nonlinear aeroelastic analysis using rom/rom methodology ... · danny d. liu zhicun wang shuchi...

Danny D. Liu

Zhicun Wang

Shuchi Yang

Chunpei Cai

Nonlinear Aeroelastic Analysis using

ROM/ROM Methodology:

Membrane-on-Wedge with Attached Shock

9489 E. Ironwood Square Drive, Scottsdale, AZ 85258-4578, Tel (480) 945-9988, Fax (480) 945-6588, E-mail: [email protected]

Marc Mignolet

Presented at the Bifurcation and Model Reduction Techniques for Large Multi-Disciplinary Systems Meeting at the University of Liverpool 26-27 June, 2008

22

This present work is under the support of a

NASA SBIR Phase I contract, with Dr. Robert

Bartels as the Technical Monitor.

Acknowledgement

3

Reentry aeroshell A conceptual design

for balluteInflatable ballute

entry

• Clamped Ballute• Trailing/Torroid Ballute

Different Ballute Types

4

*Parameters: ballute mass =220 lb, ballute diameter: 92 ft, Initial Mach number: 20;

Gravity acceleration: Mars:10.40 ft/s2, Earth 32.1740 ft/s2. Initial altitude: Mars= 660

kft, Earth: 900 Kft

Martian Entry*Earth Entry*

Earth/Mars Entry Profiles

Knudson number: , a GasKinetic parameterM

KnRe

55

Overview• Ballute aeroelastic problem requires Gaskinetic

(microscopic) aerodynamics in the rarefied hypersonic flight

regime

– Boltzmann/BGK method (time accurate) is adopted.1

• Ballute is an inflatable (nonlinear) structure

– Nonlinear structural ROM (ELSTEP) is adopted.2

• Ballute flutter/LCO computation procedure needs to be

expedited

– ZONA’s nonlinear/linear ROM-ROM procedures are

adopted.3

• Membrane-on-Ballute with Bow-Shock is modeled first by a

2D membrane-on wedge with attached shock- thus the

present study

Supported by: 1. AFOSR/Schmisseur; 2. NASA/Rizzi; 3. AFOSR/Fahroo.

66

• Introduction: Ballute Systems

• Nonlinear Structural ROM (ELSTEP)

• Boltzmann Unsteady Aerodynamics: Time-

Accurate BGKX

• Nonlinear Aeroelastic Static Deformation

Analysis

• Aerodynamic ROM (Sys. Id. + ARMA)

• ROM/ROM Time-Domain Flutter Analysis for

Undeformed/Deformed Mean Configuration

• Concluding Remarks

Outline

7

M∞

Rigid ring

Oscillatory shock

Vibrating membrane

Mean shock

x

r

Rigid nose

Reflected wave trains

Modeled (Axisymmetric) ballute system with

nonlinear structural-aerodynamic interactions.

M∞L

θVibrating membrane

Wedge angle

Oscillatory Shock

Mean Shock

Characteristics

(Mach Waves)

Membrane-on-wedge in hypersonic/supersonic flow.

Modeled Ballute System

8

NL Structural

Solver

Aerodynamic

Solver

NL Structural

ROM

Aerodynamic ROM

NL AE Static

Analysis

Around Mean Config.

Flutter/LCO

Analysis

Present Nonlinear Aeroelastic Methodology

9

ZTRAN

Linear Flutter

Modes

Linear Transverse

Modes (Nastran)

SOL 103

In-Plane Static

Response to

Transverse Loads

SOL 106

NONLINEAR ROM

STRUCTURAL MODEL

Aerodynamic Forces Structural Response

Generation of Nonlinear Structural ROM

10

• ELSTEP/FAT = Equivalent Linearization Stiffness

Evaluation Procedure/Fatigue (due to) Acoustics

and Thermal Gradients

• An advancement of ELSTEP code previously

developed by Steve Rizzi/NASA Langley and Alex

Muravyov/MSC

• ZONA/ASU R&D efforts of ELSTEP/FAT supported

under several AF/SBIR’s and NASA/SBIR’s from

1999 ~ 2005

About ELSTEP/FAT

11

Large Deformation

Mechanics; Lagrangian

Formulation Assumed Displacement

Field, Basis Funct

( ) ( ) ( )=

=M

n

nini XUtqtXu

1

)(,

)3()2()1(ipljijlpljijljijjijjij FqqqKqqKqKqCqM =++++ &&&

Linear

stiffnessQuadratic

stiffness

Cubic

stiffness

EXACT FORMULATION

)(miU

( ) ( ) ( ) ( )1

,M

n

n

n

t q t=

=u X U X

GAF

)(t

FLin + FNL : Run Sol #106

Sol #103+

Nonlinear ROM Procedure: ELSTEP

12

FEM model

(Nastran)

Runs in static

nonlinear

Evaluate coefficients of the ROM model(1)

ijK )2(ijl

K, ,)3(

ijlpK

Solve for Nonlinear Equation of Motion & Boundary Conditions

Time Histories (Displacements)

– Flutter/LCO Analyses

– Stresses/Fatigue Life Prediction

)3()2()1(ipljijlpljijljijjijjij FqqqKqqKqKqCqM =++++ &&& )(t

ELSTEP Nonlinear Structural ROM

Sol103

Sol106

13

• Impose a series of static deflections and determine (e.g. from

finite element model) the forces required and the stresses.

• Then, identify the coefficients of the reduced order model.

( )j

jq=u U ( ) (1) (2) 2 (3) 3

i ij j ijj j ijjj jaF K q K q K q= + +

( ) ( )(2)

22

i ia bijj

j

F FK

q

+=

(1) ,ijK

(2) ,iljK )3(iljpK

)3(ijjjK

(3) ,iljjK

( )jjq= u U

( )ˆj

jq=u U

( ) ( )j l

j lq q= +u U U

( ) ( )j l

j lq q= u U U

( ) ( ) ( )j l p

j l pq q q= + +u U U U

( ) (1) (2) 2 (3) 3

i ij j ijj j ijjj jbF K q K q K q= +

Cond. (a)

Cond. (b)

Inner terms

Cross terms

Cond. (c)

Procedure to Evaluate Nonlinear Stiffness Terms

14

Significant nonlinearity

first peak = 154 Hz

first nat. freq. = 110 Hz

1.E-16

1.E-15

1.E-14

1.E-13

1.E-12

1.E-11

1.E-10

1.E-09

1.E-08

0 200 400 600 800 1000 1200 1400

Hz

PS

D (

DIS

P^2

/Hz)

Nastran NL

20 Linear Modes

5 Linear + 5 Duals

Example of Application: Fully clamped; no temperature

Acoustic excitaiton 135dB

0 200 400 600 800 1000 1200 1400

Frequency (Hz)

PS

D E

xc

ita

tio

n

ELSTEP Past Validation

15

NL Structural ROM for the Membrane-on-Wedge

• The NL structural ROM for the

flexible panel uses the first 6

transverse modes and 11 dual

modes

• Deformation solutions under

constant pressure agree

excellently with Nastran nonlinear

solutions

0.01

0.1

1

10

100

1 10 100 1000 10000 100000 1000000

Pressure (Pa)

Tra

ns.

Dis

p./

Th

ick

nes

s

NASTRAN

ELSTEP (6T11D)

u lp p p=

lp p=Assuming:

uu l pp p p q C= =

16

NS

level

Burnett

Level(Kn>0)

Time –

Accurate

Aerothermo-

dynamics

Unsteady

Motion

Local

Features

Compu.

Speed

CFL3D -- -- Deforming

mesh

Embedded

Mesh Faster

FUN3D -- Deforming

mesh

Adaptive

Mesh Slowest

BGKX Moving

mesh

Adaptive

Mesh Slower

= yes; -- = not available

High level CFD methods

2D Inviscid methods

• ZPEC (Zona Perturbed Euler Characteristics)

• Piston Theory

ZONA Capability of Hypersonic Flow Solvers

* *

* Under development

• CFL3D

• BGK

17

Rarefied Hypersonics:

Microscopic versus Macroscopic Approaches• Macroscopic approaches (Continuum)

• Flow parameters: Mach no., Reynolds no.

• All continuum methods: Euler, N-S, etc.

• Microscopic approaches (Gaskinetic)

• Flow parameter: Knudsen number

• DSMC (direct simulation Monte Carlo) First-principal

particle collision approach; no governing PDE

• Boltzmann Eqn. (Integro-Differential Eqn.)

• BGK Eqn. (approximation of Boltzmann)

1 2 1 2 u f I f f , f ft x

+ =

g fu f

t x

+ =

18

• Boltzmann/BGK uses distribution function (f) as a

single dependent variable with (7) independent

variables (t, xi, vi)

• Euler/N-S have 5 prime variables (P, U, V, W, ρ)

with (4) independent variables (t, xi)

• Potential flow uses velocity potential function (Φ)

as a single dependent variable with (4)

independent variables (t, xi).

• To recover solution from f and from Φ to prime

variables (P, U, V, W, ρ) requires respectively to

integrate f and to differentiate Φ.

Boltzmann/BGK vs. Classical Eqns.

19

• For BGK equation, the right hand side (RHS) of collision terms is

simplified as one relaxation term between equilibrium state, g, and

instantaneous distribution, f, and is the characteristic relaxation

time:

• For BGKX equation, Xu adopts modern CFD kinetic flux for the left

handside (LHS) terms, for the RHS, Xu replaces the relaxation time

τ by a strained relaxation time τ*, which allows for extended

Knudsen number (Kn) range from 0 towards 1.0, thus covering the

continuum to transient flow regime up to the order of BGKX-Burnett.

Note that tackling this flow regime with DSMC would overburden its

computing cost and with continuum CFD would be pushing its

capability; whereas the BGKX–Burnett is a proper one.

f f g fu

t x

+ =

BGK and BGKX Equations

1Kn

,

20

• BGK eqn. is a higher level one than continuum Euler/N-S eqns.

• BGKX covers wide range of Knudsen number (Kn); it unifies

continuum flow (Kn~0) with transition flow (0<Kn<1.0).

• BGK solver is time-accurate, hence most suitable for unsteady

aerodynamic applications.

• One-step computational procedure for pressure and heat flux

solutions.

• Single gas distribution function, f, simplifies the flux algorithm.

• Consistent and unified procedure to handle equilibrium,

equilibrium and chemically reacting flows.

Merits of the BGKX Method

• ZONA has been supported by AFOSR/STTR on the BGKX Solver

development since 2004. For publications see: Cai, C., Liu, D.D., and Xu,

K., “A One-Dimensional Multi-Temperature Gaskinetic BGK Scheme for

Planar Shock Wave Computation,” AIAA Journal, Vol. 46, No. 5, May 2008.

21

Shock-shock interaction. Surface pressure and heat flux distributions.

Shock-Shock Interaction by BGKX

22

MHD actuator effects on the

shock stand off distance.

MHD actuator effects on heat flux from

the cylinder.

Heat-Rate Reduction by BGKX/MHD

23

BGK simulation results of Cp, heat flux over a spherically headed 15 degree cone.

Ma =10.6, Re =1.1e5, T∞= 85R, R= 1.1 inch, Pr=0.72, Tw =540R.

Surface Cp and Mach number contours. Surface heat flux and pressure contours.

BGKX for Blunted Cone/Cylinders:

Cp and Heat flux along the Surface

24

a) Pressure, Re =1.835 105, b) Pressure, Re=1.835 104

c) Heat Flux, Re =1.835 105, d) Heat Flux, Re =1.835 104

M=16.03, T∞ =124.93 K, Tw=294.4 K

Pressure Heat Flux

b) a) c) d)

BGKX Simulation Results of a

Hypersonic Flow over a Paraboloid

25

NL Aeroelastic Static Deformation

Start

Run CFD Solver

Solve NL Structural

ROM Equation

Converge

End

Initial Configuration

GAF

Deformed Configuration

Yes

No

•CFL3D

•BGK

26

M∞

Rigid ring

Oscillatory shock

Vibrating membrane

Mean shock

x

r

Rigid nose

Reflected wave trains

M∞L

θVibrating membrane

Wedge angle

Oscillatory Shock

Mean Shock

Characteristics

(Mach Waves)

Membrane-on-Wedge

27

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

t(s)

q

Mach= 5, Altitude= 100000 ftq1q2q3q4q5q6q7q8q9q10q11q12q13q14q15q16q17

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-2000

-1500

-1000

-500

0

500

1000

t(s)

GA

F

GAF1GAF2GAF3GAF4GAF5GAF6GAF7GAF8GAF9GAF10GAF11GAF12GAF13GAF14GAF15GAF16GAF17

NL AE Static Def. Solutions for Membrane-on-Wedge

• Comparison of Cp distribution

along the undeformed wedge

surface by BGKX and CFL3D

(Mach = 5)

• Numerical simulation

process (Alt = 100 Kft, Mach

= 5)

xa

ya

Cp

0 0.5 1 1.5 2

0

0.5

1

1.5

2

0

0.05

0.1

0.15

Y

Cp

Cp (BGKX)

(CFL3D)

ya

28

• Nonlinear aeroelastic static deformed

shapes for the flexible membrane at

various altitudes represented in the

structural coordinate system

• Nonlinear aeroelastic static deformed

shapes for the flexible membrane at

various altitudes represented in the

aerodynamic coordinate system

NL AE Static Deformed Shapes for

Membrane-on-Wedge

-0.5 0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

xa

ya

NL Aeroelastic Static Deformed Shape

UndeformedAlt = 0 KftAlt = 20 KftAlt = 50 KftAlt = 70 KftAlt = 90 KftAlt = 100 Kft

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

xs

zs

NL Aeroelastic Static Deformed Shape


29

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

xs

zs

Comparison of Static Deformed Shape

UndeformedZPECCFL3DBGKX

-0.5 0 0.5 1 1.5 2-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

xa

Cp

Comparison of Cp Distribution Alt= 0ft

Undeformed-BGKXZPECCFL3DBGKX

Solution Comparison Using Different Aerodynamic Solver

Mach = 5; Alt = 0 ft

• Comparison of the statically deformed

shapes using various aerodynamic

solvers (represented in the structural

coordinate system)

• Comparison of Cp distributions along

the statically deformed wedge surface

using various aerodynamic solvers

30

xy

0 0.5 1 1.5-0.5

0

0.5

1

1.5

P

7.5

7

6.5

6

5.5

5

4.5

4

3.5

3

2.5

2

1.5

Alt = 0 ft, Mach = 5

• Pressure contour plot on deformed

configuration (Alt = 0 ft; Mach =5)

BGKX Solutions on the Statically Deformed Wedge

• Cp distribution along the deformed

wedge surface (Mach =5)

-0.5 0 0.5 1 1.5 2-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

Cp


31

ROMs

( )q t ( )GAF tAerodynamic System

(CFD Solver)

System Inputs System Outputs

Aerodynamic ROM

32

Unsteady BGK Solver

33

• Various aerodynamic ROM methods:

• Present approach: System Identification technique,

specifically, Auto-Regressive-Moving-Average (ARMA) model.

Aerodynamic ROM Approaches

− POD/ROM-HB: Dowell (1998)

− Volterra: Silva (1993)

− ERA: Kim (2004)

− POD/ROM-Time Domain: Beran (2003)

− ARMA: ZONA* (2008)

− NNet: ZONA

* Z. Wang, et al., “Flutter Analysis with Structural Uncertainty by Using CFD-based

Aerodynamic ROM”, presented in 49th AIAA/ASME/AHS/ASC SDM, 7-10 April, 2008,

Schaumburg, IL

34

• Filter Impulsive Method (FIM) signals are chosen as excitation signals for their

Broader range of frequency with concentration on the frequency of interest.

Symmetric about zero axis.

• A FIM signal is given by:

( ) ( ) ( )2

0 0

0 0

0

sin when

0 when

a t tu t Ae t t t t

t t

=

=

• A staggered sequence FIM input of modal coordinates is employed.

• Each mode uses its own natural

frequency as the ω.

• The lowest order goes first.

u(t)

Freq(Hz)

t(s)

PSD

• Input FIM Signals

• FFT of the Input

Aerodynamic ROM Training Excitations

35

• With the prescribed staggered FIM excitations to the modal coordinates, a

special run of CFD soler is carried out at the specified Mach number. The time

histories of the normalized (nondimensionalized) generalized aerodynamic

forces are recorded. Therefore, the complete set of training data is available.

• ROM are sought to define the relationship between modal coordinates (serving

as the System Inputs, u) and GAFs, (serving as the System Outputs, y).

• If m structural modes are used, there will be m ROMs identified.

• Auto-Regressive-Moving-Average (ARMA) model is used:

( ) ( ) ( )1 1

1a bn n

i j

i j

y t a y t i t j nk= =

= + + b u

where na-nb-nk, the so-called delay order are found by a trial and error

procedure

Aerodynamic ROM: ARMA

36

• Nonlinear aerodynamic ROM is represented by neural network model.The modeled plant output at time t by the neural network would be givenin a concise notation as:

• Using the training data, an optimization procedure is implemented tosearch for the best parameters and by minimizingthe mean square of the error between model output and targeted output orthe generalized mean square error.

( ) ( ) ( )1,1 1 2,1, , W b W ( )2

b

( ) ( ) ( ) ( ) ( ) ( )( ) ( )2 2,1 1,1 1,2 1 2tanh py t a W W U W y b b= = + + +

...

.

.

....

.

.

.

U

yp

n1

(1)

n2

(1)

nS

(1)

a1

(1)

a2

(1)

aS

(1)

n(2)

a(2)

Inputs Input Layer Output Layer

wij

(1,1)

wi

(2,1)

...

wij

(1,2)

bS

(1)

b(2)

b2

(1)

b1

(1)

Two-layer Feed-Forward Neural Network

Aerodynamic ROM: NNet

37

Linearized Equation of Motion around

Statically Deformed Configuration

(1)

NL aM q K q F F+ + =0q q q= +

( ) ( ) ( )

( ) ( ) 0

1 2

0 0 0

1 2

1:

2

1:

2

NL

NL

q

Static K q F q V GAF q

FDynamic M q K q V GAF q

q

+ =

+ + =

38

Aero ROM Training for Undeformed Mean/Wedge Configuration

• Aero ROMs are developed

for the first 6 transverse

modes using CFL3d

• Staggered FIM signals are

shown in the first sub-figure

• ARMA models for the 6

normalized GAF are

obtained by optimization

procedure using the training

data set

• GAFs for the other 11 dual

modes are assumed zero

• Aero ROM predictions agree

well with direct CFL3D

outputs

0 20 40 60 80 100 120 140-5

0

5x 10

-3

q

q1q2q3q4q5q6

0 20 40 60 80 100 120 140-2

0

2x 10

-3

N. G

AF

1 CFL3DROM Sim.

0 20 40 60 80 100 120 140-5

0

5x 10

-3

N. G

AF

2 CFL3DROM Sim.

0 20 40 60 80 100 120 140-0.01

0

0.01

N. G

AF

3 CFL3DROM Sim.

0 20 40 60 80 100 120 140-5

0

5x 10

-3

N. G

AF

4 CFL3DROM Sim.

0 20 40 60 80 100 120 140-5

0

5x 10

-3

N. G

AF

5 CFL3DROM Sim.

0 20 40 60 80 100 120 140-0.01

0

0.01

N. G

AF

6

Nondimensional Time

CFL3DROM Sim.

39

• The first sub-figure is the

time histories of the modal

coordinates providing inputs

to both the aerodynamic

ROMs and the direct CFL3D

solver

• Specifically, only the second

modal coordinate assumes

a sinusoid time history while

others are kept zero.

• Aero ROM predictions agree

well with direct CFL3D

outputs

• The exceptions are for the

fourth and sixth GAFs, but

these two are very small,

two-order smaller the others

Validation of Aero ROMs for Undeformed Mean/Wedge

0 20 40 60 80 100 120 140-5

0

5x 10

-3

q

q1q2q3q4q5q6

0 20 40 60 80 100 120 140-2

0

2x 10

-3

N. G

AF

1

CFL3DROM Sim.

0 20 40 60 80 100 120 140-2

0

2x 10

-4

N. G

AF

2

CFL3DROM Sim.

0 20 40 60 80 100 120 140-5

0

5x 10

-3

N. G

AF

3

CFL3DROM Sim.

0 20 40 60 80 100 120 140-5

0

5x 10

-5

N. G

AF

4

CFL3DROM Sim.

0 20 40 60 80 100 120 140-2

0

2x 10

-3

N. G

AF

5

CFL3DROM Sim.

0 20 40 60 80 100 120 140-5

0

5x 10

-5

N. G

AF

6

Nondimensional Time

CFL3DROM Sim.

40

• Conventional type of flutter

analysis: linear structural

EOM unchanged as altitude

change

• Under our dynamic

simulations, the first modal

coordinate is given a small

initial value; all the other

initial conditions are zeros

• By varying the altitude

(consequently, the free-

stream speed and the

dynamic pressure, i.e., the

match-point methodology),

one explorer the decaying,

near neutral, and diverging

time responses.

ROM-ROM Flutter Analysis: Undeformed Mean/Wedge

0 0.2 0.4 0.6 0.8 1 1.2

-2

0

2

4

x 1031

q

Dynamic Simulation Alt= 0 ft q 1q 2q 3q 4q 5q 6

0 0.2 0.4 0.6 0.8 1 1.2

-5

0

5

x 1015

q

Dynamic Simulation Alt= 20 Kft q 1q 2q 3q 4q 5q 6

0 0.2 0.4 0.6 0.8 1 1.2

-1

0

1

x 105

q


0 0.2 0.4 0.6 0.8 1 1.2-1

0

1

q


0 0.2 0.4 0.6 0.8 1 1.2

-1

0

1

x 10-3

q


0 0.2 0.4 0.6 0.8 1 1.2

-5

0

5

10x 10

-4

t(s)

q


41

Linearized Stiffness of Deformed Mean/Wedge

• Change of the natural frequencies around the

deformed mean wedge configuration at various

altitudes

0 10 20 30 40 50 60 70 80 90 1000

100

200

300

400

500

600

700

800

900

1000

Altitude (Kft)

Fre

q. (H

z)

Mode 1Mode 2Mode 3Mode 4Mode 5Mode 6

42

ROM-ROM Flutter Analysis: Deformed Mean/Wedge

0 0.05 0.1 0.15 0.2 0.25-1

0

1x 10

-3

q

Dynamic Simulation Alt= 0 ft q1q2q3q4q5q6

0 0.05 0.1 0.15 0.2 0.25-1

0

1x 10

-3

q

Dynamic Simulation Alt= 20 Kft q1q2q3q4q5q6

0 0.05 0.1 0.15 0.2 0.25-1

0

1x 10

-3

q


0 0.05 0.1 0.15 0.2 0.25-2

0

2x 10

-3

q


0 0.05 0.1 0.15 0.2 0.25-2

0

2x 10

-3

q


0 0.05 0.1 0.15 0.2 0.25-1

0

1x 10

-3

t(s)

q


43

Conclusions

• Ballute aeroelastic problem requires Gaskinetic (microscopic)

aerodynamics in the rarefied hypersonic flight regime.

– Boltzmann/BGK method (time accurate) is adopted

• Ballute is an inflatable (nonlinear) structure

– Nonlinear structural ROM (ELSTEP) is adopted

• Ballute flutter/LCO computation procedure needs to be expedited

– ZONA’s nonlinear/linear ROM-ROM procedures are adopted.

• Membrane-on-Ballute with Bow-Shock is modeled first by a 2D

membrane-on-wedge with attached shock-- thus the present

study

• For a wedge with a mean deformed membrane, its stiffness

increases with decreasing altitude, thus it becomes dynamically

more stable – contrary to the outcome of undeformed membrane

• Axisymmetric membrane-on-ballute model aeroelastic study is in

progress

nonlinear aeroelastic analysis using rom/rom methodology ... · danny d. liu zhicun wang shuchi...

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