NASA Technical Memorandum 100538 

. 

i 

STRUCTURAL SHAPE OPTIMIZATION IN 

M U L T I D I S C I P L I N A R Y S Y S T E M S Y N T H E S I S 

Jaroslaw Sobieszczanski-Sobieski 

[lASA-Ttl-100538) STRUCTURAL SHAPE N86-196 50 

OPTIIIZBTION IN BULTlDISCIPLINABY SYSTEM 

SPlTEESIS (NASA) 10 p CSCL 20K 

Unclas 

63/37 0130326 

January 1988 

N;ilioniil Aeronirutics and 

Sprrx: ArJminislration 

Langley Research Center 

Hampton, Virginia 23665-5225 

.. 
STRUCTURAL SHAPE OPTIMIZATION IN MULTIDISCIPLINARY SYSTEM SYNTHESIS 

Jaroslaw Sobieszczanski-Sobieski 

NASA Langley Research Center, MS246 

Hampton, Virginia 23669, U.S.A. 

ABSTRACT. Structural shape optimization couples with other discipline 

optimization in the design of complex engineering systems. For 

instance, the wing structural weight and elastic deformations couple to 

aerodynamic loads and aircraft performance through drag. This coupling 

makes structural shape optimization a subtask in the overall vehicle 

synthesis. Decomposition methods for optimization and sensitivity 

analysis allow the specialized disciplinary methods to be used while the 

disciplines are temporarily decoupled, after which the interdisciplinary 

couplings are restored at the system level. Application of decomposition 

methods to structures-aerodynamics coupling in aircraft is outlined and 

illustrated with a numerical example of a transport aircraft. It is 

concluded that these methods may integrate structural and aerodynamic 

shape optimizations with the unified objective of the maximum aircraft 

performance . 

Nomenclature. 

acce(Y,,X) - 

F(X) - 

G(X) - 

Jash1- 

MsuU(X) - 

ui - 

xYi - 

Introduction. 

subscript referring to aerodynamics. 

airfoil chord. 

cumulative constraint. 

vector of equality constraints. 

system objective function. 

vector of system inequality constraints. 

airfoil depth. 

J(Ua,Us), jacobian matrix of partial derivatives of Ua 

with respect to Us. 

subscript indicating lower bound. 

Mach number. 

subscript referring to structures. 

subscript indicating upper bound. 

vector of system behavior variables. 

vector of behavior variables output from analysis of the 

i-th discipline. 

vector of system design variables. 

vector of i-th subsystem design variables. 

Structural optimization for shape has a long history of theoretical 

developments and successful applications, e.g., ref.1. It encompasses 

both the overall geometry of assembled structures, e.g., ref.2 (Ch.4 and 

161, and the shape of individual components as in ref.1 (Ch.9). Its tool 

inventory includes a rich variety of methods ranging from the classical 
variational approaches reviewed in ref.2 (Ch.5) to sophisticated finite 

element-based techniques, e.g., ref.1 (Ch.5 and 6). These developments 

have reached the point where it becomes practical for an engineer to use 

shape optimization methods not only in design of a structure treated as 

an isolated object, but also in design of structure whose shape 

determines its interaction with the environment. 

where structures and the surrounding airflow interact through the shape 

of the structure boundary. Similar examples may also be drawn from 

naval architecture and automotive industry. Based on aeronautical 

experience, this paper presents the structural shape optimization as 

part of the overall aircraft system synthesis in which the structural 

discipline must interact with aerodynamics and other disciplines in 

search for the shape that maximizes the flight vehicle performance. 

Aircraft structural design deals with probably the most difficult cases 

Structures - Aerodynamics Coupling. 

EXAMPLES OF COUPLING 

One mechanism that couples structures and aerodynamics in near-sonic 

cruise aircraft is the wave drag which causes the wing drag to rise as 

the flight Mach number approaches unity. The drag rise shown in Fig.1 

shifts toward lower M and becomes steeper as the h/c ratio increases. On 

the other hand, the structural weight of a bending-dominated wing 

generally decreases with the increase of h/c. Consequently, a 

conventional shape optimization performed for minimum weight within the 

discip.line of structures would tend to increase h/c, while an 

aerodynamic optimization for minimum drag at a constant M would tend to 

lower h/c. However, both weight and drag are detrimental to flight 

performance, such as payload delivered over a given range for a constant 

gross weight at take-off. Thus, a compromise between the two 

conflicting trends must be sought, by formulating a unified optimization 

problem, drawing its objective function from the system performance, and 

including constraints from aerodynamics and structures. 

The optimization task is further complicated because the wing 

structural and aerodynamic behavior are coupled not only through the 

system performance but also directly via aerodynamic loads whose 

magnitude and distribution on the wing depend on the wing planform and 

airfoil shapes and on the wing structural deformations. Dependence of 

the deformations on the loads and wing shape completes the coupling. 

Swept wings provide a second example of the effect of this type of 

coupling on the metalic wing structural weight plotted in Fig.1 as a 

function of sweep angle. Lift in a forward-swept wing increases the 

streamline airfoil angle of attack, that generates more lift, and so on 

- a positive feedback that may result in a wing divergence. A 

supercritical airfoil wing is a third example of the structuresaerodynamic coupling, again, through the loads and deformations. As 

shown in Fig.2, the supercritical airfoil's lift resultant is shifted 

aft relative to that of a conventional airfoil. As a result, torque on 

the wing box is increased and this may require more structural material 

for strength, and also for stiffness necessary to keep the wing from 

twisting excessively which redistributes the lift inboard with an 

attendant increase of induced drag. 

METHODS FOR COUPLED, STRUCTURES - AERODYNAMIC OPTIMIZATION 

The structures-aerodynamics-performance interactions may be conveniently 

presented in the form of a directed graph (a means widely used in 

Operations Research (OR) system analysis, e.g., ref.3) shown in Fig.3. 

Each discipline on the graph represents a disciplinary analysis which 

2 
from a system's perspective is simply a black box transforming input 

into output. For the wing design, the inputs and outputs are inscribed 

by the arrows in the figure. Assuming that structure must fit inside of 

aerodynamic envelope, the external shape geometry is shown as an input 

from aerodynamics to structures. The loads and deformations add to the 

direct coupling of the two disciplines, the remainder of the coupling is 

through performance analysis. This representation of a wing as a system 

corresponds to the aspect decomposition defined in OR (ref.4) as one in 

which the system itself remains an indivisible object but each of its 

aspects (disciplines) is represented by a black box - as opposed to the 

object decomposition applicable when a black box representation may be 

assigned to each physically separable subsystems. Without 

decomposition, the structural shape optimization and aerodynamic design 

methods (e.g., ref.5) cannot be used to their full potential in design 

of a wing as a coupled system because neither design method accounts for 

the mutual influence of the two disciplines, and, therefore, the 

individual optimizations may work on cross-purposes. Further, 

optimization without decomposition requires repetition of the entire 

system analysis for every design variable perturbation (a 

multidimensional parametric study) to determine the full effect of the 

variables on the system performance. In contrast, the system 

decomposition approach temporarily decouples the disciplines and permits 

the application of specialized methods inside each black box. This is 

crucially important to being able to use the disciplinary advanced 

methods in design of coupled systems. 

System Optimization Algorithms. 

There are several ways the system optimization algorithm may be 

organized and the distinguishing element is usually the means of 

restoring the coupling to the temporarily decomposed system. Two 

recently developed algorithms are outlined. 

HIERARCHAL, TOP-DOWN DECOMPOSITION. 

When system Optimization by hierarchal decomposition (ref.6, 7, and 8) 

is applied, the design variables are divided into the system variables 

that directly affect aerodynamics, structures, and performance, and 

the local (subsystem) variables which directlx affect only aerodynamics, or structures. By these criteria, the wing planform and airfoil shape 

variables are system variables because they affect all three analyses. 

The structural weight is also a system variable since it enters the 

performance analysis. Amplitudes of basis functions representing the 

wing structural deformation are also variables at the system level due 

to the influence they have on aerodynamics and structural stiffness. In 

contrast, the structural cross-sectional dimensions are local variables 

affecting only the structure, and the variables defining the wing shape 

outside of the structural box are local variables affecting only 

aerodynamics. 

The system optimization proceeds top down. The optimization problem 

solved at the system level in the i-th iteration is: 

find X, within move limits 

such that F(X) is minimum, subject to constraints 

x1 <= x <= xu, 

G(X) <= 0, Ca < = 0, Cs<= 0. (1) 

Examples of F(X) and G(X) are the take-off gross weight, and the 

required climb rate. The constraints C are cumulative constraints 

3 
formulated using the Kreisselmeier-Steinhauser envelope function 

(ref.9). They are differentiable functions of the constraints local to 

aerodynamics and structures, and approximate the largest (most violated) 

constraint providing a single measure of feasibility of each subsystem 

design. In eq. 1, the cumulative constraints are approximated as 

functions of X by linear extrapolation from which the approach takes its 

name. For example, the extrapolation for Cs is 

‘s I ‘smin + vxcs Ax 

The derivatives forming the vxCs are carried from the aerodynamic and 

structural optimizations executed in the previous iteration. These 

subsystem optimizations may be generalized by a single definition, using * as a substitute for a and s: 

find Y, such that C,(Y,,X) is minimum, subject to constraints 

The purpose of the optimization is to proportion the wing shape and 

structure cross-sections to minimize the aerodynamic and structural 

constraint violations represented by C. The equality constraints, e, 

enforce compliance with the prescribed structural weight and structural 

deformation. The aerodynamic loads used in the optimization of the 

structure in the i-th iteration are computed in iteration i-1. The 

aerodynamic and structural optimizations are, in effect, decoupled and 

may be carried out concurrently. 

algorithm described in ref.10 or 11 yields derivatives of C,min with 

respect to X to form the vxC, used in eq.2. 

decisions made at the system level to improve the system performance 

include an approximate knowledge of their effects on aerodynamic and 

structural constraints. 

Optimum sensitivity analysis using the 

Thus, the shape design 

NON-HIERARCHAL DECOMPOSITION. 

Under this approach, decomposition is introduced only for the purposes 

of the system behavior sensitivity analysis while optimization is 

carried out for the entire system. The sensitivity analysis yields the 

derivatives used in linear extropolations of the system behavior which 

substitute for system analysis in optimization performed in a piecewiselinear manner. The algorithm for sensitivity analysis by decomposition 

from ref.12 yields the derivatives of the system response as a solution 

of a set of simultaneous, linear, algebraic equations 

(4) 

The Jacobians in the matrix of coefficients in eq. 4 represent 

sensitivity of the output from one disciplinary analysis to the inputs 

received from the other, e.g., partial derivatives of aerodynamic 

pressure with respect to the structural deformations. Similarly, the 

partial derivatives on the right hand side measure sensitivity of the 

output from aerodynamic and structural analyses to the design variables, 

. 

I 

4 
e.g., partial derivatives of structural deformation to the wing sweep 

angle as a shape variable. The partial derivatives, by definition, may 

be computed within each decoupled disciplinary analysis, so that 

specialized methods of disciplinary sensitivity analysis may be used, 

such as structural analysis for sensitivity with respect to shape, ref.1 

(Ch.31, and the corresponding analysis in CFD formulated in ref.13. The 

method is new and experiences with its use in optimization is limited to 

ref. 14. 

Numerical Example. 

An optimization study in which hierarchal, top-down decomposition of the 

type outlined in the foregoing was applied to a transport aircraft shown 

in Fig.4 was reported in ref.15. The objective was to minimize fuel 

consumption subject to constraints on the aircraft performance, 

aerodynamics, and structures, including the detailed local buckling 

constraints of the stiffened panel in the wing covers. The design 

variables governed the airfoil shape and wing structure cross-sectional 

dimensions, down to the stringer detail level. Individual cover panel, 

ribs, and spars were represented in the finite element model. The 

procedure performance was satisfactory; a typical histogram for the fuel 

objective and structural weight is shown in Fig.5. It demonstrated that 

a large number of design variables (1303), including shape variables, 

and constraints (1950) representing diverse disciplines may be included 

in a large engineering system optimization, and that the system 

performance may be linked mathematically to the design details. 

Conclusions 

When shape optimization is applied to a structure interacting with the 

surrounding airflow, the optimization task becomes multidisciplinary and 

has to account for the coupling of structural mechanics and aerodynamics 

involving the mutual dependence of structural deformations and 

aerodynamic pressures. 

of a large-scale engineering problem where the above occurs and where 

the shape optimization becomes a subtask in a system optimization 

because both structures and aerodynamics have a strong impact on 

performance. 

Methods for system optimization by decomposition have been developed 

for such multidisciplinary applications. They allow use of specialized 

methods for optimization and sensitivity, by separating the disciplinary 

optimization tasks while accounting for the interdisciplinary couplings. 

As illustrated by a numerical example for a large transport aircraft, 

these methods have begun to approach the level of maturity required for 

effective applications in large scale engineering design. 

Aircraft design for optimum shape is an example 

References. 

1. The Optimum Shape, Automated Structural Desiqn ; editors: 

Bennett, J. A., and Botkin. M. E.: Plenum Press, N.Y.. 1986. 

2. New Directions. in Optimum Structural Design ; ehitorsi Atrek, E.; 

Gallagher, R. H.; Ragsdell, K. M.; Zienkiewicz, O.C.; John Wiley - & 

Sons, 1984. 

3. Steward, D. V.; Systems Analysis and Manaqement ; PBI, N.Y., 1981. 

4. Archer, B.:'The Implication for the Study of Design Methods of 

Recent Developments in neighbouring Disciplines'. International 

5 
5. 

6. 

7. 

8. 

9. 

Conference on Engineering Design, Hamburg, Aug. 1985, Proceedings 

of, vol.1, pp.833-840, publ. Heurista, Zurich. 

Shankar, V.: 'A Full Potential Inverse Method Based on a Density 

Linearization Scheme for Wing Design'. AIM 14th Fluid and Plasma 

Dynamics Conference, Palo Alto, Cal., June 23-25, 1981. 

Sobieszczanski-Sobieski, J.: A Linear Decomposition Method for 

Large Optimization Problems - Blueprint for Development. NASA 

TM 83248, Feb. 1982. 

Sobieszczanski-Sobieski, J.; Barthelemy, J. M.; and Giles, G. L.: 

'Aerospace Engineering Design by Systematic Decomposition and 

Multilevel Optimization'; Paper No ICAS-84-4.7.3; 14th Congress of 

the International Council of the Aeronautical Sciences, Toulouse, 

France, Sept. 1984. 

Sobieszczanski-Sobieski, J.; and Barthelemy, J. M.: 'Improving 

Engineering System Design by Formal Decomposition, Sensitivity 

Analysis, and Optimization; International Conference on Engineering 

Design', Hamburg, Aug. 1985; Proceedings of, pp.314-321. 

Kreisselmeier, G; and Steinhauser, R.: 'Systematic Control Design by 

ODtimizina a Vector Performance Index'; International Federation of 

ALtive CoAtrol Symposium on Computer-Aided Design of Control 

Systems, Zurich, Switzerland, Aug. 29-31, 1979. 

10. Sobieszczanski-Sobieski, J.; and Barthelemy, J. M.; and Riley, K. 

M.: 'Sensitivity of Optimum Solutions to Problem Parameters'; 

AIAA J, VOl.20, Sept. 1982, pp. 1291-1299. 

11. Sobieszczanski-Sobieski, J.: and Barthelemy, J. M.: 'Optimum 

Sensitivity Derivatives of Objective Functions in Nonlinear 

Programing'; AIAA J, Vo1.21, No.6, June 1983, pp.913-915. 

12. Sobieszczanski-Sobieski, J.: On the Sensitivity of 

Complex, Internally Coupled Systems, NASA TM 100537 January 1988. 

13. Yates, E. C., Jr.: Aerodynamic Sensitivities from Subsonic, 

Sonic,and Supersonic Unsteady, Nonplanar Liftinq-Surface Theory; 

NASA TM 100502, Sept. 1987. 

14. Sobieszczanski-Sobieski, J.; Bloebaum, K.; and Hajela, P.: 

Sensitivity of Control-Augmented Structure Obtained by a System 

Decomposition Method, NASA TM 100535 January 1988. 

15. Wrenn, G. A.; and Dovi, A. R.: 'Multilevel Decomposition Approach to 

the Preliminary Sizing of a Transport Aircraft Wing'; 

AIAA/ASME/ASCE/AHS 28th Structures, Dynamics, and Materials 

Conference, Monterey, Cal., April 6-8, 1987, AIAA Paper 

NO. 87-0714-CP. 

1 I I 'M 

.7 .8 .9 1 40 -20 0 20 40 60 80 

Sweep angle, deg 

Figure 1. 

right- Structural weight of a wing vs. the wing sweep 

angle(positive = aft). 

left- Drag rise as a function or M and h/c; 

6 
Figure 2. Aft shift of the lift resultant on supercritical airfoil 

(right) vs. conventional airfoil (left). 

Figure 3. 

Flig 

, rWeight 

\ \ 

I * \ 

- 

Structures I 

I 

Aerodynamics 7 7 

I 1 

Deformations J Loads 

Disciplinary analyses coupled in wing design. 

Figure 4. 

its finite element model. 

Transport aircraft wing optimization, ref 15.:aircraft, and 

7 
3 9ox 10 

oCase 1 80 oCase 1 

oCase2 Wing 70 oCase 2 

weight 

Ibs. 60 

50 

40 

l7OO 2 4 6 8 10 12 14 

Cycles Cycles 

Figure 5. 

(left) and wing structural weight (right) for transport aircraft wing (case 1, 2: initial design infeasible and feasible, respectively). 

Histograms for the fuel consumption as objective function 

8 
Report Documentation Page 

1 Report No 

NASA TM-100538 - - - 

2 Government Accession No. 

Structural Shape Optimization in 

Multidisciplinary System Synthesis 

_____- 

7 Kov Words (Suggcstod by Authodsl) 

. __ - - . - . - 7. Authorlsl 

18. Distribution Statement 

Jaroslaw Sobieszczanski-Sobieski 

I 

9 Security classif (of this-report) -2Eecurity Classif. (of this page) 21. No. of pages 

UNCLASSIFIED UNCLASSIFIED 9 

-. __ - . . 

9 Performing Organiiation Name and Address 

22. Price 

A02 

NASA Langley Research Center 

Hampton, VA 23665-5225 

- - 12. Spoiysoring Agency Name and Address 

National Aeronautics and Space Administration 

Washington, DC 20546 

- - - - .. . - _. - .- 

I5 Supplementary Nott!~ 

3. Recipient's Catalog No. 

5. Report Date 

January 1988 

6. Performing Organization Code 

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8. Performing Organization Report No. 

10. Work Unit No. 

505-63-1 1 

11. Contract or Grant No. 

13. Type of Report and Period Covered 

Technical Memorandum 

14. Sponsoring Agency Code 

Presented at the International Union of Theoretical and Applied Mechanics 

(IUTAM) Symposium on Structural Optimization, February 9-13, 1988, Melbourne, 

Austral i a. 

Structural shape optimization couples with other discipline optimization in 

the design of complex engineering systems. 

weight and elastic deformations couple to aerodynamic loads and aircraft 

performance through drag. This coupling makes structural shape optimization 

a subtask in the overall vehicle synthesis. 

zation and sensitivity analysis allow the specialized disciplinary methods to 

be used while the disciplines are temporarily decoupled, after which the interdisciplinary couplings are restored at the system level. 

composition methods to structures-aerodynamics coupling in aircraft is outlined 

and illustrated with a numerical example of a transport aircraft. It is 

concluded that these methods may integrate structural and aerodynamic shape 

optimizations with the unified objective of the maximum aircraft performance. 

For instance, the wing structural 

Decomposition methods for optimiApplication of deOptimization 

Sensitivity 

Design 

Unclassified - Unlimited 

Star Category - 31