Abstract Design Optimization &amp; MDO studies carried out at CASDE, IIT Bombay are summarized. MDO architectures using WingOpt, effective use of low fidelity design thumb rules to shrink design space for S-Duct for a combat aircraft are briefly touched upon. Robust design of systems using low fidelity analysis tools and characterization of fidelity uncertainty using sparse high fidelity evaluations is discussed in detail

Amitay Isaacs

K Sudhakar

P M Mujumdar

Umakant Jyotsula

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SAROD-2005
Fidelity Uncertainty Characterization Leading to Robust Design
Umakant Jyotsula † K Sudhakar ‡ P M Mujumdar ‡ Amitay Isaacs ‡
Abstract
Design Optimization & MDO studies carried out at CASDE, IIT Bombay are summarized. MDO
architectures using WingOpt, effective use of low fidelity design thumb rules to shrink design
space for S-Duct for a combat aircraft are briefly touched upon. Robust design of systems using
low fidelity analysis tools and characterization of fidelity uncertainty using sparse high fidelity
evaluations is discussed in detail.
Key Words: Robust design, Design Optimization, MDO
1 INTRODUCTION
Center for Aerospace Systems Design & Engineer-
ing (CASDE) was established in 1999 and initiated
work in the area of Design Optimization and Multi-
disciplinary Design Optimization (DO/MDO) in 2000.
CASDE has gained insights into issues and tools in
DO/MDO [1] and also conducted several professional
development courses in the area for practicing engi-
neers [2]. Current research includes uncertainty char-
acterization for robust design and systems design of
aerospace vehicles.
2 AEROELASTICITY BASED MDO OF
AIRCRAFT WING
Initial studies at CASDE were aimed at understand-
ing of various generic MDO architectures discussed
in the literature through a relatively complex Multi-
disciplinary aerospace sub-system. Aero-elastic de-
sign of wings offered an interesting problem for the
same that generated insight into various MDO ar-
chitectures [3]. The studies involved reformulation
of the wing design problem through several different
variants of existing single level generic architectures,
specifically for coupled aerodynamic and structural
optimization of wing, focused on static aeroelasticity.
The design problem involved simultaneous optimiza-
tion of the wing aerodynamic plan-form and section
†Defence Research and Development Laboratory, Hyder-
abad, India
‡Indian Institute of Technology Bombay, Mumbai, India
c©Umakant Jyotsula, K Sudhakar, P M Mujumdar & Ami-
tay Isaacs
SAROD-2005
Published in 2005 by Tata McGraw-Hill
variables along with its structural sizing variables for
minimum load carrying structural weight subjected to
structural, aerodynamic, performance and geometric
constraints. The associated Multi-Disciplinary Anal-
ysis (MDA) problem is a coupled solution of the state
equations of the aerodynamic and the structural dis-
ciplines by nested iterations. The Multi-disciplinary
Design Optimization (MDO) problem is posed as a
three discipline coupled problem, with the trim (ma-
neuver) process required to define structural design
loads considered as a separate discipline. This lead
to a number of interesting reformulations of the MDO
problem based on (i) the reordering of the nested it-
erations and (ii) decoupling the nested iterations at
different levels through the introduction of pseudo de-
sign variables and pseudo constraints. Formulation of
six variants of the MDO problem and their implemen-
tation was presented along with computational issues
related to convergence of the iterative processes. A
special constraint based on a divergence control pa-
rameter has been formulated to indirectly handle the
aeroelastic instability, without an explicit divergence
eigenvalue constraint. Optimization results from the
different formulations were compared to study their
computational performance and bring out the impact
of aeroelasticity on the design of the flexible wing.
3 S-DUCT DESIGN
S-Ducts are commonly employed in combat aircraft
as part of engine air induction systems. They are re-
quired to meet air mass flow demands of engine with
high pressure recovery and low distortion. While good
estimates of pressure recovery is possible using low fi-
delity engineering methods, distortion requires high fi-
delity CFD analysis. Short design cycle for CFD based
design of 3D Ducts is hence extremely useful and forms
the motivation for this work. Design methodology [4]
2 Josyuala Umakant, K Sudhakar, P M Mujumdar and Amitay Isaacs
for 3D-Ducts uses a mixture of design thumb rules,
low fidelity analysis (LFA) and high fidelity analy-
sis (HFA). Thumb rules identify good ducts that are
likely to be free of pockets of flow separation and LFA
estimates pressure recovery. A simple optimization
problem that maximizes LFA pressure recovery sub-
ject to thumb rules that ensure separation free ducts
defines a base line duct. Commercial CFD software
FLUENT is used for HFA analysis. A grid sensitivity
analysis is performed on the baseline duct to deter-
mine the minimum grid size required to capture the
flow solution and distortion coefficient, DC60 appro-
priately. This is essential to strike a proper balance
between the conflicting requirements of lower simula-
tion times and the desired accuracy levels. 60 longi-
tudinal sections with 6634 cells at each section that
have a total of 398,040 cells is chosen. FLUENT and
its pre-processor GAMBIT are harnessed to provide
requisite automation. Difficulties related to good gra-
dients do not allow direct coupling of the automated
CFD analysis to a gradient based optimizer, and in-
stead surrogate based approach is used. Design space
is reduced by exploiting LFA thumb rules to identify
good ducts. Sampling is performed in the reduced de-
sign space only. Initially 12 samples are chosen for
CFD analysis. A DACE [5] model is built for DC60
values at these 12 points. Further sampling is driven
by the need to reduce relative error (prediction er-
ror/value) everywhere to less than 5%. An additional
9 samples were required to realize this. DACE surro-
gate model using 21 samples was used for defining the
HFA based optimum duct. LFA based optimum duct
that formed the baseline had DC60 of 6.91 and HFA
optimization reduced this to 1.70. A Linux based clus-
ter of 8 nodes (each node with PENTIUM IV 1.6GHz
processor with 1 GB RAM) was used for this study.
A single run of FLUENT using 4 nodes took around
30 hours for each run. Overall design took 630 hours
of CPU time.
4 FIDELITY UNCERTAINTY
CHARACTE-RIZATION LEADING TO
ROBUST DESIGN
Computer simulation based design processes are be-
ing extensively used in preliminary design phase of
complex aerospace vehicles like scramjet powered hy-
personic vehicles. Analysis tools of varying fidelity
are generally used to assess the system performance
metrics. Often there is a constraint on the number of
simulations using high fidelity analysis tools to pre-
dict the performance metrics, due to attendant com-
putational demands. Hence the designer is confronted
with the challenge of taking decisions in an environ-
ment wherein uncertainty is ever present. It is impor-
tant here to distinguish between variability and un-
certainty. Variability [6] is inherent randomness in
the system. The use of probability theory to rep-
resent variability is well established. Uncertainty is
defined as a potential deficiency in any phase or ac-
tivity of the modeling process that is due to lack of
knowledge. Uncertainty may also arise when there is
a scarcity of high fidelity information. This is true
when new classes of systems like scramjet powered
hypersonic vehicles are being developed and no his-
torical database exists. Probabilistic approaches to
handle uncertainty in lieu of replacing the expensive
high fidelity analysis with low/medium fidelity analy-
sis and their application in design scenarios have been
demonstrated recently. Quantification of uncertainty
using a Bayesian approach to update the uncertainty
model was proposed by Mantis [7] in the context of
an aerospace design. Charania et al [8] used engi-
neering methods for various participating disciplines
in the design of Reusable Launch Vehicle together
with a multiplier coefficient that is characterized by
assuming a probability distribution. A probabilistic
design approach for hypersonic vehicle has been also
demonstrated by Umakant et al [9]. In this study, the
effect of fidelity uncertainty on a disciplinary metric
namely mass flow capture of air was propagated onto a
system metric namely thrust deliverable and a design
that maximized the system metric was sought through
formal optimization. However, there are several issues
that remain to be addressed. Basically the develop-
ment of a non-deterministic design process involves
three major steps, namely:
1. Based on the information available, construct an
input uncertainty model
2. Propagate the effect of disciplinary uncertainty onto
system performance metrics
3. Assess the system performance and take design de-
cision under uncertainty
In most of the studies, discussed above, the focus
has been on the last two steps. An uncertainty model
was assumed based on the disciplinary experts recom-
mendation regarding the prediction accuracy of the
low fidelity analysis or based on evaluating the anal-
ysis tool with respect to similar applications. For ex-
ample, if the disciplinary expert declared that the low
fidelity analysis was accurate within ±10%, then a
normal distribution with normalized mean µ = 1 and
standard deviation σ = 3.33% is assumed. Based on
four high fidelity observations for a disciplinary met-
ric, namely mass flow capture of air, a Weibull distri-
bution was constructed by the authors of this paper to
represent the uncertainty in its estimation using the
low fidelity tool. These four observations correspond
Fidelity Uncertainty Characterization Leading to Robust Design 3
to arbitrary points in the design space. Therefore it
is required to ascertain the validity of the assumed
probability distribution if another set of observations
were used. The present study seeks to bridge this
gap. A novel approach based on ranks is proposed by
Umakant et al [10] to aggregate high fidelity informa-
tion in a sequential and cost effective manner. Rather
than trying to model the uncertainty for the entire
design support, a high fidelity sample is sequentially
aggregated from the regions where the expensive func-
tion is potentially attractive. Design points with low
response value receive higher ranks while those with
higher response values receive lower ranks. Based on
this information, residue is estimated as the difference
between the high fidelity response value and the cor-
responding low fidelity response value. Based on this
data, a probabilistic model for the residues is then
constructed. Uncertainty model for the estimation of
expensive function is now defined as the low fidelity
model complemented with probabilistic model of the
residue and the model may be used to take robust de-
sign decisions. It may be noted that in the context
of optimization (for minimization), the inaccuracy of
the model in the regions where the function is rela-
tively higher is not of much interest. Rank transfor-
mation of the response enables to introduce the pref-
erential characteristics in the uncertainty model. The
approach has been demonstrated to quantify the un-
certainty in estimation of a typical disciplinary metric
for hypersonic vehicle design, namely mass flow cap-
ture of air. The low fidelity tool is based on oblique
shock theory and the high fidelity tool is based on
inviscid CFD computations.
5 CONCLUSION
CASDE is engaged in Design Optimization/MDO
studies related to aerospace vehicles. Several studies
in this area have been successfully carried out in the
last 5 years. Uncertainty characterization and robust
design form the current research interests.
Acknowledgment
REFERENCES
[1] Centre for Aerospace Systems De-
sign & Engineering, IIT, Mumbai.
http://www.casde.iitb.ac.in/MDO/specific-
studies/index.html.
[2] Centre for Aerospace Systems De-
sign & Engineering, IIT, Mumbai.
http://www.casde.iitb.ac.in/MDO/sig-
mdo/index.html.
[3] Ajmera, H., Mujumdar, P. M., and Sudhakar, K.,
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45th AIAA/ASME/ASCE/AHS/ASC Structures,
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[4] Ghate, D., Isaacs, A., Sudhakar, K., Mujum-
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[5] Schonlau, M., “Computer Experiments and
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[6] Oberkampf, W. L., Bibliography for Verifica-
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[7] Mantis, G. C., Quantification and Propagation of
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[8] Charania, A. C., Bradford, John E., Olds, John
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[9] Umakant, J., Sudhakar, K., Mujumdar, P.M.,
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Fidelity Uncertainty”, AIAA-2004-4543, 10th
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[10] Umakant, J., Sudhakar, K., Mujumdar, P.M.,
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Specialist Conference and 1st AIAA Non-
Deterministic Approaches Conference, 47th
AIAA/ASME/ASCE/AHS/ASC Structures,
Structural Dynamics, and Materials Conference,
May 2006 Rhode Island, USA.


Fidelity Uncertainty Characterization Leading to Robust Design

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Fidelity Uncertainty Characterization Leading to Robust Design

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