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Nonlinear Aeroelastic Optimization

The Cultural and Convention CenterMETU

Inonu bulvariAnkara, Turkey

Sponsored by:RTA-NATO

The Applied Vehicle Technology Panel

presented byR.M. Kolonay Ph.D.

General Electric Corporate Research & Development CenterAnkara, Turkey Oct.. 1-5, 2001

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• Introduction

• Nonlinear Unsteady Aerodynamic Approximations

• Nonlinear Unsteady Aeroelastic Analysis for Design

• Nonlinear Unsteady Aeroelastic Sensitivity Analysis

• Nonlinear Unsteady Aeroelastic Optimization (Transonic)

• Nonlinear Static Aeroelasticity Analysis for Design (Transonic)

Presentation Outline

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Goal of Computational Aeroelastic DesignMethods

• To accuratelypredict static and dynamic response/stability at areasonable computational cost

• Methods need to be suitable for incorporation into MDA and MDOEnvironments

Introduction

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Discretization of EOM

• Structures - Typically, although not necessarily, repre-sented by Finite Elements in either physical or generalized coordi-nates. Derived in aLagrangian frame of reference.

• External Loads - Aerodynamic loads. Representationsrange from Prandtl’s lifting line theory to full Navier-Stokes withturbulence modeling. Represented in physical and generalizedcoordinates in a (usually)Eulerian frame of reference.

K B M, ,

F u u t, ,( )

Introduction

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Fluid-Structural Coupling Requirements

• Must ensure spatial compatibility - proper energy exchange acrossthe fluid-structural boundary

• Time marching solutions require proper time synchronizationbetween fluid and structural systems

• For moving CFD meshes GCL [1] must be satisfied

Introduction

If coupling requirements for time-accurate aeroelastic simula-tion are not met then dynamical equivalence cannot beachieved. That is, regardless of the fineness of the CFD/CSMmeshes and the reduction of time step to 0, the scheme may con-verge to the “wrong” equilibrium/instability point.[2]

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Nonlinear Aerodynamics• Assume a continuum (Conservation of Mass, Momentum, Energy)

Decreasing Degree of Approximation Equations

Complete Navier-Stokes EquationsNo Viscosity Euler EquationsIrrotational, Isentropic Full Potential EquationsSmall Perturbations Transonic Small DisturbanceLinearize Linear Lifting Surface Theory

Introduction

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Nonlinear Flow Conditions

• High angles of attack• Large control surface deflections• Transonic Speeds• . . .

Nonlinear Flow Characteristics• Attached flows with shocks• Mixed attached and separated flows• Mixed attached and separated flows with shocks• Fully separated flows• Vortex flows• . . .

Introduction

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General Modeling Comments

• Use appropriate theory to capture desired phenomena

- Fluids - Navier-Stokes vs. Prandtl’s’ lifting line theory- Structures - Nonlinear FEM vs. Euler beam theory

• Model the fluid and structure with a consistent fidelity

- For a wing do not model the fluid with NS and the structure with beam theory

• For design methods remember:- Total Load = Weight*Nz- Stability is determined by a perturbation about a steady state

Introduction

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Aeroelastic Phenomena

Introduction

Static Aeroelastic Phenomena

• Lift Effectiveness

• Divergence

• Control Surface Effective-ness/Reversal

• Aileron Effectiveness/Reversal

Dynamic Aeroelastic Phenomena

• Flutter

• Gust Response

• Buffet

• Limit Cycle Oscillations (LCO)

• Panel Flutter

• Transient Maneuvers

• Control Surface Buzz

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The e es as

(1)

Wher and aerodynamic forcesrespe

- .

-

And t

(2)

q{ } ] q t( ){ }

u t( ){

Φ[ ]

)dS

Equations of Motionquations of motion can be expressed in generalized coordinat

e are generalized mass, stiffness, damping,ctively.

set of independent generalized coordinates defined by

- spatially discretized structural dof.

Modal transformation matrix

he generalized aerodynamic forces can be represented as

M[ ] q{ } B[ ] q{ } K[ ] q{ }+ + F{ }=

M[ ] K[ ] B[ ] F{ }, , ,

u t( ){ } Φ[=

}

Fi t( )ρ∞U∞

2

2-----------------CR

2 CpLx y t, ,( ) CPU

x y t, ,( )– Φi x y,(S∫∫=

Introduction

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Some Approximation Methods

• Reduced Order Unsteady Aerodynamics

• Pulse Transfer Function (Linear Impulse Function)

• Indicial Response Method

• Discrete-Time Linear and Nonlinear Aerodynamic Impulses (VolterraTheory)

Unsteady Aerodynamic Approximations

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Reduced Order Unsteady Aerodynamics[3]• Use Karhunen-Loeve (K-L) Modes as basis for fluid system

K-L eigenvectors are determined from the eigenvalue problem

- the deviation of each instantaneous flow field from the mean flow.

- an instantaneous flow field vector retained atJ discrete times.

- mean flow field.

• Use to reduce the aerodynamic system from to whereR<< N.

• Demonstrated on 2-D Euler.

V[ ] Φ[ ] v{ } λ v{ }=

Φ j k, qcj

T

qck

=

qcj

qj{ } q'˜{ }–≡

qj{ } j 1 2 3… J,, ,=( )

q'˜{ }

V[ ] NxN RxR

Unsteady Aerodynamic Approximations

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Pulse Transfer Function[4]• Determine the response due to a smoothly varying pulse (exponential

pulse) for each structural mode participating in the analysis.

• Unsteady generalized aerodynamic forces infrequency domain aredetermined by dividing the Fast Fourier Transform (FFT) of thetime domain generalized displacements with the FFT of the timedomain generalized aerodynamic forces (Transfer Function)

• Requires one time domain solution for each mode in analysis

• Assumes dynamic response linear about a nonlinear steady state

• Demonstrated on 3-D Euler and Navier-Stokes

Unsteady Aerodynamic Approximations

14

D

ar Aerody-ory)[5]

lt ar system can bel convolution inte-

- - S

F

S

th

2 …+

imations

iscrete-Time Linear and Nonlinenamic Impulses (Volterra The

erra theory of nonlinear systems - Any nonlinemodeled as an infinite sum of multidimensionagrals of increasing order.

Response of the nonlinear system toAn arbitrary inputteady state value about which the response is computed

irst-order kernel (linear unit impulse response)

econd-order kernel

order kernel

y t( ) ho h1 t τ–( )u τ( ) τd

0

∞

∫ h2 t τ1–( )u τ1( )u τ2( ) τ1d τd

0

∞

∫0

∞

∫

… hn t τ1– … t τn–, ,( )u τ1( )…u τn( ) τ1d … τnd

0

∞

∫0

∞

∫ …

+ +

+ +

=

u t( )

Unsteady Aerodynamic Approx

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D

• Vo

-

-

-

- n

y t( )u t( )ho

h1

h2

hn

15

D

r

pulses, with the number

of the kernel of interestnd .

e er order kernels

e

p r during pertur-

imations

nel Identification- - Impulse response

- - Responses of the nonlinear system to multiple unit im

of impulses applied to the system equal to the order - - Response to two unit impulses applied at times a

quires more time integrations to determine high

neralization of Impulse Function Technique

roximationcan capture nonlinear effects that occubation

h1

hn

h2 t1 t2

Unsteady Aerodynamic Approx

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• Ke

• R

• G

• Ap

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Scope of Remaining Discussions

• To develop a transonic unsteady aeroelastic design methodology forpreliminary designat a reasonable computational cost.

• Such a methodology could be incorporated into the multidisciplinaryanalysis and design optimization environment (MDAO)

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Definition of Preliminary Design

• Geometry assumed fixed (wing planform, airfoil shape, #spars, #ribs,spar spacing etc.)

• Designed components consist of structural elements (thicknesses ofskins, ribs and spars, cross sectional areas of posts, spar and ribcaps, and concentrated masses etc.)

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Critical Issues/Requirements• Transonic aerodynamics are nonlinear

• Optimization problem is nonlinear

• Coupling two nonlinear problems is not realistic

Preliminary Design MDDA Requirements

• Structural and mass distribution down to substructure level

• F.E.M 50k to 100k d.o.f maximum

• Aerodynamic loads normal to lifting surface

• Small disturbance theory (location and strength of weak shocks)

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Solution Approach• Formulate as nonlinear mathematical programming problem

• Solve by gradient based optimization

• Analysis- F.E.M

- TSD with Indicial Response Method to approximate the unsteady aerodynamic forces

* Modal basedp-Method for flutter analysis

• Constraint/Constraint gradients- Constraint on modal damping rather than on flutter velocity

- Semi-analytic gradients (some assumptions can produce fully analytic gradients)

• Redesign- Use First Order Taylor Series Approximations for functional values and constraints

- Solve approximate problem by Method of Modified Feasible Directions

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10]

• Ass ar about some non-l perturbations).

• Allo pproximateperposition).

• IRM that participates inns (Mach number,

• Sm r significantly move

• Imp se derivative of

imations

Indicial Response Method[6],[

ume the unsteady aerodynamic forces are linelinear static aeroelastic solution (valid for smal

ws use of anIndicial Response Method (IRM) to aunsteady aerodynamic forces (limited linear su

requires one nonlinear solution for each modeflutter analysis for a given set of initial conditioinitial angle of attack, Reynolds #, etc.).

all perturbations - can neither create, destroy noa shock.

ulse Function and IRM related (Impulse responStep response)

Unsteady Aerodynamic Approx

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Indicial Response Method (IRM)

Unsteady Aerodynamic Approximations

A

B

tt t–

t

α t( )

α t( )StepStep resp.

Smoothlyvarying

Time (t)

Cl

(3)Cl t( ) Clα t i

t( )∆α t i( )i 0=

n

∑= or Cl t( ) Clαt0

t t–( )∆α t( )0

t

∑=

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In the ) becomes

(4)

Eq (4) e response of a system toan arb . It assumes that linearsuperp

Now fo as

(5)

Transf nvolution integral gives

(6)

Eq (6) in

tions (IRM)

limit and also with a change of variable Eq (3

is a form of Duhamel’s formula. It enables the calculation of thitrary input by knowing only the indicial response of the systemosition applies.

r a generalized force , step Eq can be written

orming from the time domain to the Laplace domain via the co

represents generalized unsteady aerodynamics forces in thes-doma

∆ti 0→ τ t t–=

Cl t( ) Clαt0

t( )α 0( ) Clαt0

τ( )α t τ–( ) τd

o

t

∫+=

Fi t( ) qj t( ) Qij t( )

Fi t( )

12---ρ∞U∞

2 CR2

----------------------------- Qij t( )qj 0( ) Qij τ( )q j t τ–( ) τd

0

t

∫+=

Fi s( ) 12---ρ∞U∞CR

2sQij s( )qj s( )=

Unsteady Aerodynamic Approxima

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Defini o thes-domain yields

(7)

The L - ex - os

• Sol ot-locus etc.d not be har-

σω

U ons (IRM)

ng and transforming the rest of Eq. (1) int

aplace variable is in general complex and is defined asponential dampingcillatory frequency of the system

utions to Eq (7) for flutter can be found byp-k, p, roProvided is a continuous function ofs but neemonic.

Q s( )[ ] CR2

s Q s( )[ ]=

M[ ]s2 K[ ] B[ ]s 12---ρ∞U∞

2 Q s( )[ ]–+ + q s( ){ } 0=

s s σ iω+=

Q s( )

nsteady Aerodynamic Approximati

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• As

(8)

• De anner

ondition

place transformable

e the “best” approxi-

matin s( )qSjs( )

s (IRM)

sume

termine the coefficients in the following m

- Determine the static aeroelastic response for a given flight c

- Determine in the generalized force due to a La

step input for the th mode.

- Once is obtained are selected to produc

g curve fit between and where

- with

Qij s( ) sCR2

Cr ijs br ij

+------------------

r 1=

n

∑=

Cr ijbr ij

,

∆Fij t( )N Fi t( )

qSjt( ) j

Fij t( )∆ N Cr ijbr ij

,

Fij t( )∆ N Fij t( )∆ A Fij t( )∆ A ℑ 1– Qij

=

qj s( ) AjΦ j 10ta---

3e

st–15

ta---

4e st– 6

ta---

5e st–+–

td0

a

∫ e st–

a

∞

∫+

=

Unsteady Aerodynamic Approximation

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Let Eq. (7)

beco

(9)

• Sol

• p-m flutter constraints)

s

p( )} 0=

U ons (IRM)

, define , ,

mes

ve Eq. (9) by thep-method

ethod damping valid away from axis (nice for

ω γ i+( )= p k γ i+( )≡ k ωCR( ) 2U∞( )⁄= bCR2

--------=

U∞b

---------2

p2 M[ ]U∞b

--------- p B[ ] K[ ] 1

2---ρ∞U∞

2 Q p( )[ ]–+ + q{

γ

nsteady Aerodynamic Approximati

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Indicial Response Flutter AnalysisAGARD 445.6 Wing (Weak Model 3) - TSD Aerodynamics (CAP-TSD)

Nonlinear Unsteady Aeroelastic Analysis

X

Y

Z X

Y

Z

X

Y

Z X

Y

Z

14.5”

46.3°

22.”

30.”

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CRD Analysis

X

Y

Z

31.38

27.48

23.57

19.67

15.77

11.86

7.958

4.055

.1511

-3.753

-7.656

-11.56

-15.46

-19.37

-23.27

-27.17

X

Y

Z

X

Y

Z

71.52

65.25

58.97

52.69

46.42

40.14

33.87

27.59

21.31

15.04

8.761

2.485

-3.791

-10.07

-16.34

-22.62

X

Y

Z

= 89.94 Hz.

X

YZ

X

YZ

X

YZ

X

YZ

= 50.50 Hz.

s

Nonlinear Unsteady Aeroelastic

Mode Shapes and frequencies

X

YZ

X

YZ

X

YZ

X

YZ

Mode 4,ω

X

Y

Z

25.09

20.38

15.68

10.97

6.269

1.565

-3.139

-7.843

-12.55

-17.25

-21.96

-26.66

-31.36

-36.07

-40.77

-45.48

X

Y

Z

Mode 2,ω = 37.12 Hz.

X

Y

Z

27.92

26.05

24.19

22.32

20.45

18.59

16.72

14.85

12.98

11.12

9.250

7.383

5.516

3.649

1.782

-.08551

X

Y

Z

Mode 3,ωMode 1,ω = 9.63 Hz.

445.6 Flutter Analysi

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445.6 Time Integration Response

Nonlinear Unsteady Aeroelastic Analysis

0.0 0.1 0.2 0.3 0.4

-0.0015

-0.0010

-0.0005

0.0000

0.0005

0.0010

0.0015

Time (sec)

Gen

eral

ized

Dis

plac

emen

t

γ1= 13.0682ω1= 88.352(Hz)

γ2= 5.9408ω2= 52.285(Hz)

γ3= 24.1002ω3= 30.268(Hz)

γ4= .1687ω4= 15.574(Hz)

DATA

FIT

ERROR

0.0 0.1 0.2 0.3 0.4

-0.0020

-0.0010

0.0000

0.0010

0.0020

0.0030

Time (sec)

Gen

eral

ized

Dis

plac

emen

t

γ1= 5.7635ω1= 52.269(Hz)

γ2= 25.9159ω2= 30.188(Hz)

γ3= .0396ω3= 15.564(Hz)

γ4= 12.4610ω4= 88.338(Hz)

DATA

FIT

ERROR

0.0 0.1 0.2 0.3 0.4

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

Time (sec)

Gen

eral

ized

Dis

plac

emen

t

γ1= -.0366ω1= 15.561(Hz)

γ2= 32.0916ω2= 30.286(Hz)

γ3= 445.6037ω3= .612(Hz)

γ4= 4.7738ω4= 52.054(Hz)

DATA

FIT

ERROR

0.0 0.1 0.2 0.3 0.4

-0.0040

-0.0030

-0.0020

-0.0010

0.0000

0.0010

0.0020

0.0030

0.0040

Time (sec)

Gen

eral

ized

Dis

plac

emen

t

γ1= .0282ω1= 15.569(Hz)

γ2= 25.8733ω2= 30.095(Hz)

γ3= 5.7688ω3= 52.242(Hz)

γ4= 12.0791ω4= 88.547(Hz)

DATA

FIT

ERROR

(c)

(d)

(a)

(b)

Time response for 445.6 wing (a) mode 1 (b) mode 2 (c) mode 3 (d) mode 4 (M∞ = 0.901,α0 = 0.0˚,q∞ = 0.67 psi,U∞ = 11998.75 in/sec)

qf 0.667 psi,U f 11 971 in/sec ω f 15.45 Hz≈,,≈≈

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445.6 Indicial Responses∆F j1

Nonlinear Unsteady Aeroelastic Analysis

0.0 5.0 10.0 15.0 20.0 25.0 30.0

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

Time (non-dimensional)

Gen

eral

ized

For

ce (∆

F 11)

Analytic ∆F11

Numeric ∆F11

0.0 5.0 10.0 15.0 20.0 25.0 30.0

0.0

2.0

4.0

Time (non-dimensional)

Gen

eral

ized

For

ce (∆

F 21)

Analytic ∆F21

Numeric ∆F21

0.0 5.0 10.0 15.0 20.0 25.0 30.0

-2.0

0.0

Time (non-dimensional)

Gen

eral

ized

For

ce (∆

F 31)

Analytic ∆F31

Numeric ∆F31

0.0 5.0 10.0 15.0 20.0 25.0 30.0

0.0

Time (non-dimensional)

Gen

eral

ized

For

ce (∆

F 41)

Analytic ∆F41

Numeric ∆F41

M∞ = 0.901,α0 = 0.0˚

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6000 8000 10000 12000 14000

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

MODE 1MODE 2MODE 3MODE 4

445.6 Indicial Response Flutter Analysis

Nonlinear Unsteady Aeroelastic Analysis

6000 8000 10000 12000 14000

100

200

300

400

500

MODE 1MODE 2MODE 3MODE 4

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2

100

200

300

400

500

MODE 1MODE 2MODE 3MODE 4

445.6 Wing D amping R atio versus Fr equency

(M = .9 01 ,α = 0.0 0°, ρ∞ = 9 .307 E-09 sl ug s/ in )

V vs. g V vs. ω g vs. ω

V (in/sec) V (in/sec)

Dam

ping

Rat

io (g

)

Frequency (radians)

ω(r

adia

ns)

ω(r

adia

ns)

qf 0.650 psi,U f 11 810 in/sec ω f 15.35 Hz≈,,≈≈