1

Star Grain Optimization using Genetic Algorithm

Ali Kamran1, Liang Guozhu

2, Amer Farhan Rafique

3, Saima Naz

4, Qasim Zeeshan

5

School of Astronautics, Beijing University of Aeronautics and Astronautics (BUAA), 37 Xue Yuan Road, Beijing

China, 100191

The research effort optimizes Star grain configuration with geometric and ballistic objective functions for

neutral and non-neutral configuration thru genetic algorithm. Latin Hypercube Sampling provides efficient

design space filling for geometry of Star configuration. CAD module performs parametric modeling of the

entire family of convex Star grain. Equilibrium pressure method calculates the internal ballistics. Two

configurations are used as test cases. Results indicate that ballistic objective function is appropriate as

compared to geometrical objective functions. In addition, the non-neutral configuration provides higher

initial pressure, thereby realizing significant increase in web-time total impulse as compared to neutral

configurations.

Nomenclature

Ab = Burning area (mm2)

At = Area of throat (mm2)

a = Burning rate exponent (mm/s/Pan )

Cf = Thrust coefficient

c* = Characteristic velocity (m/s)

F = Thrust (kN)

Fav = Average thrust (kN)

I = Action time total impulse (kN-s)

Isp = Specific impulse (s)

It = Web time total impulse (kN-s)

L = Length of grain (mm)

mp = Mass of propellant (kg)

N = Number of star points

Neu = Neutrality

n = Pressure exponent

pamb = Ambient pressure (bar)

Pav = Average pressure (bar)

pc = Chamber pressure (bar)

pe = Nozzle exit pressure (bar)

pmax = Maximum pressure (bar)

R = Grain outer radius (mm)

r1 = Fillet radius (mm)

r2 = Cusp radius (mm)

t = Action time (s)

tb = Web burn-out time (s)

Vp = Volume of propellant (m3)

ΔV = Volume Change

wf = Web fraction

Δw = Web change

ρp = Propellant density (kg/m3)

η = Valley angle (deg)

ε = Angular fraction

1 Ali Kamran, PhD candidate, School of Astronautics, Student member AIAA, a1k1s1@yahoo.com

2 Liang Guozhu, Prof, School of Astronautics, lgz@buaa.edu.cn

3 Amer Farhan Rafique, PhD candidate, School of Astronautics, Student member AIAA, afrafique@sa.buaa.edu.cn

4 Saima Naz, Mathematician, Institute of Space Technology, Karachi Campus

5 Qasim Zeeshan, PhD candidate, School of Astronautics, Student member AIAA, qasimzeeshan@sa.buaa.edu.cn

51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference<BR> 18th12 - 15 April 2010, Orlando, Florida

AIAA 2010-3084

Copyright © 2010 by Ali Kamran. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

2

= Area ratio

σ = Sliver fraction

ηv = Loading fraction

I. INTRODUCTION

rain design is to evolve burning surface area and develop the relationship with web burnt1. Grain design can

prove to be vital in terms of minimizing inert mass by adopting a high volumetric efficiency with minimum

possible sliver.

Star grain, being well known for its simplicity, reliability, neutral burning and efficiency, has been graded

amongst the best and widely used grain configurations for more than sixty years. It can be applied to a reasonable

range of web fraction thus encompassing range of volumetric loading fraction. Certain disadvantages like sliver and

undesirable tail-off are objectionable but affects of the same can be reduced by optimization.

Previous design work of star grain can be summarized as follows: Ref.2 has given non-dimensional equation for

the design of two dimensional star grain. Ref.3 describes the complete set of analytical expressions for star grain

geometries including convex and concave star geometries. Ref.4 provides generalize approach to determine the

geometry parameters required for star grain analysis. Though, there is enough research done in the design of under

study grain configuration, but optimization area is still lacking in both depth and breadth. Previous research uses a

parametric approach with different number of variables in the optimization process. Following lines summarizes the

research on optimization of star configuration: Ref.5 uses four design variables, generates plots of web fraction,

progressivity ratio, and sliver fraction by varying volumetric loading with a fixed value of star points. Ref.6

considers five design variables, selects a dimensionless form of the objectives such as sliver, volumetric loading and

burning perimeter plotted against web fraction. It generates carpet plots for fix value of star points with cusp radius

of zero. Ref.7 uses six design variables. The approach identifies, for a given volumetric loading fraction, the most

neutral-burning star with given web fraction, symmetry number and two small radii. A parametric analysis

investigates the effects of design variables. The following observations can be concluded regarding the above

described work:

1. The above methods are graphical in nature.

2. Set of equations for commonly used star configuration are used for optimization.

3. Geometrical parameters are considered in detail with limited treatment of ballistic constraints.

4. All methods design a star configuration which is close to a neutral trace in search of the best design.

The current research expands the design space by taking all possible six variables, entire convex star faimly

3

with and without taking neutrality into consideration. Three different types of objectives are sought; Maximum

volumetric loading fraction, minimum sliver fraction and the web time total impulse. Grain geometry is a parametric

model with dynamic variables. Surface offset simulates grain burning regression and calculates subsequent volume

G

CAD

based Grain Design Module

Grain

Boundary Star parameters

Geometric

calculation

Design Variables (X)

OPTIMIZATION

______________________ Find: Optimum Design

Variables (X*)

Satisfy: Constraints

Geometry

Ballistics

Design of Experiments

___________________

Latin Hypercube

Sampling

Optimal Design (X*)

Visual Basic

____________________ Read: Design Variables (X)

Update Variables

Satisfy: Calculation to maximum

Web

Write: Output

Fig.1 Design and Optimization Process

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at each step. Meta-heuristic approach is adopted to optimize the star configuration using Latin Hypercube Sampling

(LHS) to create efficient initial population of the design variables and thus reducing time of the overall process. Fig.

1 presents the flow chart of the process.

II. GEOMETRIC MODEL AND PERFORMANCE PREDICTION

Ref. 3 explains eight different types of configuration with seven different zones of burning. In all cases neutral

burning in the second zone is a function of star points and valley angle. For neutral burning the equation for valley

angle is as under:

22tan

N (1)

In the present study CAD modeling is adopted instead of programming a large number of equations for

considering all convex configurations. Equation (1) calculates valley angle for neutral burning in the second zone.

Fig. 2 presents the geometry, different zones of burning and design variables of Star configuration.

Following steps are taken for constructing the parametric geometric model for grain configuration:

Grain boundary is a solid and constructed by revolve protrusion with no burning surface and length equal to

grain length.

Grain bore (star) is constructed by extruded surface and all surfaces burning.

Surface offset function available in CAD software is used to simulate burning, by offsetting the surface by

a web increment equal and orthogonal in all direction.

Boolean function is used at each web increment to subtract the solid within grain bore to create new

geometry.

Offsetting and Boolean operations are repeated till web is completely burnt.

The grain regression is achieved by equal web increment in all directions. At each step new grain geometry is

created automatically thereafter volume at each web increment is calculated. A decreasing trend is obtained for

volume of the grain. Burning surface area is calculated as:

w

VAb

(2)

Propellant Mass is calculated as:

ppp Vm (3)

r2

r1

w

R

η

ζ

Number of star points N

Web fraction wf

Grain outer radius R

Fillet radius r1

Cusp Radius r2

Valley angle η

Angle coefficient ζ=επ/N

Web burnt

P

e

r

i

m

e

t

e

r

4

1

2

3

Zone 1: Burns till r2 vanishes

Zone 2: Progressive, regressive or neutral

depending upon η and N

Zone 3: Typically progressive

Zone 4: Sliver

Fig.2 Star grain geometry description

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Table.1 gives the descriptions of design variables for grain burning regression.

Table 1 Design Variables for Grain Geometry

Design Variables Units Symbol

Grain length mm L

Grain outer radius mm R

Web fraction - wf

Number of star points - N

Valley angle deg η

Angular fraction deg ε

Fillet radius mm r1

Cusp radius mm r2

The grain length to diameter ratio is less than 4 with port-to-throat area ratio more than 2, suggesting lower gas

velocity in port thus allowing the use of equilibrium pressure ballistic model. Steady state pressure is calculated by

equating mass generated in chamber to mass ejected through nozzle throat8,9

. The chamber pressure is calculated as:

)1(1* n

pc Kacp

(4)

Where K =Ab / At

The thrust and thrust coefficient are calculated as

tcF ApCF (5)

c

ambe

c

e

fp

pp

p

pC

11

12

11

2

1

2

(6)

Web time total impulse is calculated as

bt FtI (7)

Action time total impulse is calculated as

FtI (8)

Specific impulse is calculated as

gm

II

p

sp (9)

Neutrality is defined as

avp

pNeu max

(10)

III. OPTIMIZATION

A. Design Objective

These objective functions are chosen to investigate the effects of ballistic and geometrical objective functions on

design and performance of star grain with fixed length and diameter of grain while adhering to stringent constraints

of burning time, grain mass, neutrality and maximum pressure. The mathematical description of design objectives

are as follows:

Max It (X) ,

Max ηv (X) ,

Min σ (X) (11)

Where the design variable (X) is:

X = f (R, L, wf, N, r1, r2, η, ε)

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B. Design Constraints

Present study identifies two types of star in terms of non neutral and neutral burning in second zone with same

constraints. Eq. 12 presents the set of constraints.

2000:

2119:

670650:

20.1:

65:

272:

as;given is Where

6.....,2,1 ,0

6

5

4

3

max2

1

LC

tC

mC

NeuC

pC

RC

C

iC

b

p

i

(12)

C. Design bounds

Table.2 shows the values of lower and upper bounds of star grain design variables.

Table. 2 Bound of design variables

*Only for non-neutral star.

D. Optimization Method

The most popular methods that go beyond simple local search are GAs. Heuristic methods are able to handle

both discrete and continuous variables, making them well suited to large, multidisciplinary design problems. Among

the heuristic search methods, there are the ones that apply local search (e.g., hill climbing) and the ones that use a

non convex optimization approach, in which cost-deteriorating neighbors are accepted also.

Genetic algorithm is capable of examining historical data from previous design attempts to look for patterns in

the input parameters which produce favorable output. GA uses neither sensitivity derivatives nor a reasonable

starting solution and yet proves to be a powerful optimization tool. Being a non-calculus, direct search based global

search method, it allows to be applied in the design phase, which traditionally has been dominated by qualitative or

subjective decision making.

To perform its optimization-like process, the GA employs three operators to propagate its population from one

generation to another. The first operator is the “Selection” operator that mimics the principal of “Survival of the

Fittest”. The second operator is the “Crossover” operator, which mimics mating in biological populations. The

crossover operator propagates features of good surviving designs from the current population into the future

population, which will have better fitness value on average. The last operator is “Mutation”, which promotes

diversity in population characteristics. The mutation operator allows for global search of the design space and

prevents the algorithm from getting trapped in local minima. Details of GA are found in literature

10-12. Pseudo-code

of the optimization is listed as under;

Design Variables Symbols Units LB UB

Web fraction wf - 0.38 0.50

Number of star point

N - 3 11

Valley angle*

η deg 20 45

Angular fraction ε - 0.5 1

Fillet radius r1 mm 5 50

Cusp radius r2 mm 5 50

6

Optimization Routine

Initialize

Set population size

Set total number of generation

Set stopping criteria

While (stopping criteria NOT achieved)

Create public-board to store information

Generate population (random)

For i = 1 to total generations

For j = 1 to population size

Call VISUAL BASIC

Arrange INPUT data for CAD

Call CAD

For k = 1 to web

(a) Make Grain Geometry (b) Calculate physical properties

(c) Write OUTPUT data

End

End

Call BALLISTICS

Evaluate Constraints

Evaluate Fitness

CALL CROSSOVER

Check crossover rate

Create new off-springs

CALL MUTATION

Mutate prescribed amount of individuals (random)

Send information to Public-board

End

End

E. Design of Experiments Module

Design of Experiment (DOE) strategies help to explore the design space of interest to generate approximate

model of each of the decision variables. Sampling, a statistical procedure, involving the selection of a set of

individuals to represent and deduce appropriate knowledge about a solution space of concern.

Latin Hypercube Sampling

Latin hypercube sampling is a stratified random procedure that provides an efficient way of sampling variables

from their multivariate distributions. LHS is better than random sampling for estimating the mean and the population

distribution function. LHS is asymptotically better than random sampling in that it provides an estimator with lower

variance. LHS yields biased estimates of the variance of the output variables13-15

. LHS follows the concept of a Latin

square where there is only one sample in each row and each column. Latin hypercube generalizes this concept to an

arbitrary number of dimensions. In LHS of a multivariate distribution, a sample size m from multiple variables is

drawn such that for each variable the sample is marginally maximally stratified. A sample is maximally stratified

when the number of strata equals the sample size m and when the probability of falling in each of the strata is m-1.

Given k variables X1; . . . ;Xk the range of each variable X is divided into m equally probable intervals (strata), then

for each variable a random sample is taken at each interval (stratum). The m values obtained for each of the

variables are then paired with each other either in a random way or based on some rules. Finally we have m samples,

where the samples cover the m intervals for all variables. This sampling scheme does not require more samples for

more dimensions (variables) and ensures that each of the variables in X is represented in a fully stratified manner.

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The LHS algorithm is as follows: divide the distribution of each variable X into m equiprobable intervals; for the ith

interval, the sampled cumulative probability is:

Probi = (1/m) ru + (i - 1) / m (13)

Where ru is a uniform random number ranging from 0 to 1; transform the probability into the sampled value x

using the inverse of the distribution function DF-1

:

X= DF-1

(Prob) (14)

The m values obtained for each X are paired randomly or in some prescribed order with the m values of the other

variables. LHS “space filling” design strategy is used to treat all regions of the design space equally.

IV. RESULTS

Neutral burning and non-neutral burning star configurations are studied with same constraints to infer affects of

burning zones w.r.t. different objectives. This study estimates two geometrical objectives (loading fraction, sliver

fraction) and one ballistic objective (Web-time total impulse) for both the case. These objective functions are chosen

to investigate the effects of ballistic and geometrical objective functions on design and performance of star grain.

The diameter of star grain is fixed at 544 mm and length at 2000 mm.

A. Neutral burning

The first case has neutral profile in the second zone of burning. The valley angle in this case has a fixed value

depending upon the number of star points and is calculated using Eq. 1. Table.3 lists the values of optimum design

variables for all objective functions. It is evident that the ballistic objective has the highest web fraction with the

maximum number of star points; consequently it has the maximum web burn-out time leading to highest total

impulse of all cases of objective functions. Table 4 lists the values of ballistic and geometrical parameters for all

objective functions.

Table. 3 Optimum values of design variables

As already noted, the ballistic optimum star provides about 0.2 % higher web-time total impulse as compared to

geometrical optimum, with an overall higher specific impulse.

Table. 4 Ballistic Performance

Parameter Symbol Unit Min. Sliver Max. Total Impulse Max. Loading

Web time average thrust Fav kN 68.61 68.55 68.92

Mass of propellant mp kg 651.2 656.4 669.99

Burning time tb s 20.014 20.294 19.762

Web time average pressure Pav bar 51.22 50.73 52.02

Maximum Pressure Pmax bar 60.59 60.52 60.58

Web time neutrality Neu - 1.1829 1.1924 1.1644

Web time total impulse It kN-s 1513.08 1520.4 1517.1

Action time total impulse I kN-s 1562 1580 1610

Specific impulse Isp s 244.5 245.4 245.0

Sliver fraction σ - 0.04284 0.04633 0.06385

Loading fraction ηv - 0.8004 0.8069 0.8236

Design Variables Symbols Units Min. Sliver Max. Total Impulse Max. Loading

Web fraction wf - 0.42065 0.42572 0.41749

Number of star points N - 6 8 7

Valley angle η deg 33.53 37.31 35.56

Angular fraction ε deg 0.7403 0.9254 0.9633

Fillet radius r1 mm 5.412 6.063 11.672

Cusp radius r2 mm 11.394 9.303 7.118

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Fig. 3 depicts the pressure time histories clearly suggesting burring of geometrical objective stars at slightly

higher pressure and more importantly in lesser web burn-out time. Fig. 4 presents the dimensions and sketches of

optimum star grain configuration.

Figure. 3 Pressure time trace

B. Non-Neutral burning

Second case has non-neutral profile in second zone of burning. The valley angle in this case follows the value

stated in Table 2. Table 5 lists the values of optimum design variables for all objective functions. It is evident that

the ballistic objective has the highest angular fraction with maximum number of star points; consequently it has the

maximum initial burning area, thus higher average pressure with approximately same web burn-out time leading to

highest total impulse of all cases of objective functions. Table 6 lists the values of ballistic and geometrical

parameters for all objective functions.

0 5 10 15 20 250

10

20

30

40

50

60

70

Time(s)

Pre

ssu

re(b

ar)

Max. Impulse

Max. Loading

Min. Sliver

Fig. 4 Optimum Star Configurations

9

Table. 5 Optimum values of design variables

As already noted, the ballistic optimum star provides about 0.9 % higher web time total impulse compared to

geometrical optimum, with an overall higher specific impulse.

Table. 6 Ballistic Performance

Parameter Symbol Unit Min. Sliver Max. Total Impulse Max. Loading

Web time average thrust Fav kN 72.35 73.65 67.11

Mass of propellant mp kg 653.73 665.93 669.92

Burning time tb s 20.369 20.178 20.382

Web time average pressure Pav bar 51.88 52.74 50.77

Maximum Pressure Pmax bar 60.54 61.86 60.07

Web time neutrality Neu - 1.167 1.172 1.183

Web time total impulse It kN-s 1557.8 1571.9 1525.1

Action time total impulse I kN-s 1588.2 1621.1 1599.7

Specific impulse Isp s 247.6 248.1 243.4

Sliver fraction σ - 0.0249 0.0344 0.0591

Loading fraction ηv - 0.8036 0.8186 0.8235

Fig. 5 depicts the pressure time histories clearly indicating, burring of ballistic objective stars at higher initial

pressure and contributing to a higher total impulse. Fig. 6 presents the dimensions and sketches of optimum star

grain configuration.

Figure. 5 Pressure time trace

0 5 10 15 20 250

10

20

30

40

50

60

70

Time(s)

Pre

ssure

(bar)

Max. Impulse

Max. Loading

Min. Sliver

Design Variables Symbols Units Min. Sliver Max. Total Impulse Max. Loading

Web fraction wf - 0.42992 0.43036 0.43109

Number of star points N - 9 10 5

Valley angle η deg 22.093 25.722 21.922

Angular fraction ε - 0.826963 0.9512 0.7530

Fillet radius r1 mm 5.9762 6.1413 17.239

Cusp radius r2 mm 22.111 18.847 27.252

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V. CONCLUSION

A technique for design and optimization of star grain has been conducted employing geometrical and ballistic

objectives under specific constraints for neutral and non-neutral star configurations. The inclusion of genetic

algorithm in liaison with CAD modeling of geometry, catering for entire convex star family, with different objective

functions is a decisive result of this work.

The results illustrate the ballistic objective function to be more useful as compared to pure geometrical objective

functions. For the case of neutral burning star web burn-out time and for non-neutral one web time average pressure

proves to be driving variables. In addition, the non-neutral configuration provides higher web time total impulse

compared to neutral star configuration. Ballistic objective function proves superior over geometrical objective

function.

References

1Khurram, N. Grain Design and Performance Optimization of Solid Rocket Motor. Ph.D. dissertation; Beihang University

Beijing, Number , 2008. 2Marcel Barrere, et al. “Rocket Propulsion”. Amsterdam, Elsevier Publishing Company , 1960 3Ricciardi A. Generalized geometric analysis of right circular cylindrical star perforated and tapered grains. Journal of

Propulsion and Power. Vol.8, No.1, pp51-58, 1992. 4Wang Guanglin, Cai e., The design of Solid rocket motor, Published by Northwestern Polytechnical University Press, 1994. 5Stone M W., A Practicle Mathematical Approach to Grain Design. . ARS Semi- Annual meeting San Francisco 1957. 6Jean A. V., Recent Advances in Solid Propellant Grain Design. ARSJ ,July 1959. 7Brooks W T. Ballistic Optimization of the Star Grain Configuration . Journal of Space craft, Vol. 19, No. 1, Jan–Feb, 1982. 8Sutton P, Oscar B. “Rocket Propulsion Elements”. Seventh edition. Wiley-Interscience, 2001. 9Davenas A. “Solid Rocket Propulsion Technology” . Elsevier Science & Technology, 1993. 10Goldberg, David, E., Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley, 1989. 11Coly, D. A., An Introduction to Genetic Algorithms for Scientists and Engineers, World Scientific, Singapore, 1999. 12Murray B., Anderson, Genetic Algorithms In Aerospace Design: Substantial Progress, Tremendous Potential, Sverdrup

Technology Inc. TEAS Group Eglin Air Force Base, FL 32542, USA 13Pebesma, E.J., Heuvelink, G., 1999.” Latin hypercube sampling of Gaussian random fields”. Technometrics 41, 303–312. 14Zhang, Y., Pinder, G.F., 2004. “Latin-hypercube sample-selection strategies for correlated random hydraulic-conductivity

fields”. Water Resources Research 39 (Art. No. 1226). 15Iman, R.L., Conover, W.J., 1982. A distribution-free approach to inducing rank correlation among input variables.

Communications in Statistics B11, 311-334.

Fig. 6 Optimum Star Configurations