A Compact CFD-Based Reduced Order Modeling for Gust Analysis

Z. Wang1, Z. Zhang2, P.C. Chen3, and D. Sarhaddi4 ZONA Technology, Inc., Scottsdale, AZ, 85258

A general unsteady CFD/Aeroelastic code, ZEUS, is enhanced with arbitrary discrete gust profile analysis capability. Typical type of discrete gust profile such as one-minus-cosine gust is easily defined in the ZEUS input, and the sharp edge gust and any other type of gust can be simply defined using the table input format. The aerodynamic response under the sharp edge gust combined with the Proper-Orthogonal-Decomposition (POD) method allows us develop a compact and efficient reduced-order model (ROM), which can be used to rapidly predict pressure coefficient solutions on the aerodynamic surface under any type of discrete gust profile other than the sharp edge gust. The gust ROM developed at the pressure efficient level makes it independent of the structural model, and readily applicable for the framework of Dynamic Flight Simulation (DFS) tool which combines a flight dynamics model with an add-on aeroelastic model in a Matlab/Simulink environment. The AGARD 445.6 wing configuration is used as a numerical example to demonstrate the present gust ROM methodology. It has been shown that the gust ROM with the convolution law and POD method is proper over various Mach numbers considered.

I. Introduction UST analysis plays an important role in modern aircraft design. The Federal Aviation Regulations (FAR) [1] in particular specifies the discrete gust and continuous gust design criteria to which the aircraft must be subjected.

Conventionally, the dynamic response of an aircraft subject to a discrete gust has been analyzed or predicted by panel methods in the frequency domain [2, 3]. If time domain solutions are required, then some numerical technique such as the rational function approximation method may be employed to convert from frequency domain to time domain. In the transonic regime, the computational fluid dynamics (CFD) methodology obviously possesses advantages over the potential flow panel methods in terms of shock wave capturing and compressive flow solution accuracy. Extending the CFD method for discrete gust analysis has been relatively new in the aerospace industry and academia. Raveh and her colleagues work in this field [4-6] are the inspiration for our current work. In their CFD framework, the so called “Field Velocity” method, in which the gust velocities are assigned to the CFD grids’ vertical velocity components without actually moving the grids. Considering the fact that the discrete gust is modeled as a traveling gust, it would mean that, at every moment, the vertical gust velocity component at each CFD grid would need be updated according to the gust profile at current time. The number of CFD grids can easily reach to thousands or tens of thousands, time consuming could be an issue for “Field Velocity” method during gust analysis. In present paper, the gust capability is enabled towards the CFD code employed here, ZEUS (ZONA’s Euler Unsteady aerodynamic Solver) [7], by adding the downwash due to gust into the boundary conditions to satisfy the no-penetration condition of the Euler’s equations at the aerodynamic surface grids. Since the number of grids at the aerodynamic surface is much less than the total number of grids in the CFD computational domain, the question of the efficiency achieved here is not an issue.

G

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On the other hand, the time-domain reduced order modeling (ROM) for gust loads is also desired here for the work of a dynamic flight simulation tool called the Dynamic Flight Simulation (DFS) tool which combines a flight dynamics model with an add-on aeroelastic model in a Matlab/Simulink environment [8], where the gust ROM is needed to quickly provide incremental aerodynamic forces besides the ones due to the aeroelastic deformations. Raveh [5, 6, 9] has also investigated a couple of gust ROM’s such as the convolution method and the auto regressive

1 R&D Engineering Specialist, Member AIAA 2 R&D Engineering Specialist 3 President, Associate Fellow AIAA 4 General Manager

52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference<BR> 19th4 - 7 April 2011, Denver, Colorado

AIAA 2011-2041

Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

moving average (ARMA) model. The convolution method requires the CFD gust analysis under the sharp edge gust; while the Gaussian filtered white noise data is used as the representative gust profile, and the results are used for the identification of ARMA model. In Raveh’s gust ROM models, the system outputs are either the total lift/moment or generalized aerodynamic forces (GAF). The GAF’s are dependent upon the structural mode shapes. In the present work, we intend to develop a gust ROM which is independent of the structural properties, and it is only relying on the aerodynamic geometrical configuration. To do so, the system outputs we choose for the gust ROM are directly on the pressure coefficient level. And, instead of developing the gust ROM for each surface panel individually, we further employ the proper orthogonal decomposition (POD) method [10] to reduce the number of ROMs to only a few for the whole aerodynamic surface. This technique can be applied to the whole computational domain as well if desired.

POD method has been broadly used for the model reduction. It also known as Karhunen-Loeve decomposition. Given a set of solutions or snapshots, the POD method calculates a set of basis solutions that hold the most energetic modes of the system. With POD modes available, different approaches have been applied to obtain the reduced order model in place of the original dynamic system. Among them include the Galerkin projection method and the response surface method (RSM). POD/Galerkin approach has also been used in the frequency domain [11], while POD/RSM is largely used in the optimization field to serve as a surrogate model in order to achieve computational efficiency [12, 13]. In this paper, POD method is used in a novel way to reduce the number of ROMs required to represent the dynamic system.

The organization of this paper is as follows: First, the aerodynamic/aeroelastic solver, ZEUS, and the procedure to include discrete gust consideration are briefly introduced. Then the method for the gust reduced order modeling is described. In Section IV, the results on the numerical example, the AGARD 445.6 wing, are presented. Finally some remarks/conclusions are given in the end.

II. Aerodynamic Solver – ZEUS ZEUS is a ZONA Euler Unsteady aerodynamic Solver that integrates the essential disciplines required for

aeroelastic design/ analysis. It uses the Cartesian Euler flow solver with a boundary-layer option to include viscous effects as the underlying aerodynamic force generator coupled with structural finite element modal solution to solve various aeroelastic problems, such as flutter, maneuver loads, store ejection loads, gust loads, and static aeroelastic/trim analysis.

ZEUS is driven by the need of higher fidelity CFD methods for aeroelastic analysis, yet the aerospace industry is still accustomed to using panel code such as the doublet lattice method and ZAERO [14]. This is because the surface meshing scheme employed by these panel codes, is far simpler than the grid generation procedure required by the CFD methods. To this end, ZEUS is developed to use the bulk data input format that is very similar to that of Nastran and ZAERO. In fact, the majority of the input bulk data cards of ZEUS are identical to those of ZAERO. And most impressively, ZEUS is equipped with an automated mesh-generation scheme that can automatically generate a flowfield mesh by extending from the surface mesh. This automated mesh-generation greatly relieve users from the sometimes tedious CFD grid generation procedure, and is one of the advantages of ZEUS over other CFD codes in the market. In addition, ZEUS has an overset mesh capability to handle very complex aircraft configurations such as a complete aircraft with external stores.

In present work, we will focus on the Euler solver of ZEUS. Viscous effects through the boundary layer are not considered unless indicated.

A. Time-Accurate Euler Method The three-dimensional unsteady Euler equations in conservative integral form over a fixed control volume V

enclosed by the surface S are as follows:

0V S

dV dSt∂

+ =∂ ∫ ∫W G ni (1)

where

(2) [ Tu v w Eρ ρ ρ ρ ρ=W ]

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ρ

( )

u pv pw p

E p u v w

ρρρ

ρ

⎡ ⎤⎢ ⎥+⎢ ⎥⎢ ⎥+=⎢ ⎥+⎢ ⎥⎢ ⎥+ + +⎣ ⎦

x

y

z

z y z

q eq eGq e

q e e e

q

u v w

(3)

= + + zq e e ex y (4)

( 2 2 2

1 2E u v

γ ρ= + + +

−)1 1p w (5)

Applying Equation (1) to each cell we obtain a set of ordinary differential equations of the form

( ) ( ), , , , , , 0i j k i j k i j kd Vdt

+ =W R W (6)

where is the volume of the (i,j,k) cell and the residual , ,i j kV ( ), ,i j kR W is obtained by evaluating the flux integral.

Furthermore, the ddt

operator is approximated by an implicit backward difference formula of second-order

accuracy in the following form (dropping the subscripts for clarity): , ,i j k

( )1 13 2 1 02 2

n n n nV V Vt t t

+ −⎡ ⎤ ⎡ ⎤ ⎡ ⎤− + +⎣ ⎦ ⎣ ⎦ ⎣ ⎦Δ Δ ΔW W W R W 1+ = (7)

Equation (7) can be solved for at each time step by solving the following steady-state problem in a peudo time :

1n+W*t

( )1

* 1* 0n

nddt

+++ =

W R W (8)

where

( ) ( ) ( ) ( ) (* 13 2 12 2

nV Vt t t

−= + − +Δ Δ Δ

R W R W W W W )n V (9)

Details of the integration scheme for Equation (6) can also be found in Zhang [15]. Notably, the local time stepping, residual smoothing, and multigrid technique are implemented to accelerate the convergence of the solution.

B. Approximate Boundary Condition and Discrete Gust Implementation A thin wing moving slightly or deforming about its mean position is considered. The mean position of the wing

chord plane lies in the horizontal plane z=0. The shape of the wing is described by z=f(x,y) and g(x,y) for its upper and lower surfaces, respectively, then the instantaneous position of the wing is described by z=F(t,x,y) and z=G(t,x,y) for the upper and lower surfaces. Under the assumption 1F , the first order approximation of the wall velocity boundary conditions on the upper surface of the wing at instant t reads:

(10) ( , , ,0 ) ( , , ,0 ) ( , , ,0 ) ( )w t x y u t x y F v t x y F F O Fx y t+ + += + + +

where the subscripts x, y, and t denote the partial derivatives with respect to x, y and t, respectively; O(F) represents terms of the same order of magnitude as F or higher. The normal velocity boundary condition on the lower surface is treated similarly.

For a traveling gust with an arbitrary profile, its time-domain representation can be illustrated in and expressed as a gust magnitude multiplied by a time function T as Equation

Figure 1(11):

oG G

C

x xw W T tV

⎛ ⎞−= −⎜

⎝ ⎠⎟ (11)

where t is time, 0x is the reference point for the gust, VC is the free-stream speed, x is a point on the aircraft, and is the magnitude of the gust. GW

z

xV

x0

⎟⎠⎞

⎜⎝⎛ −⋅

VxxtTWG

0

- z

xV

x0

⎟⎠⎞

⎜⎝⎛ −⋅

VxxtTWG

0

-

Figure 1. Time-domain representation of a traveling gust with an arbitrary profile

For a one-minus-cosine gust, its equivalent form of Equation (11) can be expressed as:

( )1 21 cos for 0 /

, 2 /0 for /V or 0

GG G C

G C

W Lw x t L V

L

πτ τ

τ τ

⎧ ⎛ ⎞− ≤⎪ ⎜ ⎟= ⎨ ⎝ ⎠

⎪ > <⎩

G CV≤ (12)

where o

C

x xtV

τ −= − , and is the length of the gust profile. GL

Numerical implementation of the gust effects due to a traveling discrete gust profile into computation is performed as follows. The gust is assumed to be uniform along span-wise direction, and only has vertical component. For a given time (t), the flow velocity due to gust at any point, (xi, yi) can be immediately calculated using Equation (11) or Equation (12) for a one-minus-cosine gust. This component is then normalized by the free stream speed and inserted into the boundary condition (10) to satisfy the no-penetration condition at the aerodynamic surface:

( )( , , ,0 ) ( , , ,0 ) / ( , , ,0 ) ( )x G C y tw t x y u t x y F w V v t x y F F O F+ + += − + + + (13)

Note-worthily ZEUS not only can handle the one-minus-cosine discrete gust, but also is enhanced to handle any arbitrary gust profiles, which may be inputted in a table format.

III. Reduced Order Model of Gust Aerodynamics Our objective in ROM is to find a simplified mathematical model representing the dynamics of a dynamic

system in terms of the relationship between inputs and outputs by means of system identification terminology. As for the gust aerodynamics, one easy choice for the system input is the gust velocity observed at the most forefront location of the aerodynamic configuration. Since we only consider span-wise uniform gust profile, all the locations

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at the aerodynamic surface will observe a similar gust profile in a staggered fashion. The system outputs will be the pressure coefficients, Cp, at all the aerodynamic surface panels.

A. Convolution Method One of most straightforward models for such a mathematical formulation is the convolution method. The

convolution method requires the aerodynamic solution under a sharp edge gust. A sharp edge gust input profile is defined as a traveling gust of constant vertical velocity as:

( ) for 0,

0 for 0SEG

G

Ww x t

ττ≥⎧

= ⎨ <⎩ (14)

where C

xtV

τ = − , and is the gust velocity strength. SEGW

During the analysis the time histories of the Cp solution with the steady component subtracted out can be recorded. If the convolution method is applied to each aerodynamic panel individually, we would have the solution for any other gust profile using the following formula:

0

( ) ( ) ( )t

p G p SEGC t w C t dτ τ τ= ∫ − (15)

where is the time derivative of the gust velocity, and Gw p SEGC represents the solution under the sharp edge gust.

B. Proper-Orthogonal-Decomposition (POD) on Cp Solutions Although one could carry out Equation (15) for each location, there is a far more efficient way to fulfill the task.

We demonstrate that the POD method can be well adopted to couple with the convolution approach. Consider a system where the state variable, e.g., Cp, at all N locations are measured at n time steps. They can be

arranged in a matrix, S: N n×

( ) ( ) ( )1 2, , , nS x x x⎡ ⎤= ⎣ ⎦ (16)

An approximation of the system dynamics is obtained by projecting the original N-dimensional state space onto an m-dimensional subspace:

[ ] [ ]N m m nS β

× ×= Φ (17)

where each column of [ ]Φ represents a POD modes, and [ ]β consists of the corresponding coefficients at each time step. The POD modes can be found through the eigenvalue analysis to the matrix of A:

TA S S= (18)

The eigenvectors of A matrix are the POD modes of S matrix. The eigenvalues of A express the energy level of each POD mode. Based on the accumulative energy, we could choose suitable number, m, of POD modes so that sufficient energy is remained even limited number POD modes are used.

The POD modes on the Cp flowfiled time histories for the training are firstly identified. The corresponding β coefficients will be used as the system outputs instead. Thus, if m POD modes are used, there will be m ROMs identified.

This POD analysis has been automated in the RETINAS sub-module of ZEUS. The results of RETINAS directly output the POD β coefficients, which together with POD modes consists of the gust ROM.

C. Prediction of Cp under Arbitrary Gust Profile using ROM

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With the POD modes and POD coefficients under the sharp edge gust available, the m POD coefficients for any other gust profile ( )Gw t will be computed similarly as Equation (15):

(19) 0

( ) ( ) ( ) , 1, ,t

i G i SEGt w t d iβ τ β τ τ= − =∫ m

m

Equation (19) in the discrete time format (at time n) will be:

(20) ( ) [ ]1

( ) ( ) , 1, ,n

i G iSEGk

n w k n k dt iβ β=

= − =∑

where dt is the time step size. Note that iSEGβ should be normalized by the sharp edge gust strength when applying Equation (20).

Then the time histories of Cp can be obtained by simply multiplying the POD modes with the 'sβ :

[ ] [ ] 1( )p N m mC n β× ×

= Φ (21)

IV. Numerical Example Results The AGARD 445.6 wing model is used as the numerical example to demonstrate the proposed gust ROM

methodology. The wing has a sweptback angle, and the cross section is NACA 65A004 airfoil. Figure 2 shows the aerodynamic surface mesh, and the automatically generated Cartesian grid by ZEUS based on the surface mesh plus some CFD grid control parameters. The aerodynamic surface mesh consists of total 2400 panels with upper and lower surface each of 1200 panels (60 chord-wise × 20 span-wise).

A. Validation of ZEUS’s Direct Gust Analysis Capability Before the work on the gust ROM, we would like to validate ZEUS’s gust analysis capability by checking

against ZAERO solutions. An one-minus-cosine gust with strength of , and length of is considered.

/ 0.00G CW V =

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The gust ROM is developed for the rigid aerodynamic configuration, i.e., the effects of elastic deformation is not considered. By doing so, the structural effects are eliminated. Three Mach number are investigated: 0.2, 0.678, and

/ 0.0G CL V = Figure 3 presents the comparison of solutions of the lift and moment by ZEUS and ZAERO on the rigid AGARD wing configuration. The gust velocity observed in the most forefront point is plotted in Figure 3(a) as well. The pitching moment is with respect to the leading edge point at the root. As we can see that, the solutions by ZEUS are somehow greater than those by ZAERO. The difference may come from the effects of the airfoil thickness. But in general speaking both solutions are in good agreement. Note that the ZAERO solutions are obtained from frequency domain solutions then converted into time domain.

ZEUS is also an aeroelastic analysis software system. The transferring of displacements and aerodynamic forces between the structural and aerodynamic grids in ZEUS is accomplished by a spline approach. This spline approach consists of four spline methods that jointly assemble a spline matrix. These four spline methods include: 1) thin plate spline; 2) infinite plate spine; 3) beam spline; and 4) rigid body attachment methods. The spline matrix can transfer the x, y, and z displacements and slopes from the structural grids to all aerodynamic grids as well as the forces at the structural grids to the aerodynamic grids. For the transient response analysis, the Euler solver is coupled with a state-space equation that involves the generalized mass, damping, and stiffness of the structures. At each time step, the state-space solution is first solved, then the solution of the generalized coordinates are applied to the boundary condition of the Euler equations to compute the aerodynamic forces for the next time step, providing a closely coupled aeroelastic simulation for nonlinear aeroelastic analysis.

The response of the flexible AGARD wing under the same one-minus-cosine gust previously used for rigid configuration is analyzed. The first five elastic mode shapes’ projection on the aerodynamic mean surface are plotted in Figure 4. ZEUS’s infinite plate spine method has been used to generate the spline matrix. The wing tip deflection time history solution by ZEUS is compared to that by ZAERO in Figure 5. A good correlation between the two solutions can be clearly seen from the plots. Again, the amplitude of the tip deflection by ZEUS is slightly higher than that by ZAERO.

B. Application of the Gust ROM

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0.9

he AGARD 445.6 wing configuration under a sharp edge gust with the strength

0. In all three Mach number cases, the air density is chosen to be 2.43e-9 slin/in3, and the free stream speed/flight speed is 2687.0, 9109.0, and 11680.8 in/s, respectively.

1. Mach 0.2 Case

T 3/ 1.G CW V e−= is firstly prise more tha

per e and the fis

arily chosen locations, and the Cp solutions by direct ZEUS ana

analyzed using ZEUS. ZEUS RETINAS module determines that the first 13 POD modes com n 99.9 cent of the total energy. The time histories of the gust velocity at the leading edg t 13 POD

coefficients are presented in Figure 6. Apparently the first two are the most dominant ones. Figure 7 shows the contour plots of the first 13 POD modes. Note that the z-coordinates have been exaggerated in the plots in order to differentiate the upper and lower surfaces. Somehow in similar to the structural free vibration modes, the complex level becomes higher for the higher order POD modes.

Using the convolution law and POD modes, we can predict the Cp solutions for any other gust profile. Figure 8 shows the predictions by the gust ROM on several arbitr

lysis is presented as well for comparison. The gust profile to which the wing is subject is the one-minus-cosine gust with the strength 3/ 1.G CW V e−= , and / 0.06G CL V = . A good agreement between the direct and by ROM is clearly seen from the figure. Figure 9 shows again good comparison for the one-minus-cosine gust with a narrower length, / 0.01G CL V = .

The gust ROM approach also does a very good job for a random gust profile, whose results are given in Figure 10. The random gust strength has exceeded 0.02, which is much larger than the sharp edge gust strength at /W V

way fro

G C31.e− . Probably it is because of this larger gust strength of the random gust profile that the deviation of the Cp

predictions by the gust ROM a m the direct solution at some locations, e.g., Panel # 1000 and 2200, begins to up. It’s stipulated at larger gust strength, the flow nonlinearity may have kicked in. Another comparison is

made for an OMCOS gust with / 0.02G CW V = , and / 0.06G CL Vshow

= in Figure 11. Still a very good correlation is observed.

2. Mach 0.678 Case

he similar procedure is applied for the case when Mach number is 0.678. Figure 12 presents the time histories d the first two POD coefficients. At this Mach number, POD analysis determines the first two

PO

0.90 Case this section we move to a higher Mach number in the transonic range, Mach = 0.90. Again the POD analysis

irst two POD modes of the solution under the sharp edge gust are sufficient. The evolution pat

arks We have successfully demonstrated t under the sharp edge gust can be used to

predict the aerodynamics under the convolution method. The adoption of the

One drawback of the convolution method involves the integration from the zero moment, hence for longer period of time of interests, it might become less efficient towards the end. In light of this, future work may investigate

Tof the gust velocity an

D modes, rather than 13 in the case of Mach=0.2, has already occupied more than 99.9 percent of the total energy. Therefore, we only use the first two POD modes and coefficients to predict Cp solutions. The Cp prediction by the gust ROM for a couple of various gust profiles and their comparison with the direct solutions by ZEUS are given in Figure 13 - Figure 15. Obviously, the present gust ROM approach works just fine for this medium Mach number.

3. Mach

Indetermines that the f

tern of the first two POD coefficients is much like those at Mach number of 0.2 and 0.648 cases as shown in Figure 16. Figure 17- Figure 19 presents the comparison between the direct solution and by the gust ROM under two one-minus-cosine and one random gust profiles. Once more, the gust ROM has done a good job. However, notably at some locations, the deviation of the Cp solution by the ROM away from the direct solution becomes more evident. A closer look at the results indicates that Cp solutions at those locations are actually at smaller amplitudes than other locations. It is believed that such deviations at those lesser locations are indeed due to the POD method because POD methods pick up the solutions with higher energy thus those locations with higher amplitudes of Cp have higher chances to be represented in the first couple of POD modes. Nevertheless, the predictions of Cp at all the locations are agreeably in phase with the solutions by direct.

V. Concluding Remhat the aerodynamic solutions

any other traveling gust excitations with Proper Orthogonal Decomposition method makes such a reduced order model even more compact. Thus

expedient solutions can be obtained with drastically reduced time compared to direct CFD analysis.

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onses under a rep

alternative reduced order modeling methods, e.g., the auto regressive moving average (ARMA) [16] model or the neural network (NNet) model. ARMA or NNet model would require the aerodynamic resp

resentative gust excitation other than the sharp edge gust. One of such representative gust profiles could be the Gaussian filtered white noise signal as used by [5].

(a) (b)

Figure 2. (a) AGARD wing surface mesh; (b) Aerodynamic grid

TIME

Nor

mal

Forc

e

wG/V

C

0 0.02 0.04 0.06 0.08 0.10

0.05

0.1

0.15

0.2

0

0.0004

0.0008

0.0012

ZAERO (Mach=0.2)ZEUS (Mach=0.2)

Gust

ZAERO (Mach=0.678)ZEUS (Mach=0.678)

TIME

Pitc

hing

Mom

ent

0 0.02 0.04 0.06 0.08 0.1

-3

-2

-1

0

ZAERO (Mach=0.2)ZEUS (Mach=0.2)ZAERO (Mach=0.678)ZEUS (Mach=0.678)

(a) (b)

Figure 3. Comparison of solutions between ZEUS and ZAERO on the rigid AGARD wing: (a) normal force; (b) pitching moment

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Figure 4. Mode shape projections on the aerodynamic mean surface mesh by ZEUS spline module for the AGARD 445.6 wing

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TIME

Tip

Def

lect

ion

0 0.2 0.4 0.6 0.8 1 1.2-0.002

-0.001

0

0.001

0.002

ZAEROZEUS

(a)

TIME

Tip

Def

lect

ion

0 0.2 0.4 0.6 0.8 1-0.02

-0.01

0

0.01

0.02

ZAEROZEUS

(b)

Figure 5. Comparison of tip deflection solutions by ZAERO and ZEUS

for the flexible AGARD 445.6 wing subject to an one-minus-cosine gust: (a) Mach = 0.2; (b) Mach=0.678

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0 0.02 0.04 0.06 0.08 0.1 0.120

0.2

0.4

0.6

0.8

1

1.2x 10

-3

t(s)

wG

0 0.02 0.04 0.06 0.08 0.1 0.12-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

t(s)

PO

D C

oef.

Coef 1Coef 2Coef 3Coef 4Coef 5Coef 6Coef 7Coef 8Coef 9Coef 10Coef 11Coef 12Coef 13

Figure 6. Time histories of the gust and POD coefficients of Cp solutions for the AGARD 445.6 wing

(Mach=0.2, AOA=0, Sharp Edge Gust)

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Figure 7. POD modes for the AGARD 445.6 wing under the sharp edge gust

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0 0.02 0.04 0.06 0.08 0.10

2

4

6

8

x 10-4

t(s)

wG

0 0.02 0.04 0.06 0.08 0.1-6

-5

-4

-3

-2

-1

0x 10

-3

t(s)

Cp a

t #1

DirectROM

0 0.02 0.04 0.06 0.08 0.1-15

-10

-5

0x 10

-4

t(s)

Cp a

t #50

0

DirectROM

0 0.02 0.04 0.06 0.08 0.1

-5

-4

-3

-2

-1

0x 10

-4

t(s)

Cp a

t #10

00DirectROM

0 0.02 0.04 0.06 0.08 0.10

1

2

3

4

5

6x 10

-3

t(s)

Cp a

t #12

01

DirectROM

0 0.02 0.04 0.06 0.08 0.10

1

2

3

4

5

x 10-4

t(s)

Cp a

t #22

00

DirectROM

Figure 8. Comparison of Cp solutions between by direct and by ROM for the AGARD 445.6 wing

(Mach=0.2, AOA=0, OMCOS Gust with =0.001 and =0.06) /G CW V /G CL V

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0 0.02 0.04 0.06 0.08 0.10

2

4

6

8

x 10-4

t(s)

wG

0 0.02 0.04 0.06 0.08 0.1

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0x 10

-3

t(s)

Cp a

t #1

DirectROM

0 0.02 0.04 0.06 0.08 0.1

-8

-7

-6

-5

-4

-3

-2

-1

0x 10

-4

t(s)

Cp a

t #50

0

DirectROM

0 0.02 0.04 0.06 0.08 0.1

-2.5

-2

-1.5

-1

-0.5

0x 10

-4

t(s)

Cp a

t #10

00DirectROM

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2

2.5

3

3.5

4

x 10-3

t(s)

Cp a

t #12

01

DirectROM

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2

2.5

x 10-4

t(s)

Cp a

t #22

00

DirectROM

Figure 9. Comparison of Cp solutions between by direct and by ROM for the AGARD 445.6 wing

(Mach=0.2, AOA=0, OMCOS Gust with =0.001 and =0.01) /G CW V /G CL V

American Institute of Aeronautics and Astronautics

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0 0.02 0.04 0.06 0.08 0.1

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

t(s)

wG

0 0.02 0.04 0.06 0.08 0.1-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

t(s)

Cp a

t #1

DirectROM

0 0.02 0.04 0.06 0.08 0.1

-8

-6

-4

-2

0

2

4

6

8

x 10-3

t(s)

Cp a

t #50

0

DirectROM

0 0.02 0.04 0.06 0.08 0.1

-2

-1

0

1

2

x 10-3

t(s)

Cp a

t #10

00DirectROM

0 0.02 0.04 0.06 0.08 0.1

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

t(s)

Cp a

t #12

01

DirectROM

0 0.02 0.04 0.06 0.08 0.1

-2

-1

0

1

2

x 10-3

t(s)

Cp a

t #22

00

DirectROM

Figure 10. Comparison of Cp solutions between by direct and by ROM for the AGARD 445.6 wing

(Mach=0.2, AOA=0, Random Gust)

American Institute of Aeronautics and Astronautics

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0 0.02 0.04 0.06 0.08 0.10

0.005

0.01

0.015

0.02

t(s)

wG

0 0.02 0.04 0.06 0.08 0.1-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

t(s)

Cp a

t #1

DirectROM

0 0.02 0.04 0.06 0.08 0.1

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

t(s)

Cp a

t #50

0

DirectROM

0 0.02 0.04 0.06 0.08 0.1

-10

-8

-6

-4

-2

0x 10

-3

t(s)

Cp a

t #10

00DirectROM

0 0.02 0.04 0.06 0.08 0.10

0.02

0.04

0.06

0.08

0.1

0.12

t(s)

Cp a

t #12

01

DirectROM

0 0.02 0.04 0.06 0.08 0.10

2

4

6

8

10

x 10-3

t(s)

Cp a

t #22

00

DirectROM

Figure 11. Comparison of Cp solutions between by direct and by ROM for the AGARD 445.6 wing

(Mach=0.2, AOA=0, OMCOS Gust with =0.02) /G CW V

American Institute of Aeronautics and Astronautics

17

0 0.02 0.04 0.06 0.08 0.1 0.120

0.2

0.4

0.6

0.8

1

1.2x 10

-3

t(s)

wG

0 0.02 0.04 0.06 0.08 0.1 0.12-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

t(s)

PO

D C

oef.

Coef 1Coef 2

Figure 12. Time histories of the gust and POD coefficients of Cp solutions for the AGARD 445.6 wing

(Mach=0.648, AOA=0, Sharp Edge Gust)

American Institute of Aeronautics and Astronautics

18

0 0.02 0.04 0.06 0.08 0.10

2

4

6

8

x 10-4

t(s)

wG

0 0.02 0.04 0.06 0.08 0.1-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0x 10

-3

t(s)

Cp a

t #1

DirectROM

0 0.02 0.04 0.06 0.08 0.1

-15

-10

-5

0x 10

-4

t(s)

Cp a

t #50

0

DirectROM

0 0.02 0.04 0.06 0.08 0.1

-5

-4

-3

-2

-1

0x 10

-4

t(s)

Cp a

t #10

00DirectROM

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2

2.5

3

3.5

4x 10

-3

t(s)

Cp a

t #12

01

DirectROM

0 0.02 0.04 0.06 0.08 0.10

1

2

3

4

5

x 10-4

t(s)

Cp a

t #22

00

DirectROM

Figure 13. Comparison of Cp solutions between by direct and by ROM for the AGARD 445.6 wing

(Mach=0.648, AOA=0, OMCOS Gust with =0.001 and =0.06) /G CW V /G CL V

American Institute of Aeronautics and Astronautics

19

0 0.02 0.04 0.06 0.08 0.10

2

4

6

8

x 10-4

t(s)

wG

0 0.02 0.04 0.06 0.08 0.1-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0x 10

-3

t(s)

Cp a

t #1

DirectROM

0 0.02 0.04 0.06 0.08 0.1-15

-10

-5

0x 10

-4

t(s)

Cp a

t #50

0

DirectROM

0 0.02 0.04 0.06 0.08 0.1

-4

-3

-2

-1

0x 10

-4

t(s)

Cp a

t #10

00DirectROM

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2

2.5

3

3.5x 10

-3

t(s)

Cp a

t #12

01

DirectROM

0 0.02 0.04 0.06 0.08 0.10

1

2

3

4

x 10-4

t(s)

Cp a

t #22

00

DirectROM

Figure 14. Comparison of Cp solutions between by direct and by ROM for the AGARD 445.6 wing

(Mach=0.648, AOA=0, OMCOS Gust with =0.001 and =0.01) /G CW V /G CL V

American Institute of Aeronautics and Astronautics

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0 0.1 0.2 0.3-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

t(s)

wG

0 0.1 0.2 0.3-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

t(s)

Cp a

t #1

DirectROM

0 0.1 0.2 0.3-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

t(s)

Cp a

t #50

0

DirectROM

0 0.1 0.2 0.3-0.05

0

0.05

t(s)

Cp a

t #10

00DirectROM

0 0.1 0.2 0.3-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

t(s)

Cp a

t #12

01

DirectROM

0 0.1 0.2 0.3-0.1

-0.05

0

0.05

0.1

t(s)

Cp a

t #22

00

DirectROM

Figure 15. Comparison of Cp solutions between by direct and by ROM for the AGARD 445.6 wing

(Mach=0.648, AOA=0, Random Gust)

American Institute of Aeronautics and Astronautics

21

0 0.02 0.04 0.06 0.08 0.1 0.120

0.2

0.4

0.6

0.8

1

1.2x 10

-3

t(s)

wG

0 0.02 0.04 0.06 0.08 0.1 0.12-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

t(s)

PO

D C

oef.

Coef 1Coef 2

Figure 16. Time histories of the gust and POD coefficients of Cp solutions for the AGARD 445.6 wing

(Mach=0.90, AOA=0, Sharp Edge Gust)

American Institute of Aeronautics and Astronautics

22

0 0.02 0.04 0.06 0.08 0.10

2

4

6

8

x 10-4

t(s)

wG

0 0.02 0.04 0.06 0.08 0.1

-3

-2.5

-2

-1.5

-1

-0.5

0x 10

-3

t(s)

Cp a

t #1

DirectROM

0 0.02 0.04 0.06 0.08 0.1

-2

-1.5

-1

-0.5

0x 10

-3

t(s)

Cp a

t #50

0

DirectROM

0 0.02 0.04 0.06 0.08 0.1

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0x 10

-4

t(s)

Cp a

t #10

00DirectROM

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2

2.5

3

x 10-3

t(s)

Cp a

t #12

01

DirectROM

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2

2.5

3

3.5

x 10-4

t(s)

Cp a

t #22

00

DirectROM

Figure 17. Comparison of Cp solutions between by direct and by ROM for the AGARD 445.6 wing

(Mach=0.90, AOA=0, OMCOS Gust with =0.001 and =0.06) /G CW V /G CL V

American Institute of Aeronautics and Astronautics

23

0 0.02 0.04 0.06 0.08 0.10

2

4

6

8

x 10-4

t(s)

wG

0 0.02 0.04 0.06 0.08 0.1

-2.5

-2

-1.5

-1

-0.5

0x 10

-3

t(s)

Cp a

t #1

DirectROM

0 0.02 0.04 0.06 0.08 0.1

-15

-10

-5

0x 10

-4

t(s)

Cp a

t #50

0

DirectROM

0 0.02 0.04 0.06 0.08 0.1

-4

-3

-2

-1

0

x 10-4

t(s)

Cp a

t #10

00DirectROM

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2

2.5

x 10-3

t(s)

Cp a

t #12

01

DirectROM

0 0.02 0.04 0.06 0.08 0.1

0

1

2

3

4

x 10-4

t(s)

Cp a

t #22

00

DirectROM

Figure 18. Comparison of Cp solutions between by direct and by ROM for the AGARD 445.6 wing

(Mach=0.90, AOA=0, OMCOS Gust with =0.001 and =0.01) /G CW V /G CL V

American Institute of Aeronautics and Astronautics

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0 0.1 0.2 0.3-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

t(s)

wG

0 0.1 0.2 0.3-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

t(s)

Cp a

t #1

DirectROM

0 0.1 0.2 0.3-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

t(s)

Cp a

t #50

0

DirectROM

0 0.1 0.2 0.3-0.05

0

0.05

t(s)

Cp a

t #10

00DirectROM

0 0.1 0.2 0.3-0.5

0

0.5

t(s)

Cp a

t #12

01

DirectROM

0 0.1 0.2 0.3-0.1

-0.05

0

0.05

0.1

t(s)

Cp a

t #22

00

DirectROM

Figure 19. Comparison of Cp solutions between by direct and by ROM for the AGARD 445.6 wing

(Mach=0.90, AOA=0, Random Gust)

American Institute of Aeronautics and Astronautics

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American Institute of Aeronautics and Astronautics

26

References [1] Code of Federal Regulations, "Sec. 25.341: Gust and Turbulence Loads". 2009. [2] Chen, P.C. "Nonhomogeneous State-Space Appraoch for Discrete Gust Analysis of Open-Loop/Closed-Loop

Aeroelastic Systems". in 43rd AIAA/ASME/ASCE/AHS/ASC Structure, Structural Dynamics, and Materialx Conference. 2002. Colorado, Denver.AIAA 2002-1715.

[3] Karpel, M., Moulin, B., and Chen, P. C., "Dynamic Response of Aeroservoelastic Systems to Gust Excitation". Journal of Aircraft, 2005. 42(5): p. 1264-1272.

[4] Zaide, A., and Raveh, D., "Numerical Simulation and Reduced-Order Modeling of Airfoil Gust Response". AIAA Journal, 2006. 44(8): p. 1826-1834.

[5] Raveh, D. "CFD-Based Models of Aerodynamic Gust Response". in 47th AIAA/ASME/AHS/ASC Structures, Structural Dynamics, and Materials Conference. 2006. Newport, Rhode Island.AIAA 2006-2022.

[6] Raveh, D. "Gust Response Analysis of Free Elastic Aircraft in the Transoinc Flight Regime". in 51st AIAA/ASME/AHS/ASC Structures, Structural Dynamics, and Materials Conference. 2010. Orlando, Florida.AIAA 2010-3050.

[7] Chen, P.C., Zhang, Z., Sengupta, A., and Liu, D., "Overset Euler/Boundary-Layer Solver with Panel-Based Aerodynamic Modeling for Aeroelastic Applications". Journal of Aircraft, 2009. 46(6): p. 2054-2068.

[8] Baldelli, D.H., Chen, P. C., and Panza, J., "Unified Aeroelastic and Flight Dynamics Formulation Via Rational Function Approximations". Journal of Aircraft, 2006. 43(3): p. 763-772.

[9] Raveh, D., "CFD-Based Gust Response Analysis of Free Elastic Aircraft". ASD Journal, 2010. 2(1): p. 23-34. [10] Lucia, D.J., Beran, P. S., and Silva, W. A. "Aeroelastic System Development Using Proper Orthogonal Decomposition

and Volterra Theory". in 44th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Conference. 2003. Norfolk, VA.AIAA 2003-1922.

[11] Lieu, T., and Farhat, C., "Adaptation of Aeroelastic Reduced-Order Models and Application to an F-16 Configuration". AIAA Journal, 2007. 45(6).

[12] Chen, P.C., Liu, D. D., Chang, K. T. , et al. "Aerothermodynamic Optimization of Hypersonic Vehicle TPS Design by POD/RSM-Based Approach". in 44th AIAA Aerospace Sciences Meeting and Exhibit. 2006. Reno, Nevada.AIAA 2006-777.

[13] Cai, X., and Ladeinde, F. "A Comparison of Two POD Methods for Airfoil Design Optimization". in 35th AIAA Fluid Dynamics Conference and Exhibit. 2005. Toronto, Ontario, Canada.AIAA 2005-4912.

[14] ZONA, "ZAERO". 2007, ZONA Technology, Inc.: Scottsdale, AZ. [15] Zhang, Z., Liu, F., and Schuster, D. M. "An Efficient Euler Method on Non-Moving Cartesian Grids with Boundary-

Layer Correction for Wing Flutter Simulations". in 44th AIAA Aerospace Sciences Meeting and Exhibit. 2006. Reno, Nevada.AIAA 2006-0884.

[16] Wang, Z., Zhang, Z., Lee, D. H., Chen, P. C., Liu, D. D., and Mignolet, M. P. "Flutter Analysis with Structural Uncertainty by Using CFD-based Aerodynamic ROM". in 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. 2008. Schaumburg, IL.AIAA 2008-2197.