This paper investigates the interior ballistic propelling charge design using the optimization methods to select the optimum charge design and to improve the interior ballistic performance. The propelling charge consists of a mixture propellant of sevenperforated granular propellant and onehole tubular propellant. The genetic algorithms and some other evolutionary algorithms have complex evolution operators such as crossover, mutation, encoding, and decoding. These evolution operators have a bad performance represented in convergence speed and accuracy of the solution. Hence, the particle swarm optimization technique is developed. It is carried out in conjunction with interior ballistic lumpedparameter model with the mixture propellant. This technique is applied to both singleobjective and multiobjective problems. In the singleobjective problem, the optimization results are compared with genetic algorithm and the experimental results. The particle swarm optimization introduces a better performance of solution quality and convergence speed. In the multiobjective problem, the feasible region provides a set of available choices to the charge’s designer. Hence, a linear analysis method is adopted to give an appropriate set of the weight coefficients for the objective functions. The results of particle swarm optimization improved the interior ballistic performance and provided a modern direction for interior ballistic propelling charge design of guided projectile.
Recently, study of the propelling charge design becomes very essential to achieve the interior ballistic performance and assure the safety firing. This study is known as design of the interior ballistic which is considered as a crucial branch of gun system design. There are a number of computerbased interior ballistic models with different capabilities. These models allow the researchers of interior ballistic to predict the interior ballistic performance of a particular gun, charge, and projectile combination. The classic interior ballistics models including the characteristics of the gun, charge, and projectile are utilized to predict the muzzle velocity and the peak pressure. But these models cannot provide the best design of the propelling charge that gives the optimum solution of the interior ballistic performance.
Optimization techniques allow the designers to evaluate a large number of design alternatives in a systematic and efficient manner to find the best design. The solution of an interior ballistic model is considered improved if there is a net increase in muzzle velocity without violating gun constraints. The optimization techniques are used to improve the gun performance and/or decrease the design cost while meeting the constraints appropriate to the problem. Therefore, some researchers tried to apply the optimization techniques with the interior ballistic models. A numerical optimization method called augmented Lagrange multiplier was carried out and coupled to a classic interior ballistic model. This method was used to design the parameters of sevenperforation propellant and to obtain the best muzzle velocity for the projectile fired from 120 mm tank cannons [
Particle swarm optimization (PSO) was firstly developed by Kennedy and Eberhart [
In this work, PSO technique is coupled with the interior ballistic model to optimize the propelling charge design. The utilized charge is a mixture charge which consists of two different propellants, granular sevenperforated propellant and tubular propellant. PSO technique is applied to the classic interior ballistic model for 76 mm naval gun with guided projectile to improve the interior ballistic performance. Two types of objective functions are used with PSO techniques, singleobjective function and multiobjective function. Through this work, it is found that the optimization results improved the interior ballistic performance and firing safety with a high quality and quick convergence of the optimum solution.
The optimization of the charge design in interior ballistic process is considered as nonlinear optimization, multiconstrained problem. It consists of multiobjective function and multiple design variables. The design variables have different effects on the performance of interior ballistic. In the interior ballistic process coupled with the optimization technique, there are restrictions between the objective functions. For example, the designer of the interior ballistic intends to obtain the maximum muzzle velocity as a main objective. Maximum muzzle velocity requires a high muzzle pressure, while one of the objective functions is to minimize the muzzle pressure. Hence, the restriction is clear between the muzzle velocity objective function and muzzle pressure objective function. The objective functions should be weighted according to the advantages and disadvantages. The general optimization problem for the classical interior ballistic process can be described as follows:
The relation between the optimization technique and the interior ballistic model is the objective functions that are calculated from the interior ballistic model and then optimized by the optimization model.
There exist some different tools to simulate the interior ballistic process such as lumpedparameter model and twophase flow model. In this work, lumpedparameter model is utilized to simulate the interior ballistic process and calculate the objective values that will be optimized by using the PSO technique. The lumpedparameter model can be written as follows [
According to the form shape function of the sevenperforated propellant,
In the literature there exist numerous methods of optimization that study and analyze the optimization problems under different conditions. PSO is considered as one of the best computationally efficient optimization techniques. It converges to the optimal solution in many problems where most analytical methods fail to converge. PSO has some advantages over other similar optimization techniques, namely, the following [
PSO has few and simple parameters. Hence, it is easy to implement.
It has a more effective memory capability than the GA as every particle remembers its own previous best value as well as the neighborhood’s best value.
PSO is more efficient in maintaining the diversity of the swarm, since all the particles use the information related to the most successful particle in order to improve themselves, whereas, in GA, the worse solutions are discarded and only the good ones are saved; therefore, in GA the populations evolve around a subset of the best individuals.
Initialize the swarm by assigning a random position in the problem hyperspace to each particle (the swarm composed of population of random solutions called particles).
Evaluate the fitness function for each particle (fitness function is the objective function obtained from the interior ballistic simulation).
For each individual particle, particle’s fitness value is compared with its
Identify the particle that has the best fitness value in the swarm. The value of its fitness function is identified as
Update the velocities and positions of all the particles.
Repeat steps 2–5 until the condition of stopping is met (e.g., maximum number of iterations or a sufficiently good fitness value).
PSO technique is coupled with the lumpedparameter model to improve the interior ballistic performance via optimizing the propelling charge design. This model is applied to 76 mm naval gun with guided projectile utilizing mixed propellant. Two different approaches will be investigated through this present work:
singleobjective function PSO method,
multiobjective function PSO method.
Firstly, the singleobjective PSO method is carried out in order to predict the optimum charge design and to improve the performance of interior ballistic process. The singleobjective optimization problem contains objective function, design variables, and constraints. Through this problem the muzzle velocity is considered as the objective function.
The muzzle velocity is considered as the most important parameter in the interior ballistic process. Hence, the only objective function in this method is maximizing the muzzle velocity.
Selection of the design variables is considered very crucial to obtain the optimum solution. Half web thickness and loading density of the propellant are considered the main characteristic parameters of the propellant [
The penalty method is utilized to treat the constraints of the optimization problem. The original constrained problem is replaced by a sequence of unconstrained problems [
The selection of the constraints limits is considered according to the launch safety, the technical requirements, the experimental work for charge design, and the data of the typical gun.
The details of the constraints can be described as follows.
The loading density limits depend on the launch safety and the required interior ballistic performance. The limits of this constraint can be considered as follows:
Low relative burnout point means that the propellant was burnt out early while the projectile was still inside the bore. Hence, some of the kinetic energy of the projectile will be lost to overcome the engraving force. Large relative burnout point means that the projectile exited from the muzzle and the propellant was not completely burnt. Hence, the remaining amount of the propellant is considered as a waste propellant. Due to charge design experience, the limits of this constraint can be considered as follows:
The limits of
The nonoptimized parameters of the interior ballistic process for the 76 mm gun are tabulated in Table
Nonoptimized parameters.
Parameter  Value  Unit  Parameter  Value  Unit 

Gun caliber  0.076  m  Chamber volume  0.00354  m^{3} 
Tube length  4.045  m  Impetus force  980000  J/Kg 
Chamber length  0.38  m  Ignition temp.  615  K 
Projectile mass  5.9  Kg  Covolume  0.001  m^{3}/Kg 
In this section, the general optimization model can be written in (
In the singleobjective problem, two different optimization techniques, GA and PSO, are carried out in conjunction with the lumpedparameter model. The values of the main parameters utilized in GA and PSO algorithm are tabulated in Tables
Parameters of GA.
Parameters  Value  Parameters  Value 

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

Population size  40  Number of constrains  6 
Number of generations  500  First acceleration coefficient 
1.8 
Number of objectives  1  Second acceleration coefficient 
1.8 
The optimization results for GA and PSO algorithms compared with the experimental data are tabulated in Table
Results of the different optimization techniques compared with the experimental date.








Experimental  983.27  344.3  0.595  0.695  0.705  0.057 
GA  985.80  347.2  0.6086  0.6105  0.6933  0.0608 
PSO  992.52  345.9  0.6102  0.7078  0.7013  0.0618 
Convergence graph of different algorithms for singleobjective problem.
From the optimization results illustrated in Table
Figure
Convergence process of different interior ballistic parameters for singleobjective problem by PSO technique.
In the actual design process, the designers have different interests in the objective function. Hence, the multiobjective model should be considered to satisfy the different requirements. This section deals with multiobjective optimization model for interior ballistic propelling charge design. Three objective functions, the muzzle velocity
The design variables and the constraints are the same as in the singleobjective problem, but the objective function is not the same. The objective function contains the following objectives.
The multiobjective model for the specific gun can be described as follows:
Figure
Convergence graph of the three objective functions.
Figure
Values of the objective functions according to the design points.
Design points  A  B  C 


988.12  970.26  980.21 

86.132  82.192  83.372 

0.2508  0.2536  0.2634 
Feasible region of design variables corresponding to multiobjective function.
Muzzle velocity
Muzzle pressure
Energy efficiency
Based on the optimization results at point A, Figures
Pressuretime curve for different schemes.
Velocitytime curve for different schemes.
The feasible region shown in Figure
According to the feasible region obtained by the multiobjective function, it is crucial to investigate the effect of changing the weight coefficients on the optimal solution. Hence, the linear analysis method is utilized to obtain a suitable combination of the weight coefficients. Three cases are tabulated in Table
Linear analysis method coefficients for the three cases.
Runs  Case 
Case 
Case 










 
1  0  0.325  0.675  0.713  0  0.287  0.823  0.177  0 
2  0.2  0.648  0.152  0.535  0.2  0.265  0.641  0.159  0.2 
3  0.4  0.246  0.354  0.422  0.4  0.178  0.355  0.245  0.4 
4  0.6  0.023  0.377  0.213  0.6  0.187  0.241  0.159  0.6 
5  0.8  0.132  0.068  0.101  0.8  0.099  0.102  0.098  0.8 
6  1  0  0  0  1  0  0  0  1 
It indicates that
It indicates that
It indicates that
According to the multiobjective PSO method, the three objective values,
Optimal solution of different six weights of Case
Through the observations of Figures
Optimal solution of different six weights of Case
Optimal solution of different six weights of Case
This paper has given an account of and the reasons for the widespread use of the particle swarm optimization. PSO algorithm was developed in conjunction with interior ballistic lumpedparameter model utilizing mixed propellant. The mixed propellant is composed of granular sevenperforated propellant and tubular onehole propellant. PSO was applied to optimize the interior ballistic propelling charge design in order to improve the interior ballistic performance. The following conclusions can be drawn from the present study.
Based on the characteristics of interior ballistic propelling charge design, particle swarm optimization technique was developed with two different approaches: singleobjective problem and multiobjective problem.
GA and PSO techniques were applied to the single objective problem of interior ballistic propelling charge design. The optimization results were compared with the experimental data. The optimized solution showed that PSO technique has a better performance than GA. The better performance was represented in the speed of convergence and the quality of the solution.
PSO with multiobjective problem provided a good opportunity for the charge’s designers. The best parameters can be selected through the feasible region to attain the optimum solution according to the tactical requirements. Utilizing the multiobjective method improved the interior ballistic performance and assured the launch safety of the guided projectile.
The linear analysis method was developed to present the most appropriate set of the weight coefficients of objectives.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The research was supported by the Research Fund for the Natural Science Foundation of Jiangsu province (BK20131348) and the Key Laboratory Fund (Grant no. 9140C300103140C30001), China.