multidisciplinary aerodynamic-structural shape …shape optimization using deformation (massoud)...

(c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

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8th AIAA/USAF/NASA/ISSMOSymposium on Multidisciplinary

Analysis and Optimization6-8 Sept. 2000 Long Beach, CA

AOO-40152 AIAA-2000-4911

MULTIDISCIPLINARYAERODYNAMIC-STRUCTURALSHAPE OPTIMIZATION USING

DEFORMATION (MASSOUD)Jamshid A. Samareh*

NASA Langley Research Center, Hampton, VA 23681

This paper presents a multidisciplinary shape parameterization approach. The ap-proach consists of two basic concepts: (1) parameterizing the shape perturbations ratherthan the geometry itself and (2) performing the shape deformation by means of the softobject animation algorithms used in computer graphics. Because the formulation pre-sented in this paper is independent of grid topology, we can treat computational fluiddynamics and finite element grids in the same manner. The proposed approach is simple,compact, and efficient. Also, the analytical sensitivity derivatives are easily computedfor use in a gradient-based optimization. This algorithm is suitable for low-fidelity (e.g.,linear aerodynamics and equivalent laminated plate structures) and high-fidelity (e.g.,nonlinear computational fluid dynamics and detailed finite element modeling) analysistools. This paper contains the implementation details of parameterizing for planform,twist, dihedral, thickness, camber, and free-form surface. Results are presented for amultidisciplinary application consisting of nonlinear computational fluid dynamics, de-tailed computational structural mechanics, and a simple performance module.

Nomenclaturewing areawing aspect ratioBernstein polynomialwing spanchordcoefficients of drag and liftcamberdegreescale factor for twist and shearingB-spline basis functionnormal vectororigin of parallelepipedcoordinates of NURBS control pointcoordinates of the solid elementscoordinates of deformed modelcoordinates of baseline modelshearing vectortwist planethicknessparameter coordinateMASSOUD design variable vectorNURBS weightsdesign variable vectorCartesian coordinates of deformed modelCartesian coordinates of baseline model

a angle of attack, degA total deformation6 deformation9 twist angle, degA leading edge sweep angle, degA wing taper ratio£, rj, £ coordinates of deformation objectp twist radius

Subscripts

ca camber/, J, K total numbers of control pointsi, j, k indices for NURBS control pointid, jd design variable indices

'Research Scientist, Multidisciplinary Optimization Branch,Mail Stop 159, -<j.a.samareh©larc.nasa.govX, AIAA SeniorMember.

This paper is a work of the U.S. Government and is not subject to copy-right ;

LlemoutPPi1rshtetthtwU

innerwing lower surfaceleading edgemidlineouterdegree of B-spline basis function in i directionplanformdegree of B-spline basis function in j directionrootsheartrailing edgetipthicknesstwistwing upper surface

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AMERICAN INSTITUTE OP AERONAUTICS AND ASTRONAUTICS


Fuel tankStiffeners

Spar*

Fig. 1 Internal components of a wing.

SuperscriptsT transpose of the matrix

Introduction

MULTIDISCIPLINARY design optimization(MDO) methodology seeks to exploit the

synergism of mutually interacting phenomena tocreate improved designs. An MDO process commonlyinvolves sizing, topology, and shape design variables.Multidisciplinary shape optimization (MSO) finds theoptimum shape for a given structural layout. Perform-ing MSO for a complete airplane configuration is achallenging task with high-fidelity analysis tools. Theanalysis models, also referred to as grids or meshes,are based on some or all of the airplane components,such as skin, ribs, spars, and the stiffeners. Theaerodynamic analysis uses the detailed definitionof the skin, also referred to as the outer mold line(OML), whereas the computational structural me-chanics (CSM) models use all components. Generally,the structural model requires a relatively coarsegrid, but it must handle very complex internal andexternal geometries. In contrast, the computationalfluid dynamics (CFD) grid is a very fine one, butit only needs to model the external geometry. TheMSO of an airplane must treat not only the wingskin, fuselage, flaps, nacelles, and pylons, but alsothe internal structural elements such as spars andribs (see Fig. 1). The treatment of internal structuralelements is especially important for detailed finiteelement (FE) analysis. For a high-fidelity MSOprocess to be successful, the process must be basedon a compact and effective set of design variablesthat yields a feasible and enhanced configuration.For more details, readers are referred to an overviewpaper by this author on geometry modeling and gridgeneration for design and optimization.1

Model parameterization is the first step for anMSO process. Over the past several decades, shapeoptimization has been successfully applied for two-dimensional and simple three-dimensional configura-tions. Recent advances in computer hardware andsoftware have made MSO applications more feasiblefor complex configurations. An important ingredientof aerodynamic shape optimization is the availabilityof a model parameterized with respect to the aerody-

namic design variables, such as planform, twist, shear,camber, and thickness. The parameterization tech-niques can be divided into the following categories:discrete, polynomial and spline, computer-aided de-sign (CAD), analytical, and deformation. Readers arereferred to reports by Haftka and Grandhi,2 Ding,3

and Samareh4 for surveys of shape optimization andparameterization.

In a multidisciplinary application, the parameter-ization must be compatible with and adaptable tovarious analysis tools ranging from low-fidelity tools,such as linear aerodynamics and equivalent laminatedplate structures, to high-fidelity tools, such as nonlin-ear CFD and detailed CSM codes. Creation of CFDand CSM grids is time-consuming and costly for a fullairplane model: detailed CSM and CFD grids basedon a CAD model take several months to develop. Tofit the MSO process into product development cycletimes, the MSO must rely on parameterizing the anal-ysis grids. For a multidisciplinary problem, the processmust also use a geometry model and parameterizationconsistently across all disciplines. Gradient-based op-timization requires accurate sensitivity derivatives ofthe analysis model with respect to design variables.

This paper presents a shape parameterization ap-proach suitable for MSO as part of a multidisciplinarydesign optimization application. The approach con-sists of two basic concepts. The first concept is basedon parameterizing the shape perturbations rather thanthe geometry itself. The second concept is based onusing the soft object animation5 (SOA) algorithmsfor shape parameterization. The combined algorithminitially introduced by this author1 was successfullyimplemented for aerodynamic shape optimization withanalytical sensitivity for structured grid6'7 and un-structured grid8 CFD codes. This algorithm has alsobeen used for multidisciplinary application of a high-speed civil transport (HSCT).9'10

Parameterizing the Shape PerturbationsAt first sight, parameterization by splines may seem

to be a viable approach for shape parameterization.The spline representation uses a set of control points,which could define any arbitrary shape. These controlpoints could be used as design variables for optimiza-tion. Typically, over a hundred control points arerequired to define an airfoil section, and over 20 airfoilsections to define a conventional wing. This require-ment results in over two thousand control points (i.e.,six thousand shape design variables) for a simple wing.The number of control points is even larger for a com-plete airplane model created with a commercial CADsystem. The large number of control points is neededmore for accuracy than for complexity.

Even if using a large number of design variables werefeasible, the automatic regeneration of analysis models

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(e.g., CSM and CFD grids) is not possible with cur-rent technology. For example, it takes several monthsto create an accurate CSM model of an airplane. Also,traditional shape parameterization processes parame-terize only the OML and are ineffective in parameteriz-ing internal components such as spars, ribs, stiffeners,and fuel tanks (see Fig. 1).

It is possible to use any shape (e.g., a sphere) asthe initial wing definition, allowing the optimizer tofind the optimum wing shape; however, this option isnot commonly practiced. Typically, the optimizationstarts with an existing wing design, and the goal isto improve or redesign the wing performance by us-ing numerical optimization. The geometry changes(perturbations) between initial and optimized wing arevery small,11'12 but the difference in wing performancecan be substantial. An effective way to reduce thenumber of shape design variables is to parameterizethe shape perturbations instead of parameterizing theshape itself. Throughout the optimization cycles, theanalysis grid can be updated as

R(v) = (1)

For multidisciplinary aerodynamic-structural shapeoptimization using deformation (MASSOUD), thechange, A_R, is a combination of changes in thickness,camber, twist, shear, and planform:

(2)

Far fewer design variables are required to parameterizethe shape perturbations A/? than the baseline shapef itself.

Figures 2 and 3 depicts the typical MSO and MAS-SOUD processes. In a typical MSO process (Fig. 2),a geometry modeler perturbs the baseline geometrymodel. Because automatic grid generation tools arenot available for all disciplines, automating this MSOprocess would be very difficult. In contrast, the MAS-SOUD process (Fig. 3) relies on parameterizing thebaseline grids and avoids the grid generation process,hence automating the entire MSO process.

Soft Object AnimationThe field of SOA in computer graphics5 provides

algorithms for morphing images13 and deforming mod-els.14'15 These algorithms are powerful tools for mod-ifying the shapes: they use deformation of a high-levelshape, as opposed to manipulation of lower level ge-ometric entities. Hall presents an algorithm and pro-vides computer codes for morphing images.13 The de-formation algorithms are suitable for deforming mod-els represented by either a set of polygons or a set ofparametric curves and surfaces. The SOA algorithmstreat the model as rubber that can be twisted, bent,

/ tmizer -^ —— ( g

r

aseline Xsometrymodel ,/-~ __ ^^

(^Design variables_^>

rGeometry modeler

i •( Geometry >»x^rnodel .)

I ————— i —— -

Grid— generationuC3. 1'

Q Analysis

1

i '

1I Grid1 cs generation

1 cj f ' r

1 -a1 3 Analysis1

... 1 ————— r

V 4

S GridS generation<u

^3 | '

Q Analysis

i '

Fig. 2 A typical MSO process.

n variables^}

i

Disc

iplin

e #1

'

Griddeformation

1

Analysis

i

Dis

cipl

ine

#2Grid

deformation

1

Analysis

i

Dis

cipl

ine

#Max Grid

deformation

1 '

Analysis

1

Fig. 3 The MASSOUD process.

tapered, compressed, or expanded, but will still retainits topology. This technique is ideal for parameteriz-ing airplane models that have external skin as well asinternal components (e.g., see Fig. 1). The SOA al-gorithms link vertices of an analysis model (grid) to asmall number of design variables. Consequently, theSOA algorithms can serve as the basis for an efficientshape parameterization technique.

Barr presented a deformation approach in the con-text of physically based modeling.14 This approachuses physical simulation to obtain realistic shape andmotions and is based on operations such as transla-tion, rotation, and scaling. With Barr's algorithm,the deformation is achieved by moving the vertices ofa polygon model or the control points of a paramet-ric curve and surface. Sederberg and Parry presenteda variant15 of the free-form deformation (FFD) algo-rithm, which operates on the whole space, regardless ofthe representation of the deformed objects embeddedin the space. The algorithm allows a user to manip-

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ulate the control points of trivariate Bezier volumes.The disadvantage of FFD is that the design variablesmay have no physical significance for the design en-gineers. This drawback makes it difficult to select aneffective and compact set of design variables. This pa-per presents a set of modifications to the original SOAalgorithms to alleviate this and other drawbacks.

For the modified SOA algorithms presented in thenext several sections, implementation will include thefollowing common set of steps:

1. Select an appropriate deformation technique andobject. This step defines the forward mappingfrom the deformation object coordinate system(£> f?) C) to the baseline grid coordinate system( x , y , z ) .

2. Establish a backward mapping from the baselinegrid coordinate system (x,y,z) to the deforma-tion object coordinate system (£, 77, Q. The £, 77, £mapping parameters are fixed and are indepen-dent of the shape perturbations. This preprocess-ing step is required only once.

3. Perturb the control parameters (design variables)of the deformation object.

4. Evaluate the grid perturbation (A.R) and shapesensitivity derivatives (dR/dv) with the £, 77, £ pa-rameters.

The following sections provide recipes for using SOAalgorithms for parameterizing airplane models forthickness, camber, twist, shear, and planform changes.

Thickness and CamberWe used a nonuniform rational B-spline (NURBS)

representation as the deformation object for thicknessand camber parameterization. The NURBS represen-tation combined the desirable properties of NationalAdvisory Committee for Aeronautics (NACA) defini-tion16 and spline techniques, and it did not deterioratenor destroy the smoothness of the initial geometry.

The changes in thickness and camber were repre-sented by

(3)

(4)

s=0 j=0

i=0 j=0

j=0

Figures 4 and 5 show the NURBS control points in(£,77) and (x, y, z) coordinate systems, respectively.

Fig. 4 Thickness and camber definitions in wingcoordinate system.

' Z

Fig. 5 Thickness and camber definitions in x, y,and z coordinate system.

The control points and weights could be used as designvariables.

The NURBS representation had several importantproperties for design and optimization. A NURBScurve of order p, having no multiple interior knots,is p — 2 differentiable. As a result, the NURBS rep-resentation was able to handle a complex deformationand still maintain smooth surface curvature. Read-ers are referred to the textbook by Farin for details onthe properties of NURBS representation.17 The controlpoints were the coefficients of the basis functions, butthe smoothness was controlled by the basis functions,not by the control points. The NURBS representa-tion was local in nature, allowing the surface to bedeformed locally, hence leaving the rest of the surfaceunchanged. Equations (3-4) served as the forwardmapping between the thickness and camber designvariables and the grid perturbation (5R^, 5Rca).

The next step was to establish the backward map-ping from the deformation object (i.e., the NURBSsurface) coordinates (£, 77) to the baseline model coor-dinates (x,y,z). The percentage of chord, %C, wasused for £, and the spanwise location, y, was used for

(5)

To calculate %C, we needed to determine the wing

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the B-spline basis functions.

dR __ dR Nidi

dPC;

Fig. 6 Curves defining the backward mapping.

chord at each y station. The baseline CAD modelprovided the leading edge Rie(rf), trailing edge R t e ( r j ) ,wing midline_-Rm (77), and normal vector defining theairfoil plane T(rj), as shown in Fig. 6. The curve denn-ing the wing midline did not have to be at the centerof the wing, but needed to be somewhere between theupper and the lower wing surfaces. The Rie(rf), Rte(rf)}

and Rm(r)) were used to separate points on the uppersurface from points on the lower surface.

Because we knew 77 for each grid point, we were ableto define a plane that passed through the grid pointby means of a normal vector defined by T(ij). We thenused the following equations to find the intersection ofthis plane and the curves shown in Fig. 6.

f(r,).[f-Rle(r,)]T =T(r1)-[f-Rte(r1)}T =

f(r,).[f-Rm(r,)]T =

(6)

(7)

(8)

Equations (6-8) must be solved for each grid point inthe model. For a high-order NURBS curve, Eqs. (6-8) are nonlinear and can be solved by the Newton-Raphson method. The solution to Eqs. (6-8) for each77 was a set of three points located at the leading edge,the trailing edge, and the center. The %C was cal-culated based on the leading and trailing edge points.Next, we needed to separate the grid points definingthe wing model into upper and lower surfaces. Weconnected the three points obtained from Eqs. (6-8) to form a curve that separated the upper surfacefrom the lower surface. This curve need not representthe camber line accurately, and a wing with droopingleading edge or with highly cambered airfoil sectionsmay require more than one Rm(rf) to define the curve.With this approach, the deformation may be localizedto a specific design area by setting allowable %C'min,/oOmaxi ^?min > and ?7max'

As the design variables (control points Pij)changed, we calculated the contribution to the thick-ness and camber by Eqs. (3-4). The advantage of thisprocess was that the sensitivity of grid point locationwith respect to design variables was only a function of

J

3=0

(9)

Consequently the sensitivity, as shown in Eq. (9), wasindependent of the design variables (Pjdjd) and thecoordinates (x, y, z). Thus, we needed to calculate thesensitivity with respect to thickness and camber onlyat the beginning of the optimization.

Twist and ShearThe twist angle is defined as the difference between

the airfoil section incident angle at the root and eachairfoil section incident angle. Similarly, the shear (di-hedral) is defined as the difference between the airfoilleading edge z coordinate for the root and the z co-ordinate at each airfoil section. If the twist angle atthe tip is less than the twist at the root, the wing issaid to have a washout, which could delay the stall atthe wing tip. Also, as the wing washout increases, thewing load shifts from outboard to inboard. As a re-sult, the spanwise distribution of the twist angle playsan important role in the wing performance.

The SOA algorithms are used to modify the wingtwist and shear distribution. Alan Barr presented aseries of SOA algorithms for twisting, bending, andtapering an object.14 Watt and Watt referred to thesealgorithms as nonlinear global deformation.5 Seder-berg and Greenwood extended Barr's ideas to handlecomplex shapes.18 Modified versions of these algo-rithms are presented in this paper.

To modify the twist and shear distributions, thewing was embedded in a nonlinear deformation ob-ject referred to as a twist cylinder, shown in Fig. 7.The center of the cylinder was defined by a NURBScurve, Rm(rj). The effect of deformation was confinedto a section of a wing by limiting the parameter 77 tovary between ?7min and ?7max. The ?7m;n could be ex-tended to the wing root, and the T7max went beyond thewing tip. The cylinder could be twisted and shearedonly in a plane (twist plane) defined by a point along-Rm('j) with a normal vector of f(r]). The Pm^) andPout (il) were the radii of the inner and outer cylinders,respectively (see Fig. 7). The deformation had no ef-fect on grid points located outside the outer cylinder,and the effect of deformation was scaled linearly fromthe outer cylinder to the inner cylinder. This linearblending allowed us to blend the deformed region withthe undeformed region in a continuous manner.

The 0(r/) and 5(f?) variables are defined by theNURBS representations:

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AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS


Fig. 7 Inner twist cylinder.

i=0

5(1,) =

(10)

(11)

Similarly to thickness and camber algorithms, we used

r] = y, f(T1) = (Q,y,Q)T (12)

The second step for twist and shear deformation wasto establish the forward mapping from the deformationobject (twist cylinder) coordinate system (77) to themodel coordinate system (x, y, z). We used Eq. (8) todetermine 77. Once 77 was determined, we calculatedthe local p(rj), />;„(??), Pout(*?), T(rj), 0(rj),jaid 5(r?).The point f was rotated 9(r\) deg about Rm(rj) andsheared S(TJ).

SR^) = e(p)p(rj)[Sme(T,),G,cosd(r1)}T (13)Jflshfo) = e(p)S(r,) (14)

where e(rj) was a scale factor that diminished the effectof deformation as we approached the outer cylinder.

Pout (15)

The sensitivity of a grid point with respect to thetwist and shear design variables was

OR 55(77)~

(17)

The term dO(rf)/dOi was independent of the twist de-sign variables #,- (see Eq. (10). However, sin#(77) and

Fig. 8 Twist definition for a transport.

Fig. 9tip.

Result of 45 deg twist on a transport wing

Fig. 10 Planform of a generic HSCT.cos #(77) depended on the twist design variables andwere updated for every cycle of the optimization. Incontrast, the term dS(ri)/dSi was independent of sheardesign variables 5j (see Eq. (11)).

Figure 8 shows the inner twist cylinder for a com-mercial transport. Figure 9 shows the result of twist-ing the wing 45 deg at the tip. This amount of twist islarge and unrealistic, but demonstrates the effective-ness of the SO A.

Planform ParameterizationThe wing planform is typically modeled with a set of

two-dimensional trapezoids in the x-y plane. Figure 10shows the planform of a generic HSCT that uses twotrapezoids. As shown in Fig. 11, each trapezoid isdenned by the root chord (Cr), tip chord (Ct), span(6), and sweep angle (A). From these values, otherplanform parameters such as area (A), aspect ratio

), and taper ratio (A), are defined:

A=-(Cr •Ct),b2-, = £ (18)

The FFD algorithm described by Sederberg andParry15 is ideal for deforming the polygonal models.

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Span, b

Fig. 11 Planform definition.


Figure 12 shows a general parallelepiped defined bya set of control points forming three primary edgesor directions along £, 77, and £. The relation for aparallelepiped is defined as

(19)

where O is the origin of the parallelepiped, and ?\, n^,and n^ are the unit vectors along the parallelepipedprimary edges in the £, 77, and C directions, respec-tively. Equation (19) defines a mapping betweenthe deformation object (parallelepiped) and the gridpoints. The grid points, f, are mapped to the coordi-nates of the parallelepiped, £, 77, and £, as

x nc • (r - PQ)

(f — PQ)Fig. 12 Parallelepiped volume for FFD.

C =

nf x rif • (n,,)fit: x n,, • (f - PQ)

fif. x n,, • (r\)

(20)

Fig. 13 NURBS volume for FFD.

Like other SOA algorithms, this algorithm maintainsthe polygon connectivity, and the deformation is ap-plied only to the vertices of the model. The FFDprocess is similar to embedding the grid inside a blockof clear, flexible plastic (deformation object) so that,as the plastic is deformed, the grid is deformed as well.Deformation of complex shapes may require severaldeformation objects. The shapes of these deformationobjects are not arbitrary. In fact, the shapes must bethree-dimensional parametric volumes and could rangefrom a parallelepiped as shown in Fig. 12 to a generalNURBS volume as shown in Fig. 13. The block isdeformed by perturbing the vertices that control theshape of the deformation block (e.g., corners of theparallelepiped). For parametric volume blocks, param-eters controlling the deformation are related throughthe mapping coordinates (£ ,?? ,C)- These coordinatesare used in both forward and backward mapping.

A grid point is inside the parallelepiped if 0 < £, 77, C <1.

The FFD technique based on the parallelepiped isvery efficient and easy to implement. This techniqueis suitable for local and global deformation. The onlydisadvantage is that the use of the parallelepiped lim-its the topology of deformation. To alleviate thisdisadvantage, Sederberg and Parry proposed to usenonparallelepiped objects.15 They also noted that theinverse mapping would be nonlinear and would requiresignificant computations.

Another popular method to define FFD is to usetrivariate parametric volumes. Sederberg and Parryused a Bezier volume.15 Coquillart at INRIA ex-tended the Bezier parallelepiped to nonparallelepipedcubic Bezier volume.19 This idea has been fur-ther generalized to NURBS volume by Lamousin andWaggenspack.20 The NURBS blocks are defined as

E Efc=0

i=0 j=0(21)

Lamousin and Waggenspack20 used multiple blocks tomodel complex shapes. This technique has been usedfor design and optimization by Yeh and Vance21 andalso by Perry and Balling.22

The common solid elements used in FE analysis(Fig. 14) can be used as deformation objects as well.

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Fig. 14 FE analysis solid elements.

The mapping from the solid element coordinates is de-fined23 by

(22)

where Ni are the FE basis functions and Qi are thenodal coordinates of deformation objects, which arerelated to the design variables. The equations for in-verse mapping are nonlinear for all solid elements withthe exception of tetrahedron solid elements. The solidelements provide a flexible environment in which to de-form any shape. Complex shapes may require the useof several solid elements to cover the entire domain.

To model the planform shape, we used hexahedronsolid elements with four opposing edges parallel to thez coordinate. Then, the planform design variableswere linked to the corners of the hexahedral elements.Figure 15 shows the initial and deformed models fora transport configuration. The solid lines representthe controlling hexahedron solid elements. The base-line model is on the left-hand side, and the deformedshape is on the right-hand side.

As with the camber and thickness algorithms, thesensitivity of grid point coordinates was independentof the design variables (P) and coordinates ( x , y , z ) .Thus, we needed to calculate the sensitivity only once,at the beginning of the optimization.

ImplementationFigure 16 shows the implementation diagram for the

combined algorithm. The implementation started witha CAD model that defined the geometry. The first twosteps were implemented in parallel. The first step wasto determine the number and the locations of the de-sign variables with the aid of the CAD model. In thesecond step, the grids were manually generated for allinvolved disciplines. In the third step, the forward andbackward mappings described in the previous sectionswere calculated for each grid point. In the fourth step,the new grid was deformed in response to the newdesign variables, and the sensitivity derivatives were

Defoimcd

Fig. 15 Planform deformation of a transport.

Slep 2:Create baselineanalysis models or^rids

Number andlocations of D.V.s

s.

Baseline analysismodels or grids

i.

Desijin variables

Fig. 16 MASSOUD Implementation diagram.

computed as well. The third and fourth steps werecompletely automated. The first three steps were con-sidered preprocessing steps and needed to be done onlyonce.

Parameterizing CSM ModelsParameterizing CFD and CSM models appears to

be similar in nature, but the CSM model parameter-ization has two additional requirements. First, theCSM model parameterization must include not onlythe OML but also the internal structural elements,

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such as spars and ribs. Second, the deformed CSMmodel must be a valid design. For example, the sparsmust stay straight during the optimization. The algo-rithms presented in this paper can easily handle thefirst requirement. However, if the planform designvariables are not selected with care, the second re-quirement could easily be violated. To avoid creatingan invalid CSM model, the model must be parame-terized with few hexahedron solid elements, and thoseused must be aligned with major structural compo-nents such as spars and ribs.

Sensitivity AnalysisSensitivity derivatives are defined as the derivatives

of the coordinate locations with respect to the designvariables. The previous sections present a formula-tion for shape parameterization based on a specificset of design variables (£>,-, i = l , im ax)- It is pos-sible to introduce a new set of design variables (u>j,j = l,jmax). The sensitivity derivatives with respectto Wj were computed based on the chain rule differen-tiation as

OR OR(23)

The previous sections provide techniques to computethe first term on the right-hand side. The second termis defined in a matrix form where the matrix has ?max

rows and jmax columns.

du>2

(24)

Design-Variable SequencingIn a typical optimization problem, the number of

design variables is determined a priori. However, theNURBS representation algorithms permits the use ofan adaptive algorithm to determine the number of de-sign variables. The design variables are control pointsof a NURBS curve or surface. Optimization of a wingsection could start with as few as three design variables(see Fig. 17). Then, the number of design variables canbe increased to five by enriching the NURBS curve orsurface. Enrichment is accomplished by inserting ad-ditional knots into the design NURBS curve or surface.

This method is similar to mesh sequencing andmultigrid methods used in CFD to accelerate the con-vergence. Multigrid methods exhibit a convergencerate independent of the number of unknowns in thediscretized system.24 Using MASSOUD for design-variable sequencing needs to be investigated further.

3DVs

5DVs

9DVs

17DVs

33DVs

Fig. 17 A sequence of design-variable sets.

Deformed

Baseline

Fig. 18 Baseline and deformed CSM models of anHSCT.

Results and ConclusionsThe algorithms presented in this paper have been

applied for parameterizing a simple wing, a blendedwing body, and several HSCT configurations. Fig-ure 18 shows the baseline and deformed CSM grids foran HSCT. The solid lines represent the initial positionof the hexahedron solid elements controlling the plan-form variation. The parameterization results from thisresearch have been successfully implemented for aero-dynamic shape optimization with analytical sensitivityfor structured6 and unstructured8 CFD grids.

An aerodynamic optimization of an ONERA M6wing was performed6 using a sequential linear pro-gramming technique. The objective of the optimiza-tion was to minimize the drag while maintaining thesame lift as the baseline design. Figure 19 shows thedesign cycle history for both lift and drag. In thisoptimization, the angle of attack is fixed, and it wasfound that in order to move away from the currentdesign, the constraint on the lift coefficient had to berelaxed temporarily. This is shown clearly in the fig-ure: for the first 19 design cycles, CL was allowed todeviate by up to 0.01 from the desired value. Afterdesign cycle 19 the tolerance on the lift constraint wastightened to 10~6. The net result was approximately

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0.300

0.290

0.280

0.270

0.260

0.250

CL constraint = 0.29176 +/-1 .e-6

CL constraint = 0.29176 +/-1 .e-2

10 15 20

Design Cycle25

0.013

0.012

0.011

0.010

0.009

0.0085 10 15 20

Design Cycle25

Fig. 19 Design cycle history of ONERA M6 wingoptimization for coefficient of drag.

29 counts of drag reduction at the baseline lift. Fig-ure 20 shows comparisons of the solutions computedon the baseline and final designs. The results indicatea significant reduction in the shock strength at mostspanwise stations.

This approach has also been applied to multidisci-plinary optimization of a HSCT.9>1° Figure 21 showsthe design cycle history for aircraft drag, as measuredrelative to the baseline values. The figure shows thedrag has been reduced by 7.5 % relative to the base-line. Although the optimizer has not fully convergedfor this case, the convergence history from 20 designcycles suggests that little additional reduction in dragwould be obtained from additional design cycles. Fig-ure 21 shows a comparison of the baseline and finalsurface pressures on both the upper and lower sur-faces of a HSCT. The planform changes that occurredbetween the initial and final design cycles are also rep-

Flnal DesignC0 = 0:00891

Baseline Design= 0.01185

Fig. 20 Comparison of the M6 wing final designand baseline surface pressures.

1.05

ri.oo

ua

Q 0.95

0.90

7.5% reduction

10 20Design Cycle

30

Fig. 21 Design cycle history of HSCT optimizationfor drag.

resented. The primary effect on the planform has beento increase the span and aspect ratio slightly and tomove the outer wing leading edge break to a more in-board spanwise location. Although not evident in thefigure, the wing thickness has been slightly reduced.

The parameterization algorithm presented in thispaper is easy to implement for an MDO applicationfor a complex configuration. The resulting parame-terization is consistent across all disciplines. Becausethe formulation is based on the SOA algorithms, theanalytical sensitivity is also readily computed. Thealgorithms are based on parameterizing the shapeperturbations, thus enabling the parameterization ofcomplex existing analysis models (grids). Anotherbenefit of parameterizing the shape perturbation isthat the process requires few design variables. Use ofNURBS representation provides strong local control,and smoothness can easily be controlled.

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Upper surface

Fig. 22 Comparison of the HSCT final design andbaseline surface pressures.

AcknowledgmentsThe author would like to thank Drs. Perry Newman

and James Townsend of Multidisciplinary Optimiza-tion Branch at NASA Langley Research Center andSusan Costello of NCI Information Systems for theirassistance with this paper. The author would also liketo thank Dr. Robert Biedron for providing figures 19and 20 and Joanne Walsh for providing figures 21 and22.

References1Samareh, J. A., "Status and Future of Geometry Modeling

and Grid Generation for Design and Optimization," Journal ofAircraft, Vol. 36, No. 1, 1999, pp. 97-104.

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4Samareh, J. A., "A Survey of Shape ParameterizationTechniques," AIAA/CEAS/ICASE/NASA-LaRC InternationalForum on Aeroelasticity and Structural Dynamics Conference,Williamsburg, Virginia, 1999, pp. 333-343.

5Watt, A., and Watt, M., Advanced Animation and Ren-dering Techniques, Addison-Wesley Publishing Company, NewYork, 1992.

6Biedron, R. T., Samareh, J. A., and Green, L. L., "Par-allel Computation of Sensitivity Derivatives With Applicationto Aerodynamic Optimization of a Wing," 1998 ComputationalAerosciences Workshop, NASA CP-20857, Jan. 1999, pp. 219-224.

7Green, L. L., Weston, R. P., Salas, A. O., Samareh, J. A.,Townsend, J. C., and Walsh, J. L., "Engineering Overview ofa Multidisciplinary HSCT Design Framework Using Medium-Fidelity Analysis Codes," 1998 Computational AerosciencesWorkshop, NASA CP-20857, Jan. 1999, pp. 133-134.

8Nielsen, E. J., and Anderson, W. K., "AerodynamicDesignOptimization on Unstructured Meshes Using the Navier-StokesEquations," 7th AIAA/USAF/NASA/ISSMO Symposium onMultidisciplinary Analysis and Optimization Proceedings, Sept.1998, pp. 825-837, also AIAA-98-4809-CP.

9 Walsh, J. L., Townsend, J. C., Salas, A. 0., Samareh, J. A.,Mukhopadhyay, V., and Barthelemy, J. F., "MultidisciplinaryHigh-Fidelity Analysis and Optimization of Aerospace Vehicles,Part 1: Formulation," Paper AIAA-2000-0418, Jan. 2000.

10Walsh, J. L., Weston, R. P., Samareh, J. A., Mason, B. H.,Green, L. L., and Biedron, R. T., "Multidisciplinary High-Fidelity Analysis and Optimization of Aerospace Vehicles, Part2: Preliminary Results," Paper AIAA-2000-0419, Jan. 2000.

11 Hicks, R. M., and Henne, P. A., "Wing Design by Numer-ical Optimization," Journal of Aircraft, Vol. 15, No. 7, 1978,pp. 407-412.

12Cosentino, G. B., and Hoist, T. L., "Numerical Optimiza-tion Design of Advanced Transonic Wing Configurations," Jour-nal of Aircraft, Vol. 23, No. 3, 1986, pp. 193-199.

13Hall, V., "Morphingin 2-D and 3-D," Dr. Dobb's Journal,Vol. 18, No. 7, 1993, pp. 18-26.

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15Sederberg, T. W., and Parry, S. R., "Free-Form Deforma-tion of Solid Geometric Models," Computer Graphics, Vol. 20,No. 4, 1986, pp. 151-160.

16 Abbott, I. A., and Von Doenhoff, A. E., Theory of WingSections, Dover Publications, New York, 1959.

17Farin, G., Curves and Surfaces for Computer Aided Geo-metric Design, Academic Press, New York, 1990.

18Sederberg,T. W., and Greenwood, E., "A Physically BasedApproach to 2-D Shape Blending," Computer Graphics, Vol. 26,No. 2, 1992, pp. 25-34.

19Coquillart, S., "Extended Free-Form Deformation: ASculpturing Tool for 3D Geometric Modeling," SIGGRAPH,Vol. 24, No. 4, 1990, pp. 187-196.

20Lamousin, H. J., and Waggenspack, W. N., "NURBS-Based Free-Form Deformation," IEEE Computer Graphics andApplications, Vol. 14, No. 6, 1994, pp. 95-108.

21Yeh, T.-P., and Vance, J. M., "Applying Virtual Real-ity Techniques to Sensitivity-Based Structural Shape Design,"Proceedings of 1997 ASME Design Engineering Technical Con-ferences, No. DAC-3765 in DETC97, Sept. 1997, pp. 1-9.

22Perry, E., and Balling, R., "A New Morphing Method forShape Optimization," Paper AIAA-98-2896, June 1998.

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Hirsch, C., Numerical Computation of Internal and Exter-nal Flows, John Wiley fc Sons, New York, 1988.

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