Engineering Optimization Multidisciplinary design and ...

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Multidisciplinary design and optimization of an air launched satellite launch vehicle using a hybrid heuristic search algorithm a

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A. F. Rafique , L. S. He , Q. Zeeshan , A. Kamran & K. Nisar

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School of Astronautics , Beijing University of Aeronautics and Astronautics (BUAA) , Beijing, People's Republic of China Published online: 11 Oct 2010.

To cite this article: A. F. Rafique , L. S. He , Q. Zeeshan , A. Kamran & K. Nisar (2011) Multidisciplinary design and optimization of an air launched satellite launch vehicle using a hybrid heuristic search algorithm, Engineering Optimization, 43:3, 305-328, DOI: 10.1080/0305215X.2010.489608 To link to this article: http://dx.doi.org/10.1080/0305215X.2010.489608

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Engineering Optimization Vol. 43, No. 3, March 2011, 305–328

Multidisciplinary design and optimization of an air launched satellite launch vehicle using a hybrid heuristic search algorithm Downloaded by [INASP - Pakistan (PERI)] at 02:28 05 August 2014

A.F. Rafique*, L.S. He, Q. Zeeshan, A. Kamran and K. Nisar School of Astronautics, Beijing University of Aeronautics and Astronautics (BUAA), Beijing, People’s Republic of China (Received 30 April 2009; Revision received 30 September 2009; final version received 19 April 2010 ) A multidisciplinary design and optimization strategy for a multistage air launched satellite launch vehicle comprising of a solid propulsion system to low earth orbit with the implementation of a hybrid heuristic search algorithm is proposed in this article. The proposed approach integrated the search properties of a genetic algorithm and simulated annealing, thus achieving an optimal solution while satisfying the design objectives and performance constraints. The genetic algorithm identified the feasible region of solutions and simulated annealing exploited the identified feasible region in search of optimality. The proposed methodology coupled with design space reduction allows the designer to explore promising regions of optimality. Modules for mass properties, propulsion characteristics, aerodynamics, and flight dynamics are integrated to produce a high-fidelity model of the vehicle. The objective of this article is to develop a design strategy that more efficiently and effectively facilitates multidisciplinary design analysis and optimization for an air launched satellite launch vehicle. Keywords: multidisciplinary design optimization; satellite launch vehicle; genetic algorithm; simulated annealing; hybrid heuristic search algorithm

1.

Introduction

Launch vehicle design involves many interrelated design parameters; therefore, the designer has to understand the effect of each parameter on the whole system. A designer can improve the design of one subsystem and simultaneously affect another subsystem, which may lead to an overall enhancement or deterioration at the system level. Therefore, the need arises for a multidisciplinary design optimization (MDO) and the use of an artificial intelligence learning tool, or their combination, that can control the design of each component simultaneously. With the MDO process, the design engineer can set explicit system-level goals and then turn the design optimization process entirely over to the optimizer. This distinctive approach for launch vehicle system design relieves engineers from tedious, iterative tasks and enables them to improve their component level models. The computers tirelessly perform thousands of design iterations while learning which design features work and which ones do not for a given set of constraints. *Corresponding author. Email: [email protected], [email protected]

ISSN 0305-215X print/ISSN 1029-0273 online © 2011 Taylor & Francis DOI: 10.1080/0305215X.2010.489608 http://www.informaworld.com

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Literature is in abundance with rocket-based vehicle design optimization using various optimization techniques to advance the overall design, analysis and optimization of these complex aerospace systems. Bayley et al. (2008) designed and optimized three and four-stage solid propellant launch vehicles to reach the low earth orbit (LEO). Vehicle weight and ultimately vehicle cost is minimized by using genetic algorithm (GA). Bayley and Hartfield (2007) presented the design and optimization of liquid-fuelled two- and three-stage space launch vehicles for minimum weight and cost per launch with the application of GA. Conceptual design and optimization of the solid and liquid propellant satellite launch vehicle using the search properties of GA and sequential quadratic programming is presented in Rafique et al. (2008). Conceptual design analysis for multistage launch vehicles using rocket based or air-breathing propulsion systems through probabilistic methods is presented by Villeneuve et al. (2004). A multidisciplinary design optimization model is developed to address both cost and performance aspects for the launch vehicle design. The performance estimating relationships are based on fundamental analytical techniques pertinent to the launch vehicle system analysis, using modern historical data (LeMoyne 2008). The conceptual design of two-stage reusable rocket vehicle including trajectory optimization is given by Takeshi and Takashige (2004). The above-mentioned literature and many more give insight that GA has been the most attractive choice of designers for MDO of SLVs. However, in this research effort, a new and innovative strategy of combining the advantages of GA and simulated annealing (SA) with the introduction of design space reduction (DSR) to improve the convergence is proposed. This proposed optimization method is aimed at combining the benefits of both GA and SA and at the same time catering for built in variations in solution quality of GA. A fundamental feature of this study is to develop a design methodology that more efficiently and effectively facilitates integrated design analysis and optimization, when considering the immensely complex problem of air launched satellite launch vehicle (ASLV). The remainder of this article is structured in four major sections. The multidisciplinary design analysis and optimization module of an ASLV is described in Section 2. Analysis of all the integrated subsystems along-with interdisciplinary design variables, design objective and constraints are presented in this section. Heuristic optimization methods implemented in this study are summarized and their advantages, disadvantages and application in ASLV design problem are discussed in the Section 3. Hybrid heuristic search algorithm (HHSA) is also explained in this section. Performance results and conclusions are presented in the Sections 4 and 5.

2.

Multidisciplinary design analysis and optimization of an ASLV

The multidisciplinary design process is highly coupled and, therefore, requires substantial data exchange and iteration among disciplines and disciplinary codes (engineering models). The diversity of characteristics of the individual disciplinary codes requires a variety of optimization approaches capable of treating discrete or continuous design variables, few or many coupling variables and constraints, single solution or multiple minima, and simple or computationally intensive analyses. Many practical MDO methods have been widely applied in the aerospace industry for overall designs of aircraft and space transportation vehicles (Sobieszczanski-Sobieski et al. 1997 and Haftka 1985). Rahn et al. (1996) and Wolf (1994) have proposed design tools and integrated computer software for space transportation systems. Multidisciplinary feasible design formulation (MDF) (Hulme and Bloebaum 1998, Bartholomew 1998) proved to be an efficient and effective optimization strategy for the launch vehicle conceptual design problem. MDF offers following advantages:

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(1) No auxiliary design variables are necessary to ensure interdisciplinary convergence. (2) Interdisciplinary convergence is automatically ensured within the multidisciplinary analysis system. (3) No additional constraints are required except those of the original problem. The optimization module (optimizer) controls all the interdisciplinary design variables, X, and modifies them to search for the optimum of the objective function while satisfying constraints (Figure 1). Multidisciplinary design of an ASLV is an iterative process, requiring a large number of design iterations to obtain a balance of emphasis from the diverse inputs and outputs. Simplicity and transparency are the two fundamental advantages of MDF which thus becomes an attractive choice for use in the complex design and optimization problem of an ASLV. Equation (1) summarizes the optimization problem of ASLV based on MDF. Min

GLW(X)

(Objective Function) = f (x)

Subject to gj (X) ≤ 0

(1)

hk (X) = 0 upper

xilower ≤ xi ≤ xi

where GLW is gross launch weight and is taken as objective function for the design problem of an ASLV, gj (x) are inequality constraints, hk (x) are equality constraints and xi are side or boundary constraints. X is a vector of design variables. X = [x1 , x2 , x3 , . . . , xn ]

(2)

Figure 2 illustrates a flow chart of the multidisciplinary design analysis approach followed in this research. Figure 3 depicts inter-disciplinary couplings.

Figure 1.

Multidisciplinary feasible design architecture.


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Figure 2.

Multidisciplinary design approach.

Figure 3.

Design structure matrix for ASLV.

2.1. Vehicle definition The ASLV design is a highly integrated process requiring synergistic compromise and tradeoffs of various design parameters. The synthesis of an effective compromise among design parameters requires balanced emphasis in subsystems, unbiased tradeoffs, and at the same time evaluation of many alternatives. Starting with a well-defined baseline that has similar performance parameters expedites the design convergence. It provides a more accurate design and boosts the speed of the design process. The baseline design is launched from 40,000 feet (12,192 metres) at Mach 0.8. This is intended as a representative number taken from launch conditions of similar vehicles (Isakowitz 1999). The mission of the ASLV is to deliver a 200 kg payload (satellite) to the LEO. The propulsion system is a solid fuelled solid rocket motor (SRM). Though the number of stages may also be


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one of the design variables, in this research the proposed ASLV is fixed as a three-stage vehicle. The payload (satellite) is enclosed in a fairing whose shape is known beforehand. The payload weight and volume requirements are specified in the problem definition before the optimization is computed. The payload fairing has a length of 2 metres, whereas the diameter of the fairing is same as that of the first and second stages.


2.2. Propulsion analysis The propulsion analysis describes fundamental parameters like thrust, burn time, mass flow rate and nozzle parameters (Sutton and Biblarz 2001). The chamber pressure (pc ) is a vital design variable having a momentous influence on SRM specific impulse (Isp ), burning rate of the propellant (ui ), size of expansion nozzle and thickness of casing materials to withstand the pressure stresses. Douglass et al. (1970, 1971, 1972) have discussed the design and performance of SRM and grains. However, the present analysis is not restricted to a particular shape of grain, rather a variable grain shape factor (ψi ) is used to represent the burning surface area (Sri ) of the grain as a function of grain length (Li ) and diameter of stage (Di ), as given by Equation (3): ψsi = Sri /Li Di

(3)

The mass of grain (mgni ) is calculated from the design variables and propulsion analysis. Equations (4)–(6) compute burn time (tbi ) and grain consumption rate (λgni ), as described by Sutton and Biblarz (2001) and He (2004): π ρgni ψi λgni Di3 4 πηvi Di tbi = 4ui ψi

mgni =

(4) (5)

m ˙ gni = ρgni ui Sri = ρgni ui ψi λgni Di2

(6)

where ρgni is density of grain, λgni is fineness ratio of grain (grain length/diameter) and ηvi is volumetric loading fraction. Nozzle throat area (At ), expansion ratio (ε) and nozzle exit area (Ae ) are calculated as: ρgni ui Ab,max At = (7) Rc Tc o pc,max ε=

o

(pe /pc )1/γ 2γ /γ − 1 1 − (pe /pc )γ −1/γ

Ae = At ε √ o = γ (2/γ + 1)γ +1/2(γ −1)

(8) (9) (10)

where Ab is burning area of grain, Rc is the gas constant, Tc is the temperature in the combustion chamber, pe is exit pressure and γ is the specific heat ratio of gas. The following relations vac ) and thrust (F ): calculated vacuum specific impulse (Isp vac F1..N = Isp m ˙ gni − pa Aei vac Isp

=

a Isp

+ (pe /pc )

γ −1/γ

(11) a Rc Tc /go2 Isp

(12)

a is average specific impulse where N is the number of stages, pa is the atmospheric pressure, Isp and go is the acceleration due to gravity.

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2.3. Weight analysis Using a combination of physics-based methods and empirical data, the weight of the foremost components for the solid stages is determined from He (2004). The total mass of a multistage ASLV includes the masses of propellants and their tanks, related structures and payload mass. The GLW (m01 ) for a multistage ASLV is the cumulative sum of various components, as shown below: m01 = msat +

N

mgni + msti + mai + mf i

(13)


i=1

where m01 is the gross mass of the ith stage vehicle, mgni is the mass of the ith stage SRM grain, msti is the mass of the ith stage SRM structure, mai is the total mass of the ith stage aft skirt and mf i is total mass of the ith stage forward skirt. The design mission dictates the mass of the satellite (msat ). The total mass of the ith stage aft and forward skirt are simplified as: mai = msvi + masi

(14)

mf i = mf ei + mf si

(15)

mai + mf i = Ni moi

(16)

where msvi is the mass of the control system, safety self-destruction system, servo, and cables inside the ith stage aft skirt; masi is the mass of the ith stage aft skirt including shell structure, equipment rack, heating protector structure, and directly subordinate parts for integration; mf ei is the mass of equipment and cables inside the ith stage forward skirt; mf si is the mass of the ith stage forward skirt including shell structure, equipment rack, and directly subordinate parts for integration; and moi is the mass of ith stage SRM. The skirt mass ratio or structural coefficient (Ni ) has small dispersions and can be selected from the statistical data (He 2004, Sutton and Biblarz 2001). The simplified form of the N -stage launch vehicle mass is: m0i = N

i=1 [1

msat − Ni − Kgni μki (1 + αsti )]

(17)

where Kgni is propellant reserve coefficient (sliver). Relative mass coefficient of effective grain (μki ) and structure mass fraction (αsti ) are given by following relations: uki = αsti

megni

moi msti = mgni

(18) (19)

The main parameter for designing a multistage ASLV is the structure mass fraction. It is dependent upon structural material and grain shape, as well as the parameters of internal ballistics of SRM. The mass of the ith stage SRM structure, as shown in Equation (20), is comprised of the mass of the motor cylinder (mcyi ), motor dome ends (mc1i and mc2i ), forward and aft skirt (mqi ), forward and aft attachment (mj 1i andmj 2i ), forward and aft insulation liner (min,c1i and min,c2i ), cylindrical section insulation liner (min,cyi ), nozzle expansion cone (mnoz,eci ), nozzle spherical head (mnoz,shi ), nozzle insulation (mnoz,ini ), ignitor (migi ), thrust vector control (mT V Ci ), cables (mcabi ) and attachment parts (mapi ). The mass of the structure will be used to calculate αsti and so GLW of the ASLV. msti = mcyi + mc1i + mc2i + mqi + mj 1i + mj 2i + min,c1i + min,c2i + min,cyi + mnoz,eci + mnoz,shi + mnoz,ini + migi + mT V Ci + mcabi + mapi

(20)


311

The mass of the motor cylinder, motor dome ends, forward and aft skirt, forward and aft attachment, forward and aft insulation liner, cylindrical section insulation liner, nozzle expansion cone, nozzle spherical head, nozzle insulation, igniter, thrust vector control, cables and attachment parts are given in Equations (21)–(38).


mcyi =

3 3Kcy πffp Dch pc λgni

8σb

(21)

where Kcy is the ratio of the cylindrical length to grain length, f is factor of safety, fp is the relative pressure in chamber (pcmax /pc ); Dch is the diameter of chamber and σb is the ultimate strength of the chamber material.

3 πλ2e Dch ffp pc 1 mc1i = 1 − (22) 8(λ2e − 1)σ cos θ2 1 + (λ2e − 1) sin2 θ2 where λe is the ellipsoid ratio, θ2 is normally 60◦ ∼ 65◦ and σ is strength ratio (σb /ρcl ).

2 2 3 πλ2e Dch ffp pc λe Dch − (λ2e − 1)d 2 1 mc2i = − 2 8(λ2e − 1)σ cos θ2 λ2e Dch 1 + (λ2e − 1) sin2 θ2

(23)

where d is the diameter of the rear end opening of the nozzle. 2 mqi = πDch ρq δq

lq1 + lq2 Dch

(24)

where ρq is the density of skirt material, δq is the thickness of skirts and lq1 and lq2 are lengths of forward and aft skirts.

1 mj 1i = πρj ri2 δ1i − yi2 (δ1i + δ2i ) + (25) (2b02 + ri2 ) b02 − ri2 − (2b02 + Ri2 ) b02 − Ri2 6 where ρj is the density and δj is the ultimate stress of the joint material. mj 1i = mj 2i min,c1i =

1 2 Ra tb ρin π Dcy 4

(26) (27)

where ρin is the density of insulation material, Dcy is diameter of case cylindrical section and Ra is the rate of ablation.

2 2 Ra tb λ2e Dcy − (λ2e − 1)d 2 ρin πDcy min,c2i = (28) 2 4(λ2e − 1) Dcy εin cin ρin + αgi ccy ρcy δcy min,cyi = πDcy Lcy ρin αgi cin ρin

2εin (ln θp ccy ρcy δcy + αgi tb )αgi cin ρin × 1− (29) 2 − 1 ln θp εin cin ρin + αgi ccy ρcy δcy θp =

Tg − Tcy Tg − T i

(30)

where Lcy is length of the case cylindrical section, εin is the heat transfer coefficient of insulation material, cin is the specific heat capacity of insulation, αgi is the heat transfer coefficient from

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gas to insulation, ccy is the specific heat capacity of cylindrical section, ρcy is the density of case cylindrical section material, δcy is the thickness of case cylindrical section, Tg is the temperature of the gas, Tcy is the allowable temperature of the cylindrical section and Ti is the initial temperature of the cylindrical section.

π Ae 6 4/3 pc max σec mnoz,eci = ρec (31) − 1 dt f × 0.67 de /dt − 1(1 − S) 4 sin βn At Eec


where ρec is the material density of the expansion cone, βn is the expansion half angle, dt is the diameter of the throat, de is the nozzle exit diameter, S is submerged coefficient of the nozzle, σec is the ultimate strength and Eec is Young’s modulus of the expansion cone material. mnoz,shi = 3.656ρsh dt3

(32)

where ρsh is the material density of the spherical head of the nozzle. mnoz,ini = ρinnz Snz δinnz

(33)

where ρinnz is the material density and δinnz is thickness of the insulation. Snz is the surface area of nozzle. migi = 1.454(7π dt2 /4)1.2 mT V Ci = (1.3 ∼ 1.7)mnz

(34) (35)

where mnz is the total mass of nozzle. mnz = mnoz,shi + mnoz,eci + mnoz,ini mcabi = (1 + 3Lcy )ρci

(36) (37)

where Lcyi is the ith stage SRM length and ρci is the ith stage cable density. 1.148 mapi = (6.13 × 10−7 )Di2 Lcyi

(38)

2.4. Aerodynamics analysis In the preliminary design phase of an ASLV, speedy and cost-effective estimations of aerodynamic stability and control characteristics are frequently required. Thus, a need arises for use of time-efficient computer software that can predict aerodynamic properties over a range of flight conditions. US Air Force Missile DATCOM 1997 (digital) (Blake 1998) has been widely used in the aerospace industry for this purpose. DATCOM is capable of quickly and economically estimating the aerodynamics of a wide variety of design configurations and in the different flow field regions that the ASLV encounters during atmospheric flight. DATCOM is used in the present research for quick and economic estimation of coefficient of lift (CL ) and coefficient of drag (CD ). Force coefficients were calculated for 18 Mach numbers in the specified range, at 14 angles of attack for each Mach number, ranging from −4◦ to +22◦ . 2.5.

Trajectory analysis

The three degree-of-freedom (3DOF) approach is time efficient and diminishes the requirements of a large number of input variables calculated at a lesser accuracy due to unavailability of a detailed shape of the structure and other aerodynamic parameters in the conceptual design phase.


313


Therefore, in this research effort 3DOF trajectory analysis is preferred over 6DOF model (Zipfel 2007, Xiao 2005). A 3DOF model was developed and simulated in SIMULINK to analyse the flight path of the ASLV. Trajectory simulation obtained from 3DOF model is computationally efficient and serves the purpose at the conceptual design level. The trajectory analysis depends on inputs from the aerodynamics, mass and propulsion modules. The flight program and results obtained from the other disciplines computes the flight trajectory. This investigation treats ASLV as a point-mass and flight in 2D over a spherical and non-rotating earth is assumed. This implies that the Coriolis and centrifugal pseudo forces are negligible. Figure 4 shows forces acting on the ASLV and Equation (39) presents the set of governing equations of motion. dv T cos α − D = − go sin ϑ dt m dϑ T sin α + L go cos ϑ v = − + cos ϑ dt mv v Re + h dh = v sin ϑ dt dl Re = v cos ϑ dt Re + h α =η+ϕ−ϑ η=

l Re

α = αpro (t) 1 L = CL ρv 2 Sref 2 1 D = CD ρv 2 Sref 2

(39)

where V is the velocity, m is the mass of vehicle, T is the thrust force, go is the acceleration due to the gravity, ϕ is the flight path angle, α is the angle of attack, η is the range angle, θ is the

Figure 4.

Forces acting on ASLV.

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trajectory angle, Re is the radius of the Earth, h is the height about ground, l is the range, L is the lift force, D is the drag force and Sref is the surface area. The powered flight trajectory of the ASLV is composed of several distinct phases. After a short, horizontal launch phase, it undergoes a launch manoeuvre by introducing a carefully selected angle of attack (AOA) profile. The flight program is optimized under the constraints of the maximum allowed AOA (αmax ), lateral (ny ) and axial overloads (nx ). The lateral and axial overloads are limited to ensure the structural integrity. Each stage shuts down one after another and separates to shed inert mass. Because the equations of motion change discontinuously at the shutdown points, the trajectory must be divided into intervals, the number of which corresponds to number of stages. The phases are explained in the following sub-sections. 2.5.1. Horizontal launch phase The launch sequence of the proposed ASLV begins with the release from the carrier aircraft at an altitude of approximately 12,195 metres (40,000 feet) at Mach 0.80. Approx 5 seconds after drop, when the proposed ASLV has cleared the carrier aircraft, Stage 1 ignition occurs. An initial condition is that the flight path angle is zero deg. Aerodynamic lifting surfaces (wings, horizontal and vertical tail) provide stability and lift after launch and further within atmospheric flight. 2.5.2. Powered ascent phase This phase starts from the end of transonic phase (0.8 ≤ Ma ≤ 1.3) up to the shutdown of the second stage motor. The angle of attack must be zero when the flight Mach is in the transonic phase. This is because aerodynamic forces vary acutely in transonic speeds and the large angle of attack would yield strong disturbance to the control system and servo overload to the body of ASLV. Furthermore, drag loss will be more in the transonic region. The angle of attack must approach zero at the time of the first-stage separation. However, there is no such restriction on separation of upper stages. In the rocketry, the flight program means the prescribed variation of the rocket pitching angle during the flight which has substantial influence on reachable altitude, injection accuracy in the orbit and aerodynamic loading etc. The flight angle of attack is programmed using the following relations: αprog (t) = am sin2 f (t) π(t − t1 ) k(t2 − t) + (t − t1 ) tm − t1 k= t2 − t m

f (t) =

(40) (41) (42)

where am is launch manoeuvre variable, t is the time of flight, t1 is the time of start of power ascent phase (turning), t2 is the time at end of turning and tm is the time corresponding to the maximum angle of attack (αmax ). Figure 5 explains the powered ascent phase of ASLV. Axial and lateral overload coefficients should not be exceeded from allowable ranges. Otherwise, the control system, the strength of grain, and the bond strength between grain and motor case will be destroyed. Constraints on both these overloads are enforced in the optimization problem. The following relations calculate axial and lateral overloads: nx =

T + L sin α − D cos α ≤ nx max mgo

(43)



315

Figure 5. AOA programming with time.

ny =

T sin α + L ≤ ny max mgo

(44)

2.5.3. Coasting phase The final stage carrying the satellite separates and flies up with no thrust in an elliptical orbit at the end of the second stage. 2.5.4. Kick phase Finally, an apogee kick puts the satellite into a circular low earth orbit. The constraint conditions in these phases are the flight path angle of zero degrees, dynamic pressure, normal acceleration, and the body axial acceleration. 2.6. Design objective The ASLV design can pose different objective functions for the optimization problem depending upon mission requirements. For example, one could minimize cost, maximize payload for a fixed launch weight, maximize injection accuracy in orbit, and minimize launch weight for placing a fixed payload in a particular orbit. Traditionally, minimum take-off weight concepts have been sought in the launch vehicle design. This is because weight (or mass) is a strong driver on vehicle performance and cost, so take a central role in the vehicle design process. The design objective for the ASLV optimization problem is to minimize the gross launch weight (GLW) of the entire vehicle. From Equation (17) it can be seen that GLW of the ASLV can be written as: m0i = m0i (N, mP AY , Kgni , μki , αsti )

(45)

By further exploring αsti and introducing propulsion variables (pci , pei and ui ), trajectory variables (αmax and am ) and removing independent variables (N, mP AY and Kgni ), set of the design variables (X) can be written as: X = f (μki , Di , pci , pei , ui , ψi , αmax , am ) 2.7.

(46)

Design variables

Table 1 lists the system design variables for each stage with respective discipline. There are total of 19 design variables that govern the integrated design and optimization problem of the ASLV.

316 Table 1.

A.F. Rafique et al. Discipline-wise design variables.

Discipline Structure + Propulsion Structure + Propulsion + Aerodynamics Structure + Propulsion Structure + Propulsion + Trajectory Structure + Propulsion Propulsion + Trajectory Aerodynamics + Trajectory Aerodynamics + Trajectory

Design variable

Symbol

Units

Relative mass coefficient of grain (Stage 1, 2, 3) Body diameter (Stage 1, 2, 3) Chamber pressure (Stage 1, 2, 3) Exit pressure (Stage 1, 2, 3) Coefficient of grain shape (Stage 1, 2, 3) Grain burning rate (Stage 1, 2, 3) Max angle of attack (Stage 1) Launch manoeuvre variable (Stage 1)

μki Di pci pei ψi ui αmax am

Ratio m bar bar mm/s deg


2.8. Design constraints The mission velocity (V ) and corresponding altitude (altf ) are the trajectory constraints. The overall structure of the system should be extremely strong to survive the high g-loads. Therefore, an axial overload constraint is implemented to restrict it below 12 g for the first and second stage and the lateral overload to restrict below 2 g for the first stage. During the launch manoeuvre, the maximum angle of attack is constrained to be below 22◦ and to ensure that it is zero during the transonic phase and at the first stage separation. The flight path angle is constrained to be zero at orbit insertion (third stage). The nozzle exit diameters are constrained to be less than the stage diameters. The first and second stage diameters are constrained to be equal. If any of these conflicts occur, the program is set to send back extremely poor performance values in each goal area so that it will learn not to evaluate these designs in the future. Constraints are formulated as under: Ci ≥ 0,

i = 1, 2

Ci ≤ 0,

i = 3...8

Ci = 0,

i = 9, 10, 11

(47)

where C is given as C1 = vf ≥ 7600 C2 = altf ≥ 450 C3 = nx1 ≤ 12 C4 = nx2 ≤ 12 C5 = ny1 ≤ 2 C6 = αmax ≤ 22 C7 = De2 ≤ D1 C8 = De3 ≤ D1 C9 = D1 = D2 C10 = α = 0

(for 0.8 ≤ Ma ≤ 1.3)

C11 = ϕ = 0

(at orbit insertion)

Constraints are included in the objective function and are handled by dynamic penalty function. Equation (48) represents the symbolic problem statement: min GLW (x) = f (x) + h(k)

m i=1

max{0, gi (x)}

(48)


317

where f (x) is the fitness function, h(k) is modified penalty value through optimization and k is the current iteration number of the algorithm. The function gi (x) is a relative violated function of the constraints (Crossley and Williams 1997).


3.

Optimization approach

Optimization deals with betterment and improvement, and with advancement of computer technology, optimization and MDO have received new attention. MDO gives the engineer an opportunity to find the optimal solution of the whole system accounting for interactions between different disciplines. It should be noted that a multidisciplinary solution might not be the solution for any one discipline analysed separately from other disciplines, but is the best solution accounting for interactions. 3.1. Heuristic optimization methods Heuristic optimization methods are well suited to be applied in large, multidisciplinary problems as they are able to handle both discrete and continuous variables. The most popular heuristic optimization methods that go beyond local search are GAs and other evolutionary techniques, such as evolutionary programming, evolutionary strategies, SA, and Tabu Search. Particle swarm optimization is another optimization technique that has shown enormous potential. Several heuristic tools have evolved in the past decades to facilitate optimization of difficult and complicated problems. Even though heuristic methods do not guarantee the exact optimal solution, they are frequently used because of availability of fast computational resources and easy applicability (Reeves 1993). The main objective in heuristic search is to construct a model that is implicit, and that provides reliable solutions in a reasonable amount of computing time. Developing solutions with these tools offers two main advantages: (1) Development time is much less as compared to traditional approaches. (2) The system design is more robust, and is relatively less sensitive to noisy and/or missing data. 3.2. Genetic algorithm (GA) Conventional optimization techniques start with a starting candidate and search iteratively for the optimal solution by applying static heuristics. On the other hand, GA uses a population of candidates to explore several areas of a solution space, simultaneously and adaptively. Holland (1975) developed genetic algorithms, which are capable of finding the global-optimal solution (or acutely near solutions) in complex multi-dimensional search spaces. Details on GA can be found in Holland (1975), Goldberg (1989), Beasley et al. (1993) and Beasley et al. (1993). GA applications have gained an enormous popularity among aerospace professionals in the last decade. This is due to the ease with which GA can be implemented and its exceptional ability to solve difficult complex problems more efficiently. Some of the novel ideas using GA include spacecraft design, aircraft design, system modelling, airfoil design, satellite components design, and launch vehicle design. Figure 6 illustrates the flow chart of GA. 3.3. Simulated annealing (SA) SA was originally proposed by Metropolis in the early 1950s as a model of the crystallization process. It was only in 1980s that independent research, done by Kirkpatrick et al. (1983) and Cerny


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Figure 6.

Flow chart of genetic algorithm.

(1985), noted similarities between the physical process of annealing and some combinatorial optimization problems. The main advantage of SA is that it can be applied to large problems regardless of the conditions of differentiability, continuity, and convexity that are normally required in conventional optimization methods. The algorithm starts with an initial design; new designs are randomly generated in the vicinity of the current design. The change of the objective function value, (E), between new and current design is calculated as a measure of the energy change of the system. At the end of search, when the temperature is low, the probability of accepting worse designs is remarkably low. The best solution from the optimization is the solution after the system is in the equilibrium state (Chattopadhyay et al. 1994, Aarts et al. 1988, Aarts and Korst 1989). Equation (48) presents the probability of acceptance of new solution. P = exp(Ei − Ej /kB T ) (49) where Ei is initial state of the energy, Ej is next state of the energy, kB is the Boltzmann constant and T is the temperature. SA, as powerful stochastic search method, is applicable to a wide range of problems (Osman and Laporte 1996, Tekinalp and Utalay 2000, Tekinalp and Bingol 2004), but application of SA on MDO of launch vehicles is extremely rare. Figure 7 gives an outline of the basic SA algorithm. 3.4.

Hybrid heuristic search algorithm (HHSA)

There are many instances of using different types of hybrid optimization methods in recent times. Misra (2001) developed one of the most basic hybrid optimization methods. This hybrid method involves maintaining a population of points, as with the GA. However, during the selection process, less-fit points are allowed to proceed into the next generation if they meet an acceptance criterion. This method exploited parallel processing, to which SA is not well suited. Another hybrid method,



Figure 7.

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General outline of simulated annealing.

developed by Boseniuk and Ebeling (1988), involved running a few instances of SA. Another variation of this method, involved postponing the selection process until the system has cooled down, and then reheating the system after selection has been done. Ackley (1987), Sirag and Weisser (1987), Whitley and Kauth (1988), Goldberg and Segrest (1987) and many others have developed hybrid methods that use either of GA, SA, or both these methods. The optimization problem of an ASLV is solved by using the hybrid heuristic search algorithm (HHSA). The HHSA aims to integrate GA and SA in order to blend their advantages and minimize their disadvantages. HHSA allows global search to be performed using a cascaded architecture (Figure 8). GA identifies the promising and feasible neighbourhood of solutions while SA explores these promising regions to reach to an optimal solution. A set of design variables (X), with lower (LB) and upper bounds (UB), is passed to the optimizer which creates initial random population and performs its further operations. These candidates are then passed to multi-disciplinary design and analysis modules. The algorithm executes iteratively in a closed loop until an optimal solution is obtained. The design space reduction (DSR) is implemented to utilize the most promising solutions during first optimization phase. GA may need a small number of generations to enter into the feasible region of the design space; once in this domain GA requires a large number of generations to reach the global optimum, and thus may prove to be a time consuming process and computationally inefficient. The type of initial population, mutation and crossover functions affect the solution quality and have a significant impact on the computational cost and number of function evaluations to achieve the optimal solution. The solution process is highly dependent on management of the above-mentioned parameters. In the case of ASLV, where design space is broad, a time efficient and effective strategy must be used in search of optimality. In the present research, HHSA performed a small number of GA runs with a different mechanism for population, mutation and crossover to obtain solution sets in the feasible region. DSR is performed to reduce the search space and modified bounds of


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Figure 8.

Hybrid heuristic optimization strategy.

Figure 9.

Feasible and optimality region by genetic algorithm.

the design variables now reside near the line of optimality. SA is then incorporated to search for the space defined by new bounds thus giving computational efficiency and enhancing the quality of solution. Figure 9 provides a symbolic description of the feasible region which fulfils all the requirements of constraints obtained by GA and the line of optimality. The shaded portion, , shows the feasible region for the constrained problem (bounded by constraints). The line in the shaded region shows all the sets of design variables from which the optimal objective function is achieved, i.e. line of optimality. The whole problem was solved using GA for 10 times using different type of population, mutation and crossover function. This exercise has provided the large pool of near optimal solutions.


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These solutions may exist in proximity of optimal solutions but are not same in terms of quality. This procedure is extensive but not futile and explored the better region of optimality for the complex design space under consideration. The best of the optimal solutions from the GA pool was then taken as an initial guess for SA. The reduced lower (LBR ) and upper-bounds (UBR ) in this phase are selected in the vicinity of the optimal solutions obtained from 10 runs of GA, thus reducing the design search space. Implementation of this strategy has:


(1) Decreased the computational time to reach a near-optimal solution. (2) Obtained the region of optimality and performed better exploration of the said region. (3) Mitigated inherited difference in optimal solutions of GA. ∗ ∗ ∗ The pool of all the optimized design variables (Xm,1 , Xm,2 , . . . , Xm,n ) created ⎤ ⎡ ∗ ∗ ∗ . X1,n X1,1 X1,2 ⎢ X∗ . . . ⎥ 2,1 ⎥ P OOL = ⎢ ⎣ . . . . ⎦ ∗ ∗ . . Xm,n Xm,1

where, m = 1, 2, . . . , 10 n = 1, 2, . . . , 19 Table 2.

Parameters for HHSA.

Genetic algorithm Maximum generations: 100 Population size: 20 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: 2000

Figure 10.

Simulated annealing Optimization type: Fast annealing Maximum iterations: 1000 Function tolerance: 10−6 Temperature function: Exponential Maximum function evaluations: 1000 Initial temperature: 100

Convergence of genetic algorithm and simulated annealing.

(50)

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and LBR and UBR are devised through: for ∀n LBR = min(Xn,m )

m = 1, 2, . . . , 10

U BR = max(Xn,m ) m = 1, 2, . . . , 10

(51)


where n is the total number of design variables for the ASLV design problem, m is the number of optimized designs of GA to find the min and max of each design variable to give as new LBR and UBR for SA.

Figure 11. Annealing schedule of SA.

Table 3.

Optimum values of design variables. Initial

DSR

X

Symbol

Units

LB

UB

XGA

LBR

UBR

XH H SA

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

μk1 μk2 /μk1 μk3 /μk2 D1 D3 pc1 pc2 pc3 pe1 pe2 pe3 u1 u2 u3 ψ1 ψ2 ψ3 αmax am

Ratio Ratio Ratio m m bar bar bar bar bar bar mm/s mm/s mm/s

0.60 1.00 1.00 1.20 0.80 55.00 55.00 55.00 0.10 0.08 0.08 5.00 5.00 5.00 1.50 1.50 1.50 1.00 0.01

0.75 1.10 1.10 1.40 1.00 75.00 75.00 75.00 0.16 0.16 0.16 8.00 8.00 8.00 2.30 2.30 2.30 22.00 0.10

0.6965 1.0014 1.0026 1.2967 0.8542 63.6260 67.4910 61.1130 0.1404 0.1484 0.0998 6.0632 6.8872 5.2907 2.2808 2.2727 2.2790 21.9320 0.0168

0.6905 1.0000 1.0000 1.2888 0.8500 60.7100 60.0600 60.0600 0.1400 0.1077 0.0963 5.2700 6.5400 5.2500 1.7200 2.1200 1.7600 21.0000 0.0124

0.7105 1.0206 1.0100 1.3000 0.8781 68.6400 67.5000 61.5000 0.1542 0.1495 0.1000 6.9500 7.0000 6.7300 2.2900 2.3000 2.2900 22.0000 0.0221

0.6999 1.0008 1.0002 1.2999 0.8503 63.9100 65.2100 60.0600 0.1403 0.1427 0.0998 5.7900 6.9900 5.4800 2.2100 2.2900 2.2200 21.9900 0.0157

deg



Figure 12.

Performance graphs of optimum configuration of ASLV.

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Continued.


3.4.1.

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Optimization parameters


In HHSA, the maximum number of function evaluations along with tolerance on the design variables and the objective function are set to be the convergence criteria. Column one of Table 2 shows the optimization parameters for one case of genetic algorithm and column two shows the optimization parameters of simulated annealing in HHSA. Figure 10 illustrates the convergence bound obtained depicted by max and min solution of GA and single run of SA with minimum GA solution as starting solution. It also shows single case of GA with minimum value of the objective function which was evaluated for 3000 function evaluations. This was done to make computational comparison between GA and HHSA. Figure 11 shows the annealing schedule of SA.

4.

Performance results of the optimized ASLV

Multi-disciplinary design and optimization using hybrid heuristic search algorithm is successfully implemented for the complex design problem of ASLV under stringent mission objectives and performance constraints. Genetic algorithm has identified the most promising regions of the optimal solution while simulated annealing, coupled with design space reduction, has done efficient and effective exploration of these regions of the feasible solution to achieve better of the optimal solution. This strategy proved to be more effective in terms of exploring the design space and fulfilling design objectives of the air launched satellite launch vehicle design problem. HHSA also reduced stochastic error that can generally be present in single execution of GA. The best feasible fitness value is the minimum GLW ever encountered that does not violate the constraints. Figure 10 presents convergence patterns of GA and HHSA. It reveals the fact that GA is initially fast in search of feasible optimal region but proved to be computationally expensive in search of an optimal solution. SA seems to be very fast in search of optimal solution when in

Figure 13.

Flight profile of optimized ASLV through GA and HHSA.


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a feasible region. Moreover, global optimum attained by HHSA has better quality (less GLW) compared with GA when executed for a same number of function evaluations. Table 3 contains optimal values from HHSA for all the design variables. The optimized design variables, XGA andXH H SA , of the design space of an ASLV lie between their upper and lower bounds. XH H SA proved to be better than XGA . Figure 12(a)–(f) illustrates the performance graphs of optimized configurations of the ASLV. The required orbit injection velocity (≥7600 m/s) and altitude (≥450 km) are perfectly achieved. Axial overloads are within the allowable ranges and required thrust is achieved to complete the mission of ASLV. Figure 13 depicts the flight profile of the proposed ASLV, and it is been learnt that the ASLV is capable of completing the specified mission through both GA and HHSA. Table 4 demonstrates the stage layout and propulsion parameters of the optimal configurations achieved through GA and HHSA. The reduction in GLW achieved by using hybrid heuristic search algorithm is about 1431.4 kg which is about 6.52%. This reduction in gross launch weight is quite remarkable at the conceptual design level where quality matters the most. Figure 14 contains orthogonal views of the optimized vehicle. Table 4.

Optimal configurations of genetic algorithm and hybrid heuristic search algorithm.

GA optimized configuration

HHSA optimized configuration

Gross launch mass: 21,928.7 kg Payload: 200 kg

Gross launch mass: 20,497.3 kg Payload: 200 kg

STAGE III Gross mass: 981.96 kg Propellant mass: 831.44 kg Thrust: 46.36 kN burn time: 44.51 s

STAGE III Gross mass: 953.33 kg Propellant mass: 808.13 kg Thrust: 48.17 kN burn time: 43.93 s

STAGE II Gross mass: 4351.87 kg Propellant mass: 3906.69 kg Thrust: 171.93 kN Burn time: 52.70 s

STAGE II Gross mass: 4004.04 kg Propellant mass: 3613.01 kg Thrust: 179.02 kN Burn time: 51.65 s

STAGE I Gross mass: 16,394.90 kg Propellant mass: 15,340.73 kg Thrust: 568.20 kN Burn time: 63.33 s

STAGE I Gross mass: 15,339.96 kg Propellant mass: 14,347.10 kg Thrust: 540.00 kN Burn time: 68.51 s

Figure 14.

Orthogonal views of air launched satellite launch vehicle.


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The computational efficiency is something which can never be ignored in the conceptual design and optimization of any aerospace system. The conceptual design phase requires that various configurations can be tested in a remarkably short span of time. The proposed 3DOF simulations gives ASLV optimization routine the computational efficiency required for the conceptual design.


5.

Conclusion

The present study devised a strategy to overcome inherent disadvantages of GA, which requires huge number of function evaluations to obtain near optimal solutions, and SA, which requires feasible direction at initial stage. In this particular test case of the design and optimization of an air launched satellite launch vehicle, hybrid heuristic search algorithm outperformed the genetic algorithm with a larger differential in the solution quality and it proved to be more efficient in catering for variations present in the solution of genetic algorithm. The mathematical codes for genetic algorithm have been developed and matured over the years. Their implementation in the design and optimization of rocket-based systems in general and satellite launch vehicles in particular has been successful since long. In the same time, neither genetic algorithm nor their hybrid modes have been tested before for the design and optimization of air launched satellite launch vehicles. The results of this preliminary design can be used as a basis for detailed design. The optimization results and performance are to be considered as preliminary (proof-of-concept) only, but they can be compared to existing systems (Isakowitz 1999), and can be applied in conceptual design of comparable space launch systems. Acknowledgements A.F. Rafique wishes to thank Higher Education Commission (HEC) of Pakistan for award of scholarship for PhD studies and would also like to thank Mr Sadique Ahmed for fruitful discussions and suggestions about programming.

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