Prof. P. M. Mujumdar, Prof. K. Sudhakar H. C. Ajmera, S. N. Abhyankar, M. Bhatia

5-7-2003 WingOpt WingOpt - 1

WingOpt - An MDO Tool for Concurrent Aerodynamic Shape and Structural Sizing Optimization

of Flexible Aircraft Wings.

Prof. P. M. Mujumdar, Prof. K. SudhakarH. C. Ajmera, S. N. Abhyankar, M. Bhatia

Dept. of Aerospace Engineering, IIT Bombay


Aims and Objectives

• Develop a software for MDO of aircraft wing

• Aeroelastic optimization

• Concurrent aerodynamic shape and structural sizing optimization of a/c wing

• Realistic MDO problem


Aims and Objectives

• Test different MDO architectures

• Influence of fidelity level of structural analysis

• Study computational performance

• Benchmark problem for framework development


Features of WingOpt

• Types of Optimization Problems– Structural sizing optimization– Aerodynamic shape optimization– Simultaneous aerodynamic and structural

optimization


Features of WingOpt

• Flexibility– Easy and quick setup of the design problem– Aeroelastic module can be switched ON/OFF– Selection of structural analysis (FEM / EPM)– Selection of Optimizer (FFSQP / NPSOL)– Selection of MDO Architecture (MDF / IDF)– Design variable linking


Architecture of WingOpt

Optimizer( )f x

)(xh)(xg

xAnalysis

Block

I/P

O/P

I/Pprocessor

MDOControl

O/Pprocessor

INTERFACE

ProblemSetup History


Test Problem

• Baseline aircraft Boeing 737-200

• Objective min. load carrying wing-box structural weight

• No. of span-wise stations 6

• No. of intermediate spars (FEM) 2

• Aerodynamic meshing 12*30 panels

• Optimizer FFSQP


Test Problem

Design Variables

• Skin thicknesses - S

• Wing Loading

• Aspect ratio

• Sweep back angle

• t/croot

} A


Test Problem

Load Case 1 (max. speed)• Altitude = 25000 ft• Mach no.= 0.8097 (*1.4)• ‘g’ pull = 2.5

• Aircraft weight = Wto

Load Case 2 (max. range)• Altitude = 35000 ft• Mach no.= 0.7286• ‘g’ pull = 1

• Aircraft weight = Wto


Test Problem

Constraints

• Stress – LC 1

• fuel volume – LC 1

• MDD – LC 1

• Range – LC 2

• Take-off distance

• Sectional Cl – LC 1} A

S-


Test Problem

• Structural Optimization (with and w/o aeroelasticity)

• Aerodynamic Optimization• Simultaneous structural and aerodynamic

optimization without aeroelasticity• Simultaneous structural and aerodynamic

optimization with aeroelasticity (6 MDO architectures)


Test Cases

Cases D.V. & C. S.M. AE MDO

1 S EPM No -

2 S EPM Yes MDF1

3 A EPM No -

4 S + A EPM No -

5 S + A EPM Yes MDF1



8 S + A EPM Yes MDF-AAO

9 S + A EPM Yes IDF1


Results

Case

Active Constraints Objectivenf ng time

StressesFuel

volumeMdd Range

Take-off distance

ClmaxWeight

(kg)1 - - - - - 696.372 - - - - - 580.79

3 - 24.08 (20.29)

4 - 576.145 493.98 176 5651 57686 494.14 143 4530 89037 495.05 154 4889 94668 494.02 301 11805 92039 490.78 4943 279499 61654


Results

Case

Skin thickness (mm) Wing loading (N/m2)

Sweep angle (deg.)

t/c ratio

Aspect ratio1 2 3 4 5 6

1 6.25 3.36 5.03 2.46 2.0 2.0 5643 25 0.16 8.83

2 5.26 2.77 3.84 2.0 2.0 2.0 5643 25 0.16 8.83

3 5.26 2.77 3.84 2.0 2.0 2.0 5995 24.74 0.159 13.0

4 5.49 2.87 3.86 2.03 2.0 2.0 5840 31.33 0.20 8.18

5 4.67 2.42 2.88 2.0 2.0 2.0 5840 31.34 0.20 8.13

6 4.67 2.42 2.89 2.0 2.0 2.0 5840 31.34 0.20 8.13

7 4.66 2.41 2.91 2.0 2.0 2.0 5840 31.34 0.20 8.13

8 4.67 2.42 2.89 2.0 2.0 2.0 5840 31.34 0.20 8.13

9 4.66 2.37 2.79 2.0 2.0 2.0 5818 31.27 0.20 8.14


Summary

• Software for MDO of wing was developed

• Simultaneous structural and aerodynamic optimization

• Focused around aeroelasticity

• Handles internal loop instability

• MDO Architectures implemented


Future Work

• Further Testing of IDF

• Additional constraints– Buckling– Aileron control efficiency

• Extension to full AAO


Thank You


Problem Formulation

• Aerodynamic Geometry

• Structural Geometry

• Design Variables

• Load Case

• Functions Computed

• Optimization Problem Setup Examples


Aerodynamic Geometry

• Planform• Geometric Pre-twist• Camber• Wing t/c

y

x

• single sweep, tapered wing

• divided into stations

• S, AR, λ, Λ

citp

b/2

Λ

croot

AR = b2/S

λ = citp/croot

Wing stations




y

x

• constant α' per station

• α'i , i = 1, N




• formed by two quadratic curves

• h/c, d/c

c

h

d

First curve Second curve

Point of max. camber




• linear variation in wing box-height

t

stations


Structural Geometry

Cross-section Box height Skin thickness Spar/ribs

yA

A

A

x

A

• symmetric • front, mid & rear boxes• r1, r2

r1 = l1/cr2 = l2/c

l1

c

l2

Front box

Mid box

Rear box

Structural load carrying wing-box


Structural Geometry


• linear variation in spanwise & chordwise direction• hroot , h'1i , h'2i ; where i = 1, N

A

yA

A

x

Ahfront hrear

h'1 = hrear / hfront


Structural Geometry


• Constant skin thickness per span• tsi , where s = upper/loweri = 1, N

AA

tupper

tlower

yA

A

x


Structural Geometry


• modeled as caps• linear area variation along length• Asjki , where s = upper/lowerj = cap no.; k = 1,2; i = 1, N

A

2

Aupper12

1

yA

A

x

rib

front spar rear sparintermediate spar

spar cap


Design Variables

• Wing loading• Sweep• Aspect ratio• Taper ratio

• t/croot

• Mach number• Jig twist*• Camber*

• Skin thickness*• Rib/spar position*• Rib/spar cap area*• t/c variation*• wing-box chord-wise

size and position

Aerodynamics Structures

* Station-wise variables


Load Case Definition

• Altitude (h)

• Mach number (M)

• ‘g’ pull (n)

• Aircraft weight (W)

• Engine thrust (T)


Functions Computed

• Aerodynamics– Sectional Cl

– Overall CL

– CD

– Take-off distance– Range– Drag divergence Mach number

• Structural– Stresses (σ1 , σ2)– Load carrying Structural Weight (Wt)– Deformation Function (w(x,y))

• Geometric– Fuel Volume (Vf)


Optimization Problem Set Up

• Select objective function• Select design variables and set its bound• Set values of remaining variables (constant)• Define load cases• Set Initial Guess• Select constraints and corresponding load case• Select optimizer, method for structural analysis,

aeroelasticity on/off, MDO method.


Design Case – Example 1

tsi

Wtσ ---VfW(x,y)--MddVstallCLCDiClF

Asjkih'2i h'1hrootr2r1d/ch/cα'iΛλARSX

StructuralAerodynamic

ConstraintObjective Desg. Vars.

Structural Sizing Optimization: Baseline Design


Design Case – Example 2

Cl CDi

AR

---VfW(x,y)Wtσ--MddVstallCLF

Asjkitsih'2i h'1hrootr2r1d/ch/cα'iΛλSX

StructuralAerodynamic

ConstraintObjective Desg. Vars.

Simultaneous Aerod. & Struc. Optimization


Optimizers

FFSQP• Feasible Fortran

Sequential Quadratic Programming

• Converts equality constraint to equivalent inequality constraints

• Get feasible solution first and then optimal solution remaining in feasible domain

NPSOL• Based on sequential

quadratic programming algorithm

• Converts inequality constraints to equality constraints using additional Lagrange variables

• Solves a higher dimensional optimization problem


History

• Why ?– All constraints are evaluated at first analysis

– Optimizer calls analysis for each constraints

– !! Lot of redundant calculations !!

• HISTORY BLOCK– Keeps tracks of all the design point

– Maintains records of all constraints at each design point

– Analysis is called only if design point is not in history database


History

• Keeps track of the design variables which affect AIC matrix

• Aerodynamic parameter varies calculate AIC matrix and its inverse


VLM

EPM/

FEM{α}str. stresses

Aerodynamic mesh, M, Pdyn

Aerodynamic pressure

Structural deflections

Cl

Structural Loads Deflection Mapping

Structural Mesh, Material spec.,

Pressure Mapping

Analysis Block Diagram

non.–aero Loads

To MDO Control

{α}rigid+{α}str.

Trim ( L-nW = From MDO Control

To MDO Control


Aerodynamic Analysis

• Panel Method (VLM)

• Generate mesh

• Calculate [AIC]

• Calculate [AIC]-1

• {p}=[AIC]-1{}

• Calculate total lift, sectional lift and induced drag


Structures

• Loads– Aerodynamic pressure loads– Engine thrust– Inertia relief

• Self weight (wing – weight)

• Engine weight

• Fuel weight


Inertia Relief

• Self-weight calculated using an in-built module in EPM

• Engine weight is given as a single point load

• Fuel weight is given as pressure loads

• Self-weight is calculated internally as loads by MSC/NASTRAN

• Engine weight is given as equivalent downward nodal loads and moments on the bottom nodes of a rib

• Fuel weight is given as pressure loads on top surface of elements of bottom skin

EPM FEM


Aerodynamic Load Transformation

• Transfer of panel pressures of entire wing planform to the mid-box as pressure loads as a coefficients of polynomial fit of the pressure loads

• Transfer of panel pressures on LE and TE surfaces as equivalent point loads and moments on the LE and TE spars

• Transfer of panel pressures on the mid-box as nodal loads on the FEM mesh using virtual work equivalence

EPM FEM


Deflection Mapping

• EPM w(x,y) is Ritz polynomial approx.

• FEM w(x,y) is spline interpolation from nodal displacements

, , 1, 2,.., no. of panels,

, panel collocation point

i ii

i i

w x yi

xx y


Equivalent Plate Method (EPM)

• Energy based method

• Models wing as built up section

• Applies plate equation from CLPT

• Strain energy equation: 1

2 x x y y z z

0

0

, ,

dwu u z

dxdw

v v zdx

w x y w x y


Equivalent Plate Method (EPM)

• Polynomial representation of geometric parameters• Ritz approach to obtain displacement function

• Boundary condition applied by appropriate choice of displacement function

• Merit over FEM– Reduction in volume of input data– Reduction in time for model preparation– Computationally light

i i i iW C X x Y y


Analysis Block (FEM)

NASTRAN Interface

CodeWing Geometry

Meshing Parameters

Load Transformation

Input file for NASTRAN

Output file of NASTRAN

MSC/ NASTRAN

Loads Transferred on FEM Nodes

FEM Nodal Co-ordinates

Aerodynamic Loads on Quarter Chord points of

VLM Panels

Max Stresses, Displacements, twist and Wing Structural Mass Nodal displacements

Panel Angles of Attack

DisplacementTransformation

(File parsing)

(Auto mesh & data-deck

Generation)


Need for MSC/NASTRAN Interface Code

• FEM within the optimization cycle

• Batch mode

• Automatic generation– Mesh– Input deck for MSC/NASTRAN

• Extracting stresses & displacements


Flowchart of the MSC/NASTRAN

Interface Code


Meshing - 1


Meshing - 2

Skins – CQuad4 shell element


Meshing - 3

Rib/Spar web – CQuad4 shell element


Meshing – 4

Spar/Rib caps – CRod element


Loads and Boundary Condition


Deformation transformation

• w = displacements (know on nodal coordinates)

• w(x,y) = a0 + axx + ayy + aii (Interpolation function)

– where ai is interpolation coefficient

i(x,y) are interpolation functions

are displacement function solution of the equation

for a point force on infinite plate

• ai are calculated using least square error method

4D w q


Deformation Transformation (contd..)

• In matrix notation {w} = [C]{a} where [C] represents the co-ordinates where w is known.• This gives {a}=[C]-1{w}• At any other set of points where w is unknown {w}u

is given by

{w}u = [C]u[C]-1{w}• ie. {w}u = [G]{w} where [G] = transformation matrix


Deformation Interpolation (contd..)

• {w}a = [G]as {w}s

• Panel angle of attack calculated as:

aa x

w

}{


Load Transfer Method

• Transformation based on the requirement that– Work done by Aerodynamic forces on quarter chord

points of VLM panels

=

Work done by transformed forces on FEM nodes


{ua} = [Gas] {us}

{ua}T {Fa}= {us}T {Fs}

{ua}T ([Gas]T {Fa} - {Fs}) = 0

{Fs} = [Gas]T {Fa}

Load Transfer Formulation

Displacement Transformation

Virtual Work Equivalence

Force Transformation

[Gas] Transformation Matrix obtained using

Spline interpolation


Load Transfer Validation - 1


LE control surfaces

TE control surfaces

Wing box FEM model

Wing span divided into 6 stations

Wing Topology

Aerodynamic pressure on the entire planform to be transferred to the load-carrying structural wing box


Loads Transferred From VLM Panels of Entire Wing Planformto the FEM Nodes of the Wing-box Planform


Loads Transferred From VLM Panels of Wing-box Planformto the FEM Nodes of the Wing-box Planform


VLM – Elemental Panels and Horseshoe Vortices for Typical Wing Planform


VLM – Distributed Horseshoe Vortices Lifting Flow Field


MDO Control

Tasks• Carry out aeroelastic iterations

j = iteration number; i = node number;

N = number of node

while satisfying = L – nW = 0

2

11( )

N

j j ii

w w

wN


MDO Control

Issues• Handling aeroelastic loop

– Stable/unstable

– Asymptotic/oscillatory behavior

• Ways of satisfying L=nW (also aerodynamics and structures state equations)

• Ways of handling inter disciplinary coupling

1. Six methods of handling MDAO evolved

2. Special instability constraint evolved


Divergence Constraint Parameter


MDO Architectures

Analysis 1

Iterations till convergence

Analysis 2


Multi-Disciplinary Analysis (MDA)

Interface

Optimizer

12y

21y

z hgf ,,

1 2z z 1 2s s

Analysis 1


Analysis 2


Disciplinary Analysis

Interface

Optimizeryz , yhgf ,,,

121, yz 212 , yz 211, ys 122 , ys

Evaluator 1

No iterations

Evaluator 2

No iterations

Disciplinary Evaluation

Interface

Optimizerysz ,, ryhgf ,,,,

111 ,, ysz 222 ,, ysz1r 2r

Individual Discipline Feasible (IDF)

All At Once (AAO)

1. Minimum load on optimizer2. Complete interdisciplinary

consistency is assured at each optimization call

3. Each MDA i Computationally expensive ii Sequential

1. Complete interdisciplinary consistency is assured only at successful termination of optimization

2. Intermediate between MDF and AAO

3. Analysis in parallel

1. Optimizer load increases tremendously

2. No useful results are generated till the end of optimization

3. Parallel evaluation4. Evaluation cost relatively

trivial

Iterative; coupled

)0( r)0( r

Multi-Disciplinary Feasible (MDF)

Uncoupled Non-iterative; Uncoupled


Variants of MDF


MDF - 1

AerodynamicsStructuresaeroloads

To optimizer From optimizer .

, , , /req

jigLinitial

x C w x , ,f g h

{(w)<)}?

Update root

Update panel

Yes

No

displacement (w)

,. riridreq elasticroot L L L

panel root jig

C C C

w

x

Aerodynamics

0

Update root

panel

panel jig

w

x

elasticLC


MDF - 2


To optimizer

From optimizer.

, , req

jigLx C

, ,f g h{(= 0 ) and (w)<)}?

Update root

Update panel

Yes

No

displacement (w)

0Compute

elasticLC

w,

xinitialrootinitial

0.

1

0

1

req elastic

i i

i elastic

i

L L

root root

L L

panel root

i

C C

C C

w

x


MDF - 3

Aerodynamics Structuresaeroloads

To optimizer

From optimizerx

, ,f g h=0 ?

Update panel

Yes

No

displacement (w)

Update root

(w)<?

No

Yes



To optimizer

From optimizer*root,x

*, ,f g h

MDF - AAO

(w)<?

Update panelNo

displacement (w)

Yes

*root

*

design variable

includes h L nW


IDF - 1

Aerodynamics

Structures

To optimizer

From optimizer*,x

*, ,f g h

Update rootNo

Yes

*

*

pseudo design variables

includes ICCsh

= 0 ?

1

* *

1

*

*

( , ) ( , )

( , ) ( , )

,

ICCs :

m

k kk

m

k kk

i ii

i

k k

w x y x y

w x y x y

w x y

x

Calculate {panel

Calculate & ICCs


IDF - 2

Aerodynamics

Structures

To optimizer

From optimizer*

root, ,x

*, ,f g h

*

*

pseudo design variables

includes ICCs and 0h

1

* *

1

*

*

( , ) ( , )

( , ) ( , )

,

ICCs :

m

k kk

m

k kk

i ii

i

k k

w x y x y

w x y x y

w x y

x

Calculate {panel

Calculate k,ICCs,


p

Analysis v/s Evaluators

*Solving pushed to optimization level

Conventional approach:

INTERFACE

Solve

z

z p

hgf ,,

design variables

pressure load

objective function

nequality constraints

equality constraints

z

p

f

g

h

OPTIMIZER

0p AIC 2. Calculates

1AIC

3. Calculates

1p AIC

Evaluator:Does not solve Evaluates residues for given Computationally inexpensive

, z pOPTIMIZER

INTERFACE

, z p rhgf ,,,

EVALUATOR

, z p r

, design variables

residue

objective function

equality constraints

, equality constraints

z p

r

f

g

h r

A different approach*:

r p AIC

Analysis:Conservation laws of systemIf nonlinear, iterativeMultidisciplinaryTime intensive

1. Generates AIC

z p

2. Calculates r p AIC

r

0r


MDO Architectures

Analysis 1


Analysis 2


Multi-Disciplinary Analysis (MDA)

Interface

Optimizer

12y

21y

z hgf ,,

1 2z z 1 2s s

Analysis 1


Analysis 2


Disciplinary Analysis

Interface

Optimizeryz , yhgf ,,,

121, yz 212 , yz 211, ys 122 , ys

Evaluator 1

No iterations

Evaluator 2

No iterations

Disciplinary Evaluation

Interface

Optimizerysz ,, ryhgf ,,,,

111 ,, ysz 222 ,, ysz1r 2r

Individual Discipline Feasible (IDF)

All At Once (AAO)

1. Minimum load on optimizer2. Complete interdisciplinary

consistency is assured at each optimization call

3. Each MDA i Computationally expensive ii Sequential

1. Complete interdisciplinary consistency is assured only at successful termination of optimization

2. Intermediate between MDF and AAO

3. Analysis in parallel

1. Optimizer load increases tremendously

2. No useful results are generated till the end of optimization

3. Parallel evaluation4. Evaluation cost relatively

trivial

Iterative; coupled

)0( r)0( r

Multi-Disciplinary Feasible (MDF)

Uncoupled Non-iterative; Uncoupled


Overview

• Aims and objective• WingOpt

– Software architecture– Problem setup– Optimizer– Analysis tool– MDO architecture

• Results• Summary and Future work


Inference

• History block reduces computational time to 1/10th

• FEM requires substantially more time than EPM• dcp constraint fails in some cases to give optimum

results whenever aeroelastic iterations are oscillatory

• MDF-1 fails occasionally without dcp constraint• MDF -3 fails to find feasible solution• More robust method for load transfer is required

Prof. P. M. Mujumdar, Prof. K. Sudhakar H. C. Ajmera, S. N. Abhyankar, M. Bhatia

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Transcript of Prof. P. M. Mujumdar, Prof. K. Sudhakar H. C. Ajmera, S. N. Abhyankar, M. Bhatia