Transcript Multidisciplinary Design optimization incorporating Robust

Multidisciplinary Design optimization incorporating Robust Design
Approach to tackle the uncertainties in the design of a Reusable Aerospace
Vehicle
Multidisciplinary Robust Design Optimization for a
Reusable Aerospace Vehicle
1st Progress Seminar after 6 months
(Roll No. 02401701)
Under the guidance of
Prof. K. Sudhakar
Prof. P.M. Mujumdar
FULLY REUSABLE TSTO
TYPICAL FLIGHT PROFILE
DEORBIT
SATELLITE
DEPLOYMENT
RE-ENTRY
DOWN/CROSS RANGE
MANEUVERS
RE-ENTRY
SEPARATION AT
80-100 KM, M 10-12
PARACHUTE DEPLOYMENT
LANDING MANEUVERS &
LANDING ON AIR BAGS
MANOEUVERS
TURN
HORIZONTAL
LANDING
CRUISE AT M0.8
& H=10-12 KM
SHYAM / LVDG
Expendable Launch Vehicle
• Participating disciplines are..
Aerodynamics, Structures, Aero-thermodynamics,
Navigation Guidance & Control, Propulsion and
Mission & Trajectory.
Reusable Launch Vehicle
• Complexity = Launch vehicle + Space plane
• Should also account for Life cycle Disciplines…..
….. Economics, Reliability, Manufacturability,
Safety & Supportability
• And should account for uncertainties
The Design Problem is…
• Design of a reusable technology demonstrator
for the First stage of Two Stage to Orbit fully
reusable Launch Vehicle.
For which the mission is defined as…
TYPICAL MISSION PROFILE OF RLV-TD
ALT = 150 km
RE-ENETRY / GLIDE /
RANGE MANOEUVERS
2G TURN
ALT = 35 km
S12 SEPARATION
ALT = 60 km
M = 10.0
FLYBACK
CRUISE ALT = 12 km
M=0.8
LIFT-OFF
LANDING
The Traditional Design Process
Mission
Requirements
Configuration
Concepts
Historical Data base
Engineering knowledge base
Trade off based on Figures of Merit
Historical Data base
Concept Design &
Vehicle Sizing
Aerodynamics
No
Constraints
met
yes
Eg: Space Shuttle, X-34,
HYFLEX, ALFLEX, CRV
Low fidelity Analysis
Weight
Propulsion
Structures
Estimation,
CG
Trajectory Analysis
Stability
&
Control
Yes
No
Apply Small Perturbations in design variables
Constraints met
Low fidelity Analysis on Aero, Propulsion
Weight Estimation & CG, Stability & Control
and Trajectory
Select the best
No
High fidelity analysis on Aerodynamics,
Constraints met
Structures, Weight & Cg estimation, Aerothermal
Propulsion, Stability & Control and Trajectory
yes
Detailed Design
The methodology proposed in this research work
Vehicle Conceptual Design using engineering methods (low fidelity)
Aerodynamics, sizing, Propulsion, Stability& Control and Trajectory
Design
Noises
variables factors
Multidisciplinary Analysis (High fidelity)
Sizing
Optimiser
Cost Model
 f,  f
Objective
Functions Constraints
INSCOST
Aerodynamics
CFD
Propulsion
Structure
Empirical
FEM
Aerothermal
MINIVER
The Configuration Concept selected
Parameterization of the vehicle
configuration concept
Airfoil 2for vertical
tail
X5
X3
X5
X3
X7
For wing
2
X14
1
• X1 to X17 – Length parameters
• t1,t2 – Thickness parameters
• 1, 2 – Angular variables
• R1,R2 – radii
• di – internal diameter
• X3 to X8 and 1, 2 and the airfoil 1 are exclusively
for the wing design.
• Since accommodating the 24 variables for
multidisciplinary robust design optimization, will
be computationally expensive, the wing can be
optimized separately for its intended performance
and to take care of the variations due to operational
& manufacturing uncertainties
The Constraints
1. The take off weight should not exceed 2000Kg  GTOW  2T
2. The maximum diameter of the fuselage should be between 0.9 to 1.2 m
3. The volume requirement inside the fuselage for the avionics boxes,
propulsion modules, landing gear wells and other auxiliary system are
estimated as 3.0m3
4. The nose cone length (x1) is estimated as 2250 mm for the scramjet
propulsion module performance point of view.  x1 = 2.250m
5. di, the internal diameter of the top half of the fuselage = 1.00 m
(considering the interfacing requirement with the solid boosters of 1 m
diameter)
6.
The bottom surface of the demonstrator should be flat so that the
scramjet modules integration as well as the inlet conditions are satisfied.
Also for easy mounting of the Thermal Protection System tiles. This
will compel the selection of an airfoil with flat bottom for the wings.
7.
8.
The hypersonic L/D  1.5
Subsonic L/D max = 4.5
9.
The landing weight of the vehicle also should be taken as 2.00 T to take
care of abort scenario.
10. For subsonic cruise, Lift L = 2000 kg, for Mach 0.8 at 12 km altitude
11. Wing leading edge sweep  45 and leading edge radius should be for
minimum re-entry heating.
12. For the subsonic cruise, the drag (D) of the vehicle should be equal to
the thrust deliverable
14. The cruise range  800 km.
15. The touch down speed  80 m/sec and the landing angle of attack will be
15 deg.
16. The sink rate at touch down  4.5 m/sec.
17. The runway roll  1.8 km.
18. During the ascend boost phase the maximum dynamic pressure should be
120 KPa and angle of attack should be less than 3 degree and Maximum
longitudinal acceleration should be 10 g
19. During the re-entry phase the dynamic pressure should be less than 20
KPa and the maximum lateral acceleration should be 3 g
20. During re-entry the overall heat flux should be less than 50 Watts/sq-cm
The objective function Maximize the cruise range
Uncertainties Expected Manufacturing Uncertainty – which will result in
surface roughness and the excrescence effects, like
mismatches ,gaps, contour deviation and fastners
flushness (rivets, etc) on the wetted surface.
Operational Uncertainty – like uncertainty in cruise
Mach number.
Literature Survey
Parametric Optimization of Manufacturing Tolerances at the aircraft
surface - A.K. Kundu, John Watterson, and S. Raghunathan, Journal of
Aircraft, Vol.39, No.2, March-April 2002.
 Aimed at reducing life cycle cost of the passenger aircraft by relaxing
the manufacturing tolerances on 11 key features in the nacelle.
Parasite drag increase resulted by the degradation of the surface
smoothness qualities, for example, the discrete roughness on the component
parts and at their subassembly joints. These are seen as aerodynamic
defects, collectively termed as one of the excrescence effects, typically,
i) mismatches (steps etc.)
ii) gaps,
iii) contour deviation and
iv) fastners flushness (rivets, etc) on the wetted surface.
The four types of surface excrescence at the key manufacturing
features
.
Findings
 The results show that feature by feature percentage changes for one
nacelle with a drag coefficient increment of 0.824% and a reduction of
2.26% on the nacelle cost.
 This will result to 0.421% overall reduction in DOC (Direct Operating
Cost) of the transport aircraft.
 Tolerance relaxation tradeoff study between drag increase
quality function) and manufacturing cost reduction (gain).
(loss of
 Further research work is planned by the same group to extend the
study to wing and fuselage.
Probabilistic Approach to Free-Form Airfoil Shape Optimization Under
Uncertainty - Luc Huyse, AIAA Journal, Vol. 40, No.9 September, 2002
Luc Huyse, R. Michael Lewis, “Aerodynamic Shape Optimization of Twodimensional Airfoil Under Uncertain Conditions” [16], NASA/CR-2001210648
 This work is on the operating uncertainties which will
affect the performance of an aircraft. The airfoil shape
optimization is addressed.
 In airfoil design, the objective is to minimize drag with
the specified cruise Mach number and target lift coefficient
 Robust design of airfoils for a transport aircraft. Here
robust design technique accounts for the variation in cruise
Mach number.
The objective is lift constrained wave drag minimization over the Mach range
M  [0.7,0.8]:
min Cd (d,M)
dD
Sub to Cl (d,M)  Cl*
over M  [0.7,0.8]
Where d is the vector of design variables and D is the design space. Cl* is the
minimum lift corresponds to typical values found for commercial transport
airliners. In this study, the Mach number is the only uncertain parameter.
This deterministic optimization model is not necessarily an
accurate reflection of the reality. The formulation contains no
information regarding off-design condition performance. So the
drag reduction is achieved only over a narrow range of Mach
numbers. This is of concern if substantial variability is associated
with operating condition.
Consider different Mach numbers and to generalize the objective
function to a linear combination of flight conditions:
m
min  Wi Cd (d,Mi)
dD
i=1
Sub to Cl (d,Mj)  Cl*
For j = 1,2,…..m
Practical problems arise with the selection of flight conditions
(Mi) and with the specification of the weights Wi. There are no
clear theoretical principles to guide the selection, which is in
fact, largely left to the designer’s discretion. With multipoint
formulation, Cd can be realized over a wide range of Mach
numbers M, however this formulation is still unable to capture
the full range of uncertainty .
Nondeterministic Approach
M is now treated as a random variable and the optimization problem is now
interpreted as a statistical decision making problem.
So using the probability density function of the Mach number, the objective
function is stated as
min  Cd (d, M) fM(M) dM
Sub to Cl (d,Mj)  Cl* for all M, where fM(M) is the probability density
function of the free flow Mach number M.
The practical problem is that integration is required in each of
the optimization steps.
Theoretically sound but
computationally expensive. This work is an example of using
probabilistic approach in achieving robustness, provided the
distribution pattern of the noise variable is known.
* Robert H. Sues, Mark A.Cesare, Stephan S. Pageau, & Justin Y.-T.Wu,
“Reliability – Based Optimization Considering Manufacturing and Operational
Uncertainties”[13], Journal of Aerospace Engineering, October, 2001
Discuss about the approach of integrating MDO and probabilistic methods to
perform reliability based MDO.
RBMDO Demonstrated on – Passenger Airplane Wing Design problem
•Objective : Maximize expected cruise range
•Subjected to constraints
•1. P (upper surface root stress 1 ≤  y ) ≥ 99.0 %
••-6. P (take off distance ≤ 3000 ft ) ≥ 99.0 %
3 case studied
Case A: Six deterministic constraints with no safety factor on yield stress
Case B : Six deterministic constraints with safety factor of 1.5 on yield stress
Case C: Six probabilistic constraints
• Manufacturing uncertainties are simulated on design variables
•Operational uncertainties are considered.
The results show that:
RBMDO ( case C ) gives optimum solution that balances performance
and reliability
Range : (NM)
Reliability: (%)
Case A
Case B
1024.7
984.7
38
96
Case C
974.9
99
* P.B.S.Reddy and K.Nishina, Dr. Subash Babu “Taguchi’s methodology
for multi-response optimization- A case study in the Indian plastic industry”[6]International journal of Quality & Reliability Management, Vol.15, No.6, 1998,
pp.646-668
•Taguchi’s methodology for carrying out a robust design is
narrated in this.
•The salient features of robust design are presented and the
robust design methodology applied to the case having multi
responses (for an injection moulding process for the agitator of
washing machine) is presented.
•The output responses considered were outer diameter, height &
pull out force.
• The goal was minimizing the variance of the height,and outer
diameter of the agitator while keeping the mean on target and pull
out strength <1.8 kg/sq-cm
• Based on cause-effect diagram seven factors were identified
• Three noise factors identified were - change of machine operators,
variation in raw material quality and change in temperature &
environmental conditions.
•By performing robust design in the specific case of
injection moulding process the rejection rate could be
reduced from 20 % to zero percent which helped the
company in many ways related to cost, delivery,
quality & productivity.
* K.K.Choi, B.D.Youn, “Issues Regarding Design Optimization
Under Uncertainty”, http://design1.mae.ufl.edu/~nkim/indexfiles/choi4.pdf
Discusses the mathematical formulation of robust
design problems
The conventional optimization model is defined as
Minimize
OBJ (d)
Sub. to
Gi (d)  0,
i = 1,2,….NC
dL  d  dU
where OBJ is the objective function, Gi is the ith constraint function, NC is
the number of constraints, d is the design variable vector, dL & dU are the
lower & upper bounds of d.
In robust design,
Minimize
OBJ [  R,  R ]
sub.to
Gi ( ) + k  Gi  0 ,
i = 1,2,….NC
d L  d  dU
where  R,  R are the mean & standard deviation of the response
R, Gi ( ) and  Gi are the mean & standard deviation respectively
of the ith constraint function, k is the penalty function decided by
the designer, d is the design variable vector, dL & dU are the lower
& upper bounds of d.
NPR
OBJ [  R,  R ] = 
[ w1j ( Rj – Rj t ) 2 + w2j  2 Rj ]
J=1
sub.to
Gi ( ) + k  Gi  0 ,
i = 1,2,….NC
dL  d  dU
Where, w1j is the weight parameter for mean on target, w2j that for the jth
performance to be robust, Rj and Rj the mean and standard deviation of the
jth performance, Rj t is the target value of the jth performance and NPR is the
number of performances to be robust.
For the robust design for best overall performance over the entire
life time the objective function is based on the joint probability
density function of the random variable x .
NPR
OBJ ( d, x ) =  x  wj Rj (d,x) f x (x) dx
J=1
s.t
Gi ( ) + k  Gi  0 ,
i = 1,2,….NC
dL  d  dU
Where f x (x) is the joint probability density function of the random variable x,
Rj (d,x) is the jth performance function to be minimized and wj is the weight
parameter for the jth performance to be robust.
Mathematical formulation of reliability based design (RBDO)
problem
Minimize
OBJ (d)
s.t
P { ( Gi (d )  c }  CFLi , i = 1,2,….NC
dL  d  dU
where CFLi is the confidence level associated with the ith constraint, P denotes
the probability, Gi (d ) is the ith constraint function and c is the limiting value.
Example:
P ( stress 1   y )  99.0 % , where  y is the yield stress. (ie) Since there are some
uncertainty in the material properties, instead of stating the constraint as, stress 1   y,
it is stated as the probability of stress 1   y, is greater than or equal to 99.0%.
Robust & Reliability Based Design.
When the objective function is based on the robust design principle, focusing
on making the response insensitive to the variations in the design variables and
the constraints are modified to probabilistic constraints with the assigned
probability of each constraint function, the result is a Robust & Reliability
Based Design (RRBDO)
The mathematical formulation of such a method is given below.
Minimize
OBJ [  R,  R ]
s.t
P { ( Gi (d )  c }  CFLi ,
i = 1,2,….NC
dL  d  dU
where NC is the number of constraints and the objective function is defined as
NPR
OBJ [  R,  R ] =  [ w1j ( Rj – Rj t ) 2 + w2j  2 Rj ]
J=1
Plan of Action for the Next Two Years
Robust & Reliability
Final
based MDO of the
thesis
Integration of MDO
Aerospace vehicle
submission
architecture with
design
robust design
techniques &
Draft thesis
probability analysis
preparation, review &
modifications
Understanding, practising &
applying stochastic techniques
Exploring the options for probabilistic design
& applying to the problem
Activity
Identifying & understanding the
uncertainty analysis methods
Identifying a
proper
strategy to
attack the
problem
Review of the
strategy &
modifications
Defining the design
problem, identifying the
constraints, control
variables, noise factors &
objective functions.
2nd Progress
seminar
Literature Survey & exploring in-depth information on robust & reliability based
design practices, techniques & the related research works around the globe
August, 03
Month / Year
August, 04
August, 05
Acknowledgement
I would like to express my sincere thanks to
Prof. K. Sudhakar and Prof. P.M. Mujumdar
of Aerospace Engineering Department
for their continuous guidance,
encouragement and support.
Methods of simulating the variation in noise
factors
• Monte Carlo Simulation
• Taylor Series Expansion
• Orthogonal Array based
Simulation
• Monte Carlo Simulation
A random number generator is used to simulate a large number
of combinations of the noise factors called testing conditions.
The value of the response is computed for each testing
conditions and the mean and variance of the response are then
calculated.
For obtaining accurate estimate of mean & variance, the Monte
Carlo method requires evaluation of the response under a large
number of testing conditions. This can be very expensive,
especially if we also want to compare many combinations of
Methods of simulating the variation in noise factorscont’d
• Taylor Series Expansion
The mean response is estimated by setting each noise factor equal to its
nominal value. To estimate the variance of the response, the derivatives of
the response with respect to each noise factor is found out.
Let R denote the response and 12 , 22,………n2 denote the variance of n
noise factors. The variance of R is then computed by the formula:
R2
n
=  (R / xi)2 i2 ,
i=1
where xI is the ith noise factor.
Methods of simulating the variation in noise factorscont’d
• Orthogonal Array based Simulation
Orthogonal arrays are used to sample the domain of noise
factors. For each noise variable different levels are taken.
The advantage of this method over the Monte Carlo method is
that it needs a much smaller number of testing conditions; yet
the accuracy will be excellent. The orthogonal array based
simulation gives common testing conditions for comparing two
or more combinations of control factor settings.
* Amy E. Kumpel, Peter A. Barros Jr., and Dimitri N. Mavris
“Quality Engineering Approach to the determination of the Space Launch
capability of the peace keeper ICBM utilizing probabilistic methods” [1] AIAA 2002-10-6
Discuss about the use of a comprehensive and robust methodology for the
conceptual design of an expendable launch vehicle employing the existing
Peacekeeper ICBM This methodology includes an Integrated Product and
Process Development (IPPD) approach, coupled with response surface
techniques and probabilistic assessments. It also provides a probabilistic
framework to address the inherent uncertainty in vehicle requirements in an
analytical manner by representing payload, mission, and design requirements as
distributions instead of point values.
The three primary objectives identified are (i) to design for the minimization of
the time-to-launch, (ii) minimization of development and production costs, and
(iii) the maximization of useable payload
The first step in the design process was to define the problem by mapping the
customer requirements to engineering characteristics. A Quality Function
Deployment approach, utilizing a House of Quality, was employed. Possible
engine and propellant types, as well as staging arrangements, were organized in a
Morphological Matrix of design alternatives. Several vehicle concepts from the
Morphological Matrix were then evaluated in terms of performance, cost,
availability, reliability, safety, commonality with existing space systems, and
compatibility with various launch sites.
Ranges were assigned to several significant design variables, and a sensitivity
analysis was performed on the responses to see how small perturbations in the
design variables would affect the outcome. A parametric study was also performed
on some of the assumptions made in the design process so that the exact effects of
the estimates on the vehicle concept could be determined. A Response Surface
Methodology (RSM) in conjunction with a Monte Carlo simulation was used for
these tasks. This methodology was an iterative process and was repeated until both
technical feasibility and economic viability were achieved.
The tool employed in the problem definition stage of this study was the Quality
Function Deployment (QFD) process which is a "planning and problem solving tool
that is finding growing acceptance for translating customer requirements into the
engineering characteristics of a product”. The broad requirements of the
engineering characteristics are transformed into an interrelationship digraph (ID).
A statistical analysis software package called JMP is used to create the DoEs. A
Monte Carlo simulation is used in conjunction with response surface equations in
order to model thousands of designs in seconds. The software package Crystal Ball
by Decisioneering® is used for this task. Crystal Ball is a risk analysis software
package and an add-in to Microsoft Excel. It allows for the definition of design
variables as probability functions bounded by a range or a set of values. It then uses
the defined ranges in a Monte Carlo simulation. For each uncertain design variable,
a probability distribution is used to define the possible values. Distribution types
include normal, triangular, uniform, logarithmic, etc. The Monte Carlo simulation
creates Probability Distribution Functions (PDFs) and Cumulative Distribution
Functions (CDFs), in order to illustrate the probability of success for a response.
A PDF is the mathematical function that maps the frequency of the response to
metrics within the given range. The PDF is then integrated to determine a CDF.
The CDF is the mathematical function that maps the probability of obtaining a
response to the metric within the given range. If the amount of feasible design
space is unacceptable, three options exist for the designer/decisionmaker:
1. Modify the design variable ranges;
2. Relax the constraints;
3. Select a different alternative concept.
At this point in the design process, the system is evaluated to check if the
responses satisfy the customer requirements as established in the problem
definition phase. If any of the requirements are violated at any point in this
iterative process, the design process will be repeated.
* Brent A. Cullimore , “Reliability Engineering & Robust Design : New Methods for
Thermal / Fluid Engineering”[14], C & R White Paper, Revision 2, May 15, 2000
Mention that overdesign provides robustness but it is costly in areas such as
aerospace. He modified SINDA/FLUINT, the thermodynamic analyzer software
to make the design robust by statistically integrating the probabilities of design
variables to control the probability of response.
The paper dwells on the
SINDA/FLUINT software and also mention about the add on software named
Relaibility Engineering module, which will estimate the reliability of a point
design based on uncertainties in the dimensions, properties, boundary conditions
etc.
Few possible capabilities of Reliability Engineering Module in SINDA/FLUINT
(1)
A design can be selected using the solver and then (in the same or later
run) the reliability of that design can be estimated.
(2)
The reliability of a design can be used as an objective (maximize
reliability or minimize the chances of failure).
(This feature can be useful to the present problem).
(3)
The reliability of a design can be used as an optimization constraint (find
the minimum mass design that achieves a reliability of at least 99%).
(4)
The range or variance of a random variable can be used as a design
variable What variations can be tolerated : how tight must tolerance be ?.
The paper also mention about a commercial tools named Engineous
iSIGHT ® that can perform optimization, reliability estimation and robust design
generation.
* Wei Chen, Kemper Lewis, “A Robust Design Approach for Achieving Flexibility
in Multidisciplinary Design”[2], http:/www.uic.edu/labs/ideal/pdf/Chen-Lewis.pdf
(2001)
Explained the two types of robust design. In type 1, the robust design concept is
applied to the early stages of design for making decisions that are robust to the
changes of downstream design considerations (called Type I robust design).
Furthermore, the robust design concept is extended to make decisions that are
flexible to be allowed to vary within a range (called Type II robust design) [2,18]. In
Type II robust design, the performance variations are contributed by the
deviations of control factors (decision variables) rather than the noise factors. The
concept behind Type II robust design for determining flexible design solutions is
represented in Figure below.
For a typical optimization model that is stated below
The robust optimization can be formulated as a multiobjective optimization
problem shown as the following:
f and f
are the mean and the standard deviation of the objective
function f (x), respectively. In the above equation, the mean locations and
the range of design solutions are identified as x and ?x. To study the
variation of constraints, the worst case scenario is used, which assumes
that all variations of system performance may occur simultaneously in the
worst possible combination of design variables. To ensure the feasibility
of the constraints under the deviations of the design variables, the original
constraints are modified by adding the penalty term to each of them,
where kj are penalty factors to be determined by the designer. The bounds
of design variables are also modified to ensure the feasibility under
deviations. Depending on the computation resource, f and f could be
obtained through simulations or analytical means such as Taylor
expansions.
Improving the quality
of a product through
minimizing the effect
of the causes of
variation without
eliminating the causes
To assure proper levels of safety
(Probability of being safe) for the
system designed
* Wei Chen, Xiaoping Du, “Efficient Robustness and Reliability
Assessment in Engineering Design”,
www.icase.edu/colloq/data/colloq.Chen.Wei2001.5.9.html (2001)
Narrates the difference between Robust Design & Reliability based Design and
Integrated Robust & Reliability Assessments and schematically presented the
procedure for optimization under uncertainty.
Uncertainty
Classification
Catastrophe
Risk analysis
Reliability based
Design & optimization
Performance
loss
Cost benefit analysis
Robust Design &
optimization
Everyday
fluctuations
Extreme events
Frequency of
events
And the difference in problem formulation for Conventional
Optimization model & Robust Design model is given as below.
Integrated
Robustness &
Reliability
Assessments
* Stephen M. Batill, John E. Renaud, Xiaoyu Gu “Modeling &
Simulation Uncertainty in Multidisciplinary Design Optimization” –
AIAA-2000-4803
Dwell on the technical risk & uncertainty in the model
based design of physical artifacts. The issues of physical process
variability, information uncertainty and the influence of the use of
models & simulations on the design decision process are
discussed. This paper only qualitatively addresses these issues. It
suggests Monte Carlo simulation for uncertainty analysis in which
variations in the design variables & parameters are selected from
an appropriately selected population of random numbers.
* Daniel P.Schrage “Technology for rotorcraft
affordability through Integrated Product/Process Development
(IPPD)” [19]– 55th Annual Forum of American helicopter
Society, May 25th –29th, 1999.
Highlights the Robust Design Simulation as the main
approach in the roadmap to Affordability. He defines the
benefit-cost ratio (BCR) as an objective and defines robust
design as the systematic approach to find optimum values of
design factors which results in economical designs which
maximize the probability of success. The steps in economic risk
analysis & other research activities in the related areas at
Georgia Tech are mentioned.
Focus on to the specific Aerospace Vehicle Design Problem
• To design a reusable technology demonstrator ( for the First stage of
Two Stage to Orbit fully reusable Launch Vehicle.)
TSTO Features
• 10 T to LEO (Low earth Orbit) payload capability
• Vertical take off
• Semicryogenic booster stage with Isp of 330 sec (mission average) &
cryogenic orbiter stage Isp 400 sec (mission average)
• Total lift off weight < 700 tons
• Winged body booster which should boost the orbiter to Mach 10 at
altitude of 80-100 km then separate, return to launch site land
horizontally in a conventional runway of 2.5 km stretch.
• The vehicle structures should be designed for 100 flights & engine/
stage systems should be for 50 flights
• Turn around time should be 30 days
• The service life life of the vehicle should be 15 years
• Ideal velocity at 400 km circular orbit =9.8 km/sec
• Number of missions < Ten per year
• Payload fraction = 2% (Measure of efficiency of the vehicle)
• Cost effectiveness < $1000/kg for the LEO payload
• Realizability – Total development time <10 years
• Reliability >0.995
(Other features which are not relevant to the specific design problem
are not mentioned)
• The mission profile of the TSTO vehicle is shown in the next slide.
Flying regimes and related parameters of the TSTO first stage
• Vertical lift off with T/W of 1.3 (GLOW < 700 tons)
• Maximum acceleration (longitudinal) during ascend phase < 10g
• During the ascend boost phase the maximum dynamic pressure should be
120 KPa and angle of attack should be less than 3 degree
• Separation altitude & velocity =80-100 km & Mach 10
• Re-entry altitude = 100-80 Km
• Angle of attack during re-entry 40 deg
• During the re-entry phase the dynamic pressure should be less than 20
KPa and the maximum lateral acceleration should be 3 g
• Maximum Heat flux during re-entry < 50 Watts /sq-cm
• Turn around manoeuvres by aero control surfaces – Elevons & rudders
•
•
•
•
•
•
•
Cruise Mach number =0.8
Cruise altitude = 12 km
Down range at cruise start = 800 km
Angle of attack at touchdown < 15 deg
Horizontal velocity at touch down = 80-100 m/sec
The sink rate at touch down < 4.5 m/sec
The landing roll shall be < 2 km
So the task is to design a Technology demonstrator for the first stage of this TSTO
Design Guidelines
1.
2.
3.
4.
5.
6.
7.
Maximum landing speed (horizontal) is  85 m/sec
Subsonic L/D max = 4.5.
The vehicle should have near neutral stability at subsonic
speeds.
The vehicle should be trimmable at all Mach numbers.
The vehicle takes off vertically and lands horizontally
hence the wing design should be for landing and checked
for the other flight regimes.
It should have minimum pitching about CG for mated
configuration, with the solid boosters.
The hypersonic L/D shall be  1.5 for better cross range.
– cont’d
8. The landing weight of the vehicle is taken as equal to the
take off weight to take care of the abort missions.
9. The wing loading should be selected in such a way that the
structural weight is minimum.
10. Wing plan form should be selected for the best
performance in hypersonic as well as subsonic regimes.
11. Wing should provide a lift equal to the GTOW during
cruise at subsonic speed (Mach 0.8) at altitude of 12 km.
12. The airfoil should be selected to have maximum Cl at
landing and lower heating at hypersonic speed as well as
for mounting of tiles, the bottom flat airfoil is preferred.
13. The control surfaces should be sized to trim the vehicle
with minimum force and maintain the attitude at
hypersonic speed.
– cont’d
14. Fuselage fore body should be shaped so as to minimize
the re-entry heating, to have minimum drag and for good
directional stability.
15. Wing leading edge sweep shall be not less than 45 deg.
16. Wing leading edges shall be blunted to reduce re-entry
heating.
17. Low aspect ratio wing with highly swept leading edge
angle shall be considered to reduce heating and drag.
18. Thickness of airfoil shall be selected based on lift, drag
& leading edge bluntness requirements.
19. Vertical tail area should be designed to provide positive
directional stability and rudder shall be designed for
landing conditions at high angle of attack in cross-wind
and aft C.G conditions.
– cont’d
20. The gaps between control surfaces and aerosurfaces shall
be kept minimum to reduce the heating problems & to
improve the performance.
21. The fuselage should be sized considering the volume
requirements for accommodating the avionics systems,
propulsion modules, landing gears and other auxiliary
systems and also for better interfacing with the carrier
solid boosters.
22. The vehicle should be capable of testing the scramjet
module in a dedicated mission.
23. The vehicle should have a supersonic cruise capability at
Mach 3.0 using the ramjet engine.
After analysing the additional requirements in terms of
technology, from all the entities the major objectives of the
RLV-TD are listed as follows [23].
• (i) To evaluate the aero-thermo dynamic characteristics of wing
body vehicle and associated control surface effectiveness at various
flight regimes i.e. from sub-sonic to hypersonic zones.
• (ii) To assess the autonomous navigation, control and guidance
schemes to function in the demanding environment of re-entry, cruise
flight and auto-landing phase with constraints on loads and thermal
environment.
• (iii) To demonstrate the auto-landing technologies including landing
gear, aerodynamic control, deceleration systems etc.
• (iv) To evaluate the thermal protection system (TPS), re-usable light
weight structures for multiple missions (say about 100), evaluation of
air-breathing propulsion system, to design and validate redundant
electro-mechanical actuators to control the vehicle at severe
environment condition.
• (v) To evaluate the integrated flight management and ground
operation requirements.
• (vi) To demonstrate the scramjet propulsion module.
The Challenges
• The multidisciplinary nature
• The coupling with different disciplines and a large design
variable set
• Uncertainties of parameters and model structure
• The computational burden
The best solution to tackle the uncertainties is the robust design
solution (ie) by making a design insensitive to the variations in
noise factors and to tackle the variation in the control factors a
reliability based solution in which the constraints are stated
probabilistically to get the probability of an objective function for a
known deviations in the control factors.
Information flow of a Multidisciplinary System
Complexities
of MDO
under
Uncertainties
Xs- the sharing
variables
Xi-the design
variables of
subsystem
(discipline) i,
Yij-linking
variables of
subsystem
Zi-output of
subsystem i
A Typical model with uncertainties
The major disciplines involved in the design of the aerospace
vehicle are Aerodynamics, Structures, Aero-thermodynamics
Propulsion and control.
(1) Aerodynamics
F1 (Xs, X1,Y21----Y51)
1 (Xs, X1,Y21----Y51)
(2) Structures
F2 (Xs, X2,Y12--Y52)
2 (Xs, X2,Y12--Y52)
(3) Propulsion
F3 (Xs, X3,Y13--Y53)
3 (Xs, X3,Y13--Y43)
RMDO
(4) Control
F4 (Xs, X4,Y14--Y54)
4 (Xs, X4,Y14—Y54)
(5) Aero-thermodynamics
F5 (Xs, X5,Y21----Y51)
5 (Xs, X5,Y21----Y51)
* K.K.Choi, B.D.Youn, “Issues Regarding Design Optimization
Under Uncertainty”, http://design1.mae.ufl.edu/~nkim/index-files/choi4.pdf
Emphasizes the different ways of applying the term robustness as,
Definition 1- Identify designs, which minimize the variability of the performance
under uncertain (manufacturing or operation) conditions, Definition 2 -Provide
the best overall performance over the entire lifetime of the structure or device
and Definition 3- Mitigate the detrimental effects of the worst-case performance.
The design with the “best” worst-case performance is selected as the robust
solution as per definition 3.
The conventional optimization model is defined as
Minimize
OBJ (d)
s.t
Gi (d)  0,
i = 1,2,….NC
dL  d  dU
where OBJ is the objective function, Gi is the ith constraint function, NC is the
number of constraints, d is the design variable vector, dL & dU are the lower &
upper bounds of d.
In robust design, of definition 1 the objective is to keep the mean on target &
minimize the variation. So the mean & standard deviation of the response will
constitute the objective function. So the formulation will be
Minimize
OBJ [  R,  R ]
s.t
Gi ( ) + k  Gi  0 ,
i = 1,2,….NC
dL  d  dU
where  R,  R are the mean & standard deviation of the response R, Gi ( ) and
 Gi are the mean & standard deviation respectively of the ith constraint function,
k is the penalty function decided by the designer [5], d is the design variable
vector, dL & dU are the lower & upper bounds of d.
Here the objective function takes care of the signal & noise factors
and the constrained functions are modified such that the allowed
variation in them are limited by the sigma bounds.
NPR
OBJ [  R,  R ] =
 [ w1j ( Rj – Rj t ) 2 + w2j  2 Rj ]
J=1
s.t
Gi ( ) + k  Gi  0 ,
i = 1,2,….NC
dL  d  dU
Where, w1j is the weight parameter for mean on target, w2j that for the jth
performance to be robust, Rj and Rj the mean and standard deviation of the jth
performance, Rj t is the target value of the jth performance and NPR is the number
of performances to be robust
And for the robust design for best overall performance over the entire life time
(definition2) the objective function is based on the joint probability density
function of the random variable x . According to the theory of probability &
statistics, integral of the probability density function will give the probability and
when this is multiplied by the performance function, the expected value
corresponding to that probability will be obtained. The expression given below is
based on this theory.
NPR
OBJ ( d, x ) =  x  wj Rj (d,x) f x (x) dx
J=1
s.t
Gi ( ) + k  Gi  0 ,
i = 1,2,….NC
dL  d  dU
Where f x (x) is the joint probability density function of the random variable x,
Rj (d,x) is the jth performance function to be minimized and wj is the weight
parameter for the jth performance to be robust.
In the Reliability Based Design Optimization (RBDO) problems, the objective is
to maximize expected system performance while satisfying constraints that ensure
reliable operation. Because the system parameters are not necessarily
deterministic, the objective function & constraints must be stated probabilistically.
For example RBDO can determine the manufacturing tolerance required to
achieve a target product reliability because the method considers the
manufacturing uncertainties, such as dimensional tolerance as probabilistic
constraints.
RBDO will ensure proper levels of safety & reliability for the system
designed. The mathematical formulation for RBDO is shown below[18].
Minimize
s.t
OBJ (d)
P { ( Gi (d )  c }  CFLi , i = 1,2,….NC
dL  d  dU
where CFLi is the confidence level associated with the ith constraint, P denotes
the probability, Gi (d ) is the ith constraint function and c is the limiting value.
The following example [8] will clear the concept of probabilistic constraint.
P ( stress 1   y )  99.0 % , where  y is the yield stress. (ie) Since there
are some uncertainty in the material properties, instead of stating the constraint
as, stress 1  y, it is stated as the probability of stress 1  y, is greater than or
equal to 99.0%.
When the objective function is based on the robust design principle (with
mean & standard deviation of the response), focusing on making the response
insensitive to the variations in the design variables and the constraints are
modified to probabilistic constraints with the assigned probability of each
constraint function, the result is a Robust & Reliability Based Design
(RRBDO)
The mathematical formulation
next slide.
[18]
of such a method is given in the
Minimize
OBJ [  R,  R ]
s.t
P { ( Gi (d )  c }  Poi ,
i = 1,2,….NC
dL  d  dU
where NC is the number of constraints and the objective function is defined as
NPR
OBJ [  R,  R ] =  [ w1j ( Rj – Rj t ) 2 + w2j  2 Rj ]
J=1
The different parameters in the above definition are already explained in the
formulation for robust design. This approach will yield a design whose
response is insensitive to the effects of noises & whose reliability can be
predicted based on the reliabilities apportioned to the different constraints.