OPTIMIZATION OF AN EXTERNAL FUEL TANK OF A
SPACE SHUTTLE By
Raghunath Sai Katragadda, Mechanical Graduate Student
Padmaja, Mechanical Graduate Student
Sophia Guntupalli, Aerospace Graduate Student
ME 555-09-01
Winter 2009 Final Report
UNIVERSITY OF MICHIGAN, ANN ARBOR, MICHIGAN
ABSTRACT
In this project, the design of an external fuel tank of a space shuttle is optimized for minimum
weight and first natural frequency. In this optimization project, the objective is to be attained
while ensuring structural and thermal integrity, aerodynamic drag and volume of the fuel carried.
The optimum design of external fuel tank is an iterative process involving several conflicting
objectives and constraints. Every subsystem has an optimum design satisfying the individual
requirements and the designs are ultimately integrated to determine the overall optimum design.
Within the limitations of the model, the problem is shown to exhibit a modest coupling.
Optimization is viewed not only as a method to determine the best design, but, equally
importantly, as a process through which understanding the system under investigation and
unveiling its intrinsic trade-offs. This study explored the multidisciplinary system design
optimization of the External Fuel Tank of the Space-Transportation-System.
TABLE OF CONTENTS
1. INTRODUCTION
2. LIQUID OXYGEN TANK---Raghunath Sai Katragadda
2.1 Design problem statement
2.2 Nomenclature
2.3 Mathematical model
2.4 Model analysis
2.5 Optimization study
2.6 Parametric study
2.7 Results
3. INTER TANK---Padmaja
3.1 Design problem statement
3.2 Nomenclature
3.3 Mathematical model
3.4 Model analysis
3.5 Optimization study
3.6 Results
4. LIQUID HYDROGEN TANK---Sophia Guntupalli
4.1 Design problem statement
4.2 Nomenclature
4.3 Mathematical model
4.4 Model analysis
4.5 Optimization study
4.6 Parametric study
4.7 Results
5. SYSTEM INTEGRATION
5.1 Design problem statement
5.2 Nomenclature
5.3 Mathematical model
5.4 Optimization study
5.5 Results and Conclusion
6. REFERENCES
7. APPENDIX
1. INTRODUCTION
The Space Shuttle External Fuel Tank (SSEFT) is the largest single element and the only major
non-reusable component of the Shuttle system. The SSEFT is 46.88 meters long, 7.8 meters in
diameter and its empty weight is 29,937 kg. It carries more than 2 million liters of cryogenic
propellants that are fed to the orbiter's three main engines during powered flight.
A design optimization study of the Space Shuttle External Fuel Tank (SSEFT) is performed with
a model which although simplified captures some of the important attributes of the system’s
behavior. The goal of this project is to minimize the weight and the first natural frequency of the
SSEFT by varying its geometric characteristics, while keeping in mind the constraints imposed
on it by the materials chosen and the launching conditions. Gradient based method is applied on
the model to reach optimum points. The endeavor of spaceflight in the United Stated is becoming
increasingly scrutinized in terms of both safety and cost effectiveness. The space agency is
aiming towards maximizing the payload by reducing the system weight. Usage of less material
reduces the cost associated with it. The structural integrity of the system is equally important in
spacecraft voyage. The frequency of vibrations of the system should be as low as possible to
avoid amplitude factors and resonance.
The model is divided into three sub systems, namely,
1. The Liquid Oxygen tank
2. The Intertank
3. The Liquid Hydrogen tank
The rationale behind choosing these subsystems is the fact that the SSFET is composed primarily
of these three components which are welded together. Analyzing individual components,
keeping in mind the influence of one on the other is a good way to proceed, since it will help for
smooth system integration.
The study addresses the following issues:
(1) The simplified model of the external tank,
(2) Single objective optimization using gradient based method of subsystems,
(3) Multi objective optimization using gradient based method of the complete system
The system comprises of seven design variables, all of which are geometric in nature. One of the
variables namely the radius of the cylinder is taken to be the same for all the three subsystems
during system integration. This is considered to be a tradeoff owing to the complexities
associated with structuring the model in ANSYS. The integrated system is governed by six
constraints which are multidisciplinary in nature. The constraints include internal and external
stress constraint, buckling stress constraint, thermal stress constraint, coefficient of aerodynamic
drag constraint, volume of liquid oxygen fuel carried and volume of liquid hydrogen fuel carried.
The material of the system, generally Aluminum Lithium alloy 2090, is taken to be the
parameter. Material properties such as elasticity modulus, Poisson ratio, and thermal
conductivity are implicit parameters as they are included in the Ansys Parametric Design
Language (APDL). The entire system and the subsystems were modeled and analyzed using
ANSYS. The optimization exercise was performed using the optimization tool, ISight, using the
NLPQL algorithm.
Figure 1: Exploded view of SSEFT
2. THE LIQUID OXYGEN Tank
2.1 Design Problem Statement
The liquid oxygen tank is also referred to as the “nose cone” of the system. The nose cone forms
the tip of the external fuel tank. This component is structurally as well as aerodynamically very
crucial to the fuel tank. The liquid oxygen fuel in the tank exerts pressure on the internal walls of
the tank. The external walls of the tank are subjected to aerodynamic pressure which is because
of the aerodynamic force that opposes the movement of the system. The shape of the nose cone
should be aerodynamically very efficient since the drag experienced by the tank is primarily
dependent on the shape of the nose cone. The main objective is to minimize the volume of the
nose cone structure (as the aluminum lithium alloy family has the same density, the objective is
to minimize the volume of material used), accommodating internal stress constraint due to fuel
pressure, external stress due to wind pressure, thermal stress constraint due to difference in the
internal and external temperatures, coefficient of aerodynamic drag constraint and the constraint
involving the volume of liquid oxygen fuel carried. The material used for modeling the structure
is Aluminum Li alloy 2090. The volume of the nose cone is related to the geometry of the
structure. Therefore, the design variables in the problem statement are the geometric dimensions
of the structure.
Design Variables:
The design variables comprise of the geometric dimensions of the nose cone. They are the height
of the cone (H), Radius of the cone (Rcone), Height of the Rim (D) and the thickness of the cone
(tcone).
Parameters:
The explicit parameter is the material of the structure. This parameter actually includes sub-
parameters such as elasticity modulus, Poisson ratio, thermal conductivity, yield strength, etc.
2.2 NOMENCLATURE
H: Height of cone in centimeters
D: Height of the Rim in centimeters
: Radius of the cone in centimeters
: Skin thickness of the cone in centimeters
Fstructallowed : Allowable Structural Stress,190e6
Fthermalallowed : Allowable Thermal Stress, 190e6
: Coefficient of Aerodynamic Drag
: Volume of Liquid Oxygen Fuel, 550
Figure 2: Liquid Oxygen Tank
2.3 MATHEMATICAL MODEL
The objective function of the subsystem is to minimize the volume of the cone which depends on
the geometric design variables of the cone.
x = [H, D, , ]
In order to minimize Volume(x)
As the density is constant the objective modifies to Minimization of Volume of material used.
The volume of material, in terms of design variables is given as
Volume =
The objective function is subjected to the following constraints
Structural Stress Constraint:
The structure need to sustain certain stress levels. The structure is subjected to external wind
pressures and internal fuel pressure. The geometry of the structure should be such that, for a
given material, it used should be able to bear the stress levels. The stresses that are developed in
the structure because of these pressures should not exceed the yield strength of the material
employed in the structure. Therefore a stress constraint is included such that the stresses
developed in the structure should always be less than or equal to the yield strength of the
material.
The stresses developed because of the pressure on the walls are determined using simulation
software ANSYS. The stresses at each node of the mesh should be less than or equal to yield
strength of the material used.
g1: Stresses attained from Simulation - Fstructallowed ≤ 0
Here, the factor of safety has not been considered. Maximum allowable stress can be taken as
the limit which provides some leeway during the optimization process.
Thermal Stress Constraint:
The structure gets exposed to harsh temperatures. The external walls and the internal walls of the
structure experience extreme temperatures. The external walls are exposed to the atmospheric
temperatures and the internal walls are exposed to the fuel temperatures. The difference in the
temperatures builds thermal stresses in the structure. The thermal stresses developed within the
structure should be within the yield strength of the material.
The stresses developed because of the temperature on the walls are determined using ANSYS.
The stresses at each node of the mesh should be less than or equal to the allowable stress of the
material used.
g2: Stresses attained from Simulation - Fthermalallowed ≤ 0
Here, the factor of safety has not been considered. Maximum allowable stress can be taken as
the limit which provides some leeway during the optimization process.
Aerodynamic Constraint
The aerodynamic constraint involves the drag coefficient of the areas concerned with the cone.
The aerodynamic drag coefficient for a regular cone is around 0.5. The relation to express the
coefficient of aerodynamic drag is simplified by substituting the standard density of air, take off
velocity and frictional force of air. The values of these factors are taken from NASA website.
The coefficient of drag is mathematically expressed as follows:
The drag coefficient of the nose cone should be less than or equal to the drag coefficient of a
regular cone to reduce aerodynamic frictional forces.
g3:
Fuel Volume Constraint: The volume of Liquid Oxygen carried in the Nose cone should not be
affected by optimization process of the structure. The internal volume available to store the fuel
should be more than or equal to the volume of the fuel carried.
g4: ≤ 0
The geometric design variables are chosen to be within the limitations recommended by NASA
300≤D≤500
1150≤H ≤1350
350≤ ≤550
0.9≤ ≤3
All the units are in terms of centimeters
Figure 3: Internal and External stresses
Summary
Min Volume =
Subjected to
g1: Stresses attained from Simulation - Fstructallowed ≤ 0
g2: Stresses attained from Simulation - Fthermalallowed ≤ 0
g3:
g4: ≤ 0
300≤D≤500
1150≤H ≤1350
350≤ ≤550
0.9≤ ≤3
2.4 MODEL ANALYSIS
Finite element analysis
The structure is modeled and analyzed in ANSYS. Owing to the huge structure of the nose cone,
axisymmetric model is constructed. This reduces the computation time and cost. Aluminum-Lithium
alloy 2090 is used as the material of the structure. The material properties include elasticity modulus
which is 75 Giga Pascal, Poisson’s ratio of 0.34, thermal conductivity of 88 w/m-k, density of
2.59g/cubic cm. For the structural stress analysis, opposing force due to wind is 29929 Newtons and
the internal fuel pressure is 240875 N/ sq.m. The external temperature of the nose cone is provided as
283 K whereas the internal fuel temperature is around 50 K. An initial starting point is considered
within the constraint region of the geometric variables.
Figure 4: Axisymmetric Model of the Nose Cone
Monotonicity Analysis
The constraints g1 and g2 cannot be studied using monotonicity analysis because of the
nonexistence of the mathematical models for these constraints. The objective function and
constraints g3, g4 involve mathematical relationship.
D H f + + + +
g3 - g4 - - - -
Table 1:Monotoncity analysis with two constraints
The monotonocity analysis involving the objective function and two constraints yield that g3 is
active with respect to . The constraint activity can be precisely estimated because of the
non existence of comprehensive mathematical models.
2.5 OPTIMIZATION STUDY
The subsystem is modeled in ANSYS using APDL. The input files for structural stress analysis
and thermal stress analysis are developed using APDL. The nose come subsystem optimization is
carried out using optimization software (ISight FD3.1) linked with ANSYS. The optimization
model was executed based on a model by Christopher Michael Lawson of MIT. In this approach,
the optimization process requires ANSYS to act as a server for generating input and output files
while ISight runs the variable optimization process. The starting points of the design variables
are randomly chosen within the geometric constraints. The initial runs were performed on
Optimus. But due to the incapability of ANSYS to generate simulations within a desirable time
span, ISight is was used. The gradient-based algorithms i.e. Nonlinear Programming Quadratic
Line Search (NLPQL) is used to obtain results. The Simcode palette and the calculator are used
in the optimization process. The arrangement of the system is as follows:
Figure 5: Optimization model in ISight
Two simcode palettes and two calculators are arranged in parallel for the optimization procedure.
The ANSYS input files and output files are incorporated in the simcode. The calculator includes
mathematical relations of the fuel volume and drag coefficient. The optimization run failed when
performing the experiment for the first few times because of improper mapping of the variables.
Different initial starting points within the constraints have been employed over various runs. At
every instance, the solution was converging to the same values. There existed only one active
constraint with positive lagrangian multiplier of value 0.18D3. The solution is being converged
to the same points for different starting points within the constraints, so it can be concluded as
global optimum. The aerodynamic coefficient of drag constraint is the active constraint. A result
could not be attained when the system was scaled. The runs yielded only infeasible solutions. For
the unscaled model, the KKT conditions are satisfied such that the lagrangian multipliers are
zero for inactive constraints and positive for an active constraint. This constraint activity is
consistent with the monotonic analysis. The volume attained from ANSYS is minimized.
Figure 6: Inputs and Constraints employed in ISight
2.6 PARAMETRIC STUDY
The material of the subsystem incorporates many parameters. During the construction of the
model in ANSYS, material properties such as Young’s Modulus, Poisson Ratio, Thermal
Conductivity, Coefficient of Thermal of Expansion, Density, Thermal Stress, and Yield Stress
are used. The material used during optimization study is Aluminum Lithium alloy 2090. A
change in the material brings about change in all the above mentioned parameters. These
parameters are changed in the APDL code of ANSYS. Different compositions of Aluminum
Lithium alloy material are used for the parametric study. The results attained with materials from
the Aluminum Lithium alloy family yielded similar results with a variation of +/- 10 cubic
centimeters. Substantial difference in the results can be visualized when materials way too
different in characteristics are used.
2.7 RESULTS
In optimizing the subsystem for weight reduction, certain tradeoffs were taken into account.
There is a tradeoff between the volume and stress constraints. As the volume of the material
reduces, the structural and thermal stability of the structure varies. An approximation of the true
geometry of the subsystem is considered. Incorporating the true geometric details led to complex
meshing causing ANSYS to crash. So a simplified version of the structure within the true
dimensional range has been considered. The subsystem is designed in ANSYS using ANSYS
Parametric Design Language (APDL). Geometrical tradeoffs are inevitable because APDL is
highly sensitive towards mesh generation. The main objective in optimizing the subsystem was,
to minimize the volume of a given material used in the structure, at the same time withstanding
structural and thermal stresses, vibrations and providing sufficient volume to carry the requisite
amount of liquid Oxygen. The NLPQL algorithm was used because the problem is a
continuously differentiable optimization problem. It works with both feasible and infeasible
initial design points and has a superior rate of convergence. A total of 21 feasible designs are
identified after undergoing 26 design evaluations. The g3 constraint is the active constraint. A
volume percentage reduction of 69% was attained after running the optimization process.
Realistic analysis can be done by accommodating the complexity of the geometry. The radius of
the cone is the design variable which is in common with the other two sub systems.
Optimization Results
Optimization Technique: NLPQL
Failed Run Objective Value = 1.0E30
Failed Run Penalty Value = 1.0E30
Max Failed Runs = 5
Max Iterations = 10
Min Abs Step Size = 1.0E-4
Rel Step Size = 0.01
Save Technique Log = false
Termination Accuracy = 1.0E-6
Use Central Differences = false
Starting design point:
d = 400 [300.0 < x < 500.0]
h = 1260 [1150.0 < x < 1350.0]
r = 420 [350.0 < x < 550.0]
t = 2 [1.0 < x < 3.0]
Total design evaluations: 26
Number of feasible designs: 21
NLPQL termination reason: LINE SEARCH REQUIRED TOO MANY FUNCTION EVALUATIONS
Optimum design point:
Run # = 7
Objective = 1834050.0
Penalty = 0.0
ObjectiveAndPenalty = 1834050.0
d = 406.0
h = 1326.0
r = 460.0
t = 1.0
cd = 6.04E-5
stress = 2642.6
ther = 738740.0
vf = 5.5824E8
vol = 1834050.0
Calculated constraint values at the optimum:
cd (Upper Bound Constraint) = -0.4999395085066163 (satisfied)
ther (Upper Bound Constraint) = -1161260.0 (satisfied)
stress (Upper Bound Constraint) = -1897357.4 (satisfied)
vf (Lower Bound Constraint) = -8241445.700000048 (satisfied)
3. THE INTERTANK
3.1 Design Problem Statement
The Intertank acts as a link between the LH2 and the LO2 tanks of the Space Shuttle. It is
basically a thin walled right circular cylinder and acts as a main load bearing element of the
SSEFT. It transmits the weight of the fuel, the External Tank structural weight and the Orbiter
weight to the Solid Rocket Boosters and Thrust loads from the Solid Rocket Boosters to the
Orbiter. It is also subjected to very high temperature gradients due to the fuel tanks on either
side. First, a simplified geometry is considered to represent the Intertank structure. The
simplicity of the geometry enables us to represent the model with pre defined mathematical
equations to conduct initial studies. The intertank was approximated to a right circular cylinder
of Diameter, D, Length, L and thickness, t. D, L and t are the variables in the optimization
problem. The objective is to minimize the volume of the cylinder by changing these variables.
Since the material we use is constant, we are actually optimizing for minimum weight.
Figure 7: Parts of the Intertank
3.2 NOMENCLATURE
E, Youngs modulus of Aluminum Lithium alloy 2090, Pa
L, Length of the intertank, cm
D, Diameter of intertank, cm
P, Internal Pressure in the intertank, MPa
F, Buckling forces on the Intertank, N
t, Skin thickness of intertank, cm
I, Moment of inertia of a cylinder,
σ, Internal stress, Pa
σy, Max allowable stress of Al-Li alloy
fnat, natural frequency of Aluminum-Li alloy, Hertz
f, Frequency of due to applied loading, hertz
3.3 MATHEMATICAL MODEL:
OBJECTIVE:
minimize f=π(D2-(D-t)2)L
SUBJECT TO:
• g1: An Internal Pressure
Since the geometry is approximated to a cylinder, the hoop stress, σ caused due to the internal
pressure, P is given by the mathematical formula, σ=PD/2t.
This stress must be less than the yield strength of Aluminum7075, σy=96.5MPa
g1: PD/2t < σy
• g2: Buckling stress due to launch forces
Since the geometry is approximated to a cylinder, the buckling stresses, σb caused by the launch
forces, F are given by the the formula, σb=My/I
.
Moment of Inertia, I= [3]
This stress must be less than the yield strength of Aluminum7075, σy=96.5MPa
g2: < σy
• g3: Vibrations at Launch conditions
Since the geometry is approximated to a cylinder, the natural frequency is given by
f= [3]
Moment of inertia, I= [3]
Young’s modulus Aluminum 7075, E=71.7GPa
This frequency must be less than the first natural frequency of aluminum7075, fnat=806Hz
g3: < fnat
• Constraint g1: An internal pressure
Interface forces F are caused at the weld-land between the intertank and the two fuel
tanks.
The average pressure, P caused due to these forces was found to be 833.52e5N per unit
area. This pressure causes hoop stresses inside the intertank. These stresses must be less
than the yield strength of material of Intertank
Figure 8: Internal Pressure
• Constraint g2: Buckling due to the launch forces
The average launch forces acting on the Intertank were found to be F=6200.791e2 N per
unit area from the top and a self equilibrated force from the bottom [2]. Due to these
forces, there can be buckling of the thin walled Intertank. Buckling stresses thus
produced must be less than the yield strength of material of Intertank. Aluminum7075,
σy=96.5MPa
Figure 9: Buckling Forces
• Constraint g3: Vibrations at launch conditions
The natural frequency of the structure depends on the geometry. Since we are changing
the dimensions of the system, it must be kept in mind that the frequency of the system
does not exceed the first natural frequency of Aluminum7075, fnat=806Hz. This avoids
failure due to resonance.
Summary
Minimize f= π(D2-(D-t)2)L
Subject to
g1: PD/2t < σy
g2: < σy
g3: < fnat
Further, due to the manufacturing limitations and space constraints of the launch site, the geometric dimensions were given upper and lower bounds
Geometric Constraints
g4: 400<L<800
g5: 600<D<1000
g6: 0.8<t<1
g7: 0.5< (L/D) <3
3.4 MODEL ANALYSIS
MONOTONICITY ANALYSIS:
Monotonicity analysis was conducted on the above model to determine active and inactive
constraints. The following Monotonicity Table summarizes the analysis.
D T L f + + +
g1 - g2 - g3 -
Table 2: Monotonicity Table
From the monotonicity table, the objective function, f is monotonically increasing in all the
variables. The variables need an upper bound in order to obtain a feasible solution space. On
conducting monotonicity analysis, by MPI, all the constraints are active.
g1 is the stress constraint which is active with respect to the variable, t. It is understood that,
as the thickness decreases, the greater is the stresses produced on the walls of the intertank
g2 is the buckling constraint which is active with respect to the variable, L. The smaller the
length of the intertank, lesser the chances of reaching critical buckling stress.
g3 is the frequency constraint which is active with respect to the variable, D.
3.5 OPTIMIZATION STUDY
The summary model was solved using Excel Solver to obtain initial feasible points. The design
space was sampled at 4 different starting points. All minimized the objective function to
863751.3mm3
S.No f* mm3
Starting Point Optimum Point R=D/2 (mm)
L (mm)
t (mm)
R*=D*/2 (mm)
L* (mm)
t* (mm)
1(random point)
863751.3 2 2 2 357.6304 476.9605 0.806329
2(lower bound)
863751.3 600 400 1 355.729 471.9228 0.819292
3(middle) 863751.3 800 600 1 358.5706 479.444 0.8 Table 3: Optimization Points
The optimum points lie around the same value no matter where we approach it from. A Feasible region can thus be established for the Geometric Constraints.
The lower and upper bounds on the variables were narrowed down using the Excel Solver results
as
300<L<400
800<D<1000
0.7<t<0.9
These bounds also ensure a good L/D ratio.
Now that a rough idea is obtained on the feasible regions, we can model the problem on its actual
geometry. This model has a more complicated geometry since it simulates the existing intertank.
Due to this complication, no mathematical formulae exist to determine stresses and frequencies
produced in the model.
In order to overcome this problem, a Finite Element Model of the intertank was generated in
ANSYS using the ANSYS Parametric Design Language (APDL) in order to run it in Batch
mode. The FEA model was constructed in terms of the variables in the geometry. The model is
then analysed using Finite Elements to obtain the stresses which cannot be otherwise calculated
using simple mathematical equations. The results are written in output files.
The intertank was modeled as a cylinder of length L, thickness, t and radius D. Further, five
stiffener panels of radius H were added at equal distances along the length of the intertank. The
stiffeners suppress Lateral Torsion Deflection of the intertank. Two flanges at the top and bottom
edges also make the intertank more stable.
Figure 10: Actual Geometry of the Intertank
The geometry was then divided into 156 areas and 12 volumes. These volumes were meshed
using the SOLID92 element. The volume mesh generated 23481 nodes. The model’s external
surface was completely constrained so that its outer geometry does not change and influence the
Orbiter’s aerodynamics. The model was then simulated for stress conditions in the Internal
Pressure (g1) and Buckling stress ( g2) constraints.
Figure 11: Meshed Model
Figure 12: Initial Stress and Buckling Analysis
iSIGHT –FD-3.1 optimization software was used to link the FEA analysis using its SIMCODE
palette. The objective function and the modal analysis (g3 constraint) were inputted using the
calculator palette.
Figure 13: Design Gateway Setup
A Gradient Based Algorithm analysis was done, using NLPQL.
A starting point was chosen as, which is close to the Excel Solver’s optimum values.
L=400cm
D=320cm
H=300cm
t=0.7cm
The Results of the Optimization are attached.
3.6 RESULTS AND DISCUSSIONS
• The Optimum Dimensions were found to be
L=434cm
D=370cm
H=336cm
t=0.78cm
• Two constraints were found to be active, namely, the Stress due to Internal Pressure(g1)
and the Buckling Stress(g2). Both of them had Positive Lagrange Multipliers
• The Volume of the Intertank was reduced by 5.4%
Optimization Results
Started on Mon Apr 13 17:37:43 EDT 2009
Optimization Technique: NLPQL
Failed Run Objective Value = 1.0E30
Failed Run Penalty Value = 1.0E30
Max Failed Runs = 5
Max Iterations = 10
Min Abs Step Size = 1.0E‐4
Rel Step Size = 0.1
Save Technique Log = false
Termination Accuracy = 1.0E‐6
Use Central Differences = false
Starting design point:
D = 320.0 [300.0 < x < 400.0]
H = 300.0 [300.0 < x < 400.0]
L = 400.0 [400.0 < x < 500.0]
t = 0.7 [0.7 < x < 0.9]
Completed on Mon Apr 13 20:39:08 EDT 2009
Total design evaluations: 69
Number of feasible designs: 35
NLPQL termination reason: MAXIMUM NUMBER OF ITERATIONS REACHED
Optimum design point:
Run # = 66
Objective = 835016.590120464
Penalty = 0.0
ObjectiveAndPenalty = 835016.590120464
D = 370.08399746107386
H = 336.3079643620858
L = 434.76903526584186
t = 0.7817971312272031
f = 835016.590120464
freq = 0.36965429106139847
strbuc = 88.197
stress = 95.731
Calculated constraint values at the optimum: (constraint value is a difference between the bound or target and the output parameter value, scaled and weighted)
stress (Lower Bound Constraint) = ‐192.231 (satisfied)
stress (Upper Bound Constraint) = ‐0.7690000000000055 (satisfied)
freq (Lower Bound Constraint) = ‐0.36965429106139847 (satisfied)
freq (Upper Bound Constraint) = ‐607.6303457089386 (satisfied)
strbuc (Lower Bound Constraint) = ‐184.697 (satisfied)
strbuc (Upper Bound Constraint) = ‐8.302999999999997 (satisfied)
4.LIQUID HYDROGEN TANK
4.1 Design Problem Statement
The LH2 tank is the bottom portion of the ET. The tank is constructed of four cylindrical barrel
sections, a forward dome, and an aft dome. The barrel sections are joined together by five major
ring frames. These ring frames receive and distribute loads. The forward dome-to-barrel frame
distributes the loads applied through the intertank structure and is also the flange for attaching
the LH2 tank to the intertank. The aft major ring receives orbiter-induced loads from the aft
orbiter support struts and SRB-induced loads from the aft SRB support struts. The remaining
three ring frames distribute orbiter thrust loads and LOX feedline support loads. Loads from the
frames are then distributed through the barrel skin panels. The LH2 tank has a volume of 53,488
cubic feet (1,514.6 m³) at 29.3 psig (3.02 bar absolute) and −423 °F (20.3 K, cryogenic).
The forward and aft domes have the same modified ellipsoidal shape. For the forward dome,
mounting provisions are incorporated for the LH2 vent valve, the LH2 pressurization line fitting,
and the electrical feed-through fitting. The aft dome has a manhole fitting for access to the LH2
feedline screen and a support fitting for the LH2 feedline.
In our project we simplify the structure of the LH2 tank. We consider a cylindrical tank with
hemi spherical domes on both ends. We analyze this modified structure for the same loads which
are present in the actual tank. The main aim is to obtain an optimum design for the given loading
conditions i.e the internal pressure and temperature. We consider stress analysis and vibration
analysis. The optimization is done using ISIGHT which in turn uses ANSYS for the simulation.
The objective is to increase the payload of the tank without compromising on the stability and
strength. We do this by optimizing the weight of the tank. The external fuel tank is not reusable,
so care should be taken such that the material used for the tank is inexpensive and the machining
time is less.
4.2 NOMENCLATURE
M: Mass
R: Radius of the cylinder
H: Height of the cylinder
T: Thickness of the cylinder
σH: Hoop Stress
σc: Critical load
E: Young’s Modulus
I: Moment of Inertia
4.3 MATHEMATICAL MODEL
Objective Function
The objective of the project is to maximize the payload carried by the tank for the same amount
of fuel. This is done by minimizing the weight of the structure. The model is optimized based on
the maximum stress criteria. The design of the structure is changed and the weight is reduced by
keeping the material properties as constant.
For a cylinder with hemispherical ends:
where:
• R is the radius
• L is the middle cylinder length only, and the overall length is L + 2R
• M is mass
• P is the pressure difference from ambient, i.e. the gauge pressure
• ρ is the density of the pressure vessel material
• σ is the maximum working stress that material can tolerate.
Constraints
The constraints can be classified into two types
Physical constraints: These are constraints which define natural laws and engineering
specifications. In this problem the physical constraints are:
Stress analysis: The structural and thermal loads are applied on the model to obtain the
maximum and minimum stress.
Figure 14: Internal Pressure
Figure 14 shows a hollow cylinder, which is subjected to a uniformly distributed internal pressure P.
The figure details an element of material at some radius r, contained within an elemental cylinder.
Due to the design of the thick cylinder the longitudinal stress σL may be ignored and only a bi-axial
system be considered. Hence the stress formulas are shown below:
Figure 15: Stress variation
Buckling analysis: In addition to the stresses in the cylinder, We have to take care of the buckling
loads of the system. Considering a linear relation between stress and strains, we have the following
Euler load for a beam.
The buckling load in the cylinder should be less than this critical load.
Modal analysis: Since the cylinder has some vibration will it is under motion, there might be a
possibility of resonance occurring in the cylinder if the frequency of vibration of the cylinder
exceeds its natural frequency. A Free-Free modal analysis is carried out to make sure that the
given model does not exceed the natural vibration of the material.
Freq(cylinder) ≤Freq(material)
Freq(cylinder), f=
Practical constraints: These constraints provide bounds to the variables used. This includes that
all variables are positive and non zero. For a light weight model we arbitrarily assign bounds on
the variables to find the optimized result.:
Diameter: The diameter of the tank can vary between 5 to 10 meters (actual diameter = 8.4m).
Length: The length of the tank can vary between 10 to 40 meters (actual length = 29.5 m)
Thickness: The thickness of the tank should be positive and non zero.
Design variables and Parameters
Design variables are quantities that specify different states of a system by assuming different
values. Parameters are quantities that are given a specific value in any particular model
statement. They are fixed by the application of the model, rather than by the underlying
phenomenon.
The variables for this problem are:
• Radius of the cylinder
• Length of the cylinder
• Thickness of the cylinder
The parameters are the material properties of the cylinder and the operating conditions. Here we
use Al/Ti alloy (Al 2090-T83). The properties of this alloy at the operating temperature of the
cylinder are:
• Density: 2.81 g/cc
• Tensile strength, Ultimate: 221 MPa.
• Tensile strength, Tensile: 190MPa.
• Elastic modulus: 71.7 GPa
• Poisson’s ratio: .330
Operating conditions are:
• Pressure acting in the cylinder: 302Pa
• Operating temperature: -252.85 C (20.3 K)
Summary Model
Min W = f(R,H,T)
Subject to
G1: abs(σH (min, max)) ≤ abs(σY)
G2: σc ≤σY
G3: freq(cylinder) ≤ freq(natural)
G4: 100 ≤ L ≤400
G5: 50 ≤ D ≤100
G6: T ≥ 0
4.4 MODEL ANALYSIS
Using Monotonicity rules we try to understand the boundness and behavior of the function.
D H T
f + + +
g1 + -
g2 - +
g3 - -
Table 4: Monotonicity Table
The monotonicity analysis concludes that the constraint g1 is active with respect to T, g2 is active with
respect to D and g3 is active with respect to H. This is an approximation of the true model as the stresses
are calculated using ANSYS rather than the approximated mathematical relations.
4.5 OPTIMIZATION STUDY
Using the above problem formulation an optimization study is carried out. One constraint is active and it
has a positive lagrangian multiplier. The result is given below.
Optimization Results
Optimization Technique: NLPQL
Failed Run Objective Value = 1.0E30
Failed Run Penalty Value = 1.0E30
Max Failed Runs = 30
Max Iterations = 80
Min Abs Step Size = 1.0E-4
Rel Step Size = 0.0010
Save Technique Log = false
Termination Accuracy = 1.0E-6
Use Central Differences = false
Starting design point:
h1 = 147.0 [140.0 < x < 155.0]
h2 = 144.0 [140.0 < x < 155.0]
h3 = 143.0 [140.0 < x < 155.0]
r1 = 40.0 [30.0 < x < 50.0]
r2 = 39.0 [30.0 < x < 50.0]
r3 = 20.0 [0.0 < x < 50.0]
Total design evaluations: 71
Number of feasible designs: 42
NLPQL termination reason: LINE SEARCH REQUIRED TOO MANY FUNCTION
EVALUATIONS
Optimum design point:
Run # = 54
Objective = 1229.62101
Penalty = 0.0
ObjectiveAndPenalty = 1229.62101
h1 = 147.3390518942312
h2 = 144.02302300752893
h3 = 143.04498455423982
r1 = 39.39530717495598
r2 = 38.80942412484222
r3 = 20.27829688424548
mass = 1229.62101
max_stress = 21509.7183
min_stress = 12053.4611
ux_max = 0.107045183
uy_max = -0.0810246568
uz_max = 0.107045183
Calculated constraint values at the optimum:
(constraint value is a difference between the bound or target
and the output parameter value, scaled and weighted)
max_stress (Lower Bound Constraint) = -43609.7183 (satisfied)
max_stress (Upper Bound Constraint) = -590.2816999999995 (satisfied)
min_stress (Lower Bound Constraint) = -34153.4611 (satisfied)
min_stress (Upper Bound Constraint) = -10046.5389 (satisfied)
ux_max (Lower Bound Constraint) = -5.107045183 (satisfied)
ux_max (Upper Bound Constraint) = -4.892954817 (satisfied)
uy_max (Lower Bound Constraint) = -4.9189753432 (satisfied)
uy_max (Upper Bound Constraint) = -5.0810246568 (satisfied)
uz_max (Lower Bound Constraint) = -5.107045183 (satisfied)
uz_max (Upper Bound Constraint) = -4.892954817 (satisfied)
4.6. PARAMETRIC STUDY
Parametric study is done by varying the parameter value and studying the sensitivity of the
model. Here the only parameter used is the material properties.
We consider the following materials for our parametric studies:
• Al 7075
o Modulus of Elasticity : 71.5 Gpa
o Poisson’s ratio : 0.33
o Tensile strength : 96.5 Mpa
o Density : 2.81 g/cc
• Al 7175-T66
o Modulus of Elasticity : 72 Gpa
o Poisson’s ratio : 0.33
o Tensile strength : 520 Mpa
o Density : 2.80 g/cc
• Al 7475 – T61
o Modulus of Elasticity : 70.3 Gpa
o Poisson’s ratio : 0.33
o Tensile strength : 490 Mpa
o Density : 2.81 g/cc
There are no design modifications obtained by slight changes in the material properties.
4.7. Discussion of Results
The optimum values are given by:
h1 = 147.33
h2 = 144.02
h3 = 143.04
r1 = 39.395
r2 = 38.809
r3 = 20.278
mass = 1229.62
All dimensions in centimeters.
5. SYSTEM INTEGRATION
5.1 Design Problem Statement
The integration of the three subsystems was modeled in ANSYS. The common variable which is
the radius of the nose cone/radius of the intertank/radius of the liquid hydrogen tank is taken to
be the same for all the three subsytems. Equating all the three radii is important because the
geometry of the structure demands the need to have the same radii for all the sub systems. This
variable is the linking variable between the systems. The entire system is optimized to minimize
the weight of the structure and the first natural frequency. The material of the structure is taken
as Aluminum Lithium alloy which is a parameter. Considering the usage of this material, the
objective function is modified to minimize the volume of the system. The optimization model of
the system is as follows:
Figure 16: System Model
There are three sim code palettes accommodating structural stress constraints, thermal stress
constraints and modal analysis to attain the first natural frequency. The three calculators include
aerodynamic coefficient of drag and the constraints in regard to the volume of fuel carried in the
liquid oxygen tank and the liquid hydrogen tank. The system involves seven design variables and
six constraints.
5.2 NOMENCLATURE
- Coefficient of aerodynamic drag
- Internal volume of liquid oxygen tank, cubic centimeter
- Internal volume of liquid hydrogen tank, cubic centimeter
Cyr- Radius of the cylinder in centimeters
Cyh- Height of the cylinder in centimeters
Cyt- thickness of the cylinder in centimeters
It- intertank thickness in centimeters
Coh- height of the rim in centimeters
Col- length of the cone in centimeters
Cot- thickness of the cone in centimeters
5.3 MATHEMATICAL MODEL
The problem is a multi objective optimization problem where the volume of the system and the
first natural frequency of the system are minimized. These two objectives are calculated within
the ANSYS software. The objectives are subjected to following constraints:
g1: Internal Stresses attained from Simulation - Fstructallowed ≤ 0
g2: External Stresses attained from Simulation - Fstructallowed ≤ 0
g3: Stresses attained from Simulation - Fthermalallowed ≤ 0
g4:
g5: Volume of liquid oxygen carried - ≤ 0
g6: Volume of liquid hydrogen carried - ≤ 0
Coh = 300 [300.0 < x < 500.0] col = 1150 [1150.0 < x < 1350.0] cot = 1 [0.9 < x < 3.0]
cyh = 2800 [2800.0 < x < 3100.0] cyr = 350 [350.0 < x < 550.0] cyt = 1 [0.9 < x < 3.0] it = 1 [0.9 < x < 3.0]
5.4 OPTIMIZATION STUDY
The system is modeled in ANSYS using axisymmetric elements to reduce computation time and
cost. The system is analyzed for structural stresses, thermal stresses and the modal frequency.
The results attained for system integration are different from the results attained from each
subsystem. Individual subsystems were optimized for a single objective. The integrated system is
optimized for multiple objectives. The second objective which is the minimization of the first
natural frequency has a substantial influence over the first objective. It more or less behaves like
an additional constraint. Also the structural and thermal stress constraints where checked over
the entire subsystem. On the system level, the maximum stresses of the entire system are selected
for optimizing rather than the maximum stresses of every sub system. The input files are
included in the appendix. The model of the system in ANSYS using axisymmetric elements is as
follows:
Figure 17: ANSYS Model
Two constraints are shown to be active for the system optimization. The lagrangian multipliers
are positive (0.23D1 and 0.11D1) and satisfy the KKT conditions. The NLPQL algorithm is used
to optimize the model. The optimization runs failed when the model was scaled because of
continuous generation of infeasible points which violated the constraints. The same solution was
reached irrespective of the starting point. Hence, it is a global solution.
5.5 RESULTS and CONCLUSION
Some of the results from system optimization were consistent with those of the sub system
optimization. The variation among other variables is because of the stricter constraint imposed
by the second objective of system optimization. Also the geometry of the sub systems is not
similar to the geometry incorporated in the system. The fasteners in the intertank are removed
and the curved edges of the liquid hydrogen tank are replaced by sharp edges. One design
variable is taken to be common for all the three subsystem and to eliminate complexities some
variables such as the height of the inter tank are taken as constants. All these approximations and
the inclusion of second objective function vary the system results from those of the subsystems.
The model underwent 27 design evaluations and had two feasible designs. The volume of the
material of the system was reduced by 7.2% and the first natural frequency was reduced to 82726
hertz. As a subsystem, the nose cone had a major impact but when included in to the system its
effect is restricted. The project showcased the difficulties in integrating ANSYS with ISight.
Multi Disciplinary optimization problems include more complexity. Multi objective optimization
problems may have the cases where one objective function behaves as a constraint over the other
objective function.
Optimization Results
Optimization Technique: NLPQL Failed Run Objective Value = 1.0E30Failed Run Penalty Value = 1.0E30Max Failed Runs = 5 Max Iterations = 10 Min Abs Step Size = 1.0E-4Rel Step Size = 0.0010Save Technique Log = false
Termination Accuracy = 1.0E-6Use Central Differences = false
Starting design point:
coh = 300 [300.0 < x < 500.0] col = 1150 [1150.0 < x < 1350.0] cot = 1 [0.9 < x < 3.0] cyh = 2800 [2800.0 < x < 3100.0] cyr = 350 [350.0 < x < 550.0] cyt = 1 [0.9 < x < 3.0] it = 1 [0.9 < x < 3.0] Total design evaluations: 27 Number of feasible designs: 2
Optimum design point:
Run # = 21 Objective = 1.2925326E7Penalty = 0.0 ObjectiveAndPenalty = 1.2925326E7coh = 324.0 col = 1350.0 cot = 2.0 cyh = 2800.0 cyr = 350.0 cyt = 1.0 it = 1.0 freq = 82726.0 stress = 442110.0 thestr = 7576300.0 volume = 1.28426E7
6. REFERENCES
1. Heppenheimer, T. A., 2002, Development of the Space Shuttle, 1972-1981, Smithsonian Institution Press, Washington, D.C.
2. Jenkins, Dennis R., 2001, Space Shuttle: The History of the National Space
Transportation System: The First 100 Missions, Specialty Pr Pub & Wholesalers.
3. http://science.ksc.nasa.gov/shuttle/technology/sts-newsref/et.html#et
4. http://www.engineous.com/index.htm
5. Warren C. Young , Richard Budynas “Roark's Formulas for Stress and Strain” McGraw Hill.
6. http://www.uni-bayreuth.de/departments/math/~kschittkowski/nlpql.htm
7. Panos Y. Papalambros and Douglass J. Wilde, “Principles of Optimal Design – Modeling and Computation”, 2nd edition, ISBN 0 521 62727 3, (paperback), Cambridge University Press, 2000
8. R. D. Young, M. P. Nemeth, T. J. Collins, and J. H. Starnes, Jr., NASA Langley Research Center - Hampton, Virginia “Nonlinear Analysis of the Space Shuttle Superlightweight External Fuel Tank”, NASA Technical Paper 3616, December 1996
9. SPACE SHUTTLE EXTERNAL FUEL TANK DESIGN OPTIMIZATION, Massimo
Usan, MIT
10. Multidisciplinary Design Optimization of the Space Shuttle External Tank, Mo-Han Hsieh and Christopher Michael Lawson, MIT
11. insideracingtechnology.com/tech102drag.html
12. Strength of Materials, G.H.Ryder
13. ieeexplore.ieee.org/iel1/20/12212/00560140.pdf?arnumber=560140
14. ANSYS APDLanguage
7. APPENDIX
Input file of System for structural stress and volume
/prep7 cyr=420 cyh= 2950 cyt=2 it=2 coh=400 col= 1260 cot=2 k,1,0,0 k,2,cyr,0 k,3,cyr,cyh k,4,0,cyh k,5,0,cyh-cyt k,6,cyr-cyt,cyh-cyt k,7,cyr-cyt,cyt k,8,0,cyt k,9,cyr,cyh-360 k,10,cyr+it,cyh-360 k,11,cyr+it,cyh+330 k,12,cyr,cyh+330 k,13,0,cyh+80 k,14,cyr,cyh+80 k,15,cyr,cyh+80+coh k,16,0,cyh+80+coh+col k,17,0,cyh+80+coh+col-cot k,18,cyr-cot,cyh+80+coh k,19,cyr-cot,cyh+80+cot k,20,0,cyh+80+cot l,1,2 l,2,3 l,3,4 l,4,5 l,5,6 l,6,7 l,7,8 l,8,1 l,9,10 l,10,11 l,11,12 l,12,9 l,13,14 l,14,15 l,15,16 l,16,17 l,17,18 l,18,19 l,19,20 l,20,13
a,1,2,3,4,5,6,7,8 a,9,10,11,12 a,13,14,15,16,17,18,19,20 et,1,plane42,0,,1 mp,ex,1,75e7 mp,prxy,1,0.34 aesize,2 type,1 mat,1 amesh,all finish /solu antype,0 dl,1,1,all,0 dl,2,1,all,0 dl,3,1,all,0 dl,4,1,all,0 dl,8,1,all,0 dl,9,2,all,0 dl,10,2,all,0 dl,13,3,all,0 dl,14,3,all,0 dl,16,3,all,0 dl,20,3,all,0 sfl,5,pres,2272.63 sfl,6,pres,2272.63 sfl,7,pres,2272.63 sfl,7,pres,(104135780)/(3.14*cyr*cyr) sfl,11,pres,(165354400)/(3.14*it*(2*cyr+it)) sfl,12,pres,416.76e7/(6.28*cyr*cyh) sfl,17,pres,2408.75 sfl,18,pres,2408.75 sfl,19,pres,2408.75 sfl,15,pres,2992900/(9.93*cyr*cyr) solve finish /post1 /OUTPUT,modelstress,dat,, etable,evolume,volu ssum prnsol,s,prin
Input file of System for thermal stress /prep7 cyr=420 cyh= 2950 cyt=2 it=2 coh=400 col= 1260 cot=2 k,1,0,0 k,2,cyr,0 k,3,cyr,cyh k,4,0,cyh k,5,0,cyh-cyt k,6,cyr-cyt,cyh-cyt k,7,cyr-cyt,cyt k,8,0,cyt k,9,cyr,cyh-360 k,10,cyr+it,cyh-360 k,11,cyr+it,cyh+330 k,12,cyr,cyh+330 k,13,0,cyh+80 k,14,cyr,cyh+80 k,15,cyr,cyh+80+coh k,16,0,cyh+80+coh+col k,17,0,cyh+80+coh+col-cot k,18,cyr-cot,cyh+80+coh k,19,cyr-cot,cyh+80+cot k,20,0,cyh+80+cot l,1,2 l,2,3 l,3,4 l,4,5 l,5,6 l,6,7 l,7,8 l,8,1 l,9,10 l,10,11 l,11,12 l,12,9 l,13,14 l,14,15 l,15,16 l,16,17 l,17,18 l,18,19 l,19,20 l,20,13 a,1,2,3,4,5,6,7,8
a,9,10,11,12 a,13,14,15,16,17,18,19,20 et,1,plane77,,,1 mp,kxx,1,8800 aesize,2 type,1 mat,1 amesh,all physics,write,thermal physics,clear etchg,tts mp,ex,1,75e7 mp,prxy,1,0.34 mp,alpx,1,23.6e-7 physics,write,struct physics,clear finish /solu antype,0 physics,read,thermal dl,1,1,temp,283 dl,2,1,temp,283 dl,3,1,temp,270 dl,5,1,temp,20 dl,6,1,temp,20 dl,7,1,temp,20 dl,13,3,temp,270 dl,14,3,temp,283 dl,15,3,temp,283 dl,17,3,temp,50 dl,18,3,temp,50 dl,19,3,temp,50 dl,9,2,temp,283 dl,10,2,temp,283 dl,11,2,temp,283 dl,12,2,temp,270 solve finish /solu physics,read,struct ldread,temp,,,,,,rth dl,all,1,all,0 dl,all,2,all,0 dl,all,3,all,0 solve finish /post1 /OUTPUT,modelthermalstress,dat,,
prnsol,s,prin Input file of system for Modal Analysis /prep7 cyr=420 cyh= 2950 cyt=2 it=2 coh=400 col= 1260 cot=2 k,1,0,0 k,2,cyr,0 k,3,cyr,cyh k,4,0,cyh k,5,0,cyh-cyt k,6,cyr-cyt,cyh-cyt k,7,cyr-cyt,cyt k,8,0,cyt k,9,cyr,cyh-360 k,10,cyr+it,cyh-360 k,11,cyr+it,cyh+330 k,12,cyr,cyh+330 k,13,0,cyh+80 k,14,cyr,cyh+80 k,15,cyr,cyh+80+coh k,16,0,cyh+80+coh+col k,17,0,cyh+80+coh+col-cot k,18,cyr-cot,cyh+80+coh k,19,cyr-cot,cyh+80+cot k,20,0,cyh+80+cot l,1,2 l,2,3 l,3,4 l,4,5 l,5,6 l,6,7 l,7,8 l,8,1 l,9,10 l,10,11 l,11,12 l,12,9 l,13,14 l,14,15 l,15,16 l,16,17 l,17,18 l,18,19 l,19,20 l,20,13
a,1,2,3,4,5,6,7,8 a,9,10,11,12 a,13,14,15,16,17,18,19,20 et,1,plane183,0,,1 mp,ex,1,75e7 mp,prxy,1,0.34 mp,dens,1,2590e-6 aesize,2 type,1 mat,1 amesh,all finish /solu antype,modal modopt,subsp,5 dl,all,1,all,0 dl,all,2,all,0 dl,all,3,all,0 MXPAND,5 solve FINISH /POST1 /OUTPUT,modelfrequency,dat,, set,list set,first pldisp