largercells more heavily than small cells, and drives theadaptation process closer toward global refinement.

6. Results6.1 Validation of Cokriging Method

CD Optimization for 2-Dimensional Design CaseTwo-dimensional Cokriging models were created for

the CD of the supersonic business jet test problem us-ing sample data obtained from CFD analyses. Theintent is to validate and investigate the ability of thesemodels to approximate the results of the original CFDcode. The two design parameters chosen as were thewing streamwise location along fuselage and the radiusof the fuselage at its mid-point. 400 CFD calculationswere performed by varying the design variables to ob-tain a graphical representation of the actual objectivefunctions. The results are shown in Figure 7 (a). Thedesign variables were chosen so as to generate a re-alistic test function having multiple local extrema tocheck the ability of Cokriging models to simulate thisfeature. One important point to note from the Figureis that the actual drag coefficient varies smoothly withrespect to the geometric design variables. The selec-tion of Gaussian exponential correlation function forthe Kriging method was based on the assumption that

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the response function to be modeled was very smoothin nature. Thus, Figure 7 (a) provides the validation ofthe assumption and the rationale for using the Gaus-sian correlation function in this problem.

A total of 9 sample points were used to generate theoriginal Kriging model, and the 9 sample points to-gether with gradient information at these points wereused for the Cokriging models. From the comparisonsin Figure 7, it is clear that Cokriging models performedmuch better than the Kriging model in predicting boththe general shape and magnitude of the objective func-tion. The discrepancies in the original Kriging modelare caused by under-sampling.

In order to investigate the applicability of the Cok-riging models in 2-dimensional design spaces, CD op-timizations were performed using each of the approx-imation models generated above. Figure 8 (a) showsthe CD optimization results using the database con-structed from 400 CFD calculations over the designspace. The optimization used a gradient-based methodfrom Matlab33 and was repeated five different times bychanging the starting points. Four of them convergedto one local minimum point and one converged to adifferent one. The optimizations were again repeatedusing the different approximation models generatedfrom Kriging and Cokriging methods, and the resultsare compared in the subsequent figures. As shown, thepredicted optimum design point and optimum value ofCD for these cases were nearly identical. The predictedCD from the CFD database was 0.0059708 whereasthose for the Cokriging models with 5 and 9 samplepoints (and gradients) were 0.0059759 and 0.0059758respectively with almost the same optimum designpoint locations. The relative error for the optimizedCD value is within 0.086%. As we can observe fromFigure 8 (b), the ability of the original Kriging methodto simulate the unknown function was clearly limitedwithout an extensive set of sampled data. The opti-mum point and the predicted CD value were far offfrom the actual ones.

Boom Optimization for 2-Dimensional Design Case

The procedure used in the previous section was re-peated for ground boom minimization with a differentset of design variables. This time, the design variablesand their range of variation were specifically selectedto present a challenging boom minimization case. Thetwo design variables selected represent the radii of twofuselage sections located at 10% and 20% of the lengthof the fuselage. A total of 121 CFD calculations werecarried out and the results of the boom strength vari-ation over the design space are shown in Figure 9 (a).Unlike the CD objective function case, the values of theinitial shock jump at the ground vary almost linearlywith each design variable until sharp discontinuitiesare found for lower values of both design variables.This kind of functional variation is considered to be

quite difficult to capture with a small number of sam-ple data points. The subsequent Figures show theapproximations of the exact boom overpressure valuesusing both Kriging and Cokriging models with 5 sam-ple data points. Similarly to the previous test case, theaccuracy of the Kriging model was improved by the ad-dition of gradient information, although the functionfits were not as good as in the coefficient of drag case.

Even though Cokriging models had some difficultyto capture the exact variation of the overpressuresover the entire design space, three successive designiterations using an empirical trust region methodol-ogy produced quite satisfactory results. Figure 10 (a)shows the history of sample points computed for allthree design iterations as well as the evolution of theoptimum design point superimposed on the exact CFDcontour plot (for reference purposes only). Figures 10(b),(c),(d) represent Cokriging models and the loca-tion of the sample points for each design iteration aswell as the true and estimated optima for the Cokrig-ing models. After three design iterations the actuallocation and the function value of the local minimumwas achieved fairly accurately.

6.2 Validation of Gradient Enhanced micro-GA

The results of tests of the new hybridization strat-egy are presented in this section. In Figure 11 (a) (b)we compare the results of the pure micro-GA and gra-dient enhanced micro-GA algorithms applied to singleobjective optimization problems on 2-dimensional testfunctions. The extension of the idea to multiobjec-tive optimization was also tested on 2-D and 10-D testcases and the results are presented in Figure 11 (c)(d). The use of gradient information inside of the GAclearly improves the overall efficiency of the processin terms of the required number of function evalu-ations to obtain an optimal or sub-optimal solution.GAs have the merit of robustness, implying that theycan be used in a wide variety of application problems,but has somewhat poor efficiency in that they requiremany function evaluations. Gradient-based optimiz-ers, on the other hand, have very high convergencerates but somewhat limited applicability in boom op-timization problems. The new idea of hybridizationmakes it possible to achieve a high level of robustnessand efficiency at the same time.

6.3 15-Dimensional Design Problem with QSP107

Multiobjective Optimization using Kriging-Basedmicro-GAs

Using a latin hypercube sampling technique, 150sample points around the baseline design point wereselected and their performance values were computedusing QSP107 calculations. A Kriging model wasthen generated based on the sampled data and usedfor the function evaluation routine within the MOGAsearch. The estimated Pareto set from the Kriging-

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based MOGA search procedure is plotted as black cir-cles whereas the CFD validation results of the Paretoset are shown as red asterisks in Figure 12 (a). Theresults of the MOGA search directly connected toQSP107 for 120 generations are also plotted as bluedots for comparison purposes. The CD values of theestimated Pareto set from the Kriging-based MOGAand their CFD validation calculations were surpris-ingly well matched while some of the values of theboom estimations were far off from their CFD counter-parts. The results demonstrate the fact that the boomdesign space may have discontinuous or non-smoothregions and cause difficulties in generating accurateKriging models. However, the estimation producedsome good design points in terms of both design crite-ria.

One drawback of the Pareto design points was thatthe ground boom signatures had shapes such that thedistance (or time) between the first and second shockswere very small. The outcome implied that the ob-tained design points were very sensitive to changes inthe design variables and would therefore not be accept-able to designers in search of robust designs. The firstdesign iteration described above was repeated with aconstraint imposed on the distance between the firstand the second peaks of the ground boom signatureand its results are presented in Figure 12 (b). Im-proved Pareto designs (in terms of robustness) wereobtained even though the actual values of design cri-teria did not improve as much as in the previous case.

The second design cycle was performed by collectinganother 150 samples around the best designs identifiedfrom the previous design cycle, and generating Krigingmodels for the MOGA search procedure. The resultsare shown in Figure 12 (c). The estimated Pareto frontand the CFD validation results became much closer toeach other, as expected, since the design space wasrestricted for this second iteration.

The resulting design configuration of a point on thePareto front is compared with that of baseline designin Figure 13. The wing sweep angle, leading edge ex-tension, and dihedral angle increased while the wingposition along the fuselage and wing aspect ratio de-creased. The nose section was deformed such that itdecreased the initial shock strength from the nose andgenerated an expansion region by inducing a bump-likeshape at the lower fuselage section. The effect of thisexpansion wave is to both weaken the initial shock andto prevent the first and second shocks from coalescing.

Gradient Enhanced Multiobjective Genetic Algorithms(GEMOGA) with Kriging Models

The idea of using Kriging models to provide gradientinformation to the QSP107-based MOGA in order toaccelerate the search procedure was tested using theKriging models obtained from the first design cycle.The results are presented in Figure 12 (d). As has been

demonstrated with analytic test functions, implement-ing gradient information inside of GA runs greatlyimproves the convergence rate of the MOGA searchand much better Pareto optimal sets could be iden-tified with a smaller number of function evaluations.Furthermore, one can see in this test that Kriging mod-els can provide fairly accurate gradient information atleast to assist in the robust search techniques used inthe hybrid GAs.

6.4 15-Dimensional Design Problem with QSP-UA

The integration of approximation methods withGAs results in a very robust design optimizationframework suitable for typical MDO problems. Thetechnique can effectively be automated and easilyadapted to a variety of problems. The bottleneck ofthe procedure, however, occurs when the mesh per-turbation routine fails for configurations requiring alarge deviation from the baseline configuration. In ad-dition, the amount of work required to include complexgeometry features such as engine nacelles, pylons anddiverters, in this environment is rather large. Full au-tomation of the procedure in the context of large geom-etry perturbations can be achieved with unstructuredmesh re-generation and perturbations procedures. Ifthe geometry changes by an amount that would de-crease the quality of the volume mesh to the point thatit becomes unusable, the mesh can simply be regener-ated from scratch. Several measures of mesh qualityare readily available in the literature.

The 15-dimensional design problem presented before(with QSP107) was repeated with the QSP-UA en-vironment to demonstrate this new, fully automated,design framework. Function evaluations are computedusing the Centaur mesh generation system, the AERO-SURF geometry engine, and the AirplanePlus flowsolver. Typically, four levels of adaptation are usedin each of the function evaluations with mesh sizes inthe neighborhood of 3 million nodes.

A total of 140 sample points were initially com-puted using a Latin Hypercube sampling distribution.The results of these calculations are used to constructKriging response surfaces which are then fed to ourmultiobjective GA formulation to arrive at the Paretofront for this problem. The results are very similar tothose obtained with the QSP107 multiblock structuredapproach: the CFD validations of the approximatedpoints on the Pareto front once again appear to showreasonable agreement in the coefficient of drag, butshow discrepancies for the calculation of the sonicboom. Once again, this is evidence of the fact thatthe boom design space is much less smooth than thatof the coefficient of drag and, therefore, either largernumber of sampling points, or more concentrated sam-pling locations are needed to improve the accuracy ofthe fit. Additional iterations with a trust region ap-proach (not computed here) should provide the added

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accuracy to achieve closer agreement between the com-puted and estimated Pareto fronts.

7. ConclusionsThe following conclusions can be drawn from the

work in this paper:

• Integrated tools for the analysis of sonic boomand aerodynamic performance based on fully non-linear CFD analyses for both structured and un-structured meshes (QSP107, QSP-UA) have beendeveloped and validated in their applicability todesign optimization of low-boom supersonic busi-ness jets.

• The applicability and efficiency of approximationmodel-based GAs has been demonstrated.

• A new hybridization strategy implementing gra-dient information inside of the GA optimizationswas proposed and its advantage to accelerate theGA search procedure has been validated usingsome preliminary test cases.

8. AcknowledgmentsThe authors wish to acknowledge the support of the

NASA Langley Research Center under Stanford con-tract number 2DBA401 and the help of Dr. PeterCoen.

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Ground Plane

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(a) Mesh after first adaptation with number of mesh nodes=562,057

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Fig. 6 Unstructured Mesh Adaptation for 15-Dimensional SBJ Design Problem

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Fig. 8 CD Optimization Results for 2-D SBJ Test Case

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Fig. 10 Boom Overpressure Optimization Results using Indirect Cokriging Method for 2-D SBJ DesignProblem

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(d) Design Results using GEMOGA with Kriging Models

Fig. 12 Multiobjective Optimization Results

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Fig. 13 Configuration Comparison between Final and Baseline Designs (red:baseline, green:Desinged)

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Fig. 14 Results of MOGA using QSP-UA analyses and a Kriging approximation model.

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