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Surrogate Based Design Optimization of Aerostat Envelope
Darshit M. Mehta1 and Rajkumar S. Pant2
Indian Institute of Technology Bombay, Mumbai, India, 400076
To improve performance, decrease costs and enhance safety in aerospace systems, typical design
processes involve time consuming and computationally expensive high fidelity models. In this
context, use of surrogate-based approach can play a vital role in analysis and optimization.
Surrogates are models constructed through regression or interpolation using results from high
fidelity models at certain test points. They provide fast approximations of objective functions at new
points thereby making sensitivity and optimization studies much more feasible. This paper presents
an overview of surrogate-based analysis and design (SBAO) and its application to aerostat envelope
optimization. We generated aerostat shapes using parameters created by Optimal Latin Hypercube
Sampling and passed them through a CFD solver. We then constructed various surrogate models and
found Kriging to be a suitable surrogate model because of its accuracy, as proved by the cross
validation tests. Using Kriging, the envelope is optimized for drag using the Efficient Global
Optimization algorithm.
Nomenclature
Ai , Bi = coefficients of cubic splines
Cp = pressure coefficient
CD = drag force coefficient
CDv = drag force coefficient based on volume
EGO = Efficient Global Optimization
k = number of design variables
PRS = Polynomial Response Surface
PRESS = Predicted Sum of Squares (Cross Validation Error)
RBF = Radial Basis Functions
Re,v = Reynolds Number based on Volume
X = Points in design space
Y = Responses at design points
I. Introduction
HERE have been only few attempts at aerostat optimization using surrogate-based approach and possibility of
use of various surrogate types for this purpose has been largely unexplored. The aim of this investigation is to
find an optimum shape of the aerostat envelope (with fins added for stability), which results in the greatest payload
capacity for a given volume of the aerostat, using robust design of experiments along with different surrogates. The
accuracy of each surrogate model will be measured and validated. The attempt is directed towards arriving at a
simple but generic methodology for estimation of coefficient of drag as a function of envelope geometry. This can
be coupled to multi-disciplinary optimization algorithms to determine the optimum envelope shape from various
considerations.
II. Historical development of Surrogate Methodology
A surrogate can be considered to be a model of a model, i.e., it mimics the behavior of the system model while
being computationally cheaper. Surrogate Based Analysis and Optimization (SBAO) involves choosing few design
points intelligently, evaluating responses at those points experimentally or computationally, fitting a surrogate on
those points, identifying new design points for analysis, analyzing at those points and reiterating. Surrogate
modeling can be seen as a non-linear inverse problem, for which one aims to determine a continuous function (f) of
1Graduate Student, Aerospace Engineering Department, IIT Bombay, Student Member, AIAA. 2Professor, Aerospace Engineering Department, IIT Bombay, Member, AIAA.
T
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a set of design variables from a limited amount of available data (f). To obtain an accurate approximation, it is
necessary to have a good training set and a proper surrogate. Much of the research on surrogates has been done in
these two fields, viz., design of experiments and selection and validation of surrogate models. Queipo et al.1 and
Forrester and Keane2 have reviewed the state of the art in SBDO in general; Ahmed and Qin3 have presented a
review of literature related to its application in the field of aerospace engineering.
The development of RSM was initiated by Box and Wilson4, followed by a comprehensive account of the
methods of using RSM by Myers and Montgomery5. Polynomial Response Surface (PRS) is the most widely used
surrogate model, since it is computationally inexpensive, and easy to implement. However, it has been shown
unsuitable for highly non-linear and irregular performance problems6, for which alternative surrogates like Kriging
and Radial Basis Functions were proposed.
Kriging was formally developed by Matheron7, and was named after a South African mining engineer. In
Kriging, the error at an un-sampled is not treated as a random variable, but is a function of its distance from sampled
points. Kriging considers the value of a function as sum of a general trend (could even be a constant) and a
systematic departure. Sacks et al.8, and Jones and Schonlau9 have demonstrated the efficiency of Kriging for
modeling and optimization of deterministic functions. Attempts have also been made to increase the performance of
Kriging by using approaches like Analysis of Variance (ANOVA)10 and gradient data (co-Kriging)11. Radial Basis
Functions (RBFs) such as Gaussian, use linear combination of radially symmetric functions to approximate response
functions12. RBFs have a special feature that their response decreases (or increases) monotonically with distance
from a central point. Michler and Heinrich13 have used RBFs to create surrogate models for simulating a fighter
aircraft in trimmed state.
Artificial neural network model, a nonparametric regression method, utilizes the functional concept of neurons in
the brain.14A neural network is composed of neurons which can be represented as nonlinear transfer functions of the
inputs. Support Vector Regression (SVR) is a relatively new method for surrogate modeling, which is based on work
by Vapnik at Bell Labs15. In SVR, a model is produced by only using points which lie on or outside a specified error
region. Analogously, the model produced by SVR only depends on a subset of the training data, because the cost
function for building the model ignores any training data that is close (within a threshold ε) to the model
prediction16.
III. Optimization of Aerostat Envelope Shape
For optimization of airship shape, a large body of literature is available for drag calculation and shape
optimization of axi-symmetric bodies of revolution submerged in incompressible flow. Parsons and Goodson17 were
the first to report application of numerical optimization techniques; they represented the rounded-nose tail boom
bodies by eight parameters, and coupled a boundary-layer method to a panel code. By exploiting laminar flow while
avoiding turbulent separation, they obtained a body with a drag coefficient one-third below the best existing laminar
design. Pinebrook18 has suggested a method that calculates momentum deficit in the boundary layer at the trailing
edge, and uses Evolution Strategy for finding the minimum drag.
Lutz and Wagner19 have developed a tool for numerical optimization of airships. They modeled the body contour
by specifying a source distribution on the body axis and changed the body shape by changing the strength of the
sources. They have obtained optimized hull shapes for different Reynolds number regimes. Using the semi empirical
en – method based on linear stability to find the transition point, they found that the transition criterion has a high
impact on the optimization result. Kanikdale et al.20 used a composite function that included drag, surface area and
the maximum hoop stress for multi-disciplinary optimization.
IV. Application of Surrogate Methodology for optimization
Kale et al.21 attempted aerostat optimization using a Polynomial Response Surface and obtained a co-relation
between CDV and some geometry related parameters. The aim was to eliminate the need of using a flow solver for
determination of CDV in all iterations of the optimization process. They used a shape generation algorithm to
generate around 600 feasible shapes, and computed CDV using FLUENTTM. A correlation was obtained by fitting a
quadratic response surface, but the use of other surrogate models was not explored.
The aim of this study is finding an optimum envelope shape using appropriate surrogate methods that minimizes
volumetric drag coefficient CDV of an envelope of a given volume. Surrogate based optimization eliminates the
need for running CFD codes each time the shape is altered. Parametrically varied aerostat shapes are provided as
input to CFD analysis, which determines aerodynamic coefficients as function of the shape parameter. Surrogate
models can then be fitted on the CFD results to identify optimum aerostat shape, thus considerably simplifying the
design process. Aerostat shapes were parameterized in MATLABTM 24. The design of experiments, fitting of
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surrogate models, cross validation, and optimization were all carried out using the SURROGATES Toolbox for
MATLABTM by Viana23. Generated shapes were meshed in GAMBITTM and drag forces were calculated in
FLUENTTM 24. The basic flowchart of the surrogate based design optimization suggested by Viana et al.25 is shown
in Fig. 1
Figure 1. Surrogate Based Design Optimization25
Figure 2. Shape Parametrization20
A. Shape parameterization of aerostat envelope
Based on the work by Kanikdale et al.20, the aerostat shape was defined as a generic envelope profile in terms of
a combination of two cubic-splines, with a spherical cap in the front portion, and a parabolic shape in the end, as
shown in Fig. 2.
The selection of a spherical cap for the nose portion enables the shape to be compatible with the spherical
mooring cups that are in use in the winching and mooring systems of aerostats. A parabola was selected for the rear
portion to make attachment of fins easier. The origin is at the leading edge of the envelope, ordinate x is along the envelope length, and abscissa y is
normal to it. The equations describing each part of the geometry are as follows
Sphere (Circle in 2-D): y2=2xR-x2 (1)
Spline I: y= a1x3+b1x
2+ c1x+d1 (2)
Spline II: y=a2x3+b2x
2+c2x+d2 (3)
Parabola: y2=an(L-x) (4)
The total number of unknowns in Eq. 1-4 are eleven, namely (R,a1,b2,c1,d1,a2,b2,c2,d2,an,L). The size of the
design vector can be found out after specifying constraints in the form of boundary conditions for the governing
equations. The various constraints and conditions imposed on the geometry are as follows
Volume of airship = 2000 m3
Slope continuity at point (x1, y1)
Slope continuity at point (x2, y2)
Slope continuity at point (x3, y3)
Slope at maximum diameter location (x2, y2) = 0
Using these five conditions, the size of the design vector can be reduced from eleven to six as given by Eqn 5.
XD= (x1, x2, y2, x3, x4, R) (5)
B. Design of Experiments (DOE)
DOE is an important part of surrogate modeling. Having a large number of points increases accuracy but is also
expensive. The points should be space filling to give good global results while being in a higher concentration at
places where there is significant variation and curvature. To address such trade-offs, different designs exist including
classical methods like full-factorial design (FFD), partial factorial design (PFD), face-centered cubic (FCC), central
composite design (CCD) and D-optimal design.
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These schemes are easy to implement, but number of points increases rapidly with factors and levels. To counter
these difficulties, Space Filling Designs are used where points tend to uniformly cover the entire design space. They
are used where deterministic errors are expected such as in computer experiments. Latin Hypercube Sampling
(LHS)26 and Orthogonal arrays (OA)27 employ space filling (See Fig. 3).
OA produces uniform design but can generate particular
forms of point replication and is sometimes inflexible while
LHS does not produce point replicates but is not uniform.
To address these concerns, OA-based LHS 28, 29 and other
optimal LHS schemes30 have been proposed.
Using a thumb rule of 10k design points9, an Optimal
LHS design (OLHS) consisting of 60 training points was
generated. Table 1 lists the details of the design space
considered, and lists the parameters of first four training
shapes.
Table 1 Sample points in the Design Space considered
Design
Parameter x1 x2 y2 x3 x4 R
Minimum 1.02 8.07 4.70 18.89 24.82 1.67
Maximum 2.13 25.50 6.33 36.93 46.96 4.02
Example
Training
Shapes
1 1.83 17.06 5.5 23.8 34.82 2.87
2 1.06 16.14 4.82 35.32 43.94 2.14
3 1.22 17.1 5.28 31.86 38.26 1.85
4 1.37 14.54 5.07 30.46 38.93 2.17
Figure 4 shows the profiles of the four training shapes listed in Table 1, along with the location of the maximum
diameter point.
Figure 4. Example Training Shapes
C. Aerodynamic solution
The generated shapes were passed onto GAMBITTM for meshing. Since the drag coefficient was measured for only
zero angle of attack, the axi-symmetric solver in FLUENTTM was used. Each mesh had 19200 cells with at least 10
cells in the boundary layer found by the empirical relation for turbulent flow, i. e.,5/1Re/382.0 xx . Using
Spallart-Allarmas viscous model, the CDV obtained for GNVR shape was 0.029 which was a good match with the
experimental results20, and was used for the calculation of the drag of the training shapes. The simulations were
carried out at reference atmospheric pressure of 0.85 bar and Mach. No. 0.15
Figure 3. Design of Experiments (DOE)
(a) Classical Design. (b) Space Filling Design
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a) Grid with boundary conditions
b) Detailed grid near aerostat
c) Static pressure distribution
d) Boundary layer visualization
Figure 1. Sample results of CFD calculations
D. Surrogate Fitting
Kriging, PRS and Radial Basis Functions surrogate models were fitted on the responses generated by the test
shapes. Table 2 lists the standard equations that were used for these surrogate models.
Table 2: Standard Equations of various surrogate methods
Model Kriging PRS RBF
Standard
Equations
∑ | |
is the correlation equation
Response at an unknown x is
given by:
is the Gram Matrix defined as
(‖ ‖)
And is the mean often treated as
a constant
and p are the additional
parameters of Kriging which can be
used to control the „smoothness‟
and „activity‟
∑
∑∑
Where k = Number of design
parameters
= Regression parameters
xi = Design variables
is estimated as
[
]
and n = Number of training points
∑
Where w = Weights
= Radial Basis Functions
= Errors with variance σ2
is estimated as
[ (
) ( ) ( )
( ) (
) ( )
( ) (
) ( )]
and n = Number of training points
Cross
Validation
Error
0.0150 0.0165 0.0148
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The accuracy of fit of each surrogate model was calculated using cross validation analysis31, which involves
division of data into k subsets of approximately equal size. A surrogate model is constructed k times, each time
leaving out one of the subsets from training, and using the omitted subset to compute the error measure of interest. It
gives an unbiased estimation of the generalization error.
Kriging has cross validation error slightly higher than that of RBF. But due to suitability for combining with the
EGO algorithm, it was chosen as the surrogate model.
E. Optimization
One way to use surrogate model for optimization would be to test the point where the predicted response is
minimum. But this would only lead to local search and does not address the issues of poor sampling or bad surrogate
fitting.
The Efficient Global Optimization (EGO) algorithm proposed by Jones and Schonlau9 provides a solution to this,
by introducing a new parameter called the Expected Value of Improvement (E[I(x)]). E[I(x)] is the expected value
of Probability of Improvement P[I(x)](shown as the shaded region in Fig. 6). In this algorithm, new design points
are added where there is a high E(I). E(I) increases at points near the minimum of the predictor (local exploitation)
and where there is a high predictor error (global search) thus leading to global convergence (Refer Fig. 7). The
Expected Improvement is calculated in the entire design space and the location where it is maximized, is selected for
further testing.
Figure 2. Graphical representation of Probability of
Improvement2
Figure 3. Predicted response and E(I)
9
E(I) is high near x=2.2 where there is a minimum of the
surrogate and at x=8 where sampling is poor and hence
predictor error is higher.
F. Results
For optimization in our case, EGO was run for 29 iterations and was stopped because E(I) dropped down to
0.0019 i.e. 10% of the present best solution. The iterations at which there was an improvement are listed in Table 3.
Table 3: Iterations at which improvements were observed
Iteration
Number CD,min
Design vector
x1 x2 y2 x3 x4 R
0
(initial) 0.0257 1.85 25.50 4.70 35.03 46.27 2.36
11 0.0242 1.23 14.72 4.70 36.60 46.96 2.35
26 0.0205 1.81 23.08 4.70 35.51 45.04 2.61
It can be seen from Table 3 that the optimum design was driven to minimum y2 and maximum x4 of the design
space as that would make the body streamlined. For comparison, the CP distributions of the three shapes with lowest
drag are shown in Fig. 8.
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Figure 8. Cp distribution of shapes obtained through optimization
Optimization was achieved using Kriging and EGO with only 29 more runs of the high fidelity model. The
computational time breakdown of various tasks is given in Table 4. The time taken for aerodynamic simulation
would depend on the configuration of the system, quality of mesh, solver model, and the discretization scheme used,
but the table gives a rough idea of the time scales.
Table 4: Computational time taken for each task
DOE Shape Generation Meshing Aerodynamic solution Surrogate Fitting Optimization
20s 1s 95s/iteration 180s/iteration 3s 40s/iteration
Thus despite the fact this was an axisymmetric flow analysis, a major chunk of time is spent on solving the flow
equations, while surrogates themselves hardly take any time. Higher fidelity codes such as CFD in 3-dimensions or
Finite Element Analysis of various structures can take hours or even days to complete and hence surrogates are so
useful since they reduce the number of runs of these codes.
V. Conclusion
An overview of the surrogate methodology along with its application for optimization of aerostat envelope has
been provided in this study. Effectiveness of surrogates has been demonstrated for optimization of time consuming
and computationally expensive problems. Surrogate Based Design Optimization can be used for a wide range of
design problems and its quick and easy to implement nature makes it very attractive to use. Future work could
include exploring more surrogate schemes such as Support Vector Regression (SVR), Neural Networks (NN), and
the Weighted Average Surrogate, which uses a combination of surrogate models.
For the optimization of aerostat shape, the entire body including the fins and tether could be considered instead
of just the envelope. The objective would then be a composite function such as maximizing the payload and
minimizing the cost.
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