Content uploaded by Anirban Chaudhuri

Author content

All content in this area was uploaded by Anirban Chaudhuri on Feb 08, 2023

Content may be subject to copyright.

Data-driven Model Reduction via Operator Inference for

Coupled Aeroelastic Flutter

Benjamin G. Zastrow∗, Anirban Chaudhuri†, Karen E. Willcox‡

University of Texas at Austin, Austin, TX, 78712

Anthony Ashley§, Mike Henson¶

Lockheed Martin, Fort Worth, TX, 76101

This paper presents a data-driven, physics-informed model reduction technique applied

to two large-scale applications: parametric nonlinear aerodynamics and coupled aeroelastic

ﬂutter. We ﬁrst develop a reduced-order model (ROM) parameterized by Mach number for the

compressible Navier-Stokes equations solved via a computational ﬂuid dynamics code. We use

the non-intrusive operator inference scientiﬁc machine learning method as our model reduction

technique. We then extend this technique to the coupled aeroelastic case by incorporating

a modal decomposition model obtained from a ﬁnite element analysis code. We use NASA’s

FUN3D software to run the high-ﬁdelity simulations. We demonstrate the parametric ROM

for nonlinear subsonic ﬂuid ﬂow over the VAT aircraft and the coupled ROM for ﬂutter of the

AGARD wing. We show that compared to the high-ﬁdelity FUN3D simulations, parametric

operator inference speeds up computation time by approximately three orders of magnitude,

and coupled operator inference speeds up ﬂutter computation time by approximately two orders

of magnitude. ∗∗

I. Introduction

Finite element analysis (FEA) and computational ﬂuid dynamics (CFD) are widely used in aircraft design but their

usefulness is limited by their high computational cost, which limits the degree to which the design space can be explored.

CFD models can routinely have tens of millions of degrees of freedom, requiring signiﬁcant simulation time on high

performance computing clusters to make useful predictions. This precludes them from use in many-query problems,

such as uncertainty quantiﬁcation, ﬂutter analysis, design optimization, and integration into digital twins for online

control. Model reduction techniques lead to cheaper, physics-informed surrogates that can be used in many-query

problems. This paper presents a data-driven model reduction technique applied to two large-scale applications in

parametric nonlinear aerodynamics and coupled aeroelastic ﬂutter analysis.

Model reduction aims to represent high-dimensional dynamics with a lower-dimensional system while maintaining

a suﬃcient level of predictive accuracy. The reduced-order models (ROMs) can be run at orders of magnitude lower

computational cost compared to the full-order models (FOMs). Projection-based approaches like the proper orthogonal

decomposition (POD) [

1

–

4

] use the trajectory data from a FOM to derive a reduced basis, then the full-order operators

of the governing equations are projected onto the reduced basis and the dynamics are integrated forward in time [

5

,

6

].

Another technique, dynamic mode decomposition (DMD) [

7

,

8

], uses the full-order trajectory data to construct a linear

reduced operator to approximate the system, thus enabling analysis of eigenmodes and eigenvalues that often can

reasonably characterize even nonlinear full-order dynamics. The governing equations of many physical systems of

interest contain parametric dependencies, and thus they require a decision about whether to deﬁne the ROM using a

local or global basis, with implications for both the accuracy and the computational performance of the ROM. In [

5

], the

eﬀectiveness of rational interpolation methods, balanced truncation, POD, and the reduced basis method are compared

from a parametric ROM perspective.

∗Graduate Student, Department of Aerospace Engineering and Engineering Mechanics, AIAA Student Member.

†Research Associate, Oden Institute for Computational Engineering and Sciences, AIAA Member.

‡Director, Oden Institute for Computational Engineering and Sciences, AIAA Fellow.

§Aeronautical Engineer Senior, CFD, AIAA Member

¶Aeronautical Engineer, Senior Staﬀ, Transformational Solutions/AeroIT, AIAA Associate Fellow

∗∗

This version corrects a few minor errors in the manuscript published in the AIAA SciTech proceedings, notably in Figures 12 and 13, and it corrects

Section IV.B to state that the AGARD ﬂuid ﬂow model is compressible.

©2022 Joint copyright of Lockheed Martin Corporation and the University of Texas at Austin, all rights reserved

1

Aeroelastic ﬂutter was originally analyzed using linear methods that focused on the generation of V-g (velocity

vs. damping) and V-f (velocity vs. frequency) plots to characterize the boundaries of the safe operational range for

an aircraft [

9

,

10

]. A number of model reduction techniques have been proposed as a way to bring higher ﬁdelity

aerodynamic and structural information into aeroelastic computations. In [

11

], a vortex lattice model of ﬂow over a

2D airfoil is used to demonstrate that unsteady ﬂow solutions can be characterized with a small number of dominant

eigenmodes of a modal decomposition. A review of several modal reduction methods for unsteady aerodynamics,

including eigenmodes, POD modes, and balanced modes, is given in [

12

], along with a discussion of implementing

these methods for aeroelastic models. A parametric reduction method for ﬂutter that interpolates between ROMs at

diﬀerent free stream Mach numbers using the tangent space of a Grassman manifold is applied to both the F-16 and

F-18/A full-aircraft conﬁgurations in [

13

]. In another approach, Walsh functions are used to simultaneously excite

multiple impulse responses in a CFD model [

14

]. The eigensystem realization algorithm (ERA) [

15

] is then used to

convert these impulses into an unsteady aerodynamic ROM that can be coupled to a structural model to create a coupled

aeroelastic ROM [

16

]. The Walsh function technique from [

14

] has been extended to a POD methodology in [

17

] that

builds the basis via an incremental approach to avoid handling the full snapshot matrix all at once. In this work, we

build non-intrusive ROMs for coupled multi-physics problems such as aeroelastic ﬂutter by extending the operator

inference method [18].

Operator inference is a scientiﬁc machine learning approach that replaces the intrusive Galerkin projection step with

a non-intrusive linear least squares problem, thus combining the beneﬁts of data-driven learning and physics-informed

modeling [

18

]. An additional “lifting” step is formally suggested in [

19

] to expose the desired polynomial ordinary

diﬀerential equation structure via introduction of auxiliary state variables. The parametric modeling capability of

operator inference is detailed for an aﬃne parametric structure in [

20

]. We build on the data-driven operator inference

method [

18

] to learn non-intrusive ROMs for two real-world applications: a parametric ﬂuid ﬂow model of the Validation

of Aeroelastic Tailoring (VAT) aircraft [

21

], and a coupled aerostructural ﬂutter model of the Advisory Group for

Aerospace Research and Development (AGARD) wing [

22

]. We generate high-dimensional simulation data by using

NASA’s FUN3D software [23] for the aerodynamic analysis of the VAT aircraft and the AGARD wing. For the ﬂutter

analysis, we use FUN3D’s aeroelasticity capability to couple the structural and ﬂuid dynamics and generate snapshot

training data for the AGARD wing. The main contributions of this paper are:

(i)

developing a non-intrusive parametric ROM for large-scale aerodynamics analysis: We develop a non-aﬃne

parametric data-driven operator inference ROM for aerodynamic analysis under varying Mach numbers. We

build the ROMs via reduced-state solution interpolation with global basis vectors.

(ii)

developing a non-intrusive coupled ROM for ﬂutter analysis: We develop coupled data-driven operator

inference ROMs for ﬂutter analysis. We exploit existing knowledge of the structure of the coupled system by

identifying basis functions for the aerodynamic states separately from the structural basis functions obtained

via modal decomposition. We then combine the structural and ﬂuid reduced states to learn a single coupled

ROM.

The remainder of this paper is organized as follows. Section II describes the operator inference method as initially

presented in [

18

], which will be used to construct the parametric reduced-order models. Section III applies the operator

inference method to the coupled (aerostructural) setting. Section IV presents the application of the operator inference

method to the VAT aircraft subsonic ﬂuid ﬂow problem and to the AGARD wing aerostructural ﬂutter problem.

Section V provides concluding remarks.

II. Non-intrusive learning of reduced-order models using operator inference

Operator inference is a scientiﬁc machine learning method that uses knowledge of the structure of the full-order

governing equations to specify a matching structure for the reduced-order model. In Section II.A, we present a summary

of the standard non-parametric operator inference framework. Section II.B presents an approach for parametric operator

inference ROMs via reduced-state solution interpolation.

A. Non-parametric operator inference

This work targets systems governed by nonlinear partial diﬀerential equations. We consider a semi-discrete form of

the governing equations after spatial discretization that results in a polynomial ordinary diﬀerential equation structure as

d

d𝑡x(𝑡)=c+Ax(𝑡) + H(x(𝑡) ⊗ x(𝑡))+Bu(𝑡)(1)

©2022 Joint copyright of Lockheed Martin Corporation and the University of Texas at Austin, all rights reserved

2

where x

(𝑡) ∈ R𝑛

is the semi-discrete state vector at time

𝑡

,u

(𝑡) ∈ R𝑚

is the input vector at time

𝑡

with

𝑚

inputs, c

∈R𝑛

are the constant terms, A

∈R𝑛×𝑛

is the discretized linear operator, H

∈R𝑛×𝑛2

is the discretized quadratic operator,

and B

∈R𝑛×𝑚

is the input operator. The dimension

𝑛

of the state xis given by

𝑛=𝑑𝑛𝑥

, with

𝑑

as the number of state

variables and a spatial discretization with

𝑛𝑥

cells. For dynamical systems that are not initially in this polynomial form,

we can expose such structure via lifting variable transformations as shown in [19].

We build a projection-based ROM that can preserve the polynomial structure of Eq.

(1)

. To build a ROM, we need to

derive a reduced basis which deﬁnes a low-dimensional subspace in which the dynamics are approximated. We obtain

this basis from the full state trajectory via the proper orthogonal decomposition (POD). The POD basis vectors are

obtained by taking the singular value decomposition (SVD) of the snapshot matrix X

=[x1, . . . , x𝑘]∈R𝑛×𝑘

, where

each column x

𝑖=

x

(𝑡𝑖), 𝑖 =

1

, . . . , 𝑘

is the full-order state solution at a given time

𝑡𝑖

, and

𝑘

is the number of snapshots in

X. Typically

𝑘≪𝑛

for large-scale applications. The thin SVD of the snapshot matrix is X

=

V𝚺W

⊤

, where V

∈R𝑛×𝑘

and W

∈R𝑘×𝑘

are orthogonal matrices, and 𝚺

∈R𝑘×𝑘

is a square diagonal matrix consisting of the singular values of

the snapshot matrix. Our reduced basis for projection V

𝑟∈R𝑛×𝑟

consists of the ﬁrst

𝑟

columns of V, where

𝑟≪𝑛

(and

𝑟 < 𝑘 ). The approximation of the full state in terms of reduced state b

x(𝑡) ∈ R𝑟is given by x(𝑡) ≈ V𝑟b

x(𝑡).

In an intrusive projection-based model reduction approach, Galerkin projection is used to obtain the reduced matrix

operators and requires access to source code of the simulation software. However, operator inference is a non-intrusive

projection method that does not require access to the full-order operators of Eqn.

(1)

, instead learning the reduced

operators via the solution of a linear least squares problem [18]. We wish to learn reduced-order dynamics that match

the form of the full-order dynamics in Eq. (1), leading to the reduced system

d

d𝑡b

x(𝑡)=b

c+b

Ab

x(𝑡) + b

Hb

x(𝑡) ⊗ b

x(𝑡)+b

Bu(𝑡),(2)

where b

c∈R𝑟,b

A∈R𝑟×𝑟,b

H∈R𝑟×𝑟2,and b

B∈R𝑟×𝑚are the reduced operators.

The reduced operators are obtained by solving the regularized linear least squares problem

min

b

c,b

A,b

H,b

B(𝑘

𝑖=1b

c+b

Ab

x𝑖+b

H(b

x𝑖⊗b

x𝑖) + b

Bu𝑖−¤

b

x𝑖

2

2+𝜆1b

c2

2+b

A

2

𝐹+b

B

2

𝐹+𝜆2b

H

2

𝐹),(3)

where

𝜆1>

0and

𝜆2>

0are the regularization hyperparameters,

b

x𝑖

is the reduced state vector of

𝑖

th snapshot, and

¤

b

x𝑖

is

the reduced state time derivative estimated via ﬁnite diﬀerence approximation. The regularization hyperparameters are

split into two separate parts: (1)

𝜆1

penalizes the constant, linear, and input operators, and (2)

𝜆2

penalizes the quadratic

operators. This was done because the quadratic operators are typically of signiﬁcantly diﬀerent order of magnitude, thus

requiring a diﬀerent scale of penalization. Regularization in the operator inference learning problem promotes stability

of the ROM and helps avoid overﬁtting. Selecting good values for the hyperparameters 𝜆1and 𝜆2used in the learning

step in Eqn.

(3)

is the key to eﬀective operator inference regularization. In this work, we follow the regularization

hyperparameter selection algorithm described in [20].

B. Parametric operator inference

For many systems of interest, a key question is how the dynamics are inﬂuenced by the variation of system parameters.

This parametric dependence is incorporated into operator inference by modifying Eqn. (2) to yield

d

d𝑡b

x(𝑡;𝜇)=b

c(𝜇) + b

A(𝜇)b

x(𝑡;𝜇) + b

H(𝜇)b

x(𝑡;𝜇) ⊗ b

x(𝑡;𝜇)+b

B(𝜇)u(𝑡),(4)

where

𝜇∈R𝑝

is the vector of

𝑝

parameters. The parametric case is addressed in the original operator inference paper [

18

]

by learning a ROM for each parameter value and interpolating the reduced model operators

b

c(𝜇),b

A(𝜇),b

H(𝜇),and b

B(𝜇)

via a cubic spline, for a total of 𝑟+𝑟2+𝑟3+𝑟𝑚 interpolations.

We employ a related approach here, except rather than interpolating between reduced model operators, we directly

interpolate between the reduced-state solutions. Let the training data for learning the ROM consist of

𝑞

parameter values

𝜇1, . . . , 𝜇𝑞. Then the reduced-state interpolation for parametric operator inference consists of the following steps:

•

Assemble the concatenated snapshot matrix

[

X

(𝜇1), . . . ,

X

(𝜇𝑞)] ∈ R𝑛×𝑘𝑞

that contains all the snapshots from the

FOM results at all 𝑞training parameter values.

•

Obtain the POD basis by taking the SVD of the concatenated snapshot matrix in the same manner as in Section II.A

to get the global reduced basis 𝑉𝑟.

©2022 Joint copyright of Lockheed Martin Corporation and the University of Texas at Austin, all rights reserved

3

•

Learn

𝑞

local operator inference ROMs at training parameters

𝜇𝑖, 𝑖 =

1

, . . . , 𝑞

using snapshot matrix X

(𝜇𝑖)

and

global reduced basis 𝑉𝑟.

•

Integrate the

𝑞

local ROMs to predict the reduced-state solutions

b

X(𝜇𝑖) ∈ R𝑟×𝑙, 𝑖 =𝑖, . . . , 𝑞

for the desired number

of time steps 𝑙≥𝑘involving extrapolation in time.

•

Interpolate over the dataset

n𝜇𝑖,b

X(𝜇𝑖)o𝑞

𝑖=1

using

𝑟𝑙

cubic spline interpolations (other regression techniques can

also be used) and use it to predict the reduced-state solution

b

X(𝜇) ∈ R𝑟×𝑙

at any new parameter value

𝜇

for

𝑙

time

steps.

•Use the global reduced basis to reconstruct to the full-order space V𝑟b

X(𝜇).

In a related work in [

24

], the reduced-state solution is ﬁt using a Gaussian process regression that takes parameters and

time as inputs. However, in this paper we do not have time as an input to our regression model. Instead we use the

operator inference ROM to extrapolate in time at each training parameter sample, then we interpolate at each time step.

Our parametric ROM method has two speciﬁc diﬀerences compared to the operator interpolation method in [

18

].

First, reduced-state solution interpolation can be accomplished with a cubic spline after integration of the ODEs and

thus without sacriﬁcing stability. In contrast, attempting to interpolate via the reduced model operators directly tends

to produce unstable ROMs for large-scale problems. We conclude that to correctly interpolate between operators,

manifold interpolation techniques such as interpolation on a Grassman manifold [

13

,

25

] should be investigated. Second,

reduced-state solution interpolation never obtains the reduced model operators at the new parameter value, and thus we

cannot test many typical stability metrics such as inspecting the sign of the real part of each eigenvalue of the linear

operator.

III. Coupled operator inference for ﬂutter

Systems with coupled physics present a particularly challenging problem for reduced-order modeling. We develop a

non-intrusive coupled ROM using operator inference by taking advantage of prior knowledge of the form of the coupled

system’s governing equations. In this paper, we consider the form of the coupled aerostructural equations that arise

from combining the governing equations of structural dynamics and ﬂuid dynamics. For full-order solution generation,

we use FUN3D’s aeroelasticity capability [

26

] to model the ﬂuid-structure interaction (FSI), and we summarize that

method in this section. Then, we describe the process of learning a coupled operator inference ROM speciﬁcally for the

ﬂutter FSI problem.

The structural dynamics behavior is described by the semi-discrete second-order linear diﬀerential equation

𝑀¥

𝛿+𝐶¤

𝛿+𝐾𝛿 =𝐹A,(5)

where

𝑀

is the mass matrix,

𝐶

is the damping matrix,

𝐾

is the stiﬀness matrix,

𝛿

is the vector of nodal displacements,

and

𝐹A

is the aerodynamic forcing applied to the structure. Using modal decomposition to deﬁne a reduced basis

for the structure uncouples the degrees of freedom. The structural state vector in the reduced space is then given by

b

xS=h𝜂¤𝜂i⊤

∈R2𝑟s

, where

𝜂

and

¤𝜂

are the modal displacements and modal velocities of the reduced-order structural

system, and

𝑟s

is the number of reduced-order structural DOFs (modes) retained in the reduced basis. Then we rewrite

the reduced structural dynamics equations in the block matrix format as a system of ﬁrst-order equations,

¤

b

xS(𝑡)="0𝐼

Ω𝑍#"𝜂(𝑡)

¤𝜂(𝑡)#+"0

b

𝐹A(𝑡)#,(6)

where

Ω = diag 𝜔2

1, . . . , 𝜔2

𝑟s

is a diagonal matrix consisting of the squares of the natural frequencies

𝜔𝑖

for structural

modes

𝑖=

1

, . . . , 𝑟s

,

𝑍=diag 2𝜔1𝜁1, . . . , 2𝜔𝑟s𝜁𝑟s

, and

𝜁𝑖

is the damping ratio of structural mode

𝑖

. The reader is

referred to the FUN3D exposition of the aeroelastic modeling capability [

26

] and to standard texts on structural dynamics

[27] for further details.

The ﬂuid dynamics behavior is described by the viscous, compressible, three-dimensional Navier-Stokes equations.

The Navier-Stokes equations are typically discretized using conservative variables as the states. Deriving a ROM using the

conservative variables does not lead to a quadratic form. However, [

28

] details how we can use a lifting transformation by

introducing the speciﬁc volume variable

𝜉=1

𝜌

, where

𝜌

is the density, to expose quadratic structure for the Navier-Stokes

equations. The lifted aerodynamic system has ﬁve state variables as deﬁned by x

A=hpuvw𝜉i⊤

∈R5𝑛𝑥

for a

spatial discretization with

𝑛𝑥

cells, where p

∈R𝑛𝑥

is the pressure, u

∈R𝑛𝑥

,v

∈R𝑛𝑥

, and w

∈R𝑛𝑥

are the ﬂuid velocities

4

in each direction, and

𝜉∈R𝑛𝑥

is the speciﬁc volume. The non-intrusive nature of the operator inference method allows

us to post-process the simulation data to the required form without needing to modify the high-ﬁdelity solver.

In FUN3D, aeroelasticity is implemented by integrating the structural dynamics model in Eqn.

(6)

via a predictor-

corrector scheme [

26

] coupled with the typical integration of the ﬂuid dynamics. Note that the software assumes

small deﬂections, an acceptable detail given that ﬂutter detection only requires assessment of the damping of an initial

transient, not accurate modeling of large-displacement dynamics.

We are now ready to consider the construction of a coupled ROM to represent this ﬂutter model using operator

inference. We generate training snapshots from a solve of the full-order FUN3D ﬂutter simulation at a given operating

condition. These snapshots consist of full-order ﬂuid state data and reduced structural state (modal displacement) data

because the FUN3D ﬂutter simulation uses the reduced-order structural dynamics. After generation of the training

snapshots, we project the full-order ﬂuid state x

A(𝑡)

to the reduced state,

b

xA(𝑡)

, using the reduced POD basis obtained

via SVD of 𝑋A.

The reduced state vector for the coupled FSI model,

b

xFSI (𝑡)

, is deﬁned as the concatenation of the reduced ﬂuid

states b

xA(𝑡)and the reduced structural states b

xS(𝑡)as

b

xFSI (𝑡)="b

xA(𝑡)

b

xS(𝑡)#.(7)

Then we learn a linear coupled operator inference ROM of the form

d

d𝑡b

xFSI (𝑡)=b

cFSI +b

AFSIb

xFSI (𝑡)(8)

where

b

cFSI

and

b

AFSI

are the reduced operators for the coupled aerostructural ﬂutter system. We perform a single least

squares calculation for the entire coupled operator inference ROM. In future work, we plan to extend this methodology

by embedding the block sparsity of Eqn.

(6)

and our knowledge of the quadratic behavior of the ﬂuid dynamics into

Eqn. (8).

IV. Applications of parametric and coupled operator inference

We present the application of the operator inference method to the construction of a nonlinear aerodynamic ROM

and a coupled aerostructural ROM. Section IV.A presents a parametric operator inference ROM for subsonic ﬂow over

the VAT aircraft. Section IV.B presents a coupled operator inference ROM for predicting ﬂutter in the AGARD wing.

A. VAT aircraft: aerodynamic analysis with oscillating angle of attack

The validation of aeroelastic tailoring (VAT) aircraft, shown in Figure 1, is a full-aircraft, 3D model created by

General Dynamics in the 1980s for carrying out ﬁghter airplane research and testing [21, 29]. The VAT wing’s airfoil

transitions from the NACA 64A003.5 at the root to the NACA 64A004 at the tip. The wing has a root chord of 42 inches,

a tip chord of 10.75 inches, and a half-span of 39.59 inches as shown in Figure 1c. We use an oscillating angle of attack

around 2

°

with an amplitude of 0

.

25

°

as an input to the dynamical system to induce similar unsteady aerodynamics as

seen in aircraft ﬂutter. The angle of attack is prescribed by

𝛼(𝑡)=2+0.25 sin(200𝜋∗𝑡),

where 200

𝜋rad

s

−1

corresponds to 100 Hz. This input is applied as a rigid body pitching motion of the VAT aircraft.

The ﬂuid ﬂow is at a Reynolds number of 1.5 million. We study the problem for a range of Mach numbers in

[

0

.

5

,

0

.

8

]

.

The full-order model of the VAT aircraft uses NASA’s FUN3D software [

23

] for CFD, along with a mesh discretization

generated by Pointwise v18.4R3

∗

. The mesh is constructed with a spherical far-ﬁeld of increasing coarseness to

reduce computational cost and with a radius of 350 times the root chord length. The unstructured mesh, shown in

Figure 2, consists of

𝑛𝑥=

2

,

321

,

731 cells, each of which will represent ﬁve ﬂuid ﬂow state variables leading to

2

,

321

,

731

∗

5

=

11

,

608

,

655 degrees of freedom. We use NASA’s FUN3D software to run a viscous, compressible

CFD simulation with the one-equation Spalart-Almaras turbulence model. The time step size is 50

µs

, providing a

temporal resolution of 200 time steps per oscillation cycle of the angle of attack. Figure 3 shows a representative surface

pressure contour over the VAT aircraft and an associated slice of the far-ﬁeld pressure around the wing, demonstrating

the kind of full-order snapshot data that will be generated for use in training and validating the operator inference ROM.

5

(a) VAT aircraft isometric view. (b) VAT wing top view. (c) Dimensions of the VAT wing [29].

Fig. 1 VAT aircraft geometry.

(a) VAT aircraft surface mesh. (b) VAT wing mesh. (c) VAT far-ﬁeld mesh.

Fig. 2 VAT aircraft meshes.

Fig. 3 Pressure contour from full-order FUN3D solution at Mach 0.6, 100 Hz oscillation of angle of attack.

We use the simulations at Mach 0.5, 0.7, and 0.8 as training data for the parametric ROM and the simulation at

Mach 0.6 as testing data for assessing the ROMs’ predictive capabilities. The snapshots are saved for the lifted state

variables

{𝑝, 𝑢, 𝑣, 𝑤, 𝜉 , }

as described in Section III. The training data consists of simulating from 0.25-0

.

27 s, capturing

around two oscillations of the data, for a total of 400 snapshots for each parameter sample in the training set. The

size of the global snapshot matrix constructed using concatenated data from the three training parameter samples is

∗Pointwise User Manual, v18.4R3

6

11

,

608

,

655

×

1200. The global POD basis vectors are obtained by taking the SVD of the global snapshot matrix. The

singular value decay plot for the global snapshot matrix is shown in Figure 4a. We use a basis size of

𝑟=

16, capturing

more than 99.999% of the total energy as shown in Figure 4b.

(a) Singular value decay, normalized by largest singular value (b) Cumulative energy retained for a given reduced dimension

Fig. 4 Singular value decay and cumulative energy of VAT global snapshot matrix for Mach ={0.5,0.7,0.8}.

We train a parametric operator inference ROM using the reduced-state solution interpolation method described in

II.B. We ﬁrst train the local operator inference ROMs at the three training parameter samples using the local snapshot

matrix of size 11

,

608

,

655

×

400 consisting of 400 time steps. Note that the global reduced basis is used during the

learning process. Then we integrate the three local ROMs to predict for a total of 1000 time steps from 0.25-0

.

3 s,

capturing ﬁve oscillations of the data while extrapolating in time. The reduced-state solutions of size 16

×

1000 obtained

from each of the three local ROMs are interpolated across the parameter space to capture predictions both in parameter

space and in time for the parametric ROM. The prediction at the test parameter of Mach 0.6 for 1000 time steps is

obtained from the trained parametric ROM and reconstructed to the full-order solution using the global reduced basis.

We look at pressure traces at four diﬀerent locations around the airfoil in one slice of the VAT wing as shown in

Figure 5. Figure 6 compares the full-order and reduced-order pressure traces at the four diﬀerent locations for Mach

0.6. The parametric operator inference ROM yields accurate predictions of the dominant periodic behavior of all four

pressure traces for the unseen parameter with small discrepancy in the pressure oscillation amplitude showing good

approximation of the FOM. This shows the usefulness of the scientiﬁc machine learning method that embeds physics

into the learning process as compared to a black-box surrogate method that might not have the ability to learn the

oscillatory behavior. Our parametric operator inference ROM actually learned the underlying dynamic feature of the

responses in a non-intrusive way

To summarize, a single FUN3D VAT simulation takes about 23 hours to run on 64 cores, with an additional 26 hours

needed to write the complete state outputs to ﬁle, for a total of about 2 days to complete a single full-order aerodynamic

solution at a single operating condition. Once the training snapshots are generated from the FOM, the operator inference

method requires approximately 1 hour to complete the POD and learning steps that produce the parametric ROM. After

the ROM is built, an additional computational expense on the order of seconds is required to predict the reduced-order

aerodynamic solution at a new parameter value (Mach number) over the same temporal domain as the FOM solutions.

We see that the expense of generating full-order training snapshots can be on the order of days for large-scale CFD

problems, but once the parametric ROM is built, we see speed-ups of at least three orders of magnitude for predictions

of ﬂuid state outputs at new parameter values.

B. AGARD wing: aeroelastic ﬂutter

The AGARD wing originated from a NATO working group and is used in this paper to demonstrate coupled ROM

ﬂutter prediction capabilities. Flutter occurs when the structural dynamics (i.e. the resonance) of an aircraft’s wings and

the ﬂuid ﬂow over the surface of the wings become unstably coupled, typically leading to catastrophic vehicle failure.

This makes avoiding ﬂutter a critical consideration in aircraft design. However, it is expensive both to test and to model

ﬂutter accurately, due to the inherent multiphysics coupling of the problem. We show the potential of the coupled

ROM method discussed in Section III to accurately predict the onset of ﬂutter in an aerostructural model at much lower

7

(a) Spanwise location. (b) Location relative to spanwise slice of airfoil.

Fig. 5 Locations of pressure traces for VAT aircraft model.

(a) Trace location 1 (b) Trace location 2

(c) Trace location 3 (d) Trace location 4

Fig. 6 Comparison of pressure traces through time for Mach 0.6 obtained from the FOM and ROM (

𝑟=

16)

shown at four diﬀerent positions in an airfoil slice of the wing.

computational cost than a full-order, nonlinear CFD solver (in our case, FUN3D).

The AGARD wing has an aspect ratio of 1.6525, a taper ratio of 0.6576, and a sweepback angle of 45 degrees [

22

].

Similar to the VAT aircraft wings, the AGARD wing has a NACA65A004 airfoil. We use the unstructured mesh of

8

the AGARD wing from the FUN3D example problem materials

†

as shown in Figure 7. FUN3D’s built-in aeroelastic

modeling feature is used to generate the full-order solutions. The aeroelastic simulation requires the results of a modal

decomposition from a linear structural dynamics model to eﬃciently represent the structural behavior of the AGARD

wing. Speciﬁcally, we use the ﬁrst four AGARD wing mode shapes provided by the NASA FUN3D developers. These

mode shapes are visualized in Figure 8 along with their associated natural frequencies. For more details about FUN3D’s

aeroelastic modeling capability, we refer the reader to the original exposition of these capabilities in [26].

Fig. 7 AGARD wing CFD mesh with pressure trace locations.

(a) Mode 1 (9.6 Hz) (b) Mode 2 (38.2 Hz) (c) Mode 3 (48.3 Hz) (d) Mode 4 (91.5 Hz)

Fig. 8 First four structural mode shapes of the ﬁxed-boundary AGARD wing, scaled independently for

visualization

To analyze a ﬂutter case in FUN3D, we select a Mach number and a dynamic pressure, then we perturb the system by

prescribing an nonzero initial generalized modal velocity for each of the structural mode shapes. Then the generalized

modal displacement is monitored over time to see if the modal displacements settle to steady-state values or continue to

increase and become unstable, which indicates that ﬂutter has occurred. In this study, we use a viscous, compressible

ﬂuid ﬂow model. The CFD problem is solved for ﬁve ﬂuid ﬂow state variables over

𝑛𝑥=

492

,

778 cells leading to

492

,

778

∗

5

=

2

,

463

,

890 degrees of freedom. The procedure is repeated at diﬀerent Mach number - dynamic pressure

combinations to characterize the ﬂutter boundary. In Figure 9, we see the modal displacement of the ﬁrst mode at Mach

0.9 and dynamic pressures

𝑄={

65

,

75

,

85

,

95

}

psf for the full-order FUN3D results. Note that the orange curve for 75

psf is nearly in a limit cycle, indicating that this operating condition is near the ﬂutter boundary. The green and red

curves are growing in amplitude over time, indicating ﬂutter.

Although it is not the key quantity of interest for ﬂutter, it is informative to look at how well the ROM can replicate

the ﬂuid dynamics response of the system as well, since it is the coupling between the structural response and the ﬂuid

response that explains why a wing ﬂutters under particular operating conditions. Figure 10 shows an example of the

kind of surface pressure proﬁle that occurs in the full-order FUN3D results.

This is an interesting demonstration case for our non-intrusive coupled ROM described in Section III because we do

not have explicit access to the original mesh of the AGARD wing or even the structural dynamics code that was used to

†FUN3D v13.4 Training - Session 16: Aeroelastic Simulations: https://fun3d.larc.nasa.gov/session16_2018.pdf

9

Fig. 9 Modal displacement of ﬁrst AGARD mode shape for Mach 0.9 and dynamic pressures

𝑄={

65

,

75

,

85

,

95

}

psf

Fig. 10 Representative pressure contour on surface of AGARD wing at Mach 0.9, dynamic pressure 75 psf.

obtain the modal results. However, we can still learn a coupled ROM since it is completely non-intrusive and we can

work with legacy data. The mode shapes can be used as additional reduced basis functions that we add directly to the

set of bases we obtain from POD after running a FUN3D coupled solve. Thus we are able to very easily incorporate

prior knowledge of the physics of the system into our model reduction method when this knowledge becomes available,

without requiring any additional work on the part of the simulation analyst. We exploit the structure of the coupled

aerostructural governing equations, which consist of linear structural dynamics equations and quadratic ﬂuid dynamics

(Navier-Stokes) equations, to learn the coupled ROM for ﬂutter prediction. In this work, we learn a coupled operator

inference ROM with linear form. We use basis functions from both the structural dynamics modal decomposition (i.e.

an eigenvalue problem) and the proper orthogonal decomposition of the ﬂuid state snapshot matrix. We will extend the

coupled ROM to include quadratic terms for the ﬂuid ﬂow equations in a future eﬀort. The ﬂuid ﬂow snapshot matrix

consists of training data from 400 time steps with a time step size of 0.277 ms, leading to a training snapshot matrix of

10

size 2

,

463

,

890

×

400. The singular value decay plot for the ﬂuid ﬂow training snapshot matrix is shown in Figure 11a.

We use a basis size of 18 for ﬂuid ﬂow POD, capturing 99

.

995% of the total energy as shown in Figure 11b . The total

size of the reduced state vector for the coupled ROM ends up being 18

+

2

∗

4

=

26 after including four structural modes

(each mode including a modal displacement and a modal velocity).

(a) Singular value decay, normalized by largest singular value (b) Cumulative energy retained for a given reduced dimension

Fig. 11 Singular value decay and cumulative energy of snapshot matrix.

Figure 12 compares the FOM and the coupled ROM predictions for a non-ﬂuttering operating condition (

𝑄=

75

psf) and a ﬂuttering operating condition (

𝑄=

95 psf). For the modal displacement plots in Figures 12a and 12c, the

dashed black lines representing the coupled ROM predictions are overlaid almost exactly on the solid lines representing

the full-order predictions for all four modes. In Figures 12b and 12d, we can see that absolute error between the FOM

and the coupled ROM predictions is below 2% for the non-ﬂuttering case of

𝑄=

75 psf and below 5% for the ﬂuttering

case of

𝑄=

95 psf. Typically, the ﬁnal step of the ﬂutter analysis would be to use a method such as the log decrement to

numerically estimate the damping of the modal displacements at a given operating condition to decide if the case is

ﬂuttering. For that conventional workﬂow, the temporal extrapolation we see in Figure 12 using the coupled ROM is

accurate for a suﬃciently long duration to calculate damping and predict ﬂutter. We also investigate the response of

ﬂuid states such as the pressure during aerostructural coupling. In Figure 13, we see the pressure trace at the leading

edge and the trailing edge of the wing tip (see Figure 7) for the nonﬂuttering and ﬂuttering cases. The coupled ROM is

able to accurately track the higher frequency oscillations, especially in the trailing edge plots.

A single FUN3D ﬂutter simulation takes 25 minutes to run on 32 cores, with an additional 3 minutes for writing

data outputs to ﬁle, for a total of 28 minutes to complete a full-order ﬂutter solution at a single operating condition.

Once the training snapshots are generated from the FOM, the operator inference method requires approximately 1.5

minutes to complete the POD and learning steps that produce the ROM. After the ROM is built, an additional 3 seconds

of computation are required to predict the reduced-order ﬂutter solution over the same temporal domain as the FOM.

Thus, we see that the operator inference ROM provides a run-time speed up from minutes for the FOM to seconds for

the ROM, with the added ability to rapidly and accurately extrapolate beyond the temporal domain of the FOM results.

V. Conclusion

Operator inference is a data-driven, physics-informed model reduction method for dynamical systems. The non-

intrusive nature of the operator inference method makes it an appealing strategy for generating ROMs for commercial

solvers and for legacy solvers or data since it only requires access to the state solutions generated by the solvers and

not to the discretized operators of the underlying governing equations. In this paper, we develop a parametric ROM

for large-scale aerodynamic analysis and a coupled ROM for ﬂutter analysis based on operator inference. We lift

the Navier-Stokes equations governing the ﬂuid ﬂow problem to a quadratic form for operator inference and use a

reduced-state solution interpolation method to build the parametric ROM. We show that the parametric ROM is able to

accurately extrapolate in time and parameter space to predict the dynamical behavior for a large-scale aerodynamic

problem using the VAT aircraft at unseen Mach numbers. Then we show the coupled ROM for coupled multiphysics

applications, where we often have some type of physical insight into at least one of the governing physical regimes a

11

(a) Q=75, modal displacement (solid lines: FOM; dashed

lines: ROM) (b) Q=75, absolute error in modal displacement

(c) Q=95, modal displacement (solid lines: FOM; dashed lines:

ROM) (d) Q=95, absolute error in modal displacement

Fig. 12 Comparison of modal displacement for four modes between the FOM and the coupled ROM at Mach=0.9.

Training data and testing data split is shown by vertical black line.

priori. In the case of the ﬂutter analysis, this prior knowledge is the modal decomposition information which, similar

to many typical industry workﬂows, was provided to us without access to the initial full-order model. We are able to

learn a non-intrusive coupled ROM, which is capable of predicting the key quantity of interest of generalized modal

displacement to within less than 5% absolute error. Both the parametric ROM and the coupled ROM lead to signiﬁcant

speedups compared to the high-ﬁdelity simulations. Our future work will combine these capabilities (parametric and

coupled ROMs) for aeroelastic systems to create parametric coupled ROMs capable of predicting the onset of ﬂutter

at unseen dynamic pressures and Mach numbers. This eﬀort would be supported by more robust development of the

learning step of the operator inference method for coupled ROMs, in order to incorporate knowledge of the block-sparsity

of the second-order coupled governing equations.

Acknowledgements

This work was supported in part by Lockheed Martin University Research Agreement MRA-16-005-RPP012-001

and by the AFOSR MURI on physics-based machine learning under grant FA9550-21-1-0084.

References

[1]

Sirovich, L., “Turbulence and the dynamics of coherent structures. I. Coherent structures,” Quarterly of Applied Mathematics,

Vol. 45, No. 3, 1987, pp. 561–571.

12

(a) Q=75, Trace location 1 (b) Q=75, Trace location 2

(c) Q=95, Trace location 1 (d) Q=95, Trace location 2

Fig. 13 Operator Inference pressure predictions at Mach=0.9 at the wing tip. Training data and testing data

split is shown by vertical black line.

[2]

Berkooz, G., Holmes, P., and Lumley, J. L., “The proper orthogonal decomposition in the analysis of turbulent ﬂows,” Annual

Review of Fluid Mechanics, Vol. 25, 1993, pp. 539–575.

[3]

Lumley, J. L., “The Structures of Inhomogeneous Turbulent Flow,” Atmospheric Turbulence and Radio Wave Propagation,

1967, pp. 166–178.

[4]

Rathinam, M., and Petzold, L., “A New Look at Proper Orthogonal Decomposition,” SIAM Journal on Numerical Analysis,

Vol. 41, No. 5, 2003, pp. 1893–1925. https://doi.org/10.1137/S0036142901389049.

[5]

Benner, P., Güğercin, S., and Willcox, K., “A survey of projection-based model reduction methods for parametric dynamical

systems,” SIAM Review, Vol. 57, No. 4, 2015, pp. 483–531. https://doi.org/10.1137/130932715.

[6] Antoulas, A. C., Approximation of Large-Scale Dynamical Systems, SIAM, Philadelphia, PA, 2005.

[7]

Schmid, P. J., “Dynamic mode decomposition of numerical and experimental data,” Journal of Fluid Mechanics, Vol. 656,

2010, p. 5–28. https://doi.org/10.1017/S0022112010001217.

[8]

Tu, J. H., Rowley, C. W., Luchtenburg, D. M., Brunton, S. L., and Kutz, J. N., “On dynamic mode decomposition: Theory and

applications,” Journal of Computational Dynamics, Vol. 1, No. 2, 2014, pp. 391–421. https://doi.org/10.3934/jcd.2014.1.391.

[9]

Theodorsen, T., “General Theory of Aerodynamic Instability and the Mechanism of Flutter,” Annual Report of the National

Advisory Committee for Aeronautics, Vol. 268, 1935, p. 413.

[10] Dowell, E. H., A Modern Course in Aeroelasticity, 6th ed., Springer, Durham, NC, 2021.

13

[11]

Hall, K. C., “Eigenanalysis of Unsteady Flows about Airfoils, Cascades, and Wings,” AIAA Journal, Vol. 32, No. 12, 1994, pp.

2426–2432. https://doi.org/10.2514/3.12309.

[12]

Dowell, E. H., and Hall, K. C., “Modeling of Fluid-Structure Interaction,” Annual Review of Fluid Mechanics, Vol. 33, 2001,

pp. 445–490. https://doi.org/10.1146/annurev.ﬂuid.33.1.445.

[13]

Amsallem, D., and Farhat, C., “Interpolation Method for the Adaptation of Reduced-Order Models to Parameter Changes and

Its Application to Aeroelasticity,” AIAA Journal, Vol. 46, No. 7, 2008, pp. 1803–1813. https://doi.org/10.2514/1.35374.

[14]

Silva, W. A., “Simultaneous Excitation of Multiple-Input/Multiple-Output CFD-Based Unsteady Aerodynamic Systems,”

Journal of Aircraft, Vol. 45, No. 4, 2008, pp. 1267–1274. https://doi.org/10.2514/1.34328.

[15]

Juang, J.-N., and Pappa, R. S., “An Eigensystem Realization Algorithm for Modal Parameter Identiﬁcation and Model Reduction,”

Journal of Guidance, Control, and Dynamics, Vol. 8, No. 5, 1985, pp. 620–627. https://doi.org/10.2514/3.20031.

[16]

Silva, W. A., “AEROM: NASA’s Unsteady Aerodynamic and Aeroelastic Reduced-Order Modeling Software,” Aerospace,

Vol. 5, No. 2, 2018. https://doi.org/10.3390/aerospace5020041.

[17]

Lowe, B. M., and Zingg, D. W., “Eﬃcient Flutter Prediction Using Reduced-Order Modeling,” AIAA Journal, Vol. 59, No. 7,

2021, pp. 2670–2683. https://doi.org/10.2514/1.J060006.

[18]

Peherstorfer, B., and Willcox, K., “Data-driven Operator Inference for Nonintrusive Projection-Based Model Reduction,”

Computer Methods in Applied Mechanics and Engineering, Vol. 306, 2016, pp. 196–215. https://doi.org/10.1016/j.cma.2016.

03.025.

[19]

Qian, E., Kramer, B., Peherstorfer, B., and Willcox, K., “Lift & Learn: Physics-informed machine learning for large-

scale nonlinear dynamical systems,” Physica D: Nonlinear Phenomena, Vol. 406, 2020, p. 132401. https://doi.org/https:

//doi.org/10.1016/j.physd.2020.132401.

[20]

McQuarrie, S. A., Huang, C., and Willcox, K. E., “Data-driven reduced-order models via regularised Operator Inference

for a single-injector combustion process,” Journal of the Royal Society of New Zealand, Vol. 51, No. 2, 2021, pp. 194–211.

https://doi.org/10.1080/03036758.2020.1863237.

[21]

Rogers, W., Braymen, W., Murphy, A., Graham, D., and Love, M., “Validation of aeroelastic tailoring by static aeroelastic and

ﬂutter tests,” Tech. Rep. AFWAL-TR-81-3160, General Dynamics Corporation, Fort Worth, Texas, 1982.

[22]

Yates, E. C. J., “AGARD Standard Aeroelastic Conﬁgurations for Dynamic Response. Candidate Conﬁguration I.-Wing 445.6,”

Tech. rep., NASA Langley Research Center, Hampton, Virginia, 08 1987.

[23]

Biedron, R. T., Carlson, J.-R., Derlaga, J. M., Gnoﬀo, P. A., Hammond, D. P., Jones, W. T., Kleb, B., Lee-Rausch, E. M.,

Nielsen, E. J., Park, M. A., Rumsey, C. L., Thomas, J. L., Thompson, K. B., and Wood, W. A., FUN3D Manual 13.6, NASA

Langley Research Center, Hampton, Virginia, 2019.

[24]

Guo, M., and Hesthaven, J. S., “Data-driven reduced order modeling for time-dependent problems,” Computer Methods in

Applied Mechanics and Engineering, Vol. 345, 2019, pp. 75–99. https://doi.org/10.1016/j.cma.2018.10.029.

[25]

Absil, P.-A., Mahony, R., and Sepulchre, R., “Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic

Computation,” Acta Applicandae Mathematica, Vol. 80, No. 2, 2004, pp. 199–220. https://doi.org/10.1023/B:ACAP.0000013855.

14971.91.

[26]

Biedron, R., and Thomas, J., “Recent enhancements to the FUN3D ﬂow solver for moving-mesh applications,” 47th

AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition, 2009, p. 1360. https:

//doi.org/10.2514/6.2009-1360.

[27] Ginsberg, J. H., Mechanical and Structural Vibrations, John Wiley & Sons, Inc., New York, NY, 2001.

[28]

Swischuk, R., Kramer, B., Huang, C., and Willcox, K., “Learning Physics-Based Reduced-Order Models for a Single-Injector

Combustion Process,” AIAA Journal, Vol. 58, No. 6, 2020, pp. 2658–2672. https://doi.org/10.2514/1.J058943.

[29]

Ruhlin, C. L., Watson, J. J., Ricketts, R. H., and Doggett, R. V., “Evaluation of Four Subcritical Response Methods for

On-Line Prediction of Flutter Onset in Wind Tunnel Tests,” Journal of Aircraft, Vol. 20, No. 10, 1983, pp. 835–840.

https://doi.org/10.2514/3.44951.

14