Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc.

AIAA Meeting Papers on Disc, 1996, pp. 88-98A9638712, AIAA Paper 96-3991

Use of CAD geometry in MDO

Jamshid A. SamarehComputer Sciences Corp., Hampton, VA

AIAA, NASA, and ISSMO, Symposium on Multidisciplinary Analysis and Optimization,

6th, Bellevue, WA, Sept. 4-6, 1996, Technical Papers. Pt. 1 (A96-38701 10-31), Reston,

VA, American Institute of Aeronautics and Astronautics, 1996, p. 88-98

The purpose of this paper is to discuss the use of Computer-Aided Design (CAD) geometry in a MultiDisciplinaryDesign Optimization (MDO) environment. Two techniques are presented to facilitate the use of CAD geometry bydifferent disciplines, such as CFD and Computational Structural Mechanics (CSM). One method is to transfer theload from a CFD grid to a CSM grid. The second method is to update the CAD geometry for CSM deflection.(Author)

Page 1

AIAA-96-3991-CP96-3991

A96-38712

USE OF CAD GEOMETRY IN MDOJamshid A. Samareht

1 Abstract

The purpose of this paper is to discuss the use ofComputer-Aided Design (CAD) geometry in a Multi-Disciplinary Design Optimization (MDO) environ-ment. Two techniques are presented to facilitate theuse of CAD geometry by different disciplines, such asComputational Fluid Dynamics (CFD) and Compu-tational Structural Mechanics (CSM). One method isto transfer the load from a CFD grid to a CSM grid.The second method is to update the CAD geometryfor CSM deflection.

2 Introduction

The process of aircraft design can be broken intothree phases[?]: (1) conceptual design, (2) prelimi-nary design, and (3) detail design. The conceptualdesign process focuses on the basic design optimiza-tion of features, such as weights, sizes, and overallperformance. During the preliminary design, the fo-cus is on the mathematical modeling of the outsideskin of an aircraft with sufficient accuracy. After thisphase, the geometry is frozen, and any change couldbe costly. Detail design concentrates on the actualdesign of pieces to be fabricated.

Often an aircraft is represented by a simple modelduring the conceptual and preliminary designs. Be-cause simple models are neither accurate nor com-plete, optimization of these models could lead to animpractical design [?, ?]. This shortcoming can bealleviated by using a high fidelity model, and the in-teraction among various disciplines must be modeledaccurately. These interactions are very complicatedand important piece of MDO.

The strong interactions of CSM and CFD are verycommon in an MDO environment. Such interactionscan prompt physically important phenomena such asthose occurring in aircraft due to aeroelasticity. Cor-

rect modeling of these complex aeroelastic phenom-ena requires direct coupling of CSM and CFD. Dur-ing optimization of a flexible structure (e.g., wing),the geometry changes due to the aeroelastic effect.All disciplines share the same geometry, and must beable to consistently communicate and share informa-tion (e.g. deflection and load). The geometry repre-sentation for MDO must be accurate and suitable forgrid generation for various disciplines such as CFDand CSM. To further complicate the process, model-ing of complicated geometrical models requires use ofCAD systems.

The interactions among various disciplines requirethe manipulation of the original CAD geometry thatis stored as a set of NonUniform Rational B-Splines(NURBS). This paper describes two techniques tomanipulate the NURBS geometry. In the next fol-lowing sections there are brief discussions on NURBS,parameterization of aircraft geometry, NURBS-basedoptimization, load transfer, deflection transfer, re-sults, and conclusions.

3 NURBS

This section contains a brief overview of the NURBS,and readers should consult [?] for a detailed discus-sion. A NURBS curve, R(u), can be represented as

(1)

*This paper is declared a work of the U. S. Government andis not subjected to copyright protection in the United States.(ftp://techreports.larc.nasa.gov/pub/techreports/larc/96)

'Senior Computer Scientist (j.a.samareh@larc.nasa.gov),Computer Sciences Corporation, Geometry Laboratory (GE-OLAB, http://geolab5.larc.nasa.gov)

The parameter, u, is bounded by wmjn < « < umax.The Pi are the control points (forming a control poly-gon), and Wi are the weights. The Ni:p are the p-thdegree B-spline basis functions defined on the non-periodic and nonuniform knot vector (w)

(2)where k is the number of knots. This completesthe mapping between the one-dimensional parameterspace, it, and the three-dimensional Euclidean space,R. A NURBS curve has five important properties:

• It is invariant under linear transformation.

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• A NURBS curve of order p, having no multipleinterior knots, is p - 2 difFerentialable.

• The approximation is local in nature..

• A NURBS curve is contained in the convex hullof its control points.

• The NURBS approximation is variation dimin-ishing.

To evaluate, the three-dimensional curve NURBS iscommonly represented in homogeneous form as

R(X,Y,Z)-t=^-Rw(WX,WY,WZ,W). (3)

So, the NURBS curve can conveniently be defined asa perspective map of its nonrational counterpart infour-dimensional space as

1=0

where P™ is defined as

Pf = {XiWi,YiWitZW, Wi}. (5)

The basis functions can be efficiently computed byusing DeBoor algorithm [?] as

if «,• < u < «,+i,w,- < M,-+i, ,g,otherwise ^ ^

Ntj,(u) =

where

Li(u) =

_ Uj+p+1 - U

p-i(ti), (7)

(8)

(9)

It is agreed that 0/0 = 0.A NURBS surface is a parametric surface and is

defined as a function of two parameters as

= {X(U),Y(U),Z(U)}

where the components of vector, U, are the sur-face parameters and have no geometrical significance.However, for a constant U2, as «i increases the point,R(U) moves always from one side of the surface tothe other side. The NURBS definition for the surfaceis defined as

where Pij are control points (forming a control sur-face), Wij are the weights, and AT,-ip and Nj>q arethe p-th and q-th degree B-spline basis functions de-fined on the non-periodic and nonuniform knot vec-tor. The evaluation process is very similar to theNURBS curve evaluation.

4 Aircraft Parameterization

To use complex shapes in an MDO environment, theparameterization and geometry modeling must becompatible with existing CAD systems, and it mustbe adaptable to CFD (i.e., block-structured or un-structured grids) and CSM. The four approaches forparameterization of an aircraft geometry are basedon: analytical, semi-analytical, discrete, and CADrepresentation.

Analytical methods for optimization have beenused for a long time. These methods converts a setof design variables (e.g., wing sweep, thickness ra-tios) into a set of surfaces. Then, these surfaces canbe used to analyze and compute the objective func-tion (e.g., [?, ?, ?]). This approach is very simple andrequires a few design variables. The geometry basedon this approach is very smooth. On the other hand,the geometry can take a limited form, and it is hardto optimize existing and free-form geometry. Inter-actions among disciplines are very difficult to model.Blair and Reich [?] have implemented a Virtual De-sign Process (VDM) that is integrated with full as-sociativity within Pro/Engineer CAD/CAM software[?]•

In the second approach, semi-analytical, a set ofpoints can describe the initial geometry, and a poly-nomial can model the perturbation of geometry [?].Then, the coefficients of this polynomial are used as aset of design variables. Again, this method is simple,and it allows the designer to use existing geometry.This approach requires a few design variables, and thesmoothness of the geometry depends on the baselinegeometry. This approach is very difficult to general-ize and use in an MDO environment for a complexgeometry.

The third approach, discrete, is based on a discreterepresentation of the geometry. The baseline geom-etry creates the grids, and the position of each gridpoint becomes a design variable for the optimizer.

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This is very easy to implement, and the geometrychanges don't have a limited form. The latter couldcreate a problem in which the optimum design maynot be practical to manufacture. The number of de-sign variables often becomes very expensive which,leads to high costs and a difficult optimization prob-lem to solve. Also it is difficult to maintain a smoothgeometry.

The fourth approach, CAD, is based on the op-timization of a NURBS representation of geometry(e.g., [?]). The large number of design variables of-ten becomes very expensive which leads to high costsand a difficult optimization problem to solve. Thegeometry continuity and smoothness are guaranteed.Also, the geometry can change locally without affect-ing everything else. This type of parameterization isflexible enough to represent a wide range of geome-tries. Existing complicated CAD models can be usedas the baseline model, but modeling the interactionamong disciplines is very difficult.

These four approaches are summarized and listedin Table 1. In the next section, a NURBS-based opti-mization that is a hybrid approach based on first andfourth approaches is discussed.

5 NURBS-BasedOptimization

CAD systems have been developed very rapidly andintegrated into the design process(e.g., see [?]). Use ofCAD systems for geometry modeling in an MDQ en-vironment could potentially save development time.However, there are two drawbacks: (1) initial invest-ment (software and training), and (2) inability tocalculate analytical sensitivity. The geometry rep-resentation in these systems is complicated due toa large number of entity representations. In a tra-ditional CAD system, the geometry is representedas one of many possible mathematical forms such asBezier, Coons patch, B-Spline, surface of revolution,etc. However, one can use NURBS equations to rep-resent most parametric and implicit surfaces withoutloss of accuracy [?]. NURBS can represent quadricprimitives (e.g., cylinder, cones), as well as free formgeometry [?]. There are some CAD surfaces (e.g.,helix and helicoidal[?j) that cannot be directly con-verted to NURBS. These surfaces are not common inmost CAD system. The NURBS representation mustbe used in such as a way that it should be possible toaltered it automatically to accommodate the changesdue to: (1) design variable changes, or (2) structural

and control surface deflections.Calculating of the sensitivity of geometry with re-

spect to the design variables could prove to be verydifficult. In some instances, it is possible to relatethe NURBS control points to the design variables.Then the analytical sensitivity can be calculated out-side of the CAD system. Another way to calculatethe sensitivity is to use finite difference, as long asthe perturbed geometry has the same topology as theunperturbed one. Both methods, the analytical andfinite difference, have their pitfalls and limitations.

Implementation described here is based on theFramework for Interdisciplinary Design Optimization(FIDO) [?, ?] developed at NASA Langley ResearchCenter (LaRC). The process for geometry creation,integration, and manipulation are designed aroundthe NURBS representation of the complete geometry.To embed this process into an optimization processsuch as FIDO, the model (High-Speed Civil Trans-port) has to be parameterized with a set design vari-ables (DVs).

For each optimization cycle, the run starts witha definition for a set of design variables (see Fig-ure ??), to build a NURBS-based geometry. Pro-Engineer ([?]) is proposed for this step as the geome-try builder. Once the geometry is built, the NURBSgeometry will be deposited into a NURBS database,which will be shared among different disciplines. Thisdatabase will be maintained as the baseline geome-try changes. During the optimization process, CFDand CSM disciplines will need the complete geometrydefinition.

For each iteration of the aeroelastic loop, the CADgeometry is used to create CFD and CSM grids. Thisrequires that both disciplines use and modify thesame unique geometry in the geometry database. Ineach loop, the CFD grid is used to compute the aero-dynamic load, which is transferred to the CSM grid.This load will be converted to a NURBS definitionand deposited into the database. At this point, theCSM grid will be generated based on the NURBSdatabase. The NURBS CFD load will be mapped tothe CSM grid, which along with the CSM grid is usedto compute the resulting deflection. The next criti-cal step is to modify the original CAD geometry toaccommodate and reflect the aeroelastic deflection.

Each optimization loop can be summarized asshown in Figure ??:

• 1. The CAD system, Pro/Engineer, will converta set of design variables (D) into a NURBS geom-etry (G). Because the source for Pro/Engineer

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code is not available, the sensitivity has to becalculated outside of the system.

• 2. At this point, the geometry, G, and itssensitivity, GD, will be stored in the NURBSdatabase.

• 3. The Coordinates and Sensitivity Calcula-tor for Multidisciplinary Design Optimization(CSCMDO) ([?]) will calculate the CFD meshand its sensitivity M, MD.

• 4. The CFD code, with the capability to calcu-late the sensitivity, will obtain the load, whichwill be stored in the NURBS database.

• 5. Similarly, the CSM grid generator can cre-ate the finite element grid and its sensitivity.The load stored in the NURBS database will bemapped onto the finite element model.

• 6. The CSM code, with the capability to cal-culate the sensitivity, will obtain the deflectionwhich will be stored in the NURBS database.

• 2'. The final step is to modify the NURBSdatabase to accommodate the new structural de-flection.

The two steps that are defined and discussed inthis paper in detail are: (1) converting CFD loadto a NURBS definition, and (2) modify the NURBSgeometry to accommodate the deflection.

6 Load Transfer

To transfer the load to CSM elements, one needs tobe able to compute the load at any point on the sur-face. Initially, the load, F(K)cFD, is computed onthe CFD grid, (RCFD - {X, Y, Z}T), which could bea set of structured quadrangles or unstructured tri-angles with the appropriate connectivity. There aretwo basic problems in fitting this data with NURBS:(1) the data has four dimensions, (X,Y,Z,F), and(2) the CFD grid could be an unstructured grid.

The first problem can be solved by mapping theCFD grid to the original NURBS surfaces, hence,reducing the dimension from four, (X, Y, Z, F), tothree, (U, V, F). The U, V are the parametric coor-dinates of the original NURBS surface. This infor-mation may be available from a CFD grid generationprocess. If not, the CFD grid points can be projectedonto the original NURBS surface [?]. The processof projecting a point, f = {X, Y, Z}T, on a surface,

R(U), can be performed by finding a w such that thedistance, d, between the F and R(w) is minimal andw is constrained to G [ ( a , b ) , ( c , d ) ] . The distance, d,can be written in terms of parameters w as

d2(w) = f(w) = \R(w)-f\-\R(w)-f\. (12)

The next step is to fitting a single-value three-dimensional surface, F — F(U, V). This surface canbe fitted based on a least-squares approximation [?, ?]that minimizes the approximation error. A three-dimensional curve is used as an example of the least-squares fitting.

A set of points in three space, r(u), can be fittedby a B-Spline curve, R(u). The B-Spline equationcan be expressed at each parameter, u, as

The above equation can be expressed as

[r\ =

where

[f\T = , 2)

#„,*(«!)

(13)

(14)

(15)

(16)

(17)

where jmax is the maximum number of data points,FJ. Rn is the nth B-Spline control points. If jmax —n, the matrix [TV] is a square matrix and the controlpoints can be calculated directly by matrix inversion,

[R] = [N]-1^. (18)

In this case, the resulting B-Spline curve passesthrough each data point. However, if the numberof data points, jmax, is greater than the number ofcontrol points, n, the problem is over-specified. Aleast-square method can solve the problem as,

[R] = [[N}T[N]]-i[N}T[r\. (19)

The least-squares approximation for surfaces isvery similar to the least-squares approximation forthe curves. The minimization error can be written as

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8 Results and DiscussionsError =

pEN = s: z(20)

where F,-j are control points for the NURBS sur-face representing the CFD load, Wij are the weights,and NiiF and Njit are the p-th and q-th degree B-spline basis functions defined on the non-periodic andnonuniform knot vector. N is the number of pointsin the CSM grid. This forms a system of linearequations that can be solved for control points of aNURBS surface representing the load.

7 Deflection Transfer

As described in the previous section, the load is de-fined on the CFD grid, RCFD = { X , Y, Z}T. On theother hand, the deflection, A/fcsMi is defined on theCSM grid, RCSM — {X, Y, Z}T, which is representedby a set of polygons (e.g., triangles and quadrangles)with appropriate connectivity. The goal is to modifythe CAD geometry definition, R(U), such that it re-flects the deflection produced by CSM. The algorithmfor deflection transfer has four steps:

1. Project the RCSM onto the original NURBS sur-face.

2. Create a NURBS surface based on the deflection,&RCSM(U, V), which has the same degree as theoriginal NURBS surface.

3. Add/remove knots from the new surface to makeit compatible with the original NURBS surface.

4. Add the control points to the original NURBSsurface to form the new surface.

This algorithm has following properties:

1. As A.RCSM approaches zero, the method will re-produce the original NURBS surface, R(U).

2. Smoothness is controlled on the resultingNURBS surface.

3. The results surface is a NURBS surface with thesame degree as the original NURBS surface.

The results are presented for the load and deflectiontransfers. For the load transfer, a generic pathfindergeometry (see Figure ??) is used. The geometry isa single cubic NURBS surface with 53 by 24 controlpoints. There are two test cases for the load trans-fer. For the first test case, a Sin function is definedover a triangular mesh (see Figure ??), which cov-ers the surface of the pathfinder. The data is fittedwith a NURBS surface. The original and interpolatedcontours are shown in Figure 11. The Root MeanSquares (RMS) error for interpolation is less than twopercent, and the resulting cubic NURBS surface has15 by 15 control points. For the second test case, thepressure distribution on the surface is fitted with acubic NURBS surface (see Figure ??). The result-ing NURBS surface has 35 by 24 control points, andthe RMS error is one percent. Figure 11 shows aclose-up view near the inboard leading edge.

The next two test cases demonstrate the resultsof deflection transfer. A generic High-Speed CivilTransport (IISCT) geometry is used for both testcases. This geometry is made of three surfaces:fuselage, inboard wing, and outboard wing. Figure11 shows the original NURBS surface, the deflectedCSM grid, and the deflected NURBS surface. Totest the limits of the deflection-transfer algorithm,the CSM grid has a large and unrealistic deflection.Figure 11 shows a side view of the same test case.For the last test case, the HSCT model was deflectedby using a Sin function. Figure 11 shows the originalNURBS surface, the deflected CSM grid, and the de-flected NURBS surface. Figure 11 shows a side viewof the same test case. The resulting NURBS surfaceshave an RMS error of less than one percent. For thelast two test cases, even though the inboard and out-board were fitted separately, no gaps or overlaps wereobserved. But this could be a potential problem.

Using the techniques described here, it is possibleto include CAD geometry within an MDO environ-ment with strong interdisciplinary interactions. Thepotential drawbacks for these algorithms are: (1) cal-culating sensitivity, and (2) potential gaps and over-laps among multiple surfaces. Both need to be stud-ied further.

9 Acknowledgments

4. It is possible to maintain the same knot vector The author would like to thank Jill Vaden for proof-as the original NURBS surface. reading this paper on such a short notice.

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References

[I] Raymer, D.P. "Aircraft Design: A ConceptualApproach," AIAA, AIAA Educational Series.Washington B.C., 1989.

[2] Aidala, P.V. Davis, W.H. Mason, W.H., "SmartAerodynamic Optimization," AIAA Paper 83-1863, 1983.

[3] Hutchison, M.G., Huang, X., Mason, W.H.,Haftka, R.T., and Grossman, B. "Variable-Complexity Aerodynamic-Siructural Design of aHigh-Speed Civil Transport Wing," AIAA-92-4695, September 1992.

[4] Farin, G. "Curves and Surfaces for ComputerAided Geometric Design, " Academic Press, 1990.

[5] DeBoor,;; C. "On Calculating with B-SpUnes"Journal of Approximation Theory, no. 6, pp. 50-72- 1972. -.

[6] Bloor, M.I.G., Wilson, M.J. "Efficient Parame-terization of Generic Aircraft Geometry," Journalof Aircraft, vol. 32, no. 6, pp. 1269-1275, Decem-ber 1995.

[7] Smith, R. E., Bloor M.I.G., Wilson, M. J.,and Thomas, A.T. "Rapid Airplane Paramet-ric Input Design (RAPID)," Proceedings of 12thAIAA Computational Fluid Dynamics Confer-ence, AIAA 95-1687, San Diego, CA, June 1995.(ftp://techreports.larc.nasa.gov/pub/techreports/larc/95/NASA-aiaa-95-1687.ps.Z)

[8] Blair, M., Reich, G. "A Demonstration ofCAD/CAM/CAE in a Fully AssociativeAerospace Design Environment, " AIAA Paper96-1630, April 1996.

[9] Pro/Engineer is a licensed product of ParametricTechnology Corporation, (http://www.ptc.com)

[10] Hicks, R.M., Henne, P.A. "Wing Design by Nu-merical Optimization," Journal of Aircraft, vol.15, no. 7, July 1978.

[II] Braibant, V., Fleury, C. "Shape Optimal DesignUsing B-Spline Design, " Computer Methods inApplied Mechanics and Engineering, vol. 44, pp.247-267, 1984.

[12] Letcher, J.S., Shook, M. "NURBS ConsideredHarmful for Gridding (Alternative Offered)," 4thInternational Meshing Roundtable, pp. 253-264,Oct. 16-17, 1995.

[13] Townsend, J.C., Weston, R.P., and Bid-son, T.M. "A Programming Environment forDistributed Complex Computing. An Overviewof the Framework for Interdisciplinary De-sign Optimization (FIDO) Project, " NASATM 109058, December 1993. (http://hpccp-www.larc.nasa.gov/ fido/homepage.html)

[14] Sistla, R., Krishnan, R., Dovi, A. "A Computa-tional Framework for HSCT Design, " the NASAComputational Aero-sciences Workshop, August13-15, Moffett Field, CA, 1996.

[15] Jones, W. T., Samareh, J. "A Grid GenerationSystem for Multi-disciplinary Design Optimiza-tion," AIAA Paper 95-1689, 1995.

[16] Samareh-Abolhassani, Jamshid, "UnstructuredGrid on NURBS Surfaces," AIAA-Paper 93-3454,1993.(ftp://techreports.larc.nasa.gov/rJSib/techreports/larc/93/NASA-aiaa-93-3454.ps.Z)

[17] Hayes, J.G., Halliday, J. "The Least-Squares Fit-ting of Cubic Spline Surfaces to General DataSets, " Journal of Institute of Applied Mathemat-ics and Applications, no. 14, pp. 89-103, 1974.

[18] Rogers,D.F., Fog, N. G. "Constrained B-SplineCurve and Surface Fitting," Computer-Aided De-sign, vol. 21, no. 10, pp. 641-648, December 1989.

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i(D) 1 (1)

GeometryBuilder(Pro)

(G) (2)

(M

CSM GridGenerator

1. MD) 1 (7)

|CSM -

-1 (G, GD)

(6)

r NURBS ' (G'GD)»,Data Base (3) ^

I

(d, dD)

(8)

nCFD GridGenerator

{ (M. MD)

(L, LD) _J(5)

(4)

r

ur-u

Figure 1: NURBS Based Optimization

CriteriaDesign VariablesLevel of Complex GeometryDiscipline InteractionSmoothness of GeometryShape LimitationLocal ControlAnalytical Sensitivity

AnalyticalFewLowDifficultEasyVery LimitedNoneEasy

Semi- AnalyticalFewLowDifficultEasyLimitedSomeEasy

DiscreteManyMediumFairDifficultNo LimitsGoodDifficult

CADManyHighDifficultEasyNo LimitsExcellentVery Difficult

Table 1: Summary of Parameterization Techniques

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Figure 2: A Generic Pathfinder

Figure 3: Triangular Mesh for the Generic Pathfinder

Original Function Interpolated Function

Figure 4: Transfer of Sin Function for a Generic Pathfinder

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Original Function Interpolated Function

Figure 5: Load Transfer for a Generic Pathfinder

Original Function Interpolated Function

Figure 6: Load Transfer for a Generic Pathfinder (Close Up)

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Deflected NURBS Surf ace

Original NURBS Surface Original NURBS Surface

Figure 7: Deflection of a Generic HSCT Configurations (Front View)

Original NURBS Surface

Original NURBS Surface

Figure 8: Deflection of a Generic HSCT Configurations

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Deflected CSM Grid Deflected NURBS Surface

Original NURBS Surface Original NURBS Surface

Figure 9: Deflection of a Generic IISCT Configurations Based on a Large Deflection (Front View)

Original NURBS Surface

Deflected NURBS Surface

Original NURBS Surface

Figure 10: Deflection of a Generic HSCT Configurations Based on a Large Deflection

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