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I. Introduction

O ver the past 70 years, solid rocket motors (SRMs) proved to be a reliable and cost-constructive propulsion system for a wide range of rocket-based applications starting from small tactical weapons up to current large space boosters. Many designers and manufacturers from different organizations prefer the SRM option when compared to other types of rocket propulsion systems. This is due to their ease of industry, long-lifetime storage along with the short time needed for launching, and a great deal of chemical potential energy in a relatively small volume. SRMs are different from other types of chemical rocket propulsion (i.due east., liquid propellant rocket motors and hybrid propellant rocket motors). The propellant of SRM is stored in solid course, in which solid fuels and solid oxidizers are held together by a solid binder, forming the so-called composite propellant [1 –3].

Although SRMs are relatively simple in principle, their mod types are circuitous systems that must incorporate several technical disciplines and teams to meet stringent mission requirements and design criteria. To accomplish their objective within given overall arrangement requirements and constraints, SRM subsystems and components must be carefully designed and optimized [3,4].

All-encompassing inquiry has been conducted for decades to find the optimal blueprint for a rocket vehicle that achieves the alien objectives of safe, reliability, performance, operability, and cost [five]. Billheimer [6] was ane of the pioneers who best-selling the importance of automating the SRM design in his paper in 1968. Although his efforts were limited by the computational capabilities available at that time, he demonstrated the power of numerical optimization techniques in achieving ameliorate preliminary designs by varying specific design parameters. In 1977, Woltosz [7] used a pattern search technique to find an optimal solid grain geometry that satisfies a specified performance for a specific vehicle. In 1980, Sforzini [8] coupled the aforementioned blueprint search technique to a special internal ballistic model in an effort to automate the blueprint of SRM. In Sforzini's automated design approach, fifteen design variables of the desired SRM were optimized and so that the delivered performance is as close as much to that obtained from static tests of the start space shuttle. The optimization methods used in these studies and others [9 –11] were gradient calculus-based methods that required good starting points to initiate the solution, in add-on to their inability to handle discrete variables (eastward.g., number of star grain fins) [5,12]. As a result, the earlier designs of SRMs relied heavily on an expert'due south cognition and trial-and-error techniques that often stop with designs non too different from previous ones [13]. On the other side, parallel to the vast advocacy in computational capacity in the by two decades, new powerful, gradient-costless optimization techniques have been regularly introduced and successfully applied to diverse aerospace blueprint issues. However, the subject of SRM blueprint optimization was rarely found in the literature until 2000 when Anderson et al. [12] examined the utilize of unmarried-objective and multi-objective genetic algorithms (GAs) to design SRMs as a component within an overall missile system. In this study, the GA was employed every bit a design automation tool without prior knowledge of the problem, and information technology was able to produce a variety of reasonable solutions and an efficient grain pattern. Since so, many attempts have been made at applying modern optimization methods to facilitate the SRM design procedure.

1 of the recent studies that attempted to overview the subject area of SRM was by Geisler et al. [3] in 2010, which was devoted to embrace various historical and technical aspects of SRM. They concluded their overview that, "though many designs and practices had been proven over the terminal 60 years, much research remains to be completed toward the optimization of SRM performance," which was valid to a groovy extent. In contempo years, the field of SRM pattern optimization has gained increasing attention from researchers and designers, reflected in new approaches and methods that take been proposed to treat different scenarios of SRM design problems. Consequently, the nowadays effort aimed to survey recent publications on the field of study of SRM blueprint optimization past focusing on metaheuristic-based optimization techniques that have been applied to the SRM pattern in the by two decades.

The rest of this newspaper is divided into three sections. Section Two describes briefly the general SRM design problem and the ways in which it has been viewed in the literature. Department Iii discusses the different metaheuristic-based optimization approaches practical to SRM pattern, and explains the advantages and drawbacks of each i. The last department summarizes the concluding remarks.

II. SRM Design Problem

By and large, the blueprint process of the SRM system starts with predefined mission requirements under specified design constraints. The SRM design is a highly integrated, highly iterative procedure that comprises the designing of several subsystems, namely, combustion sleeping room, nozzle, propellant grain, and insulation [1]. The propellant blazon is usually selected as early on every bit possible according to the desired performance of the SRM nether consideration, and more than often depending on the bachelor manufacturing applied science and the organization's expertise. Typically, there exists a strong interaction between these subsystems, and they share common pattern parameters that should be adapted in accordance with overall system design requirements. Practically, proper aligning of design parameters always requires a balanced tradeoff between competing pattern objectives (eastward.k., between toll and performance, between mass and safety, etc.). For that reason, engineers and designers started early to utilize optimization techniques to perform the SRM design procedure effectively and efficiently [seven,eight,14,xv]. Despite the fact that all SRMs share the same basic subsystems, there is no universal design method or well-defined process, and diverse approaches to treat the SRM design problem have been applied by the designers' community. Approaches may vary with the mission scenario, grade of awarding, background experience, and the amount of blueprint data bachelor [16].

A. SRM as a Part of Rocket System Pattern

The term rocket generally refers to whatsoever rocket-propelled flight vehicle (east.k., unguided military missile or whatever launch vehicle that is used for upper temper and space missions [16,17]. For both applications, the optimum SRM blueprint is the i that satisfies an optimum total impulse, an optimum thrust–fourth dimension contour, an optimum nozzle configuration, an optimum chamber pressure, and a preferred solid-propellant-grain configuration [sixteen]. Major aerospace sectors, including government, academia, and commercial companies, started as early on as late of the past century to focus more on developing and applying design and optimization tools mainly to achieve a rapid, high-performance, and low-price design of rocket vehicle systems. A major NASA program, the Intelligent Synthesis Environment, was introduced to bring together sufficient resource to blueprint, build, and demonstrate new, powerful, integrated pattern tools [v]. Airbus SE, formerly known equally the European Aeronautic Defence and Space Visitor (EADS-LV), developed their optimization tool for the preliminary design of launchers incorporating figurer codes for trajectory analysis, and vehicle and SRM sizing [18]. EADS-LV used the special propulsion optimization tool PROPULSE based on gradient methods to pattern and optimize a small three-stage SRM launcher [eighteen].

Designing a missile or a launch-vehicle system is a complex procedure that demands integration of unlike technology disciplines, for example, aerodynamics, propulsion, flight dynamics and trajectory, construction and mass/weight analysis, guidance and control, etc. Typically, each subject is mathematically modeled, and its interactions with other disciplines are taken into account. A combination of two models (propulsion/aerodynamics) is common in missile blueprint systems, although a combination of three models (propulsion/aerodynamics/flight dynamics) as well exists in the literature. In contrast, the integration of three disciplinary models or more than seems to be more common in launch-vehicle blueprint.

Anderson et al. [12] and Tekinalp and Bingol [19] incorporated an SRM internal ballistic model, an external shape aerodynamic model, and a flight dynamics model to detect the optimum missile design and the respective trajectory that achieves the maximum range. The vehicle performance modeling requires that an analysis from four separate disciplines be integrated into the blueprint optimization process. Bayley et al. [20] combined four discipline models of propulsion, aerodynamics, mass properties, and flight dynamics to produce a high-allegiance system model of the unabridged vehicle. The model built was used to blueprint and analyze a multistage solid-propellant launch vehicle to reach the low Earth orbit. Another study by Rafique et al. [ii] integrated a propulsion module with aerodynamics, structure, phase layout, mass distribution, and trajectory modules to design an air-launch vehicle, as shown in Fig. 1. The propulsion system was a iii-stage SRM.

Fig. one

Multidisciplinary analysis of satellite launch vehicle [2].

The SRM blueprint in the aforementioned studies and others [21 –25] is often performed upwards to the preliminary blueprint stage, and further endeavour is e'er needed toward a more detailed blueprint of the motor, especially for the propellant-grain configuration and its burn-dorsum analysis.

B. SRM Blueprint every bit a Separate System

The optimum SRM system pattern, as in Rafique et al. [2], is the 1 that satisfies the overall rocket system requirements under specified constraints. The design procedure should be carried out in a logical sequence, that is, independent design parameters should be identified every bit the starting time step, and then dependent design parameters are evaluated accordingly. Parameters derived from the mission/vehicle assay and propellant properties are usually considered as independent design parameters. Mission analysis and vehicle definition define the ballistic performance required by the motor (average thrust, called-for time, and total impulse), and provide the geometrical limits and performance conditions that constraint the design (motor length, weight, maximum expected operating pressure, temperature range, and ambience pressure [26]). Propellant properties that represent contained design variables are specific impulse, called-for rate, temperature sensitivity, density, and specific estrus ratio. The side by side step is to evaluate the dependent design parameters, which are generally associated with nozzle configuration and propellant-grain configuration design. These parameters mainly are pressure, thrust coefficient, pharynx area, volumetric loading, web fraction, port-to-throat-area ratio, and length-to-diameter ratio [26].

Once the aforementioned two types of design parameters take been identified and evaluated, the suitable propellant-grain shape (unremarkably from available and proved shapes) can then be selected, designed, and analyzed within the internal ballistic analysis loop. The internal ballistic analysis integrates the motor subsystem designs, and produces results in terms of thrust and/or pressure–fourth dimension curves. The results are further confirmed confronting the requirements; if there is a deviation, a necessary adjustment is made, and the analysis is repeated until convergence is accomplished.

The accuracy of performance simulation/prediction results is dependent on the physical model used in the internal ballistic assay. At that place are some physical and chemical processes that occur during the motor operation still not well understood; hence, the physical models have been simplified based on several assumptions [16]. A cypher-dimensional (0-D) analytical model that is based on the mass conservation law and thermodynamic relations was ordinarily used and proved to be adequate for SRM of small-to-moderate length:bore ratios and preliminary design purposes [27 –29]. More authentic 1-D, ii-D, and three-D models, which are typically solved numerically, were used when dynamic and erosive burning is expected to have a significant influence on motor performance [29,xxx].

Iii. SRM Blueprint Optimization

The conceptual and preliminary design phases of a rocket vehicle usually dictate the major configuration and engineering science decision of the system; hence, they represent the largest percentage of the total program cost and schedules. Consequently, a significant comeback is needed in these early stages of the design process [5]. Optimization deals with betterment and comeback, and its concept was broadly described as "humanity's want to excel" [31]; therefore, extensive research has been conducted for decades to find the optimal design for a rocket vehicle that achieves the conflicting objectives of rubber, reliability, performance, operability, and cost [v].

The first step in optimization is to formulate a mathematical model of the organization to be optimized. In well-nigh SRM design optimization problems, analytical 0-D internal ballistic models and analytical grain burn-back assay models were frequently preferred, equally they provide a fast assay tool adequate for the preliminary design phase [32,33]. More accurate numerical models for both internal ballistic analysis and grain burn down-back analysis were also used. Zeping et al. [34] coupled a 1-D internal ballistic assay with a burn-back analysis model (level gear up method) to design and optimize an SRM using an innovative full general optimization framework based on performance matching (Fig. ii). The grain burn down-back analysis model is sometimes decoupled from the internal ballistic model; in this case, the grain geometry is usually modeled and optimized using a parametric modeling characteristic in CAD software [1,35]. A detailed description of internal ballistic analysis models and grain burn-back analysis models can exist found in their respective literature, as discussing them is non the purpose of this paper.

Fig. ii

SRM performance matching design framework [34].

A. Optimization/Pattern Objective

One of the early steps in the SRM blueprint optimization is to make up one's mind the design goal based on the specific mission requirements (e.yard., maximum thrust, minimum weight, matching a predefined criterion, etc.). Practically, the combination of two or more desired criteria is oft considered when setting the design goal. The pattern goal is then used to codify the optimization objective office, which provides a mathematical representation of the desired blueprint criterion. Single-objective and multi-objective optimization has been considered in various cases of SRM design optimization problems. The ballistic performance parameters tin be summed into the thrust time profile, which provides a comprehensive mensurate for the overall SRM performance. Therefore, maximizing or matching a desired thrust–time profile has long been considered every bit a design goal, especially for single-objective optimization issues under specified constraints [viii,36]. Kamran and Guozhu [37] and Nisar and Guoxhu [38] optimized 3-D grain blueprint parameters to reach the maximum average thrust while not exceeding the commanded propellant mass. Raza and Liang [39] maximized the average thrust ratio and total impulse in a dual-objective optimization of a dual-thrust SRM. Zeping et al. [34] have proposed in a contempo study a pattern framework for matching a predefined thrust–time contour based on surrogate modeling.

On the other hand, cost is ever an essential gene in developing any rocket-based vehicle. The full vehicle mass (inert mass plus propellant mass) has traditionally been viewed every bit a master commuter toward the final vehicle cost [40]. Therefore, minimizing the gross lift-off mass/weight was the chief objective in many vehicle optimization studies. The minimum launch mass was considered for the single-objective optimization of the launch vehicle in [22,41 –43]. Anderson et al. [12] and Tekinalp and Utalay [44] minimized the weight and maximized the range of a missile system in a multi-objective optimization.

In contempo years, the demand for an efficient rocket-based arrangement is not dominated only by its operation, but also past its affordability and cheapness. In this new context, enquiry has been driven toward new designs that ensure optimized functioning, substantial cost saving, and risk mitigation [21]. To do so efficiently, designers demand to consider both technical and economic drivers at the conceptual and preliminary blueprint phases of a potential system and its subsystems simultaneously. Additionally, such a blueprint concept will involve the implementation of a combined multi-objective and multidisciplinary pattern optimization (MDO) approach, which adds more than complexity to the blueprint problem and more computation cost.

A recent study that explicitly addressed this issue was presented by Dupont et al. [21], who proposed a new strategy based on the dedicated MDO of a launch vehicle, in which a cost module has been integrated to other technical modules (i.east., propulsion, structure, aerodynamics, and trajectory). The methodology described in [21] was likewise applied to the preliminary design and analysis of a real-world cost-effective micro-launch vehicle named ROXANE [45]. Tola and Nikbay [32] take attempted to design an efficient price-effective SRM by minimizing the propellant mass while still achieving the best possible performance past maximizing the motor specific impulse. Furthermore, they coupled the internal ballistic analysis with structural design in MDO framework to ensure a more reliable design.

B. Optimization Techniques in SRM Design

Equally a result of continuous research on the blueprint and optimization of rocket-based vehicles in the by few decades, significant progress has been accomplished. Despite that, information technology is still uncertain what kind of search method might be suitable for the vast range of SRM design bug [2]. The no-gratis-luncheon theorem [46] concluded that there is no supremacy for a particular algorithm over other algorithms in all classes of bug. If an algorithm outperforms other algorithms at a specific class of problems, in that location must be cases on which this algorithm will perform inefficiently. This is noticeable in practice, in which diverse methods accept been applied to different problem scenarios. Based on our literature survey, the optimization techniques that have been oftentimes applied to handle the SRM design trouble can be categorized into four broad types: metaheuristic optimization, hybrid optimization (HO), hyper-heuristic optimization (HHO), and MDO, which will be discussed accordingly.

1. Metaheuristic Optimization Approach

Metaheuristic optimization algorithms are stochastic search methods that perform random exploration of the design space, and formally defined as "an iterative generation process that guides a subordinate heuristic past combining intelligently different concepts for exploring and exploiting the search space; learning strategies are used to construction information in order to detect efficiently near-optimal solutions" [47]. Some of these algorithms that have been widely used in SRM design in recent years include GAs, fake annealing (SA), and particle swarm optimization (PSO). These algorithms are basically inspired by the natural phenomena and concrete processes, for example, natural selection and survival of the fittest in GA, the social behavior of bird flocks in PSO, and metals annealing in SA [48]. Dissimilar the traditional method, metaheuristics practice non crave slope and/or Hessian data to proceed; instead, they follow either a population-based approach (e.g., GA and PSO) or a single-solution-based approach (east.g., SA). In the population-based approach, an initial set of solutions, the and then-called population, is ordinarily randomly provided and evaluated. Constraint violations and objective function values are then used to rank the solutions, and a combination of random search and nature-based heuristics allows updating the population for the next iteration [49]. In the single-solution-based approach, a generation and replacement procedure is applied to a single solution. A set of candidate solutions is generated in the generation stage, and then a suitable solution, from the generated set up, replaces the current solution [48].

Metaheuristics can solve a variety of optimization problems, in which the objective function is non smooth and potentially discontinuous (involving discrete design variables). Moreover, metaheuristics have shown more trend toward finding a global optimum due to their capability of locating the global optimum region in a big search infinite. This inherent randomness of the search process of metaheuristics makes them suitable for handling the multi-objective nature of engineering pattern problems [49].

The literature includes many instances, in which individual metaheuristics have been successfully applied to the design optimization of SRMs and SRM-based vehicles. One of the famous metaheuristics is the GA, which has been ofttimes used (more than eight times) in different cases and trouble scenarios. Information technology was used past Anderson et al. [12] as a dual-objective and multi-objective optimizer to automate the design process of a solid-propellant missile system. In Bayley et al. [twenty], GA was used to analyze and optimize the blueprint of three- and iv-stage solid-propellant launch vehicles with the aim of minimizing the overall vehicle mass and ultimately minimizing the cost per launch. A recent study past Tola and Nikbay [32] employed a multi-objective GA to maximize the specific impulse and minimize the propellant mass of an SRM using slotted propellant grain. Some earlier applications of GA to rocket-based blueprint optimization take been reviewed and summarized in 2003 by Anderson [13].

An interesting comparative report of the three metaheuristic optimizers GA, PSO, and SA was conducted past Rafique et al. [l]. Their performance on solving an optimization problem of air-launched satellite launch vehicle (ASLV) adopting a three-stage SRM propulsion system was evaluated and compared based on computational time and accuracy of results. Some other research endeavour devoted to assessing the performance of dissimilar metaheuristic optimizers to the SRM design problem was made by Badyrka et al. [51]. The study considered four different pattern optimization techniques for developing a desired SRM propellant grain. These four techniques included a real coded GA, a binary GA, a repulsive PSO, and a direct geometric search algorithm. The four algorithms were used individually to friction match a prescribed set of parameters that define the motor'southward thrust–fourth dimension profile; after that, the algorithms were compared based on their speed and effectiveness in solving the blueprint problem. Both studies [50,51] have reached to a conclusion that GA was more than efficient in terms of the quality of solution and the ability to find a global optimum, but it took more time to converge than the other optimizers, which made it less efficient in terms of the computational price.

Although the individual metaheuristic methods take shown notable advantages over the traditional methods for solving the SRM design optimization problem, still they accept some shortfall, which tin exist summarized in the following points [52]:

1. The computation time associated with metaheuristics is longer than that of the traditional methods. Metaheuristics often crave more function evaluations to converge to a solution, specially for a single-objective optimization.
2. Considering metaheuristics are stochastic search methods that typically involve random choices, different optimal solutions may be produced from different runs of the same algorithm code.
3. Different traditional or gradient-based methods, in which at to the lowest degree a local optimum is guaranteed upon convergence, metaheuristics practise non have proof of convergence to an optimal solution, unless they are advisedly employed and their parameters are appropriately set up.

To overcome these challenges, various methods and improvements have been proposed in the literature, and the two most important were the hybrid method and the hyper-heuristic method.

two. HO Approach

The no-free-lunch theorem for optimization [46] has concluded that all algorithms accept advantages and drawbacks when applied to dissimilar types of problems. The HO arroyo combines two algorithms to perform the search in an try to exploit specific features of each, and thus, enhancing the overall optimization procedure (e.g., a metaheuristic algorithm and a gradient-based algorithm, or two metaheuristic algorithms) [53]. Either way, the motive is oftentimes to amend the quality of the solution or to subtract the computation time, or both [54,55,56].

At that place are several cases of SRM design, in which an HO approach has been adopted. Nisar et al. [57] combined GA for global solution convergence and sequential quadratic programming (SQP) for further local convergence of the solution to minimize the propellant mass of an SRM under given design constraints. They presented that the results obtained from HO were the best among those received from GA or SQP separately. The aforementioned technique (i.east., hybridization of GA and SQP) was also used by Zeeshan et al. [58] to facilitate the conceptual blueprint of a basis-based interceptor missile that uses a 3-stage solid rocket propulsion system based on the MDO approach. The simulation results in this report showed that the GA/SQP hybrid algorithm performed better than did the single GA optimizer, and information technology improved the solution past 10%. Another similar piece of work was washed past Jenkins and Hartfield [59], in which they examined the performance of a hybrid optimizer based on the integration of a blueprint search algorithm into a constrained repulsive PSO to match a thrust vs time profile for an SRM grain/nozzle combination. The results were compared with those obtained from a PSO, a binary encoded GA optimizer, and a real code GA optimizer, and it was plant that the hybrid optimizer was more than efficient in terms of convergence speed and solution quality. This innovative approach was later applied in [lx] to examine the design of an SRM to match different burn profiles using dissimilar geometries of a circular perforated (tapered) grain.

Another way to construct a hybrid optimizer is past associating two metaheuristic algorithms together (i.east., using one algorithm for global search and some other one for further local search). In this case, a scheme of HO based on the GA (for global search) and the SA (for local search) proved its usefulness and suitability in achieving loftier-speed processing and high-quality solution for various engineering design problems [54,61]. This arroyo has been adopted past Raza and Liang [39] and Rafique et al. [42]. In [42], the authors hybridized the GA and the SA to develop a design strategy that could efficiently and effectively facilitate the multidisciplinary design analysis and optimization for an ASLV. Modules for mass backdrop, propulsion characteristics, aerodynamics, and flight dynamics were integrated to produce a loftier-fidelity model of the vehicle. The propulsion module independent designing and optimizing a iii-stage SRM in an MDO environment. Meanwhile, in [39], the authors applied the GA–SA hybridization arroyo to incertitude-based robust pattern optimization of a dual-thrust SRM propulsion system. This type of SRM typically comprises two levels of thrust, namely, boost-phase thrust and sustain-stage thrust, and is usually required past an awarding, in which the want is to accelerate the vehicle up to a sure altitude from zero to a certain stabilized velocity with high Mach number in quite a brusk period of fourth dimension, and then sustain the vehicle at a abiding velocity for a longer time with low level of thrust. The traditional pattern objective of such a propulsion system is to maximize the average boost-phase-to-sustain-stage-thrust ratio and the total impulse of the motor. In both cases [42] and [39], the GA was initially used to identify a feasible region of solutions, and then the SA was used to search for local optimality inside the viable region that has already been identified past the GA.

3. HHO Approach

Metaheuristic and hybrid methods proved to be successful in solving near of the real-earth optimization problems, but they yet meet some difficulties in terms of universality and flexibility (i.eastward., the applicability to different types of issues regardless of the trouble scenario) [62]. Moreover, there is a lack of information on how to select and manipulate a big set of parameters of a item algorithm for a particular problem, as the evolution and deployment procedure of these methods require sufficient knowledge on both the problem domain and the metaheuristic method [62]. HHO is an emerging methodology in search and optimization, which aims to enhance the level of generality of optimization systems by automating the choice and adaptation of several low-level heuristic algorithms [63]. Indeed, searching infinite of metaheuristics rather than searching space of solutions increases the power to notice optimal solutions without significant tuning of the algorithm for unlike problem scenarios [64]. This feature makes HHO able to handle a wide range of problem domains rather than the current metaheuristic methods, which may non exist direct applicable to another problem domain or new instances of the same trouble [63].

There are few works establish in the literature on which an HHO has been applied to the design and optimization of an SRM-based vehicle. Rafique et al. [ii] used a nonlearning random role to control metaheuristic algorithms (GA, PSO, and SA) in serial. The adult approach was successfully applied to a complex design case report of an ASLV that uses a three-stage SRM every bit a propulsion organisation. In this study, five disciplines (vehicle configuration, propulsion, mass, aerodynamic, and trajectory) and nineteen design variables were considered in an MDO procedure to minimize the gross lift-off mass of the entire vehicle. Another attempt to utilise the HHO was made by Kamran and Guozhu [one], who implemented the same control role and low-level heuristic algorithms used in [2] to design an SRM system. The goal of this study was to minimize the gross mass of motor nether specified geometrical and ballistic constraints by optimizing eight design variables representing the motor subsystems (i.eastward., combustion chamber, nozzle, and propellant grain). The HHO arroyo adopted in the aforementioned two studies was able to accomplish the desired objective and handle the complexity efficiently without adequate tuning of algorithm parameters; however, obviously, the computation cost is loftier due to the big number of function evaluations during the search procedure.

4. MDO Arroyo

MDO is an emerging field in aerospace technology that attempts to innovate a structured methodology to locate the best possible design in a multidisciplinary environs, with the aim of acting every bit an amanuensis to bind the other disciplines together [65]. Sobieszczanski-Sobieski and Haftka [65] have described the MDO in their ain paper as "a methodology for the blueprint of systems where the interaction between several disciplines must be considered, and where the designer is free to significantly impact the system functioning in more than i subject area." The MDO involves the coordination of multiple disciplinary analyses to realize more than effective solutions during the design and optimization of circuitous systems [66].

Traditionally, disciplines have been defined in terms of knowledge areas, for example, pattern disciplines of a flight vehicle include aerodynamics, structures, propulsion, guidance and control, etc. However, a single discipline tin besides constitute an MDO problem if information technology undergoes farther decomposition into its components or subsystems [67]. The MDO process typically involves the following steps [66]: ane) decomposing the arrangement into multiple subsystems or disciplinary analyses, 2) developing mathematical models that describe the overall organization and each subsystem considering disciplinary interactions, 3) selecting an appropriate MDO conception and algorithm, and 4) solving the MDO problem to generate solutions.

Because the MDO essentially deals with the interaction of diverse disciplines of different analysis models, the demand for organizing these models efficiently and selecting the suitable optimization software represents a disquisitional job for the proper implementation of MDO. There are several terms in the literature that are used to describe the way, in which the conception and organisation are combined, such as strategy, architecture, methodology, method, algorithm, etc. [67,68]. The architecture tin can be either monolithic or distributed. In the monolithic approach, analysis models are compiled as a single executable code that contains internal modules or subroutines to each field of study, whereas in the distributed approach, the system is decomposed into smaller units that tin be individually optimized, and necessary data are externally exchanged between codes [v].

MDO is alluring more interest in the field of aerospace design optimization. A comprehensive review of different MDO approaches and techniques applied in various aerospace applications is institute in Refs. [68 –70]. The literature reveals that at least x papers take implemented the MDO approach on design and optimization of SRM-based vehicles, most of which were published in the last decade. Different optimization methods and algorithms, including private metaheuristic algorithms [12,19,32,71– 73], hybrid method [21,42,58,74,56], hyper-heuristic method [ane,2], and fifty-fifty gradient-based methods [18,22,41,43] were used to facilitate the MDO procedure.

At that place are many cases in the literature, in which an SRM design has been treated every bit a subsystem within the overall organization blueprint process [two,5,12,18 –22,24,25,42,43,50,58,75,71]. In most of these cases, the physical disciplines of weight and size analysis, propulsion, aerodynamics, and trajectory were incorporated to codify the MDO problem. Bayley et al. [twenty] demonstrated the effectiveness of MDO in analyzing and optimizing the design of 3- and four-stage solid-propellant launch vehicles (Fig. 3), with a last goal of minimizing the overall vehicle mass, later on minimizing the total toll per launch. For this purpose, they used tournament-based GA (IMPROVE 3.ane© GA) adult by Anderson [76], and built a loftier-fidelity system model based on the integration of four disciplines: propulsion, aerodynamics, mass, and flight dynamics. The analysis code used for the propulsion module was originally developed past Hartfield et al. [33]. After the specific solid-propellant backdrop are preloaded to the propulsion model, the basic SRM characteristics, such as thrust, burn down time, grain mass, and combustion pressure, are calculated, and thus, the thrust–time profile of the motor could exist obtained. Appropriately, the propulsion characteristics of each phase were adamant separately and in sequence.

Fig. 3

3-stage solid-propellant launch-vehicle schematic [20].

There are also few instances, in which the designing of an SRM has been viewed every bit an independent organisation; for case, designing of each subsystem of grain, combustion bedroom, structure, and nozzle was commonly considered as a unmarried subject area for the MDO trouble [1,32,74]. Kamran and Guozhu [1] did a comprehensive disciplinary analysis of a typical SRM arrangement (Fig. 4), and explained the interaction between the subsystems and the coupled blueprint variables in a design construction matrix (Fig. 5). They too integrated CAD software with an optimization module that used the hyper-heuristic method.

Fig. 4

SRM system design assay [i].

Fig. 5

SRM design construction matrix [1].

Although MDO allowed designers and researchers to handle the interactions between the disciplines simultaneously, and thus, achieving superior optimum solutions compared to conventional methods, incorporating all disciplines simultaneously increased the complexity of the problem significantly. This complexity tin be summarized into iii bug. The first effect was related to the problem formulation when heterogeneous disciplinary modules demand to be coupled properly and run as a single trouble. The second issue is that dealing with mathematical models involving discrete design variables ofttimes limits the selection of a suitable optimization algorithm. Finally, optimizing design variables taken from multiple disciplines results in an increased volume of the blueprint space exponentially, which makes the solution computationally expensive and strongly sensitive to different problem scenarios [ane,twenty]. Even so, to overcome these challenges, several approaches have been proposed by researchers, such as using surrogate-model-based optimization [34,77], response surface method [25], and hybridization with SQP or complex method [41,78].

Four. Conclusions

As mentioned previously, there is a common agreement among authors that SRM is a toll-effective, highly reliable, and relatively simple rocket propulsion system that has been used for decades in many aerospace applications. Typically, the optimum design of the SRM system is a tedious process that requires high integration of several subsystems and a balanced tradeoff between competing objectives, such every bit performance, reliability, cost, etc. Therefore, employing optimization tools becomes necessary to facilitate the design process efficiently and effectively. An overview of various modern metaheuristic-based optimization techniques that have been applied to the SRM design in the past 2 decades was presented in this newspaper.

At the present time, information technology tin can be ended that the GA was the almost widely used past researchers, either as an individual optimization algorithm or as a global search algorithm in the HO approach. In both cases, the GA proved its superiority at exploring large design space, subsequently increasing the potentiality of finding a global optimum solution. The hybrid arroyo seemed to exist a favorable choice in many instances to overcome the limitations of individual metaheuristic algorithms, despite the relative difficulty associated with constructing such kind of approach. The HHO was started recently to attract more involvement as a universal and automation pattern tool, generally useful in dealing with complex multidisciplinary design bug, and therefore, suitable to facilitate the MDO process of rocket-based system design. The computation price imposed by applying HHO and MDO is a challenging issue that needs to be adequately addressed and continuously reviewed due to the parallel fast development of computational tools. Although metaheuristic-based optimization methods do not guarantee optimality, they were able to produce consequent (unique in some cases) solutions in a reasonable amount of computing fourth dimension in most SRM design cases.

A disquisitional step in the SRM optimization is the selection of the design goal, based on which the objective function is formulated and mathematically represented. The surveyed literature confirmed the traditional common view that costs and operations are proportional to vehicle weight and size. Withal, this indicate of view has started recently to change toward a more comprehensive and concurrent pattern optimization based on both the maximized operation and the minimized toll. It appears that still there are a lack of SRM design models that simultaneously clarify (approximate) the performance-based and cost-based criteria in the preliminary design stage. Thanks to the new powerful computation tools, maybe edifice and validating these models can exist a potential area of improvement in SRM design optimization, leading to decreased product evolution fourth dimension and reduced costs.

In this survey, the authors have attempted to highlight the most used algorithms, the virtually mutual pattern objectives, the recent trends, and the main challenges in SRM design optimization in a simplified and organized manner. The current try was intended to serve as an initial guide for SRM designers and researchers in selecting the optimization method that well suits their problem, and to help them know where to become next.

A. D. Ketsdever Acquaintance Editor

Acknowledgment

The authors would like to thank the Universiti Sains Malaysia Enquiry University grant (grant number 1001/PAERO/8014019) for funding this work.

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