Rapid Airplane Parametric Input Design(RAPID) Robert E. SmithyMalcolm I. G. BloorzMichael J. WilsonxAlmuttil M. Thomas{

ABSTRACT

NOMENCLATURE

An ecient methodology is presented for dening aclass of airplane congurations. Inclusive in this denition are surface grids, volume grids, and grid sensitivity. A small set of design parameters and grid control parameters govern the process. The general airplane conguration has wing, fuselage, vertical tail,horizontal tail, and canard components. The wing,tail, and canard components are manifested by solving a fourth-order partial dierential equation subjectto Dirichlet and Neumann boundary conditions. Thedesign variables are incorporated into the boundaryconditions, and the solution is expressed as a Fourierseries. The fuselage has circular cross section, andthe radius is an algebraic function of four design parameters and an independent computational variable.Volume grids are obtained through an application ofthe Control Point Form method. Grid sensitivity isobtained by applying the automatic dierentiationprecompiler ADIFOR to software for the grid generation. The computed surface grids, volume grids, andsensitivity derivatives are suitable for a wide range ofComputational Fluid Dynamics simulation and conguration optimizations.

ABDNXXsurfXvolab

BtBwBcCEF1; F2H1H2K1 ; K2K3 ; K4MPRFR0; R1R2S1 ; S2TTaXt ; ZtXw ; ZwXc ; Zcarxyycyt

This paper is declared a work of the U. S. Government andis not subjected to copyright protection in the United States.y Senior Research Engineer , NASA Langley Research Center, Hampton Virginia 23681-0001, Associate Fellow, AIAA.z Professor, Department of Applied Mathematics Studies,University of Leeds, Leeds LS2 9JT, UK.x Senior Lecturer, Department of Applied MathematicsStudies, University of Leeds, Leeds LS2 9JT, UK.{ Graduate Assistant, Department of Mechanical Engineering, Old Dominion University, Norfolk, Virginia 23529-0247.

1

Vector Fourier coecientsVector Fourier coecientsVector Dirichlet boundary conditionsVector Neumann boundary conditionsSurface coordinateSurface gridVolume gridVector constant for Fourier expressionVector constant for Fourier expressionRoot chord length for tail componentsRoot chord length for wing componentRoot chord length for canard componentWing chord length at crankWingtip chord lengthParameters for airfoil denitionInboard wing span lengthOutboard wing span lengthConstants for grid spacing controlConstants for grid spacing controlMaximum wing camberLocation of maximum wing camberFuselage lengthParameters for fuselage radiusParameter for radius at rearmost pointDerivative control design parametersMaximum wing thicknessWing taper parameterCoordinates of trailing tip pointCoordinates of trailing wing pointCoordinates of trailing crank pointPDE weighting factorFuselage radiusAirfoil independent variableAirfoil dependent variableWing camberWing thicknessCoordinate weighting parameter

; ; ; ;

PK

IndicesIJKn

Computational coordinatesComputational coordinatesGrid spacing control coordinateGrid spacing control coordinateFuselage denition variableSet of design parametersSet of grid control parameters

sensitivity for automated analysis (both low-level andhigh-level) and optimization. As the geometry becomes detailed, it is imperative that a CAD model,with its general characteristics be developed, and anyparameter-dened model should be upgraded with aconventional CAD system. Alternately, it would bedesirable to incorporate a methodology like the onedescribed here in a conventional CAD system.

pointpointpointIndex for Fourier series

ithjthkth

Creating an airplane surface or any other objectsurface with design parameters implies that there isan underlining set of rules or correspondences (modelfunctions) that are driven by the parameters and independent computational variables. Surfaces gridsare discrete evaluations of the surface functions, andsurface grids can be described as organized sets ofpoints. Dierent discipline analyses and dierenttechniques within a discipline most often require different grids to be generated from the surface model4.

1. INTRODUCTION

Airplane design has historically been divided intothree phases: (1) conceptual design; (2) preliminarydesign; and (3) detailed design1;2;3. The conceptualdesign of an airplane usually begins with specications for a proposed mission and rough sketches ofthe conguration. Geometry begins to evolve in theform of sets of connected points. Usually only theminimal amount of information for low-level analysesis created. As a conguration approaches the end ofthe conceptual design phase, Computer-Aided Design(CAD) models are created. CAD models are most often derived by interpolating and rening the earlierspecied sets of connecting points used in low-levelanalyses.

High level aerodynamic analysis, such as Euler orNavier-Stokes simulation, require that volume gridsbe constructed about the conguration surface grid.Information, such as far-eld boundary surfaces andgrid spacing controls to capture anticipated physics,is required. Surface grids that are generated for lowlevel analyses usually are not suitable to directly connect to a surrounding volume grid. Interpolation orreevaluation of the surface grid is most often requiredIn the preliminary-design phase, high-level anal- before proceeding to high-level analysis5.ysis and testing of physical models are performed.Geometry for computational analysis and the con- The sensitivity of mission dependent variables withstruction of test models is extracted from the CAD respect to design variables is a desired and oftenmodel. Usually the airplane surface geometry is xed used feature in the design process. An intermediexcept for the occasional change that may result from ate requirement for many techniques is surface-gridthe new analyses, and these changes are implemented and volume-grid sensitivity with respect to the dein the CAD model. In the detailed-design phase, sign variables6.the CAD model is the central design representation,now containing detailed information for manufacturIn this paper, a methodology to dene a class ofing the airplane1.airplane congurations and directly evaluate surfacegrids, volume grids, and grid sensitivity is presented.It can be argued that a CAD model should be im- The objective of the methodology is to provide surplemented at the very earliest stage of conceptual de- face denition and grid generation for conceptual design. However, conventional CAD models and CAD sign that could be used in a wide spectrum of analysoftware are very general and very complex. Usually ses (potential ow to Navier-Stokes). The methodola CAD specialist is required to implement the soft- ogy and associated software is called Rapid Airplaneware. In an environment where the ability to quickly Parametric Input Design (RAPID). The general conchange features of the geometry is nearly as impor- guration, at this writing, has wing, fuselage, verticaltant as the geometry itself, it is desirable: (1) to have tail, horizontal tail, and canard components (Fig. 1).the geometry model specied in terms of a small number of design parameters; (2) to visualize the geometry and interact with it to explore the envelope ofpossibilities; and (3) to quickly extract grids and grid2

3. FUSELAGE SURFACE

The denition of the lifting surfaces is based on thePDE method as described by Bloor and Wilson7;8.Design parameters are incorporated into boundaryconditions for the PDE solution. The fuselage hascircular cross section and is dened with an algebraicfunction.

The fuselage denition in the RAPID methodologyis an algebraic function which creates two surfaces one above the fuselage intersection with the liftingcomponents and one below (Fig. 3). The airplaneis considered to be symmetric about the xz plane aty = 0, and only one side of the airplane surface is2. THE PDE METHODcomputed. The fuselage cross section is circular, andThe PDE method generates a Euclidean Space sur- both the upper and lower surfaces can be representedface X = (x(; ), y(; ), z (; )) transformed from as X = (x(; ); y(; ); z (; )) where(0 1) (0 1) computational space.The transformation is obtained by solving the fourth x = RF ; y = r( )cos(=2); z = r( )sin(=2);order partial dierential equationr( ) = R0 sin() + R1sin(3);

222@2 @ = ((1 R2) + R2);a+ @X = 0:(1)2@ 20 1;0 1:(3)By letting (0 1) ! (0 2), a general = 0 corresponds to the end point on the fuselage,periodic solution to Eq. 1 is:and = 0 corresponds to a point along the curve sep1Xarating the upper and lower fuselage surfaces (Fig.X(; ) = A0 + An()cos(n) + Bn()sin(n); 3).n=1(2)The parameters for the fuselage are: RF , the fusewherelage length; R0 and R1, control for the fuselage raA0 = a00 + a01 + a022 + a033;dius; and R2, a parameter to control a nite radiusat the end of the fuselage. The boundary curve sepaAn = an1ean + an2ean + an3e an + an4e an ; rating the upper and lower fuselage surfaces is a comof the fuselage intersection with the liftingBn = bn1ean + bn2ean + bn3e an + bn4e an : binationcomponents and cubic curves connecting the interan1, an2, an3, an4 and bn1, bn2, bn3, bn4 are vector- sections. The fuselage center is optionally allowed tovalued constants determined by the boundary condi- translate upward along a quadratic function from thetion imposed at = 0 and = 1. The boundary trailing wing/fuselage intersection point to the end ofconditions are:the fuselage. This creates a \duck tail" characteristicin the fuselage (Fig. 4).X(; 0) = D0(); X(; 1) = D1();A surface grid is created by evaluating the surfaceX (; 0) = N0(); X (; 1) = N1():functions at discrete (I ) and (K ). In order to conIncorporating a set of design parameters P in the centrate the grid in certain regions, such as aroundboundary conditions controls the shape of the sur- the wing/fuselage intersection, it is necessary to create control functions that map 0 ; 1 intoface.0 ; 1. The grid control functions and theFor the family of airplanes described herein, two grid control parameters used in RAPID for this purPDE surfaces dene the wing (Fig.2). Boundary con- pose are discussed in a Section 5.ditions are specied at: (1) the wing/fuselage in4. PDE BOUNDARY CONDITIONStersection; (2) the crank between the inboard-wingcomponent and outboard-wing component; and (3) Two PDE surfaces are used in RAPID to crethe wingtip. The horizontal tail, vertical tail and ate a wing. Each surface is computed in a subroucannard components are each described with a single tine. The input is an evaluation of the Dirichlet andPDE surface with boundary condition at the fuselage Neumann boundary conditions D0( (I )), N0 ( (I )),intersections and at the tips. The PDE boundary D1( (I )), N1 ( (I )), and (J ) chosen to ensure tanconditions are detailed in Section 4.gent continuity between adjoining surface. Spacingcontrol in the direction is achieved by mapping3

0 1 ! 0 1 prior to the surface evalua- the inboard wing component. The Dirichlet boundtion. Grid spacing is discussed in the Section 5. The ary condition for this component at the wing/fuselagePDE output is a surface grid X(I; J ) which can be intersection is:visualized, used for volume grid computation about23x = BCw x( ) + Xwthe airplane, or used in an analysis of the airplane.6767pin( ) = 622 7:(6)Dr(

)zy=F167The manipulation of a single airfoil section is cur45rently applied in RAPID for all of the Dirichlet condiz = y( )Ta + Zwtions. The section is governed by design parametersand is scaled, rotated and translated into dierent Bw is the wing-root chord length, Xw and Zw transboundary positions with additional parameters. The late the wing/fuselage intersection, and Ta scales theairfoil section is dened by the sum of a camber curve thickness at the wing/fuselage intersection relative toand a thickness curve (Fig 6). The airfoil equations the thickness at the crank. The -location on theare:fuselage corresponding to the intersection is:( ) = Csin;

x

( ) = yt ( ) + yc ( );

y

F

( ) = T2 (sin2 + F1 sin4 + F2sin6 );

yt

M

( ) = P 2 (2P sin (sin ) );

yc

2

x

w:= RBwC x( ) + XRF

F

The Dirichlet boundary condition for the outboardwing surface at the wing tip is:

P;

2

(1 2P + 2P sin (sin )2 ) x P;yc ( ) = M(1 P )20 P 1; 0 1:(4)

D (

out 0

x

= CE x( ) + Xt

66) = 66 y = R0 + H1 + H24

3777:75

(7)

Ez= Cy( ) + ZtThe design parameters for the section are: C , thesection chord length; T , the section manimun thick- E is the chord length at the wing tip; Xt and Ztness; F1 and F2 , Fourier coecients; M , maximum translates the wing tip in the xz -plane; and H2 is thecamber; P , location of maximum camber. The de- span length of the outboard-wing component.nition of the section starts at the trailing point, proceeds beneath the camber curve, around the leading The Neumann boundary condition for both thepoint and over the camber curve back to the trailing inboardand outboard-wing surface at the crank is:point. The location of maximum camber is measured32 @x Sfrom the trailing point.= 1 x( )

666out(

)=6164

@

2

77

7@yGiven the basic wing section, the Dirichlet boundNin0 () = N:(8)@ = S1 77ary condition for the two wing components can be5expressed. Boundary = 0 for the inboard surface@z@ = 0is at the crank, and the = 1 boundary is at thewing/fuselage intersection (Fig. 5). For the outboardwing component the = 0 boundary is at the wing S1 is a design parameter which aects the transitiontip and the = 1 boundary is at the crank. The between the inboard and outboard wing components.crank Dirichlet boundary condition is:The Neumann boundary condition applied at the23x=x( ) + Xcwing/fuselageintersection is:67672 @x367@ yDin0 () = Dout(5)1 ( ) = 6 y = R0 + H1 7 ;@ = S2 sin @4

z

= y( ) + Zc

5

N

in1

where (Xc ; Zc ) translates the crank boundary in a xzplane at y = R0 + H1 and H1 is the span length of4

66= 6664

@y@@z@

==

@x r( ) @r +z @zF @x@@

y

S2 sin @@x

7777:75

(9)

where S2 is a design parameter aecting the transi- by a Newton-Raphson process while satisfying a rstderivative continuity condition at (K3 ; K1).tion of the wing into the fuselage.The Neumann boundary condition at the wing tip The grid control parameters are distinguished fromis zero in current RAPID software.the conguration design parameters. The design parameters are referred to as the set P , and the gridThe tail and cannard components are described in parameters are referred to as the set K. K includesa similar fashion with a single surface, and the de- the grid spacing parameters described above and thetails are not presented. There are numerous choices volume grid control points discussed in the Section 6.of boundary conditions to achieve a desired eect in6. VOLUME GRID GENERATIONthe lifting surfaces. Those described here representonly one choice that is incorporated into RAPID soft- A Control Point Form/Transnite Interpolationtechnique10 is used to compute volume grids for theware.RAPID methodology. A considerable amount of information has been published on this grid generation5. GRID SPACING CONTROLThe evaluation of the equations presented in the method and its variations, and only the major stepsprevious two sections results in surface grids. An H- are presented here.type topology is chosen for the general airplane surface and volume grid denition. The proper spacing Having established a grid on the conguration surof grid points within the topology constraints is very face, the volume grid generation is accomplished inimportant for achieving acceptable accuracy in the four major steps.application of a ow analysis about the vehicle surface. A double exponential function9 which maps the Step 1 is the determination of a grid in the symcomputational variables , , and onto themselves metry plane. The basic functions used in RAPIDis used in the RAPID methodology. The grid spacing are those for Bezier curves computed with the decontrol function is:Casteljau scheme11. Control points for an intermeK2diate curve and for a far-eld curve are computed

e K31;from the dimensions of the fuselage (Fig. 8). A set of = K1 Ke 21points are distributed in the -direction on the control curves obtained from the control points. Inter0 K3 ; 0 K1 ;polation from the fuselage surface across the controlK K31e 4 1 K3curves is obtained with a de Casteljau application in = K1 + (1 K1 )the -computational direction (Fig. 9).eK41 ;K3 1; K1 1;Step 2 is the determination of a three-dimensionalD (K3)K4 chosen 3

C 1:(10)gridsurface containing the lifting components (Fig.D10).Note that in the H-topology, the top and botFig. 7 is used to help describe the grid control param- tom gridsare considered separately. A similar processeters K1 ; K2; K3, and K4 . K1 and K3 are coordinates to that usedthe symmetry grid for computingof a point in the unit square. is the independent control pointswithfromthe fuselage and lifting surfacescomputational variable and corresponds to the per- is applied.centage of grid points in a particular direction. is the dependent computational variable and corresponds to the percentage of distance in the physical Step 3 is the determination of a cap grid. Controlspace along a grid curve. K2 and K4 are coecients points are extracted from the extreme x and y gridin the exponential functions dened for a particular coordinates in the lifting surface grid and the extremepart of the unit square. Where there is low slope in z-grid coordinates in the symmetry plane grid (Fig.the control functions, there is a concentration in the 11). The de Casteljau scheme is applied with thesegrid points, and where there is high slope, there is control points (Fig. 12).dispersion in the grid points. In the RAPID methodology Eq. 10 is used several times. The approach Step 4 is the application of Transnite Interpolaspecies a desired spacings at the = 0 and/or at tion to compute the interior grid (Fig. 13).

= 1 and/or K3 . K1 ; K2,and K4 are determined5

14 for linear aerodynamics optimization. There isIt is necessary to use several grid-spacing control also a C-programming language version of ADIFORfunctions and their control parameters in addition to which has been applied to CSCMDO.the interpolation control points in order to achieve a8. RAPID EXAMPLESgood grid for a given set of design parameters. Atthis writing, this requires some trial and error before Four RAPID examples are presented to demonacceptable parameters are realized. However, once strate the range of application. The examples showan acceptable set of grid parameters K are found for only the conguration surfaces.a given set of design parameters P , small changes inP do not require changes in K. Therefore, repetitive The rst example is shown in Fig. 14. The wing issmall changes in the design parameters such as dur- high relative to the fuselage and has high aspect raing conguration optimization, does not require the tio. The fuselage is relatively short compared to theconstant modication of the grid parameters. Also wing span. The wing is relatively thin and camberednote that the volume grids obtained with this algo- at the mid chord.rithm are computed only out to the wing tip. Anadditional far-eld grid would be necessary for mostThe second example is shown in Fig. 15. It is ahigh-level uid analyses.High-Speed Civil Transport (HSCT) like conguration with a double-delta wing. The leading edge ofAn option to using the volume grid generation de- each wing segment is straight as the result of settingscribed above is to use the Coordinate and Sensitiv- the parameters S1 and S2 equal to zero.ity Calculator for Multidisciplinary Design and Optimization (CSCMDO) process described in Reference12. In CSCMDO, the RAPID surface grid or a surface The third example is shown in Fig. 16 and is simigrid from some other source is INPUT. The GRID- lar to the HSCT conguration. The dierence is thatGEN/CSCMDO software is used to establish an ini- the wing now has a single delta planform created withtial volume grid. Thereafter, the initial grid is used two components.by CSCMDO to generate a new volume grid for a newINPUT surface grid. Here again, only small changes The fourth example shown in Fig. 17 is anotherin the design parameters can be tolerated without HSCT like conguration. It represents the degree ofreestablishing the initial grid.sophistication that can be incorporated into a RAPIDmodel. The fuselage has a \coke bottle" shape. The7. GRID SENSITIVITYwing has dihedral, twist, and the planform is not rectGradient based techniques applied to aerodynamic angular. A canard is also included.congurations optimization require the determina9. SUMMARY AND CONCLUSIONSvol @ Xsurf ). Intion of grid sensitivity ( @ X@ Pvol = @@XXsurf@PThe RAPID methodology has been outlined forthe past in order to evaluate such derivatives, each creatingairplane congurations for conceptual analyexpression would have to be dierentiated and chain sis and optimization.By establishing a grid topology,ruled through out the mathematical system, either grid spacing control, semiautomaticvolume grid genby hand or with the aid of a computer-aided alge- eration, and grid sensitivity a more completeanalysisbraic manipulation system. Fortunately, today there and optimization can be carried out in the conceptualare automatic dierentiation programs that dieren- design phase without incurring expensive geometrictiate code in other programs. The automatic dier- development. The methodology has considerable verentiation program used with RAPID in called Auto- satility and is very computationally ecient.matic Dierentiation for FORTRAN (ADIFOR) developed at the Argonne National Laboratory and10. ACKNOWLEDGEMENTSRice University13. ADIFOR is a preprocessor whichdierentiates FORTRAN code. The output of ADI- The authors would like to thank Dr. Jamshid AbolFOR is another FORTRAN code containing both the hassani and Mr. Bill Jones, members of the Langleyfunction evaluation and the derivative evaluations of Research Center GEOmetry LABoratory (GEOLAB)the function with respect to specied input variables. for many helpful discussions and assistance with theADIFOR has been applied to batch versions of the application of CSCMDO. The rst author would alsoRAPID methodology and is described in Reference like to thank Dr. Gerald Farin of Arizona State Uni6

versity with whom many helpful and interesting discussions on the topic of this paper were held duringthe author's sabbatical at ASU in 1994.

11. References

Raymer, D. P., Aircraft Design: A Conceptual Approach, AIAA Educational Series, AIAA, 1989.2Nicolai, L., Fundamentals of Airplane Design, Distributed by the University of Dayton, Dayton, OH, 1975.3Roskam, J., Airplane Design, Rowskam Aviation andEngineering Co., Ottawa, KA, 1989.4 Smith, R. E., and Kerr, P. A.,\Geometric Requirementsfor Multidisciplinary Analysis of Aerospace-Vehicle Design," AIAA Paper 92-4773-CP, Sept. 1992.5Thompson, J., Warsi, Z., and Mastin, C., NumericalGrid Generation Foundations and Applications, NorthHolland, 1985.6Sobieszczanski-Sobieski, J. \The Case for AerodynamicSensitivity Analysis," Paper presented to NASA/VPISSY Symposium on Sensitivity Analysis in Engineering,September 25-26, 1986.7Bloor, M. and Wilson, M., \Generating Blend SurfacesUsing Partial Dierential Equations," CAD, 21, No. 3 pp165-171, 1989.8Bloor, M., and Wilson, M., \Using Partial DierentialEquations to Generate Free-Form Surfaces," ComputerAided Design, 22, pp 202-212, 1990.9 Smith, R. and Everton, E., \Interactive Grid Generation for Fighter Aircraft Geometries," Numerical GridGeneration in Computational Fluid Mechanics '88, pp.805-814, Pine Ridge Press Ltd., 1988.10Eiseman, P. R., and Smith, R. E., \Applications ofAlgebraic Grid Generation , Applications of Mesh Generation to Complex 3-D Congurations," AGARD-CP-464,pp. 4-1-12, 1989.11Farin, G., Curves and Surfaces for Computer-AidedGeometric Design A Practical Guide, Academic Press,1993.12 Jones, W., and Samareh-Abolhassani, J., \A Grid Generation System for Multi-disciplinary Design," AIAA Paper 95-1689, June, 1995.13Bischof, C., et. al., \Automatic Dierentiation of Advanced CFD Codes for Multidisciplinary Design," Computer Systems in Engineering No. 3, pp 625-637, 1993.14Thomas, A., Smith, R., and Tiwari, S., \Aerodynamic Shape Optimization of Blend Surfaces Representing HSCT Type Congurations," AIAA Paper 95-1826,June, 1995.1

Fig. 1 General airplane conguration

Fig. 2 Conguration surface topology

7

Fig. 4 Duck tail fuselage characteristic

Fig. 5 Wing section denition

Fig. 3 Fuselage surface and topology

Fig. 6 Wing component boundary conditions

8

Fig. 7 Grid spacing control functionFig. 9 Symmetry grid

Fig. 8 Symmetry plane control netFig. 10 Surface grid containing lifting components

9

Fig. 11 Control point for outer grid surface

Fig. 13 Sample grid surfaces

Fig. 14 High wing congurationFig. 12 Outer grid surface

10

Fig. 15 HSCT conguration

Fig. 16 Delta wing conguration

Fig. 17 Canard twisted-wing conguration

11