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Aerodynamic Data Reconstruction and Inverse Design Using Proper Orthogonal Decomposition AIAA J
OURNAL
Vol. 42, No. 8, August 2004
Aerodynamic Data Reconstruction and Inverse Design Using
Proper Orthogonal Decomposition
T. Bui-Thanh and M. Damodaran Nanyang Technological University, Singapore 639798, Republic of Singapore
and
K. Willcox Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
The application of proper orthogonal decomposition for incomplete (gappy) data for compressible external
aerodynamic problems has been demonstrated successfully in this paper for the rst time. Using this approach, it
is possible to construct entire aerodynamic owelds from the knowledge of computed aerodynamic ow data or
measured ow data specied on the aerodynamic surface, thereby demonstrating a means to effectively combine
experimental and computational data. The sensitivity of ow reconstruction results to available measurements and
to experimental error is analyzed. Another new extension of this approach allows one to cast the problem of inverse
airfoil design as a gappy data problem. The gappy methodology demonstrates a great simplication for the inverse
airfoil design problem and is found to work well on a range of examples, including both subsonic and transonic cases.
Introduction
T
HE proper orthogonal decomposition (POD), also known as
KarhunenLo`eve expansion and principle components analy-
sis, has been widely used for a broad range of applications. POD
analysis yields a set of empirical modes, which describes the domi-
nant behavior or dynamics of given problem. This technique can be
used for a variety of applications, including derivation of reduced-
order dynamical models,
1
steady analysis and design of inviscid
airfoils,
2
image processing,
3
and pattern recognition.
4
Sirovich introduced the method of snapshots as a way to ef-
ciently determine the POD modes for large problems.
5
In particular,
the method of snapshots has been widely applied to computational-
uid-dynamic (CFD) formulations to obtain reduced-order models
for unsteady aerodynamic applications.
6
9
A set of instantaneous
ow solutions, or snapshots is obtained from a simulation of the
CFD method. The POD process then computes a set of modes from
these snapshots, which is optimal in the sense that, for any given
basis size, the error between the original and reconstructed data is
minimized. Reduced-order models can be derived by projecting the
CFD model onto the reduced space spanned by the POD modes.
Everson and Sirovich
10
have developed a modication of the ba-
sic POD method that handles incomplete or gappy data sets. Given
a set of POD modes, an incomplete data vector can be reconstructed
by solving a small linear system. Moreover, if the snapshots them-
selves are damaged or incomplete an iterative method can be used
to derive the POD basis. This method has been successfully applied
for reconstruction of images, such as human faces, from partial data.
In the research described here, the gappy POD approach is ex-
tended for application to aerodynamic problems. Incomplete aero-
Received 29 April 2003; presented as Paper 2003-4213 at the AIAA 16th
Computational Fluid Dynamics Conference, Orlando, FL, 2326 June 2003;
revision received 19 February 2004; accepted for publication 13 March 2004.
Copyright
c
2004 by the authors. Published by the American Institute of
Aeronautics and Astronautics, Inc., with permission. Copies of this paper
may be made for personal or internal use, on condition that the copier pay
the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rose-
wood Drive, Danvers, MA 01923; include the code 0001-1452/04 $10.00 in
correspondence with the CCC. Graduate Research Student, School of Mechanical and Production En-
gineering, SingaporeMassachusetts Institute of Technology Alliance,
Nanyang Avenue. Associate Professor, School of Mechanical and Production Engineer-
ing, SingaporeMassachusetts Institute of Technology Alliance, Nanyang
Avenue. Associate Fellow AIAA. Assistant Professor of Aeronautics and Astronautics, Aerospace Com-
putational Design Laboratory. Senior Member AIAA.
dynamic data can arise in several situations. First, a limited set
of data might be available from experimental measurements. The
gappy POD provides a way to reconstruct full oweld information,
using a combination of the available experimental and supplemental
computational data. Second, certain data might be missing because
they are not known. For example, one can have a set of snapshots that
correspond to a set of airfoil shapes and their respective owelds.
Given a new airfoil shape, the gappy POD provides a way to quickly
estimate the corresponding oweld. Conversely, the gappy POD
can be used to solve the problem of inverse design: given a target
pressure distribution, the optimal airfoil shape can be determined
by appropriate interpolation of known designs. Finally, data might
be incomplete as a result of damage of storage facilities.
Many of the current methods for airfoil design focus on the use
of optimization. Lighthill
11
developed pioneering work using the
method of conformal mapping, which was later extended to com-
pressible ows by McFadden.
12
By introducing the nite difference
method to evaluate sensitivity derivatives, Hicks and Henne
13
rst
attempted to solve the airfoil design problem as a constrained opti-
mization. Since then, gradient-based methods have been widely used
for aerodynamic design. More recently, Jameson
14
applied control
theory to shape design optimization for Euler and NavierStokes
problems, using an efcient adjoint approach to obtain gradient in-
formation. To reduce the computational cost of solving the design
optimization problem, Legresley and Alonso
2
used the POD tech-
nique for both direct and inverse designs. The gappy POD method
presented here for the inverse design problem is fast compared with
other optimization methods, which is an advantage for routine use
in design. In addition, the gappy approach allows for both computa-
tional and prior experimental data to be utilized in the design process.
In this paper, the basic POD approach is rst outlined, followed
by a description of the gappy POD method. A series of examples
that demonstrate how the gappy POD method can be used for re-
construction of oweld data and extended for airfoil inverse design
are then presented. Finally, we present some conclusions.
Proper Orthogonal Decomposition
Theory and Extensions
Proper Orthogonal Decomposition
The basic POD procedure is summarized briey here. The optimal
POD basis vectors
are chosen to maximize the cost
1
:
max |(U, )
2
|
(, ) =
|(U, )
2
|
(, )
(1)
1505 1506
BUI-THANH, DAMODARAN, AND WILLCOX
where
(U, ) is the inner product of the basis vector with the
eld U
(x, t), x represents the spatial coordinates, t is time, and
is the time-averaging operation. It can be shown that the POD basis
vectors are eigenfunctions of the kernel K given by
K
(x, x ) = U(x, t), U (x , t)
(2)
where U denotes the hermitian of U. The method of snapshots,
introduced by Sirovich,
5
is a way of determining the eigenfunctions without explicitly calculating the kernel K . Consider an ensemble
of instantaneous eld solutions, or snapshots. It can be shown that
the eigenfunctions of K are linear combinations of the snapshots as
follows: =
m
i
= 1 i
U
i
(3)
where U
i
is the solution at a time t
i
and the number of snapshots m
is large. For uid dynamic applications, the vector U
i
contains the
ow unknowns at a given time at each point in the computational
grid. The coefcients i
can be shown to satisfy the eigenproblem
R
=
(4)
where R is known as the correlation matrix
R
i k
= (1/m)(U
i
, U
k
)
(5)
The eigenvectors of R determine how to construct the POD basis
vectors [using Eq. (3)], while the eigenvalues of R determine the
importance of the basis vectors. The relative energy (measured by
the 2-norm) captured by the i th basis vector is given by i
m
j
= 1 j
The approximate prediction of the eld U is then given by a linear
combination of the eigenfunctions
U p
i
= 1 i i
(6)
where p
m is chosen to capture the desired level of energy,
i
is the i th POD basis vector, and the POD coefcients i
must be
determined as a function of time.
The basic POD procedure just outlined considers time-varying
ows by taking a series of ow solutions at different instants in time.
The procedure can also be applied in parameter space as in Epureanu
et al.,
15
that is, obtaining ow snapshots while allowing a parameter
to vary. The parameter of interest could, for example, be the ow
freestream Mach number, airfoil angle of attack, or airfoil shape.
POD for Reconstruction of Missing Data
In CFD applications, the POD has predominantly been used for
deriving reduced-order models via projection of the governing equa-
tions onto the reduced space spanned by the basis vectors. Here, we
consider a different application of the method, which is based on the
gappy POD procedure developed by Everson and Sirovich
10
for the
reconstruction of human face ima