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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
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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
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Aims and Objectives
• Test different MDO architectures
• Influence of fidelity level of structural analysis
• Study computational performance
• Benchmark problem for framework development
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Features of WingOpt
• Types of Optimization Problems– Structural sizing optimization– Aerodynamic shape optimization– Simultaneous aerodynamic and structural
optimization
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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
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Architecture of WingOpt
Optimizer( )f x
)(xh)(xg
xAnalysis
Block
I/P
O/P
I/Pprocessor
MDOControl
O/Pprocessor
INTERFACE
ProblemSetup History
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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
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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
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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-
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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)
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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 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
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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
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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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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
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
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Aerodynamic Geometry
• Planform• Geometric Pre-twist• Camber• Wing t/c
y
x
• constant α' per station
• α'i , i = 1, N
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Aerodynamic Geometry
• Planform• Geometric Pre-twist• Camber• Wing t/c
• formed by two quadratic curves
• h/c, d/c
c
h
d
First curve Second curve
Point of max. camber
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Aerodynamic Geometry
• Planform• Geometric Pre-twist• Camber• Wing t/c
• linear variation in wing box-height
t
stations
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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
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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
yA
A
x
Ahfront 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/loweri = 1, N
AA
tupper
tlower
yA
A
x
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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/lowerj = cap no.; k = 1,2; i = 1, N
A
2
Aupper12
1
yA
A
x
rib
front spar rear sparintermediate spar
spar cap
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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
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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 (σ1 , σ2)– 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
tsi
Wtσ ---VfW(x,y)--MddVstallCLCDiClF
Asjkih'2i h'1hrootr2r1d/ch/cα'iΛλARSX
StructuralAerodynamic
ConstraintObjective Desg. Vars.
Structural Sizing Optimization: Baseline Design
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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
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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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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
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Aerodynamic Analysis
• Panel Method (VLM)
• Generate mesh
• Calculate [AIC]
• Calculate [AIC]-1
• {p}=[AIC]-1{}
• 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
• 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
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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
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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
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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: 1
2 x x y y z z
0
0
, ,
dwu u z
dxdw
v v zdx
w x y w x y
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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
i i i iW C X x Y y
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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)
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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 + 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
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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
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Deformation Interpolation (contd..)
• {w}a = [G]as {w}s
• Panel angle of attack calculated as:
aa x
w
}{
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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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{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
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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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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
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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
2
11( )
N
j j ii
w w
wN
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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
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Divergence Constraint Parameter
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MDO Architectures
Analysis 1
Iterations till convergence
Analysis 2
Iterations till convergence
Multi-Disciplinary Analysis (MDA)
Interface
Optimizer
12y
21y
z hgf ,,
1 2z z 1 2s s
Analysis 1
Iterations till convergence
Analysis 2
Iterations till convergence
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
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Variants of MDF
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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
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MDF - 2
AerodynamicsStructuresaeroloads
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
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MDF - 3
Aerodynamics Structuresaeroloads
To optimizer
From optimizerx
, ,f g h=0 ?
Update panel
Yes
No
displacement (w)
Update root
(w)<?
No
Yes
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AerodynamicsStructuresaeroloads
To optimizer
From optimizer*root,x
*, ,f g h
MDF - AAO
(w)<?
Update panelNo
displacement (w)
Yes
*root
*
design variable
includes h L nW
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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
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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,
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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
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MDO Architectures
Analysis 1
Iterations till convergence
Analysis 2
Iterations till convergence
Multi-Disciplinary Analysis (MDA)
Interface
Optimizer
12y
21y
z hgf ,,
1 2z z 1 2s s
Analysis 1
Iterations till convergence
Analysis 2
Iterations till convergence
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
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Overview
• Aims and objective• WingOpt
– Software architecture– Problem setup– Optimizer– Analysis tool– MDO architecture
• Results• Summary and Future work
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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