5-7-2003 WingOpt - 1 WingOpt - An MDO Research Tool for Concurrent Aerodynamic Shape and Structural...

5-7-2003 WingOpt WingOpt - 1

WingOpt - An MDO Research 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 - Study issues of integrating MDA for formal design optimization

• Aeroelastic optimization as an MDO problem - Concurrent aerodynamic shape and structural sizing optimization of a/c wing

• Realistic MDO problem - Showcase a reasonably complex aircraft design optimization problem with high fidelity analysis


Aims and Objectives

• Study different MDO architectures – reformulations of the optimization problem

• Influence of fidelity level of structural analysis

• Study computational performance

• Benchmark problem for MDO framework development


Design Drivers/Constraints for the WingOpt Architechture

• Definition of a meaningful overall design problem based on available analysis and optimization capability

• Limited disciplines considered: Geometry, Aerodynamics, Structures, Trim/Maneuver

• Aeroelasticity as basis for coupling disciplines

• Software integration within confines of high level programming languages (FORTRAN/C) through students

• At least one discipline taken to its highest fidelity (structures)

• Emulate some elements of a general purpose framework


Variables & Function Database

• Identify array of all variables/functions associated with the system analysis

• Identify all possible candidates for design variables/constraints

• Partition variables database to fixed and design parameters.

• Tag user codes to all variables/functions • Define subset optimization problem through tags• Create location look-up tables for selected subset

variables/constraints


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)

and their variants– Design variable linking– Load Case specification. Variables/design

constraints attached to load cases


Software modules integrated

• Gradient based optimizers– FFSQP; NPSOL (Source codes)

• Aerodynamic Analyses– VLM (source code)– Semiempirical (Raymer/Roskam) (source code)

• Structural Analyses– Equivalent Plate Method (source code)– Finite Element Method (commercial licensed

software (executable))Source code integration with minimal modifications to code through I/O files


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

Sr. no Item

Load Case

1 Structural (VDive)

2 Range (Vlong range cruise)

3 MDD (Vmax.

cruise)

1 Altitude(m) 7620 10668 7620

2 Mach No. .8097 .72864 .8097

3 Load Factor 2.5 1.0 2.5

4Fuel present: Fuel capacity

1.0 1.0 1.0

5Fuel Flow Rate

(kg/hr)2827 2827 2827

6 Pdyn. Factor 1.98 1.0 1.0


Test Problem

Constraints

• Stress – LC 1

• fuel volume

• MDD – LC 3

• Range – LC 2

• Take-off distance

• Sectional Cl – LC 1} Aerodynamic

Structural-

- Geometric


Test Cases

Cases Design Variable and Constraints

Aeroelasticity MDO Methods

1 S No Direct

2 S Yes MDF-1

3 S + A No Indirect

4 S + A No Direct

5 S + A Yes MDF-1

6 S + A Yes MDF-2

7 S + A Yes MDF-3

8 S + A Yes MDF-AAO

9 S + A Yes IDF

10 S + A Yes IDF-AAO


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 5.43 2.84 3.86 2.0 2.0 2.0 5790 31.14 0.20 8.18

4 5.49 2.87 3.88 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

10 8.70 6.99 7.35 4.12 4.11 4.11 5654 27.56 0.159 9.24


Results

Case

Active Constraints

StressesFuel

volumeMdd Range

Take-off distance

ClmaxPseudo

constraintsL=nW

1 - - - - - - -

2 - - - - - - -

3 -

4 - -

5 - -

6 - -

7 - -

8 -

9

10 ☓ ☓ ☓ ☓ ☓ ☓ ☓ ☓


Results

Case

Objective Number of Time (s)

Weight (kg)

Design variables

Cons-traints

Analysis performed

Obj func call

Const func call

AerodyStruc Total

1 696.37 6 24 175 132 3210 25 68 1112 580.79 6 25 83 70 1785 32 41 3633 576.5 13 32 2341 609 21028 12945 868 138794 576.14 10 29 651 191 5695 4335 239 46885 493.98 10 31 644 176 5651 4367 233 57686 494.14 10 31 488 143 4530 3666 4063 89037 495.05 10 31 523 154 4889 3698 4477 94668 494.02 13 34 1135 301 11805 6078 2744 92039 490.78 42 61 14466 4943 279499 50034 8959 61654

10 1131.8 45 64 1033 331 21644 1953 608 2736


Conclusions

• Aeroelasticity analysis leads to significant weight reduction

• Simultaneous structural and aerodynamic optimization significant impact on design

• IDF-AAO failed• MDF1 loop stability not related to physical

divergence• Stability information in IDF and IDF-AAO

cannot be captured


Conclusions

• In MDF1 time taken in aerodynamic very high compared to structures

• MDF1 most efficient, iteration convergence is fastest, however not fully reliable

• MDF2 and MDF-AAO are very robust and took almost same computational time

• Direct method much efficient than indirect method


Conclusions

• Simultaneous optimization are very time consuming

• With non-linearity (more time consuming analysis) IDF and AAO might be more benificial

• Maintaining history saves significant computational time


Summary

• Software for MDO of wing was developed• Simultaneous structural and aerodynamic

optimization• Focused around aeroelasticity• Handles internal loop instability• MDO Architectures formulated and implemented• Methods for accelerating convergence formulated

and implement• Multiple load case implemented• User interface improved


Future Work

• IDF and IDF-AAO for FEM• Additional features

– Buckling– composites– Aileron control efficiency

• Multilevel MDO Architectures• Non linear problem• Parallel computation• High fidelity aerodynamics analysis


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 (VLM)

– Overall CL (VLM)

– CD (VLM + empirical))

– Take-off distance– Range (Brueget)

– Drag divergence Mach number (Semi-empirical)

• 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


Interface Block

• Design Variables un-scaled • Design Variable Superset updated• Design Variable Superset partitioned • Analysis routines called through MDO control• Required function value returned to optimizer

X2

. .

. .

. .

P1

P2

P3

1

X3

Look-up Table

Selected Variables

X1

2

3

4

5

n

.

.

.

Partitioning

LogicTo input

processors


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


FE Model & Load Transfer

Figs. 1-4: Development of Wing model and loadsFigs. 5-6: Load Transformation Process

5 - Aerodynamic Loads and its Response6 - Structurally equivalent Loads and its Response FEM Model


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

• Manages analysis execution sequence control. Strings analysis modules to form MDA

• Manages iterations for coupled interdiciplinary analysis

• Manages coupling variables transfer


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


MDA

Tasks• Carry out aeroelastic iterations

z = tip deformation; j = iteration number;

while satisfying = L – nW = 0

1

1

( ) j j

j

z zz

z


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 .

, , , /reqL jig initial

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

Aerodynamics Structuresaeroloads

To optimizer

From optimizerx

, ,f g h=0 ?

Update panel

Yes

No

displacement (w)

Update root

(w)<?

No

Yes

panel root jig

w

x


MDF - 3


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



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 - AAO

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,



Asymptotic

dcp > 0divergence dcp < 0convergence

h1

h2

1 2dcp h h

h1

h2



Oscillatory

dcp > 0divergence dcp < 0convergence

1 2dcp h h

h1

h2

h1

h2


Slow Convergence


Convergence Accelerated


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

5-7-2003 WingOpt - 1 WingOpt - An MDO Research Tool for Concurrent Aerodynamic Shape and Structural...

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Transcript of 5-7-2003 WingOpt - 1 WingOpt - An MDO Research Tool for Concurrent Aerodynamic Shape and Structural...