American Institute of Aeronautics and Astronautics

1

Application of Multiple Methods for Aeroelastic Uncertainty Analysis

Brian P. Danowsky,* Jeffery R. Chrstos, PhD,† David H. Klyde,‡ Systems Technology, Inc., Hawthorne, CA 90250

Charbel Farhat, PhD§ CMSoft, Inc., Palo Alto, CA 94306

and

Marty Brenner** NASA Dryden Flight Research Center, Edwards, CA, 93560

Flutter is a potentially explosive phenomenon that is the result of the simultaneous interaction of aerodynamic, structural, and inertial forces. The explosive nature of the flutter phenomenon mandates that flutter flight testing be cautious and conservative. It is therefore clear that further investigation of uncertainty analysis methods with respect to the flutter problem is desired and warranted. The analytical prediction of flutter in the transonic regime requires high fidelity simulation models that are computationally expensive. Due to the computational demands, traditional uncertainty analysis is not often applied to flutter prediction, resulting in reduced confidence in the results. The work described herein is aimed at exploring methods to reduce the existing computational time limitations of traditional uncertainty analysis. Specifically, the coupling of Design of Experiments (DOE) and Response Surface Methods (RSM), and the application of robust stability techniques, namely µ-analysis, are applied to an example aeroelastic model. From a high fidelity nonlinear aeroelastic simulation, a linear Reduced Order Model (ROM) is produced that still captures the essential dynamic characteristics. Using ROMs, the DOE/RSM and µ-analysis approaches are compared to traditional Monte Carlo based stochastic simulation. All of these approaches to uncertainty analysis have advantages and drawbacks. The multiple methods and their robustness are compared and evaluated with a validated aeroelastic model of the AGARD 445.6 wing.

I. Introduction The potentially explosive nature of flutter mandates that flutter flight testing be cautious and conservative. The

development of a flutter stability parameter was undertaken four decades ago to give a more reliable technique to predict the onset of flutter. Other methods of predicting flutter such as damping versus velocity are known to have a number of shortcomings, one of which can be the sudden degradation in damping as illustrated in the explosive flutter example shown in Zimmerman and Weissenburger.1 In this classic paper, the flutter stability parameter was developed from the classical two degree-of-freedom (bending/torsion) model of an airfoil, and the technique was shown to be applicable for higher degree-of-freedom analysis. The flutter stability parameter (flutter margin) is calculated using the decay rate and damped frequency.

Traditionally, computational aeroelasticity for loads and flutter prediction combines a linear finite element formulation for the structure with linear aerodynamic methods. At the same time, the prediction of aerodynamic

* Senior Research Engineer, AIAA Member † Principal Research Engineer ‡ Technical Director and Principal Research Engineer, AIAA Associate Fellow § President, AIAA Fellow ** Aerospace Research Engineer, AIAA Member

AIAA Atmospheric Flight Mechanics Conference and Exhibit18 - 21 August 2008, Honolulu, Hawaii

AIAA 2008-6371

Copyright © 2008 by Systems Technology, Inc. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

American Institute of Aeronautics and Astronautics

2

performance and control surface effectiveness accounts for the effects of the structural elastic deformations on the external aerodynamics by means of correction factors applied to the results obtained when the aircraft is assumed to be rigid. Both practices are well-established in aircraft design and give accurate, reliable, and rather inexpensive predictions for static and dynamic effects at subsonic and supersonic speeds. In the transonic flight regime and for rapid maneuvering conditions where nonlinear aerodynamic effects may become non-negligible, the aircraft design, development, and certification processes rely today on expensive structural and aerodynamic experimental models and on extensive flight testing. Adoption of innovative, unconventional designs and aerodynamically unstable configurations in modern aircraft exacerbates the presence and impact of aerodynamic nonlinearities. A possible consequence of the inaccurate prediction of aerodynamic loads involving nonlinear phenomena; such as shocks, vortices and separated flows; flutter, LCO, and other adverse aeroelastic effects remain unveiled until the flight tests. Unfortunately, flight tests are expensive and can be dangerous. Perhaps for these reasons, a leading aeroelastician at Boeing's Phantom Works wrote in 2001 “The results of a finite number of [nonlinear] CFD [Computational Fluid Dynamics] solutions could be used as a replacement for wind tunnel testing, assuming a validated code was available”,2 and “Even at present, existing CFD codes should be able to obtain five flutter solutions in one year.”2 Indeed, state-of-the-art CFD-based nonlinear aeroelastic simulation technologies exemplified by the AERO code3,4 developed at CMSoft have recently become a superior choice over linear computational methods, and a viable complement (and in some cases an alternative) to scaled wind-tunnel testing for many types of aeroelastic analyses such as flutter prediction. In any case, flight testing can also benefit from high-fidelity aeroelastic numerical simulators in many ways; for example, in planning and reducing the number of sorties, anticipating critical points, and expanding flutter envelopes.

Recently, a team of investigators, including an F-16 test pilot and a flutter engineer, reported that flight test measurements recorded when different yet similar F-16 aircraft were used for the same set of flutter test missions produced the same trends but exhibited differences in response magnitude.5 Uncertainties are present in both design-related quantities and operational quantities. The work presented herein will address the first kind of uncertainty. The behavioral uncertainties relate to the characterization of damage and failure and are best treated by refining the mechanistic models used in describing them. The crucial question remains whether the discrepancies between the numerical results and the experimental data can be eliminated by refining the solution of the computational grids leading to an ever larger numerical burden, or whether the differences are within the bounds of uncertainties in the structural and flow models or the scatter in experimental data. The answer to this question will provide key knowledge needed for deciding where to spend future resources for improving prediction capabilities.

Based on previously conducted research using realistic aircraft models, as well as published literature,6,7 the following sources of structural and flow uncertainties have been identified: structural damping, mechanical properties of joints, mass distribution, influence of un-modeled structural components including structural details and flow boundary conditions (i.e., at the engine inlet and outlet), geometry and material properties, roughness of the surface, and flight conditions. Clearly, the significance of the uncertainties associated with these parameters depends in great measure on the quantities of interest in the analysis.

II. Description of the Aeroelastic Reduced Order Model (ROM)

A. Overview In this work, the aeroelastic dynamics of the model being analyzed are assumed to take on the form of a linear

Reduced Order Model (ROM). Any aeroelastic system (i.e., fixed cantilever wing or an entire aircraft) can be realized into a ROM form. The ROM has many advantages including the ability to determine stability parameters such as flutter points in a rapid manner. Rapid solutions were essential to the development of the methods but are not necessary for execution of the methods. More complex, non-linear models can be analyzed using the methods described herein.

The combined CFD/FE code AERO, developed by CMSoft Inc., has the capability of solving very high order models with both structural and aerodynamic nonlinearities. An important fluid regime that has been historically troublesome to model accurately is the transonic regime. Linear aerodynamic methods have had trouble solving problems in this regime due to the highly nonlinear nature of the flow physics. Traditionally, data is extrapolated from linear models in the subsonic and supersonic regime to predict the behavior in the transonic regime. A model that is linearized about a point in the transonic regime can be very useful if it accurately approximates the flow characteristics. The AERO code has the capability to produce a compact linear ROM of the combined fluid and structural system.8 This model is linearized around a stable operating point that can be at any flight regime, including transonic. The innovative technique to determine this ROM involves constructing a Proper Orthogonal

American Institute of Aeronautics and Astronautics

3

Decomposition (POD) basis from frequency response analysis of the full model. This ROM, once constructed, is as accurate as the high fidelity model that it approximates in a linear operating range. Since the flutter problem is, in essence, a linear instability this ROM serves as an excellent means to analyze this problem.

The ROM, as obtained by CMSoft from the combined CFD/FE code known as AERO, is non-dimensional and is supplied in the following form:

200 0s

H B C

N PI

− − −⎡ ⎤⎢ ⎥

= −Ω⎢ ⎥⎢ ⎥⎣ ⎦

(1)

where • H is an f fn n× fluid ROM matrix

• 2Ω is an s sn n× diagonal matrix storing the squares of the structural natural circular frequencies

• P is an s fn n× load transfer matrix

• B and C are f sn n× fluid/structure coupling matrices

• sI is an s sn n× identity matrix

• The non-dimensional portions of the ROM are based on a fixed constant Mach number.

N can be dimensionalized as follows:

2( , ) 00 0s

p pH B C

N p p PI

ρ ρ

ρ

∞ ∞

∞ ∞

∞ ∞ ∞

⎡ ⎤− − −⎢ ⎥

⎢ ⎥⎢ ⎥

= −Ω⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(2)

and it governs the perturbation equations about an equilibrium:

q Nq= (3)

where ( )Tm mq w u u= is the aeroelastic state vector consisting of the fluid ROM states, the structural modal displacements, and structural modal velocities. The blocks of N govern the following system of coupled, dimensional fluid/structure equations:

2

0m m

s m m

p pw Hw Bu Cu

I u u p Pw

ρ ρ∞ ∞

∞ ∞

∞

+ + + =

+ Ω =

(4)

where p∞ and ρ∞ denote the free-stream pressure and density of interest, respectively. The ROM matrix N can be exploited in at least the following ways:

American Institute of Aeronautics and Astronautics

4

• The eigenvalue analysis of N determines the stability of the aeroelastic system for the atmospheric flight conditions defined by p∞ and ρ∞ . This is effectively defining the altitude if a standard atmosphere model is utilized.

• The aeroelastic ROM can then be used for time-domain simulations provided that initial conditions are specified for mu and w.

The matrix blocks of N may be extracted to form a system of coupled, dimensional fluid/structure equations (Eq. (4)). Again, the flight conditions can be specified by choosing appropriately the free-stream pressure and density.

According to the 1976 Standard Atmosphere,9 the pressure and density are both directly related to the altitude, h. By this fact, Eq. (2) can be reduced to be simply a function of one variable: h. The relationship to the altitude has been derived using the standard atmosphere relations (Eq. (5)):

0 0

0 01 2

0

(1 ) (1 )

( ) (1 ) 00 0s

p h p hH B C

N h p h PI

η

α αρ ρ

α +

⎡ ⎤− −− − −⎢ ⎥

⎢ ⎥⎢ ⎥

= − −Ω⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

, (5)

where: 0

3 4 40

6

14.70 [psi]

0.002377 [slug/ft ] ( 1/12 1.146e-7 [slug-ft/in ])

6.87(10 )4.2561

p

ρ

αη

−

=

= × =

==

Based on the constant values used, it was noted that the equations that are a function of altitude can be approximated as much simpler polynomials with respect to the altitude for values between 30,000 ft below sea level and the stratosphere (-30,000 < h < 36,089 ft). These polynomials are given below:

200 1 2 0 1 2

0

(1 ), 11326, -0.0391, -6.9686e-8

p hf f h f h f f f

αρ−

≅ + + = = = (6)

1 2 3

0 0 1 2 3 0 1

2 3

(1 ) , 14.7, -5.3273e-4, 7.9248e-9, -5.4093e-14p h g g h g h g h g g

g g

ηα +− ≅ + + + = =

= = (7)

B. Aeroelastic ROM in Terms of Structural Parameters and Approximate Relationship The ROM parameters are also nonlinear functions of structural parameters. Assume the structural parameters

that vary, or have uncertainty, are collected into the vector β. The components of the ROM non-dimensionalized to atmospheric parameters (Eq. (1)) are functions of these parameters. The matrix B, for instance, can be written as an arbitrary function of the Mach number and the β vector.

( , )B f M β∞= (8)

Just as was done for the atmospheric parameters, this can be approximated as a polynomial in β. A second order polynomial is displayed below.

American Institute of Aeronautics and Astronautics

5

0 1 1 2 2

1,2 1 2 1, 12 2

11 1

... (linear terms)... (interaction terms)

... (squared terms)

n n

n n n n

nn n

B B B B BB B

B B

β β ββ β β β

β β

− −

≅ + + + +

+ + +

+ + +

(9)

Approximating this polynomial is dependant on the data available for B as a function of the parameters β. The polynomial approximation (or response surface equation) for B is determined by a least squares fit to available data. These data are a series of B matrices as functions of different β parameter sets. If β is large it may take many sets of parameters to determine an adequate approximation for B. Design of Experiments (DOE) can, and will, be used to determine the parameter sets to be used that will give the “best” fit to B while limiting the amount of parameter sets to a minimum.

C. Flutter Point for the ROM and Model Definition The parameter h will be used to define the stability. The flutter point for a specific ROM is defined as the

altitude (h) for which one eigenvalue of the ROM matrix N is directly on the imaginary axis and all other eigenvalues are in the left half plane (Eq. (10)).

( )( ) min max[ , ] : max Re : ( , , ) 0 0 ( , )flutth h h h s sI N h M f Mβ β∞ ∞= ∈ − = = = (10)

All other parameters that define the flutter point (i.e., Vflutt, ωflutt, etc.) can be found using the Standard Atmosphere, the Mach number and the value of the destabilizing eigenvalue. If the polynomial approximations are defined, the range of h is defined based on how the parameters in Eqs. (6) and (7) have been derived. These were defined based on a range of altitudes below the stratosphere (h < 36089 ft). The minimum value of the altitude will be allowed to be less than zero as it is anticipated that for low Mach numbers the nominal flutter point for certain aircraft configurations will be below sea level.

For all analysis pertaining to uncertainty, this parameter, hflutt, will be used as the dependent parameter for comparison. This parameter is a function of the structural parameters.

The model (Figure 1) is defined as that which produces the flutter altitude for a given set of structural parameters at a specified Mach number. In the case of using the linear aeroelastic ROM, this is determined using Eq. (10). For a more general case there are many methods that can be used to determine the dependent flutter altitude.

III. General Overview of Uncertainty Analysis Methods Applied to the Flutter Problem Traditional flutter prediction uses a deterministic aeroelastic simulation model with parameters representing the

“as designed” aircraft to calculate “the” structural damping ratio or flutter margin as a function of airspeed to create the predicted flutter boundary. While not incorrect, outside of the simulation environment, nothing is exactly “as designed.” Airfoil skin thickness varies, fastener clamping force varies, etc., all of which add uncertainty to the flutter boundary. Thus, flutter margin becomes a confidence interval, rather than an absolute number.

At its most basic level, the purpose of uncertainty analysis is the estimation of the “degree-of-confidence” of the output of a system or model based on known uncertainties in the inputs and system characteristics or model parameters. Input and parameter uncertainties are propagated through a model and the statistical distribution of the model outputs constitute the uncertainty.

Most uncertainty analysis techniques require making multiple, and often many simulation runs. The number of required runs typically scales by a power law based on the number of inputs and parameters whose uncertainty is to be propagated through the simulation. Analysis quickly becomes time prohibitive for all but the shortest simulation runs. Therefore, a first order goal is to reduce the number of required simulation runs.

It is the goal of this project to significantly reduce the time required to perform the uncertainty analysis using both Design of Experiments\Response Surface Methods and Robust Stability\µ-analysis. Comparison to a traditional stochastic approach (Monte Carlo analysis) will validate the use of these two methods as much less time prohibitive and computational intensive alternatives that still provide a comprehensive analysis with the same or better level of confidence.

( , )N M β∞ flutth

1β 3β2β nβ...

M∞

Figure 1: Model

definition.

American Institute of Aeronautics and Astronautics

6

A. Overview of Monte Carlo Methods Monte Carlo methods describe a class of computational algorithms that use repeated random sampling to

compute results. These methods primarily involve simulating a physical system repeatedly while randomly changing parameters that the system is dependant on for each simulation. Results are collected and can be analyzed statistically to determine means, standard deviations, maxima, minima and other statistical parameters. With several simulation runs, the results of Monte Carlo approach a continuous surface. The optimal number of runs is that which is a minimum number but produces relatively identical statistical results if more runs are made. By this manner, that minimal amount of runs represents the results as if infinite runs were made. Monte Carlo analysis is used when it is impossible or infeasible to compute exact results with a deterministic approach.

Monte Carlo methods provide a reliable means to analyze problems that are concerned with uncertainty. Uncertain parameters of a physical system are randomly sampled over several runs and the results represent system behavior subject to these uncertainties. Sensitivity to the uncertain parameters is realized by the statistical analysis of the Monte Carlo results. The drawback of these methods is the large amount of runs can be computationally burdensome. If the physical system being modeled is complex and takes a relatively large amount of computational time to run this burden is amplified by the large (1E3 - 1E5) amount of times that are required to run in order to obtain meaningful statistical results.

B. Overview of Design of Experiments and Response Surface Methods (DOE/RSM) Any system can be described as an input/output relationship. The outputs of a system are dependant on the

system’s inputs and independent parameters. With highly complex or empirical systems that feature large amounts of uncertainty, the input/output relationship of the system in question may be difficult to determine. Response Surface Methods (RSM) are used as an accurate approximation to characterize a system’s outputs based on variations of the system’s inputs and parameters. The DOE/RSM technique is used to determine an accurate approximate model of a system with minimal sets of input required. The resulting analytical model described by the RSE (usually a polynomial) is, in most cases, in a simpler form than the original model and is thus more efficient while retaining a level of accuracy.

Design of Experiments (DOE)10 and Response Surface Methods (RSM)11 will be used to significantly reduce the number of simulation runs. While Monte Carlo generates random values within the range of input and parameter values, DOE’s purposely select input and parameter values to maximize the information available from the output.

Response Surface Equations (RSE) are used to characterize a system’s outputs (called “targets”) based on variations in its inputs and parameters (called “factors”). The DOE/RSM technique chooses a RSE that is appropriate for characterizing a particular system, and then designs a set of experiments that will yield maximal information for the regression analysis to fit the RSE to the data. This is the fundamental difference between DOE/RSM and Monte Carlo: Monte Carlo makes no assumption about the relationship between the model output and its inputs/parameters, while DOE/RSM does. The down side for DOE/RSM is when the form of the RSE is poorly chosen, it yields poor or misleading information. The up side is that, with the properly chosen RSE, the number of model runs required is cut by an order of magnitude or more.

C. Overview of µ-analysis Methods

1. The Small Gain Theorem Robust stability deals with the stability of the interconnections of stable operators. The Small Gain Theorem

serves as a basis for the determination of stability of the interconnections of stable operators. The small gain theorem states that a closed-loop feedback

system of stable operators is internally stable if the loop gain of those operators is stable and bounded by unity (Figure 2).12

In Figure 2, the operators P and ∆ are stable transfer function operators. The closed loop system is well-posed and internally stable if 1P ∞∆ < . This condition also ensures that a unique output, y ∈L2 will exist for any input, u ∈ L2.††

By direct use of the small gain theorem and properties of

†† L2 represents the space of all finite energy signals and H∞ represents the space of all transfer functions that are linear time invariant and stable.

:,∆P L2 ⎯→⎯ L2

∈∆,P H∞ P

u y

∆

Figure 2: Block Diagram of the small gain

theorem.

American Institute of Aeronautics and Astronautics

7

the norm it can be shown that the interconnection is robustly stable if Equation (11) is satisfied.

1P −∞ ∞< ∆ (11)

2. Uncertainty Modeling Robust stability is analyzed with respect to a set of perturbations. The stability of the interconnection involves a

known nominal plant and unknown bounded perturbations to that plant. Figure 2 represents a feedback interconnection of stable operators. Without loss of generality, the known

nominal plant dynamics can be represented by P and the unknown perturbations to that plant can be represented by ∆. The true plant is assumed to be a known nominal plant value with perturbations that represent the plant’s uncertainty. The small gain theorem can be used to analyze the stability of the interconnection since the operators are stable and bounded by unity.

Although the small gain theorem guarantees stability of the system it is overly restrictive since the structure of the uncertainty is not defined.

The ∆ block (Figure 2) representing the perturbations to the nominal plant has a known structure. It generally has a block diagonal structure consisting of full block uncertainty and repeated scalar block uncertainty. These uncertainty block structures are described by the following two equations.

: | , [1, ], [1, ]n mfull ij r i n j m×= ∆ ∆ ∈ ∆ < ∈ ∀ ∈ ∀ ∈∆ (12)

: | , [1, ] | 0,n nrs ii ijr i n i j×= ∆ ∆ ∈ ∆ < ∈ ∀ = ∆ = ∀ ≠∆ (13)

Equation (12) defines the set of all full block uncertainty elements and Eq. (13) defines the set of all repeated scalar block uncertainty elements. The ∆ block belongs to the set which is a block diagonal structure composed of full and repeated scalar blocks (Eq. (14)).

,1 , ,1 , ,1 ,

, , ,

, , , , , , , ,

: Re( ), ,

rs rs m rs rs n full full p

rs i rs rs i rs full i full

diag⎧ ⎫∆ = ∆ ∆ ∆ ∆ ∆ ∆⎪ ⎪= ⎨ ⎬∆ ∈ ∆ ∈ ∆ ∈⎪ ⎪⎩ ⎭

∆∆ ∆ ∆

… … … (14)

Superscripts and on the repeated scalar blocks represent real and complex, respectively. They are separated for clarity.

Full block uncertainty typically measures complex uncertainty. These blocks model magnitude and phase so they are useful in modeling signal variations. Repeated scalar block uncertainty is typically used to measure parametric uncertainty in physical parameters in the equations of motion of a physical system. These blocks are typically real and constant and are the basis for uncertainty in the work presented here.

Given this knowledge of the uncertainty structure a less conservative measure of the robust stability of the system, which is based on the small gain theorem, can be formulated.

3. µ: The Structured Singular Value Referring to the framework in Figure 2 the plant, P∈H∞ represents the stable aeroelastic wing dynamics. The

uncertainty in physical parameters in the nominal plant P is represented by ∆∈∆. The stability of the interconnection can be analyzed using the small gain theorem as described above. While stability is guaranteed by this condition it may be overly conservative. Since knowledge of the uncertainty structure is known, a less conservative measure of robustness can be used which is referred to as µ: the structured singular value (Eq. (15)).

1( )min ( ) : det( ) 0

PI P

µσ

=∆ − ∆ =

∆∈∆ (15)

American Institute of Aeronautics and Astronautics

8

This is an exact measure of robust stability for any system with a known structured uncertainty since it only considers uncertainty of the form defined by ∆. Given the system in Figure 2, the plant P is robustly stable with respect to the set ∆ if and only if Eq. (16) is satisfied.

1( )Pµ −∞< ∆ (16)

It is evident that µ (Eq. (16)) is equivalent to the small gain theorem (Eq. (11)) if the uncertainty is unstructured. The problem with µ is the fact that it is difficult to compute. Closed form solutions exist for only a small number of uncertainty structures. Upper and lower bounds are used to compute worst and best case µ values for generalized uncertainty structures.

It is necessary to utilize the upper bound on µ as a basis for analyzing the smallest ∆ matrix that drives the plant, P, unstable since the upper bound represents the worst case. The lower bound, if used, may produce a ∆ matrix that drives the plant unstable. Computing µ is an optimization problem and is described in a number of texts.13,14,15

IV. Uncertainty Analysis Methods Applied to the AGARD 445.6 Wing In this section the aeroelastic uncertainty analysis methods have been applied to a computational model of the

AGARD 445.6 Wing (shown in Figure 3 and herein referred to as the AGARD Wing). This model has been computationally constructed using the coupled Computational Fluid Dynamics (CFD) and Structural Finite Element (FEM) code AERO,4 developed at CMSoft, Inc. This section summarizes the analysis applied to the AGARD Wing at two separate Mach numbers. The model is a 2.5 foot half-span, swept, fixed cantilever wing. Several configurations of this wing with differing half spans and structural strengths were built and tested to their flutter points to compile a comprehensive database of aeroelastic data.16 Several computational aeroelastic codes have used this data as a benchmark for validation. For this analysis the “weakened 3” model was used. The versions of the models used for this analysis are the Reduced Order Models (ROMs). These ROMs are reduced order, linearized models of the full aeroelastic dynamic wing system. Although the model order has been greatly reduced and it is linearized around a defined operating point (Mach number, orientation) the validity pertaining to the flutter point is completely retained. Flutter is a linear instability and can henceforth be analyzed accurately with a linear model representation. As long as the linear ROM representation can adequately predict the flutter point, it is completely valid for use in this uncertainty analysis. Also, the ROM can be analyzed in a relatively rapid manner when compared to the full model. Upon demonstration of the feasibility of the methods currently under development, a much more complex model (i.e., a full order, non-linear model of a complete aircraft) can be used.

A. Aeroelastic ROMS for the AGARD Wing The AGARD ROMs are organized in matrix form according to Eq. (17).

3 3

4 4

23 11 2 3 4

22( , , , , , ) 0

0 0s

p pH B C

pN p P

I

β ββ ρ β ρ

β βρ β β β β

ββ

∞ ∞

∞ ∞

∞∞ ∞

⎡ ⎤− − −⎢ ⎥

⎢ ⎥⎢ ⎥⎢ ⎥= − Ω⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(17)

This is slightly different from the ROM defined in Eq. (2) as it includes structural parameter dependence directly in the ROM definition. The parameters β1 and β2 are the two structural parameter multipliers.17 They represent changes in the structural elastic moduli (E1, E2, G12) and structural density, respectively. The eigenvalues of the ROM matrix N display the stability of the aeroelastic system. These eigenvalues are dependant, non-linearly, on the atmospheric pressure and density (p∞, ρ∞) and on the structural

Table 1: Nominal flutter altitudes for the AGARD Wing.

ROM Mach Number Nominal Flutter Altitude

0.901 1,257 ft

0.499 -46,205 ft.

American Institute of Aeronautics and Astronautics

9

parameters (β1, β2). The other parameters, β3 and β4, represent variations in the atmospheric pressure and density respectively. In total there are four uncertain parameters: β1, β2, β3 and β4.

Nominally, the structural parameter multipliers (β1, β2) and the variation in local atmospheric pressure in density multipliers (β3, β4) are each equal to one.

Eigenvalues of N determine the stability of the AGARD Wing. The ROM is a function of several parameters, consisting of the atmospheric pressure and density (p∞, ρ∞), structural parameters and atmospheric variations (β1, β2, β3, β4). The structural parameters and atmospheric variations for which N is dependant on can be grouped into a vector β. The density and pressure can be defined as functions of the altitude. N is then a function of h and β ( ( , )N f h β= ). The matrix N is still dependant on a fixed constant Mach number so entirely different ROMs will result from different Mach numbers. The stability at a given set of values for the structural parameters is governed by the altitude. The flutter altitude (i.e., the altitude at which N is neutrally stable) is a function of the structural parameters (β) and is governed by Eq. (10).

The nominal flutter altitude is found using Eq. (10) and setting β1 = β2 = β3 = β4 = 1. Table 1 provides the nominal flutter altitudes.

B. Sensitivity Analysis for the AGARD ROMs: Screening RSE A screening response surface equation was generated to observe the sensitivity of the four β parameters.

Although some parameters were shown to be less significant all were retained for subsequent analysis since only four parameters were present. Figure 4 displays the coefficient relative magnitudes of the quadratic response surface equation fit to the flutter altitude (Eq. (10)) as a function of the β parameters. These fits were determined when the β parameters were allowed to vary by no more than 10% of their nominal value for the Mach 0.901 ROM and 3% for the Mach 0.499 ROM. A full factorial design was selected for each RSE since it only required 81 runs. The coefficients are based on the quadratic equation in (18).

0 1 1 2 2 3 3 4 4

5 1 2 6 1 3 7 1 4 8 2 3 9 2 4 10 3 42 2 2

11 1 12 2 13 3

(5 constant and linear terms)

(6 interaction terms)flutth A A A A A

A A A A A A

A A A

β β β β

β β β β β β β β β β β β

β β β

≅ + + + +

+ + + + + +

+ + + + 214 4 (4 squared terms)A β

(18)

Figure 3: AGARD Wing planform view and panel

dimensions (Ref 16).

1 2 3 4 5 6 7 8 9 10 11 12 13 140

0.2

0.4

0.6

0.8

1

Non-dimensional RSE coefficient number

Rel

ativ

e C

oeffi

cien

t Val

ue (n

orm

aliz

ed to

larg

est)

Coefficient Relative Magnitudes (A0 = 1259.1954)

Quadratic Model, Total Coefficients = 15

10% of max value

1 2 3 4 5 6 7 8 9 10 11 12 13 14

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Non-dimensional RSE coefficient number

Rel

ativ

e C

oeffi

cien

t Val

ue (n

orm

aliz

ed to

larg

est)

Coefficient Relative Magnitudes (A0 = -45668.9235)

Quadratic Model, Total Coefficients = 15

10% of max value

Mach 0.901 Mach 0.499

Figure 4: Coefficient relative magnitudes of a quadratic response surface equation fitted to the flutter altitude for the AGARD Wing ROMs.

American Institute of Aeronautics and Astronautics

10

Observing Figure 4 it is evident that the flutter altitude is very nearly linearly related to the structural parameters in this 10% range for the Mach 0.901 ROM. The Mach 0.499 ROM differs and it is apparent that a quadratic RSE is required.

C. Baseline Monte Carlo Analysis for the AGARD ROMs A baseline Monte Carlo case was run

with the four β parameters varying uniformly with a deviation of ±1% of their nominal value. For this case, 5,000 runs were made for both Mach numbers. With the full model this Monte Carlo analysis took 7 minutes and 3 seconds for the Mach 0.901 ROM and 10 minutes and 53 seconds for the Mach 0.499 ROM.‡‡ The resulting output of the Monte Carlo analysis for the Mach 0.901 and 0.499 ROMs are summarized in Figure 5 and Figure 6 respectively.

The distribution for the Mach 0.499 ROM is unusual and does not appear to be normally distributed at the lower altitudes. The important parameter is the least stabilizing altitude, which is the maximum altitude in the distribution (-41,069 ft.). This parameter, representing the least destabilizing robust change in altitude, can be compared to the other analysis methods.

D. DOE/RSM Analysis for the AGARD ROMs A quadratic design for the RSE was selected. With four parameters, the RSE for the flutter altitude takes on the

form in Eq. (18). A quadratic design with four terms requires 15 runs (experiments) at a minimum and 34 = 81 runs at a maximum.

A full factorial design with 81 terms was selected since the number of runs required is relatively small. The non-dimensional coefficient relative magnitude has already been shown in Figure 4 above.

-4.9 -4.8 -4.7 -4.6 -4.5 -4.4 -4.3 -4.2 -4.1

x 104

0

10

20

30

40

50

60

Freq

uenc

y

Value

Normal Distribution Fit: Mean = -46125.746, σ = 1511.2676

600 800 1000 1200 1400 1600 1800 20000

10

20

30

40

50

60

Freq

uenc

y

Value

Normal Distribution Fit: Mean = 1258.6764, σ = 232.4569

Figure 6: Histogram and resulting normal fit to Monte Carlo results for the Mach 0.499 ROM of the

AGARD Wing.

Figure 7: Histogram and resulting normal fit to Monte Carlo analysis using the RSE for the Mach

0.901 ROM. It is evident from the non-dimensional RSE magnitudes that the first 3 linear terms are dominant for the Mach

0.901 ROM. This suggests that a linear RSE including only these terms is most likely adequate. However, maintaining a level of conservatism and considering the fact that a quadratic RSE does not contain many terms, the full quadratic model is retained. Although the interaction and quadratic terms are smaller than the linear terms; keeping the quadratic model will maintain the term’s influence at a relatively inexpensive cost computationally.

A Monte Carlo case was run using this quadratic RSE, where a 5,000 case simulation was again used, so that comparisons can be made with the Monte Carlo analysis of the full model. ‡‡ On a Dell Precision PWS390, Intel® Core TM2 CPU 6300 @ 1.86 GHz.

600 800 1000 1200 1400 1600 1800 20000

10

20

30

40

50

60

Freq

uenc

y

Value

Normal Distribution Fit: Mean = 1258.5579, σ = 231.1473

Figure 5: Histogram and resulting normal fit to Monte Carlo results for the Mach 0.901 ROM of the AGARD Wing.

American Institute of Aeronautics and Astronautics

11

The results of this analysis is summarized in Figure 7 for the Mach 0.901 ROM and in Figure 8 for the Mach 0.499 ROM.

Comparing the results of this analysis with that of the full model, it is evident that the RSE accurately approximates the full model in the range of the input parameters (factors) tested. The RSE for the Mach 0.901 ROM produces almost identical results, while the RSE for the Mach 0.499 ROM is not as accurate, but very close. The mean and standard deviation of the results in Figure 5 and Figure 7 are virtually indistinguishable. The RSE is a much simpler function and therefore runs much faster than the full model. For this model, the Mach 0.901 RSE Monte Carlo analysis took only 8 seconds to run, which is a 50 times as fast as the Monte Carlo analysis with the full model! Although the full model did not take long to run itself, this factor of time improvement is impressive and the benefit of an accurate RSE is displayed. The time benefits will be realized when more complex aeroelastic models are utilized.

E. µ-analysis Approach for the AGARD ROMs For the AGARD Wing model presented here, the instability is reached with a decrease in altitude. At altitudes

below the flutter altitude the system is unstable and contains poles in the right half plane. For altitudes above the flutter altitude the system is stable and contains all left half plane poles.

First, the system must be described as a polynomial function of perturbations to the stability parameter (δh) and perturbations to the uncertain parameters (δβ). The ROM (N, Eq. (17)) is assumed dependant on the altitude by a 3rd order polynomial and assumed dependant on the uncertain parameters (β) linearly (Eq. (19)). The 3rd order polynomial dependence on the altitude is not exact, but a very good approximation (Eqs. (6) and (7)).

The dependence on the β parameters is not exactly linear but since the parameters are assumed to vary by a small degree a linear approximation to the perturbation in β is assumed accurate.

It is noted that the RSE technique can be used to generate polynomial fits for the ROM matrix and will benefit the µ-analysis technique.

Before applying the uncertainty a nominal model without uncertainty (Eq. (20)) was constructed. The flutter altitude can be found with this model and should match that obtained above for each ROM (Table 1). This serves as a useful first check to validate the methods are being applied correctly.

The M-∆ model is formulated from Eq. (20). The M-∆ feedback configuration in general is displayed in Figure 9.

2 300 10 20 30

01 1 0

11 1 12 2

21 1 23 3

31 1 3

( , )...

...

...

...

n n

n n

n n

n n

N h N N h N h N hN NN h N h

N h N h

N h N h

δ δβ δ δ δδβ δβδ δβ δ δβ

δ δβ δ δβ

δ δβ δ δβ

= + + +

+ + +

+ + +

+ + +

+ + +

(19)

2 300 10 20 30( )N h N N h N h N hδ δ δ δ= + + + (20)

-5 -4.9 -4.8 -4.7 -4.6 -4.5 -4.4 -4.3 -4.2 -4.1 -4

x 104

0

10

20

30

40

50

60

70

Freq

uenc

y

Value

Normal Distribution Fit: Mean = -46084.2597, σ = 1341.4286

Figure 8: Histogram and resulting normal fit to Monte Carlo

analysis using the RSE for the Mach 0.499 ROM.

yw yq

q w q q

M M

M M′ ′

∆

qq

yw

Figure 9: M-∆ feedback configuration.

American Institute of Aeronautics and Astronautics

12

The matrix coefficients N00, N10, N20 and N30 can be determined completely analytically utilizing Eqs. (5), (6) and (7). A nominal stable altitude is defined as h0. A model was constructed in the form of Figure 9 and an iterative algorithm13 was used to find the nominal flutter altitudes for each ROM (Table 2).

yw yq

q w q q

M M

M M′ ′

∆

qq

yw

∫

∆

Pyw

Figure 10: Formulation of P-∆ model.

Table 2: Nominal flutter altitudes for the AGARD Wing found using the µ method.

ROM Mach Number Nominal Flutter Altitude

0.901 1,257 ft.

0.499 -46,205 ft.

These values differ slightly (less than 1 ft.) from those listed in Table 1, validating that the M matrices are being

determined correctly and the polynomial approximation is valid. The M structure was determined for the case involving the uncertain parameters β. The coefficient matrices in

Eq. (19) were initially determined using the RSM method. There are 20 coefficient terms in Eq. (19) and that defines the minimum number of cases to generate a fit. With a maximum power of δh being 3 and the number of β parameters being 4, a full factorial experiment will be 64.

Table 3: Robust Flutter Altitudes determined using the µ method.

ROM Mach Number

Nominal stable altitude

(hnom [ft])

Nominal flutter frequency

(ωnom [rad/s])

Robust Flutter Altitude (hrob [ft])

Robust Flutter Frequency

(ωrob [rad/s]) 0.901 3,000 112.92 1,980 110.42

0.499 -38,000 599.30 -39,545 596.80

In order to determine µ the bottom loop

of the model in Figure 9 was closed with the integral operator to form the P-∆ frequency dependant model (Figure 10). The unity norm bound condition was applied ( ( ) 1Pµ < ). Due to this condition, the plant must be scaled to account for δh variation greater than unity.

The structured singular value is determined as a function of frequency. An iterative method to determine the robust flutter boundaries was used.13 A dense grid of frequency points was selected around the frequency of the nominal flutter velocity. In this initial case the range of frequencies was ωflutt ± 5 rad/s with 5 frequencies evenly spaced. The robust flutter altitudes are displayed in Table 3, along with the nominal stable altitudes and the nominal flutter frequencies.

By observing Figure 5 and Figure 6, it is apparent that these robust altitudes are in the neighborhood of the maximum flutter

106 108 110 112 114 116 1180

0.2

0.4

0.6

0.8

1µ vs. frequency with a change in δh of: 1020.5192

frequency, rad/s

µ bo

unds

µ upper bound

µ lower bound

Figure 11: ( )Pµ vs. frequency at the robust flutter altitude for the Mach 0.901 ROM.

594 596 598 600 602 604 6060

0.2

0.4

0.6

0.8

1µ vs. frequency with a change in δh of: 1545.1991

frequency, rad/s

µ bo

unds

µ upper bound

µ lower bound

Figure 12: ( )Pµ vs. frequency at the robust flutter altitude for the Mach 0.499 ROM.

American Institute of Aeronautics and Astronautics

13

altitudes found using the Monte Carlo analysis, providing validation of this estimate. The calculated µ bounds vs. the frequencies sampled when the robust flutter margin is reached are displayed in Figure 11 for the Mach 0.901 ROM and in Figure 12 for the Mach 0.499 ROM. The fact that the peaks are within the range of frequencies tested confirms that the range of frequencies tested is adequate.

F. Comparison of the Uncertainty Methods The robust flutter boundaries vs. Mach number determined from all of the methods are collectively displayed in

Figure 13 and Table 4. Table 4: Robust Flutter Altitudes Determined from the various methods.

ROM Mach number

Nominal Flutter Altitude (ft)

hrob

Monte Carlo (ft)

hrob

DOE/RSE (ft)

hrob

µ Analysis (ft)

-46,205 -41,069 -40,834 -39,545 0.499

run time§§ → 10 mins 53 secs 27 secs 14 hrs 24 mins

1,257 1,884 1,887 1,980 0.901

run time§§ → 7 mins 3 secs 8 secs 8 hrs 5 mins

0.4 0.5 0.6 0.7 0.8 0.9 1-50,000

-40,000

-30,000

-20,000

-10,000

0

10,000

Mach number

Alti

tude

, ft

Nominal Flutter BoundaryRobust Flutter Boundary determined using Monte Carlo AnalysisRobust Flutter Boundary determined using DOE/RSE AnalysisRobust Flutter Boundary determined using µ Analysis

No Flutter

Flutter

Figure 13: Robust Flutter Boundaries Determined from the various methods.

At first glance it is apparent that all three methods are producing robust flutter altitudes in the same

neighborhood of values. This is supporting evidence that each method is producing accurate results. The Monte Carlo analysis method, which is the less elegant but thorough and reliably tested brute force approach, can be defined as the baseline to compare the other methods against. By this criterion, it is evident that the other two methods; DOE/RSE and µ-analysis, are producing accurate and representative results. In fact, the other two methods are producing slightly more conservative flutter boundaries with the µ-analysis method producing the most conservative robust flutter altitudes. This supports the fact that µ is determined over the entire space of all possible uncertainty values that conform to the defined structure. Essentially, µ should produce results that are representative of a Monte Carlo analysis with infinite runs. By this definition, the robust flutter boundaries determined using µ should always be either equal to or more conservative than the Monte Carlo analysis. The above results suggest that

§§ On a Dell Precision PWS390, Intel® CoreTM2 CPU 6300 @ 1.86 GHz.

American Institute of Aeronautics and Astronautics

14

this is the case. Of course, in order to utilize the µ-analysis approach the system must be approximated as a linear operator with polynomial dependency on the uncertain parameters. These approximations could lead to robust flutter altitudes that are less conservative than those determined using Monte Carlo analysis since the system is an approximation of the full system. The fact that the above results produced robust flutter boundaries that are more conservative than the Monte Carlo analysis approach strongly supports the claim that the approximation is valid and accurately represents the full system.

V. Conclusions

A. Summary The three uncertainty methods (Monte Carlo, DOE/RSM and µ-analysis) have been effectively demonstrated for

use with an aeroelastic wing model. The model type utilized for this work is a linear reduced order model (ROM). This type of aeroelastic model has several advantages. One advantage is that flutter solutions are found relatively rapidly. This ability to produce flutter solutions in a rapid manner was essential to the development of the methods. Since the methods are now established utilizing this rapid model, a much more complex model (i.e., a detailed aircraft model) could be analyzed with the elements that have been established under this work.

The validity of the DOE/RSM method has been effectively demonstrated as the results are almost identical to the baseline full Monte Carlo analysis. The computational time advantage of the DOE/RSM method has also been successfully illustrated by the fact that the number of runs of the full model has been reduced by two orders of magnitude. This advantage will be much more realized if higher order models with much more complexity are used.

The µ-analysis method has been shown to produce valid results. Robust flutter boundaries found using the structured singular value have been shown to compare with the Monte Carlo results very well. In fact, the results obtained utilizing µ should, in effect, be better since µ represents an infinite run Monte Carlo case.

The µ-analysis method works very well with uncertainty if the system is dependant on the uncertain parameters in a polynomial form. The DOE/RSM method can be used to effectively determine approximate system models that are dependant on parameters in a polynomial form. Due to these facts, the DOE/RSM method can be used to complement the µ-analysis method very well and the work displayed herein has effectively shown the union of these two techniques.

Three aeroelastic uncertainty methods with varying degrees of robustness have been established and tested utilizing an accurate model of the AGARD 445.6 wing. The AGARD wing is a benchmark model and has extensive experimental data to be utilized for comparison. The methods established in this work can be applied to a much more complex model. For example, the robust flutter boundaries for an entire aircraft can be rapidly determined using the DOE/RSM and µ-analysis methods established herein.

B. Areas of Future Investigations Inclusion of polynomial based uncertainty in the β parameters is desirable. Currently, for µ-analysis, variation in

the flutter parameter (altitude for the AGARD ROM) is allowed to be polynomial dependant to any order but the β parameters are only allowed linear dependence. If polynomial uncertainty is integrated, then large variations on the β parameters can be incorporated. The effect of large changes in parameters (i.e., burning fuel mass or a large change in stiffness due to aerothermoelastic effects) can be included to determine robust stability boundaries.

For the AGARD ROM model utilized here, the dependence on the β parameters (Eq. (17)) is relatively straight forward. The ROM itself is defined with this dependence directly. For more complex problems, the uncertainty may not be as easily understood. To incorporate more specific uncertainty, an approach that incorporates DOE/RSM is proposed. As established earlier, a ROM can be constructed for a model of any complexity and will be representative as being a linear model around a stable operating point. The ROM that is constructed is dependant on parameters that may be uncertain to the model (β). Using the AERO code, separate ROMs can be constructed for differing values of the uncertain β parameters. These ROMs can be effectively fit to a polynomial equation, or a matrix RSE. DOE can be used to effectively select the experiments (or β sets) that are used for the polynomial fit so that the number of sets is a minimum. The matrix RSE that represents the ROM has multiple uses as an approximate model representation that incorporates parameter dependence. It can be used for Monte Carlo analysis or can be directly applied to the µ method with ease. An established method such as this would allow for models of any complexity to be incorporated in a relatively straightforward manner.

An area of future investigation is a refinement of the µ calculations. According to Table 4 it is noted that the robust flutter boundaries calculated with µ-analysis took a large amount of time to be determined. This is due to size of the matrix for which the structured singular value was computed. For the AGARD Wing model, the frequency

American Institute of Aeronautics and Astronautics

15

dependant P matrix, for which µ was computed, was in excess of 800 by 800 elements. The singular value computations, which are compounded by the µ optimization are computationally intensive and the computational burden grows exponentially with the size of the matrix P. If the matrix P is reduced by just half (400 by 400) it is anticipated that the time of the robust flutter boundary computation could be reduced by an order of magnitude or more. Effectively minimizing the size of this matrix would be extraordinarily beneficial.

Incorporation of feedback design is a desired next step as well. Feedback controllers that are robust to uncertainties can be designed to include aeroservoelasticity. Established techniques, such as µ-synthesis, would fit well into a toolbox that includes feedback controller design.

VI. Acknowledgements The work described herein was conducted as part of a Phase I Small Business Innovation Research program

sponsored by NASA Langley Research Center in conjunction with NASA Dryden Flight Research Center. The authors acknowledge the National Aeronautics and Space Administration for valuable resources and funding to make this project possible.

References 1. Zimmerman, N., and J. Weissenburger, “Prediction of Flutter Onset Speed Based on Flight Testing at

Subcritical Speeds,” Journal of Aircraft, Vol. 1, No. 4, July-August 1964, pp. 190-202. 2. Yurkovich, R. N., Liu, D. D., and Chen, P. C. , The State-of-the-Art of Unsteady Aerodynamics for High

Performance Aircraft, AIAA Paper 2001-0428. 3. Geuzaine, P., Brown, G., Harris, C., and Farhat, C., Aeroelastic Dynamic Analysis of a Full F-16

Configuration for Various Flight Conditions, AIAA Journal 41, 2003, pp. 363-371. 4. Farhat, C., Geuzaine, P., and Brown, G., Application of a Three-Field Nonlinear Fluid-Structure Formulation

to the Prediction of the Aeroelastic Parameters of an F-16 Fighter, Computers & Fluids 32, 2003, pp. 3-29. 5. Sussingham, C., DeJoannis, J., and Sansone, A., F-16 Aeroelastic Flight Testing: Impact on the Operational

Fleet, June 9 2000. 6. Kuttenkeuler, J., A Finite Element Based Modal Method for Determination of Plate Stiffnesses Considering

Uncertainties, J. of Composite Materials 33, 1999, pp. 695-711. 7. Kuttenkeuler, J. and Ringertz, U., Aeroelastic Tailoring Considering Uncertainties in Material Properties, Str.

Optimization 15, 1998, pp. 157-162. 8. Lieu, T., C. Farhat and M. Lesoinne, Reduced-Order Fluid/Structure Modeling of a Complete Aircraft

Configuration, Computer Methods in Applied Mechanics and Engineering, Vol. 195, pp. 5730-5742, 2006. 9. Anon., U.S. Standard Atmosphere, 1976, National Technical Information Service, U.S. Department of

Commerce, Washington, DC, Oct. 1976. 10. Montgomery, D. C., Design and Analysis of Experiments, 5th Edition, John Wiley & Sons, Inc., 2001. 11. Myers, R. H. and Montgomery, D. C., Response Surface Methodology: Process and Product Optimization using

Designed Experiments, 2nd Edition, John Wiley and Sons, Inc., 2002. 12. Lind, R. and M. J. Brenner, Robust Flutter Margin Analysis That Incorporates Flight Data, NASA/TP-1998-

206543, March, 1998, pp. 12-13, 22. 13. Lind, R. and M. Brenner, Robust Aeroservoelastic Stability Analysis, Springer-Verlag London Limited, 1999. 14. Zhou, K., J. C. Doyle, and K. Glover, Robust and Optimal Control, Prentice Hall, NJ, 1996. 15. Zhou, K. and J. C. Doyle, Essentials of Robust Control, Prentice Hall, NJ, 1998. 16. Yates, E. Carson Jr., AGARD Standard Aeroelastic Configurations for Dynamic Response. Candidate

Configuration I.-Wing 445.6, NASA-TM-100492, August, 1987. 17. Farhat, C., Parameterized Reduced-Order Structural and Aerodynamic Models of the AGARD Wing 445.6 for

Real-time Non-deterministic Aeroelastic Analyses, Report 1, Purchase Order C100027/1379/DHK