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American Institute of Aeronautics and Astronautics

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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 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)

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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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