Advanced Algorithms for Design and Optimization of Quiet...

AIAA 2002–0144Advanced Algorithms for Designand Optimization of QuietSupersonic PlatformsJuan J. Alonso, Ilan M. Krooand Antony JamesonStanford University, Stanford, CA 94305

40th AIAA Aerospace Sciences Meeting and ExhibitJanuary 14–17, 2002/Reno, NV

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AIAA 2002–0144

Advanced Algorithms for Design andOptimization of Quiet Supersonic Platforms

Juan J. Alonso∗, Ilan M. Kroo†

and Antony Jameson‡

Stanford University, Stanford, CA 94305

This paper describes the work carried out within the Stanford University group aspart of the DARPA-funded Quiet Supersonic Platform (QSP) project. The objective ofour work was to develop advanced numerical methods to facilitate the analysis and de-sign of low sonic boom aircraft. The focus of the boom reduction activities was placed ontwo main ideas: the shaping of the configuration and a multidisciplinary design approachto minimize its total empty weight. Accurate and efficient tools are needed to achievethese tasks. Our approach was meant to enhance the existing state-of-the-art which wasdeveloped in the 1970s and during the early stages of the High Speed Research (HSR)program. For that purpose, we designed tools that would ultimately become fully non-linear (in contrast with current design efforts based on linear theories of the 1970s).These tools are also able to account for important tradeoffs between boom reduction andaerodynamic perormance, as well as other disciplines. Because of the difficulty of theproblem, the tools were designed to support a combination of gradient and non-gradientautomatic optimization techniques. As a result of our efforts, rapid turnaround boomanalysis and design methods were developed. These methods permit the investigation ofradical configuration changes and their effect on the ground boom signature and the L/Dratio of the aircraft. We have also created an environment for the automatic nonlinearanalysis of QSP configurations based on the multiblock flow solver FLO107-MB. In ad-dition, we have extended the adjoint-based design method to treat remote sensitivitiesto near-field pressure signatures, allowing for a very large number of parameters to beused in modifications of the aircraft geometry. All tools were carefully validated againstexisting experimental data, other boom prediction programs, and with systematic meshrefinement studies.

Introduction

THE Quiet Supersonic Platform (QSP) programwas created by DARPA in the fall of 2000 in or-

der to reassess the available technologies necessary todevelop small supersonic aircraft with sufficiently lowsonic boom that they may be allowed to fly super-sonically over land. The program requirements statedthat such an aircraft would have to be in the 100,000lbs class, fly at a cruise Mach number of 2.4 with arange of 6,000 nautical miles, and produce an initialoverpressure of less than 0.3 psf. These design require-ments were issued as guidelines and were subsequentlyrevised so that the goal would be more realistic.Within the DARPA QSP program, a number of

groups were funded to investigate topics in both thecontributing technologies (RA-00-47) and in airframeintegration (RA-00-48). An important part of this ef-fort, the design of acceptable propulsion systems for

∗Assistant Professor, Department of Aeronautics and Astro-nautics , AIAA Member

†Professor, Department of Aeronautics and Astronautics,AIAA Fellow

‡Thomas V. Jones Professor of Engineering, Department ofAeronautics and Astronautics, AIAA Fellow

Copyright c© 2002 by the authors. Published by the AmericanInstitute of Aeronautics and Astronautics, Inc. with permission.

this aircraft, was also included, although it had ex-isted previously.The Stanford University Group decided to apply

its expertise in Multidisciplinary Design Optimization(MDO), advanced nonlinear analysis and design meth-ods, and adjoint-based design methodologies to theQSP problem. The objective of our work became todevelop the necessary methods and tools to facilitatethe high-fidelity, multidisciplinary design of low sonicboom aircraft that can fly supersonically over landwith negligible environmental impact. Although thefundamental challenge was to devise a configurationthat satisfied strict sonic boom requirements, care wastaken so that other mission requirements were met.Two fundamental ways of reducing the magnitude of

the sonic boom of a supersonic aircraft were consideredin this research. First, a new adjoint-based, inversedesign methodology for the tailoring of the footprintof the overpressures was derived and implemented totreat surface modification of complex aircraft con-figurations. Second, multidisciplinary optimizationconcepts were applied to configurations of interest toreduce the weight and boom intensity while maintain-ing vehicle performance constraints. This MDO en-vironment can take advantage of accurate CFD-based

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American Institute of Aeronautics and Astronautics Paper 2002–0144

aerodynamic computations and efficient gradient esti-mation to achieve design improvements that would notbe possible with conventional methods.Analytic or semi-analytic formulations of the sonic

boom minimization problem can provide invaluable in-sight into the development of new concepts, theoreticallimits for sonic boom improvements, and promisingcandidate configurations. However, alone they maynot be sufficient to achieve ground signatures withinitial shock strengths lower than 0.3 lbs/ft2. De-tailed numerical simulations are necessary to accountfor the non-linear effects present in this problem. For-tunately, due to the nature of the physics involved inthis problem, the solution of the Euler equations hasbeen shown to be sufficient at a preliminary designlevel. Therefore, an Euler model was used in the ma-jority of the design calculations in this work. Since thenonlinear methods developed in this work were Navier-Stokes based, it would be easy to use RANS solutionsof candidate configurations to verify that the differ-ences with the inviscid model are negligible.For typical cruise altitudes required for aircraft effi-

ciency, the distance from the source of the acoustic dis-turbance to the ground is typically greater than 50,000ft. A reasonably accurate propagation of the pressuresignature can only be obtained with small computa-tional mesh spacings that would render the analysisof the problem (let alone the design calculations) in-tractable for even the largest parallel computers. Anapproach that has been used successfully in the pastis the use of near- to far-field extrapolation of pressuresignatures based on principles of geometrical acousticsand non-linear wave propagation. These approaches,such as the F-function method of Whitham1 or theequivalent wavefront parameter method of Thomas,2

are based on the solution of simple ordinary differentialequations for the propagation of a pressure signaturefrom the near-field of the aircraft to the ground (far-field). They are able to take into account variableatmospheric properties and their cost of computationis practically negligible.While these relatively simple wave propagation

methods were used during the first phase of the work,more advanced propagation methods can easily be in-corporated into our framework. This possibility was tobe investigated during the second phase of the work.The calculation of a detailed near-field CFD solu-

tion, has been shown to require extremely fine meshes3

in order to obtain accurate far-field results. The num-ber of mesh points required for these calculations istypically of the order of 5− 20 times larger than thatrequired for a typical aerodynamic performance calcu-lation. For a complete configuration supersonic air-craft, detailed studies have shown that around 5− 10million mesh points are necessary. Depending on thenumber of design variables that are used to describethe shape of the configuration, the cost of optimization

based on these large meshes can be prohibitive unless acombination of advanced adjoint-based algorithms andefficient parallel computing implementations is used.The objective of our work was to ensure that the theturnaround of these calculations fitted within a realis-tic design environment.While a principal focus of our work was on sonic

boom minimization, this can actually be accomplishedonly in the context of vehicle design. Vehicle shapemodification that leads to reduced boom overpressuresfor a specific design often results in degraded aerody-namic performance4 and so boom reduction must beconsidered as one aspect of a multidisciplinary designproblem. Our group has had considerable experiencein aerodynamic shape optimization and multidisci-plinary aircraft design,5–15 and the combination ofthese two research areas represents an opportunity forfurther advances.Historically, a large body of work in sonic boom min-

imization already existed. In particular, most of thesonic boom propagation algorithms that are used to-day were developed during the 1960s and 70s. Duringthe late 80s and early 90s, the HSR program revivedthe interest in low sonic boom aircraft, to the extentthat quietness was emphasized during the first phase.Once it became apparent that the weight of the air-craft was such that reasonably low noise levels wouldbe impossible to achieve, the low boom component wasdropped from the HSR requirements. The work doneduring this period, however has been extremely helpfulto jump start our efforts.

Linear Analysis MethodsIn parallel with the development of high-fidelity non-

linear methods for aerodynamic analysis and design,a suite of linear methods was identified, developed,and integrated into a design framework. These linearmethods, including classical axisymmetric approachesto boom analysis along with fully 3-D surface panelmethods, are important to the present study for sev-eral reasons: they serve as fast surrogates for theemerging nonlinear methods, allowing testing of designapproaches and a better understanding of the designspace; they are dissipation-free, providing importantdata for studies of required survey locations withoutthe complicating issues of grid resolution and shocksmoothing; they provide a means for direct compari-son with previous work; and they are useful for conceptevaluation. During the course of last year, three as-pects of the linear method development were pursued.A classical boom analysis method (equivalent area

plus propagation code) was combined with a drag esti-mation code and graphical interface to form an interac-tive QSP design tool. This rapid design tool was usedto evaluate and recommend possible conceptual vehicleapproaches for other QSP contractors including DTI,Northrop-Grumman, and Boeing. It has also been pro-

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vided to Lockheed, Gulfstream, and Raytheon, for useas an exploratory tool. This analysis code was also in-tegrated into an optimization framework as describedlater and proved useful in evaluating strategies for nu-merical optimization. Figure 1 shows the interactiveversion of this tool applied to the design of a genericlow boom QSP vehicle that was later used for testingand validation of other codes.

Fig. 1 jBoom interactive design code includesaero cruise performance, boom propagation, centerof pressure tracking, and many useful parameterswith nearly instant feedback to a designer.

The second area of linear method development in-cluded the integration and application of the highorder panel code, A502. After substantial testing ofseveral approaches to boom estimation that includedthe use of programs ranging from the Harris wave dragprogram, to constant pressure methods (Woodward-Carmichael) such as WingBody, it was determinedthat true 3D effects required the use of a full surfacepanel method with off-body surveys used to provideinputs to the propagation code. The A502 programwas integrated into the Caffe design framework, pro-viding rapid communication between the conceptualdesign tool, this more accurate aerodynamic analysistool, and a suite of optimizers. This integration in-volved the development of several pieces of softwarefor automated paneling of designs with multiple non-planar lifting surfaces and extraction of results for thepropagation code and for visualization of results. Fig-ure 2 shows the web-based implementation of A502.The final area pursued as part of the linear analysis

work was undertaken as an initial check of the linearcodes and to provide a comparison with the nonlinearresults. These validation cases included comparisonsof several nonlinear and linear codes on configurationsranging from simple bodies of revolution to more com-plete lifting and nonlifting designs representative of thediverse set of aircraft that might be of interest in theQSP program. Near field pressures and propagatedsignatures using two different propagation codes werecomputed and compared with previous results, includ-

Fig. 2 A502 integration in Caffe

ing some experimental data.Figure 3 shows a comparison between the axisym-

metric code, A502, and Euler calculations for a simpleparabolic body of revolution. The character and mag-nitude of the near field pressure distributions agreedquite well although some differences exist between thecodes. This is due not only to differences in the flowphysics, but also the fidelity of the geometric model-ing.

Fig. 3 Comparison of near field pressures usingvarious aerodynamic analysis codes.

Similar comparisons with more complex configu-rations whose area distribution was developed us-ing jBoom were included in the TEM-4 presentation,showing excellent agreement. With nonaxisymmetriclifting cases, the agreement is less compelling, Resultscan be quite sensitive to section camber and twist andto the lift carry-over across the fuselage. For low boomconfigurations most of the high frequency, low ampli-tude oscillations in the near field pressures disappearin the Euler results and are sometimes quite sensitiveto paneling detail in the surface panel code.In Figure 4 a more complete configuration was

analyzed using A502, jBoom, and the Euler codeCART3D. This example shows that while certain crit-ical aspects of the aerodynamics may be captured byeach of these methods, differences do exist that areimportant to low boom design.Several experimental results were available to com-

pare with these codes as well. One of these cases is

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−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

X/L

DP

/P

Comparison of Near Field Pressure: A502 vs. Euler

Fig. 4 Generic low boom QSP configuration. Nearfield signatures from linear and nonlinear codes

shown in Figure 5. The experimental data from NASATND-7160 is compared with Euler, A502, and equiva-lent body computations. Agreement here is reasonablealthough some differences among the codes appear andthe Euler result requires very fine meshes to properlycapture the peak amplitudes.

Fig. 5 Comparison of linear and Euler codes withexperimental data.

Nonlinear Integrated Boom AnalysisEnvironment

A major outcome of our work for the QSP programhas been the development of a fast integrated tool,QSP107, for sonic boom prediction based on fully non-linear CFD analyses. This tool couples the multiblockEuler and Navier Stokes flow solver FLO107-MB16 toan H-mesh generator adapted from the HFLO4 code ofJameson and Baker,17 and to the PC Boom softwarefor far field propagation developed byWyle Associates.The H-type mesh, which is generated automaticallyfrom the geometric definition, can handle arbitrarywing-fuselage-canard configurations. The grid is fur-ther adjusted to have higher resolution in the areaswhere shock waves and expansions are present, andits grid lines slanted at the Mach angle to maximize

the resolution of the pressure signature at distances ofthe order of one body length. The user may specifythe location of an arbitrary cylindrical surface wherethe near field signature is provided as an input to amodified version of PC Boom which propagates a fullthree-dimensional signature to the ground along allrays that reach the ground. This allows for the calcula-tion of arbitrary cost functions (not only ground-trackinitial overpressure) that involve weighted integrationof the complete sonic boom footprint. The flow solvercombines advanced multigrid procedures and a precon-ditioned explicit multistage time stepping algorithmwhich allows full parallelization. Consequently the in-tegrated tool provides fully nonlinear simulations withvery rapid turn around time. Using a mesh with 3million mesh points we can obtain a complete flow so-lution and ground signature prediction in 15 minutes,using 16 processors on a Beowulf cluster made up of1.2Ghz AMD Athlon processors.Additionally, a geometry generation module for

QSP107 has been developed. This module allows forthe generation of a complete sonic boom footprint us-ing a number of parameters to describe the geometry.This is the only input required to QSP107; all otherprocedures are fully automated. Figure 7 shows aflowchart with the various modules that form part ofQSP107. Figure 8 below presents a view of the typicaloutcome of QSP107 including a bottom view of theMach number distribution and the symmetry planewave pattern.During the development of our integrated fast non-

linear boom prediction tool, QSP107, we carried outextensive validation studies in order to obtain a betterunderstanding of the sensitivity of the result to param-eters such as the number of mesh points, flow solutionconvergence, and location of the cutting plane for thenear field signature. These studies were reported onduring TEM-1 and TEM-2. We became aware thatthese sensitivities vary substantially with different con-figurations, and it seemed extremely desirable to vali-date QSP107 against an alternative nonlinear method.We identified the MIM3D software developed by MikeSiclari, formerly of Northrop-Grumman, as the mostadvanced such method.Accordingly we asked Mike Siclari to join our pro-

gram as a consultant, and obtained MIM3D fromNASA. MIM3D (Multigrid Implicit Marching 3D) as-sumes a fully supersonic flow field, and obtains a so-lution by marching in the streamwise direction withan implicit scheme based on trapezoidal integration.This requires the simultaneous solution of the flow fieldat all the points in each new cross-plane as the solu-tion is advanced downstream. A multigrid methoddeveloped in collaboration with Jameson is used tocalculate the cross-plane solutions, typically requiringabout 100 iterations for each marching step. MIM3Dhas an integrated mesh generator using a conical topol-

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ogy designed to maximize the resolution away fromthe body, and it also provides coupling to the NFBoom propagation code. The space marching proce-dure yields solutions with quite fast turnaround, of theorder of 1-2 hours on a workstation, but limits MIM3Dto fully supersonic flow fields, whereas QSP107 cantreat mixed supersonic and subsonic flows. Moreover,the cost of the time marching scheme in QSP107 is notmuch greater than the space marching procedure inMIM3D, because the multigrid procedure in QSP107produces adequately converged solutions in 200-300time steps, while MIM3D requires of the order of 100iterations in each cross-plane. When run on a paral-lel computer QSP107 can in fact provide substantiallyfaster turnaround.However, MIM3D has provisions in its mesh gener-

ation procedure for including engines, while QSP107could not automatically treat engines unless it wereprovided with an externally generated mesh. Thus weregard the tools as complementary, and for that pur-pose we are using MIM3D to compare the results ofthe two codes on a suite of representative test cases.At the same time we also plan to pursue further workon the use of unstructured meshes for the analysis ofcomplete configurations, and in order to address theproblem of engine integration.

Optimization Problem FormulationThe first step in addressing QSP design as an MDO

task involves the formulation of the problem. Thisincludes identification of suitable design variables, con-straints, and objective(s). In an initial attempt at un-derstanding some of the issues involved in the problemformulation for this program, the rapid turnaround lin-ear analysis methods were integrated with several op-timization methods and various design problems wereinvestigated. Variables associated with fuselage tailor-ing, wing location, and planform shape were selectedas design parameters with bounds on their values andother constraints imposed on the problem. In order tosimplify computation of near-field pressures and vol-ume wave drag, the equivalent area distribution wasrepresented with a Fourier series, the coefficients ofwhich were taken as design variables. This producedsmooth, rapid function evaluations, but since these de-sign variables were related in a complex way to thepayload constraints, and since the sensitivity of otherconstraints was also related in very nonlinear ways tothese coefficients, the optimization problem was verydifficult to solve. Alternatively, when specific stationson the fuselage were selected as control points in anAkima spline representation of the fuselage radius dis-tribution, the problem became much more tractable.This is simply an example of the sensitivity of op-timization to problem formulation and the need forexperience in this problem domain to achieve an effec-tive formulation.

Similar difficulties were encountered in the selectionof objective and constraints. Initially a composite ob-jective function consisting of a weighted combinationof maximum overpressure and L/D was selected alongwith constraints of cabin dimensions, c.g. location,and other parameters. The optimizer achieved a multi-shock solution with 4-5 shocks, each with a maximumoverpressure of 0.5 psf. This indicated that the schemewas functioning correctly, but this was not consistentwith the QSP goals. When maximum overpressurewas replaced by initial boom overpressure in excessof 0.3 psf, the optimizer produced a design with thedesired 0.3 psf initial overpressure and a very reason-able L/D. Closer examination showed that the initialshock was followed only a few milliseconds later by amuch stronger shock. The objective was changed to in-clude a measure sensitive to overpressure in the first 10milliseconds of the signature. This too was exploitedby the optimizer, and after some experimentation anobjective that included maximum overpressure, initialshock strength, aft shock strength, and finally somemeasure of noise (dbA) produced more acceptable op-timal solutions.

Fig. 6 Iteration history for evolutionary algo-rithm. Composite objective includes metrics re-lated to boom and aero performance.

Figure 6 illustrates progress made by the optimizerusing this composite objective in 8 design variables,including fuselage, wing planform, and wing location.

Optimization MethodologyThe minimization of sonic boom with simple con-

straints is a problem that is usually handled by asimplified parameterization of the f-function and in-verse design. Because other design considerations areimportant to this problem and are not easily incorpo-rated into an inverse problem, an obvious alternativeis to employ nonlinear optimization and direct anal-ysis. This has been undertaken in the past, but hasgenerally led to unspectacular reductions in signature.Indeed, in many of these references, it is difficult to seesignificant improvement in the optimized designs. Oneof the reasons for frequent failure of search methodsin boom minimization is the highly nonlinear charac-ter of the relation between geometric design variables

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and signature-based objective. Experience with theinteractive design code provides a clear indication ofdifficulties with conventional optimization. The sud-den appearance of multiple shocks, the coalescence ofshocks, and the change in sensitivity of shock peaksdepending on the strength of other peaks, leads to oneof the most difficult aerodynamic optimization prob-lems that the authors have encountered. The designspace is generally filled with many local optima and,depending on the details of the aero analysis and prop-agation code, may exhibit discontinuities in objectiveor constraint gradients.

To better understand the character of the designspace, and search methods that are effective for thisproblem, several optimization problems were investi-gated in the first phase of the program. As describedin the previous section several problem formulationswere investigated, but several different search methodswere also employed. The conclusion of this study wasthat for the complete design problem (including geo-metric design variables, multidisciplinary constraints,and boom/performance objectives), the presence ofmultiple local minima, created by the physics of theproblem itself or introduced by noise in the computa-tions, prevented successful application of conventionalgradient-based search methods. Optimization codeswith which the authors are quite familiar, includingworkhorse SQP methods such as SNOPT, achieved lit-tle progress on this problem, for the reasons mentionedabove.

Alternative optimization methods that do not re-quire gradient information (e.g. nonlinear simplex)showed somewhat better performance, but consis-tently converged on local minima. This led to explo-ration of less efficient but more robust schemes suchas genetic algorithms and other stochastic approaches.The GA was most effective, although very inefficient,consistently finding a better solution than could beachieved by the other optimizers, but at the cost ofmany thousands of function evaluations even for 8-10variable problems. The results shown in figure 6 aretypical of those achieved with this method. The figureshows the best and average fitness in a population of50 individuals, with convergence usually achieved afterabout 30-50 generations. Since these nondeterministicalgorithms must generally be run several times, suchdesign problems are feasible only with very rapid anal-ysis codes or with highly parallel implementations.

This suggests that one approach to these problems isto combine robust but inefficient search methods withpowerful parallel computational facilities. This mayprove useful, but remains limiting as the number ofdesign variables is increased. A more subtle approachis to combine rapid linear methods and robust searchstrategies with higher fidelity nonlinear analysis andgradient based design.

Coupled Adjoint Design Method

This section presents an overview of the coupled ad-joint design method for sonic boom reduction. Themain objective of this derivation is to formally presentthe method that allows for the calculation of sensitivi-ties of cost functions based on the ground signature toan arbitrarily large number of design parameters witha single flow and adjoint solution. The discussion inthis section uses the nomenclature presented in Fig-ure 9. The Figure clearly shows the aircraft flying ata cruise altitude h, the ground plane, and an arbitrar-ily located near-field plane which creates the interfacebetween the CFD solution and the far-field propaga-tion method. Note that the external boundary of theCFD mesh may extend beneath the near field planefor numerical convergence issues.Assume that the external shape of the configura-

tion is given by F(�b), where �b is the vector of designvariables such that �b ∈ RN . These variables caninclude anything from global surface arrangement, po-sition, and shape details. The three-dimensional spacearound the configuration is discretized using a mesh ofappropriate density. This mesh will be used for boththe flow and adjoint solutions that are detailed below.At a pre-specified distance below the aircraft, and

still within the CFD mesh, the location of a near-fieldplane can be seen. This plane is the effective interfacebetween the CFD solution and the wave propagationprogram. At the near-field plane, the flow solution w0

can be extracted. Given these initial conditions, w0,the propagation altitude, and the altitude-dependentatmospheric properties ρ(z), p(z), T (z), the propaga-tion method produces a flow solution at the groundplane we are interested in, wG, which can be used todetermine any of a variety of measures of sonic boomimpact such as overpressure, rise time, impulse, per-ceived noise level, etc.18

Regardless of the type of cost function that we de-cide to optimize, we will be looking to minimize ascalar function I = I(wG) using any of a variety ofgradient-based optimization algorithms. For this pur-pose, it is necessary to obtain sensitivities of the costfunction, I , to the design variables in the vector �b, ∂I

∂�b.

There are several ways in which the gradient vectorcan be obtained, but their dependence on the numberof design variables in the problem, N , can be quitedifferent. Our objective is to find ∂I

∂�bin a manner which

is independent of N so that design iterations can beperformed at a minimum computational cost.In order to emphasize the power of the coupled

adjoint procedure, we analyze the calculation of allsensitivities using finite differencing first.

Finite Difference Method

Given that an analysis capability of the sort we havedescribed above is readily available, the most straight-

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forward way of computing design sensitivities is viathe traditional finite difference method.The simplest approach to optimization is to define

the geometry, F , through a set of design parameters,which may, for example, be the weights bi applied toa set of shape functions Si(x) so that the shape isrepresented as

f(x) =∑

biSi(x).

Then, a cost function I is selected which might, forexample, be the drag coefficient, the lift to drag ra-tio, or the impulse of the sonic boom signature, andI is regarded as a function of the parameters bi. Thesensitivities ∂I

∂bimay now be estimated by making a

small variation δbi in each design parameter in turnand recalculating the flow to obtain the change in I .Then

∂I

∂bi≈ I(bi + δbi)− I(bi)

δbi.

The gradient vector ∂I

∂�bmay now be used to determine

a direction of improvement. The simplest procedureis to make a step in the negative gradient direction bysetting

�bn+1 = �bn − λ∂I∂�b,

so that to first order

I + δI = I +∂IT

∂�bδ�b = I − λ∂I

T

∂�b

∂I

∂�b.

If we assume that the computational cost of a singleCFD solution is represented by tCFD, while the costof the near- to far-field propagation procedure is givenby tWAV E, the total cost of computation of a singleevaluation of the gradient vector, CFD, is given by

CFD = (N + 1)(tCFD + tWAV E)

Clearly, this cost is directly proportional to the to-tal number of design variables in the problem. Al-though the procedure is quite simple to implement,the sonic boom minimization problem requires hun-dreds of design variables to represent the shape of theconfiguration and therefore this approach is not re-alistic. However, the finite difference method is oftenused in our work to validate the results of more sophis-ticated sensitivity analysis methods, and in situationsin which the total cost of the analysis is virtually neg-ligible.

Remote Sensitivity Method

As a first step towards decreasing the cost of op-timization, we can use existing adjoint technology oneach of the modules that make up the sonic boom anal-ysis procedure. Note that the cost function of interestis given by

I = I(wG),

and therefore, the total sensitivity vector can be con-structed as follows:

∂I

∂�b=

∂I

∂wG

∂wG

∂w0

∂w0

∂�b. (1)

The first term in this expansion can be easily obtainedanalytically. The second and third terms in Equation 1are slightly more complicated since they represent theJacobian matrices of the flow solution and signaturepropagation subproblems.The calculation of the last term in Equation 1 us-

ing an adjoint approach is both more involved andcomputationally expensive than the signature prop-agation procedure. Firstly, the governing equationsthat relate variations in �b to variations in w0 are par-tial differential equations, not ODEs. Secondly, thecomputational cost of solving the direct CFD problemis orders of magnitude larger than for the wave prop-agation method, requiring high-performance parallelcomputers to fit within the preliminary design cycle.During the last few years7, 9, 19, 20 our group has de-

velop adjoint methodologies for the computation ofaerodynamic shape sensitivities with very promisingresults. The methods developed have been success-fully used in studies of new aircraft, some of which areflying today. Before proceeding, a short description ofthe adjoint method, its requirements and advantagesare given below.In order to reduce the computational costs, there

are advantages in formulating both the inverse prob-lem and more general aerodynamic problems withinthe framework of the mathematical theory for the con-trol of systems governed by partial differential equa-tions.9, 21, 22 A wing, for example, is a device to pro-duce lift by controlling the flow, and its design can beregarded as a problem in the optimal control of theflow equations by variation of the shape of the bound-ary. If the boundary shape is regarded as arbitrarywithin some requirements of smoothness one must usethe concept of the Frechet derivative of the cost withrespect to a function. Using techniques of controltheory, the gradient can be determined indirectly bysolving an adjoint equation which has coefficients de-fined by the solution of the flow equations. The costof solving the adjoint equation is comparable to thatof solving the flow equations. Thus the gradient canbe determined with roughly the computational costof two flow solutions, independently of the number ofdesign variables, N .For the flow about an airfoil or wing, the aerody-

namic properties which define the cost function arefunctions of the flow-field variables (w) and the physi-cal location of the boundary, which may be representedby the function F , say. Then

I = I (w,F) ,

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and a change in F results in a change

δI =∂IT

∂wδw +

∂IT

∂F δF , (2)

in the cost function. Using control theory, the gov-erning equations of the flowfield are introduced as aconstraint in such a way that the final expression forthe gradient does not require re-evaluation of the flow-field. In order to achieve this δw must be eliminatedfrom (2). Suppose that the governing equation Rwhich expresses the dependence of w and F withinthe flowfield domain D can be written as

R (w,F) = 0. (3)

Then δw is determined from the equation

δR =[∂R

∂w

]δw +

[∂R

∂F]δF = 0. (4)

Next, introducing a Lagrange Multiplier ψ, we have

δI =∂IT

∂wδw +

∂IT

∂F δF − ψT([∂R

∂w

]δw +

[∂R

∂F]δF

)

=

{∂IT

∂w− ψT

[∂R

∂w

]}δw +

{∂IT

∂F − ψT[∂R

∂F]}

δF .

Choosing ψ to satisfy the adjoint equation[∂R

∂w

]T

ψ =∂I

∂w(5)

the first term is eliminated, and we find that

δI = GδF , (6)

where

G =∂IT

∂F − ψT

[∂R

∂F].

The advantage is that (6) is independent of δw, withthe result that the gradient of I with respect to an ar-bitrary number of design variables can be determinedwithout the need for additional flow-field evaluations.In the case that (3) is a partial differential equation,the adjoint equation (5) is also a partial differentialequation and appropriate boundary conditions mustbe determined.After making a step in the negative gradient direc-

tion, the gradient can be recalculated and the processrepeated to follow a path of steepest descent until aminimum is reached. In order to avoid violating con-straints, such as a minimum acceptable wing thickness,the gradient may be projected onto the allowable sub-space within which the constraints are satisfied. In thisway one can devise procedures which must necessarilyconverge at least to a local minimum, and which canbe accelerated by the use of more sophisticated descentmethods such as conjugate gradient or quasi-Newtonalgorithms.

As was mentioned above, the cost of obtaining thegradient of I with respect to an arbitrary number ofdesign variables, N , is independent of the number ofdesign variables. However, notice that this is only truefor a single scalar function, I .Since the last term in Eq. 1 refers to the sensitivity

of cost functions in the near-field plane to changes onthe surface of the aircraft, we typically refer to thisimplementation of the adjoint as the Remote Sensitiv-ity Method. This method has been implemented andtested and is the subject of an accompanying paper.23

In particular, this method allows for efficient modifi-cations of the surface geometry to recover a near fieldpressure distribution which is known to translate intodesirable shapes of the farfield signature.If this remote sensitivity method is to be used for in-

verse design work, it is important to be able to specifyrealistic near field target pressure distributions thatresult in ground signatures with the necessary prop-erties. For this purpose, we have developed a finite-difference based optimization loop around the wavepropagation algorithm that allows us to recover thenear field signature that results in the closest agree-ment with a specified target at the ground.A realistic example of this procedure involved a

generic QSP configuration obtained from our lineardesign methods that produced a ramp shape groundboom with the first shock strength at less than 0.3 psfand the second shock strength at less than 0.5 psf .The flight conditions were M∞ = 2 , flight altitude= 55, 000ft, angle of attack = 3.5◦, and R/L = 0.5.As shown in Figure 10b, due to nonlinear effects thatare accounted for by the Euler equations, the initialground boom signature in blue is no longer a rampshape. In this design example, instead of achievinga complete optimization, the current boom has beenimproved to satisfy the criterion used for the lineardesign. Figure 10b shows a reduction in the shockstrength and the ramp shape (approximated with mul-tiple shocks) was recovered in 500 wave propagationdesign iterations. As illustrated in Figure 10a, thiscase has produced a resulting near field pressure sig-nature which could be achievable by modifications ofthe surface shape.

Coupled Adjoint Sensitivity Method

The ideal situation for the design of low sonic boomaircraft is one where the computation of the completegradient vector, ∂I

∂�bhas a cost which is independent of

N , the number of design variables. The main focusof our work has been to develop such an algorithm.The fundamentals of this approach are outlined in thissection.We proceed by developing separate adjoints for both

the CFD and wave propagation modules and we con-clude by showing how these two adjoint problems canbe coupled and solved without need for an inner iter-

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ation.Consider the near- to far-field wave propagation

problem first. Let’s consider a coordinate along the di-rection of information propagation, τ , such that τ = τ0corresponds to the location of the near-field plane,while τ = τg coincides with the ground plane. Alongthis coordinate, the governing equations of the wavepropagation can be conceptually expressed as

dw

dτ= f(w), with initial conditions: w(τ = τ0) = w0

(7)

where w are the flow variables that participate in thepropagation problem, and f(w) can be a complicatedfunction, similar in nature to a Whitham F-function,that depends on the particular method of propagationchosen.The first variation of Equation 7 can be derived as

follows:

d

dxδw =

df

dwδw, subj. to ic. δw(τ = τ0) = δw0. (8)

Let’s assume a cost function of the form

I = I(wG) = I(w(τ = τG)), (9)

which can easily represent all of the potential costfunctions in the sonic boom minimization problem.The first variation of the cost function can be ex-pressed as

δI =∂I

∂xδwG, (10)

which can be augmented by the variation of the gov-erning equations of propagation 8 without change toproduce

δI =∂I

∂wδwG −

∫ τG

τ0

φT

(d

dτδw − df

dwδw

)dτ, (11)

and where φ is a costate variable or Lagrange multi-plier that can take on any value along the wavefrontpropagation direction. Integrating Equation 11 byparts we obtain

δI =∂I

∂wδwG − φT (τG)δwG + φT (τ0)δw0

+∫ τG

τ0

(dφT

dτδw + φT df

dwδw

)dτ. (12)

Since φ is completely arbitrary, we are free to chooseit so that it satisfies the adjoint equation of the wavepropagation method

dφ

dτ+

[df

dw

]T

φ = 0, (13)

subject to boundary conditions

φ(τG) =[∂I

∂w

]T∣∣∣∣∣τ=τG

, (14)

and then, the expression for the gradient in Equa-tion 12 simplifies to

δI = φT (τ0)δw0. (15)

We still require the values of the variation of the flowsolution, δw0, at the near-field plane with respect tothe shape design variables, δb. Here we apply theadjoint method for the flow calculation in a mannersimilar to that described in the previous section.For flows governed by the Euler equations of fluid

motion, it proves convenient to denote the Cartesiancoordinates and velocity components by x1, x2, x3 andu1, u2, u3, and to use the convention that summationover i = 1 to 3 is implied by a repeated index i. Then,the three-dimensional Euler equations may be writtenas

∂w

∂t+∂fi

∂xi= 0 in D, (16)

where

w =

ρρu1

ρu2

ρu3

ρE

, fi =

ρui

ρuiu1 + pδi1ρuiu2 + pδi2ρuiu3 + pδi3

ρuiH

(17)

and δij is the Kronecker delta function. Also,

p = (γ − 1) ρ{E − 1

2(u2

i

)}, (18)

andρH = ρE + p (19)

where γ is the ratio of the specific heats.Consider a transformation to coordinates ξ1, ξ2, ξ3

where

Kij =[∂xi

∂ξj

], J = det (K) , K−1

ij =[∂ξi∂xj

],

andQ = JK−1.

The elements of Q are the coefficients of K, and ina finite volume discretization they are just the faceareas of the computational cells projected in the x1, x2,and x3 directions. Also introduce scaled contravariantvelocity components as

Ui = Qijuj.

The Euler equations can now be written as

∂W

∂t+∂Fi

∂ξi= 0 in D, (20)

whereW = Jw,

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and

Fi = Qijfj =

ρUi

ρUiu1 +Qi1pρUiu2 +Qi2pρUiu3 +Qi3p

ρUiH

.

In order to link the near-field CFD solution with thewave propagation problem, let’s define a cost functionfor the CFD solution as the following “boundary” in-tegral

J =∫BφTw0 dξB,

such that its first variation coincides with the expres-sion for the gradient of the propagation algorithm inEquation 15. That is

δJ =∫BφT δw0 dξB.

Define the Jacobian matrices

Ai =∂fi

∂w, Ci = QijAj . (21)

The Euler equations (20) in the steady state can bewritten as

∂

∂ξiQijfj = 0, (22)

and their first variation, an equation for δw, can thenbe obtained as

∂

∂ξi

(δQijfj +Qij

∂fj

∂wδw

)= 0. (23)

The cost function can then be easily augmented byEquation 23 to produce

δJ =∫BφT δw0 dξB −

∫DψT ∂

∂ξiδQij fj dξ

−∫DψT ∂

∂ξiQij

∂fj

∂wδw dξ, (24)

whose last term can be integrated by parts to yield

δJ =∫BφT δw0 dξB −

∫DψT ∂

∂ξiδQij fj dξ

−∫BψTniQij

∂fj

∂wδw0 dξB

+∫D

∂ψT

∂ξiQij

∂fj

∂wδw dξ, (25)

Since Equation 24 should hold for an arbitrary choiceof the costate vector ψ, we can choose it as the solutionof the adjoint equation

∂ψ

∂t− CT

i

∂ψ

∂ξi= 0 in D, (26)

subject to “boundary” conditions

ψTniQij∂fj

∂w= φ.

These choices eliminate all but the second term inEquation 25 which does not depend on the variationsof either the CFD flow solution or the wave propaga-tion program, thus rendering the computation of thesensitivities of the overall cost function I with respectto an arbitrary number of design variables completelyindependent of N .Notice that the “boundary” conditions on the CFD

adjoint are not true boundary conditions. The near-field plane, as mentioned earlier, is inside the extentof the CFD mesh. Therefore, the boundary conditionswe have been discussing will appear as source termsto the CFD adjoint problem. The location of thesesource terms can be obtained with simple interpolationtechniques.In sum, since δJ = δI , the expression for the gradi-

ent can be shown to be

δJ = −∫DψT ∂

∂ξiδQij fj dξ,

which requires the solution of the CFD adjoint equa-tion, ψ. In order to obtain ψ, we must have solved theadjoint of the wave propagation problem subject to ap-propriate boundary conditions so that φ can be usedin the “boundary” conditions of the CFD adjoint.

ConclusionsAfter one year of effort, we have developed a vari-

ety of linear and nonlinear methods for the analysis ofaircraft with low sonic boom. These include very fastturnaround linear methods and fast turnaround non-linear methods that must complement each other in aneffective design framework. In addition, we have devel-oped multidisciplinary optimization methods based onthe fast linear analysis techniques, which can be usedin combination with non-gradient based optimizationalgorithms to determine the overall configuration ofthe aircraft. In the process, we have learned importantlessons regarding the setup of the optimization prob-lem, the parameterization of the design space, and theability of various optimization algorithms to succeedin such a difficult design problem. Finally, we havealso developed an efficient, accurate, nonlinear cou-pled adjoint method for the inexpensive calculation ofthe sensitivity of ground signatures to modifications inthe aircraft shape. The intention is to use this methodafter the overall aircraft configuration has been opti-mized using linear methods to recover the shape thatproduces the results originally expected. Much workremains to be done to make this type of design envi-ronment a reality.

AcknowledgmentsThe authors acknowledge the support of the

DARPA Tactical Technology Office under the direc-tion of Dr. Richard Wlezien for the completion of thiswork.

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Commun. Pure Appl. Math, V(3):301–348, 1952.2C. Thomas. Extrapolation of sonic boom pressure signa-

tures by the waveform parameter method. NASA TN D-6832,1972.

3S. E. Cliff and S. D. Thomas. Euler/experiment correla-tions of sonic boom pressure signatures. AIAA paper 91-3276,September 1991.

4K. J. Plotkin. Review of sonic boom theory. AIAA pa-per 89-1105, 12th AIAA Aeroacoustics Conference, San Anto-nio, TX, April 1989.

5J. J. Reuther, A. Jameson, J. J. Alonso, M. Rimlinger, andD. Saunders. Constrained multipoint aerodynamic shape opti-mization using an adjoint formulation and parallel computers:Part I. Journal of Aircraft, 36(1):51–60, 1999.

6J. J. Reuther, A. Jameson, J. J. Alonso, M. Rimlinger, andD. Saunders. Constrained multipoint aerodynamic shape opti-mization using an adjoint formulation and parallel computers:Part II. Journal of Aircraft, 36(1):61–74, 1999.

7A. Jameson, L. Martinelli, and N. Pierce. Optimum aero-dynamic design using the navier-stokes equations. AIAA pa-per 97-0101, 35th Aerospace Sciences Meeting and Exhibit,Reno, Nevada, January 1997.

8A. Jameson. Optimum Aerodynamic Design Using ControlTheory, Computational Fluid Dynamics Review 1995. Wiley,1995.

9A. Jameson. Aerodynamic design via control theory. Jour-nal of Scientific Computing, also ICASE Report No. 88-64,3:233–260, 1988.

10I. M. Kroo. Decomposition and collaborative optimizationfor large scale aerospace design. In N. Alexandrov and M. Y.Hussaini, editors, Multidisciplinary Design Optimization: Stateof the Art. SIAM, 1996.

11R. D. Braun and I. M. Kroo. Development and applica-tion of the collaborative optimization architecture in a multi-disciplinary design environment. In N. Alexandrov and M. Y.Hussaini, editors, Multidisciplinary Design Optimization: Stateof the Art, 1996.

12I. M. Kroo. The aerodynamic design of oblique wing air-craft. AIAA paper 86-2624, 1986.

13I. M. Kroo, J. Gallman, and S.C. Smith. Aerodynamic andstructural studies of joined wing aircraft. Journal of Aircraft,1991.

14A. Van der Velden and I. M. Kroo. The sonic boom of anoblique flying wing. Journal of Aircraft, 1994.

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16J. Reuther, A. Jameson, J. J. Alonso, M. J. Rimlinger, andD. Saunders. Contrained multipoint aerodynamic shape opti-mization using an adjoint formulation and parallel computers.AIAA paper 97-0103, 35th Aerospace Sciences Meeting and Ex-hibit, Reno, Nevada, January 1997.

17A. Jameson and T.J. Baker. Multigrid solution of the Eu-ler equations for aircraft configurations. AIAA paper 84-0093,AIAA 22th Aerospace Sciences Meeting, Reno, Nevada, January1984.

18R. Seebass and B. Argrow. Sonic boom minimizationrevisited. AIAA paper 98-295, 2nd AIAA Theoretical FluidMechanics Meeting, Albuquerque, NM, June 1998.

19A. Jameson. Automatic design of transonic airfoils to re-duce the shock induced pressure drag. In Proceedings of the31st Israel Annual Conference on Aviation and Aeronautics,Tel Aviv, pages 5–17, February 1990.

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Fig. 7 A Flowchart of the Various Modules of QSP107

a) Bottom View of Mach Number Distribution b) Symmetry Plane Wave Pattern

Fig. 8 Results of QSP107 for a Generic QSP Configuration

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Wo

F(�b)

Near Field Plane

τ

Global CFD Mesh

Atmospheric Properties

ρ(z)T (z)p(z)

Wg

h

Fig. 9 Schematic of Sonic Boom Minimization Setup with Nomenclature.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

X/L, L(150 ft)

dP

/P

Initial Near−field PressureDesigned Near−field Pressure

a) Near Field Pressure

0 20 40 60 80 100 120 140 160−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

t(msec)

P(p

sf)

Initial BoomCurrent BoomTarget Boom

b) Ground Pressure

Fig. 10 Generic QSP Configuration Nearfield to Ground Inverse Design Procedure.

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