1

Reduced Order Model Based Simulation of Aeroelastic(AE) /Aeroservoelastic (ASE) Dynamics Using ASTE-P Toolset

Patrick Hu1, Kan Ni2, Liping Xue2, Hongwu Zhao3

Advanced Dynamics Inc. Lexington, KY, USA

Marty Brenner4 NASA Dryden Flight Research Center, Edwards, CA 93524

Advanced Dynamics Inc. (ADI) has developed an integrated variable-fidelity toolset, “ASTE-P”, for modeling and simulation of aeroservothermoelasticity-propulsion (ASTE-P) effects of aerospace vehicles ranging from subsonic to hypersonic flights, which enables the accurate integration and tight/loose coupling of the fluid, structural and control field simulation with variable fidelity options available. The ASE module of ASTE-P toolset has comprehensive capability for modeling and simulation of multi-fidelity Aeroelastic (AE) / Aeroservoeastic (ASE) Dynamics of aerospace vehicles. The multi-fidelity ASE dynamic modeling and simulation environments in ASTE-P toolset include: (1) the high-fidelity and full-order AE/ASE dynamics modeling and simulation environment, (2) fast ASE dynamics modeling and simulation environment that is based upon reduced order models (i.e., POD-ROM, Volterra-ROM, etc.). Recently, the ASE module of ASTE-P has been successfully integrated with Matlab/Simulink to enable various AE/ASE dynamic analyses and control system design, including prediction of flutter/LCO and suppression. In this paper, we will present the detailed theory and numerical procedure for (1) how to construct the ROM models in ASTE-P and (2) how to perform various AE/ASE modeling and simulation in ASTE-P running environment with Matlab scripts. Two examples will be presented to illustrate the numerical procedure and demonstrate the capability of the user-friendly platform built upon the integrated ASTE-P and Matlab/Simulink environment.

I. Introduction eroservoelastic (ASE) dynamics of an aircraft involve aerodynamics, structure dynamics and control, therefore is a comprehensive and multidisciplinary research area. The analysis and evaluation of aeroservoelastic dynamics is very

important for performance, control and stability analysis of aircraft. Many of the methods that have been developed over the years for simpler aeroelastic models that use, for example, doublet lattice aerodynamics can be adopted for this purpose. However, these models are based on potential flow theory and cannot capture the nonlinear system dynamics in transonic flight regime. High-fidelity model do exist, but if high-fidelity computational fluid dynamics (CFD) and computational structure dynamics (CSD) approaches are used, the large degree-of-freedom, nonlinear fluid and structural system may take days to weeks to finish the computation and, thus are cost prohibitive. Fortunately, reduced order model (ROM) that computes full system dynamics in high-fidelity provides an alternative approach and is extremely useful for such purpose. Dowell and Hall1 presented a comprehensive review of reduced order models, and in recent years Advanced Dynamics Inc. has developed such models and solvers in its commercial software ASTE-P 2,3. Recently, by support of NASA Phase III project, Advanced Dynamics Inc. has integrated ASTE-P with Matlab/Simulink, enabling the capability for performing various AE/ASE modeling and simulation of aerospace vehicles under ASTE-P running environment with Matlab/Simulink. This enabled capability provides a convenient solution for Matlab/Simulink users to conduct AE/ASE modeling and simulation and control system design using the powerful functionality of Matlab/Simulink.

II. Construction of ROM Models

A. POD/ROM The use of proper orthogonal decomposition (POD) to construct reduced-order models (ROM) for the highest fidelity

1 President and Principal Scientist, Senior Member AIAA; 2 Principal Scientist, Senior Member AIAA; 3 Senior Scientist, Member AIAA; 4Aerospace Engineer, Senior AIAA Member

A

49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition4 - 7 January 2011, Orlando, Florida

AIAA 2011-1227

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

2

demonstrated by Thomas, Dowell and Hall1,5-6,13 and more recently by Lucia and Beran4,7-8 , Slater and Beran9, and Lieu, Farhat and Lesoinne10. These codes have been used to compute transonic aeroelastic response in order to determine flutter boundaries and limit cycle oscillation (LCO). For example, Mortara, Slater and Beran9 have computed LCO for skin panels using such methods and Thomas, Dowell and Hall1,5-6,11 have recently constructed the nonlinear reduced-order aerodynamic models for an oscillating airfoil in two-dimensional flow, which would be needed for an LCO analysis. Most recently, Hu, Bodson and Brenner12 used this method for near-real and real time simulation of aeroservoelastic dynamics of aircraft, Chen, Li and Hu13 utilized POD/ROM for design of active control law for aeroelastic system. Yet to be demonstrated is the use of such POD methods as sufficiently efficient computational model for aeroelastic flutter/LCO modeling of complex configuration and aeroservoelastic modeling for flutter/LCO suppression of aircraft.

A.1 Construction of POD-ROM For a full-coupled nonlinear aeroelastic system of a flexible aircraft, the fluid flow over the aircraft can be described as

Euler or Navier-stokes equation in a finite volume as bellow:

,, , 0

tA u w F w u v (1)

where w is the conservative flow variables, F is the flux, A is a fluid cell volume u is the structure general displacement, and v is the derivative of the structure general displacement.

Solving Eq. (1) at a nominated condition ( 0 0 0, ,w u v ), we can obtain the following steady state solution:

0 0 0, , 0F w u v (2)

Supposing that ( , ,w u v ) is a small disturbance around the steady state ( 0 0 0, ,w u v ), we can obtain the following

linearized equation:

0 0A w Hw E C v Gu (3)

0 0 0

0 0 0

0

0 0 0

, ,

, ,

, ,

FH w u v

wF

G w u vuA

E wuF

C w u vv

(4)

where 0A is the fluid cell volume in steady state. In order to simply the notation of the linearized equation, , ,w u v is used to represent the perturbation , ,w u u , respectively.

The structural dynamic equation without damping can be written as follows:

int , ( , )extMv f u v f u w (5)

int0

t0 0 0 0

( , )

( , ) , ,ext ext

ex

f u v K u

f ff u w u w u u w w

u w

(6)

Let0 0,

extfP u w

w,

0 0 0,ext

s

fK K u w

u, the following fluid system equation can be obtained:

T

w Aw B v u

F Cw (7)

where 10A A H 1

0B A E C G C P

3

Combine the structure and fluid system equations, the following full-coupled linearized aeroelastic system equation can be obtained.

1 1 10 0 0

1 10

0 0s

w A H A E C A G w

v M P M K v

u I u

(8)

CFD-based solution of Eqn. (8) is too large for real time simulation of aeroservoelastic dynamics and controller design for a flexible aircraft. Therefore, the full-order system has to be reduced. For illustration purpose, the POD approach is used

to reduce the aeroelastic system. Using one series data ,k k nx x in n-dimension space n n , POD searches a

m -dimension proper orthogonal child space n m to minimize the mapping errors from to :

1 1

min ,m m

k H k k H k H

k k

x x x x I (9)

Eqn. (9) is equivalent to: 2 2

2 21 1

, ,max ,

k km m

H

k k

x xI (10)

Suppose that 1 2 mX x x x is the snapshot matrix, then solving equation (10) is equivalent to solving the equat

ion of 0HXX I .Then the problem is transformed to find the eigenvalue of POD kernel HK XX . For high ord

ers of n nK , it is not easy to solve the problem. Consider that HXX and HX X have the same eigenvalue, we can obtain as follows:

HX X (11)

1/ 2X (12)

where 1 2 m , 1 2 mdiag , 1 2 m .

Truncate to r-order vectors, then the system represented by Eq. (8) is reduced to r-order system:

T Tr r r r r

r r

x A x Bu

y C x (13)

and the reduced r-order aeroelastic system can be obtained as follows:

( )

−−Ψ

Ψ−+Ψ−ΨΨ−

= −−−∞∞

−−−

u

v

w

I

KMCMPMV

GACEAHA

u

v

w r

Sr

Tr

Trr

Trr

002

1 1112

10

10

10

ρ (14)

A.2 Matlab Implementation of Flutter Boundary Prediction based on POD-ROM The flutter analysis is performed by computing the eigenvalues of the state matrix of the coupled aerodynamic/structural system in Eq. 14. If one eigenvalue has a positive real part, the system is not stable. For an aeroelastic system, the stability

changes with the dynamic pressure q∞ which depends on freestream velocity ∞V and density ∞ρ . By changing q∞ and

evaluating the eigenvalues, when the real part of an eigenvalue approaches 0, the flutter velocity is found. In order to find the flutter velocity automatically, the binary search method is applied. The following matlab script shows the procedure. 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

4

2 % Search for flutter velocity 3 [AA1, Reig1] = flutter_pod(vmin); 4 [AA2, Reig2] = flutter_pod(vmax); 5 6 len1 = length(Reig1); 7 len2 = length(Reig2); 8 9 % Eigenvalue 10 Reig2; 11 Reig1; 12 13 % Input the valid velocity 14 for i = 1:len1 15 if Reig1(i) > 0 16 fprintf('Please Input a lower Minmum Velocity Again\n'); 17 return 18 end 19 end 20 21 sum = 0; 22 for i=1:len2 23 if Reig2(i) < 0 24 sum = sum + 1; 25 26 end 27 end 28 29 if(sum == len2) 30 fprintf('Please Input a Higher Maximum Velocity Again\n') 31 return 32 end 33 34 % search_pod for flutter velocity 35 Vf = search_pod(vmin, vmax); 36 fprintf('Flutter Velocity found: Vf = %6.4f \n', Vf) 37 [AA, Reig] = flutter_pod(Vf); 38 Vd = Vf/(b_ref*2.0*3.14159*f_ref*sqrt(rho_ref/Rinf)); 39 fprintf('Dimensionless Flutter Velocity found: Vd = %6.4f \n', Vd) 40 u0 = 0.1 * ones(MA + 2*umode,1); 41 42 tic 43 fprintf('Solving the Differential Equation, Please wait... \n'); 44 % ut generalized displacement and velocity 45 [t, ut] = ode45(@seq, [0, t_end], u0); 46 toc 47 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

This is the main program of the matlab script for flutter analysis using POD ROM based method. Lines 3~32 evaluate the

eigenvalues. When the ∞V is set to vmin and vmax, respectively, judge if the range of the velocity is suitable or not. The variable “vmin” is the lower bound, and “vmax” is the upper bound. Lines 35~36 search the flutter velocity using binary search method. Lines 37~39 calculate the dimensionless flutter velocity -Vd. Lines 40~46 solve the differential equations for the coupled aeroelastic system to obtain the generalized displacement and generalized velocity for each structural mode.

B. Volterra-ROM Based on the first order Volterra kernel, the study of the stability of the coupled aeroelastic systems can be carried out.

Moreover, this methodology can encompass the case of an arbitrary number of degree of freedom and at the same time is conceptually clearer, computationally simpler and can provide more accurate and realistic results as compared to the conventional techniques used in nonlinear aeroelastic systems that are based on perturbation and multiple scale methods. The

5

Volterra's series approach provides a firm basis of nonlinear subcritical aeroelastic response, in the sense that it supplies an explicit relationship between the input pulses and its response.

B.1 Constructing Volterra ROM Since time domain is discretized in CFD simulation, the nonlinear system of CFD is a nonlinear discrete system. And

usually, it is time-invariant. Thus, its response [ ]x n due to an arbitrary input [ ]u n can be expressed as a Volterra series

[ ] [ ] [ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

1 2

1

0 1 2 1 2 2 1 20 0 0

1 10 0m

n n n

k k k

n n

m m m mk k

n h n k u k h n k h n k u k u k

h n k h n k u k u k

= = =

= =

= + − + − −

+ + − −

x h (15)

where 0,1,2,n = is discrete time variable, 0h is the steady state response, and ih is the Volterra kernel with order i . The convergence of Volterra series is determined by the input amplitude and the nonlinearity degree of the system. When

the maximum of the input is out of the threshold that is determined by the system itself, the convergence is insecurei. It is very important that the convergence for Volterra series can be guaranteed. If and only if the infinite series on the right hand side of the equality is convergent, the Volterra series can be used for analyzing real system.

For a linear system or a very weak nonlinear system, the Volterra series can be accurately truncated beyond the first-order:

[ ] [ ] [ ]0 10

n

k

n h n k u k=

= + −x h (16)

This is the convolution expression for linear system, hence, Volterra series can be viewed as the generalization of convolution.

The first-order Volterra kernel is just the impulse response of the CFD system which can be obtained by recording the

output signals of the CFD excited with an impulse signal. Alternatively, [ ]1h n can be converted from the step response,

which is more robust than impulse exciting. This formula is just that a linear system response to any arbitrary input can be

obtained from a linear combination of impulse responses. So, [ ]1h n is one kind of formula of a linear system. But usually,

the state-space form is preferred. This is done by the eigenvalue realization algorithm (ERA) method from the Hankle matrix

assemble with [ ]1h n .

The identification of linearized and nonlinear Volterra kernels is an essential step in the development of ROMs based on Volterra theory. These functional kernels can be transformed into linearized and nonlinear (bilinear) state-space systems that can be easily implemented into aeroelastic analysis. The eigenvalue realization algorithm (ERA) was successfully used to generate a linear, state-space ROM for an aeroelastic application by Silva and Raveh14.

ERA method was applied to identify a discrete, linear, time-invariant state-space realization of the form. [ ] [ ] [ ]

[ ] [ ]

1X n AX n Bu n

Y n CX n

+ = +

= (17)

The systems realization procedure takes measurement data [ ]Y n from the free response of the system and produces a

minimal state-space model A , B and C in a way to make Y accurately reproduced. The free pulse response of linear, time-invariant, discrete systems is given by a function known as the Markov parameter,

[ ] 1nmY n CA B−= (18)

At the same time, according to the superposition principle, a system response to any arbitrary input can be obtained from a linear combination of impulse responses. The generalized Hankel matrix of impulse responses is related to the Markov parameter by the superposition principle. The Hankel matrix is formed by windowing the impulse response data. A total of K data points are provided at discrete time steps 1n K= , and the α β× matrix Hαβ is formed as follows

[ ] [ ] [ ][ ] [ ] [ ]

[ ] [ ] [ ]

1

1 1

1 1 1 1 1

1 1 1 1 1

m m m

m m mn

m m m

Y n Y n Y n

Y n Y n Y nH

Y n Y n Y n

αβ

β

β

α α α β

−

+ + −

+ + + + + −=

− + − + + − + + −

(19)

6

where α is the total size of the data window, and β is the number of time steps used to shift the data window. The choice

of α and β is arbitrary as long as 1 1n Kα β− + + − ≤ . For a MIMO system, the dimension of the Markov parameter mY

is M L× , where M is the number of outputs and L is the number of inputs. The ERA method eliminates redundant data by using Singular Value Decomposition (SVD) on 0Hαβ ,

0 TH UDVαβ = (20)

By truncating the elements of U , D and V associated with very small singular values of 0Hαβ , the state dimensionality can

be decreased. Thus the number of states is reduced to a minimal number q . Algebra is used to recast in terms of the time

shifted Hankel matrix 1Hαβ , and the elementsU , D and V . The state-space realization flows from this manipulation, and is

as follows:

1/2 0 1/ 2

1/2

1/2

T

TL

TM

A D U H VD

B D V E

C E UD

αβ− −

−

−

=

=

=

(21)

and TME are defined below:

[ ] ( )

[ ] ( )

0 0 , dimesion is

0 0 , dimesion is

TM M M M

TL L L L

E I M M

E I L L

α

β

= ×

= × (22)

where 0M and 0L are the null matrices of order M and L respectively, and MI and LI are the identity matrices of order

M and L .

B.2 Matlab Implementation of Flutter Boundary Prediction based on Volterra-ROM Unlike the POD-ROM based method, there is no reliable criterion found for judging if the generalized

displacement/velocity is convergent or not, when performing the flutter analysis using Volttera ROM method. In the current implementation, we change the free-stream velocity and observe the variation trend of the generalized displacement/generalized velocity for each structural mode, then tune the value of the free stream velocity, until the flutter velocity is found. The following matlab script shows the procedure. 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2 % Load the data file A, B, C, D matrix for construct control system 3 A = load(path_A); 4 B = load(path_B); 5 C = load(path_C); 6 D = load(path_D); 7 8 % Read GM.txt, 9 % Obtain the Generalized Mass Matrix, M 10 % Obtain the Generalized Stiffness Matrix, K 11 % Obtain the Generalized Damping Matrix, Z 12 [M, K, Z] = ReadGM(path_GMKD); 13 14 nmode = nemode; % elastic mode num 15 Rinf = AirDensity; % reference density, 1.25 16 dt = dtao/V_inf; 17 nstep = floor(t_end/dt); % number of steps 18 [MA, NA] = size(A); 19 bsize = size(B); 20 csize = size(C); 21 s = bsize(1); % Dimension of state of the ROM 22 nin = bsize(2); 23 q = 1.0/2.0 * Rinf * V_inf * V_inf; 24 I = eye(nin); 25 % As matrix

7

26 format('short', 'e'); 27 As = zeros(2*nin, 2*nin); 28 As(1:nin, nin+1:2*nin) = I; 29 As(nin+1:2*nin, 1:nin) = -inv(M)*K; 30 As(nin+1:2*nin, nin+1:2*nin) = -inv(M)*Z; 31 32 % Bs matrix 33 Bs = zeros(2*nin,nin); 34 Bs(nin+1:2*nin, 1:nin) = inv(M); 35 36 % Cs matrix 37 Cs = zeros(nin, 2*nin); 38 Cs(1:nin, 1:nin) = I; 39 40 % Compute the discrete forms of the two matrices As and Bs 41 I1 = eye(2*nin); 42 As1 = I1; 43 j = 1; 44 for i = 1:100 45 j = j*i; 46 As1 = As1 + (As*dt)^i/j; 47 end 48 Bs1 = inv(As)*(As1 - I1)*Bs; 49 50 % simulation 51 xs = zeros(2*nin,1); 52 xs(nin+1, 1) = 0.01; 53 xs(nin+2, 1) = 0.01; 54 xs(nin+3, 1) = 0.01; 55 xs(nin+4, 1) = 0.01; 56 57 Fa = zeros(nin, 1); 58 xa = zeros(s, 1); 59 60 for j = 1:nstep 61 output(j, 1) = (j-1)*dt; 62 output(j, 2:2*nin+1) = xs(1:2*nin, 1); 63 Fa = (C*xa+D*Cs*xs)*q; 64 xa = A*xa + B*Cs*xs; % xa[n+1] 65 xs = q*Bs1*C*xa + (As1+q*Bs1*D*Cs)*xs; % xs[n+1] 66 end 67 68 V = V_inf/(b_ref*2.0*3.14159*f_ref*sqrt(rho_ref/Rinf)); 69 fprintf('Dimensionless Velocity: V = %6.4f \n', V) 70 71 AA = zeros(2*nin+s, 2*nin+s); 72 AA(1:2*nin, 1:2*nin) = As1+q*Bs1*D*Cs; 73 AA(1:2*nin, 2*nin+1:2*nin+s) = q*Bs1*C; 74 AA(2*nin+1:2*nin+s, 1:2*nin) = B*Cs; 75 AA(2*nin+1:2*nin+s, 2*nin+1:2*nin+s) = A; 76 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

This is the main program of the matlab script for flutter analysis using Volttera ROM based method. Lines 3~6 load the coefficient matrices of aerodynamic system A, B, C and D. These matrices are created by running the module, Volttera-ROM in ASTE-P. Line 12 loads the three matrices, generalized mass matrix, M, generalized stiffness matrix, K and the generalized damping matrix, Z. Lines 14~23 set the flight condition for flutter analysis, and these parameters will be loaded from the case file defined in ASTE-P. Lines 24~38, set the coefficient matrix for the CSD system. Lines 41~48 discretize the CSD system by modifying its coefficient matrices. Lines 51~58 initialize all the state variables. Lines 60~66 start the aeroelastic

8

simulation. Lines 68~69 display the current free stream velocity. Lines 71~75 calculate the system matrix of the open-loop system of the aircraft.

III. Results A. AGARD 445.6 Wing The aerolastic modeling of AGARD 445.6 wing15 was conducted by using the computational framework presented in this paper. The wing is modeled with plate elements as a single layer orthotropic material. The model consisted of 231 nodes and 200 elements. The thickness distribution was governed by the airfoil shape. The material properties used in this paper were tuned to produce the same natural frequencies of the 1st and 2nd modes as the experiment15. In present study, E1 = 3.1511 GPa, E2 = 0.4162 GPa, = 0.31, G = 0.4392 GPa, and E1 and E2 are the moduli of elasticity in the longitudinal and lateral directions, is Poisson's ratio, G is the shear modulus. Table 1 compares the measured and calculated first four natural modes and Figure 1 shows the CFD mesh, and Figure2 shows the calculated structural mode shapes.

Table 1: Comparison of Modal Frequencies for AGARD 445.6 Wing.

Mode1(HZ) Mode2(HZ) Mode3 Mode4 Experiment 9.60 38.10 50.70 98.50 Calculated 9.60 38.20 49.13 92.94

Figure 1. Computational Grid of AGARD 445.6 Wing.

(a) Mode 1, Frequency=9.60Hz (b) Mode 2, Frequency=38.20Hz

(c) Mode 3, Frequency=49.13Hz (d) Mode 4, Frequency=92.94Hz

Figure 2. The First Four Structural Modes of AGARD 445.6 Wing.

The matrices of the state-space form (A,B,C matrixes with POD-ROM method, A,B,C,D matrixes with the Volttera ROM method ) are calculated at different Mach points. At each Mach point the air density is given as the experimental measurement 15. The order number of POD-ROM is 300 in present study. For the Volttera ROM, the sampling scale 110=α , the number of time step for moving the sampling widow 35=β . The Flutter velocity and the flutter speed index for POD-ROM, Volttera-ROM and fully-coupled CFD/CSD are calculated and presented in the Table 2. As is shown in Figure 3, the calculated results agree with the experiment result very well in the subsonic speed, but in the transonic and supersonic speed all computational results have some degrees of differences from the experimental measurements.

9

Table 2: The Flutter Boundary Comparison for AGARD445.6 Wing

Mach Number(Ma) 0.499 0.678 0.901 0.96 1.072 1.141

Density (kg/m3) 0.42791 0.20829 0.0995 0.06341 0.05516 0.07836

Experiment

Flutter velocity(m/s)

172.46 231.37 296.69 309.01 344.73 364.33

Flutter speed index

0.4459 0.4174 0.3700 0.3059 0.3201 0.4031

CFD/CSD

Flutter velocity(m/s)

179.00 243.00 279.00 251.00 498.00 595.00

Flutter speed index

0.46192 0.43749 0.34718 0.24934 0.46141 0.65706

V-ROM

Flutter velocity(m/s)

187.675 249.235 292.2 301.94 274.5 436.3

Flutter speed index

0.4843 0.4487 0.3636 0.2999 0.2543 03633

POD-ROM

Flutter velocity(m/s)

180.50 246.38 286 308.3 506.124 602.4

Flutter speed index

0.46579 0.44357 0.3558 0.3062 0.4689 0.66523

Compared with full-order coupled CFD/CSD simulation, the ROM based methods are much faster (several hours vs. several seconds on a PC with Pentium quad-core) and require less computer memory. Therefore, they are more suitable for aeroelastic/aeroservoelastic analysis and design. Also, since time domain state-space POD- and Volterra-ROMs are used, the nonlinear aeroelastic analysis in transonic flight regimes can be performed. In addition, our study indicates the computational accuracy of Volttera-ROM based method is slightly better than that of POD-ROM based method in flutter analysis.

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.30.2

0.3

0.4

0.5

0.6

0.7

Ma

Flu

tterI

ndex

ExperimentCFDV-ROMPOD-ROM

Figure 3. The Flutter Boundary Comparison for AGARD 445.6 Wing.

B. Efficient AE/ASE Modeling for Advanced Fighter Aluminum (AFA) Aircraft Model Recently we studied Advanced Fighter Aluminum (AFA) Aircraft Model and we applied both full-order, full-coupled CFD/CSD and POD/ROM aeroelastic solvers in ASTE-P 2,3 to conduct the aeroelastic modeling of AFA. Then, we used Matlab scripts to conduct the flutter /LCO analysis and control design for flutter/LCO suppression. The geometric model of AFA is shown in Figure 4. The structure model of AFA was provided as a NASTRAN finite-element half model shown in Figure 5. The half model was duplicated into a full model. At the two ends of the body, there are not enough nodes, which

10

can result in abnormal mode shapes of the CFD surface grid, thus a few more points were manually added. The beam element of the body was duplicated around the body axis in order to keep a better shape of the body CFD grid. The resulting model consists of thousands of FEM nodes of which 525 points are extracted to construct the RBF (Rational Radial Base Function) interpolator for CFD/CSD data transferring. The details of RBF interpolator for CFD/CSD data transfer can be referred to Ref. 3. Figure 6 shows the transferring of the four symmetric mode shapes from CSD points to CFD surface grid and Figure 7 presents the shapes of the four anti-symmetric modes.

The CFD grid of the full AFA model consists 8 blocks with about 1,540,000 grid points as shown in Figure 8. The flow condition was Mach 0.95 with a dynamic pressure of 153,125 Pa, which was used as the nominated steady state condition. The aeroelastic simulation used a steady solution as the initial field to start the unsteady computations with the time step of 0.001s.

Figure 4. Geometric Model of AFA.

Figure 5. FEM Model of a Half AFA.

Three simulations were performed with the symmetric modes, the anti-symmetric modes and the hybrid modes. For all three simulations, the Mach number is 0.95 and mean angle of attack is 0o , while the free stream gas density is fixed at 1.225 kg/m3, which means large dynamic pressure. A snapshot of the flutter for this AFA aircraft is shown in Figure 9. In Figure 9, the complex moving shock waves over the surface of the main wings are clearly shown by the surface pressure contours and the strong counter-rotating vortices over the two main wings are clearly illustrated by the red color streamlines and the vorticity contours on the cutting plane. Figure 10 presents the general displacement and the general aerodynamic forces for the 8 structural modes (4 symmetric modes plus the 4 anti-symmetric modes) of the AFA model aircraft. From Figure 10 we can see that the flutter occurred and the amplitude of the flutter increased exponentially, and finally the aircraft ran into LCO.

What followed we conducted the POD/ROM based aeroelastic and aeroservoelastic modeling for AFA model with only 4 symmetric modes. The state-space form of this simpler model was constructed based upon the POD method. In this study, a 125-order aeroelastic POD/ROM model was constructed first and then was further reduced to 40 order BT/ROM using Balance Truncation approach. Again, in this POD/ROM simulation the AFA aircraft produced flutter at Mach = 0.95, and Figure 11(a) shows that the system went to divergence very quickly, and finally ran into LCO.

Note that the instability of the AFA aircraft is caused primarily by the first two modes. Thus, the simplest approach to stabilizing the system would be to weaken the coupling of the two modes by adding a control flap on the trailing edge of the main wing and letting it move according to the control law that is specifically designated to suppress the flutter/LCO of the aircraft. In this study, an output feedback control law is used to stabilize the aircraft while avoiding the need of observation.

The control law may be represented as 2Kδ ξ= (K is the control gain and 2ξ is the general displacement of

the second structural mode of the AFA aircraft). From the simulation based upon the ROM model of the AFA model, we selected -0.05 as the control gain. Similarly, we may select other control gains according to the performance requirement.

The active flutter control system was activated after the system ran into LCO. The control law restricts the deflection of the control flap so that it does not exceed 10 degrees. This means that if the deflection is larger than 10 degrees, it will hold on 10 degrees. The general displacements of the first 3 structural modes are presented in Figure11 (b). From Figure 11(b), it can be seen that the active control law based upon the POD/ROM model has successfully suppressed the LCO of the AFA aircraft.

11

Figure 6. Transfer Mode Shapes from CSD Points (Blue Points) to CFD Surface Grid. The Green Grids Are the Original Grid, the Red Grids Are Deformed Grids (Scaled for Better Visualization).

Figure 7. Mode Shapes of CFD Surface Grid of Anti-symmetric AFA Model. The Green Grids Are the Original Grid, the Red Grids Are Deformed Grids (Scaled for Better Visualization).

12

Figure 8. CFD Grid of AFA Model.

Figure 9. Snapshot of Flutter of AFA Aircraft.

(a) (b)

Figure 10. Full-order and Full-coupled CFD/CSD Aeroelastic Modeling of AFA Model Aircraft: (a) General Displacement (Left); and (b) General Aerodynamic Forces (right).

Figure 11. Structural Response of the AFA Aircraft of Only 4 Symmetric Modes

without and with Active Control.

IV. Conclusion In present study we have successfully generated the ROM models based upon POD and Volterra series methods

for an AGARD 445.6 wing and an advanced fighter plane by using ASTE-P tool and imported them into Matlab/Simulink to conduct AE/ASE dynamics modeling. By using these high-fidelity computational models in reduced order form, computational solutions are obtained that are not only the most accurate possible with the current state-of-the-art, but are also computationally sufficiently rapid for the AE/ASE modeling and simulation of aircraft. Many of the methods that have been developed over the past decades for simpler aeroelastic models, that use for example doublet lattice aerodynamics, can also be adopted for this purpose, but these simpler models fail in the transonic flow regime. Therefore, in the present work, we have explored new approaches using high fidelity computational models based upon POD-ROM and Volterra-ROM of Euler and Navier-stokes solutions for the fluid dynamics model, and used the available wind tunnel flutter data and other computational results for the AGARD

13

445.6 wing in our assessment of this new approach. Additionally, these validated ROM models to be generated in ASTE-P tool have been successfully integrated with Matlab/Simulink working environment, leading to user-friendly platform for AE/ASE modeling and simulation of aerospace vehicles.

References 1Dowell, E.H. and Hall, K.C., “Modeling of Fluid-Structure Interaction,” Annual Review of Fluid Mechanics, 2001, Vol. 33,

pp. 445-490. 2Hu, P. G., Xue, L., Ni, K., and Brenner, M., “Integrated Variable-Fidelity Tool Set for Modeling and Simulation of Aeroservothermoelasticity-Propulsion (ASTE-P) of Aerospace Vehicles from Subsonic to Hypersonic flight,” AIAA 2009-6141 , AIAA Atmospheric Flight Mechanics Conference, Chicago, Illinois, Aug. 10-13, 2009. 3Hu, P. G., Xue, L., Qu, K., Ni, K., Dittakavi, N., Zhao, H., Kamakoti, R., and Brenner, M.J., “Multi-Fidelity Modeling and Simulation Tool for Aero-Servo-Thermo-Elasticity and Propulsion (ASTE-P) of Aerospace Vehicles,” AIAA 2010-2966, 51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 12 - 15 April 2010, Orlando, Florida.

4Beran, P.S., “A Reduced Order Cyclic Method for Computation of Limit Cycles,” Nonlinear Dynamics, 2005, Vol. 39, No. 1-2, pp. 143-158.

5Hall, K.C., Thomas, J.P. and Dowell, E.H., “Proper Orthogonal Decomposition Technique for Transonic Unsteady Aerodynamic Flows,” AIAA Journal, 2000, Vol. 38, No. 10, pp. 1853-1862.

6Thomas, J.P., Dowell, E.H., and Hall, K.C., “Three-Dimensional Transonic Aeroelasticity Using Proper Orthogonal Decomposition-Based Reduced-Order Models,” Journal of Aircraft, 2003, Vol. 40, No. 3, pp. 544-551.

7Lucia, D.J., and Beran, P.S., “Reduced-Order Model Development Using Proper Orthogonal Decomposition and Volterra Theory,” AIAA Journal, 2004, Vol. 42, No. 6, pp. 1181-1190.

8Lucia, D.J. and Beran, P.S., “Aeroelastic System Development Using Proper Orthogonal Decomposition and Volterra Theory,” Journal of Aircraft, 2005, Vol. 42, No. 2, pp. 509-518.

9Mortara, S.A, Slater, J., and Beran, P.S., “Analysis of Nonlinear Aeroelastic Panel Response Using Proper Orthogonal Decomposition,” Journal of Vibration and Acoustics – Transactions of the ASME, 2004, Vol. 126, No. 3, pp. 416-421.

10Lieu, T., Farhat, C., and Lesoinne, M., “POD-Based Aeroelastic Analysis of a Complete F-16 Configuration: ROM Adaptation and Demonstration,” AIAA Paper 2005-2295.

11Thomas, J.P., Dowell, E.H. and Hall, K.C., “Using Automatic Differentiation to Create a Nonlinear Reduced Order Model of a Computational Fluid Dynamic Solver,” AIAA Paper 2006-7115, AIAA MAO Conference, Portsmouth, VA, September 2006. 12Hu, P.G., Bodson, M. and Brenner, M., “Towards Real-time Simulation of Aeroservoelastic Dynamics for a Flight Vehicle from Subsonic to Hypersonic Regime,” AIAA-2008- 6375, AIAA Atmospheric Flight Mechanics Conference and Exhibit, 18 - 21 August 2008, Honolulu, Hawaii. 13Chen, G., Li, Y, and Hu, P.G., “Design of Active Control Law for Aeroelastic System Based on Proper Orthogonal Decomposition Reduced Order Model,” AIAA 2010-2624, 51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 12 - 15 April 2010, Orlando, Florida.

14Silva WA, Raveh DE. Development of aerodynamic/aeroelastic state-space models from CFD-based pulse responses. AIAA 2001-1213, 42nd AIAA/ASME/ASCE/AH Structures, Structural Dynamics, and Materials Conference, Seattle, WA, 2001.

15 Yates, E. C., “AGARD Standard Aeroelastic Configurations for Dynamic Response I-Wing 445.6,” AGARD Report 765, NATO Group for Aerospace Research and Development, 1988.