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Aeronautics

Chinese Journal of Aeronautics 21(2008) 481-487

www.elsevier.com/locate/cja

A Hybrid Optimization Approach for SRM FINOCYL Grain Design

Khurram Nisar*, Liang Guozhu, Qasim Zeeshan

School of Astronautics, Beijing University of Aeronautics and Astronautics, Beijing 100191, China Received 19 February 2008; accepted 2 September 2008

Abstract

This article presents a method to design and optimize 3D FINOCYL grain (FCG) configuration for solid rocket motors (SRMs). The design process of FCG configuration involves mathematical modeling of the geometry and parametric evaluation of various independent geometric variables that define the complex configuration. Virtually infinite combinations of these variables will satisfy the requirements of mass of propellant, thrust, and burning time in addition to satisfying basic needs for volumetric loading fraction and web fraction. In order to ensure the acquisition of the best possible design to be acquired, a sound approach of design and optimization is essentially demanded. To meet this need, a method is introduced to acquire the finest possible performance. A series of computations are carried out to formulate the grain geometry in terms of various combinations of key shapes inclusive of ellipsoid, cone, cylinder, sphere, torus, and inclined plane. A hybrid optimization (HO) technique is established by associating genetic algorithm (GA) for global solution convergence with sequential quadratic programming (SQP) for further local convergence of the solution, thus achieving the final optimal design. A comparison of the optimal design results derived from SQP, GA, and HO algorithms is presented. By using HO technique, the parameter of propellant mass is optimized to the minimum value with the required level of thrust staying within the constrained burning time, nozzle and propellant parameters, and a fixed length and outer diameter of grain.

Keywords: FINOCYL grain; internal ballistics; optimization; solid rocket motor

Nomenclature

Ab — burning area, Ae — nozzle exit area

Ap — port area, Api — initial port area

At — area of throat, BR — burning rate

C* — characteristic velocity

CF — thrust coefficient, de — nozzle exit diameter

F — motor thrust, Fav — average thrust

Fmax — maximum thrust, Is — specific impulse

It — total impulse, L — length of grain

mp — mass of propellant, n — pressure exponent

N — number of fins

Pav — average chamber pressure

Pc — chamber pressure

Pmax — maximum pressure

Corresponding author. Tel.: +86-10-82339944. E-mail address: khurram_nisar6@yahoo.com

ELSEVIER

tb — burning duration, Vp — volume of propellant W — web thickness, pp — propellant density

1 Introduction

The essence of grain design is to evolve burning surface area and develop its relation with web burnt. 3D grains are usually very intricate in shape, which makes the design method so complicated that it should have the capability of handling any type of shape and geometry[1]. The FINOCYL (fin in cylinder) is a 3D grain configuration, especially with relatively low fineness ratios (L/D) requiring internal grains burning for relatively long duration and with large thrust. In most solid rocket motors (SRMs), 3D grain design processes, final designs and ballistic analysis of grains are conducted by

using computer codes[2-4].

A number of algorithms and strategies are now being used to solve complex optimization SRM design problems, which are used to be treated by classical or heuristic optimization methods. The method to design 3D FINOCYL grain (FCG) for SRMs, which incorporates heuristic optimization associated with local optimization techniques for obtaining the optimal solutions, is rarely found in literatures, but this kind of technique has already found application in solving other mathematical problems. A contribution to handle this kind of problems by successive implementation of a genetic algorithm (GA) and a sequential quadratic programming (SQP) is presented in Refs.[5-6], where GA serves to perform a preliminary search in the solution space for locating the neighborhood of the solution before the SQP method to refine the best solution thus obtained.

Unlike 2D grain geometries, the 3D FCG geometry is very complicated and has about seventeen independent design variables, which need to be optimized in order to attain the best possible solution. This greatly complicates optimization of the 3D FCG grain. As it is difficult to locate local convergence region and make a reasonable initial guess in applying local optimizer, a hybrid optimization (HO) method could be applied herein using GA associated with SQP. The HO method is responsible for designing and optimizing 3D FCG as it addresses global as well as local feasible regions, and eliminates the possibility of wrongly falling prey to local optima, thereby converging to the finest optimal solution.

2 Geometric Model and Performance Prediction

Simulation is accomplished using various combinations of some basic shapes inclusive of ellipsoid, cone, cylinder, sphere, torus, and inclined plane to describe the grain geometry. Fig.1 shows an FCG which has been used to define basic segments of 3D grain. It uses seventeen basic shapes to define the grain initial void. Fig. 1(a) shows the 3D FCG parametric view. Fig.1(b) gives out the inner void,

whereas Figs.1(c) and 1(d) show the enlarged views of fins and the cone.

(a) 3D parametric view

(b) Inner void of 3D FINOCYL grain

(c) Fins enlarged view

(d) Cone enlarged view

1—Ellipsoid 2—Cylinder x-axis 3—Ellipsoid 4—Cylinder _y-axis 5—Ring _y-axis 6—Inclined plane 7—Cylinder x-axis 8—Cylinder _y-axis 9—Inclined plane 10—Cylinder x-axis 11 —Cylinder x-axis 12—Cylinder x-axis 13—Cylinder x-axis 14—Cone 15—Ring x-axis 16—Cylinder x-axis 17—Symmetric plane

Fig.1 3D FINOCYL grain.

Fig.2 shows the y-z plane that contains lines parallel to z.

Fig.2 Cross-over points at ith step.

The differential area AA between two parallel z-lines is calculated as follows:

A ^ ^+1 ^

Ay = y+1 - yi

where i is step size on y-axis and Azi the distance of cross-over points, i.e. the distance between the points where z-lines enter the geometric shape and the points where they leave the shape at ith step. The process repeats until the cross-sectional area of propellant in this plane and at current burning distance is obtained in the form of Aj

Propellant volume is calculated along the longitudinal stations over the entire length of motor:

vk=x 2( a+A+1)(xj+1 ~ xj )

where j is the step size on x-axis and k the step size of web increment.

The propellant mass is calculated by

mp = PpVk

The burning surface area is calculated by

a _ VA+1 - VA bk -

wk+1 - wk

The port area Apk can be calculated by subtracting the grain cross-sectional area Ak from the

motor cross-sectional area Amk at that station.

Apk = Amk — Ak

The performance prediction of SRMs has obvious importance during their design. Definition of the ballistic performance parameters is related to thrust - time and pressure - time curves of the motor. Performance requirements that satisfy the mission objectives are specified in terms of chamber pressure, thrust, and time level. Therefore, typical specifications include pav, pmax, Fav, Fmax, and tb. This study also calculates other important parameters that affect the performance. As mentioned earlier, the lumped parameter method for performance prediction assumes the combustion products to be ideal gases and the chamber conditions to be uniform meaning that the pressure and temperature in chamber remain constant.

The chamber pressure is calculated by

Pc = (Ppac K)1/(1-n)

where K =Ab/At.

The thrust is calculated through pressure:

F = CfPc At

The values of L, L1, L2, L3, L4, L5, L6, F1, F2, F3, F4, F5, H1, H2, H3, and N define the

burning figures, their cross-over points, and domains. With L and F2 fixed, the number of design variables could be reduced. Fig.3 gives the grain geometry and performance module.

Fig.3 Grain geometry and performance module.

3 Optimization Approach

3.1 Optimization model

SRM grain design optimization is contingent upon geometrical and performance requirements, which limit the available optimization variables. Let minimum mp be the objective function, then

min mp (X)

where X=f (L1, L2, L3, L4, L5, L6, F1, F2, F3, F4, F5, R1, R2, R3, R4, H1, H2, H3, N).

The main system constraints to the current FCG design by using hydroxy-terminated polybuta-diene (HTPB) propellant are as follows: L = 2 395 mm, th = (73 ± 3) s, Fav = (150 ± 10) kN, F2 = 1 382 mm, de = 540 mm, dt = 135 mm, mp < 5 040 kg.

Table 1 lists the range of independent design variables in the present study.

Table 1 Range of design variables

Parameter Range/mm Parameter Range/mm

Li 100-120 F5 200-300

L2 300-370 Hi 450-500

L3 200-270 H2 500-600

L4 130-150 H3 25-50

L5 340-400 Ri 50-i00

L6 300-360 R2 50-i00

Fi 250-300 R3 5-i8

F3 340-400 R4 5-i5

F4 500-540 N 5-i5

3.2 Optimization method

The searching methods to solve continuous constraint problems can be roughly categorized into: methods of global searching and local searching. Global searching method entails a great amount of efforts to explore the global searching space, whereas, local searching method lays focus on converging to local optimal solutions. Practices have evidenced that either of them works well in many instances though, there exist lots of methods that combine local with global searching strategies and have advantages in solving problems. In this article, we try to incorporate existing local and global searching methods into cooperative solvers to solve problems.

(1) ga

GA is capable of examining historical data from previous design attempts to look for patterns in the input parameters which produce favorable outputs. As a powerful optimization tool, GA uses neither sensitivity derivatives nor a reasonable starting solution. As a non-calculus and a direct-search-based global searching method, it can be applied in the design stage instead of traditionally dominant qualitative or subjective decision making. It has several advantages in design including the ability to combine discrete and continuous variables, provision of population-based searching without requirement for an initial design solution, and the power to address nonconvex, multimodal and discontinuous functions[8-9]. Hence in our study, while conducting optimization using GA, one of the gueatest advantage has been no requirement of finding an initial starting solution. Moreover as GA can handle continuous as well as discrete variables, so the discrete variable number of fins N for all the grain configurations under study were easily incorporated and optimized along with the continuous variables.

(2) SQP

It is a calculus-based gradient decent, local searching method, whose function solves a quadratic programming (QP) subproblem in each phase of iteration. An estimate of the Hessian of the La-grangian is updated in each phase of iteration so as to calculate the positive definite quasi-Newton approximation of the Hessian of the Lagrangian function. Once the searching direction has been chosen, the optimization function determines how far to move in the searching direction. As SQP uses sensitivity derivatives in the immediate vicinity of the current solution, it may come in for local optima from which it cannot easily recover. To avoid these local optima and increase the opportunity to achieve an acceptable solution, SQP requires a reasonable starting solution. Another handicap of SQP is that it can handle continuous variables only. Hence while conducting optimization in this study, by using SQP, the discrete variable N has been handled by at-

tempting number of optimization cycles over the whole range of N, one by one, by keeping its values constant and the best value selected for getting the optimal solutions. For detailed exposition of SQP, readers can refer to Refs.[10-11].

(3) HO

Associating GA with SQP, GA links the FCG design and performance code. It passes down the values of initial design parameters to the design code which calculates the burning areas, web areas, and port areas. These parameters are passed onto performance code where internal ballistic parameters including mp, Fav, and tb are calculated. GA code passes back information about how well the design works in terms of the achieved minimum mp while remaining within the imposed constraints. The elite solution from GA as a nearly optimal guess is passed on to SQP and becomes the initial guess of SQP. The SQP then performs local convergence of the solution and calculates the minimum mp while remaining within the same set constraints. SQP finds local solution through exact analysis, thus providing the optimal result of the problem. Fig.4 shows the flow chart of HO technique adopted in this study, which converges to the best possible optimal solution by addressing global as well as local feasible regions.

Table 2 Initial guess

Design variables X min mp(X)

GA global optimizer I

Grain geometry and performance module

Solution from GA as initial guess to SQP

SQP local optimizer i

Optimal grain design

Fig.4 HO algorithm.

4 Optimization Results

Table 2 lists the initial guess for optimization by using SQP only[7].

Table 3 lists the values of design variables derived from SQP, GA, and HO.

Table 4 lists the performances of optimization by way of SQP, GA, and HO.

Parameter Result/mm Parameter Result/mm

Li i20 F5 200

L2 340 Hi 500

L3 200 H2 515

L4 i 50 H3 30

L5 360 Ri 90

L6 340 R2 70

Fi 300 R3 i7

F3 350 R4 6

F4 500 N i5

Table 3 Values of design variables

Design variables

SQP/mm

i Li i00.0 i00.90 i00.02

2 L2 300.0 323.47 348.65

3 L3 269.9 248.99 232.0i

4 L4 i49.9 i47.75 i44.29

5 L5 370.0 369.5 i 370.02

6 L6 340.0 337.89 340.0i

7 Fi 299.9 2i4.78 206.2i

8 F3 390. i 390.40 390.0i

9 F4 539.9 525.99 5i9.64

i0 F5 299.9 269.05 242.80

ii Ri 50.0 92.53 i00.00

i2 R2 i00.0 5 i .03 99.90

i3 R3 17.3 5.7i 5.0i

i4 R4 7.03 i4.94 5.0i

i5 Hi 499.9 474.29 455.47

i6 H2 515. i 50i.72 500.0i

i7 H3 29.7 27.43 25.0i

i8 N i5 i5 i5

Table 4 Performances of optimization

Parameter SQP GA HO

Fav/kN i53.80 150. i0 i57.30

mp/kg 4 875.20 4 857.50 4 832.40

tb/s 7i.80 72.27 7i .85

Abi/m2 6.72 6.35 7.59

BR/(mm-s-i) 6.55 6.47 6.6i

Pav/MPa 6.88 6.69 7.0i

The required mp has been successfully reduced after optimization. The required mass is < 5 040 kg, whereas, the calculated mass from SQP and GA are 4 875.2 kg and 4 857.5 kg, respectively. The mp can be further reduced to 4 832.4 kg by using HO technique. It is evident that results meet all the design constraints. Fig.5 demonstrates the comparison between optimal results derived from SQP, GA, and HO.

SQP -------

GA ------

GA+SQP-

0 10 20 30 40 50 60 70

Time/s (a)

300 250 * 200 g 150 H 100 50

0 10 20 30 40 50 60 70 Time/s

ч _____

GA-----

GA+SQP-

0 100 200 300 400 500

Web/mm

0 100 200 300 400 500

Web/mm

Fig.5 Comparison between SQP, GA, and HO.

Each of the design variables exerts influences on the characteristics of FCG design and optimization. According to the analysis made in Ref.[7], the initial guess can be determined when SQP alone is

used. From the initial results acquired from either SQP or GA, as illustrated in Table 4, it is noted that although the objective function has been attained while remaining within the design constraints, the minimum values of mp secured from both optimization techniques are greater than those from HO. Consequently, HO holds clear superiority in achieving the optimal solution. This is true especially in tackling the problems that possesses a large number of design variables, which are so complex as to locate local convergence region and to select a reasonable initial guess in applying a local optimizer.

5 Conclusions

With the help of HO method, a complicated 3D FCG grain has been successfully designed and optimized. Values of optimization technique and initial guesses play a critical role in achieving optimal solution especially in fulfilling design projects having many design variables. To begin with, an effort is made to optimize the mass of FCG grain by using a single local optimizer SQP. The results derived from SQP are satisfactory and the design objective is attained within the desired constraints. Next the global optimizer GA is applied. As the results from GA are globally converged, in order to further converge the solution, the results from GA are then used as initial guess to SQP. Thereafter, SQP provides further local convergence to identify the minimum mass of the grain. This HO technique removes the possibility of falling prey to wrong local convergence. This technique of optimization successfully optimizes the mass of propellant of FCG and reduces it to the minimum while meeting all desired design constraints.

References

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NASA SP-8076, 1972.

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Biographies:

Khurram Nisar Born in 1967 at Kharian, Pakistan. In 1991, he received B.S. in mechanical engineering from University of Engineering and Technology, Lahore, Pakistan. In 1998, he received M.S. in the field of solid rocket motor design from Beijing University of Aeronautics and Astronautics (BUAA), China. Presently he is a Ph.D. candidate at BUAA, in the specialty of Propulsion Engineering of Aeronautics and Astronautics. His main research interest includes design and optimization of SRM grain configurations. E-mail: khurram_nisar6@yahoo.com

Liang Guozhu Born in 1966. Ph.D. and Professor, Department of Space Propulsion, School of Astronautics, BUAA. His specialty is propulsion theory and engineering of aeronautics and astronautics. His current research fields are design and simulation of solid rocket motor and liquid rocket engine.

E-mail: lgz@buaa.edu.cn

Qasim Zeeshan Born in 1978 at Lahore, Pakistan. In 2000 he received his B.E. mechanical degree from NUST, Pakistan. He received his M.S. in flight vehicle design from BUAA, China in 2003. Currently he is a Ph.D. candidate at BUAA. His research interests include multidisciplinary design optimization and space mission design. E-mail: qsmzeeshan@yahoo.com