WingOpt - 1

1
WingOpt - An MDO Tool for Concurrent Aerodynamic
Shape and Structural Sizing Optimization of
Flexible Aircraft Wings.
Prof. P. M. Mujumdar, Prof. K. Sudhakar H. C.
Ajmera, S. N. Abhyankar, M. Bhatia Dept. of
Aerospace Engineering, IIT Bombay
2
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

3
Aims and Objectives

• Test different MDO architectures
• Influence of fidelity level of structural
  analysis
• Study computational performance
• Benchmark problem for framework development

4
Features of WingOpt

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

5
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

6
Architecture of WingOpt
I/P processor
Optimizer
Problem Setup
History
I/P
Analysis Block
MDO Control
O/P
O/P processor
INTERFACE
7
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 ? 1230 panels
• Optimizer ? FFSQP

8
Test Problem

• Design Variables
• Skin thicknesses - S
• Wing Loading
• Aspect ratio
• Sweep back angle
• t/croot

A
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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

10
Test Problem

• Constraints
• Stress LC 1
• fuel volume LC 1
• MDD LC 1
• Range LC 2
• Take-off distance
• Sectional Cl LC 1

-
S
A
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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)

12
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
6 S A EPM Yes MDF2
7 S A EPM Yes MDF3
8 S A EPM Yes MDF-AAO
9 S A EPM Yes IDF1
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Results
Case Active Constraints Active Constraints Active Constraints Active Constraints Active Constraints Active Constraints Objective nf ng time
Case Stresses Fuel volume Mdd Range Take-off distance Clmax Weight (kg) nf ng time
1 ? - - - - - 696.37
2 ? - - - - - 580.79
3 - ? ? ? 24.08 (20.29)
4 ? ? ? ? - 576.14
5 ? ? ? ? 493.98 176 5651 5768
6 ? ? ? ? 494.14 143 4530 8903
7 ? ? ? ? 495.05 154 4889 9466
8 ? ? ? ? 494.02 301 11805 9203
9 ? ? ? ? 490.78 4943 279499 61654
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Results
Case Skin thickness (mm) Skin thickness (mm) Skin thickness (mm) Skin thickness (mm) Skin thickness (mm) Skin thickness (mm) Wing loading (N/m2) Sweep angle (deg.) t/c ratio Aspect ratio
Case 1 2 3 4 5 6 Wing loading (N/m2) Sweep angle (deg.) t/c ratio Aspect ratio
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
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Summary

• Software for MDO of wing was developed
• Simultaneous structural and aerodynamic
  optimization
• Focused around aeroelasticity
• Handles internal loop instability
• MDO Architectures implemented

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Future Work

• Further Testing of IDF
• Additional constraints
• Buckling
• Aileron control efficiency
• Extension to full AAO

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Thank You
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(No Transcript)
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Problem Formulation

• Aerodynamic Geometry
• Structural Geometry
• Design Variables
• Load Case
• Functions Computed
• Optimization Problem Setup Examples

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Aerodynamic Geometry

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

• single sweep, tapered wing
• divided into stations
• S, AR, ?, ?

y
?
AR b2/S ? citp/croot
citp
croot
Wing stations
b/2
x
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Aerodynamic Geometry

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

• constant a' per station
• a'i , i 1, N

y
x
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Aerodynamic Geometry

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

• formed by two quadratic curves
• h/c, d/c

Point of max. camber
Second curve
First curve
h
d
c
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Aerodynamic Geometry

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

• linear variation in wing box-height

stations
t
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Structural Geometry
Cross-section Box height Skin thickness
Spar/ribs

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

y
Structural load carrying wing-box
Front box
r1 l1/c r2 l2/c
A
A
Mid box
Rear box
l1
l2
c
x
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Structural Geometry
Cross-section Box height Skin thickness
Spar/ribs

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

A
A
hfront
hrear
h'1 hrear / hfront
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Structural Geometry
Cross-section Box height Skin thickness
Spar/ribs

• Constant skin thickness per span
• tsi , where s upper/lower
• i 1, N

tupper
A
A
tlower
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Structural Geometry
Cross-section Box height Skin thickness
Spar/ribs

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

A
rib
Aupper12
A
A
spar cap
x
2
1
rear spar
intermediate spar
front spar
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Design Variables
Aerodynamics
Structures

• 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

Station-wise variables
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Load Case Definition

• Altitude (h)
• Mach number (M)
• g pull (n)
• Aircraft weight (W)
• Engine thrust (T)

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Functions Computed

• Aerodynamics
• Sectional Cl
• Overall CL
• CD
• Take-off distance
• Range
• Drag divergence Mach number
• Structural
• Stresses (s1 , s2)
• Load carrying Structural Weight (Wt)
• Deformation Function (w(x,y))
• Geometric
• Fuel Volume (Vf)

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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.

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Design Case Example 1
Structural
Aerodynamic
tsi
Asjki
h'2i
h'1
hroot
r2
r1
d/c
h/c
a'i
?
?
AR
S
X
Wt
-
-
-
Vf
W(x,y)
-
-
Mdd
Vstall
CL
CDi
Cl
F
s
Structural Sizing Optimization Baseline Design
Constraint
Objective
Desg. Vars.
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Design Case Example 2
Structural
Aerodynamic
AR
Asjki
h'2i
h'1
hroot
r2
r1
d/c
h/c
a'i
?
?
S
X
tsi
Cl
CDi
-
-
-
Vf
W(x,y)
Wt
s
-
-
Mdd
Vstall
CL
F
Simultaneous Aerod. Struc. Optimization
Constraint
Objective
Desg. Vars.
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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

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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

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History

• Keeps track of the design variables which affect
  AIC matrix
• Aerodynamic parameter varies ? calculate AIC
  matrix and its inverse

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Analysis Block Diagram
Aerodynamic mesh, M, Pdyn
Cl
Trim ( L-nW e )
From MDO Control
e
arigidDastr.
Aerodynamic pressure
To MDO Control
Pressure Mapping
Structural deflections
Structural Loads
To MDO Control
Deflection Mapping
Dastr.
stresses
Structural Mesh, Material spec.,
non.aero Loads
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Aerodynamic Analysis

• Panel Method (VLM)
• Generate mesh
• Calculate AIC
• Calculate AIC-1
• pAIC-1a
• Calculate total lift, sectional lift and induced
  drag

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Structures

• Loads
• Aerodynamic pressure loads
• Engine thrust
• Inertia relief
• Self weight (wing weight)
• Engine weight
• Fuel weight

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Inertia Relief
EPM
FEM

• 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

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Aerodynamic Load Transformation
EPM
FEM

• 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

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Deflection Mapping

• EPM ? w(x,y) is Ritz polynomial approx.
• FEM ? w(x,y) is spline interpolation from
  nodal displacements

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Equivalent Plate Method (EPM)

• Energy based method
• Models wing as built up section
• Applies plate equation from CLPT
• Strain energy equation

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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

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Analysis Block (FEM)
Aerodynamic Loads on Quarter Chord points of VLM
Panels
FEM Nodal Co-ordinates
Load Transformation
NASTRAN Interface Code
Loads Transferred on FEM Nodes
Wing Geometry Meshing Parameters
Input file for NASTRAN
(Auto mesh data-deck Generation)
MSC/ NASTRAN
Output file of NASTRAN
(File parsing)
Max Stresses, Displacements, twist and Wing
Structural Mass
Nodal displacements
Displacement Transformation
Panel Angles of Attack
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Need for MSC/NASTRAN Interface Code

• FEM within the optimization cycle
• Batch mode
• Automatic generation
• Mesh
• Input deck for MSC/NASTRAN
• Extracting stresses displacements

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Flowchart of the MSC/NASTRAN Interface Code
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Meshing - 1
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Meshing - 2
Skins CQuad4 shell element
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Meshing - 3
Rib/Spar web CQuad4 shell element
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Meshing 4
Spar/Rib caps CRod element
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Loads and Boundary Condition
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Deformation transformation

• w displacements (know on nodal coordinates)
• w(x,y) a0 axx ayy Sai?i (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

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Deformation Transformation (contd..)

• In matrix notation
• w Ca
• where C represents the co-ordinates where
  
• w is known.
• This gives
• aC-1w
• At any other set of points where w is unknown
  wu
• is given by
• wu CuC-1w
• ie. wu Gw
• where G transformation matrix

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Deformation Interpolation (contd..)

• wa Gas ws
• Panel angle of attack calculated as

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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

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Load Transfer Formulation
Displacement Transformation
ua Gas us
Gas ? Transformation Matrix obtained using
Spline interpolation
Virtual Work Equivalence
?uaT Fa ?usT Fs
?uaT (GasT Fa - Fs) 0
Force Transformation
Fs GasT Fa
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Load Transfer Validation - 1
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Load Transfer Validation - 2
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Load Transfer Validation - 3
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Wing Topology
LE control surfaces
Wing box FEM model
TE control surfaces
Wing span divided into 6 stations
Aerodynamic pressure on the entire planform to be
transferred to the load-carrying structural wing
box
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Loads Transferred From VLM Panels of Entire Wing
Planformto the FEM Nodes of the Wing-box
Planform
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Loads Transferred From VLM Panels of Wing-box
Planformto the FEM Nodes of the Wing-box
Planform
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VLM Elemental Panels and Horseshoe Vortices
for Typical Wing Planform
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VLM Distributed Horseshoe Vortices ? Lifting
Flow Field
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MDO Control

• Tasks
• Carry out aeroelastic iterations
• j iteration number i node number
• N number of node
• while satisfying ? L nW 0

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MDO Control

• Issues
• Handling aeroelastic loop
• Stable/unstable
• Asymptotic/oscillatory behavior
• Ways of satisfying LnW (also aerodynamics and
  structures state equations)
• Ways of handling inter disciplinary coupling
• 1. Six methods of handling MDAO evolved
• 2. Special instability constraint evolved

68
Divergence Constraint Parameter
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MDO Architectures
Multi-Disciplinary Feasible (MDF)
Individual Discipline Feasible (IDF)
All At Once (AAO)
Optimizer
Optimizer
Optimizer
Interface
Interface
Interface
Analysis 1 Iterations till convergence
Analysis 2 Iterations till convergence
Analysis 1 Iterations till convergence
Analysis 2 Iterations till convergence
Evaluator 1 No iterations
Evaluator 2 No iterations
Iterative coupled
Non-iterative Uncoupled
Uncoupled
Multi-Disciplinary Analysis (MDA)
Disciplinary Evaluation
Disciplinary Analysis
1. Optimizer load increases tremendously 2. No
useful results are generated till the end of
optimization 3. Parallel evaluation 4. Evaluation
cost relatively trivial

• 1. Minimum load on optimizer
• 2. 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
70
Variants of MDF
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MDF - 1
From optimizer
To optimizer
Yes
?(w)ltd )?
No
Aerodynamics
displacement (w)
Aerodynamics
Structures
aeroloads
72
MDF - 2
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MDF - 3
From optimizer
To optimizer
e 0 ?
Yes
No
Update aroot
?(w)ltd ?
Yes
No
displacement (w)
Aerodynamics
Structures
aeroloads
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MDF - AAO
From optimizer
To optimizer
?(w)ltd ?
Yes
No
displacement (w)
Aerodynamics
Structures
aeroloads
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IDF - 1
From optimizer
To optimizer
Calculate apanel
Aerodynamics
Calculate ?k ICCs
e 0 ?
Structures
Yes
No
Update
76
IDF - 2
From optimizer
To optimizer
Calculate apanel
Aerodynamics
Calculate ?k,ICCs, e
Structures
77
Analysis v/s Evaluators
3. Calculates
Solving pushed to optimization level
78
MDO Architectures
Multi-Disciplinary Feasible (MDF)
Individual Discipline Feasible (IDF)
All At Once (AAO)
Optimizer
Optimizer
Optimizer
Interface
Interface
Interface
Analysis 1 Iterations till convergence
Analysis 2 Iterations till convergence
Analysis 1 Iterations till convergence
Analysis 2 Iterations till convergence
Evaluator 1 No iterations
Evaluator 2 No iterations
Iterative coupled
Non-iterative Uncoupled
Uncoupled
Multi-Disciplinary Analysis (MDA)
Disciplinary Evaluation
Disciplinary Analysis
1. Optimizer load increases tremendously 2. No
useful results are generated till the end of
optimization 3. Parallel evaluation 4. Evaluation
cost relatively trivial

• 1. Minimum load on optimizer
• 2. 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
79
Overview

• Aims and objective
• WingOpt
• Software architecture
• Problem setup
• Optimizer
• Analysis tool
• MDO architecture
• Results
• Summary and Future work

80
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