American Institute of Aeronautics and Astronautics1

Design and Performance Optimization of Finocyl Grain

Ali Kamran1, Liang Guozhu2, Junaid Godil3, Zeeshan Siddique4, Qasim Zeeshan5, Amer Farhan Rafique6

School of Astronautics, Beijing University of Aeronautics and Astronautics (BUAA), 37 Xue Yuan Road, BeijingChina, 100191

The research work proposed herein addresses and emphasizes a design methodology todesign and optimize Finocyl Grain configurations considering particular test case for whichthe Average thrust and constraints have been given. A parametric solid model of the grainhas been developed which enables automatic volume calculation of the grain void and solidpropellant at each web increment thereafter burning area has been calculated. The motorperformance is calculated using a simplified ballistic model, steady state pressure iscalculated by equating mass generated in chamber to mass ejected through nozzle throat.Genetic algorithm has been employed for conducting optimization thereby achieving thedesign and performance objectives while adhering to design constraints. Latin hypercubesampling is used for better design space exploration and thus creating initial population todecrease computation time. Sensitivity Analysis of the optimized solution has beenconducted using Monte Carlo method to evaluate the effects of uncertainties in designparameters caused by manufacturing variations.

Nomenclature

Area Ratio ε Nozzle exit area Ae

Area of throat At Nozzle exit diameter de

Average pressure Pav Pressure exponent nAverage thrust Fav Specific impulse Is

Burning area Ab Thrust FBurning Duration tb Thrust coefficient Cf

Burning rate BR Total impulse It

Chamber pressure pc Volume of propellant Vp

Characteristic velocity C* Volume Change V∆Grain outer radius R Web thickness wLength of grain L Web change w∆

Mass of propellant mp

Propellant densitypρ

I. INTRODUCTIONrain Design is a key to complete the design of any Solid Rocket Motor (SRM), the key is to develop a relationbetween web burnt and the burning surface 1,2. Efficient designing of SRM Grains in the field of Rocketry is

still the main test for most of the nations of world for scientific studies, commercial and military applications. Thereis a strong need to enhance thrust, improve the effectiveness of SRM and reduce mass of motor.Different methods have been used to calculate the geometrical properties of grain burn-back/ regression analysis.Analytical methods though accurate but very restrictive has been used limitedly for three dimensional grainconfigurations3, 4, 5. CAD based programs are available in industry and have proved to be very useful for design andoptimization process of solid rocket motor. PIBAL 6 software uses CAD modeling for design of SRM grain.___________________________________________________________________1 Ali Kamran, PhD candidate, School of Astronautics, a1k1s1@yahoo.com, alikamran@ase.buaa.edu.cn2 Liang Guozhu, Prof, School of Astronautics, lgz@buaa.edu.cn3 Junaid Godil, Researcher, Institute of Space technology, Pakistan4Zeeshan Siddique, Researcher, Institute of Space technology, Pakistan5 Qasim Zeeshan, PhD candidate, School of Astronautics, Student member AIAA, qasimzeeshan@sa.buaa.edu.cn6 Amer Farhan Rafique, PhD candidate, School of Astronautics, Student member AIAA, afrafique@sa.buaa.edu.cn

G

AIAA Modeling and Simulation Technologies Conference10 - 13 August 2009, Chicago, Illinois

AIAA 2009-6234

Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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The methodology adopted in this work is CAD modeling of the propellant grain. A parametric model withdynamic variable is created that defines the grain geometry. A surface offset is used to simulate grain burningregression, and subsequent volume at each step is evaluated.

The program has four distinct modules; Grain design, Sampling, Optimization, and Sensitivity analysismodule. The Grain geometry is based on CAD software that has the capability of handling parametricmodeling. Grain is modeled in parts to provide ease and ensure lesser chances of surface creation failure. Asimple variable input is sufficient to create the geometry. The sampling module uses Latin hypercube sampling(LHS) to create initial population of the design variables. Genetic algorithm (GA) is implemented in theoptimization module. Sensitivity module uses Montecarlo method for uncertainty analysis.

The CAD software is linked to MATLAB which gives input variables. The output (geometrical properties) istaken by MATLAB. Ballistic performance is calculated by using a simplified model, steady state pressure iscalculated by equating mass generated in chamber to mass ejected through nozzle throat. Genetic algorithmshas been employed for conducting optimization thereby achieving desired Thrust~Time curve while adheringto required design constraints. Initial population used by GA is formed by using LHS. which provides excellentspace filling design thus reducing computational time. In depth study of the optimized solution has beenconducted using Montecarlo method thereby affects of all the independent parametric design variables onoptimal solution & design objectives have been examined and analyzed in detail. A flow chart of the process isshown in Figure.1.

II. GEOMETRIC MODEL AND PERFORMANCE PREDICTION

The Finocyl (Fin in Cylinder) is a 3D grain configuration especially employed to relatively low fineness ratios (L/D)requiring internal burning grains with relatively long duration and large thrust. It can provide a variety of thrust timetrace depending on mission requirement. Geometry of the grain is constructed in a modular manner. Separate entitiesare used for different parts thus ensuring ease of construction and lesser chances of surface creation failure. Thegrain regression is achieved by a web increment equal in all direction. A web increment is selected for which thegrain regression is performed; at each step new grain geometry is created automatically thereafter geometricalproperties are stored in a file. Burning surface area is calculated as:

CADbased

Grain DesignModule

GrainBoundary

Grain Core

Fins

Volumecalculation

Design Variables (X)

OPTIMIZATION______________________

Find: Optimum DesignVariables (X*)

Satisfy: Constraints• Geometry• Ballistics

Sensitivity Analysis___________________

Monte CarloSimulation

Design of Experiments___________________

Latin HypercubeSampling

Optimal Design (X*)

Visual Basic____________________

Read: Design Variables (X)Update Variables

Satisfy: Calculation to maximumWeb

Write: Output

Fig.1 Overall Design and Optimization Process

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w

VAb ∆

∆= (1)

Propellant Mass is calculated as:

ppp Vm ρ= (2)

The motor performance is calculated using a simplified ballistic model, steady state pressure is calculated byequating mass generated in chamber to mass ejected through nozzle throat 7, 8, 9.The chamber pressure is calculated as:

( ) )1(1* n

pc Kacp−

= ρ (3)

Where K =Ab / At

a is the burn rate coefficient

Thrust is calculated as

tcF ApCF = (4)

A detailed description of the grain modeling is shown in Figures.2- 5.

Description of input required for grain burning regression is given in Table.1.

Fig.4 Fin Axial shape

Fig.2 Grain Boundary Fig.3 Grain Bore

Fig.5 Fin Cross-section

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Table 1 Design Variables for Grain Geometry

Design Variables Units Symbol Design Variables Units SymbolGrain length mm L1 Rear Cone mm L4

Motor front opening mm F1 Rear Cylinder mm L5

Grain radius mm F2 Fillet radius mm R1

Motor rear opening mm F3 Fin taper angle deg αBore radius mm F5 Number of Fin - N

Fin straight portion mm L6 Fin height mm H1

Front Web mm L2 Half Fin thickness mm H2

Front Cone mm L3 Fin radius mm R2

III. OPTIMIZATION AND SENSITIVITY ANALYSIS

A. Design Objective

Requirements have been given for a given fixed length and outer diameter of the grain while remaining withinconstraints of burning time, grain mass, propellant and nozzle parameters. Maximization of average thrust is themajor design objective.

Max Fav (X) (5)Where the design variable (X) is:

X = f (F5, H1, H2, R1, L2, L3, L4, L5, L6, α, N)Upper and lower limits for these independent parameters for design and optimization have been shown in Table 2.

B. Design Constraints

Neutral thrust time trace can proved to be very useful in certain cases. In present study neutral time trace arecalculated. The constraint employed is to search for a perfect neutral thrust time trace.

Design Constraints for Finocyl Configuration:

The main system constraints for the configuration using HTPB propellant are shown in Table.2.

Table 2 Design constraints for Finocyl configuration

Design Variables Units Symbol ValueGrain length mm L 2395

Grain radius mm R 700

Burning time sec tb 74±3

Maximum Pressure Bar Pmax < 70

Propellant mass kg mp 5000±100

Area Ratio - ε 16

Neutrality - Neu ≤ 1.15

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C. 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 (GA) 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. The flow chart of GA is given in Figure.6.

GA provides several advantages for design including the ability to combine discrete and continuous variables,

it provides population-based search, there is no requirement for an initial design solution and has the ability to

address non-convex, multimodal and discontinuous functions. Details of GA are found in literature 10, 11, 12.

Design Variables (X)

Optimal Solution (X*)

Population Initialization

Selection

Crossover

Mutation

Insertion

Stopping

Yes

No

Fig. 6 Flow chart of Genetic Algorithm

Genetic Algorithm Parameters

Maximum generations: 30

Population size: 30

Population type: Double Vector

Selection: Stochastic uniform

Crossover: Single point, pc = 0.8

Mutation: Uniform, pm = 0.25641

Fitness Scaling: Rank

Reproduction: Elite count = 2

Function Evaluations: 900

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D. Design of Experiments Module

Design of Experiments (DOE) strategies are used to sample the design space to generate sample data to fit an

approximate model to each of the output variables (responses) of interest. Thus, sample points should be chosen to

fill the design space for computer experiments. Sampling is a statistical procedure which involves the selection of a

finite number of individuals to represent and infer some knowledge about a population of concern. Random

Sampling generated from the marginal distributions, is also referred to as pseudo random, as the random numbers

are machine generated with deterministic process. Statistically, random sampling has advantages, as it produces

unbiased estimates of the mean and the variance of the output variables.

Latin Hypercube Sampling

Latin hypercube sampling (LHS) 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

(of the expectation of the output function) with lower variance. In particular, the closer the output function is to

being additive in its input variables, the more reduction in variance. LHS yields biased estimates of the variance of

the output variables. It was initially developed for the purpose of Monte-Carlo simulation; efficiently selecting input

variables for computer models 13, 14 and has been used 15, 16. LHS follows the idea 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. 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 (6)

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) (7)

The m values obtained for each variable 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.

E. Sensitivity Analysis

Parametric analysis can prove to be restrictive as a prohibitive amount of analysis is required for a large

number of design variables, further error can arise due to relationship between different variables. The Monte Carlo

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method of statistical analysis is used to investigate the effects of various uncertainties in the design parameters on

the performance of optimized grain configuration. Significant variables like propellant characteristics other than

grain geometry parameters are also selected using random sampling technique. Uncertainties are considered for all

geometrical and propellant characteristics for the analysis. The analysis is used to predict the ballistic characteristics

of the designed SRMs. Results achieved will largely depend upon the tolerances and distribution used to define the

set of design variables.

IV. RESULTS

A. Designed finocyl Configuration

With Grain length and diameter fixed, the Fav required has been optimized while obeying the constraints .Table.3

shows the values of design variables obtained by applying Genetic algorithms.

Table. 3 Optimized values of design variables

Table.4 shows the performance parameters attained. All these values have been achieved by adhering and obeying

the limits of all design constraints as shown in Table.4. Figure.7 depicts the pressure and thrust time trace.

Table. 4 Ballistic Performance

Parameter symbol unit Optimum ResultAverage thrust Fav kN 181.4Mass of propellant mp kg 5067Burning time tb sec 74.1

Average pressure Pav Bar 60.7

Maximum Pressure Pmax Bar 64.8

Neutrality Neu - 1.106

S. No. Design Variables symbols units LB UB Optimum Result

1 Bore F5 mm 220 280 252.52 Fin thickness H2 mm 30 50 45.8 3 Fin length L6 mm 180 350 2224 Number of Fins N - 6 13 125 Fin angle α deg 25 50 41.56 Fin fillet R1 mm 25 95 347 Fin height H1 mm 500 580 548.78 Motor front opening F1 mm 80 120 1079 Motor rear opening F3 mm 350 400 390.7

10 Front Web L2 mm 80 130 93.211 Front Cone L3 mm 80 120 10012 Rear Cone L4 mm 80 120 99.313 Rear Cylinder L5 mm 180 270 185

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0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

Time(sec)

Pre

ssur

e(B

ar)

0 10 20 30 40 50 60 70 800

50

100

150

200

Time(sec)

Thr

ust(

KN

)

Figure. 7 Ballistic Performance

To investigate the effect of geometrical tolerances of grain design and practical limits of propellantcharacteristics Montecarlo simulations is performed. The tolerances chosen are shown in Table.5. A sample of 500runs is selected on random basis.

Table. 5 Optimized values of design variables

The effect on various parameters is shown in Table .6. It is evident that all the parameters are within limits.

Table. 6 Montecarlo Results

Parameter symbol unitMinimumvalue

Maximumvalue

Meanvalue

Standarddeviation

Total impulse It kN-sec 13260 13625 13442 0.0111Mass of propellant mp kg 4984.2 5029 5007 0.0201Burning time tb sec 71.85 76.45 74.13 1.754Maximum Pressure Pmax Bar 61.98 67.82 64.86 1.704Neutrality Neu - 1.105 1.108 1.1059 0.00083

S. No. Design Variables symbols units Value Tolerance

1 Bore F5 mm 252.5 ±0.22 Fin thickness H2 mm 45.8 ±0.13 Fin length L6 mm 222 ±0.24 Number of Fins N - 12 -5 Fin angle α deg 41.5 ±0.16 Fin fillet R1 mm 34 ±0.17 Fin height H1 mm 548.7 ±0.38 Motor front opening F1 mm 107 ±0.29 Motor rear opening F3 mm 390.7 ±0.3

10 Front Web L2 mm 93.2 ±0.211 Front Cone L3 mm 100 ±0.212 Rear Cone L4 mm 99.3 ±0.213 Rear Cylinder L5 mm 185 ±0.214 Area ratio ε - 16 ±0.115 Throat diameter Dt mm 150 ±0.216 Burn rate BR mm/sec 6.5 ±0.117 Characteristic velocity C* m/sec 1550 ±10.0

18 Propellant density pρ kg/m3 1750 ±7.0

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Scatter plots for various design parameters are shown in Figure.8

0 50 100 150 200 250 300 350 400 450 500170

175

180

185

190

Montecarlo runs

Thr

ust

(kN

)

0 50 100 150 200 250 300 350 400 450 500

71

72

73

74

75

76

77

Montecarlo runs

Tim

e(s

ec)

0 50 100 150 200 250 300 350 400 450 50056

57

58

59

60

61

62

63

64

65

Montecarlo runs

Ave

rage

pres

sure

(Bar

)

0 50 100 150 200 250 300 350 400 450 50061

62

63

64

65

66

67

68

69

70

Montecarlo runs

Max

imum

pres

sure

(Bar

)

0 50 100 150 200 250 300 350 400 450 5004980

4990

5000

5010

5020

5030

5040

Montecarlo runs

Pro

pella

ntm

ass

(kg)

0 50 100 150 200 250 300 350 400 450 5001.1035

1.104

1.1045

1.105

1.1055

1.106

1.1065

1.107

1.1075

1.108

1.1085

Montecarlo runs

Neu

tral

ity

Figure. 8 Scatter Plots of Performance Parameters

V. CONCLUSION

A technique for design, optimization, and sensitivity analysis for Finocyl grain has been proposed. Graingeometrical properties are calculated by using parametric modeling of grain configuration using solid modeling thatallows user to construct the geometry with simple input data. Optimization module is based on heuristicoptimization (Genetic algorithms) that not only eliminates the requirement of initial guess but also ensures a globaloptimum solution. Latin hypercube sampling is employed at initial population ensuring excellent space filling ofdesign variable and consequently reducing computation time. Montecarlo simulation has been used to investigate theperformance variation on a statistical probability basis. Sets of grain and ballistic parameters are selected using a

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random distribution function and performance is calculated for a sample size of 500. The optimal designs achievedprove to be insensitive to the uncertainties in design parameters caused by manufacturing processes. Montecarlosimulation can prove to be vital considering the production of a large number of SRMs and enlightens the necessityto acquire statistical data during manufacturing processes.

References

1Wang Guanglin, Cai e., The design of Solid rocket motor, Published by Northwestern Polytechnical University Press, 1994.2Wang Guanglin, Cai e., The design of Solid rocket motor, Published by Northwestern Polytechnical University Press, 1985.3Dunn S S, Coats D E. “3-D Grain Design and Ballistic Analysis using SPP97 Code”. AIAA-97-3340, 1997.4Dunn S S, Coats D E. “Solid Performance Program”. AIAA 87-1701. 19875Peterson E G, Nielson C C, Johnson W C, Cook K S. “Generalized coordinate grain design and internal ballistic evaluationprogram”. AIAA 68-490, 1968.6F. Dauch, D. Ribéreau. “A Software for SRM Grain Design and Internal Ballistics Evaluation, PIBAL”. AIAA 2002-4299,20027Sutton P, Oscar B. “Rocket Propulsion Elements”. Seventh edition. Wiley-Interscience, 2001.8Davenas A. “Solid Rocket Propulsion Technology” . Elsevier Science & Technology, 1993.9Marcel Barrere, et al. “Rocket Propulsion”. Amsterdam, Elsevier Publishing Company , 196010Goldberg, 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, SverdrupTechnology Inc. TEAS Group Eglin Air Force Base, FL 32542, USA13Iman, R.L., Conover, W.J., 1980. “Small sample sensitivity analysis techniques for computer models, with an application torisk assessment”. Communications in Statistics Theory and Methods A9, 1749–1874.14McKay, M.D., Beckman, R.J., Conover, W.J., 1979. “A comparison of three methods for selecting values of input variablesin the analysis of output from a computer code”. Technometrics 21, 239–245.15Pebesma, E.J., Heuvelink, G.B.M., 1999.” Latin hypercube sampling of Gaussian random fields”. Technometrics 41, 303–312.16Zhang, Y., Pinder, G.F., 2004. “Latin-hypercube sample-selection strategies for correlated random hydraulic-conductivityfields”. Water Resources Research 39 (Art. No. 1226).

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