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
Problem
Setup
Optimizer
History
I/P
MDO
Control
Analysis Block
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  12*30 panels
Optimizer  FFSQP

11. Test Problem Design Variables Skin thicknesses - S Wing Loading
Aspect ratio
Sweep back angle
t/croot A

12. 2 Range (Vlong range cruise)
Test Problem
Sr. no
Item
Load Case
1 Structural (VDive)
2 Range (Vlong range cruise)
3 MDD (Vmax. cruise) 1
Altitude(m)
7620
10668 2
Mach No.
.8097
.72864 3
Load Factor
2.5
1.0 4
Fuel present: Fuel capacity 5
Fuel Flow Rate (kg/hr)
2827 6
Pdyn. Factor
1.98

13. Test Problem Constraints Stress – LC 1 - Structural fuel volume
MDD – LC 3
Range – LC 2
Take-off distance
Sectional Cl – LC 1 -
Structural -
Geometric
Aerodynamic

14. Design Variable and Constraints
Test Cases
Cases
Design Variable and Constraints
Aeroelasticity
MDO Methods 1 S No
Direct 2
Yes
MDF-1 3
S + A
Indirect 4 5 6
MDF-2 7
MDF-3 8
MDF-AAO 9
IDF 10
IDF-AAO

15. Results Case Skin thickness (mm) Wing loading (N/m2)
Sweep angle (deg.)
t/c ratio
Aspect ratio 1 2 3 4 5 6
6.25
3.36
5.03
2.46
2.0
5643 25
0.16
8.83
5.26
2.77
3.84
5.43
2.84
3.86
5790
31.14
0.20
8.18
5.49
2.87
3.88
2.03
5840
31.33
4.67
2.42
2.88
31.34
8.13
2.89 7
4.66
2.41
2.91 8 9
2.37
2.79
5818
31.27
8.14 10
8.70
6.99
7.35
4.12
4.11
5654
27.56
0.159
9.24

16. Results Case Active Constraints Stresses Fuel volume Mdd Range
Take-off distance
Clmax
Pseudo constraints
L=nW 1  - 2 3 4 5 6 7 8 9 10 ☓

17. Results Case Objective Number of Time (s) Weight (kg) Design variables
Cons-traints
Analysis performed
Obj func call
Const func call
Aerody
Struc
Total 1
696.37 6 24
175
132
3210 25 68
111 2
580.79 83 70
1785 32 41
363 3
576.5 13
2341
609
21028
12945
868
13879 4
576.14 10 29
651
191
5695
4335
239
4688 5
493.98 31
644
176
5651
4367
233
5768
494.14
488
143
4530
3666
4063
8903 7
495.05
523
154
4889
3698
4477
9466 8
494.02 34
1135
301
11805
6078
2744
9203 9
490.78 42 61
14466
4943
279499
50034
8959
61654
1131.8 45 64
1033
331
21644
1953
608
2736

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, λ, Λ y Λ
AR = b2/S
λ = citp/croot
croot
citp
b/2
Wing stations x

25. Aerodynamic Geometry Planform Geometric Pre-twist Camber Wing t/c
constant α' per station
α'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 t
linear variation in wing box-height
stations t

28. Structural Geometry Cross-section Box height Skin thickness Spar/ribs
symmetric
front, mid & rear boxes
r1, r2 y
Structural load carrying wing-box A
Front box
r1 = l1/c
r2 = l2/c
Mid box A A
Rear box l1 l2 c x

29. 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 y A x A A
hfront
hrear
h'1 = hrear / hfront

30. Structural Geometry Cross-section Box height Skin thickness Spar/ribs
Constant skin thickness per span
tsi , where s = upper/lower
i = 1, N y A x
tupper A A
tlower

31. 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 y A
rib
Aupper12 A A
spar cap x 1 2
intermediate spar
rear 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 Structural Geometric
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)

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.

36. Structural Sizing Optimization: Baseline Design
Design Case – Example 1
Aerodynamic
Structural X S AR λ Λ
α'i
h/c
d/c r1 r2
hroot
h'1
h'2i
tsi
Asjki F Cl
CDi CL
Vstall
Mdd - - σ Wt
W(x,y) Vf - - -
Structural Sizing Optimization: Baseline Design
Objective
Desg. Vars.
Constraint

37. Simultaneous Aerod. & Struc. Optimization
Design Case – Example 2
Aerodynamic
Structural X S AR λ Λ
α'i
h/c
d/c r1 r2
hroot
h'1
h'2i
tsi
Asjki F Cl
CDi CL
Vstall
Mdd - - σ Wt
W(x,y) Vf - - -
Simultaneous Aerod. & Struc. Optimization
Objective
Desg. Vars.
Constraint

38. 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 ? HISTORY BLOCK
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 X2 P1 P2 P3 1 X3
Look-up Table
Selected Variables X1 2 3 4 5 n .
Partitioning
Logic
To input processors

42. Analysis Block Diagram
Aerodynamic mesh, M, Pdyn
VLM Cl
Trim ( L-nW = e )
From MDO Control e
{α}rigid+{Dα}str.
Aerodynamic pressure
Pressure Mapping
To MDO Control
EPM/
FEM
Structural deflections
Structural Loads
To MDO Control
Deflection Mapping
{Dα}str.
stresses
Structural Mesh, Material spec.,
non.–aero Loads

43. Aerodynamic Analysis Panel Method (VLM) Generate mesh Calculate [AIC]
{p}=[AIC]-1{a}
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) NASTRAN Interface Code Load Transformation
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. Skins – CQuad4 shell element
Meshing - 2
Skins – CQuad4 shell element

55. Rib/Spar web – CQuad4 shell element
Meshing - 3
Rib/Spar web – CQuad4 shell element

56. Spar/Rib caps – CRod element
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} = [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

60. Deformation Interpolation (contd..)
{w}a = [G]as {w}s
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
{ua}T {Fa}= {us}T {Fs}
{ua}T ([Gas]T {Fa} - {Fs}) = 0
Force Transformation
{Fs} = [Gas]T {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 Planform to the FEM Nodes of the Wing-box Planform

69. Loads Transferred From VLM Panels of Wing-box Planform to 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. while satisfying  = L – nW = 0
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 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

76. Divergence Constraint Parameter

77. MDO Architectures Optimizer Optimizer Optimizer Interface Interface
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
Uncoupled
Non-iterative; 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 Yes No Aerodynamics Update aroot Update Structures
From optimizer
To optimizer
Yes
{(w)<d )}? No
Aerodynamics
displacement (w)
Update aroot
Update
Structures
Aerodynamics
aeroloads

80. MDF - 2 e =0 ? Update aroot Update Aerodynamics Structures
From optimizer
e =0 ?
To optimizer
Yes No
Update aroot
(w)<d ?
Yes No
Update
displacement (w)
Aerodynamics
Structures
aeroloads

81. MDF - 3 Aerodynamics Structures Update aroot Update Yes No
aeroloads
To optimizer
From optimizer
{(e = 0 ) and
(w)<d )}?
Update aroot
Update
Yes No
displacement (w)

82. MDF - AAO Update Structures Aerodynamics From optimizer To optimizer
(w)<d ?
Yes
To optimizer No
Update
displacement (w)
Structures
Aerodynamics
aeroloads

83. IDF - 1 Calculate {a}panel Calculate k & ICCs Aerodynamics e = 0 ?
From optimizer
To optimizer
Calculate {a}panel
Aerodynamics
Calculate k & ICCs
e = 0 ?
Structures
Yes No
Update

84. IDF - AAO Calculate {a}panel Aerodynamics Calculate k,ICCs, e
From optimizer
Calculate {a}panel
To optimizer
Aerodynamics
Calculate k,ICCs, e
Structures

85. Divergence Constraint Parameter
Asymptotic h1 h1 h2 h2
dcp > 0divergence
dcp < 0convergence

86. Divergence Constraint Parameter
Oscillatory h2 h1 h1 h2
dcp > 0divergence
dcp < 0convergence

87. Slow Convergence

88. Convergence Accelerated

89. Analysis v/s Evaluators
Conventional approach:
INTERFACE
Solve
OPTIMIZER
Analysis:
Conservation laws of
system
If nonlinear, iterative
Multidisciplinary
Time intensive
1. Generates
Evaluator:
Does not solve
Evaluates residues
for given
Computationally
inexpensive
OPTIMIZER
INTERFACE
EVALUATOR
A different approach*:
2. Calculates
2. Calculates
3. Calculates
*Solving pushed to optimization level

90. MDO Architectures Optimizer Optimizer Optimizer Interface Interface
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
Uncoupled
Non-iterative; 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 Results Summary and Future work
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