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
5-7-2003
WingOpt - 1
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
5-7-2003
WingOpt - 2
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
5-7-2003
WingOpt - 3
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
5-7-2003
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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
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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)
and their variants
– Design variable linking
– Load Case specification. Variables/design
constraints attached to load cases
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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
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Architecture of WingOpt
I/P
processor
x
Problem
Setup
Optimizer
History
MDO
Control
I/P
O/P
Analysis
Block
f (x )
h (x)
g (x)
O/P
processor
INTERFACE
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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
A
• Sweep back angle
• t/croot
}
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Test Problem
Load Case
Sr. no
5-7-2003
Item
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
4
Fuel present:
Fuel capacity
1.0
1.0
1.0
5
Fuel Flow Rate
(kg/hr)
2827
2827
2827
6
Pdyn. Factor
1.98
1.0
1.0
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Test Problem
Constraints
• Stress – LC 1
• fuel volume
• MDD – LC 3
• Range – LC 2
• Take-off distance
• Sectional Cl – LC 1
5-7-2003
- Structural
- Geometric
}
Aerodynamic
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Test Cases
5-7-2003
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
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Results
Skin thickness (mm)
Case
1
2
3
4
5
6
Wing Sweep t/c Aspec
loading angle ratio t ratio
(N/m2) (deg.)
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
5-7-2003
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Results
Active Constraints
Case Stresses
Fuel
volume
Mdd Range
Take-off
distance
Clmax
Pseudo
constraints
L=nW
1

-
-
-
-
-
-
-
2

-
-
-
-
-
-
-
3




-

4




-
-
5




-
-
6




-
-
7




-
-
8




-

9






10
☓
☓
☓
☓
☓
☓
5-7-2003
☓
☓
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Results
Objective
Number of
Case Weight Design Cons- Analysis Obj
func
(kg)
variables traints performed
call
1
696.37
6
24
175
132
2
580.79
6
25
83
70
3
576.5
13
32
2341
609
4
576.14
10
29
651
191
5
493.98
10
31
644
176
6
494.14
10
31
488
143
7
495.05
10
31
523
154
8
494.02
13
34
1135
301
9
490.78
42
61
14466
4943
10
1131.8
45
64
1033
331
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Time (s)
Const
func Aerody Struc Total
call
3210
25
68 111
1785
32
41 363
21028 12945 868 13879
5695 4335 239 4688
5651 4367 233 5768
4530 3666 4063 8903
4889 3698 4477 9466
11805 6078 2744 9203
279499 50034 8959 61654
21644 1953 608 2736
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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
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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
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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
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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 formulated and implemented
• Methods for accelerating convergence formulated
and implement
• Multiple load case implemented
• User interface improved
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Future Work
• IDF and IDF-AAO for FEM
• Additional features
– Buckling
– composites
– Aileron control efficiency
•
•
•
•
5-7-2003
Multilevel MDO Architectures
Non linear problem
Parallel computation
High fidelity aerodynamics analysis
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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, λ, Λ
y
Λ
croot
citp
AR = b2/S
λ = citp/croot
b/2
Wing stations
x
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WingOpt - 24
Aerodynamic Geometry
• Planform
• Geometric Pre-twist
• Camber
• Wing t/c
• constant α' per station
• α'i , i = 1, N
y
x
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WingOpt - 25
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
5-7-2003
c
WingOpt - 26
Aerodynamic Geometry
• Planform
• Geometric Pre-twist
• Camber
• Wing t/c
• linear variation in wing
box-height
stations
t
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WingOpt - 27
Structural Geometry
Cross-section
Box height
Skin thickness
Spar/ribs
A
• symmetric
• front, mid & rear boxes
• r1, r2
y
Structural load carrying wing-box
Front box
A
Mid box
Rear box
A
x
5-7-2003
A
l1
r1 = l1/c
r2 = l2/c
l2
c
WingOpt - 28
Structural Geometry
Cross-section
Box height
Skin thickness
Spar/ribs
A
• linear variation in spanwise &
chordwise direction
• hroot , h'1i , h'2i ; where i = 1, N
y
hfront A
A
hrear
A
h'1 = hrear / hfront
x
5-7-2003
WingOpt - 29
Structural Geometry
Cross-section
Box height
Skin thickness
Spar/ribs
A
• Constant skin thickness
per span
• tsi , where s = upper/lower
i = 1, N
y
tupper
A
A
A
tlower
x
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Structural Geometry
• modeled as caps
• linear area variation along
length
• Asjki , where s = upper/lower
j = cap no.; k = 1,2; i = 1, N
Cross-section
Box height
Skin thickness
Spar/ribs
A
y
rib
A
A
x
1
spar cap
2
front spar
5-7-2003
Aupper12
intermediate spar
rear spar
WingOpt - 31
Design Variables
Aerodynamics
• Wing loading
• Sweep
• Aspect ratio
• Taper ratio
• t/croot
• Mach number
• Jig twist*
• Camber*
•
•
•
•
•
Structures
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
•
•
•
•
•
5-7-2003
Altitude (h)
Mach number (M)
‘g’ pull (n)
Aircraft weight (W)
Engine thrust (T)
WingOpt - 33
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)
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Optimization Problem Set Up
•
•
•
•
•
•
•
5-7-2003
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.
WingOpt - 35
Design Case – Example 1
Aerodynamic
X
S
AR
λ
F
Cl
CDi
CL
Λ
α'i
Vstall Mdd
Structural
h/c
d/c
r1
r2
hroot
h'1
h'2i
tsi
Asjki
-
-
σ
Wt
W(x,y)
Vf
-
-
-
Structural Sizing Optimization: Baseline Design
Objective
5-7-2003
Desg. Vars.
Constraint
WingOpt - 36
Design Case – Example 2
Aerodynamic
X
S
AR
λ
Λ
α'i
F
Cl
CDi
CL Vstall Mdd
Structural
h/c
d/c
r1
r2
hroot
h'1
h'2i
tsi
Asjki
-
-
σ
Wt
W(x,y)
Vf
-
-
-
Simultaneous Aerod. & Struc. Optimization
Objective
5-7-2003
Desg. Vars.
Constraint
WingOpt - 37
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
5-7-2003
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
5-7-2003
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History
• Keeps track of the design variables which
affect AIC matrix
• Aerodynamic parameter varies  calculate
AIC matrix and its inverse
5-7-2003
WingOpt - 40
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
1
Look-up Table
X1
P1
3
X2
P2
4
X3
.
P3
.
5
.
.
.
.
.
.
.
Selected
Variables
5-7-2003
2
Partitioning
Logic
To input
processors
n
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Analysis Block Diagram
Aerodynamic mesh,
M, Pdyn
Cl
From MDO Control
Trim ( L-nW = e )
VLM
{α}rigid+{Dα}str.
Aerodynamic
pressure
Pressure
Mapping
To MDO Control Deflection
Mapping
{Dα}str.
Structural
deflections
EPM/
FEM
e
To MDO
Control
Structural Loads
stresses
Structural Mesh,
Material spec.,
non.–aero Loads
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Aerodynamic Analysis
•
•
•
•
•
•
5-7-2003
Panel Method (VLM)
Generate mesh
Calculate [AIC]
Calculate [AIC]-1
{p}=[AIC]-1{a}
Calculate total lift, sectional lift and induced
drag
WingOpt - 43
Structures
• Loads
– Aerodynamic pressure loads
– Engine thrust
– Inertia relief
• Self weight (wing – weight)
• Engine weight
• Fuel weight
5-7-2003
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Inertia Relief
EPM
• 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
5-7-2003
FEM
• 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
WingOpt - 45
Aerodynamic Load Transformation
EPM
• 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
5-7-2003
FEM
• 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
WingOpt - 46
Deflection Mapping
ai 
 w  xi , yi )
x
, i  1, 2, .., n o . o f p an els,
 xi , yi ) 
p an el co llo ca tio n p o in t
• EPM  w(x,y) is Ritz polynomial approx.
• FEM  w(x,y) is spline interpolation from
nodal displacements
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WingOpt - 47
Equivalent Plate Method (EPM)
• Energy based method
• Models wing as built up section
• Applies plate equation from CLPT

0
dw

0
dw
u  u
v  v
w
 x,
y
)
 w
z
dx
z
dx
 x,
y
)
• Strain energy equation:
5-7-2003
 
1


2
x
e x   ye y   ze z )
WingOpt - 48
Equivalent Plate Method (EPM)
• Polynomial representation of geometric parameters
• Ritz approach to obtain displacement function
Wi  Ci X i
 x ) Yi  y )
• 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
Wing Geometry
Meshing
Parameters
NASTRAN
Interface
Code
(Auto mesh &
data-deck
Generation)
Max Stresses,
Displacements,
twist and Wing
Structural Mass
(File parsing)
FEM Nodal Coordinates
Loads Transferred
on FEM Nodes
Input file for
NASTRAN
MSC/
NASTRAN
Output file of
NASTRAN
Nodal displacements
Panel Angles of
Attack
5-7-2003
Load
Transformation
Displacement
Transformation
WingOpt - 50
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
5-7-2003
WingOpt - 52
Meshing - 1
5-7-2003
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Meshing - 2
Skins – CQuad4 shell element
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Meshing - 3
Rib/Spar web – CQuad4 shell element
5-7-2003
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Meshing – 4
Spar/Rib caps – CRod element
5-7-2003
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Loads and Boundary Condition
5-7-2003
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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
D w  q
for a point force on infinite plate
• ai are calculated using least square error method
4
5-7-2003
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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
5-7-2003
WingOpt - 59
Deformation Interpolation (contd..)
• {w}a = [G]as {w}s
• Panel angle of attack calculated as:
 w 
{a } a   

 x  a
5-7-2003
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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
{ua}T {Fa}= {us}T {Fs}
{ua}T ([Gas]T {Fa} - {Fs}) = 0
Force Transformation
{Fs} = [Gas]T {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
5-7-2003
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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
5-7-2003
FEM Model
WingOpt - 66
Wing Topology
LE control
surfaces
Wing box
FEM
model
TE control
surfaces
Wing span divided into 6 stations
5-7-2003
Aerodynamic pressure on the entire planform to be
transferred to the load-carrying structural wing box
WingOpt - 67
Loads Transferred From VLM Panels of Entire Wing Planform
to the FEM Nodes of the Wing-box Planform
5-7-2003
WingOpt - 68
Loads Transferred From VLM Panels of Wing-box Planform
to the FEM Nodes of the Wing-box Planform
5-7-2003
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VLM – Elemental Panels and Horseshoe Vortices
for Typical Wing Planform
5-7-2003
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VLM – Distributed Horseshoe Vortices
 Lifting Flow Field
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WingOpt - 71
MDO Control
• Manages analysis execution sequence control.
Strings analysis modules to form MDA
• Manages iterations for coupled interdiciplinary
analysis
• Manages coupling variables transfer
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MDO Control
Tasks
• Carry out aeroelastic iterations
N
 w
 (w) 
j
 w j 1 )
i 1
2
i
N
j = iteration number; i = node number;
N = number of node
while satisfying e = L – nW = 0
5-7-2003
WingOpt - 73
MDA
Tasks
• Carry out aeroelastic iterations
(z) 
z j  z j 1
z j 1
z = tip deformation; j = iteration number;
while satisfying e = L – nW = 0
5-7-2003
WingOpt - 74
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
5-7-2003
WingOpt - 75
Divergence Constraint Parameter
5-7-2003
WingOpt - 76
MDO Architectures
Multi-Disciplinary Feasible
(MDF)
Individual Discipline Feasible
(IDF)
Optimizer
z
z, y
Interface
Analysis 1
z, s, y
f , g,h, y
Interface
s1  s 2
y 21
Optimizer
Optimizer
f , g,h
z1  z 2
All At Once (AAO)
Analysis 2
Iterations till y Iterations till
12
convergence
convergence
Iterative; coupled

z 1 , y 12
s 1 , y 21
s 2 , y 12
f , g , h , y, r
Interface

z 2 , y 21
Analysis 1
Analysis 2
Iterations till
convergence
Iterations till
convergence
Uncoupled
z 1 , s1 , y 1
r1
r2
z2 , s2 , y2
Evaluator 1
Evaluator 2
No iterations
No iterations
Non-iterative; Uncoupled
Multi-Disciplinary Analysis
(MDA)
Disciplinary Analysis
Disciplinary Evaluation
1. Minimum load on optimizer
2. Complete interdisciplinary
consistency ( r  0 ) is
assured at each
optimization call
3. Each MDA
i Computationally expensive
ii Sequential
1. Complete interdisciplinary
consistency ( r  0 ) 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 evaluation
4. Evaluation cost relatively
trivial
5-7-2003
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Variants of MDF
5-7-2003
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MDF - 1
From optimizer x , C L , a jig ,
To optimizer
f , g,h
req .
  w /  x ) initial
Yes
{(w)< )}?
a ro o t  0
No
U p d ate a
 w
a panel  
 x
p a n el
a

jig 

Aerodynamics
displacement (w)
CL
elastic
Update aroot
Update a panel
Structures

a root  C L
req .
 CL
elastic

w

x
a panel  a root 
)  C
a
La , ririd
)

jig 

Aerodynamics
aeroloads
5-7-2003
WingOpt - 79
MDF - 2
From optimizer
x
Yes To optimizer
e =0 ?
f , g,h
No
Update aroot
Yes

a panel  a root 

w
x
a

jig 

(w)< ?
No
Update a
p a n el
displacement (w)
Aerodynamics
5-7-2003
aeroloads
Structures
WingOpt - 80
MDF - 3
From optimizer
x , C Lreq . , a
C om pute C L0
a root
jig
elastic
 w 
,


initial
  x  initial
{(e = 0 ) and
(w)< )}?
No
Update aroot
Update a panel
a root
a
displacement (w)
Structures
Yes To optimizer
i 1
C


paneli 1

L
req .
f , g,h
 C L0
C Li  C L 0
elastic
elastic
)
)a
root i

w 
 a root 


x

i
Aerodynamics
aeroloads
5-7-2003
WingOpt - 81
MDF - AAO
From optimizer
x , a root
*
Yes
(w)< ?
To optimizer
f , g,h
*
No
Update a
p a n el
displacement (w)
Structures
aeroloads
Aerodynamics
*
a ro
 d esig n variab le
ot
h
5-7-2003
*
 in clu d es L  n W
WingOpt - 82
IDF - 1
m
w( x, y ) 

k
From optimizer
k ( x, y )
k 1
x ,
m
w ( x, y ) 
*
  k k ( x, y )
To optimizer
*
k 1
ai 
*
w
*
 xi , y i )
Calculate {a}panel
 xi
IC C s :  k   k
*
f , g,h
*
Aerodynamics
Calculate k & ICCs
e=0?
Structures
Yes
No
Update
a root
  pseudo design variables
*
h  includes IC C s
*
5-7-2003
WingOpt - 83
IDF - AAO
m
w( x, y ) 

k
From optimizer
k ( x, y )
k 1
x ,  , a root
*
m
w ( x, y ) 
*

*
k
w
 xi , y i )
k ( x, y )
k 1
ai 
*
Calculate {a}panel
To optimizer
f , g,h
*
 xi
IC C s :  k   k
*
Aerodynamics
Calculate k,ICCs, e
Structures
  pseudo design variables
*
h  includes IC C s and e  0
*
5-7-2003
WingOpt - 84
Divergence Constraint Parameter
dcp  h1  h2
Asymptotic
h1
h1
h2
h2
dcp > 0divergence
5-7-2003
dcp < 0convergence
WingOpt - 85
Divergence Constraint Parameter
dcp  h1  h2
Oscillatory
h2
h1
h1
h2
dcp > 0divergence
5-7-2003
dcp < 0convergence
WingOpt - 86
Slow Convergence
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WingOpt - 87
Convergence Accelerated
5-7-2003
WingOpt - 88
Analysis v/s Evaluators
Conventional
approach:
OPTIMIZER
z
f , g, h
Analysis:
Evaluator:
Conservation laws of
system
If nonlinear, iterative
Multidisciplinary
Time intensive
Does not solve
Evaluates residues
for given z , p
Computationally
inexpensive
p
z
 A IC 
p
2. Calculates
 A IC 
z  d esig n variab les
z, p
a
Solve
1
3. Calculates
2.
1
 AIC
r p}a } A IC
 ap}}
rp
f  o b jective fu n ctio n
g  n eq u ality co n strain ts
5-7-2003
r
EVALUATOR
r  a }   AIC p}
z , p  design variables
r  residue
p  p ressu re lo ad
h  eq u ality co n strain ts
f , g, h , r
INTERFACE
1. Generates
a }   A IC   p }  0
OPTIMIZER
z, p
INTERFACE
z
A different
approach*:
*Solving
r  0 pushed to optimization level
f  objective function
g  equality constraints
h , r  equality constraints
WingOpt - 89
MDO Architectures
Multi-Disciplinary Feasible
(MDF)
Individual Discipline Feasible
(IDF)
Optimizer
z
z, y
Interface
Analysis 1
z, s, y
f , g,h, y
Interface
s1  s 2
y 21
Optimizer
Optimizer
f , g,h
z1  z 2
All At Once (AAO)
Analysis 2
Iterations till y Iterations till
12
convergence
convergence
Iterative; coupled

z 1 , y 12
s 1 , y 21
s 2 , y 12
f , g , h , y, r
Interface

z 2 , y 21
Analysis 1
Analysis 2
Iterations till
convergence
Iterations till
convergence
Uncoupled
z 1 , s1 , y 1
r1
r2
z2 , s2 , y2
Evaluator 1
Evaluator 2
No iterations
No iterations
Non-iterative; Uncoupled
Multi-Disciplinary Analysis
(MDA)
Disciplinary Analysis
Disciplinary Evaluation
1. Minimum load on optimizer
2. Complete interdisciplinary
consistency ( r  0 ) is
assured at each
optimization call
3. Each MDA
i Computationally expensive
ii Sequential
1. Complete interdisciplinary
consistency ( r  0 ) 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 evaluation
4. Evaluation cost relatively
trivial
5-7-2003
WingOpt - 90
Overview
• Aims and objective
• WingOpt
–
–
–
–
–
Software architecture
Problem setup
Optimizer
Analysis tool
MDO architecture
• Results
• Summary and Future work
5-7-2003
WingOpt - 91
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 - 92
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WingOpt: An MDO Tool for Concurrent Aerodynamic …