MDO Architectures for Coupled Aerodynamic
and Structural Optimization of a Flexible Wing
H C Ajmera*, P M Mujumdar† and K Sudhakar‡
Center for Aerospace Systems Design and Engineering, Aerospace Engineering Department
Indian Institute of Technology Bombay, Mumbai - 400076, India.
This paper deals with the formulation and testing of different variants of some existing
single level generic architectures, specifically for coupled aerodynamic and structural
optimization of wing, focused on static aeroelasticity. The design problem involves
simultaneous optimization of the wing aerodynamic plan-form and section variables along
with its structural sizing variables for minimum load carrying structural weight subjected to
structural, aerodynamic, performance and geometric constraints. The associated MultiDisciplinary Analysis (MDA) problem essentially involves coupled solution of the state
equations of the aerodynamic and the structural disciplines by nested iterations. The Multidisciplinary Design Optimization (MDO) problem is posed as a three discipline coupled
problem, with the trim (maneuver) process required to define structural design loads
considered as a separate discipline. This leads to a number of interesting reformulations of
the MDO problem based on (i) the reordering of the nested iterations and (ii) decoupling the
nested iterations at different levels through the introduction of pseudo design variables and
pseudo constraints. Formulation of six variants of the MDO problem and their
implementation is presented along with computational issues related to convergence of the
iterative processes. A special constraint based on a divergence control parameter has been
formulated to handle instability. Optimization results from the different formulations are
compared to study their computational performance and bring out the impact of
aeroelasticity on the design of the flexible wing.
I.
Introduction
D
ESIGN of a complex multidisciplinary engineering system, such as an aircraft, generally consists of a
hierarchical sequence of steps. This evolutionary process usually goes through a conceptual design phase, a
preliminary design phase and a detailed design phase followed by prototype building and testing. Merging the
conceptual design phase with the preliminary design phase with equal weightage given to the various contributing
disciplines will increase design freedom and enables a better insight of the influence of the mutual couplings
between the disciplines on the design.1 Multidisciplinary Design Optimization (MDO) is an emerging paradigm and
design enabler, which addresses the issue of carrying out formal optimization of complex, coupled, multidisciplinary
system with high fidelity analyses.2
Aeroelasticity as a platform for integrating disciplines for design of an aircraft is gaining importance in the MDO
context. Many software tools to achieve this have been developed in recent years. These include PrADO,3
WingMOD,4 ASDL IPPD approach for HSCT design,5 etc. MDO architectures6,7,8 can play a significant role in
improving the optimization process. Many single and bi-level MDO architectures have been been proposed in the
literature. Several single level MDO formulations can be derived from the generic concepts such as the Multiple
Discipline Feasible (MDF, also referred to as NAND-NAND), the Individual Discipline Feasible (IDF, also known
as SAND-NAND) and the All-At-Once (AAO, also called SAND-SAND). An overview of bi-level methods
involving decomposition of the design task into a set of disciplinary optimizations performed independently in each
discipline, together with a system-level coordination is given in Reference 8. However, most of the architectures
have these have been tested with small, artificial problems9,10. Comparison and testing of these architectures has
been carried out for a suite of simplified conceptual design problems 11 with low fidelity MDA. Recently, Gumbert et
*
Graduate Student, Department of Aerospace, IIT Bombay.
Professor, Member AIAA.
‡
Professor, Associate Fellow AIAA.
†
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al.12 have developed a software for wing aerodynamic optimization using SAND6 architecture with static aeroelastic
constraints and high fidelity analysis. Comparison of various MDO architectures for a realistic problem with
realistic constraints will give a better insight into the related computational processes and resulting impact on the
design.
This paper deals with the formulation and testing of different variants of some existing single level generic MDO
architectures, specifically for multidisciplinary design optimization of a subsonic transport aircraft wing, focused on
static aeroelasticity. The design problem involves simultaneous optimization of the wing aerodynamic plan-form
and section variables, along with its structural sizing variables, for minimum load carrying structural weight
subjected to structural, aerodynamic, performance and geometric constraints. The associated Multi-Disciplinary
Analysis (MDA) problem essentially involves coupled solution of the state equations of the aerodynamic and the
structural disciplines. The MDO problem, however, is posed as a three discipline coupled problem, with the trim
(maneuver) process required to define structural design loads considered as a separate discipline by itself. The
present work is limited to linear structural and aerodynamic analyses and the coupled MDA is carried out by nested
iterations within the disciplines, treating the disciplinary analyses as black boxes. This leads to a number of
interesting reformulations of the MDO problem based on (i) the reordering of the nested iterations and (ii)
decoupling the nested iterations at different levels through the introduction of pseudo design variables and pseudo
constraints, which allows the optimizer to be used for satisfying the interdisciplinary consistency. This is the focus
of the paper. Six variants of the MDO problem formulation are evolved. An important issue studied is the handling
of convergence of nested iterative loops and their impact on the computational efficiency of the MDO process. A
special constraint based on a divergence control parameter has been formulated to handle instability. Special
relaxation features are implemented to accelerate convergence.
The implementation of the architectures and the subsequent design optimization is carried out within the
framework of a software code, named “WingOpt”, developed by the authors. An overview of WingOpt is given in
the next section, followed by the definition of the specific design problem considered. The formulation and
implementation of MDO architectures is presented next, followed by some typical results. Numerical results are
obtained with reference to the B737-200 wing, taken as a typical example baseline configuration. Optimization
results from the different formulations are compared to study their computational performance. The results
generated also help to bring out the impact of aeroelasticity on the wing design.
II.
Overview of WingOpt
WingOpt is designed to carry out simultaneous structural and aerodynamic optimization of an aircraft wing
focused around static aeroelasticity. The software framework is designed to have complete flexibility in setting up
wing optimization sub-problems by permitting choices for design variables, constraints and objective function
within the context of an overall definition of the wing design problem. The architecture also permits options for
selection of optimizer, structural analysis method, method for handling aeroelastic couplings etc. To define the
overall optimization problem the wing is divided into a number of spanwise stations. The number of stations is input
driven. Each spanwise station consists of three segments, namely, the leading and the trailing edge control segments
and the mid-box segment. The mid-box segment is the structural multi-cell box, which bears the entire loading,
consisting of aerodynamic pressures, engine thrust and inertia relief due to engine, fuel and wing weights.
A. Optimization Problem
Design variables can be selected from a superset of candidate variables. Design variables can be either global or
local depending on whether they are independent of the station, or they define an attribute of the wing specific to a
particular station, respectively. Wing loading, aspect ratio, taper ratio, sweep-back angle, structural wing-box size
and location (normalized with respect to the wing chord), root angle of attack, cruise Mach no. and thickness to
chord ratio at the root section form the global variable set. The station-wise local variables set consists of skin, sparweb and rib-web thicknesses, thickness to chord ratio at the end of each station, spar and rib cap areas, parameters
defining camber of the airfoil shape, jig twist, etc. . Values of the variables which are not chosen as design variables
are set to desired values, which remain constant throughout the optimization process.
Similar flexibility is available in the selection of the constraint set from a superset of functions. Both equality
and inequality constraints are permitted. There is also provision to define multiple load cases. Constraint functions
include principal stress constraints, load carrying structural weight constraint, cruise range constraint, L/D
constraint, takeoff distance constraint, drag divergence constraint, sectional Clmax constraint, etc. It may be noted that
there are no explicit deformation or divergence constraints in the problem definition. Aeroelastic deformations are
limited indirectly by sectional Clmax constraints. Any of the above constraints can be set as an objective function. For
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example, the objective can be minimization of load carrying structural weight, maximization of cruise range,
maximization of L/D of the wing etc, subject to various aerodynamic and structural constraints. However, at the
present level of development, buckling constraints are not yet supported. The optimization problem setup is
completely input driven.
B. Aerodynamic and Structural Analysis
Aerodynamic computations are performed using an in-house developed code which is based on the Vortex
Lattice Method (VLM). Structural analysis is carried out using either the Equivalent Plate Method (EPM)13 or the
Finite Element Method (FEM). These medium/high fidelity methods are integrated with WingOpt. Further
aerodynamic modules for functions/evaluations such as drag divergence Mach number, takeoff velocity, range, Cdo,
etc. based on low fidelity, empirical/semi-empirical methods14 are also incorporated.
The aerodynamic mesh for the VLM analysis is automatically generated as a 2-dimensional parametric mesh
with control over the mesh panel size along the chord-wise and span-wise direction. Aerodynamic Influence
Coefficient (AIC) matrix is computed by the VLM code for given mesh and Mach number. Pressure distribution is
calculated as shown in Eq. (1)
1
(1)
{ p}  q AIC
{ }
p
where p is the panel pressure vector, q is the dynamic pressure and p is panel angle of attack vector given by



(2)
    
p
r
c
t
e
In equation (2), r is the aircraft angle of attack, c, t, e are the vectors of the additional panel angle of attack due
to camber, jig twist and structural deformation, respectively. Along with the pressure distribution, the VLM code
also calculates the Cl distribution, Cdi and overall CL of the aircraft and other such parameters of use in aerodynamic
performance evaluation.
Structural analysis is carried out either by EPM or FEM. MSC/NASTRAN is used in batch mode for Finite
Element Analysis (FEA). 4-noded 24-dof plate elements are used to model wing skin, spar-web and rib-web. 2noded 2-dof rod elements are used for modeling spar caps and rib caps. The pressure loads are mapped on to the
nodal points of the FE mesh using the principle of the energy equivalence. Finite Element Analysis (FEA) is carried
out using MSC/NASTRAN. Elastic deformation is calculated by solving Eq. (3)
K d  f 
(3)
where f is the nodal force vector, [K] is the structural stiffness matrix and d is the vector of structural deformation at
the FE nodes. Since FEA is to be carried out automatically within optimization cycles, it becomes necessary to run
MSC/NASTRAN in batch mode. Special wrapper modules have been written for automatically generating the finite
element mesh and the input deck required by MSC/NASTRAN to run in batch mode and to extract the relevant
stress and deformation data by parsing its output file.
In EPM the wing is modeled as an equivalent plate based on classical thin plate theory. The lateral deformation
of the wing as an equivalent plate is represented in the form of Ritz series as
wx, y   i 1 ii ( x, y)
n
(4)
where w(x,y) is the lateral displacement function, i represents an assumed mode and i is the corresponding
generalized coordinate. The equivalent wing stiffness matrix is computed by minimization of the potential energy
with respect to the generalized coordinates.
C. Trim Equations
The longitudinal equations of flight mechanics for a quasi-steady pull-up manuever are
L( r )  n W ; M  0
(5)
z
where L is lift, nz is the normal acceleration load factor, W is weight of aircraft and M is the pitching moment at the
cg of aircraft. For satisfying moment equilibrium, details of the tail configuration are required. Currently, WingOpt
deals only with wing analysis and optimization. Therefore only the lift equilibrium equation is satisfied in the
present design optimization. A constraint on the pitching moment can be given.
The lift is a function of dynamic pressure, wing configuration and aircraft angle of attack. Since aerodynamic
analysis is linear and lift generated L is a linear function of the aircraft angle of attack r, the trim equation can be
solved exactly for a given dynamic pressure and wing configuration, in principle, provided the elastic lift-curve
slope and aeroelastic lift at zero aircraft angle of attack are known.
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D. Aeroelastic Equilibrium
The aeroelastic pressure distribution depends on e as per equations (1) and (2). The e can be calculated from
the structural deformation through an interpolation process. Structural deformation is itself a linear function of the
pressure distribution, thus closing the loop. This closed loop system is solved iteratively starting from e = {0}. At
the ith iteration the following computations are carried out
1
i
i
{ p}  (1 / q ) AIC
{ p }
i
1
i
(6)
{ f }  [T ] { p}
1
i
1
i
{d }  [ K ] { f }
i 1
1
i
{ p }
 [T ] {d }
2
where [T1] and [T2] are appropriate transformation matrices. The dimension and values of [T1], [T2] and [K]
matrices are different for EPM and FEM. Combining all the equations in (6) above, we get
i 1
i
(7)
{ p }
 [ A]{ p }
1
1
[ A]  (1 / q )[T2 ][ K ] [T1 ][ AIC ]
where
The form of the [A] matrix as shown in equation (7) is only notional. The actual computations involve a
sequential solution of equations (1) and (3). The properties of the [A] matrix will govern the convergent or divergent
behavior of iterations. This is discussed in later in section IV. There are different ways in which the satisfaction of
the trim equation and the aeroelastic equations can be simultaneously achieved, at convergence of the iterative
MDA. This forms the basis for evolving different variants of the MDF architecture. This is discussed in detail later
in the paper.


E. Optimization in WingOpt
Optimization in WingOpt can be carried out using either FFSQP13 or NPSOL14. Both are commercially available
gradient based optimizers which use the sequential quadratic programming algorithm. Experience with initial trial
runs of WingOpt with FFSQP uncovered the presence of a number of redundant analysis calls during the
optimization cycles due to the inherent working of the optimizer. WingOpt was modified by adding an interface
module which maintains a limited history of previous analysis results to avoid redundant computations. This had a
very substantial impact on optimization time. Further, WingOpt was modified by adding an execution control switch
in such a manner that the VLM code does not calculate inverse of the AIC matrix at each call. Inversion is done only
when any of the aerodynamic parameters, which affect the AIC matrix, have been modified. Other execution control
sequences to reduce redundant computations depend upon the MDO architecture used. All gradients were computed
using finite differences in the optimizer.
III.
Design Optimization Problem Definition
To keep the computational burden manageable within the limits of the available resources and focus on the
testing of the MDO formulations, a subset optimization problem of a six station wing with 10 engineering design
variables and 89 engineering constraints is considered. The optimization problem was defined as the minimization
of the wing load carrying structural weight subject to all applicable stress constraints, drag divergence Mach number
constraint, sectional Clmax constraints, range constraint, fuel volume constraint and take-off distance constraint, with
skin thicknesses of mid box of all six stations, wing loading, sweep-back angle, root t/c ratio and aspect ratio as
design variables. Appropriate side constraints were chosen to limit the search space based on the baseline design.
The optimization problem was solved with and without the effect of aeroelasticity using both FEM and EPM, with
different MDO architectures. All optimizations were carried out using FFSQP. The size of the aerodynamic mesh as
well as the FE mesh varied with change in the design variables so as to keep the panel/element aspect ratios close to
one. The finite element structural model of the wing box was constructed using CQUAD4 plate elements for the skin
as well as spars and ribs. The wing box is a three cell structure with 2 internal spars. The spar and rib thicknesses
were kept fixed at minimum gage. Three load cases are considered for defining different engineering constraints.
These correspond to (i) maximum normal acceleration (for the stress and section Clmax constraints), (ii) long range
cruise speed (for the range constraint) and (iii) maximum cruise speed (for the drag divergence Mach number
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constraint). It should be emphasized here that although buckling constraints are an important design driver in wing
structures, the present problem does not include buckling, as the software does not yet support this feature.
IV.
MDO Implementation
In aeroelastic optimization, interaction between the disciplines can be handled r using one of the several single
level and bi-level MDO architectures8. In the present paper six single level MDO architectures have been formulated
to solve the aeroelastic optimization problem. The MDO architectures formulated here are variants of the generic
“Multi Disciplinary Feasible” (MDF) architecture and “Individual Disciplinary Feasible” (IDF) architecture
reported in the literature6,7. Elements of the All At Once” (AAO)6,7 architecture are also brought into the
formulation of the MDF and IDF variants. In MDF complete aeroelastic analysis is carried out at each optimization
call, whereas in IDF each aerodynamic and/or structural analysis are called only once in each optimization cycle.
Couplings in IDF are handled by optimizer with the help of special additional design variables and constraints also
known as coupling variables and interdisciplinary consistency constraints. In AAO, the state variables of each of the
disciplines are included as additional pseudo design variables in the design variables set and the residuals in the
corresponding state equations form additional equality constraint in the augmented optimization problem. Thus the
residuals in the state equations are driven to zero only at convergence of the optimization. This is also referred to as
Simultaneous ANalysis and Design (SAND).
A. Basis for MDO Architectures
The way in which MDA is performed forms the basis for
single level MDO architectures. There are two aspects to this,
Trim
viz.
Structures
1. How much of MDA is carried out by the optimizer? This
forms the basis for distinction between the MDF, IDF and
AAO formulations?
2. How the nested iterations are carried out in MDF
architectures?
Aerodynamic
Thus, as the number of interacting disciplines increases,
s
Structures
more number of MDO architecture variants can be
formulated. In this paper, six different MDO architectures
Trim
were formulated and implemented in WingOpt to solve the
Figure 1: Schematic representation of MDA for three discipline optimization problem considered here.
MDF1 and MDF2 architectures
Complete MDA of the current problem involves two levels of
iterations, one for solving the trim equations and other for
Aerodynamic
s
From optimizer
To optimizer
x , CLreq , t ,  c
Yes
{R(w)<)}?
No
r = 0
Update p
 p   e   t   c 
Aerodynamics
displacement (w)
CLelastic
Update r
Update p
Structures
 r  (CL  CL ) / CL
req
elastic
,rigid
 p   r   e   t   c 
Aerodynamics
aeroloads
2 MDF1
Algorithm
Figure Fig.
2: The
MDF1
Algorithm
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achieving aeroelastic equilibrium. These iterations can
be carried out along with optimization iterations to
form different MDO architectures.
From optimizer x , CLreq ,  t ,  c
Compute CL0elastic
To optimizer
Update  r
e = CLelastic - CLreq
 rj = (ej-2 rj-1 - ej-1 rj-2 )/(ej-2-ej-1 )
Yes
No
e < 
Yes
Update  p
R(w)< ?
No
Since there are two levels of iterations involved,
three variants of MDF architectures can be formulated.
These variants are termed as MDF1, MDF2 and MDF3
in this paper. Fig. 1(a) and Fig. 1(b) represent MDA for
MDF1 and MDF2 architectures respectively. The
difference arises due to the re-ordering of the nested
iterations. In MDF1, the trim equation is solved in the
inner loop and aeroelastic iterations are performed in
the outer loop.
B. MDF Architectures
The algorithm for the implementation of the MDF1
aeroloads
formulation is shown in Fig. 2. Keeping the rigid angle
of attack r zero, the lift due to camber, twist and
Structures
displacement
elastic deformation obtained in the previous iteration,
(w)
is first computed in the inner loop. The angle of attack
required to balance the lift equilibrium equation is then
Figure 4: The MDF2 Algorithm
computed from the required lift, the elastic lift and the
lift curve slope of the rigid wing. The panel angle of
attack is updated with the calculated r and the aerodynamic loads corresponding to lift balance are computed and
given to the structural analysis module. Hence during aeroelastic iterations, r varies with successive iterations so as
to always maintain lift equilibrium. Thus there are two calls to aerodynamic analysis and one to the structural
analysis. The aeroelastic iterations will terminate if iterations are found to be divergent, or if the number of iterations
reach a prescribed maximum value, or if the iterations converge. Convergence parameter (R(w))is based upon the
out of plane deformation of the trailing/leading edge at the wing tip. When the percentage increment in the
deformation from the previous iteration is less than the prescribed tolerance limit . Divergence is taken care of in a
special way as described later. On a plot of the lift versus aircraft angle of attack, this formulation is equivalent to
finding convergence by moving parallel to the line representing the rigid aircraft lift-vs-angle of attack line.
In MDF2, it is the other way around. The aeroelastic iterations are completed for a given r which is held fixed
till the aeroelastic iterations converge in the inner loop. The algorithm is shown in Fig. 3. Thus in this formulation,
instead of satisfying the trim equation in every aeroelastic iteration, it is solved after a converged aeroelastic analysis
at fixed angle of attack is completed. r is adjusted
outside the aeroelastic loop to satisfy the trim
From optimizer x , CLreq ,  t ,  c
equation. As shown in Fig. 4, the angle of attack
Compute CL0elastic
update in the current cycle is based on a linear
extrapolation/interpolation based on values of the lift
and angle of attack in the previous two cycles. Loop
To optimizer
termination criteria are the same as those for MDF1.
e < 
Additionally, the trim loop terminates when the
R(w) <  ?
displacement (w)
Yes
difference between computed elastic lift and the
required lift is within a tolerance . On a plot of the lift
No
versus angle of attack, this formulation will trace
vertical lines during the aeroelastic iterations and
e = CLelastic - CLreq
Update  r
approach the MDA solution by a trace of interpolation
 rj= (CLreq - CL0elastic)  r (j-1)
points on the horizontal line representing the lift
(CLelastic (j-1) - CL0elastic)
required.
Update  p
In MDF3, the nested loops of MDF2 are merged to
form a single loop. In this architecture, the changes in
aeroloads
the panel angle of attack due to the structural
Aerodynamics
Structures
deformation and those due to adjustment in r to
Aerodynamics
Figure 4: The MDF3 Algorithm
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satisfy the trim equation, are carried out simultaneously.
In the fourth variant of the MDF architectures, aircraft angle of attack r is chosen to be part of the design
variable set instead of computing it within the MDA. In order to ensure that trim equation is satisfied (equation (5)),
an additional equality constraint is included in the optimization problem definition. Thus the optimizer controls the
value of r so as to satisfy the trim equation eventually. This exercise is similar to that of AAO architecture.
However, unlike in full AAO, where all state variables of the MDA are augmented to the engineering design
variables set, only the trim variable and the trim equation augment the optimization problem here. This architecture
is termed as MDF4 in this paper. In this architecture, the trim block is effectively removed from the MDA, which
now returns the residual in the trim equation. Thus, MDF4 has elements of AAO in it and is no longer purely MDF
in nature. On a plot of the lift versus angle of attack, this formulation produces points which move around
unpredictably, without a trend, before settling down to the converged solution.
C. IDF Architectures
The IDF architecture is implemented in WingOpt by converting the structure-aerodynamic coupling variables
(i.e. structural deformation variables) to a set of pseudo design variables which are added to the engineering design
variables. With each pseudo variable is associated an ICC (Interdisciplinary Consistency Constraint), which is an
equality constraint to be satisfied by the optimizer through adjustment of the pseudo variables so that the final
optimized solution is consistent with aeroelastic equilibrium. In the case of EPM these pseudo variables (*) are the
counterparts or system level copies of the coefficients (generalized co-ordinates ) of the assumed modes in the Ritz
series in equation (4). The optimizer specified value of the pseudo coefficients are used to calculate angle of attack
and hence the aerodynamic loads, which are passed to the structural analysis. The structural analysis computes the
actual values of the Ritz polynomial coefficients by satisfying structural equilibrium. No iterations are required in
the disciplinary analysis. The details of the algorithm are given in Fig. 5.
The difference between pseudo and actual values is the most obvious way of formulating the ICCs. However,
this type of formulation leads to computational problems associated with constraint normalization. There will be as
many equality constraints as the number of pseudo design variables. Since all these constraints will be required to be
driven to the same tolerance during the optimization process, normalization of the constraints is a must. However,
this is difficult, since the values of the generalized co-ordinates vary by large magnitudes. Therefore, an alternate
strategy to reformulate the ICCs in terms of physical displacements was evolved while keeping the generalized
coefficient as the pseudo design variables. Using the pseudo variables, values of the normal displacement was
computed at a number of points equal to the number of pseudo variables. These were then compared with the
computed values of the actual displacements at these points to form the ICCs. Normalization of these constraints
possible since they are based on physical variables. Using the information about the gradient of the ICC, the
optimizer adjusts the pseudo variables to drive the
difference to zero, as the optimization process
converges. No iterations are required in the
From optimizer x , CLreq ,  t ,  c , *
disciplinary analysis. The additional variables and
Calculate  p  p = t +  c+ e  e =w* / x
constraints put extra burden on the optimizer, but at
the same time relieve the analysis from iterative
aeroelastic computations. In the case of FEM, a least
ICCs: * - 
Aerodynamics
squares fit using polynomial functions is generated
To optimizer
for the nodal displacements to reduce the number of
the pseudo coupling variables. The coefficients of the
polynomial functions in the fitted surface are then
From e = 0
Compute  & ICCs used as the psuedo variables in the IDF.
Calculate  r
Similar to MDF4, IDF2 was formulated wherein
the trim state variable (aircraft angle of attack) and
displacement (w)
the trim equation was also augmented to the design
variable/constraint sets respectively, in addition to the
Update  p
coupling variables and the ICCs. Thus in all six
different MDO architectures (IDF1, IDF2, MDF1,
aeroloads
MDF2, MDF3, MDF4) have been formulated. Each
architecture has different impact on the nonlinearity
Aerodynamics
Structures
of the design space. Further, each method of handling
the inter- disciplinary coupling has its own
Figure 5: The IDF1 Algorithm
convergence behavior. In turn this will affect the
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efficiency and robustness of the computational process. This is the focus of the study.
D. Handling Aeroelastic Divergence and other convergence issues in the Optimization
During the optimization, the optimizer may drive the design to minimum possible skin thickness. This may lead
to wing static divergence due to low structural stiffness. Since the wing tends to diverge at that design point,
aeroelastic iterations will not converge. Thus the iterations will have to be terminated prematurely when the number
of aeroelastic iterations crosses a certain pre-fixed maximum number of iterations. In such a condition the value of
the objective function or constraints will be inconsistent with the physics of the problem. During gradient
computations, non-convergence of the aeroelastic loop will lead to incorrect gradients, which may throw the
optimizer off-track. In optimization based on iterative aeroelastic analysis, there is no direct way to handle the
instability of the aeroelastic loop. Therefore, it is necessary to communicate the unstable behavior to the optimizer
in some way. In this paper a new methodology was evolved and implemented in WingOpt by which wing
divergence was detected early in the aeroelastic iterations, thereby stopping the iterations and saving time. The nonconvergence due to premature termination of the loop is taken care of by formulating a special constraint which
communicates this to the optimizer. This was done by defining a divergence control parameter dcp based on the
observed behavior of the wing deformation.
Let z denote a parameter which is representative of the deformation and is used to monitor the elastic
deformation during the aeroelastic iterations. For example this could be the normal displacement of the trailing edge
of the wing tip or the the wing tip section averaged angle of attack. If z is found to be strictly increasing or
decreasing (monotonic) till n m consecutive iterations, then the dcp is evaluated using the following expression
dcp  znm   znm  1  znm  1  znm  2
where z (i ) is value of z at the ith iteration, and
(8)
n m is a predefined value set for testing monotonic behavior. If the
behavior of the monitoring parameter is oscillatory, then values of z at successive peak points are detected and
stored using a peak detection algorithm. The value of dcp is evaluated after three consecutive peaks are detected
using the following expression
dcp  z p3   z p2   z p2   z p1 
(9)
where z(pi) is value of z at the i iteration and p1, p2 and p3 are iteration number corresponding to occurrence of
first, second and third peak respectively.
A value of dcp greater than zero in each case indicates that the rate of change in the value of z is increasing and
hence will lead to divergence. Therefore aeroelastic iterations are terminated on the identification of divergence. The
original optimization problem statement is modified to include an additional constraint dcp < 0. This constraint
drives the optimizer to increase the stiffness of the wing. A negative value of dcp indicates converging tendency and
the iterations are continued till convergence. A number of interesting issues related to the computational
implementation of dcp and convergence/divergence of the aeroelastic iterations were encountered during this study
and some required special conditioning methods to be implemented to achieve/accelerate convergence.
At most points in the design space the convergence/divergence behavior of the iteration error was found to be
oscillatory. Further, in all cases when the aeroelastic iterations were carried out at constant angle of attack, the
oscillatory convergence/divergence was found to be regular and the dcp criterion worked well to predict the
behavior. Oscillatory/non-oscillatory and convergent/divergent behavior can be related to the eigenvalues and
eigenvectors of the [A] matrix in equation (7), though these were not computed in this study, since the matrix could
not be explicitly evaluated. However, it was observed from the iteration histories, during initial trials, that in the case
of architectures where the angle of attack was also simultaneously varying along with the aeroelastic iterations, the
iterative behavior became irregular and unpredictable with sudden jumps in the oscillation amplitude followed by a
period of regular convergent behavior. In such cases, the dcp criterion was not found to be robust. This was
particularly so for the MDF3 and MDF1 architectures. Through numerical experimentation, it was found that this
problem could be overcome with by relaxing the requirement that the angle of attack be updated in every iteration.
Smoother behavior was obtained by updating the angle of attack in alternate iterations, instead of every iteration.
Further, in the case of regular behavior, in a few instances, the convergence was observed to be very slow. In such
cases convergence could be significantly improved by restarting from a new initial guess at the end of a block of
iterations. The new guess was taken to be the mean of the previous two consecutive peaks at the end of each block.
th
8
American Institute of Aeronautics and Astronautics
V.
Typical results
The Boeing 737-200 aircraft wing was chosen as a candidate problem (baseline) for the comparative study of the
problem formulations for multidisciplinary design optimization. Since wing box structural topology and sizing
details were not available, they were established by first performing pure structural optimization without
aeroelasticity using baseline aircraft and wing data.15 The result was considered as a baseline structural design.
Further, with different wing-box configurations, similar exercises were carried out without and with the aeroelastic
loop to establish an optimal structural box for minimum load carrying structural weight. Optimization results were
then obtained for a simultaneous structural and aerodynamic optimization problem, with and without static
aeroelasticity, for the redesign of the baseline configuration. The baseline structural optimization problem involved
minimum weight design with six skin thicknesses as design variables subject to stress constraints corresponding to a
single load case of a symmetric pull-up maneuver. The coupled aerodynamic and structural design problem involved
ten design variables, including the six skin thicknesses along with three aerodynamic planform parameters and the
wing thickness at root. Results of the optimization study are tabulated in Table 1. These results are obtained using
EPM. Satisfactory FEM results have yet to be obtained due to some numerical noise problems arising from
inadequate precision of the FE results. Further work with FEM is ongoing. The table lists the optimized values of the
design variables, shows the active constraints followed by numerical values showing the computational performance
in terms of number of analysis executions, objective and constraint function calls, and time required for the
optimization. It may be noted here that the aerodynamic analysis, in this case the VLM, is much more costlier in
execution time than the structural analysis based on EPM. The time required for FEM would be expected to be
substantially more than the aerodynamic analysis. The structural load carrying weight in Table 1, is that of the skin
alone. All optimizations in Table 1 have been carried out from exactly the same starting point and all fixed
parameters, including all optimizer related parameters have bee kept identical across all the optimizations.
Results under the structural baselines have been obtained with the aerodynamic variables fixed at their baseline
values as given in the 737-200 datasheet in Reference 17. The structural baseline with aeroelasticity on has been
evolved using MDF1. Structural baseline with aeroelasticity on shows a significantly lower weight than its
counterpart without aeroelasticity, due to the beneficial effect of the favorable pressure distribution in the flexible
case, when aerodynamic variables are held fixed. The results for the coupled aerodynamic and structural design
problem without aeroelasticity shows a favorable impact on the weight in comparison with the structural baseline,
bringing out the positive effect of MDO. However, there is a price to pay here in terms of the computational cost,
which for the small problem considered here, still shows a very large jump, even without aeroelasticity. The
aerodynamic design variables appear to have a smaller freedom than the structural variables and the aerodynamic
and performance constraints appear to govern the design tightly, as is to be expected for this class of aircraft. In fact
a comparison with the coupled design with aeroelasticity on shows that there does not seem to be any further benefit
to gain from the aerodynamics. The aerodynamic design variables have almost the same values in the two cases,
with most constraints remaining active. However, on the structural side, there is further benefit to derive from the
aeroelastic effects in the presence of aerodynamic design variables. The combined effect of simultaneous design
optimization and aeroelasticity is significant showing a combined weight reduction of about 30 %. Of course this is
not completely realistic because buckling has not been taken into the design constraints. Nevertheless, the power of
MDO comes through clearly.
A comparison across the architectures shows that all the MDFs and IDF1 give nearly the same optimum
solution. This is to be expected as they all solve the same design problem. However, IDF2 appears to have failed.
All other optimizations ended with a normal termination from the optimizer (FFSQP) except IDF2. IDF1, although
successfull, requires an order of magnitude higher time to reach the solution. In the case of IDF1 there are 32
additional equality constraints and design variables, 16 each for two load cases (one is the structural load case and
the other is the range). Thus, although the iteration time has reduced to zero, the optimization time has jumped
substantially. It may be noted that the IDF architecture has no way to capture stability information, unlike MDF.
However, the stress constraints appear to have guided the method to the same solution quite well. Further, the IDF
method failed when the ICCs were formulated in their original form. The method has worked successfully only
when the ICCs were reformulated in terms of physical displacement variables. In IDF2, three additional design
variables appear in the form of angle of attack in each load case. These variables couple the pseudo variables with
the aerodynamic constraints, which does not happen in IDF1. Thus these constraints become highly nonlinear
coupled functions of the pseudo variables. The optimizer was unable to find a feasible domain.
9
American Institute of Aeronautics and Astronautics
Amongst the MDF architectures, MDF2 was seen to be the fastest, with MDF1 being a close second. MDF1
requires a higher number of aerodynamic calls but very few structures calls. The opposite is true of MDF1. Since the
structures discipline is relatively inexpensive here, MDF2 appears to score over. However with a somewhat higher
fidelity structural analysis, MDF1 may become lower. However, MDF1 is not robust with respect to the dcp
constraint. MDF1 has shown failure to address divergence correctly from some starting points. MDF2 and MDF4
are robust with respect to dcp. However, MDF4 requires a large amount of time, because of the three additional
variables and corresponding equality constraints that the optimizer has to handle. MDF3, due to its very complicated
mechanism of managing both aeroelasticity and trim simultaneously, loses out on both performance as well as
robustness. This has been verified by numerical experiments.
Acknowledgements
This work was carried out at the Centre for Aerospace Systems Design and Engineering (CASDE), IIT Bombay.
The authors wish to acknowledge the support received from CASDE. CASDE is funded by the Aeronautics
Research and Development Board, Ministry of Defence, Govt. of India. The financial support of ARDB is gratefully
acknowledged.
References
1”Current
State of the Art of Multidisciplinary Design Optimization”, White Paper, AIAA MDO Tech. Comm., Sept. 1991
2Giesing, J. P. and Barthemy, J. M., “Summary of Industry MDO Applications and Needs," AIAA Paper 98-4737, Sept.
1998.
3Osterheld, C. M., Heinze, W., and Horst, P., Influence of Aeroelastic effects on preliminary aircraft design," Proceedings of
the ICAS 2000 Congress, Harrogare, UK, Sept. 2000, pp. 146.1-146.10.
4Wakayama, S. and Kroo, I., “The Challenge and Promise of Blended-Wing-Body Optimization," AIAA Paper 98-4736,
Sept. 1998.
5Rohl, P. J., Mavris, D. N., and Schrage, D. P., “Combined Aerodynamic and Structural Optimization of a High-Speed Civil
Transport Wing," AIAA Paper 95-1222, April 1995.
6Balling, R. and Sobieski, J. S., “Optimization of Coupled System: A Critical Overview of Approaches," AIAA J., Vol. 34,
No. 1, Jan. 1996, pp. 6-17.
7Cramer E. J., Dennis J.E., Frank P.D., Lewis R.M. and Shubin G.R., “Problem Formulation for Multi-disciplinary Design”,
SIAM J. of Optimization, Vol. 4, No. 4, November 1994, pp. 754-776.
8Kodiyalam S. and Sobieszczanski-Sobieski J.,”MDO – Some Formal Methods, Framework Requirements and Application to
Vehicle Design”, Int. J. of Vehicle Design, (Special Issue), 2001, pp. 3-22.
9Hulme, K. F. and Bloebaum, C. L., “A simulation-based comparison of multidisciplinary design optimization solution
strategies using CASCADE," Journal of Structural Multidisciplinary Optimization, Vol. 19, 2000, pp. 17-39.
10Balling, R. J. and Wilkinson, C. A., “Execution of multidisciplinary design optimization approaches on common
problems," AIAA Journal, Vol. 35, pp. 178-186.
11Alexandrova N.M. and Kodiyalam S., “Initial Results of an MDO Method Evaluation Study”, AIAA Paper No. 98-4884.
12Gumbert, C. R., Hou, G. J. W., and Newman, P. A., “Simultaneous Aerodynamic and Structural Design Optimization
(SASDO) for a 3-D Wing," AIAA Paper 2001-2527, June 2001.
13Giles, G., “Equivalent Plate Analysis of Aircraft Wing Box Structures with General Planform Geometry," Journal of
Aircraft, Vol. 23, No. 11, Nov. 1986, pp. 859-864.
14Raymer, D. P., Aircraft Design: A Conceptual Approach, AIAA Education Series, 3rd ed., 1989.
15Zhou, J. L., Tits, A. L., and Lawrence, C. T., User's Guide for FFSQP Version 3.7: A FORTRAN Code for Solving
Constrained Nonlinear (Minimax) Optimization Problems, Generating Iterates Satisfying All Inequality and Linear Constraint,
System Research Center, University of Maryland, College Pard, MD 20742.
16Gill, P. E., Murray, W., Saunders, M. A., and Margaret W., USER'S GUIDE FOR NPSOL 5.0: A FORTRAN PACKAGE
FOR NONLINEAR PROGRAMMING.
17Jenkinson L. R., Simpkin P., R. D., “Civil Jet Aircraft Design - Aircraft Data File - Boeing Aircraft,"
http://www.bh.com/companions/034074152X/appendices/dataa/table-2/table.htm.
10
American Institute of Aeronautics and Astronautics
Item Name
Structural
Baseline
Structural
Baseline
Coupled
Design
MDF1
off
on
off
6.25
5.26
5.46
3.37
2.77
2.86
5.03
3.84
3.88
2.47
2.00l
2.01
2.00
2.00l
2.00l
l
2.00l
2.00l
5645
5645
581
6
25
25
31.3
8.80
8.80
8.19
16
*
NA
NA
NA
NA
NA
16
*
NA
NA
NA
NA
NA
20
*
on
4.
66
2.
39
2.
95
2.
00l
2.
00l
2.
00l
58
40
31
.3
8.
14
20
.0u
*
*
*
*
MDF2
MDF3
MDF4
IDF1
IDF2
2.89
2.00l
on
4.
66
2.
40
2.
92
2.00l
2.00l
2.00l
on
4.
67
2.
42
2.
89
2.
01
2.00l
2.00l
2.00l
2.00l
8.13
58
41
31
.3
8.
13
20
*
20
*
58
42
31
.3
8.
13
19
.9
*
on
4.7
3
2.5
9
2.8
2
2.1
9
2.0
3
2.0
8
584
0
31.
3
8.1
3
19.
9
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
NA
10
10
10
13
42
Aeroelasticity
Skin thickness 1 (mm)
Skin thickness 2 (mm)
Skin thickness 3 (mm)
Skin thickness 4 (mm)
Skin thickness 5 (mm)
2.00
Skin thickness 6 (mm)
Wing Loading (N/m2)
l
Sweep (deg.)
Aspect Ratio
t/c ratio (%)
Stress Constraints
Fuel Volume
Max. Cl
Take-off distance
Range
Drag Diverg. Mach no.
dcp
NA
on
4.67
2.41
5841
31.3
No. of Design Variables
6
6
10
24
25
161
83
29
104
2
121
70
2965
1785
21
148
30
16
57
171
697
581
No. of Constraints
No. of Analysis runs
Objective function calls
Constraint function calls
Time (Aerodynamic)
Time (Structural)
Time (Total)
Load carrying structural
weight (kg) - Objective
31
31
31
31
61
57
16
8
402
5
87
17
16
43
288
6
121
7
4
835
56
52
16
3
27
3818
75
662
879
50
45
10
4
42
2998
97
640
12
26
38
11
202
0
1664
19
48
American
903 Institute
56 of Aeronautics and
77Astronautics
14
6
68
4954
02
977
49
49
49
577
5
494
4
5
61
370
81
118
01
731
660
981
59
118
10
106
513
507
o
n
1
6.14
1
5.58
1
5.69
8.
00
8.
00
8.
00
5
734
2
4.4
8.
87
1
5.9
*
*
*
N
A
4
5
6
4
4
23
1
36
7
643
2
050
1
34
2
345
2
255
Table 1: Optimization results
with different architectures
* Active
or near active constraint ,
# Quantities in number; l --- lower
bound;
u - ---upper
bound NA Not applicable
12
American Institute of Aeronautics and Astronautics