American Institute of Aeronautics and Astronautics

1

Particle Swarm Optimizer for Finocyl Grain Configurations

Ali Kamran1, Liang Guozhu

2, Qasim Zeeshan

3, Amer Farhan Rafique

4

Beijing University of Aeronautics and Astronautics (BUAA), 37 XueYuan Road, Beijing China, 100191

This paper presents a CAD-centric approach, a way to integrate computational tools and as

a novel way of formulating the optimization problem. A parametric solid model of the grain

automates the geometric construction, enabling volume calculation of the grain void and

solid propellant at each web increment. Simplified ballistic model calculates the motor

performance. The optimization process takes advantage of swarm intelligence optimizer’s

ability to locate global optimum solution in less computational cost. Booster motor of high

length-to-diameter ratio is the test case with the objective of maximizing average thrust

subjected to stringent performance constraints.

Nomenclature

Ab = Burning area (mm2)

At = Area of throat (mm2)

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

a1,2 = Acceleration constants (m/s2)

Cf = Thrust coefficient

c* = Characteristic velocity (m/s)

Dt = Throat diameter (mm)

F = Thrust (kN)

Fav = Average thrust (kN)

Isp = Specific impulse (s)

i = Particle index

k = Time index

L = Length of grain (mm)

mp = Mass of propellant (kg)

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)

pmin = Minimum pressure (bar)

pi = Best position found by the i

th particle (personal best)

R = Grain outer radius (mm)

tb = Burning time (s)

Vp = Volume of propellant (m3)

vp = Velocity of the ith

particle (m/s)

xp = Position of the ith

particle

ΔV = Volume Change

Δw = Web change

ρp = Propellant density (kg/m3)

ε = Area Ratio

γ1,2 = Random numbers on the interval applied to the ith

particle

Φ = Inertia function

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

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

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

AIAA 2010-3083

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

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I. INTRODUCTION

rain Design is key to complete the design of Solid Rocket Motor (SRM), the purpose is to develop relation

among web burnt and the burning surface1,2

. The fundamental driver in the design of a SRM is the choice of

general grain shape associated with a type of propellant. Different methods evolved in past to calculate the

geometrical properties of grain burn-back/ regression analysis varied from drafting techniques to the exact solution.

Analytical methods, though, accurate but restrictive will thus limit a lot of possible geometry exploration.

Generalized coordinate method has been long used for three dimensional grain configurations3-5

. The accuracy of

solution largely depends upon the web, and axial increment, chosen for volume calculation and indeed will require

certain approximation to limit computational time.

CAD based programs are available in industry and have proved to be immensely useful for the design process of

Solid Rocket Motor (SRM). PIBAL6, ELEA

7 software uses CAD modeling for design of SRM two dimensional

(2D) and three dimensional (3D) grains. Former uses a simplified ballistic model and later give a point to point

burning rate taking account of local ballistics.

Ref 8 applied Pattern Search technique for design and optimization of 3D grain configuration. The approach has

limited applicability as solution quality is heavily dependent on starting solution. The method has a tendency to fell

prey to local optima similar to any gradient method and has extreme sensitivity to the starting solution. Ref. 9

presents the design and optimization of Finocyl grain using generalize coordinate method. Ref. 10 presents Hybrid

Optimization technique for Finocyl grain configuration using the same method. Ref 11, 12 presents design and

optimization of Finocyl and slotted tube grain using CAD and genetic algorithm. The approach has the robustness of

an exhaustive search of the design space; its computational demands appear larger than alternatives.

This paper outlines the application of a CAD-centric optimization technique to the design of Finocyl grain

configuration. The design process uses commercial CAD and optimization algorithm in conjunction with a

simplified ballistic code to analyze and optimize Finocyl grain shape. The design process takes advantage of Swarm

Intelligence based optimization method for design space search; with a relatively small computational cost.

Grain is modeled in parts to provide ease and ensure lesser chances of surface creation failure. A simple

variable input is sufficient to create the geometry. Particle Swarm Optimization (PSO) controls the

optimization module. Simplified ballistic model calculates the ballistic performance, and steady-state pressure

is calculated by equating mass generated in the chamber to mass ejected through nozzle throat13-15

. Figure 1

shows the flow chart of the process.

II. GEOMETRIC MODEL AND PERFORMANCE PREDICTION

The Finocyl (Fin in Cylinder) is a 3D grain configuration especially employed for relatively long duration and

large thrust. It can provide a variety of thrust time trace depending on mission requirement. Grain geometry

construction is in a modular manner. Separate entities for different parts ensure ease of construction and lesser

G

CAD

based

Grain Design

Module

Grain

Boundary

Grain Core

Fins

Volume

calculation

Design Variables (X)

OPTIMIZATION

______________________ Find: Optimum Design Variables (X*)

Satisfy: Constraints

Geometry

Ballistics

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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chances of surface creation failure. The grain regression is achieved by a web increment equal in all direction; at

each step new grain geometry is created automatically thereafter geometrical properties are estimated. Burning

surface area is calculated as:

w

VAb

(1)

Propellant Mass is calculated as:

ppp Vm (2)

Chamber pressure is calculated as:

)1(1* n

pc Kacp

(3)

Where K =Ab / At

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)

Specific impulse is calculated as

gm

FtI

p

b

sp (7)

Neutrality is defined as:

),max( minmax

avav p

p

p

pNeu

(8)

Figures.2- 5 presents detailed description of the grain modeling.

Fig.4 Fin Axial shape Fig.5 Fin Cross-section

Fig.2 Grain Boundary Fig.3 Grain Bore

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Table.1 lists the description of input required for grain burning regression.

Table 1 Design Variables for Grain Geometry

Design Variables Units Symbol Design Variables Units Symbol

Grain 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 α

Bore 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

A. Design Objective

There can be different objective functions for grain configuration optimization problems. For the present

research effort, design objective is to maximize the average thrust that indeed would cater for a maximum specific

impulse, subject to stringent performance constraints. Mathematical description of design objective is as under:

Max

Fav (X) (9)

Where the design variable (X) is:

X = f (F5, H1, H2, R1, L2, L3, L4, L5, L6, α, N)

B. Design Constraints

Neutral thrust time trace can prove to be remarkably valuable in certain cases. Present study also calculates

neutral time trace. The main system constraints for the configuration using HTPB propellant are:

8200:

6460:

93009000:

12.1:

60:

500:

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

(10)

C. Optimization Method

Particle swarm optimization is a relatively recent evolutionary heuristic and population-based computer

algorithm for problem solving. Mechanics of PSO took inspiration from the swarming or collaborative behavior of

biological populations (flock of birds, schools of fish, and herds of animals). Social-psychological principles form

the basis of PSO, and it provides insights into social behavior, as well as contributing to engineering applications.

Ref. 16 invented PSO while attempting to simulate the choreographed, charming movement of swarms of birds.

PSO starts with propagation of randomly generated initial swarm in the design space on the way to the optimal

solution over a number of iterations. There is a lot of information sharing among all members of initial-swarm about

design space. Precision and abundance of information controls the progress of initial-swarm towards optimality. Ref.

16, 17 describe a complete chronicle of the development of PSO algorithm from simply a motion simulator to a

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worthy, heuristic optimization approach. Figure 3 describes the working of PSO algorithm and parameters for ASLV

design problem.

PSO was originally aimed at treating nonlinear optimization problems with continuous variables. Moreover, PSO

has been expanded to handle combinatorial optimization problems and both discrete and continuous variables as

well. Efficient treatment of mixed-integer nonlinear optimization problems (MINLPs) is one of the most

complicated problems in practical optimization. Moreover, unlike other optimization techniques, PSO can be

realized with only a small program and it can also handle MINLPs with only a small program. This feature of PSO

is one of its advantages compared with other optimization techniques.

A MATLAB based PSO tool box and, basic algorithm adopted from Ref. 18, 19 have been taken as the starting

point for application in current design problem. Working of PSO algorithm is summarized as under:

Define the problem to search and develop solution criteria.

Initialize population via random initial positions and random initial velocities.

Determine global best position.

Determine personal best position.

Update velocity and position equations.

PSO algorithm is given as:

)]([)]([ 2211)1(ik

gki

ik

ii

ik

ik

xpaxpavv

(11)

Velocity vector updates position of each particle as shown in Eq. 12 and illustrated in Fig. 6. These steps are

repeated until a desired convergence criterion is met.

ik

ik

ik

vxx)1()1(

(12)

PSO has high adaptability to continuous problems, and thus applied to some structural design optimization

problems20

, multidisciplinary optimization problem of Air Launched Satellite Launch Vehicle21

, and component

level design and optimization of satellites22

. However, researchers have not yet explored the potential of applying

PSO in MDO of space launch vehicles. Figure.7 outlines the flow chart of PSO and optimization parameters for

application is Finocyl grain optimization.

Current Motion Influence

g

kp

Particle Memory

Influence

Swarm Influence

i

kx 1

i

kv 1

i

kx

i

kv

ip

Figure 6. Depiction of Velocity and Position Updates

in PSO.

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IV. RESULTS

The simulated result for a booster of high length-to-diameter ratio, Finocyl grain configuration, is presented

herein. This research effort optimizes ballistic objective of average thrust. The objective function and burn time

requirement ensure motor operation at higher specific impulse. The diameter is fixed at 1000 mm and length at 8200

mm. Table 2 lists propellant and nozzle parameters used in ballistic analysis. Table.3 shows the values of design

variables obtained by applying particle swarm optimizer.

Table. 2 Propellant and Nozzle Parameters

Parameter Units Value

Dt mm 240

ε - 8

c* m/s 1550

ρp

kg/m

3 1760

n - 0.34

a

mm/s/Pan 0.0259

Propellant HTPB/AP/Al

Table. 3 Optimized values of design variables

Design Variables Symbols Units LB UB Optimum Result

Bore F5 mm 170 250 194.6

Fin thickness H2 mm 25 45 32.5

Fin length L6 mm 500 1700 1193

Number of fins N - 4 10 7

Fin angle α deg 5 20 9.7

Fin fillet R1 mm 25 70 45.6

Fin height H1 mm 410 460 447.5

Motor front opening F1 mm 80 120 116.2

Motor rear opening F3 mm 300 400 309

Front web L2 mm 80 130 89

Front cone L3 mm 150 220 202

Rear cone L4 mm 120 220 158.3

Rear cylinder L5 mm 140 250 162.6

Design Variables

Generation of Initial Condition

Evaluation of Searching Point

Modification of Each Searching Point

Reach Maximum Iteration?

Optimal Solution

YES

NO

Figure 7. Flow Chart and Optimization Parameters of PSO

Particle Swarm Optimization

Parameters

_____________________

Type: PSO with Inertia

Max Iterations:

100 Function Tolerance:

10-6

Acceleration Constant: 2

Initial Inertia Weight:

0.90 Final Inertia Weight:

0.40

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

severe design constraints. Figure.7 depicts the pressure and thrust time trace of the optimized Finocyl grain. It is

evident that the optimized configuration has excellent neutrality resulting in higher specific impulse within the

constraint of maximum pressure.

Table. 4 Ballistic Performance

Parameter Symbol Unit Optimum Result

Average thrust (vacuum) Fav kN 398.72

Mass of propellant mp kg 9221.5

Burning time tb s 62.95

Average pressure Pav bar 51.44

Maximum pressure Pmax bar 56.19

Neutrality Neu - 1.107

Figure. 7 Ballistic Performance

V. CONCLUSION

Novel methodology for design and optimization for Finocyl grain is presented. CAD-centric approach integrated

with particle swarm optimizer is applied in the process. The grain design optimization process began by (1)

determine design space, (2) creating initial population on random basis, (3) constructing geometry, (4) determine

ballistic performance, and (5) evaluate constraints. PSO creates improved design variables with successive iteration.

References

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0 10 20 30 40 50 600

10

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Time(s)

Pre

ssure

(bar)

0 10 20 30 40 50 600

50

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500

Time(s)

Th

rust

(kN

)

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