WingOpt 1

1
WingOpt - An MDO Research 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 -
  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

3
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

4
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

5
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

6
Features of WingOpt

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

7
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

8
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

9
Architecture of WingOpt
I/P processor
Optimizer
Problem Setup
History
I/P
Analysis Block
MDO Control
O/P
O/P processor
INTERFACE
10
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

11
Test Problem

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

A
12
Test Problem
13
Test Problem

• Constraints
• Stress LC 1
• fuel volume
• MDD LC 3
• Range LC 2
• Take-off distance
• Sectional Cl LC 1

-
Structural
-
Geometric
Aerodynamic
14
Test Cases
15
Results
16
Results
17
Results
18
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

19
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

20
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

21
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

22
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

23
Problem Formulation

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

24
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
25
Aerodynamic Geometry

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

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

y
x
26
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
27
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
32
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
33
Load Case Definition

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

34
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 (s1 , s2)
• Load carrying Structural Weight (Wt)
• Deformation Function (w(x,y))
• Geometric
• Fuel Volume (Vf)

35
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

39
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

40
History

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

41
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

42
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
43
Aerodynamic Analysis

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

44
Structures

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

45
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

46
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

47
Deflection Mapping

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

48
Equivalent Plate Method (EPM)

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

49
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

50
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
51
Need for MSC/NASTRAN Interface Code

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

52
Flowchart of the MSC/NASTRAN Interface Code
53
Meshing - 1
54
Meshing - 2
Skins CQuad4 shell element
55
Meshing - 3
Rib/Spar web CQuad4 shell element
56
Meshing 4
Spar/Rib caps CRod element
57
Loads and Boundary Condition
58
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

59
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

60
Deformation Interpolation (contd..)

• wa Gas ws
• Panel angle of attack calculated as

61
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

62
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
63
Load Transfer Validation - 1
64
Load Transfer Validation - 2
65
Load Transfer Validation - 3
66
FE Model Load Transfer
Figs. 1-4 Development of Wing model and
loads Figs. 5-6 Load Transformation Process 5
- Aerodynamic Loads and its Response 6 -
Structurally equivalent Loads and its Response
FEM Model
67
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
68
Loads Transferred From VLM Panels of Entire Wing
Planformto the FEM Nodes of the Wing-box
Planform
69
Loads Transferred From VLM Panels of Wing-box
Planformto the FEM Nodes of the Wing-box
Planform
70
VLM Elemental Panels and Horseshoe Vortices
for Typical Wing Planform
71
VLM Distributed Horseshoe Vortices ? Lifting
Flow Field
72
MDO Control

• Manages analysis execution sequence control.
  Strings analysis modules to form MDA
• Manages iterations for coupled interdiciplinary
  analysis
• Manages coupling variables transfer

73
MDO Control

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

74
MDA

• Tasks
• Carry out aeroelastic iterations
• z tip deformation j iteration number
• while satisfying ? L nW 0

75
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

76
Divergence Constraint Parameter
77
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
78
Variants of MDF
79
MDF - 1
From optimizer
To optimizer
Yes
?(w)ltd )?
No
Aerodynamics
displacement (w)
Aerodynamics
Structures
aeroloads
80
MDF - 2
From optimizer
To optimizer
e 0 ?
Yes
No
Update aroot
?(w)ltd ?
Yes
No
displacement (w)
Aerodynamics
Structures
aeroloads
81
MDF - 3
82
MDF - AAO
From optimizer
To optimizer
?(w)ltd ?
Yes
No
displacement (w)
Aerodynamics
Structures
aeroloads
83
IDF - 1
From optimizer
To optimizer
Calculate apanel
Aerodynamics
Calculate ?k ICCs
e 0 ?
Structures
Yes
No
Update
84
IDF - AAO
From optimizer
To optimizer
Calculate apanel
Aerodynamics
Calculate ?k,ICCs, e
Structures
85
Divergence Constraint Parameter

• Asymptotic

h1
h1
h2
h2
dcp gt 0?divergence
dcp lt 0?convergence
86
Divergence Constraint Parameter

• Oscillatory

h2
h1
h1
h2
dcp gt 0?divergence
dcp lt 0?convergence
87
Slow Convergence
88
Convergence Accelerated
89
Analysis v/s Evaluators
3. Calculates
Solving pushed to optimization level
90
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
91
Overview

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

92
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