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RAeS 3
rd
Aircraft Structural Design Conference, 9-11
th
October 2012, Delft, The Netherlands.
1
A Bi-Level High Fidelity Aero-Structural Integrated Design
Methodology - A Focus on the Structural Sizing Process
Christophe Blondeau
1
François-Xavier Irisarri
2
, François-Henri Leroy
2
Onera - The French Aerospace Lab, Châtillon, France
Itham Salah El Din
3
Onera - The French Aerospace Lab, Meudon, France
The optimization of wing-body configurations using
standard disciplinary optimization algorithms and
high fidelity numerical tools is today mature enough
to be used for real aircraft design process by aircraft
manufacturers. Among the challenges that arises
today is the capability to design optimal multi-
disciplinary feasible configurations within a single
global design process run. In the present paper a bi-
level and bi-disciplinary optimization, based on a
BLISS-like algorithm, of a transonic wing-body
configuration is presented. The aero-structure
optimization process shows that it is possible to
improve the performance of the design while
maintaining the feasibility for both aerodynamic and
structure disciplines at each step of the optimization
algorithm.
Allthough rather dedicated to the detailed description
of the high fidelity structural optimization process
(aeroelastic loads, structural sizing), this paper also
introduces the ARTEMIS (Advanced R&T Enablers
for Multidisciplinary Integrated Systems) project
which objective was to set up an original strategy to
efficiently couple a conceptual design level optimi-
zation tool and a high fidelity aero-structure bi-level
optimization capability.
I. Introduction
This paper presents recent developments and results
achieved at ONERA on multi-disciplinary wing
optimization of a wing-body configuration using a bi-
level algorithm.
______________________________________________
1
Research Engineer, Aeroelasticity and Structural Dynamics Dept.,
Christophe.Blondeau@onera.fr
2
Dr. Research Engineer, Composite Materials and Structures Dept.,
Francois-Xavier.Irisarri@onera.fr, Francois-Henri.Leroy@onera.fr
3
Dr. Research Engineer, Applied Aerodynamics Dept.,
Itham.Salah_el_Din@onera.fr
This work is part of the development of a general
framework named ARTEMIS (Advanced R&T Enablers
for Multidisciplinary Integrated Systems) in collabora-
tion between Airbus and ONERA. The ARTEMIS
project is meant to demonstrate the possibility to couple
two processes enabling the optimization of an aircraft,
considering design space at the system level (an
important number of disciplines and high level
requirements of the aircraft) along with the use of high
fidelity models and tools to achieve preliminary MDO of
the aircraft up to local sizing of components. In the
present study the two disciplines considered are the
aerodynamic of the flexible wing and the wing box
structural mechanics, the aim being to reach a feasible
optimum with respect to these two disciplines.
The problem definition, including the reference
configuration and the optimization problem formulation
will be presented. Then, a section will be dedicated to the
description of the optimization algorithm, focusing on the
composite wing structural optimization. The focus on the
aerodynamic sizing process will be the subject of a
separate communication [24]. Finally, the results of the
bi-level optimization process will be presented and
analyzed.
In the frame of the collaboration of Airbus with the R&T
community, ONERA proposed a roadmap on the
necessary steps to gradually implement the Airbus’
MDDC (Multi-Disciplinary Design Capability). This
roadmap focused on two main research axes:
• Bi-Disciplinary Process (BDP): developments concer-
ning aerodynamic / structure optimization with high
fidelity models and tools to be used at the
preliminary/detailed design level,
• Global Aircraft Process (GAP): developments concer-
ning the complete aircraft optimization (large number
of disciplines involved and lower fidelity tools) to be
used at the conceptual design level.
RAeS 3
rd
Aircraft Structural Design Conference, 9-11
th
October 2012, Delft, The Netherlands.
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This paper introduces the ARTEMIS project which
objective is to propose a methodology to couple BDP and
PAG processes, in order to achieve an optimal design
which satisfies the global aircraft performance for a
whole mission profile as well as the disciplinary
(aerodynamic, structure) constraints at the preliminary
design level (see Figure 1). This organization/architecture
is a complete breakthrough compared to the classical
sequential iterative design lines classically operated by
aircraft manufacturers.
In the present context, we will focus on the bi-
disciplinary aero-structural optimization process. A
BLISS-like approach has been chosen (Bi-Level
Integrated System Synthesis) [26]. An advantage of this
method is to preserve disciplinary design autonomy
which is of prime importance to the aircraft’s
manufacturer. This ensures that the design teams keep
responsibility of their own design methodology and
environment at the low price of changing their specific
objective function for the contribution to the global
system performance.
At the upper level, the wing planform shape is designed
(typically sweep angle, span, outer wing taper ratio,
relative thicknesses).
At the disciplinary level, the aerodynamic task is
responsible for airfoils and twist law optimal design (up
to 100 design variables, geometry and global
performance constraints) while the structural task
optimises the wingbox stiffness through composite skins
and spars physical properties (several hundreds of design
variables, up to 20000 design constraints).
Both disciplinary processes are based on up-to-date high
fidelity tools.
Additionally, the structure disciplinary design task
embeds a critical loads computation module which is
responsible for providing the correct set of aeroelastic
sizing loads. The wing box design is performed using a
bi-level scheme. At the upper level, only macroscopic
parameters and constraints are addressed while the lower
level is dedicated to the assessment of buckling criteria.
The buckling reserve factors are then approximated using
a response surface approach which is then fed back to the
macroscopic wing design level.
The link between optimal disciplinary designs and upper
level optimization problem is performed through a post-
optimal sensitivity analysis.
The process makes use of advanced numerical tools like
discrete adjoint technique for computing the flexible
aerodynamic gradients, analytic parameterisation for
CFD grid deformation and morphing techniques for FEM
mesh shape optimization.
Preliminary aero-structural optimization results for a
realistic civil aircraft transport with a composite wing
(typically A350 like) will be discussed and especially
how such an MDO approach helps in understanding the
key drivers of a candidate design.
Figure 1 – Organization of the ARTEMIS project.
II. Bi-level bi-disciplinary optimization problem
formulation
While the GAP follows a Multi-Discipline Feasible
(MDF) approach coupled with a global optimizer, the
BDP is centered around a bi-level, bi-disciplinary (aero-
structure) architecture with the aim of addressing the
issue of high fidelity modelling and numerical
computations.
The optimization problem is defined using the BLISS
MDO formulation. This approach uses two design levels
which are qualified respectively as high-level and
disciplinary-level. The high level design phase consists in
optimizing the aircraft using a common parameterization
for both disciplines. The disciplinary level optimization
is performed at fixed high level design variables
separating the aerodynamic and structures design spaces.
Disciplinary level calls high fidelity models, using
Computational Fluid Dynamics (CFD) and Compu-
tational Structural Mechanics (CSM) tools for the
analysis of the different models.
MDO methods were first used to simulate, predict and
optimise complex aerospace systems and products [28],
and has now matured into a critical aerospace technology
within Europe, see for instance [21]. Several MDO
methodologies have been proposed over the last two
decades, ranging from relatively straightforward
formulations (All-At-Once and Individual Discipline
Feasible [6], Multiple-Discipline Feasible [17]) to those
aiming at partitioning large systems into sub-system
optimization problems where a coordination algorithm
drives the sub-problem designs towards an optimal
RAeS 3
rd
Aircraft Structural Design Conference, 9-11
th
October 2012, Delft, The Netherlands.
3
solution for the overall system. These decomposition
methods exploit the hierarchy in MDO problems by
reformulating them as a set of independent disciplinary
sub-problems and introducing a master problem to
coordinate the sub-problems towards the optimal system
design. The most promising decomposition methods
include Bi-Level Integrated System Synthesis (BLISS)
and its variants, Collaborative Optimization and
Analytical Target Cascading [27, 4, 16, 31]. A classifi-
cation based on the problem formulation structure has
been proposed in [32].
The Bi-Disciplinary process, as explained earlier, aims at
optimizing the shape of the wing (planform, airfoils and
twist law) and its internal structure. The process is based
on high fidelity tools used at the detailed design level.
For CFD, the elsA/AEL aeroelastic analysis capability
[13, 5] is used while for CSM, MSC/NASTRAN
software applies. An advantage of the BLISS
methodology is the possibility to carry out the
disciplinary optimizations in an independent manner. The
different teams keep the responsibility for their design at
the low price of changing their specific objective
function for the contribution to the global system
performance [1].
A general description of BLISS is given in the next
figure knowing that:
- X
a
corresponds to aerodynamic variables,
- X
s
corresponds to structural variables,
- Z corresponds to system or shared variables.
Post-Optimal
sensitivities or RSMs
System
Optimisation / Z
X
a
+∆
∆∆
∆X
a
Z+∆
∆∆
∆Z
MDA Aero-Structure
Adjoint Aero-Structure
Gradient / X
a
, X
s
, Z
CSM
Optim.
/ X
s
CFD
Optim.
/ X
a
X
s
+∆
∆∆
∆X
s
Post-Optimal
sensitivities or RSMs
System
Optimisation / Z
X
a
+∆
∆∆
∆X
a
Z+∆
∆∆
∆Z
MDA Aero-Structure
Adjoint Aero-Structure
Gradient / X
a
, X
s
, Z
CSM
Optim.
/ X
s
CFD
Optim.
/ X
a
X
s
+∆
∆∆
∆X
s
Figure 2 – BLISS-based aero-structural optimization
algorithm.
This MDO method for decomposition based optimization
of engineering systems involves system optimization
with relatively small number of design variables and a
number of sub-system optimizations that could each have
a large number of local variables. The algorithm
alternates between disciplinary and global optimization
steps.
The first step consists in an elimination of the
interdisciplinary state variables through a Multi-
Disciplinary Analysis (MDA). Subsequently, the coupled
total derivatives of relevant state responses are computed
with respect to local and shared variables.
At the local level disciplinary parameters X
a
and X
s
, are
optimized to improve the contribution of each discipline
to the global performance. This is done through a simple
separation technique :
0 0
(1 )
d d
d d
D W
T T
a s
a s
J C W
J J
J J J J
X X
X X
α α
= + −
≈ + ∆ + ∆ = +∆
(1)
Once that the optimal X
a
and X
s
are obtained, the system
optimization is performed with respect to the shared
variables Z, while preserving the disciplinary optimality.
A trust region approach controls the validity of the
approximate models involved at the system level.
When considering high-fidelity CFD/CSM, all the steps
in an MDO process become very challenging. The
objective here is to set up a technological demonstrator
for the BLISS procedure which takes advantage of the
most recent research developments in the following
areas:
•
Bi-disciplinary aeroelasic simulation using the Onera
elsA-AEL module.
•
Computation of total flexible aero-structural gradients
with respect to shared and local disciplinary variables
through adjoint techniques. All the direct and cross
dependencies of weight and drag are to be adressed.
•
Parametric CFD grid generation using free-form
techniques and FEM mesh deformation based on
morphing techniques.
•
Identification and computation of critical transonic
load cases with advanced CFD/CSM aeroelastic
analyses.
Taking a closer look at equation (1),
J
∆
contains the
classical
D a
dC d
X
and
W s
dW d
X
terms and the
D s
dC d
X
and
W a
dW d
X
cross-terms. The former one
represents the sensitivity of the arodynamic flow with
respect to structural design parameters (i.e. flexibility),
the latter the dependency of the structural weight to the
local aerodynamic shape parameters (through critical
loads redistribution). These cross-terms have not been
explicitly computed in the present work, hence the
RAeS 3
rd
Aircraft Structural Design Conference, 9-11
th
October 2012, Delft, The Netherlands.
4
aerodynamic and structural objective functions only
depend on direct sensitivities. However, computation of
D s
dC d
X
through the discrete adjoint technique is still
an active research area [10]. This term is of prime interest
when considering the design of highly flexible wings.
The demonstrator has been applied to optimize the
Airbus XRF1 wing-body based geometry which
dimensions are typical of a transonic airliner.
Disciplinary optimization problems
The composite bi-disciplinary objective function
considered is as follows:
2
2
(1 )
D W p
J C W r Z
α α δ
= + − +
∑
(2)
The coefficient α represents the trade-off between the
disciplinary objective functions and r
p
a penalty factor
used to impact the global aircraft design phase, which is
beyond the scope of the present paper but detailed in the
companion reference [18]. The high level parameters
selected are: the wing span, the leading edge sweep
angle, the outboard wing taper ratio and the relative
thickness in two control sections.
The aerodynamic performance vs. structural integrity
compromise comes from the trade-off between
conflicting objectives of lightness and flexibility.
Aerodynamic performance is driven by large aspect ratio
and thin wing box while structural strengh requires thick
wing box and small aspect ratio.
With the increasing part of composite components in
primary structure, flexibility plays a major role in aircraft
performance. A more flexible wing yields more wash-out
(induced airfoil nose-down due to back sweep angle),
favours more inboard lift distribution (see Figure 3),
Figure 3 – Typical aerodynamic vs. aeroelastic optimum
spanwise lift distribution.
and reduce wing bending moment hence stress level and
structural weight.
The table below summarizes the differences between the
considered aerodynamic and structural optimization
problems.
Aerodynamics Structures
- Maximize aerodynamic perfor-
mance (C
L
/C
D
): Maximize Lift,
minimise Drag
- Design parameters: Twist angle
and profile shape at several span
stations
- Loadcases: Cruise segment of
the mission
- Constraints: Minimum lift coef-
ficient, bound constraints on
design variables.
- Features: Highly nonlinear cost
function (multimodal)
- Minimise weight
- Design parameters: physical
properties of structural members
(thickness, section)
- Loadcases: Critical conditions
to prevent structural failure
(+2.5G pull up, static gust,
descent, …)
- Constraints: deformations in
composite plies, bound cons-
traints on design variables and
physical properties, buckling
reserve factor, …
- Features: Linear cost function
with respect to design variables,
huge set of nonlinear constraints.
Post-optimality and system level optimization
problem
The disciplinary optimizations lead to optimal design
variables
*
a
X
and
*
s
X
, for prescribed global variables
0
Z
.
To build up the system objective function, it is necessary
to obtain the disciplinary contributions to the overall
aircraft performance, and their total derivatives with
respect to the global design variables. The system level
optimization problem is then formulated as:
* * 0 * 0
( , ) ( , ) ( , )
T
Z
dJ
min J J
d
δ
∆
= +
X Z X Z X Z Z
Z
(4)
* 0
( , ) (1 )
T T
WD
dW dC
J
d d
α δ α δ
= + − +
X Z Z Z
Z Z
0
*
. .
( , ) 0
L U
L U
st
δ
δ δ δ
≤ + ≤
≤ ≤
≤
Z Z Z Z
Z Z Z
G X Z
(4.1)
(4.2)
(4.3)
The first set of constraints corresponds to the allowed
bounds for the global aircraft geometrical modifications.
The second set manages the trust region size which
controls the validity of the linear approximation of the
objective function. The set G comprises the nonlinear
constraints that can be formulated at the overall aircraft
design level (typically general geometrical layout
constraints involving Z variables only) and also
potentially the disciplinary constraints that are still
unfeasible at the end of the disciplinary optimizations.
Introducing the prototypical parameterized optimization
problem
RAeS 3
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Aircraft Structural Design Conference, 9-11
th
October 2012, Delft, The Netherlands.
5
( , )
. . ( , ) 0,
n
m
min f p
st p
∈
≤ ∈
x
x
g x g
ℝ
ℝ
,
p
∈
ℝ
(5)
it is possible to derive a tractable expression of the total
derivative of the optimal value function
*
( ( ), )
f p p
x
.
This result is due to Fiacco [9], provided that
*
x
is a local
minimum of problem (5), sufficient optimality conditions
hold at
*
x
and gradients of the active constraints are
linearly independent (LICQ). Then, for all values of p
that do not change the set of active constraints we get :
i
i
i A
g
df f
dp p p
λ
∈
∂
∂
= +
∂ ∂
∑
(6)
Turning back to our system level problem, and recalling
that we consider a strong interaction aeroelastic problem
with
Y
denoting the interdisciplinary state variables, we
can rewrite equation (6) as :
* 0
( , )
T
Ti i
i
i A
g g
dJ J J d d
dZ Z dZ Z dZ
λ
∈
∂ ∂
∂ ∂
= + + +
∂ ∂ ∂ ∂
∑
Y Y
X Z Y Y
(7)
where p is replaced with Z and the active constraint set is
the union of the aerodynamic and structural active
constraints at the optimum
* 0 0
( , , )
a s
X X Z
for aerodyna-
mics and
0 * 0
( , , )
a s
X X Z
for structures. In this expression
the implicit dependency of
Y
and
*
X
with respect to Z is
taken into account. The first two terms on the right hand
side of equation (7) represent the total derivative of the
objective function for the coupled aeroelastic system.
Same remark applies for the gradient of the active
constraints. These terms can be obtained through a new
MDA at a perturbed configuration
0
δ
±
Z Z
followed by
a finite difference approximation or preferably directly
solving the aeroelastic discrete adjoint system. Details
about the adjoint formulation are beyond the scope of this
article but the interested reader should refer to [22, 20, 8].
At the disciplinary optimum
* 0
( , )
X Z
, the three sets of
feasible, active and violated constraints have to be
managed at the system level.
a) Feasible or inactive inequality constraints can be
used to derive the trust region size. The constraints are
linearized around the disciplinary optimum and a
perturbation vector
δ
Z
is computed such that
*
( ) 0
k k
g g Z
δ
+ ∂ ∂ =
Z
,
k I
∈
where I is the set of
inactive inequality constraints
*
{ | 0}
i
I i g
= <
. The
elements of
δ
Z
with the smallest magnitude then define
the confidence radius for constraint activation
max( | 0, )
L i
Z Z i I
δ δ δ
= ≤ ∈
Z
min( | 0, )
U i
Z Z i I
δ δ δ
= ≥ ∈
Z
(8)
Similarly, the same argument can be used to predict the
tendency of an active constraint to turn inactive by using
a linear predictor of the associated positive Lagrange
multiplier and solving for a perturbation that sets the
multiplier to zero:
*
( ) 0
k k
Z
λ λ δ
+ ∂ ∂ =
Z
. This expression
requires the sensitivity of the Lagrange multiplier with
respect to Z. Writing the stationarity condition
( , , ) 0
λ
∇ =
X
X Z
L
at
* 0
( , )
X Z
of the Lagrangian function
( , , ) ( , ) ( , )
i i
i A
f g
λ λ
∈
= +
∑
X Z X Z X Z
L
and differentiating
with respect to Z we get :
* *
Z
d d
dZ dZ
λ
∇ + = −∇
XX X
XN
L L
, with
j j
N g
= ∂ ∂
X
(9)
Solving this equation for
d dZ
λ
is practical if an
accurate Hessian matrix of the objective and constraints
is available, which is not a standard output of common
optimization algorithms.
b) The set of active constraints at the optimum
contributes to the computation of the Lagrange
multipliers [11, 12]. From the stationarity of the
Lagrangian function, the Lagrange multipliers satisfy the
over-determined set of linear equations:
,
n m
f
λ
×
= −∇ ∈
X
N N
ℝ
(10)
Assuming that a constraint qualification such as LICQ
holds, the efficient way to compute vector
λ
is based on
the QR factorization of
N
:
[ ]
1
1 2 1
T
f
λ
−
=⇒= ∇
X
R
N Q Q R Q
0
(11)
In order to build the matrix
N
of active constraints
sensitivities, let’s first recall the different types of
constraint that enter a practical structural optimization
problem such as the one considered in the ARTEMIS
project. The objective is to minimize the wing box
structural weight W
W
with respect to design parameters
X
:
min ( , ( )) with { , }
W I D
W r =
X
X X X X X
(12)
The set of design parameters is divided into independent
X
I
and dependant
X
D
design variables. This functional
dependence for a particular variable X
D
is usually defined
as a linear equation :
RAeS 3
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Aircraft Structural Design Conference, 9-11
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October 2012, Delft, The Netherlands.
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01
i
ndvi D
D M i Ii M i
iI
X
X C C C X C C
X
=
∂
= + ⇒=
∂
∑
, (13)
where ndvi is the number of independent design
variables.
These two types of design parameters are subjected to
absolute bounds and dynamic move limits (managed by a
trust region strategy) leading to the constraint set :
k k
L L U U
≤ ≤ ≤ ≤
X X X X X
, (14)
k denoting the current iteration.
Classical physical properties such as member thickness,
cross-section area or area moment of inertia, … are
derived from primal design variables through linear or
nonlinear relations. In addition to the bound constraints
that affect the primal design parameters, manufacturing
constraints can be associated directly to the physical
properties. Thus an additional set of constraints is defined
:
( , )
k k
L L I D U U
≤ ≤ ≤ ≤
p p p X X p p
(15)
In order to properly consider the contribution of these
constraints into the matrix
N
, the explicit dependence
with respect to
X
I
has to be exposed. For instance, for a
linear physical property definition we get :
0
( )
Dj
i i
i j
I I Dj I
X
dp p p
p C C X
dX X X X
∂
∂ ∂
= + ⇒= +
∂ ∂ ∂
∑ ∑
X
(16)
The structural responses r(
X
) denote the natural state
variables (or unknowns) associated to the problem
formulation and post-processing, e.g. displacements,
constraints, frequencies, temperatures … Once again
linear or nonlinear expressions of these natural outputs
can be formulated to enter the optimization problem
definition. These design responses may be used either as
constraints or as objective function. Fortunately their
derivatives are readily available as a standard post-
processing procedure of the structural optimization
solver.
Finally, gathering all contributions, the standard
structural optimization problem is formulated as :
min ( , ( )) with { , }
W I D
W r =
X
X X X X X
s.t. ( )
( )
k k
L L U U
k k
L L U U
k k
L L U U
≤ ≤ ≤ ≤
≤ ≤ ≤ ≤
≤ ≤ ≤ ≤
X X X X X
r r r X r r
p p p X p p
(17)
Thus completion of the active set information is not a
trivial task and the construction of the sensitivity matrix
N
may require a significant extra work.
In practice, the optimizer is not directly linked to the
finite element analysis solver. Rather information from
the sensitivity matrix is used to build explicit
approximations to the design responses in terms of the
independent design parameters. The resultant explicit
representation can then be used by the optimizer
whenever function or gradient evaluations are required
instead of the costly, implicit finite element structural
analysis.
Figure 4- Coupling analysis and optimization using
approximations.
The counterpart of this approach is that optimal
information at the optimizer level is computed with
respect to the approximation model. For instance, the
identification of the set of active constraints might differ
whether approximated or implicit constraint values are
considered, even if the convergence criteria are fully
satisfied at the optimum. The standard procedure is to
compute the constraint values at the optimum, identify
the active set based on the convergence criteria at the
finite element analysis level then assemble the matrix
N
from all sensitivity contributions with respect to
independent design variables. This way, it is likely that
columns of the resulting sensitivity matrix are linearly
dependent, mainly because a number of them are
numerically colinear. An additional rank revealing
technique based on a QR decomposition with pivoting
[14] has to be applied in order to restore a well posed
over-determined set of equations as in (10).
c) If any constraint is violated, it is linearized with
respect to
Z
and associated to the system level
optimization problem. This situation arises whenever a
particular set of global design variables leads to a non
feasible design at the end of the disciplinary optimization
process. For instance, it is known that an increase of the
wing box thickness induces heavier critical loads on the
structure that the current allowed stiffness budget for the
wing skins design might not be able to withstand. A way
to manage this is to bias the global layout in order to
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restore a feasible design by putting back the violated
constraints into the system level optimization problem.
As explained earlier, the shared design parameters
control the wing planform and thickness. The three
variables defining the planform (sweep angle, outer taper
ratio and span) are common to the GAP and BDP
process. Once the GAP submits a candidate design to the
BDP (the target vector
Z
GAP
), the objective function is
altered so as to avoid searching the design space in areas
that are not compatible with the results of the GAP. We
then formulate the following performance index for the
BDP :
2
* * 0
2
* 0
2
2
( , ) ( , ) (18)
( , ) (1 )
,with { , , }
TPAG
p
TT
WD
PAG
p
dJ
J X Z J X Z Z r
dZ dW dC
J X Z Z Z
dZ dZ
r Span Sweep Taper
α α
= + ∆ + −
= + − ∆ + ∆
+ − =
Z Z
Z Z Z
The penalty coefficient r
p
> 0 is monotonically increased
at each coupling step from GAP to BDP. Thus final
designs at GAP and BDP level gradually step to the same
configuration. Increase of r
p
means that each process gain
confidence in each other thanks to the data exchange and
enrichment process. Similarly, the objective function at
the GAP level is influenced by the same penalty term, but
this time
Z
BDP
being the target vector. Figure 5 below
illustrates qualitatively how the objective function of a
single parameter is modified for different values of r
p
.
A good initial choice for r
p
is is to penalize the system
level objective function such that the corresponding
optimal configuration at the outcome of the BDP first
iteration lies on the trust region boundary.
Figure 5 – Influence of the penalty factor.
III.
Wing box structural optimization process
The structural disciplinary optimization box is
detailed in Figure 6 and in particular how the high
fidelity aeroelastic computations are linked to the
structural optimization process. MSC/NASTRAN offers
some aeroelastic optimization capabilities but the
aerodynamic modeling is limited to linearized
formulations such as doublet lattice method which can
not take transonic nonlinear effects into account.
Figure 6 : Wing box structural sizing process overview.
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The objective is to set up a strategy for using
aerodynamic loads provided equally by elsA/AEL
computations or NASTRAN/Doublet Lattice Method
(DLM). In fact the CFD computations can not be
dynamically carried out during the optimization process
to cover all possible combinations of configurations and
flight conditions. They rather feed an initial Design of
Experiments database which is then combined with a
Proper Orthogonal Decomposition (POD) procedure to
derive the static load cases within the whole flight
envelope. This corresponds to the box entitled “Aero.
Tool for Static Loads” in Figure 6. For instance, the
aerodynamic field is parameterized with respect to Mach
number, angle of attack or sideslip angle.
The structural optimization process is converged for a
given set of external non-linear pressure fields. In this
optimization cycle the complete FEM is used and the
optimization parameters are the physical properties of
various structural members. An equivalent reduced order
model of the sized structural model is then automatically
derived in order to perform the adequate flexible static
aeroelastic computations at the current iteration. Once the
new flexible nonlinear loads are obtained, they are
transferred to the structural grid and a new optimization
process is performed. This organisation is very flexible as
it does not require any modification to the NASTRAN
core solver such as DMAP (Direct Matrix Abstraction
Programming) or SCA (Simulation Component
Architecture) programming.
The process depicted in Figure 6 has two levels of
convergence: one at the upper level on the critical load
cases and one at the optimiser level on physical
properties and design objective function. External sizing
criteria such as reserve factors for buckling, damage
tolerance or reparability of composite panels are included
in the form of several response surface models (RSM).
These RSM are built offline using a numerical DOE
approach coupled with dedicated expert tools.
Loads tool
CFD has improved the quality of aerodynamic design
in the industrial environment, but has not yet had as
much impact on the rest of the overall aeroplane
development process. Currently, most use of non-linear
CFD has been near the high-speed ‘cruise’ point as
illustrated in the V-n diagram in Figure 7. According to
[29], the high-speed lines development is less than 25%
of the total aerodynamics related aeroplane development
effort.
In transonic flows where strong shock wave exists at the
wing, the aerodynamic modelling based on the RANS
equations should provide more realistic load distribution
than linear DLM methods. More specifically, noticeable
differences in the required force to trim the aircraft and
differences in the ratio between fuselage lift and wing lift
are known to occur [23].
Our goal is to have a good prediction for the trimmed
critical sizing loads distribution over the entire wing. To
achieve this, we do not consider CFD simulation of the
wing-fuselage-tail assembly for time consumption
reasons, but restrict the configuration to a clean wing-
fuselage. However, this simplification will be removed in
future demonstrations that will directly trim the complete
aircraft model. Thanks to the POD based reduction tool,
it is posssible to re-build the entire flow field, pressure
distribution or any aerodynamic coefficient in the whole
flight envelope for any combination of Mach and angle
of attack. Some authors have proposed a non-Galerkin
POD based reduction coupled with a surrogate model
approximation technique, see for instance [3]. More
intrusive strategies with respect to the CFD code also
investigated Galerkin POD projections to obtain a
reduced order model of the nonlinear aerodynamic
equations [32].
“+2.5g”
“-1.0g”
CFD mostly done
near cruise point
Load factor, n
Speed
V, keas
“+2.5g”
“-1.0g”
CFD mostly done
near cruise point
Load factor, n
Speed
V, keas
Figure 7 – Typical V-n maneuver diagram and domain of
classical CFD use.
As the interpolation procedure gives also access to
the zero lift pressure distribution, it is straightforward to
obtain a rather good estimate of the penalty on the
maximum lift coefficient due to trimming. Indeed, the lift
coefficient of the horizontal tail is directly linked to the
wing-fuselage moment coefficient. Thus, knowing the
different mass configurations to be investigated during
the loads process, we have access to the effective load
factor and the correct target lift coefficient to be used for
the CFD analysis. This use of CFD for critical aeroelastic
loads prediction near the stall conditions, see Figure 8, is
currently at the cutting edge of standard aerodynamic
simulations.
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Load distributions
Balanced load cases
alpha
C
L
Aerodynamic
conditions
close to stall
CFD/CSM aeroelasticity
LOADS
TOOL
C
LMax
Load distributions
Balanced load cases
alpha
C
L
Aerodynamic
conditions
close to stall
CFD/CSM aeroelasticity
LOADS
TOOL
C
LMax
Figure 8 – Components of the CFD loads capability.
Wing box design model
Wing skins and spars are divided spanwise and
chordwise into regions which are then linked to finite
element physical properties. One design variable is
associated to each region. The demonstration presented
in this paper only considers total thickness of composite
laminates as design variables (ply proportion, stacking
sequence and material orientation were fixed).
Additionally, cross section area and first moment of
inertia of stringers were linked analytically to skin
thickness design parameters in order to preserve a
realistic material ratio between skins and stiffeners. This
allows to keep a realistic buckling behaviour with respect
to loads redistribution during the sizing process. Figure 9
shows the design variables definition.
Figure 9 – Wing box design variables definition.
A total number of 228 sizing variables was defined.
Introducing dependency between variables (symmetry
and analytical relations), a set of 66 independent
parameters was finally considered. Bound constraints on
each design variable and physical parameters were
defined. Allowables on equivalent homogeneous normal
and shear strains for skin laminates and on stringer axial
strain have been introduced at the finite element level. A
final constraint set related to the critical buckling reserve
factor computed at the panel level was added for wing
skin elements.
Component-level structural optimization
This is a brief description of the process that links
together the macroscopic wing box structural
optimization with the local stiffened composite panel
design. At the upper level, the typical characteristics of
an aeroelastic structural sizing problem for a production
type FEM are up to 400 design variables, 10 load cases,
more than 50000 design constraints (including bound
limits), about 5000 retained and up to 300 active. Thus
adding the panel design complexity to this problem is
obviously intractable. The bi-level approach adopted here
is similar to that presented in [19]. Figure 10 below
presents an overview of the process.
First, a detailed model is built for each candidate region
that is to be designed using advanced criteria. Then a
database of inputs at the panel boundary nodes is built
from a given design optimization history of the wing box
(typically 5 to 10 design cycles). This database is the
basis of the Principal Components Analysis coupled with
a numerical Design Of Experiments (DOE) that is used to
train the buckling reserve factor surogate model.
Figure 10 – The Component-level optimization process.
More details about this work are presented in [15]. The
five step procedure is depicted in Figure 11. A principal
component analysis in terms of displacements at the
boundary nodes has been selected. For this
demonstration, only the total thickness of the composite
laminate has been sized. Besides, the dimension
reduction leads to two principal components. Hence a
surrogate model with three input parameters has been
built.
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Figure 11 – Surrogate model construction for the Buckling Reserve Factor.
It is well known that the buclking reserve factor exhibits
a piecewise behaviour (depending on which buckling
mode is activated) that the surrogate model must be able
to tackle.
For the composite stiffened panel optimization task, two
operating modes can be considered:
• The first one is an analysis mode in which local
buckling reserve factors are fed back as additional
nonlinear constraints to the macroscopic wing box
optimization problem through a surrogate model. This
is the current operating mode presented in this paper.
• The second operating mode considers a true
optimization of the stacking sequence. In this case, a
bi-level scheme such as QSD has to be set up [25].
This scheme will be adopted for future
demonstrations more focused on aroelastic tailoring.
IV. Numerical experiments
GAP and BDP coupling scenario in the global process
The data flowchart between PAG and PBD is detailed
in Figure 12 below. Before running a design optimization
at the overall aircraft design (OAD) level, a number of
PBD experiments are performed around the initial
configuration to build response surfaces allowing the
PAG beforehand update of its constituent mass and drag
estimation modules.
Figure 12 – Flowchart between GAP and BDP.
Preliminary results
The BDP optimization process was run for five
iterations for each of which correspond separate
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disciplinary optimization processes with upgraded global
variables set. Hereafter we present selected results to
illustrate the BDP.
Let’s assume an optimal configuration (i.e. a set of global
variables
Z,
and a trade-off coefficient
α
) is available
from a previous run of the GAP. The first step of the
BDP procedure depicted in Figure 2 is to update CFD
and FEM meshes to match the new geometry. The
aeroelastic equilibrium is then computed using the
elsA/AEL code at cruise flight conditions, typically
Mach=0.83, altitude=35000 feet for this type of aircraft.
The CFD mesh corresponding to the flexible cruise flight
shape is then available (see Figure 13) along with the
total derivatives computed with the discrete adjoint
method. Then all information is available to build the
disciplinary objective functions and start the concurrent
optimization processes.
At the outcome of disciplinary optimizations the classical
integrated results such as drag decomposition and
structural mass breakdowns can be analyzed.
Figure 13 – 1G level flight and jig shape CFD mesh.
As an example a dedicated post-processing has been
developed to automatically derive the wing box masses
for several parts such as wing skins, stringers, ribs and
spars.
For this particular configuration, the final outer wing box
is thinner which leads to a decrease in “geometrical”
flexibility that favours an inboard aerodynamic loading.
Figure 14 –Mass breakdowns for initial and final design
at first run of the BDP (5 shared variables) .
However, the strain level in wing skins has increased
resulting in more material in panels and stringers while
spars and ribs become lighter, due to lower transverse
shear deformation.
Once an optimal set of aerodynamic and structural design
variables has been obtained, the identification of the
active constraints set is performed in order to compute
the associated Lagrange multipliers. For instance typical
results of a structural post-optimality analysis are
produced below:
For a number of 66 independent design variables, two of
them activate their bounds. 54 design responses among a
total of 3502, corresponding to equivalent strains in
composites, are active at the optimum. The optimal
designed properties, mainly manufacturing constraints
such as minimal thickness, are limited at their bounds for
280 of them. The union of active constraints related to
design responses, design variables and design properties
form the basis for the construction of the constraint
gradient matrix
N
.
A number of these constraints are linearly dependent (in
their formulation or numerically) and a preliminary
cleaning step is required in order to robustly compute the
Lagrange multipliers. Using Fiacco’s theorem, the total
derivative of the optimal structural weight with respect to
shared variables can be derived. Focusing on the
contribution of the relative thickness at wing crank and
tip we get (weight in tons) :
*
*
+
ss
i
i
i A
XX
g
dW W
dZ Z Z
λ
∈
∂
∂
=∂ ∂
∑
6.764 1.967 8.731
4.035 1.822 5.857
− −
= +
− −
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Several remarks can be formulated. Firstly, from the first
term on the right hand side, a wing box height increase
results in a structural weight increase (geometrical
effect). Secondly, the constraints satisfaction improves
when the global thickness increases. Thus there is a
competition between optimal weight reduction and active
constraints satisfaction relative to wing thickness.
Similarly it is numerically observed that a thickness
increase induces a drag rise as could be expected, along
with a lift rise at the same angle of attack. The Lagrange
multiplier related to the minimal lift coefficient
constraint,
0.50
L
C
≥
, represents the price to pay in
terms of drag degradation for a lift improvement.
The normalized structural weight and drag optimization
histories are given below. Each vertical segment
represents the step due to global variables increment
produced by a BDP system level optimization.
Figure 15 –Structural weight and total drag history.
Each oblique segment represents the variation due to a
disciplinary optimization.
The evolution of the composite objective function
(1 )
D W
J C W
α α
= + −
is given in Figure 16. The vertical
dotted line points out the best configuration according to
disciplinary experts knowledge to be fed back to the PAG
level in order to initiate a new cycle.
Figure 16 – Aircraft performance index history.
The upper curve corresponds to the system performance
computed using the aircraft total drag history while the
lower curve only considers the pressure drag con-
tribution.
It is important to consider that the aircraft configurations
produced so far are all feasible with respect to
aerodynamic and structural design criteria evaluated with
high fidelity tools.
It is thus the designer’s decision to pick up a particular
candidate configuration to be returned to the PAG for an
evaluation at the overall aircraft level. It might not be an
adequate choice to wait until the BDP process overall
convergence since the associated computational burden
would not necessarily worth the gain in performance
index.
The spanwise lift distributions for the initial and
optimized wing are given below, as well as the design
variables history.
Figure 17 – Lift distribution and shared variables
optimization history (% deviation from initial
configuration) for a BDP run.
V.
Conclusion
To answer the challenge of developing simulation-
based optimization processes aiming to deliver faster the
best aircraft performance, the ARTEMIS project has
proposed a multi-level multi-fidelity MDO capability and
demonstrated it on a realistic civil aircraft transport test
case. The optimization process has been achieved
successfully to optimize a wing-body configuration in
terms of wing drag and wing-box structural weight. The
flexible wing performance improvement obtained has
been analyzed thoroughly and shows very encouraging
results. The organization adopted here followed several
guidelines such as exploiting the state of the art
numerical optimization techniques available for CFD and
CSM along with ensuring as much as possible
disciplinary autonomy of the designer’s team and at least
fits a design process into the aircraft manufacturer’s
required timescale. To this end, a multi-site collaborative
environment based on Python language exploiting it’s
network programming interface through the valuable
PYRO package [7] has been set-up.
It must be advised that this demonstration is a first
attempt to a fully integrated multi-level multi-
disciplinary design capability that will support in the near
future many improvements in the disciplinary design
processes as well as considering several variants in the
BDP and GAP architectures.
Generally speaking, the benefits of MDO for classical
configurations (large aspect ratio wings) where a huge
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expert knowledge and efficient practices exist, lay in the
help for harmonisation of models and tools, coherent data
layout, and better understanding of interactions. Existing
aircraft manufacturer’s OAD and disciplinary design
processes do not need to be significantly altered.
For new configurations (for instance blended wing
body, joined-wing, strut-braced wing) MDO approach
helps in supporting all the interactions at the uppermost
design level. Demonstration through the ARTEMIS
project. MDO has to be considered as a great opportunity
for improving traditional design lines.
In any case, MDO, as presented in this paper, leads to
multilevel consistent and feasible designs, enables faster
and systematic parametric studies, helps in better
understanding of the disciplinary trade-offs and offers
many potential time savings for the design process.
VI.
Acknowledgments
The authors would like to thank Airbus partners for
their support as well as the French Civil Aviation
Authority (DGAC) for financing the ARTEMIS project.
VII.
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