Design Optimization at CASDE

1
Design Optimization at CASDE An Approach for
System Design Considering Fidelity
UncertaintiesApplication to Hypersonic Vehicle
Design

• K Sudhakar, PM Mujumdar, Amitay Isaacs
• Indian Institute of Technology Bombay
• J Umakant
• Defense Research Development Laboratory,
  Hyderabad
  
• CR Rao
• University of Hyderabad

2

• Presentation Outline
• Engineering Design
• Engineering Design and MDO
• Studies at CASDE-POST 2000
• Design under Uncertainty
• Synthetic Example
• Hypersonic Vehicle Design

3
Engineering Design

• During 1970s
• Evaluation using
• Empirical
• Data sheets
• Simplified analysis
• Design using
• Parametric studies

System Designer Choose X to Improve f(X)
Satisfy h(X), g(X)
4
Engineering Design
Optimizer
Optimization?
System Designer Choose X to Improve f(X) Ob
jective
Satisfy h(X), g(X) Constraints
5
Engineering Design
Optimizer
Optimization?
System Designer Choose X to Improve f(X) Ob
jective
Satisfy h(X), g(X) Constraints
6
Engineering Design
Optimizer

• 1980s
• Modeling languages
• Spreadsheet based
• Optimizers

System Designer Choose X to Improve f(X) Ob
jective
Satisfy h(X), g(X) Constraints
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Engineering Design
Optimizer

• 1990s
• Engineering Analysis
• CFD
• FEM
• - - -

System Designer Choose X to Improve f(X) Ob
jective
Satisfy h(X), g(X) Constraints
X
f, h, g
Pre-processing
Post-processing
Engineering Analysis
8
Engineering Design
Optimizer

• 1990s
• Engineering Analysis
• CFD
• FEM
• - - -

System Designer Choose X to Improve f(X) Ob
jective
Satisfy h(X), g(X) Constraints
X
f, h, g
Pre-processing
Post-processing
Engineering Analysis
9
Engineering Design - MDO
Optimizer
System Designer Choose X to Improve f(X) Ob
jective
Satisfy h(X), g(X) Constraints
1995 MDO
X
f, h, g
Pre-processing
Post-processing
Engineering Analyses
10
Engineering Design - MDO
Optimizer
System Designer Choose X to Improve f(X) Ob
jective
Satisfy h(X), g(X) Constraints
MDO Architectures How to handle couplings?
X
f, h, g
Pre-processing
Post-processing
Engineering Analyses
11
Studies at CASDE Post 2000

• WingOpt - MDO Architectures in the context of
  Wing Design for aircraft
  
• Aerodynamics VLM
• Structures - Ardema, Eq plate, FEM
• Flight mechanics
• 3D Duct (Meta) Design
• System Design through synthesis of designed
  sub-systems
  
• http//www.casde.iitb.ac.in/

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Engineering Design - MDO
Optimizer
System Designer Choose X to Improve f(X) Ob
jective
Satisfy h(X), g(X) Constraints
X
f, h, g
Pre-processing
Post-processing
Engineering Analyses
13
Design Under Uncertainty

• A1, A2, A3
• Low fidelity for design
• High fidelity for analysis
• Design decisions?
• Uncertainty due to use
• of low fidelity
• Complexity at system level?
• Complexity at sub-systems?
• Number and nature of couplings?

Optimizer
System Designer Choose X to Improve f(X) Ob
jective
Satisfy h(X), g(X) Constraints
X
f, h, g
Pre-processing
Post-processing
Engineering Analyses
14
An Approach for Robust System Design Considering
Fidelity Uncertainties Application to Hypersoni
c Vehicle Design

15
Vehicle Background
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HYPERSONIC TECHNOLOGY DEMONSTRATOR VEHICLE
(HSTDV)
Cruise Conditions

• Problem Statement
• Design an Air-Breathing Hypersonic Vehicle
  capable of cruising at M6.5 and cruise altitudes
  30-35 km
  
• Design Constraints
• Dimensional constraints on overall length, height
  and width
  
• Take-off gross weight
• Intake entry conditions
• Control deflection within allowable values
• Vehicle drag to be less than thrust deliverable

17
HSTDV Discipline Interactions

• Integrated Engine and Airframe
• Entire undersurface of the airframe forms part of
  the engine

Forebody

• Propulsion
• High Static Pressure
• Large air mass flow
• capture
• Minimum total pressure
• loss
• Allowable Intake Entry
• Mach number

• Aerodynamic Heating
• Temperature to be within
• specified limits

Forebody Design

• Sizing
• Adequate volume to
• store fuel and other
• onboard equipment

• Aerodynamics
• Lift to drag
• Longitudinal Stability

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Multidisciplinary Design Optimization for HSTDV
Optimizer

• Design Variables
• Configuration
• Mission

LEVEL 1 Low Fidelity (Engg. Methods)
Vehicle Design change
XD
f , g
Multi-disciplinary Analysis
SIZING/GEOMETRY
STABILITY CONTROL
AERO- DYNAMICS
A/D HEATING
LOADS
STRUC -TURES
PROPULSION
PERFORMANCE
19
 
Problem Statement
Minimize F -(Th_deliv/AF 1) Subject to
G1 (MI / 4.2) 1 ? 0 G2 (?trim / 20.
0) 1 ? 0 G3 (TOGW / 1200.0) 1 ? 0 G4
(L / 7.0 1) ? 0 G5 (H / 0.8 1) ? 0
Optimization variables
Min. Max. Forebody compression angle
s ?1 ,?2 , ?3 (deg.) 1.0
6.0 Wing cant angle ?w-cant (de
g.) 0.0 6.0
Wing scale factor wfac_pl
0.8 1.0
Tail scale factor tfac_pl,
0.8 1.1
Cruise altitude Hcruise (kM)
30.0 35.0
Note assume shock on lip
Parameters Lmid 2.0m Lab 1.5m ?n_pl chosen
to give body width 0.8m hintk 0.25m wintk
0.5m
a/b 2.0
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Integrated Airframe and Engine Analysis
Mo, ?, H
input
output
External Flow Analysis - Oblique shock theory
- Tangent cone/wedge
L , D, My P1 , M1 , T1
?1 ,?2 ,?nose wnose ,lmid ,ma
Intake Entry conditions
?cowl , lcowl ,lint liso , A1 /A3
Dcowl P3 , M3 , T3
Intake and Isolator Analysis -Oblique shock the
ory
Combustor Entry conditions
Dcomb , Thcomb P4 , M4 , T4
? , ma ,?comb , lcomb
Combustor Analysis -1D Heat Addition
Nozzle Entry conditions
Dbase , Thnoz Lp , Myp P6 , M6 , T6
After-body Analysis -2D CFD database
?noz ,?noz , lnoz
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• inf_con
• geom_body
• geom_wing
• geom_tail
• Internal passengers
• Engine mass

• Mass, c.g., , moments of inertia
• internal volume available
• (X,Y,Z) geom-body,geom_wing

Sizing
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Optimization Results
Note F is computed at the corresponding design
settings using High Fidelity simulation for
disciplinary metric. As expected, there is a
significant difference between the LFA based
value and the corresponding HFA based value.
Thus the designer is confronted with the
challenge to make reliable decisions in a design
environment where in uncertainty is present.
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Categorization of Uncertainty

• Error
• Recognizable
• deficiency
• Discretization
• Round-off
• Estimate of error
• is possible

• Aleatory Uncertainty
• Inherent variation
• of system
• Modeled using
• Probability theory
• Irreducible, Aleatory
• Weather conditions

• Epistemic Uncertainty
• Incomplete information or
• lack of knowledge
• Not enough data
• Reducible
• Not representing viscous
• phenomena

In this study we focus on ways to account for
epistemic uncertainty
Ref Oberkampf WL et al Mathematical
representation of uncertainty AIAA 2001
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Fidelity Uncertainty in Multidisciplinary System
Preliminary design phase - Medium / Low Fidelit
y Analysis

?Y Uncertainty due to lack of fidelity in
disciplinary response ?Z Uncertainty in system r
esponse
Design decisions -Effect of disciplinary fidel
ity uncertainty on system response
needs to be accounted

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Uncertainty Modeling - A Probabilistic Approach
Problem Statement
Maximize P( min F( Z )) subject to P(Gi(Z) ? 0)
?i i 1,2,.n ?i specified confiden
ce level
XL ? X ? XU
Steps

• Assessment of Disciplinary Fidelity
• Quantification of Uncertainty Model
• Propagation of Disciplinary Uncertainty to system
  level
  
• Taking design decisions under uncertainty

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Assessment and Quantification of Fidelity of
Disciplinary
Performance Metric
Few high fidelity observations on the
disciplinary metric are available
Calibration factors ratio of low fidelity
observation to high fidelity observation
PDF Value
CF
10Mantis Based on two observations
assumed a PDF 11 DeLaurentis Based on disci
plinary expert recommendation, ?10 accuracy as
sume N(0,3.3) 12Charania Assumed distributio
n to be available based on past data
All the CF are one sided Hence Weibull distributi
on
is assumed
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Propagation of Disciplinary Uncertainty to System
level metric
Input Histogram

• Sample the uncertainty model and do Monte Carlo
  Simulation
  
• Record the variation in system metric

Frequency
Multidisciplinary synthesis tool
Correlation Factor
?ai
AM2
Fi
AM1
a1?ai
a1
XD
i 1,2, NS
Output Histogram
Frequency
System Metric
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Taking Design Decisions
Allowing for 5 risk in the design decision, we
choose the 95 reliable value of the system metri
c.
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Transforming Probabilistic Optimization to
Deterministic Optimization
Maximize P( min F( Z )) subject to P(Gi(Z) ? 0
) ?i i 1,2,.n
XL ? X ? XU
Minimize F( Z ) subject to Gi(Z) ? 0 i 1
,2,.n
XL ? X ? XU
P(F) and P(G) with desired reliability
30
Fidelity Uncertainty
In this study we consider effect of the fidelity
uncertainty for the disciplinary metric mass flo
w capture of air on the system level metric Th
rust deliverable
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Typical High Fidelity result for Disciplinary
metric
Mass flow capture of air Euler CFD
Iso Mach Contours
Y0
Ps_entry, N\m2
Ps_entry, N\m2
Z-0.6
Top
Center
Side
Bottom
Height of intake, m
Width of intake, m
32
Optimization Results
Note F is computed at the corresponding design
settings using High Fidelity simulation for disci
plinary metric
33
Summary of Major Steps in a Probabilistic
Design Process i) Generate information
(HFA) regarding the metric ii) Based on informa
tion available construct a probability
distribution function ii) Propagate the effect
of disciplinary uncertainty onto system
performance metric iv) Assess the system perfor
mance and take design decisions under uncertainty

Issue Where to sample the observations for the p
urpose of constructing the disciplinary PDF?
Can we approach the high fidelity based solution?

34
Sensitivity of initial sample of high fidelity
observations on quantification of disciplinary fi
delity uncertainty model
Probability
Calibration Factor
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Concept of Ranks
Statistical literature provides examples on the
use of ranks when high fidelity information is s
carce. 19 Cronan use of rank transformation
to develop regression model with few
samples. Shown to yield better results than
conventional Regression. 18 Dell used judg
ement ordering when the characteristic of
interest was difficult or expensive to obtain
Ranking refers to the process of ordering a
sample, say of size N, with respect to a p
erformance metric. In the context of
minimization, the observation having lowest va
lue is given the highest rank (N)
while the observation with highest value is give
n the lowest rank (1).

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A Sequential Sampling Method for uncertainty
modeling
I. Generation of Initial sample
- Initial Sample of HFA observations, say K,
selected based on stratification. II. Genera
ting Empirical PDF for design space or support
- Augmentation of the sample with LFA responses
- Ranking the responses - Mapping the ranks ont
o design support - Computation of Empirical PDF
of design support III. Sampling from PDF - Sel
ection of new point and supplement the HFA
observations ( K1) Repeat the above for sele
ction of next point Stopping Criteria
IV. Uncertainty Model Cumulative Distribution Fu
nction for residue, e Z-z
Aggregation of High Fidelity Sample
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Step I Generation of the initial Sample
Stratification

• Divide the design space and the range of the
  response into three levels.
  
• Select one point from each
• Level of design space such that its response
  level is unique.
  
• Assignment Problem

Unlike in the DoE, the selection of the points is
not only based on the design space but also based
on the use of the LFA response
space.
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Step II Generating Empirical PDF for the design
space
The implementation is illustrated in the table
shown below
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Step III Selection of new point
CDF of design support
CDF Value
Design Variable, X
HFA observations incremented by one sample
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Effect of number of HFA observations on residue,
Z-z

The CDF exhibits nearly same variation when th
e number of
observations is more than 10.
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Aggregation of Data
The aggregation of high fidelity sample is thus
accomplished in two stages Stage I. Sel
ect K observations based on stratification
Stage II. Use the K observations as the init
ial data and select N observa
tions based on ranking
The next step is to construct probability
distribution for residue based on the NK obs
ervations
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Empirical CDF for Residue
e CDF value ---------------------
----------- -4.1954 0.00 -3.1954
0.05 -2.4794 0.15 -2.2746
0.25 -0.7712 0.35 0.1524 0.
45 0.3391 0.55 1.9132 0.65
2.8141 0.75 3.3735 0.85
7.2687 0.95 8.2687 1.00
This completes the construction of CDF of residue
in the estimation of expensive function.
Z z U(
Z)
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Results for univariate bimodal function
CDF Value
Design Variable, X
Empirical CDF for design support
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Uncertainty Bounds in the Estimation of Response
using LFA
Design Decision using the Bounds
__upper bound ___HFA ___lower bound
Response
Design Variable
95 confident that Z ?
95 confident that Z ?
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Monte-Carlo Simulation Results
CDF Value
CDF Value
Design Variable, X
Residue
46
Design Space Reduction
47
Using the HFA sample to improve the LFA model
Least Squares Regression Dependent variable Hig
h fidelity observations Regression variable Cor
responding Low fidelity response
Ranks of the HF observations are used as weights
Advantage Irrespective of the dimension o
f the design space Regression is always carried o
ut in one dimension with the Low fidelity values
as regression variable. Prediction at a new po
int
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Response
Standard Deviation
XD
XD
Current Best Minimum Design (CBM)
is considered as the Minimum of HFA
observations aggregated based on ranking
(XCBM , YCBM) Probability of improving the CBM
Pr ( Z Zo)
XD
If Pr (Z Z0) is ? 0.1 , then we perform HFA at
that point and update the regression model,
otherwise we consider the CBM as the best design
available
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Uncertainty Quantification for Hypersonic Vehicle
Design
Mass flow capture of air, ma , is a critical
disciplinary metric in the design of
propulsion flow path in Scramjet powered
Hypersonic Vehicle Design. We are interested in
higher values of ma Forebody design influence
s ma as well as body aerodynamics .
HFA Inviscid CFD (PARAS) LFA Oblique sh
ock theory M8 6.5 ? 4? H_cruise 32.
5 km 0? ? ?1, ?2, ?3 ? 6? Computational Budget
10 HFA
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Uncertainty Model for ma using the CDF
Uncertainty in estimation of ma as interval
___ lower bound ____ upper bound ____ HF r
esponse
ma (kg/s)
LFA based ma (kg/s)
51
Uncertainty Model for ma using Regression
ma-estimate

ma_LFA is a function of the 3 compression angles
The 10 Rank based HFA observations were the input
for the above model It can be observed that the m
odel, is able to provide the bounds on the
estimate better in the region of interest, namely
where the function is relatively higher.
Note To construct a surrogate model ( DACE) for
the entire support the number of
HFA observations needed were 32
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Summary

• Literature survey showed that though
  probabilistic approaches have been
  
• demonstrated for design of aerospace
  vehicles, the aspect of aggregating
  
• the HFA observation sample for the purpose of
  constructing the CDF of
  
• residues has not been rigorously studied.
• In order to bridge this gap a novel method,
  exploiting the concept of ranks,
  
• has been showcased to sequentially generate a
  high fidelity sample
  
• and construct CDF of residue in an objective
  manner.
  
• Initially the approach has been demonstrated for
  synthetic examples. The
  
• resulting CDF has been used to define the
  lower and upper bounds of
  
• uncertainty in the estimation of the
  expensive function when a LFA model
  
• is used.
• Subsequently, the approach has been adopted to
  quantify uncertainty in
  
• a typical disciplinary performance metric for
  hypersonic vehicle design.
  
• Thus with a few intelligently chosen high
  fidelity observations, it is now
  
• possible to make better and robust design
  decisions.

53
Acknowledgements Shri Prahlada, Director, DRDL D
r S Panneerselvam, Technology Director,
Hypersonics and Advanced Technologies, DRDL
Dr M Nagarathinam, Technology Director,
Aerodynamics, DRDL