Aerostructural Optimization of Aircraft Structures

Using Asymmetric Subspace Optimization

Graeme Kennedy ∗, Joaquim R.R.A. Martins †, and Jorn S. Hansen ‡

University of Toronto Institute for Aerospace Studies

Toronto, Ontario, M3H 5T6, Canada

A novel approach to the aerostructural optimization of wing-box structures called asym-metric subspace optimization, is presented and compared to the traditional multidisci-plinary feasible (MDF) approach. In the asymmetric subspace optimization approach,the analyst chooses local and global sets of design variables and constraints. The localconstraints and variables form a local optimization problem that is solved at each globaliteration. This requirement changes the sensitivities of the global objective and constraintsand requires a post-optimality sensitivity method. The main idea of this approach is tochange the path to the optimal solution, by requiring the solution satisfy certain constraintsat every global iteration without modifying the solution itself. The aerostructural problemexamined here involves a linear stress analysis while the aerodynamic analysis is performedusing a vortex lattice method.

I. Introduction

Multidisciplinary optimization (MDO) problems pose both a computational and organizational challengein that computationally complex systems must be integrated to solve both the analysis and optimizationproblems.1

Approaches in the literature for solving MDO problems generally fall within two categories: single levelarchitectures such as MDF, IDF2 and SAND and multi-level architectures such as CO,3,4 MCO,5 BLISS6,7

and CSSO.8 The survey paper by Sobieszczanski-Sobieski and Haftka9 highlights the difference between theseapproaches and many other issues faced in MDO.

One of the most common applications of MDO in the engineering literature is the aerostructural op-timization problem which involves the analysis and optimization of a flexible wing subjected to the forcesapplied by a moving fluid. This problem contains many of the important aspects of MDO and dependingon the models of each discipline, may be very computationally intensive.10–13

Chittick and Martins14,15 developed a novel architecture, called asymmetric subspace optimization, whichwas partially motivated by computational imbalances sometimes encountered in high-fidelity aerostructuraloptimization. By solving an optimization problem involving only the structural design variables at everyiteration of the global optimization problem, they were able to demonstrate that the asymmetric subspaceoptimization architecture required fewer multidisciplinary analyses to arrive at the same optimum whencompared to the MDF approach. Thus, if the aerodynamics discipline was significantly more computationallyintensive, the overall optimization might take less time.

Since the asymmetric subspace optimization approach solves a local optimization problem at every globaliteration, it does not strictly fall under either the single or multi-level approaches and is best described as anested architecture.

In this paper, Sections II and III describe the asymmetric subspace optimization architecture and differentmodifications to the original architecture that allow for greater flexibility in how constraints are handled. Theoriginal architecture and the modified architectures are applied to an aerostructural optimization problem

∗PhD Candidate, AIAA Student Member†Associate Professor, AIAA Member‡Professor, AIAA Member

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12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference 10 - 12 September 2008, Victoria, British Columbia Canada

AIAA 2008-5847

Copyright © 2008 by Graeme Kennedy. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

described in Sections V and VI. The aerostructural analysis consists of a coupled finite element and vortexlattice model. The methods used to solve this problem are described in Section VII. Section VIII describesthe specifics of the aerostructural optimization problem in greater detail and the results from the applicationof the asymmetric subspace optimization approach with and without modification. These results are thencompared to results from the MDF approach.

II. Asymmetric Subspace Optimization

The asymmetric subspace optimization architecture can be applied to optimization problems that havethe following form:

minimize f(x, xs)w.r.t. x, xs

s.t. c(x) ≥ 0cs(x, xs) ≥ 0

(1)

In this problem, the design variables are split into two distinct sets, x ∈ Rnx , the global design variables andxs ∈ Rns , the local design variables. The nonlinear optimization problem (1), is to minimize the objectivefunction f : Rnx × Rns → R, subject to the global inequality constraints c : Rnx → Rmx and the localinequality constraints cs : Rnx × Rns → Rms . It is important to note that in this formulation the globalconstraints do not depend on the local design variables while the local level constraints cs may depend uponboth the local and global variables.

The distinction between what constitutes a global or local variable or constraint is arbitrary, apart fromthe dependence shown above. However, the choice of these sets of variables and constraints affects theperformance of the architecture. In many problems, such as the aerostructural problem that is the focus ofthis paper, there is a natural composition of these sets.

A solution to the optimization problem outlined in (1), is found by determining a solution to the perturbedKarush–Kuhn–Tucker (KKT) conditions as µ→ 0,

g(x, xs, λx, λs, sx, ss;µ) =

∇xf −∇xcT λx −∇xc

Ts λs

∇sf −∇scTs λs

c− sx

cs − ss

Sxλx − µeSsλs − µe

= 0 (2)

where λx ∈ Rmx and λs ∈ Rms are Lagrange multipliers and sx ∈ Rmx and ss ∈ Rms are slack variableswith sx, ss ≥ 0.

The perturbed KKT conditions have been used here in order to present the possible use of interior pointmethods later on. Interior point methods generally solve a sequence of problems based on the system ofequations (2), for which the parameter µ ≥ 0 is decreased until a sufficiently accurate solution is found.16,17

The asymmetric subspace optimization formulation is obtained by transforming the original optimizationproblem (1), into an equivalent nested optimization problem that solves the same set of KKT conditions in asequential manner. This technique is possible due to the inter–dependence of the global and local variables.The global optimization problem works with the global variables while the local optimization problem dealsonly with the local variables and constraints. As a result, the local problem is solved more frequently thanthe global optimization problem. This approach is asymmetric in that the local constraints are satisfied atevery global iteration and so the local problem constraints and derivatives must be called more often.

The global problem is,minimize f(x, x∗s(x))w.r.t. x

s.t. c(x) ≥ 0(3)

where x∗s(x) are the design variables which are found by determining a solution to the local optimization

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problem, given a value of the global design variables x:

minimize f(x, xs)w.r.t. xs

s.t. cs(x, xs) ≥ 0(4)

As a result of this nested evaluation, the local variables are implicitly dependent on the global variables. Asolution to the local optimization problem (4), is determined by solving the local problem KKT conditionsgiven values of global design variables x,

gs(xs, λs, ss;µ, x) =

∇sf −∇scTs λs

cs + ss

Ssλs − µe

= 0 (5)

It can be shown that solving this system of equations as µ→ 0 is equivalent to solving the following systemof equations involving only the active constraints18

g∗s (xs, λs;x) =

[∇sf −∇sc

Ts λs

c∗s

]= 0 (6)

Here c∗s represent the active set of constraints at the solution of the subspace optimization problem (4). As aresult, Equation (6) must now be treated as an implicit set of residuals that are satisfied at each iteration ofthe global problem (3). Therefore, derivatives at the global problem level must now be used which representpost-optimality sensitivities,19,20

dfdx

=∂f

∂x− ∂f

∂w∗

[∂g∗s∂w∗

]−1∂g∗s∂x

(7)

where w∗ represents the combined vector w∗ =[xT

s λTs

]T

. In this case, the matrix of derivatives of g∗

with respect to w∗ is given by:∂g∗s∂w∗

=

[B −AT

A 0

](8)

where B = ∇ssf −∇sscTs λs and AT = ∇ c∗s

T are the gradients of the active set of constraints.It can be shown that the global level objective will be locally continuous and differentiable if the subspace

optimization yields a minimum, the constraints pass the LICQ and the active set of the subspace optimizationdoes not change as a function of the global variables x. These requirements raise two important issues:

1. The local optimization problems must remain feasible, otherwise the sensitivity method may fail

2. The active set within the local optimization problem must not change in the vicinity of the optimumof the global problem

The first issue relates to which variables are selected as local design variables. Specifically, the subset oflocal variables must be chosen to incorporate enough flexibility that a feasible point in the local problem canbe found. This is fundamental to the asymmetric subspace optimization approach and so must be addressedby the user in selecting the subset of local variables. If an appropriate set of local variables cannot be found,the asymmetric subspace optimization approach may not be applicable.

The second issue relates to the active set of constraints and is potentially more serious. This is similar toissues found in CO.1,4 In some problems the active set of constraints in the local problem may not changerapidly and may settle down near the solution, however this is not guaranteed. The resulting discontinuitiesmay cause poor convergence at the global level due to discontinuities or the global problem may fail toconverge at all.

In order to address this situation, a sensitivity method based on the perturbed KKT conditions for µ ≥ 0may be solved.5 In this situation the sensitivity is given by:

dfdx

=∂f

∂x− ∂f

∂w

[∂gs(µ)∂w

]−1∂gs(µ)∂x

(9)

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where w =[xT

s λTs sT

s

]T

and the matrix of partial derivatives is,

∂gs(µ)∂w

=

B −AT 0A 0 −I0 Ss Λs

(10)

III. Improving Asymmetric Subspace Optimization

A major difficulty with the asymmetric subspace optimization approach described above is that the globalconstraints can only be functions of the global variables. This is very restrictive as in many cases constraintsdepend on both the global and local variables and will therefore have to be treated as local constraints.Increasing the number of constraints in the local optimization problem is not desirable because the localproblem must remain feasible. Furthermore, decreasing the size of the local problem may be attractive sincethe sensitivities of the local problem can be expensive. Several approaches to modify the original architecturehave been undertaken to address these issues.

The modified problem with global constraints that depend on the local variables is:

minimize f(x, xs)w.r.t. x, xs

s.t. cx(x, xs) ≥ 0cs(x, xs) ≥ 0

(11)

The difference between the original formulation (1), and this modified formation (11), is the additionaldependence of the global constraints, now denoted cx, on the on the local design variables. The impact ofthis modification is significant. The perturbed KKT conditions are now:

g(x, xs, λx, λs, sx, ss;µ) =

∇xf −∇xcTx λx −∇xc

Ts λs

∇sf −∇scTx λx −∇sc

Ts λs

c− sx

cs − ss

Sxλx − µeSsλs − µe

= 0 (12)

With the additional dependence of cx(x, xs), the local optimization problem cannot be solved independentlyfor the local variables xs due to the presence of the term ∇sc

Tx λx. The local problem must now consider

the effect of the global constraints cx and their Lagrange multipliers on the local problem. However, thelocal problem in the original formulation (4), cannot know which global constraints are active, or what thevalues of their Lagrange multipliers are. Since the global constraints now directly affect the local problem,the asymmetric approach would no longer give the same optimum as the standard MDF approach.

Summarizing these observations, there are two criteria that must be met by the modified architecture:

1. It must correctly account for the presence of the global constraints

2. The local optimization problems must be constructed in a manner in which a feasible solution can befound

III.A. Augmented Lagrangian

One method that satisfies these criteria is to use an augmented Lagrangian approach for the local optimizationproblem. In this type of method an estimate of the Lagrange multipliers and an L2 penalty function areused to determine an approximate solution of the optimization problem. It can be shown that this methodconverges to the correct optimum for finite values of the penalty parameter γ.18

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Using this type of formulation the local problem is modified to account for the effect of the globalconstraints while the global problem remains unmodified,

minimize f(x, xs)− λTx (cx(x, xs)− sx) + 1

2γ(cx(x, xs)− sx)T (cx(x, xs)− sx)w.r.t. xs, sx

s.t. cs(x, xs) ≥ 0sx ≥ 0

(13)

where λx represent an estimate of the Lagrange multipliers of the global constraints cx(x, xs) at the currentiteration while sx are the associated slack variables. The local problem is separable in the slack variables,which can be eliminated from the problem by determining the minimizer with respect to sx and replacingthe values into the formulation (13).18 The value of the slack variables are,

sx = min(− λx

γ+ cx, 0

)(14)

The Lagrange multiplier estimates λx must be as accurate as possible, in order to obtain an accuratesolution of the augmented problem (13). It should be noted that these Lagrange multipliers represent thevalues from the total KKT conditions (12). In the current implementation the optimization problem (13) isapproximately solved for given values of λx and γ. Once the approximate solution has been obtained, theLagrange multipliers are updated using the least squares estimate from the global problem and the penaltyparameter is increased.

III.B. L1 Penalty Method

Another alternative for the local optimization problem that satisfies the above criteria is to use an exactpenalty function such as the L1 penalty. This can then be transformed into a continuous formulation usingan elastic-programming approach by placing a modified form of the global constraints cx(x, xs) in the localproblem constraint.

minimize f(x, xs)− λTx cx(x, xs) + γeT (t+ v)

w.r.t. x, t, v

s.t. cs(x, xs) ≥ 0cx(x, xs)− t+ v ≥ 0t, v ≥ 0

(15)

For sufficiently large, but finite values of γ, this problem has the same minimum as the exact problem.The issue of infeasible local optimization problems is again avoided since the problem simply minimizes theinfeasibility in the global constraints cx. This approach is sometimes referred to as elastic programmingand is used in SNOPT to handle infeasibility.21 As with the augmented Lagrangian approach describedabove, good Lagrange multiplier estimates are necessary for reliable convergence to the system level opti-mal solution. As before, the global optimization problem is solved by choosing a penalty parameter andLagrange multiplier estimates λx, approximately solving problem (15), updating the penalty parameter andthe Lagrange multiplier estimates λx and iterating until convergence.

With both of these approaches, the solution of the local problem x∗s(x) is treated as an implicit functionof the global variables. Thus, the sensitivities at the global level must be treated in a similar manner asdescribed above in either Equation (7) or Equation (9).

IV. Analytic Problem

A small analytic problem was developed in order to test the asymmetric subspace optimization methodsdescribed above. An extended version of the Rosenbrock function was used as the objective function, with

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three equality constraints,

minimize f(y) =5∑

i=1

[(1− yi)2 + 100(yi+1 − y2

i )2]

w.r.t. y

s.t. g1(y) =6∑

i=1

yi = 0 ; g2(y) =6∑

i=1

y2i −

32

= 0 ; g3(y) =6∑

i=5

yi −15

= 0

(16)

where y1 through y4 are the global variables and y5 and y6 are the local variables. g1 and g2 are treatedas the global constraints and g3 the local constraint. No bound constraints are placed on the variables. To

match the above notation x =[y1 y2 y3 y4

]T

, xs =[y4 y5

]T

, cs = g3 and cx =[g1 g2

]T

The three asymmetric optimization approaches described above where implemented for problem (16)using the programming language python and the nonlinear optimization library SNOPT.21 Both the aug-mented Lagrangian (ASO AL) and L1 penalty function (ASO L1) variants of the improved asymmetricsubspace optimization method were successfully applied to problem (16). Unfortunately the regular asym-metric subspace optimization approach could not be successfully applied to problem (16) as the methodentered regions which had infeasible subproblems.

Approach global DVs local DVs objectivey1 y2 y3 y4 y5 y6

NLP -0.95667616 0.68130162 0.25691996 -0.18154542 0.12856775 0.07143225 23.74911354ASO AL -0.95667623 0.68130159 0.25691988 -0.18154524 0.12856757 0.07143243 23.74911281ASO L1 -0.95667623 0.68130160 0.25691988 -0.18154525 0.12856770 0.07143230 23.74911354

Table 1. Solution to the test problem. Note that the asymmetric subspace optimization technique without modificationdid not work since the subproblem was infeasible.

Approach ASO AL ASO L1

Iteration λ1 λ2 λ1 λ2

1 0.0 0.0 0.0 0.02 31.32980309 33.75599975 31.33120893 33.756697393 31.28494981 33.57148984 31.28492214 33.571134584 31.28494910 33.57148578 31.28494742 33.57147677NLP 31.28494742 33.57148751 31.28494742 33.57148751

Table 2. Convergence of the Lagrange multipliers to the MDF values of the multipliers as the global problem isreformulated with new estimates.

Table (1) shows the solutions obtained using the ASO AL and ASO L1 variants of the architectureand compares them to the traditional optimization approach (NLP). The results agree to a high degree ofaccuracy. Table (2) shows the convergence of the Lagrange multiplier estimates λx to the NLP values ofthe multipliers after each least-squares update. The multipliers converge quickly and good results could beobtained after the second estimate. It should be noted here that using equality constraints in problem (16)fixes the active set. This avoids some difficulties that might arise in inequality constrained problems wherethe initial active set could be very different from the active set at the solution. This would result in poorLagrange multiplier estimates and poor convergence. The present results, however, demonstrate that theNLP and modified ASO approaches are equivalent for this problem.

Table (3) shows the number of function and gradient evaluations required to obtain the optimal solutions,Table (1). It should be noted that the global problem gradients for (16) are calculated using the post-optimality sensitivity equation (7) for the ASO AL and ASO L1 methods, while the regular partial derivativesare used for the NLP approach. The traditional optimization approach proves to be more efficient than eitherof the two modified approaches for the test problem (16).

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Approach global eval global grad subopt eval subopt gradNLP 27 19ASO AL 95 36 392 280ASO L1 124 50 517 363

Table 3. Comparisons of the number of gradient and function evaluations necessary to solve the problem.

V. Aerostructural Analysis

The aerostructural model used for the optimization comprises a vortex lattice aerodynamics code coupledto a finite element structural model through an aerodynamic surface whose deformation is defined by theuse of shape functions determined by using radial basis functions.22 All these components comprise theaerostructural analysis and are described briefly below.

V.A. Aerodynamics Model

The aerodynamics code is a simple, compressibility-corrected, vortex lattice code. In this code the wingis modeled as a surface which is covered by closed vortex lattice loops. At the trailing edge, a horseshoevortex, equal in strength to the final closed vortex loop on the wing surface, is propagated downstream inorder to satisfy the Kutta condition. Boundary conditions are applied on the wing to ensure that the flowis tangential to the surface. The set of aerodynamic residuals are,

RA(uA, uS ;x, xs) = 0 (17)

where RA : RNA × RNS ; Rnx × Rns → RNA . In this notation, uA ∈ RNA are the aerodynamic statevariables, which in this case represent the vortex strengths, uS ∈ RNS are the structural state variables,whose dependence comes from the deformation of the wing surface. The global and local design variables xand xs are separated from the state variables uA and uS to emphasize that they are treated as a given setof parameters during the solution of equation (17).2

The Equations (17) are linear in the state variables uA, but nonlinear in the structural displacements uS .The resulting system of equations is dense and is solved using LAPACK.

V.B. Structural Model

A finite element model is used to analyze the structural response of the wing. In this model the ribs, sparsand skins of the wing are analyzed using plates stiffened with a series beams that are designed to model sparand rib caps. This finite element model has the capability of handling composite or isotropic skins, sparsand ribs. The loads on the finite element model are determined from the output from the aerodynamicsmodel. The residuals of the finite element equations are,

RS(uA, uS ;x, xs) = 0 (18)

where RS : RNA × RNS ; Rnx × Rns → RNS . As above, uA and uS are the aerodynamic and structuralvariables respectively, while x and xs are the global and local design variables.

The system of Equations (18) is linear in both the aerodynamic and structural variables. In this case thesystem of equations is sparse and is setup in PETSc.23,24

VI. The Aerodynamic Surface

The initial wing surface is generated from planform data: root chord, span, taper ratio and sweep.Camber, based on the NACA four digit airfoil series, is added. For the optimization problem, the planformvariables: taper ratio, sweep angle and span may be included as design variables. The position of the wingsurface is modeled as the initial planform shape plus a structural deformation modeled using radial basisfunctions.

The deformation of the wing surface is given in each coordinate direction by the following interpolation:22

d(v) = aT Φ(v) + bT p(v) (19)

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where v ∈ R2 maps to d : R2 → R. The vector Φ : R2 → RN is evaluated such that

Φi(v) = φ(||v − vi||2)

The functions φ : R → R are the radial basis functions while vi ∈ R2 are the locations of the so-calledcenters: a set of N points in the domain where the deformation of the surface is known.

There are many possible radial basis functions, but the ones used here are either the Gaussian or theinverse quadratic,

φG(r) = e−r2

φIV (r) =1√

1 + r2

The function p ∈ R6 is a vector of complete polynomials of second degree that allows the forces andmoments to be consistent and conservative,

bT p(v) = b1 + b2v1 + b3v2 + b4v21 + b5v1v2 + b6v

22

The interpolation problem is to determine the weighting coefficients (a, b) of the radial basis functionsand polynomial such that the surface deformation at the center locations vi is recovered and the deformationdescribed by the polynomial is captured.

These two conditions result in the following system of N + 6 equations in the unknowns a and b:

[aT bT

] [M PT

P 0

]=

[xT 0

](20)

where x ∈ RN are the displacements at the centers. The matrices M ∈ RN×N and P ∈ R6×N are defined asfollows:

Mij = Φi(vj) Pij = pi(vj)

The deformation of the surface d(v) is found by solving the system of Equations (20) for a and b andsubstituting these values into Equation (19):

d(v) =[aT bT

] [Φ(v)p(v)

]= xT

[ (M−1 −M−1PTD−1PM−1

)M−1PTD−1

] [Φ(v)p(v)

]

where D = PM−1PT .The surface shape functions can now be written as:

Nd(v) =[ (

M−1 −M−1PTD−1PM−1)

M−1PTD−1] [

Φ(v)p(v)

](21)

VI.A. Load and Displacement Transfer

The transfer of displacements from the finite element model to the aerodynamic surface and the transfer ofloads from the aerodynamics to the structural model is one of the most critical parts of an aerostructuralanalysis. Equations (17) and (18) are usually solved using different codes, which potentially use completelydifferent solution approaches. The load and displacement transfer scheme should endeavor to keep thesecomponents as separate as possible while satisfying all requirements for a sufficiently accurate analysis.

Two conditions for any displacement and load transfer system is that the forces and moments shouldbe consistent between the aerodynamics and structures. Additionally, the work done on the aerodynamicmesh should be equal to the work done on the structure. While these conditions are necessary, they do notuniquely define a load and displacement transfer scheme.

Additional requirements for each analysis should also be taken into consideration. For instance, the dis-placements produced by most finite element models are typically C0 or at most C1 while many aerodynamiccodes will produce poor results if they are given meshes that are not smoother that this.

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In the current implementation, the aerodynamic surface is determined by adding the surface deformationto the initial surface shape. The surface deformation is determined by taking a subset of displacements fromthe finite element model, taking a weighted average and interpolating these values over the domain of thesurface using the shape functions (21). This operation is written as:

d(v) = Nd(v)TWQuS (22)

where d is the component of the surface deformation along a coordinate axis, W is a weighting matrix andQ is a projection matrix. Given this interpolation scheme, the consistent force vector can be determined byusing the method of virtual work as follows:

fc =∫

Ω

QTWTNdfdΩ =∑

i

QTWTNd(vi)fi (23)

where fc are the consistent forces on the structural degrees of freedom, fi are the point forces from thevortex lattice method and vi are the locations of these forces.

VII. Aerostructural Solution Procedure

A solution to the aerostructural problem is found when both sets of residuals in Equation (17) and (18)are satisfied simultaneously. This can be written as,[

RA

RS

]= 0 (24)

where the arguments have been omitted for simplicity of notation. An inexact Newton’s method is used tosolve this system of equations. The following linearized system of Equations (24) is solved approximatelyfor each Newton update: [

A B

C D

] [∆uA

∆uS

]= −

[RA

RS

](25)

Where A ∈ RNA×NA , B ∈ RNA×NS , C ∈ RNS×NA and D ∈ RNS×NS are the Jacobian of the residuals. TheNewton update uA ← uA + ∆uA and uS ← uS + ∆uS is applied after the Equations (25) have been solvedapproximately.

The system of equations (25) can be solved using a number of different iterative techniques, however theapproach used here is a Schur compliment method. In this approach the following two equations are solvedin sequence: (

D − CA−1B)∆uS = −

(RS − CA−1RA

)A∆uA = − (RA +B∆uS)

Iterative Krylov methods can be applied to these two equations in various ways.25 In this paper, theaerodynamic coefficient matrix A is fully populated and a full factorization is affordable for reasonably sizedproblems. The structural stiffness matrix D is sparse so a Krylov method is applied to the first equationwith the matrix D used as a preconditioner. While D is symmetric, the effect of adding −CA−1B is not, soGMRES is used to solve the system of equations. Only matrix-vector products are required with the matrix(D − CA−1B

)so the matrices B and C are not stored explicitly. The second equation can be solved after

the first using the full factorization of A.Figure (1) shows an equilibrium solution of the fully integrated aerostructural model.

VII.A. Aerostructural Sensitivities

Sensitivity calculations are performed using a semi–analytic coupled adjoint approach that involves thetranspose of the matrix in Equation (25).10 The derivative of a function f constrained by the set of residualequations (24), is determined by calculating the total derivative,

dfdx

=∂f

∂x− ψT

A

∂RA

∂x− ψT

S

∂RS

∂x(26)

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Structures: Stiffened Wing

Aerodynamics: Panel Code Lift Distribution

Figure 1. Illustration of the fully integrated model. The top two figures show the structural model of the wing: verticaldisplacement of the plates and stiffeners. The bottom figure shows the vertical force exerted by the aerodynamics.Note the curvature and displacement of the aerodynamic surface.

where x represents either the global or local design variables and ψA and ψS are the aerodynamic andstructural adjoint variables respectively. These adjoint variables are determined by solving the adjointequation, [

AT CT

BT DT

] [ψA

ψS

]=

[∂f/∂uA

∂f/∂uS

]The Schur complement approach can also be applied to this system of equations as long as the transposematrix-vector products of the off-diagonal entries BT and CT are available.

In this implementation the partial derivatives of the objective and constraints with respect to all statevariables are calculated analytically. Partial derivatives with respect to aerodynamic design variables arecalculated using a central difference method while partial derivatives of the structural design variables arecalculated analytically.

VII.B. Coupled Post-Optimality Sensitivities (CPOS)

In all of the asymmetric subspace optimization approaches described above the local variables are treated asimplicit functions of the global variables x∗s(x), defined by the solution to the subspace optimization problem(4) or (13) or (15). Thus, a post-optimality sensitivity method must be used to evaluate the derivative ofthe objective or constraint with respect to the global variables using either Equation (7) or Equation (9).This situation is further complicated, in the aerostructural problem, by the addition of the aerodynamic andfinite element residuals RA and RS and their associated state variables uA and uS that are also implicitfunctions of x and xs.

Chittick and Martins,14,15 developed a special approach to evaluating these coupled post-optimalitysensitivities (CPOS) by solving an adjoint system involving a combination of total and partial derivatives.In this paper a brute force approach is taken: the post-optimality sensitivities are evaluated using Equation(7), where the Hessian is approximated using finite differences of the total derivative of the aerostructuralproblem (26).

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VIII. Aerostructural Optimization

The aerostructural optimization problem addressed in this paper is lift constrained drag minimization.Since the aerodynamics is modeled using an inviscid vortex lattice code, the goal is to minimize the induceddrag of the wing subject to the lift constraint. The aerodynamic design variables are the angle of attack andtaper ratio. The lift constraint is imposed by setting,

L

W− 1 = 0 (27)

where L is the total lift and W is the total weight of the aircraft. The total weight of the aircraft is calculatedbased on a constant mass plus the structural mass of the wing. The constant portion of the mass representsthe payload, fuel and remainder of the structure not involved in the analysis.

Structural constraints are imposed on the aerostructural problem to ensure that the design is sufficientlysafe. These constraints are calculated based upon the assumption that the ultimate loads are 3.75 timesthe loads at the cruise condition. The failure constraints are imposed at every Gauss point within thefinite element model for maximum accuracy,26,27 however in order to reduce the number of constraints,a Kreisselmeier-Steinhauser (KS) constraints aggregation technique is used.28,29 This yields good resultswhile greatly simplifying the optimization problem.30 The structural design variables are the thickness anddiameter of the spars and the thickness of the plates.

The optimization results for three different test cases are presented below. Table (4) outlines the sizesof these test cases. Case 1 has three structural variables: the thickness of the top and bottom skins, thethickness of the ribs and the thickness of the spars. All stiffener diameters are held constant. Case 2 has 24plate thicknesses that are the thickness of each individual spar and rib as well as the thickness of the topand bottom skin for each panel of the wing. Again each stiffener is held constant. Case 3 has 42 structuraldesign variables: all the variables from Case 2 plus additional diameter variables for each stiffener. All casesinclude the angle of attack and taper ratio as the aerodynamic variables. The taper ratio is limited between0.2 to 1.0 in order to maintain a reasonable aspect ratio for the aerodynamic and structural meshes. Theconstraints in each of the cases involve the lift constraint plus a structural constraint for each stiffener, rib,spar and skin panel. If the structural design variables of multiple components are the same, then thosestructural constraints are agglomerated into a single failure constraint to reduce the size of the problem.

Problem Structure DVs Aero DVs ConstraintsCase 1 3 2 23Case 2 24 2 58Case 3 42 2 58

Table 4. Summary of the three aerostructural optimization cases

Table (5), shows the angle of attack α, the taper ratio, drag and structural mass of the MDF solutionfor each of the three problems. Figure (2) shows the optimal distribution of the thickness and diametervariables and Figure (3) shows the deflected shape of the structure in equilibrium. As the number ofstructural variables increases so does the flexibility of the structure, while the mass and drag both decrease.There is a significant reduction in the drag and structural mass between Cases 1 and 2, while the reductionis not nearly as significant between Cases 2 and 3.

Optimal values α taper drag structural massCase 1 0.41635 0.23411 1826.01 1408.69Case 2 0.11413 0.2 1741.28 1063.50Case 3 0.13006 0.2 1736.35 1020.99

Table 5. Comparison of the MDF optimization results for the three different test cases.

Table (7) shows a comparison between the solutions for Case 1 obtained from each of the asymmetricsubspace optimization approaches described above and the MDF approach. The setup for the asymmetricsubspace optimization techniques is summarized in Table (6). The original asymmetric subspace optimization

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formulation used the taper ratio as the only global design variable so that the angle of attack could be used inthe local problem to satisfy the lift constraint. This allowed the subproblems to remain feasible throughoutthe optimization. Unfortunately the modified asymmetric subspace optimization approaches did not convergeto a satisfactory tolerance. This could be for a number of reasons, however the most likely explanation isinaccurate gradients from the post-optimality sensitivity method described in Section VII.B and poor scalingof the optimization problem.

Approachglobal DVs local DVs

ASO taper ratio α, structural variablesASO AL and L1 structural variables α, taper ratio

global constraints local constraintsASO N/A lift and structural constraintsASO AL and L1 structural constraints lift constraint

Table 6. A summary of the setup of the asymmetric subspace optimization approaches

Approach DVs drag structural massα taper x1 x2 x3

MDF 0.41635 0.23411 4.07297 0.87512 0.30421 1826.01 1408.69ASO 0.41636 0.23410 4.07297 0.87512 0.30421 1826.01 1408.68ASO AL * 0.44213 0.23927 4.00497 0.88879 0.67609 1827.65 1402.64ASO L1 * 0.45181 0.25105 4.04731 0.84850 1.82362 1832.03 1450.42

Table 7. Solution summary for Case 1. * These methods did not converge, the local subspace optimization ensuredthat the structural constraints were satisfied, however the optimality tolerance could not be reached due to inaccurategradient information which lead to numerical difficulties.

IX. Conclusions

Two new asymmetric optimization approaches were developed and their solutions were shown to matchthe traditional NLP solution for an analytic test problem. These methods and the original asymmetric sub-space optimization approach were applied to an aerostructural, lift constrained drag minimization problem.The results obtained were mixed: the original asymmetric subspace optimization method worked but thenew approaches had difficulty, potentially due to gradient accuracy and scaling problems.

Further work into these approaches must address:

• How to improve the accuracy of the post-optimality sensitivities

• How to tightly integrate the Lagrange multiplier estimates

References

1ALexandrov, N. M. and Lewis, R. M., “Analytical and Computational Aspects of Collaborative Optimization for Multi-disciplinary Design,” AIAA Journal , Vol. 40, No. 2, 2002, pp. 301–309.

2Cramer, E., Dennis, J., Frank, P., Lewis, R., and Shubin, G., “Problem formulation for multidisciplinary optimization,”SIAM Journal on Optimization, Vol. 4, No. 4, 1994, pp. 754–776.

3Braun, R. D., Collaborative Optimization: An Architecture for Large-Scale Distributed Design, Ph.D. thesis, StandordUniversity, 1996.

4Braun, R., Gage, P., Kroo, I., and Sobieski, I., “Implementation and performance issues in collaborative optimization,”AIAA, , No. 96-4017, 1996.

5DeMiguel, A. and Murray, W., “An analysis of collaborative optimization methods,” Eighth AIAA/USAF/NASA/ISSMOSymposium on Multidisciplinary Analysis and Optimization, No. AIAA Paper 00-4720, 2000.

6Kodiyalam, S. and Sobieszczanski-Sobieski, J., “Bilevel integrated system synthesis with response surfaces,” AIAA Jour-nal , Vol. 38, No. 8, 2002, pp. 1479–1485.

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7Sobieszczanski-Sobieski, J., Agte, J., and Sandusky, R., “Bi-level integrated system synthesis (BLISS),” AIAA, , No.98-4916, 1998.

8Sobieszczanski-Sobieski, J., “Optimization by Decomposition: A Step from Hierarchic to Non-Hierarchic Systems,” Tech.rep., NASA, 1988.

9Sobieszczanski-Sobieski, J. and Haftka, R., “Multidisciplinary aerospace design optimization: survey of recent develop-ments,” Structural Optimization, Vol. 14, No. 1, 1997, pp. 1–23.

10Martins, J. R. R. A., Alonso, J. J., and Reuther, J. J., “High-Fidelity Aerostructural Design Optimization of a SupersonicBusiness Jet,” Journal of Aircraft , Vol. 41, No. 3, 2004, pp. 523–530.

11Martins, J. R. R. A., A Coupled-Adjoint Method for High-Fidelity Aero-Structural Optimization, Ph.D. thesis, StanfordUniversity, Stanford, CA 94305, Oct. 2002.

12K. Maute, M. Nikbay, C. F., “Coupled analytical sensitivity analysis and optimization of three-dimensional nonlinearaeroelastic systems,” AIAA Journal , Vol. 39, No. 11, 2001, pp. 2051–2061.

13Alonso, J. J., LeGresley, P., van der Weide, E., Martins, J. R. R. A., and Reuther, J. J., “pyMDO: A Framework forHigh-Fidelity Multi-Disciplinary Optimization,” AIAA Paper 2004-4480, Aug. 2004.

14Chittick, I. R. and Martins, J. R. R. A., “An Asymmetric Suboptimization Approach to Aerostructural Optimization,”Optimization and Engineering, 2008, (In preparation).

15Chittick, I. R. and Martins, J. R. R. A., “Aero-Structural Optimization Using Adjoint Coupled Post-Optimality Sensi-tivities,” Structures and Multidisciplinary Optimization, 2007, (In press).

16R.H. Byrd, M. H. and Nocedal, J., “An Interior Point Method for Large-Scale Nonlinear Programming,” SIAM Journalon Optimization, Vol. 9, No. 4, 1999, pp. 877–900.

17Byrd, R., Gilbert, J., and Nocedal, J., “A trust region method based on interior point techniques for nonlinear program-ming,” Mathematical Programming, 2000, pp. 149–184.

18Nocedal, J. and Wright, S. J., Numerical Optimization, Springer, 1999.19Braun, R., Kroo, I., and Gage, P., “Post-optimality analysis in aerospace vehicle design,” AIAA Aircraft Design, Systems

and Operations Meeting, No. 93-3932, 1993.20Sobieszczanski-Sobieski, J., “Sensitivity of complex, internally coupled systems,” AIAA Journal , Vol. 28, No. 1, 1990,

pp. 153–160.21Gill, P. E., Murray, W., and Saunders, M. A., “SNOPT: An SQP algorithm for large-scale constrained optimization,”

Tech. Rep. NA–97–2, Stanford University, San Diego, CA, 1997.22Allen, C. and Rendall, T., “Unified Approach to CFD-CSD Interpolation and Mesh Motion using Radial Basis Functions,”

25th AIAA Applied Aerodynamics Conference, No. 2007-3804, 2007.23Balay, S., Buschelman, K., Eijkhout, V., Gropp, W. D., Kaushik, D., Knepley, M. G., McInnes, L. C., Smith, B. F., and

Zhang, H., “PETSc Users Manual,” Tech. Rep. ANL-95/11 - Revision 2.1.5, Argonne National Laboratory, 2004.24Balay, S., Gropp, W. D., McInnes, L. C., and Smith, B. F., “Efficient Management of Parallelism in Object Oriented

Numerical Software Libraries,” Modern Software Tools in Scientific Computing, edited by E. Arge, A. M. Bruaset, and H. P.Langtangen, Birkhauser Press, 1997, pp. 163–202.

25Saad, Y., Iterative Methods for Sparse Linear Systems, PWS Pub. Co., 1st ed., 1996.26Bathe, K. J., Finite element procedures, Prentice Hall, 2nd ed., 1996.27Zienkiewicz, O., Taylor, R., and Zhu, J., The Finite Element Method: its basis and fundamentals, Elsevier Butterworth-

Heinemann, 6th ed., 2005.28Raspanti, C., Bandoni, J., and Biegler, L., “New strategies for flexibility analysis and design under uncertainty,” Com-

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analysis,” Structural and Multidisciplinary Optimization, 2007.

Theorem. The local variables can be written as a continuous and differentiable function of the global designvariables, xs = xs(x) if:

• The objective and constraints and their derivatives are continuous and differentiable

• The local optimization problem is at a minimum

• The local constraints satisfy the Linear Independence Constraint Qualification (LICQ)

Proof. The KKT conditions for the local level optimization form a set of residual equations with which wehope to implicitly define the relationship xs(x) by solving for g(xs, λs;x) = 0 given x. These derivativescan be shown to be continuous and differentiable under the above assumptions using the implicit functiontheorem.

The implicit function theorem requires that the objective function f(x, xs) and the residuals g∗(xs, λs;x)be continuous and differentiable in the vicinity of the solution g∗(xs, λs;x) = 0. This is satisfied if theobjective and constraints have continuous second order derivatives, since the KKT conditions involve firstorder derivatives of these functions.

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The implicit function theorem also requires that the Jacobian of g∗ with respect to xs and λ∗ be invertible.This can be shown by considering the following system:[

B −AT

A 0

](28)

where B = ∇ssf − λTs ∇sscs and AT = [∇cs]i for i in the active set.

Due to LICQ, the gradients of the active constraints are linearly independent so that has rank(AT ) = mwhere m is the number of active constraints. As a result Z = null(A), a basis for the null space of A, hasrank(Z) = nx − m. It can be shown that the matrix, Equation 28, is not singular if ZTBZ is positivedefinite. However, ZTBZ, the projection of the Hessian of the Lagrangian onto the null space of the activeconstraints, is positive definite since the current point is a minimizer. Thus, the Jacobian of g∗ is invertibleand the implicit function theorem guarantees xs = xs(x) is locally continuous and differentiable.

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(a) Case 1

(b) Case 2

(c) Case 3

Figure 2. Comparison of the optimal structural thicknesses and diameters for Cases 1, 2 and 3.

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Figure 3. Comparison of the optimal displaced wing structure for Cases 1, 2 and 3. Note that the displacementsincrease as the number of structural design variables increases.

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