James C. Ross, Ames Research. Center, Moffett Field, California. May 1997. National Aeronautics and. Space Administration. Ames Research Center.
NASA
Technical
Memorandum
112197
Neural Network Prediction of New Aircraft Design Coefficients Magnus N_rgaard, Institute of Automation, Technical University of Denmark Charles C. Jorgensen, Ames Research Center, Moffett Field, California James C. Ross, Ames Research Center, Moffett Field, California
May 1997
National Aeronautics
and
Space Administration Ames Research Center Moffett Field, California 94035-1000
NEURAL
NETWORK
PREDICTION
OF
NEW
AIRCRAFT
DESIGN
COEFFICIENTS
Magnus
NCrgaard,*
Charles
C. Jorgensen,
and James
C. Ross
Computational Sciences Division Ames Research Center
SUMMARY This paper tunnel
discusses
tests.
predictions validation,
Using
a neural a hybrid
network neural
tool for more effective
network
optimization
aircraft
method,
design
we have
evaluations produced
during
wind
fast and reliable
of aerodynamical coefficients, found optimal flap settings, and flap schedules. For the tool was tested on a 55% scale model of the USAF/NASA Subsonic High Alpha
Research
Concept
(SHARC)
lift, drag, moment
of inertia,
settings.
The latter network
optimal
flap schedules.
aircraft.
Four different
networks
were trained
to predict
and lift drag ratio (C L, C D, C M and L/D) from was then used to determine
an overall
optimal
coefficients
angle of attack
flap setting
of
and flap
and for finding
INTRODUCTION
Wind tunnel testing can be slow and costly due to high personnel utilization. Thus, a method that reduces the time spent in a wind associated with a test, are of major Modem wind tunnels have become
interest to airframe highly sophisticated
of performance
designs.
determination lift/drag
features
of aircraft
of the coefficient
ratio (L/D)
but the techniques
Currently,
a new design
researchers inspection
to interpolate of curves.
describe
the complex
approach
potentially
computing
methods
new approach earlier
calculations.
test is followed between
relationships
between
The longer
manual
evaluation
the procedure variables.
automation
term benefits
to consider
testing
of the aircraft
design
approach.
Spin-off
of measurement
are a significant
neural
benefits
processing reduction
as well. To allow
is based
on visual
expressions
networks
aerodynamic
(C u) and
we emphasize
and analysis.
is to f'md mathematical
Although
only
moment
In this paper tunnel
data fitting
this task, e.g., numerical
a very cost effective increased
chosen
and flap settings.
by extensive
and design engineers. used to measure a number
of drag (CD), pitching
to other steps of wind
measurements,
able to perform provide
In this study we have
of angle of attack are applicable
One way to automate
and include
manufacturers test facilities
of lift (C,_), coefficient
as functions
prediction,
overhead and intensive power tunnel, as well as the work load
to
are not the only
simulations,
such soft
can also result
from
and aids for checking
in costs and faster
*This author was at the NASA Ames Neuro-Engineering Laboratory in 1994 as part of a cooperative student work study program between NASA and the Institute of Automation, Electronics Institute, and Institute of Mathematical Modelling, at the Technical University of Denmark. The Danish Research Council is gratefully acknowledged for providing financial support during his stay.
a
development
of new aircraft,
automotive
or alternate
tunnel uses such as more aerodynamically
efficient
design.
This paper
is organized
and one powerful Next, we describe networks
A short introduction
to Multilayer
Perceptrons
(MLP)
is given
method we used (a variation on the Levenberg-Marquardt method) is presented. how a subset of test measurements were used with the technique to train four
to predict
settings.
as follows:
aerodynamical
We then present
coefficients
two applications.
and the L/D ratio, given
The first addresses
angle
the problem
of attack
and flap
of determining
an
"overall optimal" flap setting using a method based on integration of L/D vs. C L. The second demonstrates an easy strategy to find optimal flap schedules. Finally, details of the software tool set are given
in an appendix
as a supplement
to documentation
MULTILAYER
in the project
code.
PERCEPTRONS
The phrase "neural network" is an umbrella covering a broad variety of different techniques. The most commercially used network type is probably the MLP network. See reference 1. An example a MLP network is shown in figure 1. In this study we used a two-layer network with tangent hyperbolic activation functions in hidden layer units, and a linear transfer function in the output units. A two-layer network is not always an optimal choice of architecture (goodness measured terms of the smallest number of weights required to obtain a given precision), but it is sufficient approximate any continuous faster in this case.
function
A MLP network
type of an "all-purpose"
shown
is a special
an excellent
corresponds
ability
for function
to the following
1. A three
represent
input,
the biases.
yi(w,W)=
Here
two output, f/(x)
function,
in Sj/Sberg,
to implement
recent shown
and
situations in figure
has 1
w_
two layer MLP
Wio
network.
(ref. 4). One disadvantage
The weights
from
the inputs
set to 1
and Fj (x) = x.
_-- Fii
Wijf
j
/n
__aWjlZll-Wjo l=l
feature offered by this type of network well, without requiring an extravagant et.al.,
in many
(ref. 3). The network
k.j=l
A special functions
which
is easier
in to
form
= tanh(x)
%hj(w)+
fi
well (ref. 2), and training
approximation
functional z_
Figure
arbitrary
of
is that it can be trained amount of parameters
compared
to other network
I / "[-Wio
(1)
to approximate many (weights). This is discussed types
is that training
is slow
becausethe networkimplementsa non-linearregression,i.e.,thereis a nonlinearrelationbetween theadjustableparameters, the weights,andthe output. In this study,we wereinterestedin enhancinggenericneuralnetworksfor wind tunneltest estimation.Obtainingnetworktrainingdatais very costly,e.g.,$3,000dollarspertunnelhourfor theNationalFull-ScaleAerodynamicComplexatAmesResearchCenter.Consequently,only limited datasetswereavailableandusuallyasbyproductsof previouslyscheduledtests.The sizeof the data setimposesanupperlimit onhow manyweightsthenetworksshouldcontain.In practice,since thereis alsouncertaintyassociated with themeasurements, thenumberof datapointsmustexceed thenumberof weightsby a sufficientlylargefactorto ensurethatgoodgeneralizationmay be achieved.Trainingtime,onthe otherhand,is not of primeimportance.Many argumentscanbe madein favor of someform of MLP networksasthefight choicefor the givenproblem.The real problemis obtainingextremelyhigh accuracies criticalfor commercialviability. TRAINING Thetrainingphaseis the processof determiningnetwork measurement
data. The treatment
of different
we will consider
a generic
quantity
of three
variables:
Angle
different
aerodynamical
'y' instead. of attack
weights from a collected set of coefficients is essentially identical,
If the aircraft
(c¢), leading
flaps
edge
are coupled,
flap angle
so
y becomes
a function
(LE), and trailing
edge flap
angle (TE). y = g0 (¢p)
(2)
where
LE rE The function
'g' is unknown,
but the wind
tunnel
(3) tests provide
us with a set of corresponding
y - _0
pairs i
i
."
ZN = {[q9 ,y ],t = 1..... N} Naturally the measurements number of different sources.
(4)
of y are not exact, but will be influenced in undesired ways from a All measurement errors are grouped in one additive noise term, e y = g0(q_)+ e
The objective
is now to train the neural
network
(5)
to predict
y from
= _(tp) The predictor Expressed functions
is found
precisely, contained
from the set of measurements,
we wish to determine in the chosen network
a mapping architecture
Z N --> b
(6) Z N, from here on denoted from the set of measurements g(_0; 0)
the training
set.
to the set of
(7) 3
sothat _ is close
to the "true"
y. 0 is the parameter
this case, the network
weights).
A common
for goodness
definition
V(O)=
Thus,
training
becomes
1 --_ 2N
a flexible
training
in the neural
algorithm
containing
all adjustable
of fit in neural nets is the mean square ::
parameters
i=1
(in
error
1 N = _...._.. ___., (g(0))2
_i(0))2
(yi_
(8)
/=1
a conventional
back-propagation,
vector
unconstrained
but somewhat network
optimization
ad hoc gradient
community.
search
Ease
problem.
For various reasons
method,
has been the preferred
of implementation,
utilization
of the
inherent parallel structure, and the ability to work on large data sets are the main arguments justifying the use of this method. However, in the present case where the data sets are of limited size, backpropagation
is not the best choice.
Marquardt many
method
for solving
ways superior
Instead
we have decided
the optimization,
to back-propagation method,
optimization
packages
(ref. 5). Some important
convergence
to a (local)
necessary method
except
important
properties
our objective networks
of its neural
numerical a network
advantages
architecture.
in making
was to create
a user-friendly,
a generic
The Levenberg-Marquardt
method
has numerous
mappings variations.
guaranteed inputs
out in Mor6
Such
tool, which
for application
the complex
as pointed
is crucial
(ref. 6) the are
in this case since if in fact neural
in nonlinear
The simplest
are
advantages
use and determine
required
The
horse of many
are speed,
user-specified
these benefits.
easy-to-apply
methodology
of performing
Moreover,
to achieve
it is in
methods.
is a work
of the method
and minimal
Levenbergapproaches
search
implementation,
robustness,
free of ad hoc solutions
were capable
the original in reference
independent
for providing
is surprisingly
gradient
as well as most other gradient
Levenberg-Marquardt
minima,
to use the so-called
since like conjugate
aero design.
strategy
may be found
in
contribution of Marquardt, while one adaptation to neural network training is discussed 7. The version used here belongs to the class of trust region methods found in Fletcher
(ref. 8). Just as back propagation,
the Levenberg-Marquardt
algorithm
is an iterative
search
scheme
(9)
_k+_) = O_k)+ ld
5) V(O(k))--V(O(k)+hq'))