[american institute of aeronautics and astronautics 45th aiaa/asme/sae/asee joint propulsion...

American Institute of Aeronautics and Astronautics1

Integrated Approach for Design and Optimization of SlottedTube Grain Configurations

Ali Kamran1, Liang Guozhu2, Khurram Nisar3, Qasim Zeeshan4, Amer Farhan Rafique5

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 Slotted Tube Grain configurations considering particular test cases forwhich the average thrust and constraints have been given. A parametric solid model of thegrain has been developed which enables automatic volume calculation of grain void and solidpropellant at each web increment thereafter burning area has been calculated. The motorperformance is calculated using simplified ballistic model, steady state pressure is calculatedby equating mass generated in chamber to mass ejected through nozzle throat. Geneticalgorithm has been employed for conducting optimization thereby achieving the design andperformance objectives while adhering to design constraints Latin hypercube sampling isused for better design space exploration and thus creating initial population to decreasecomputation time. Sensitivity Analysis of the optimized solution has been conducted usingMonte Carlo method to evaluate the effects of uncertainties in design parameters caused bymanufacturing 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. INTRODUCTION

rain Design is most imperative in completing the design of any Solid Rocket Motor (SRM), the key is todevelop a relation between web burnt and the burning surface1,2. 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 militaryapplications. There is a strong need to enhance thrust, improve the effectiveness of SRM and reduce mass of motor.Different methods have been used to calculate 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 and___________________________________________________________________1 Ali Kamran, PhD candidate, School of Astronautics, Student member AIAA, [email protected] Liang Guozhu, Prof, School of Astronautics, [email protected] Khurram Nisar, PhD candidate, School of Astronautics, [email protected] Qasim Zeeshan, PhD candidate, School of Astronautics, Student member AIAA, [email protected] Amer Farhan Rafique, PhD candidate, School of Astronautics, Student member AIAA, [email protected]

G

45th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit2 - 5 August 2009, Denver, Colorado

AIAA 2009-5514

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

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optimization process of solid rocket motor. PIBAL6 software uses CAD modeling for design of SRM grain.The methodology adopted in this work is CAD modeling of the propellant grain. A parametric model with

dynamic variables 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 HypercubeSampling (LHS) to create efficient initial population of the design variables and thus reducing time of overallprocess. Genetic algorithm (GA) is implemented in optimization module. Sensitivity module uses Montecarlomethod 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 of design space thus reducing computational time. In depth study of the optimized solution hasbeen conducted using Montecarlo method thereby affects of all the independent parametric design variables onoptimal solution and 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

Geometry of the grain is constructed in a modular manner. Separate entities are used for different parts thusensuring ease of construction and lesser chances of surface creation failure. The grain regression is achieved by aweb increment equal in all directions. A web increment is selected for which the grain regression is performed; ateach step new grain geometry is created automatically thereafter geometrical properties are stored in a file. Burningsurface area is calculated as:

CADbased

Grain DesignModule

GrainBoundary

Grain Core

Slots

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 throat7, 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 2-7. Slot length may or may not have a taperportion depending upon requirement.

Fig.2 Grain Boundary Fig.3 Grain Bore

Fig.4 Partial Slot Fig.5 Full Slot

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Descriptions of design variables which are inputted for grain burning regression are given in Table.1.

Table 1 Design Variables for Grain Geometry

Design Variables Units Symbol Design Variables Units SymbolGrain length mm L1 Fillet radius mm R2

Slot length mm L2 Slot taper angle deg αSlot straight portion mm L6 Number of slots - N

Motor front opening mm F1 Slot height mm H1

Grain radius mm F2 Half Slot thickness mm H2

Motor rear opening mm F3 Slot radius mm R1

Bore radius mm F4

III. OPTIMIZATION AND SENSITIVITY ANALYSIS

A. Design ObjectiveRequirements have been given for a given fixed length and outer diameter of the grain while remaining within

constraints 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 (F4, H1, H2, R1, L2, α, N)Upper and lower limits for these independent parameters for design and optimization have been shown in Table 2.

B. Design ConstraintsNeutral thrust time trace can proved to be very useful in certain cases. In present study neutral time trace are

calculated for two cases: A high L/D ratio with full slot and low L/D with partial slot. The constraint employed is tosearch for a perfect neutral thrust time trace.

Design Constraints for Configuration 1:The main system constraints for configuration 1 using HTPB propellant are shown in Table.2.

Fig.6 Tapered Slot Fig.7 Straight Slot

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Table 2 Design constraints for configuration 1

Design Variables Units Symbol ValueGrain length mm L 3700

Grain radius mm R 145

Burning time sec tb 5±0.25

Maximum Pressure Bar Pmax < 200

Propellant mass kg mp 330±5

Area Ratio - ε 5.25

Neutrality - Neu ≤ 1.15

Design Constraints for Configuration 2:The main system constraints for configuration 2 using HTPB propellant are shown in Table.3.

Table 3 Design constraints for configuration 2

Design Variables Units Symbol ValueGrain length mm L 7100

Grain radius mm R 650

Burning time sec tb 76±3

Maximum Pressure Bar Pmax < 75

Propellant mass kg mp 14000±150

Area Ratio - ε 7.5

Neutrality - Neu ≤ 1.20

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 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 and optimization parameters of GA are

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given in Figure.8.

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 literature10, 11, 12.

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 SamplingLatin 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 (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 models13,14 and has been used15,16. 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

Design Variables (X)

Optimal Solution (X*)

Population Initialization

Selection

Crossover

Mutation

Insertion

Stopping

Yes

No

Fig. 8 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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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 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 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. Configuration 1

A full and straight slot is taken for the first case. Grain length and diameter is fixed, the Fav required has been

optimized while obeying the constraints. Figure.9 shows the population sample pass on to GA for optimization.

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0 5 10Number of slots

0 500 1000 1500 2000Slot length

40 50 60 70 80Bore Radius

10 15 20

0

5

10

Slot width

Num

ber

ofsl

ots

0

500

1000

1500

2000

Slo

tle

ngth

40

50

60

70

80

Bor

eR

adiu

s10

15

20

Slo

tw

idth

Fig. 9 Latin Hyper Cube Based Population for Configuration 1(Scale for histogram on X-axis)

Table.4 shows the values of lower and upper bounds of design variables and optimum values of design variables

achieved by applying Genetic algorithms.

Table. 4 Optimized values of design variables

Table.5 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.5. Figure.10 depicts the pressure and thrust time trace.

Table. 5 Ballistic Performance

Parameter symbol unit Optimum ResultAverage thrust Fav kN 171.01Mass of propellant mp kg 330.7 Burning time tb sec 4.93

Average pressure Pav Bar 149

Maximum Pressure Pmax Bar 155

Neutrality Neu - 1.088

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

1 Bore F4 mm 45 75 64.32 Slot thickness H2 mm 5 10 6.723 Slot length L2 mm 800 1800 1130.84 Number of slots N - 3 7 5

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

20

40

60

80

100

120

140

160

180

Time (Sec)

Pre

ssur

e(B

ar)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

20

40

60

80

100

120

140

160

180

Time (Sec)

Thr

ust(

KN

)

Figure. 10 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.6. A sample of 1000runs is selected on random basis.

Table . 6 Tolerances

The effect on various parameters is shown in Table .7. It is evident that all the parameters are within limits.

Table. 7 Montecarlo Results

Parameter symbol UnitMinimumvalue

Maximumvalue

Meanvalue

Standarddeviation

Total impulse It kN-sec 833.79 858.3 845.78 6.82Mass of propellant mp Kg 328.79 332.63 330.69 2.05 Burning time tb Sec 4.787 5.059 4.921 0.0078Maximum Pressure Pmax Bar 149.1 161.7 155.3 0.1148Neutrality Neu - 1.081 1.089 1.085 7.201 E-6

Relation of average thrust, average pressure and time is shown in Figure.11.

S. No. Design Variables symbols units Value Tolerance

1 Bore F4 mm 64.3 ±0.22 Slot thickness H2 mm 6.72 ±0.153 Slot length L2 deg 1130.8 ±34 Number of slots N - 5 -5 Area ratio ε - 5.25 ±0.1 6 Throat diameter Dt mm 95 ±0.27 Burn rate BR mm/sec 13 ±0.158 Characteristic velocity C* m/sec 1550 ±10.0

9 Propellant density pρ kg/m3 1750 ±7.0

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Figure. 11 Design Sensitivity Analysis (Scale for histogram on X-axis)

Scatter plots for various design parameters are shown in Figure.12

0 100 200 300 400 500 600 700 800 900 1000160

165

170

175

180

185

Montecarlo runs

Thr

ust

(kN

)

0 100 200 300 400 500 600 700 800 900 10004.7

4.75

4.8

4.85

4.9

4.95

5

5.05

5.1

5.15

Montecarlo runs

Tim

e(s

ec)

0 100 200 300 400 500 600 700 800 900 1000138

140

142

144

146

148

150

152

154

156

158

160

Montecarlo runs

Ave

rage

pres

sure

(Bar

)

0 100 200 300 400 500 600 700 800 900 1000145

150

155

160

165

170

Montecarlo runs

Max

imum

pres

sure

(Bar

)

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0 100 200 300 400 500 600 700 800 900 1000324

326

328

330

332

334

336

Montecarlo runs

Pro

pella

ntm

ass

(kg)

0 100 200 300 400 500 600 700 800 900 10001.065

1.07

1.075

1.08

1.085

1.09

1.095

1.1

1.105

1.11

Montecarlo runs

Neu

tral

ity

Figure. 12 Scatter Plots of Performance Parameters

B. Configuration 2

A partial & tapered slot is taken for the second case. Grain length and diameter fixed, the objective function required

has been optimized while obeying the constraints. Figure.13 shows the population sample pass on to GA for

optimization.

400 420 440Bore radius

1000 2000 3000Slot length

30 40 50Slot angle

60 80 100Slot radius

500 550 600Slot height

40 60 80Slot width

0 5 10

400

420

440

Number of slots

Bor

era

dius

1000

2000

3000

Slo

tle

ngth

30

40

50

Slo

tan

gle

60

80

100

Slo

tra

dius

500

550

600

Slo

the

ight

40

60

80

Slo

tw

idth

0

5

10

Num

ber

ofsl

ots

Fig. 13 Latin Hyper Cube Based Population for Configuration 2 (Scale for histogram on X-axis)

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Table.8 shows the values of design variables obtained by applying Genetic algorithms.

Table. 8 Optimized values of design variables

Table.9 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.9. Figure.14 depicts the pressure and thrust time trace.

Table. 9 Ballistic Performance

Parameter symbol unit Optimum ResultAverage thrust Fav kN 466.81Mass of propellant mp kg 13918.57Burning time tb sec 73.95Average pressure Pav Bar 62.53

Maximum Pressure Pmax Bar 67.88

Neutrality Neu - 1.169

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

70

80

Time(sec)

Pre

ssur

e(B

ar)

0 10 20 30 40 50 60 70 800

100

200

300

400

500

600

Time(sec)

Thr

ust(

KN

)

Figure. 14 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.10. A sample of1000 runs is selected on random basis.

S. No.Design

Variablesymbols units LB UB Optimum Result

1 Bore F4 mm 200 220 213.1532 Slot thickness H2 mm 20 35 27.153 Slot Height H1 mm 500 575 569.44 Slot length L2 mm 1200 2500 1416.75 Number of slots N - 3 7 66 Slot angle deg o 30 45 32.47 Slot radius R1 mm 60 100 86

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Table . 10 Tolerances

The effect on various parameters is shown in Table .11. It is evident that all the parameters are within limits.

Table. 11Montecarlo Results

Parameter symbol unitMinimumvalue

Maximumvalue

Meanvalue

Standarddeviation

Total impulse It kN-sec 34904.9 35908.1 35399.5 0.03099Mass of propellant mp kg 13857.2 13979.9 13919.4 0.00558Burning time tb sec 73.75 78.65 76.28 1.857Maximum Pressure Pmax Bar 62.21 70.2 67.2 0.1797Neutrality Neu - 1.187 1.192 1.19 0.0015

Relation of average thrust, average pressure and time is shown in Figure.15.

Figure. 15 Performance Curves (Scale for histogram on X-axis)

S. No. Design Variables symbols units Value Tolerance

1 Bore F4 mm 213.153 ±0.25 2 Slot thickness H2 mm 27.15 ±0.153 Slot length L2 mm 1416.7 ±34 Number of slots N - 5 -5 Slot taper angle α deg 32.4 ±0.26 Slot radius R mm 86 ±0.27 Slot height H1 mm 569.4 ±0.58 Area ratio ε - 5.25 ±0.19 Throat diameter Dt mm 95 ±0.2

10 Burn rate BR mm/sec 6.2 ±0.111 Characteristic velocity C* m/sec 1550 ±10.0

12 Propellant density pρ kg/m3 1750 ±7.0

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Scatter plots for various design parameters are shown in Figure.16.

0 100 200 300 400 500 600 700 800 900 1000420

430

440

450

460

470

480

490

500

510

Montecarlo runs

Thr

ust

(kN

)

0 100 200 300 400 500 600 700 800 900 100072

73

74

75

76

77

78

79

80

81

Montecarlo runs

Tim

e(s

ec)

0 100 200 300 400 500 600 700 800 900 100057

58

59

60

61

62

63

64

65

66

67

68

Montecarlo runs

Ave

rage

pres

sure

(Bar

)

0 100 200 300 400 500 600 700 800 900 100062

63

64

65

66

67

68

69

70

71

72

73

Montecarlo runs

Max

imum

pres

sure

(Bar

)

0 100 200 300 400 500 600 700 800 900 10001.38

1.385

1.39

1.395

1.4

1.405x 10

4

Montecarlo runs

Pro

pella

ntm

ass

(kg)

0 100 200 300 400 500 600 700 800 900 10001.18

1.182

1.184

1.186

1.188

1.19

1.192

1.194

1.196

1.198

1.2

Montecarlo runs

Neu

tral

ity

Figure. 12 Scatter Plots of Performance Parameters

Detailed analysis of the two configurations shows that the optimal solution obtained is robust and the effects oftolerances induced by manufacturing processes will not affect the ballistic performance of the rocket motor.

V. CONCLUSIONA technique for design, optimization, and sensitivity analysis for slotted tube grain has been proposed andsuccessfully implemented. Grain geometrical properties are calculated by using parametric modeling of grainconfiguration using solid modeling that allows user to construct the geometry with simple input data. Complete aswell as partial slots with and without tapered end have been modeled. Two cases are studied; one with high L/Dratio and other with low L/D ratio. Optimization module is based on heuristic optimization (Genetic algorithms) thatnot only eliminates the requirement of initial guess but also ensures a global optimum solution. Latin hypercubesampling is employed at initial population ensuring excellent space filling of design variable and consequentlyreducing computation time. Montecarlo simulation has been used to investigate the performance variation on a

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statistical probability basis. Sets of grain and ballistic parameters are selected using a random distribution functionand performance is calculated for a sample size of a thousand. The optimal designs achieved prove to be insensitiveto the uncertainties in design parameters caused by manufacturing processes. Montecarlo simulation can prove to bevital considering the production of a large number of SRMs and enlightens the necessity to acquire statistical dataduring manufacturing processes.

References

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