research article improvement of interior ballistic...

Research ArticleImprovement of Interior Ballistic Performance UtilizingParticle Swarm Optimization

Hazem El Sadek, Xiaobing Zhang, Mahmoud Rashad, and Cheng Cheng

School of Energy and Power Engineering, Nanjing University of Science and Technology, Nanjing 210094, China

Correspondence should be addressed to Hazem El Sadek; [email protected]

Received 22 January 2014; Revised 23 February 2014; Accepted 23 February 2014; Published 6 April 2014

Academic Editor: Robertt A. Fontes Valente

Copyright © 2014 Hazem El Sadek et al.This is an open access article distributed under theCreativeCommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper investigates the interior ballistic propelling charge design using the optimization methods to select the optimum chargedesign and to improve the interior ballistic performance.The propelling charge consists of a mixture propellant of seven-perforatedgranular propellant and one-hole tubular propellant.The genetic algorithms and some other evolutionary algorithms have complexevolution operators such as crossover, mutation, encoding, and decoding. These evolution operators have a bad performancerepresented in convergence speed and accuracy of the solution. Hence, the particle swarm optimization technique is developed. Itis carried out in conjunction with interior ballistic lumped-parameter model with the mixture propellant.This technique is appliedto both single-objective and multiobjective problems. In the single-objective problem, the optimization results are compared withgenetic algorithm and the experimental results.The particle swarmoptimization introduces a better performance of solution qualityand convergence speed. In themultiobjective problem, the feasible region provides a set of available choices to the charge’s designer.Hence, a linear analysismethod is adopted to give an appropriate set of the weight coefficients for the objective functions.The resultsof particle swarm optimization improved the interior ballistic performance and provided a modern direction for interior ballisticpropelling charge design of guided projectile.

1. Introduction

Recently, study of the propelling charge design becomes veryessential to achieve the interior ballistic performance andassure the safety firing. This study is known as design of theinterior ballistic which is considered as a crucial branch ofgun system design. There are a number of computer-basedinterior ballistic models with different capabilities. Thesemodels allow the researchers of interior ballistic to predict theinterior ballistic performance of a particular gun, charge, andprojectile combination. The classic interior ballistics modelsincluding the characteristics of the gun, charge, and projectileare utilized to predict the muzzle velocity and the peakpressure. But these models cannot provide the best design ofthe propelling charge that gives the optimum solution of theinterior ballistic performance.

Optimization techniques allow the designers to evalu-ate a large number of design alternatives in a systematicand efficient manner to find the best design. The solutionof an interior ballistic model is considered improved if

there is a net increase in muzzle velocity without violatinggun constraints. The optimization techniques are used toimprove the gun performance and/or decrease the design costwhile meeting the constraints appropriate to the problem.Therefore, some researchers tried to apply the optimizationtechniques with the interior ballistic models. A numericaloptimization method called augmented Lagrange multiplierwas carried out and coupled to a classic interior ballisticmodel. This method was used to design the parameters ofseven-perforation propellant and to obtain the best muzzlevelocity for the projectile fired from 120mm tank cannons[1]. Another optimization technique was carried out tooptimize the ignition system and propelling charge structurein a high muzzle velocity gun [2]. Some researchers of ourgroup tried to use the genetic algorithm (GA) [3]. Thisalgorithm is considered as one of the most currently popularintelligent optimizations to optimize the interior ballisticproblem. GA was introduced as a computational analogy ofadaptive systems. It is modeled loosely on the principles ofthe evolution via natural selection, employing a population

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 156103, 10 pageshttp://dx.doi.org/10.1155/2014/156103

2 Mathematical Problems in Engineering

of individuals that undergo selection in the presence ofvariation-inducing operators such as mutation and recom-bination (crossover). A fitness function is used to evaluateindividuals, and reproductive success varies with fitness. InGA, each genome (chromosome) contains information of theoptimization problem by encoding and then should decodethe genome information to the optimization problem afterfinding the optimum solution. According to these above trialsof optimization techniques, it is found that they have somelimitations in the accuracy of the solution and speed of theconvergence. Moreover, the interior ballistic problem cannotbe adjusted in a good manner and sometimes the optimumsolution cannot be reached.

Particle swarm optimization (PSO) was firstly developedby Kennedy and Eberhart [4]. It was considered as anexpansion of an animal social behavior simulation systemthat incorporated concepts such as nearest-neighbor velocitymatching and acceleration by distance. The PSO techniquebecomes very popular due to its ease of implementationand the quick convergence to the optimum solution [5,6]. Similarly to evolutionary algorithms, PSO exploits apopulation (a swarm) of potential solutions (particles) whichare modified stochastically at each iteration of the algorithm.However, the manipulation of swarm differs significantlyfrom that of evolutionary algorithms, promoting a coop-erative rather than a competitive model. More specifically,the other evolutionary algorithms use explicit mutation andselection operators, while PSO uses an adaptable velocityvector for each particle in order tomodify the population andfavor the best performing individuals. The velocity vector isused to update the position of the particles at each iteration.The particles are moving towards promising regions of thesearch space by exploiting information springing from theirown experience during the search, as well as the experience ofother particles. For this purpose, a separate memory is usedwhere each particle stores the best position it has ever visitedin the search space.

In this work, PSO technique is coupled with the interiorballistic model to optimize the propelling charge design. Theutilized charge is a mixture charge which consists of twodifferent propellants, granular seven-perforated propellantand tubular propellant. PSO technique is applied to theclassic interior ballistic model for 76mm naval gun withguided projectile to improve the interior ballistic perfor-mance. Two types of objective functions are used withPSO techniques, single-objective function andmultiobjectivefunction.Through this work, it is found that the optimizationresults improved the interior ballistic performance and firingsafety with a high quality and quick convergence of theoptimum solution.

2. Problem Formulation

The optimization of the charge design in interior ballisticprocess is considered as nonlinear optimization, multicon-strained problem. It consists of multiobjective function andmultiple design variables. The design variables have differenteffects on the performance of interior ballistic. In the interior

ballistic process coupled with the optimization technique,there are restrictions between the objective functions. Forexample, the designer of the interior ballistic intends to obtainthemaximummuzzle velocity as amain objective.Maximummuzzle velocity requires a high muzzle pressure, while one ofthe objective functions is to minimize the muzzle pressure.Hence, the restriction is clear between the muzzle velocityobjective function and muzzle pressure objective function.The objective functions should be weighted according tothe advantages and disadvantages. The general optimizationproblem for the classical interior ballistic process can bedescribed as follows:

Minimize 𝑓 (𝑥) = (𝑓1(𝑥) , 𝑓

2(𝑥) , . . . , 𝑓

𝑘(𝑥))

subject to 𝑔𝑖(𝑥) ≤ 0 ∀𝑖 = 1, 2, . . . , 𝑚

ℎ𝑗(𝑥) = 0 ∀𝑗 = 1, 2, . . . , 𝑝

𝑥𝐿𝑙≤ 𝑥𝑙≤ 𝑥𝑈𝑙∀𝑙 = 1, 2, . . . , 𝑞,

(1)

where 𝑥 = (𝑥1, 𝑥2, . . . , 𝑥

𝑛)𝑇 is the decision vector (design

variables), 𝑓(𝑥) is the multiobjective vector, 𝑓𝑘(𝑥) is the 𝑘th

objective function, 𝑔𝑖(𝑥) and ℎ

𝑖(𝑥) are inequality and equality

constraints, respectively, and 𝑥𝐿𝑙and 𝑥𝑈

𝑙are the lower and

upper bound of the design variables, respectively.The relation between the optimization technique and

the interior ballistic model is the objective functions thatare calculated from the interior ballistic model and thenoptimized by the optimization model.

There exist some different tools to simulate the interiorballistic process such as lumped-parameter model and two-phase flow model. In this work, lumped-parameter model isutilized to simulate the interior ballistic process and calculatethe objective values that will be optimized by using the PSOtechnique. The lumped-parameter model can be written asfollows [7]:

𝜓𝑖= 𝜒𝑖𝑍𝑖(1 + 𝜆

𝑖𝑍𝑖+ 𝜇𝑖𝑍2𝑖) ,

𝑑𝑍𝑖

𝑑𝑡=𝑢1𝑖

𝑒1𝑖

𝑝𝑛𝑖 𝑖 = 1, 2, . . . , 𝑛,

𝑑V𝑑𝑡=𝑆 ⋅ 𝑝

𝜑 ⋅ 𝑚,

𝑑𝑙

𝑑𝑡= V,

𝑆 ⋅ 𝑝 (𝑙𝜓+ 𝑙) =

𝑛

∑𝑖=1

𝑓𝑖𝜔𝑖𝜓𝑖−𝑘 − 1

2𝜑𝑚V2,

(2)

where𝜓𝑖is the relative burnt percentage of the 𝑖th propellant,

𝑍𝑖is the relative burnt thickness of the 𝑖th propellant, 𝜒

𝑖, 𝜆𝑖,

𝜇𝑖are characteristic parameters of the 𝑖th propellant, 𝑢

1𝑖is

the burning rate coefficient of the 𝑖th propellant, 𝑒1𝑖is the

half web thickness of the 𝑖th propellant, 𝑝 is the pressure inthe chamber, 𝑛

𝑖is the burning rate pressure index of the 𝑖th

propellant, 𝜑 is the coefficient accounting for the secondaryenergy losses, 𝑚 is the projectile mass, V is the projectilevelocity, 𝑆 is the barrel cross-section area, 𝑙 is the tube length,

Mathematical Problems in Engineering 3

𝑙𝜓is the ratio of chamber free volume to the bore area, 𝑓 is

the impetus force of the 𝑖th propellant, 𝜔𝑖is mass of the 𝑖th

propellant, and 𝑘 is propellant specific heat ratio.According to the form shape function of the seven-

perforated propellant, 𝜓𝑖should be changed at the start

point of slivers separation. The form shape function can bedescribed in (3) as follows:

𝜓𝑖=

{{{{{{{{{

𝜒𝑖𝑍𝑖(1 + 𝜆

𝑖𝑍𝑖+ 𝜇𝑖𝑍2𝑖) 0 ≤ 𝑍

𝑖< 1

𝜒𝑠𝑖𝑍𝑖(1 + 𝜆

𝑠𝑖𝑍𝑖) 1 ≤ 𝑍

𝑖< 𝑍𝑘𝑖

1 𝑍 ≥ 𝑍𝑘𝑖,

(3)

𝑍𝑘𝑖=𝑒1𝑖+ 𝜌𝑖

𝑒1𝑖

. (4)

3. Formulation of Particle SwarmOptimization Algorithm

In the literature there exist numerous methods of opti-mization that study and analyze the optimization problemsunder different conditions. PSO is considered as one ofthe best computationally efficient optimization techniques.It converges to the optimal solution in many problemswhere most analytical methods fail to converge. PSO hassome advantages over other similar optimization techniques,namely, the following [8].

(1) PSO has few and simple parameters. Hence, it is easyto implement.

(2) It has a more effective memory capability than theGA as every particle remembers its own previous bestvalue as well as the neighborhood’s best value.

(3) PSO is more efficient in maintaining the diversity ofthe swarm, since all the particles use the informa-tion related to the most successful particle in orderto improve themselves, whereas, in GA, the worsesolutions are discarded and only the good ones aresaved; therefore, inGA the populations evolve arounda subset of the best individuals.

The procedure of PSO algorithm can be described as follows.

(1) Initialize the swarmby assigning a randomposition inthe problem hyperspace to each particle (the swarmcomposed of population of random solutions calledparticles).

(2) Evaluate the fitness function for each particle (fitnessfunction is the objective function obtained from theinterior ballistic simulation).

(3) For each individual particle, particle’s fitness value iscompared with its 𝑝best (personal best position). If thecurrent value is better than the𝑝best value, then set thisvalue as the𝑝best and the current particle’s position,𝑥𝑖,as 𝑝𝑖.

(4) Identify the particle that has the best fitness value inthe swarm. The value of its fitness function is identi-fied as 𝑔best (global best position) and its position as𝑝𝑔𝑙.

(5) Update the velocities and positions of all the particles.(6) Repeat steps 2–5 until the condition of stopping ismet

(e.g., maximum number of iterations or a sufficientlygood fitness value).

The updating rules of velocity and position can be describedas follows:

V𝑖,𝑗(𝑡) = 𝜔V

𝑖,𝑗(𝑡 − 1) + 𝑐

1rand1(𝑝𝑖− 𝑥𝑖,𝑗(𝑡 − 1))

+ 𝑐2rand2(𝑝𝑔𝑙− 𝑥𝑖,𝑗(𝑡 − 1)) ,

𝑥𝑖,𝑗(𝑡) = 𝑥

𝑖,𝑗(𝑡 − 1) + V

𝑖,𝑗(𝑡) ,

(5)

where V𝑖,𝑗is the velocity of particle 𝑖 of the 𝑗th dimension, 𝑥

𝑖,𝑗

is the position of particle 𝑖 of the 𝑗th dimension, 𝑡 indicatesthe iteration number, 𝑖 = 1, 2, . . . , 𝑁, 𝑗 = 1, 2, . . . ,𝑀, 𝑁 isthe size of swarm,𝑀 is the dimension of the search space, 𝑐

1

and 𝑐2are the acceleration coefficients (generally 𝑐

1+ 𝑐2≤ 4),

rand1and rand

2are randomnumbers ∈ [0, 1], and𝜔 is inertia

weight which is used as a parameter to control the explorationand exploitation in the search space.

4. Application of PSO Algorithm toInterior Ballistic Model

PSO technique is coupled with the lumped-parameter modelto improve the interior ballistic performance via optimizingthe propelling charge design. This model is applied to 76mmnaval gun with guided projectile utilizing mixed propellant.Two different approaches will be investigated through thispresent work:

(1) single-objective function PSO method,(2) multiobjective function PSO method.

4.1. Single-Objective PSOMethod. Firstly, the single-objectivePSO method is carried out in order to predict the optimumcharge design and to improve the performance of interiorballistic process. The single-objective optimization problemcontains objective function, design variables, and constraints.Through this problem themuzzle velocity is considered as theobjective function.

4.1.1. Objective Function. The muzzle velocity is consideredas the most important parameter in the interior ballisticprocess. Hence, the only objective function in this methodis maximizing the muzzle velocity.

4.1.2. Design Variables. Selection of the design variables isconsidered very crucial to obtain the optimum solution.Half web thickness and loading density of the propellantare considered the main characteristic parameters of thepropellant [3]. In this work, two types of propellants areused. Hence, the design variables are considered as 𝑥 =[𝑒1, Δ1, 𝑒2, Δ2], where 𝑒

1and Δ

1are half web thickness and

loading density of the granular propellant and 𝑒2and Δ

2

are half web thickness and loading density of the tubularpropellant.


Table 1: Nonoptimized parameters.

Parameter Value Unit Parameter Value UnitGun caliber 0.076 m Chamber volume 0.00354 m3

Tube length 4.045 m Impetus force 980000 J/KgChamber length 0.38 m Ignition temp. 615 KProjectile mass 5.9 Kg Co-volume 0.001 m3/Kg

4.1.3. Constraints. The penalty method is utilized to treat theconstraints of the optimization problem. The original con-strained problem is replaced by a sequence of unconstrainedproblems [9, 10]. The unconstrained problem consists ofthe objective function and the penalty terms according toeach constraint. Penalty terms penalize the objective functionwhen its corresponding constraint is violated.

The selection of the constraints limits is consideredaccording to the launch safety, the technical requirements,the experimental work for charge design, and the data of thetypical gun.

The details of the constraints can be described as follows.

(1) Projectile Muzzle Velocity. The minimum value of muzzlevelocity 𝑉min is determined according to the sufficient per-formance required to satisfy the tactical requirements. Themuzzle velocity 𝑉

𝑚should be higher than or at least equal

to the 𝑉min:𝑉𝑚≥ 𝑉min. (6)

(2) Maximum Chamber Pressure. According to the launchsafety requirements, the strength of the gun tube has a limit.The peak pressuremust be lower than themaximumpressurevalue 𝑃

𝑚; this value is considered a design constant for each

gun caliber. Hence, the peak pressure 𝑃max should be lowerthan this value:

𝑃max ≤ 𝑃𝑚. (7)

(3) Charge Loading Density. The chamber volume is a fixedvalue. Hence the propelling charge density has a limit.Consider

0 < Δ < Δ𝑗, (8)

where Δ𝑗is considered the maximum charge density limit

calculated according to the fixed chamber volume.The loading density limits depend on the launch safety

and the required interior ballistic performance. The limits ofthis constraint can be considered as follows:

0.4 ≤ Δ ≤ 0.8 kg ⋅ dm−3. (9)

(4) Relative Charge Burnout Point. Relative burnout point𝜂𝑘is considered as an indicator that explains whether the

propellant is completely burnt or not. 𝜂𝑘can be calculated

using the following equation:

𝜂𝑘=𝑙𝑘

𝑙, (10)

where 𝑙 is the bore length and 𝑙𝑘is the projectile travel

inside the bore associated with the complete burning of thepropellant.

Low relative burnout point means that the propellant wasburnt out early while the projectile was still inside the bore.Hence, some of the kinetic energy of the projectile will belost to overcome the engraving force. Large relative burnoutpointmeans that the projectile exited from themuzzle and thepropellant was not completely burnt. Hence, the remainingamount of the propellant is considered as a waste propellant.Due to charge design experience, the limits of this constraintcan be considered as follows:

0.5 ≤ 𝜂𝑘≤ 0.8. (11)

(5) Energy Efficiency of the Charge. Energy efficiency ofthe charge 𝛾

𝑔indicates whether the charge is fully utilized.

It explains the ratio between the energy delivered to theprojectile and the energy of the charge burning. Low valueof 𝛾𝑔means small kinetic energy delivered to the projectile.

𝛾𝑔is calculated using the following equation:

𝛾𝑔=(1/2)𝑚V2

𝑚

𝑓𝜔/ (𝑘 − 1). (12)

The limits of 𝛾𝑔can be considered as follows:

0.16 ≤ 𝛾𝑔≤ 0.6. (13)

(6) Half Web Thickness. Values of the half web thicknesses𝑒1and 𝑒

2of the propellant depend on the gun caliber. The

limits of 𝑒1and 𝑒

2should be selected carefully according

to the charge manufacturing, launch safety, and technicalrequirements. These limits can be considered as follows:

0.25 ≤ 𝑒1≤ 0.9mm,

0.25 ≤ 𝑒2≤ 0.9mm.

(14)

The nonoptimized parameters of the interior ballisticprocess for the 76mm gun are tabulated in Table 1.


860

880

900

920

940

960

980

1000

Iteration number(b) PSO(a) GA

0 100 200 300 400 500

Vm

(m/s

)

860

880

900

920

940

960

980

1000

Iteration number0 100 200 300 400 500

Vm

(m/s

)

Figure 1: Convergence graph of different algorithms for single-objective problem.

Table 2: Parameters of GA.

Parameters Value Parameters ValuePopulation size 40 Number of constrains 6Number of generations 500 Probability of crossover 0.8Number of objectives 1 Probability of mutation 0.2

Table 3: Parameters of PSO.

Parameters Value Parameters ValuePopulation size 40 Number of constrains 6Number of generations 500 First acceleration coefficient 𝑐

11.8

Number of objectives 1 Second acceleration coefficient 𝑐2

1.8

In this section, the general optimization model can bewritten in (1) with the design variables 𝑥 = [𝑒

1, Δ1, 𝑒2, Δ2]:

min 𝑓 (𝑥) = −𝑉𝑚

subject to 𝑉𝑚≥ 800ms−1

𝑃max ≤ 350MPa

0.4 ≤ Δ ≤ 0.8Kg ⋅ dm−3

0.5 ≤ 𝜂𝑘≤ 0.8

0.16 ≤ 𝛾𝑔≤ 0.6

0.25 ≤ 𝑒1≤ 0.9mm

0.25 ≤ 𝑒2≤ 0.9mm.

(15)

In the single-objective problem, two different optimiza-tion techniques, GA and PSO, are carried out in conjunctionwith the lumped-parameter model. The values of the mainparameters utilized in GA and PSO algorithm are tabulatedin Tables 2 and 3, respectively.

The optimization results for GA and PSO algorithmscomparedwith the experimental data are tabulated inTable 4.The convergence of the two different algorithms for theobjective function is illustrated in Figure 1.

From the optimization results illustrated in Table 4 andFigure 1, it is clear that PSO technique is the winner. PSOprovides a better performance of interior ballistic propellingcharge design. The better performance is represented in thespeed of convergence (PSO converges after 170 iterations,while GA converges after 307 iterations) and the solutionquality (𝑉

𝑚= 992.52m/s in PSO, while 𝑉

𝑚= 985.8m/s in

GA).Figure 2 represents the convergence of different parame-

ters of the interior ballistic obtained by using PSO technique.These parameters include the design variables and someinterior ballistic performance indexes.

4.2. Multiobjective PSOMethod. In the actual design process,the designers have different interests in the objective func-tion. Hence, the multiobjective model should be considered


0.50

0.52

0.54

0.56

0.58

0.60

0.62

e 1(m

m)

e1 (mm)

Iteration number

(a) e2 (mm)(b)

(c)

(e)

0 100 200 300 400 500

0.3

0.4

0.5

0.6

0.7

0.8

e 2(m

m)


0.50

0.55

0.60

0.65

0.70

0.75

Δ1

(kg/

dm3)

Δ1 (kg/dm3) (d) Δ2 (kg/dm3)


0.040

0.045

0.050

0.055

0.060

0.065

0.070


Δ2

(kg/

dm3)

0.4

0.5

0.6

0.7

0.8

0.9

1.0

𝜂k

𝜂k (f) 𝛾g


0.20

0.21

0.22

0.23

0.24

0.25

0.26

𝛾g


Figure 2: Convergence process of different interior ballistic parameters for single-objective problem by PSO technique.


Table 4: Results of the different optimization techniques compared with the experimental date.

V𝑚(m/s) Pmax (MPa) 𝑒

1(mm) Δ

1(Kg/dm3) 𝑒

2(mm) Δ

1(Kg/dm3)

Experimental 983.27 344.3 0.595 0.695 0.705 0.057GA 985.80 347.2 0.6086 0.6105 0.6933 0.0608PSO 992.52 345.9 0.6102 0.7078 0.7013 0.0618

Table 5: Values of the objective functions according to the designpoints.

Design points A B C𝑉𝑚(m/s) 988.12 970.26 980.21

𝑃𝑔(MPa) 86.132 82.192 83.372

𝛾𝑔

0.2508 0.2536 0.2634

to satisfy the different requirements. This section deals withmultiobjective optimization model for interior ballistic pro-pelling charge design. Three objective functions, the muzzlevelocity𝑉

𝑚, the muzzle pressure 𝑃

𝑔, and the energy efficiency

of propelling charge 𝛾𝑔are considered to improve the interior

ballistic performance and to satisfy the different technicalrequirements.

The design variables and the constraints are the same asin the single-objective problem, but the objective function isnot the same. The objective function contains the followingobjectives.

Muzzle Velocity (𝑉𝑚). In gun design, the muzzle velocity is

required to be maximized as much as possible to satisfy thetactical requirements. Hence, muzzle velocity will be the keyobjective function in interior ballistic design.

Muzzle Pressure (𝑃𝑔). The muzzle pressure affects the muzzle

blast, the firing precision, and the initial dispersion. Hence,the muzzle pressure should be minimized as possible.

Energy Efficiency of the Charge (𝛾𝑔). The energy efficiency of

the charge is considered as one of the main goals in interiorballistic charge design. It should be maximized as possible.

The multiobjective model for the specific gun can bedescribed as follows:

min 𝑓 (𝑥) = (−𝜔1𝑉𝑚, −𝜔2𝛾𝑔, 𝜔3𝑃𝑔)

subject to 𝑉𝑚≥ 800ms−1

𝑃max ≤ 350MPa

0.4 ≤ Δ ≤ 0.8Kg ⋅ dm−3

0.5 ≤ 𝜂𝑘≤ 0.8

0.16 ≤ 𝛾𝑔≤ 0.6

0.25 ≤ 𝑒1≤ 0.9mm

0.25 ≤ 𝑒2≤ 0.9mm,

(16)

940

960

980

1000

1020

72

74

76

78

80

82

84

86

88

0.23

0.24

0.25

0.26

0.27

0.28

0.29

0.30

pg

(MPa

)

pg

𝛾g

𝛾g

Vm

V(m

/s)

mIteration number

0 100 200 300 400 500

Figure 3: Convergence graph of the three objective functions.

where 𝜔1, 𝜔2, and 𝜔

3are the weight coefficients of the objec-

tive functions determined by the gun designer accordingto the design requirements. For example, if the designerconsiders themuzzle velocity to be themain target, theweightcoefficient of themuzzle velocity𝜔

1should be increased at the

expense of the other coefficients 𝜔2and 𝜔

3.

Figure 3 shows the convergence graph for the threeobjective functions, muzzle velocity, muzzle pressure, andenergy efficiency of the charge. It shows a good and rapidconvergence to certain values determined by PSO technique.Figure 4 represents the feasible region of web thickness 𝑒

1

and propelling charge density Δ1with the corresponding

objective functions. Three design points marked in Figures4(a), 4(b), and 4(c) are selected to stand for the maximummuzzle velocity, minimum muzzle pressure, and maximumenergy efficiency, respectively. The values of the objectivefunctions according to the three design points are listed inTable 5.

Based on the optimization results at point A, Figures 5and 6 show the pressure-time and velocity-time curves fordifferent schemes, original scheme and optimized scheme.The original scheme refers to the lumped-parameter modeldescribed in (2) without optimization. In Figure 5, the opti-mized scheme decreases the gradient of the pressure-timecurve and the muzzle pressure. From Figure 6, the optimizedscheme increases the muzzle velocity better than the originalscheme. Therefore, the optimized scheme improves the inte-rior ballistic performance and provides a better firing safety.

The feasible region shown in Figure 4 provides a largenumber of appropriate solutions. Obviously, the optimumsolution cannot be found due to the contradictory trade-offs among the selection conditions. Hence, an appropriatemethod for adapting the weight coefficients is required


0.66

0.67

0.68

0.69

0.70

0.71

0.72

0.73

C

B

A

976

971

981

987

970

972

975

977

979

981

984

986

988

0.56 0.58 0.60 0.62 0.64 0.66

e1 (mm)

Δ1

(kg/

dm3)

Vm

(m/s

)

(a) Muzzle velocity

0.66

0.67

0.68

0.69

0.70

0.71

0.72

0.73

86

85

8384

82

83

84

85

86

87

88

C

A

B

0.56 0.58 0.60 0.62 0.64 0.66

e1 (mm)

Δ1

(kg/

dm3)

pg

(m/s

)

(b) Muzzle pressure

0.261

0.238

0.2440.247

0.2510.254

0.231

0.237

0.243

0.249

0.254

0.258

0.264

A

C

B0.66

0.67

0.68

0.69

0.70

0.71

0.72

0.73

0.56 0.58 0.60 0.62 0.64 0.66

e1 (mm)

Δ1

(kg/

dm3)

𝛾g

(c) Energy efficiency

Figure 4: Feasible region of design variables corresponding to multiobjective function.

for helping the designers of propelling charge to find theoptimum solution [11].

4.2.1. Linear Analysis Method. According to the feasibleregion obtained by the multiobjective function, it is crucialto investigate the effect of changing the weight coefficientson the optimal solution. Hence, the linear analysis method isutilized to obtain a suitable combination of the weight coef-ficients. Three cases are tabulated in Table 6. One coefficientis changed linearly in each case, and the other coefficients areselected randomly in such a way that 𝜔

1+𝜔2+𝜔3= 1, where

one has the following.

Case 1. It indicates that 𝜔1is changed linearly, and 𝜔

1

represents the weight coefficient of muzzle velocity.


2

represents the weight coefficient of energy efficiency of thecharge.

0

50

100

150

200

250

300

350

OriginalOptimized

Pres

sure

(MPa

)

Time (ms)0 2 4 6 8 10 12

Figure 5: Pressure-time curve for different schemes.


Table 6: Linear analysis method coefficients for the three cases.

Runs Case 1 Case 2 Case 3𝜔1

𝜔2

𝜔3

𝜔1

𝜔2

𝜔3

𝜔1

𝜔2

𝜔3

1 0 0.325 0.675 0.713 0 0.287 0.823 0.177 02 0.2 0.648 0.152 0.535 0.2 0.265 0.641 0.159 0.23 0.4 0.246 0.354 0.422 0.4 0.178 0.355 0.245 0.44 0.6 0.023 0.377 0.213 0.6 0.187 0.241 0.159 0.65 0.8 0.132 0.068 0.101 0.8 0.099 0.102 0.098 0.86 1 0 0 0 1 0 0 0 1

0

200

400

600

800

1000

Velo

city

(m/s

)

OriginalOptimized

Time (ms)0 2 4 6 8 10 12

Figure 6: Velocity-time curve for different schemes.


3

represents the weight coefficient of muzzle pressure.

According to the multiobjective PSO method, the threeobjective values, 𝑉

𝑚, 𝑃𝑔, and 𝛾

𝑔are illustrated in Figure 7 to

Figure 9 with the different weight coefficients in each case.Through the observations of Figures 7, 8, and 9, it is

obvious to conclude that the weight coefficients have asignificant effect on the objective values. The highest muzzlevelocity is attained at the highest value of 𝜔

1. The change

of the energy efficiency has enormous effect on the otherobjective values. Accordingly, the energy efficiency can beconsidered as crucial index of interior ballistic performance.It can be seen from the data in Figures 7, 8, and 9 that 𝜔

1

is strongly recommended to take a value greater than 0.6,𝜔2to take a value greater than 0.1, and 𝜔

3to take a value

greater than 0.1. The designers should weigh gains and lossesaccording to the change of the weight coefficients on theobjectives to obtain the optimum performance and achievethe tactical requirements and firing safety.

5. Conclusion

This paper has given an account of and the reasons for thewidespread use of the particle swarm optimization. PSO

750

800

850

900

950

1000

Runs

72

74

76

78

80

82

84

86

88

0.21

0.22

0.23

0.24

0.25

0.26

0.27

𝛾gVm pg

1 2 3 4 5 6

Vm (m/s)

𝛾g

pg (MPa)

Figure 7: Optimal solution of different six weights of Case 1.

algorithm was developed in conjunction with interior ballis-tic lumped-parameter model utilizing mixed propellant. Themixed propellant is composed of granular seven-perforatedpropellant and tubular one-hole propellant. PSO was appliedto optimize the interior ballistic propelling charge designin order to improve the interior ballistic performance. Thefollowing conclusions can be drawn from the present study.

(1) Based on the characteristics of interior ballisticpropelling charge design, particle swarm optimiza-tion technique was developed with two differentapproaches: single-objective problemandmultiobjec-tive problem.

(2) GA and PSO techniques were applied to the sin-gle objective problem of interior ballistic propellingcharge design. The optimization results were com-pared with the experimental data. The optimizedsolution showed that PSO technique has a betterperformance than GA. The better performance wasrepresented in the speed of convergence and thequality of the solution.

(3) PSO with multiobjective problem provided a goodopportunity for the charge’s designers. The bestparameters can be selected through the feasible region


750

800

850

900

950

1000

72

74

76

78

80

82

84

86

88

0.21

0.22

0.23

0.24

0.25

0.26

0.27

Runs1 2 3 4 5 6

Vm (m/s)

𝛾g

pg (MPa)

𝛾gVm pg


750

800

850

900

950

1000

72

74

76

78

80

82

84

86

88

0.21

0.22

0.23

0.24

0.25

0.26

0.27

Runs1 2 3 4 5 6

Vm (m/s)

𝛾g

pg (MPa)

𝛾gVm pg


to attain the optimum solution according to thetactical requirements. Utilizing the multiobjectivemethod improved the interior ballistic performanceand assured the launch safety of the guided projectile.

(4) The linear analysis method was developed to presentthe most appropriate set of the weight coefficients ofobjectives.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The research was supported by the Research Fund forthe Natural Science Foundation of Jiangsu province(BK20131348) and the Key Laboratory Fund (Grant no.9140C300103140C30001), China.

References

[1] J. R. Gonzalez, Interior ballistics optimization [Ph.D. thesis],Kansas State University, 1990.

[2] H. S. Zhang, The optimization studies of a new ignition systemand charge structure in high-velocity guns [Ph.D. thesis], NanjingUniversity of Science and Technology, 1998.

[3] K. Li and X. Zhang, “Multi-objective optimization of interiorballistic performance using NSGA-II,” Propellants, Explosives,Pyrotechnics, vol. 36, no. 3, pp. 282–290, 2011.

[4] J. Kennedy and R. Eberhart, “Particle swarm optimization,”in Proceedings of the IEEE International Conference on NeuralNetworks, pp. 1942–1948, December 1995.

[5] V. Kalivarapu, J.-L. Foo, and E. Winer, “Improving solutioncharacteristics of particle swarm optimization using digitalpheromones,” Structural and Multidisciplinary Optimization,vol. 37, no. 4, pp. 415–427, 2009.

[6] M. Marinaki, Y. Marinakis, and G. E. Stavroulakis, “Fuzzy con-trol optimized by a Multi-Objective Particle Swarm Optimiza-tion algorithm for vibration suppression of smart structures,”Structural and Multidisciplinary Optimization, vol. 43, no. 1, pp.29–42, 2011.

[7] Y.X. Yuan andX. B. Zhang,MultiphaseHydrokinetic Foundationof High Temperature and High Pressure, Publishing Company ofHarbin Institute of Technology, Harbin, China, 2005.

[8] Y. del Valle, G. K. Venayagamoorthy, S. Mohagheghi, J.-C.Hernandez, and R. G. Harley, “Particle swarm optimization:basic concepts, variants and applications in power systems,”IEEE Transactions on Evolutionary Computation, vol. 12, no. 2,pp. 171–195, 2008.

[9] W. Y. Sun and Y. Yuan, Optimization Theory and Methods Non-linear Programming, Springer Optimization and Its Applica-tions, Springer, 2006.

[10] A. Ravindran, K. M. Ragsdell, and G. V. Reklaitis, EngineeringOptimization: Methods and Applications, John Wiley & Sons,Hoboken, NJ, USA, 2nd edition, 2006.

[11] K. Li and X. Zhang, “Using NSGA-II and TOPSIS methods forinterior ballistic optimization based on one-dimensional two-phase flow model,” Propellants, Explosives, Pyrotechnics, vol. 37,no. 4, pp. 468–475, 2012.

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