A novel bat algorithm with double mutation operators and its application to low-velocity impact localization problem☆
Introduction
Low-velocity impacts, which are caused by collisions with floating ice, stranding, and explosion, often lead to internal damages in the plate structure of the ship. These internal damages, such as micro-cracks, are invisible, but they can reduce the structural strength (Jiang et al., 2015). Therefore, determining the locations of the low-velocity impact damages in the plate structure of the ship is necessary. The low-velocity impact signals are usually acquired using piezoelectric sensors (PZT) (Santos et al., 2013), fiber optic sensors (Pang et al., 2018), and other types of sensors (Bradshaw et al., 2006, Mahdian et al., 2017, Maleski et al., 2004). Compared with the electrical sensor, the fiber Bragg grating (FBG) sensor has advantages of flexibility, multiplexing capability, corrosion resistance, small size, and better embeddedness, and it can easily form a sensor network without damaging the structures (Wu et al., 2014), which means that the FBG sensor is more suitable for the structural health monitoring of the ship.
The impact localization methods commonly include the triangulation method (Tobias, 1976, Zhao et al., 2017), the neural network algorithm (Jang et al., 2012, Sung et al., 2000), the time reversal method (Chen et al., 2012, Ciampa and Meo, 2012), and the reference database method (Jang and Kim, 2016, Shrestha et al., 2016). Among these available methods, the triangulation method, in which the impact locations are predicted by solving a group of nonlinear equations, is widely used. This group of nonlinear equations usually consists of sensor locations, wave velocities, and time differences. Since it is more difficult to solve a group of nonlinear equations than to solve a group of linear equations, the exact solution cannot be obtained. In the low-velocity impact localization problem, the process of solving a group of nonlinear equations could be converted into a nonlinear optimization problem, and then the minimum value of the objective function could be calculated using metaheuristic algorithms, such as the particle swarm optimization (PSO) (Zhao et al., 2017), the genetic algorithm (GA) (Worden and Staszewski, 2010), and the artificial bee colony algorithm (ABC) (Kiran et al., 2015). The solutions corresponding to this minimum value are the impact locations. Nevertheless, the existing methods have some limitations when solving the impact localization problem because of their lower accuracy rates. Thus, further developing impact localization methods based on the triangulation method and metaheuristic algorithms is important for monitoring the low-velocity impact damages in the plate structure of the ship.
The metaheuristic algorithms can be divided into four categories: evolutionary algorithms (EAs), physics-based algorithms, human-based algorithms, and swarm intelligence (SI) algorithms (Faris et al., 2018). The EAs simulate the natural evolutionary process to find solutions, and representative examples include GA, evolutionary programming (EP) (Eiben and Smith, 2003), genetic programming (GP) (Koza, 2008), and differential evolution (DE) (Das and Suganthan, 2011, Hamza et al., 2018). The physics-based algorithms, such as simulated annealing (SA) (Suman and Kumar, 2006), big-bang big-crunch (BBBC) (Erol and Eksin, 2006), gravitational search algorithm (GSA) (Rashedi et al., 2009), charged system search (CSS) (Kaveh and Talatahari, 2010), and galaxy-based search algorithm (GBSA) (Shah-Hosseini, 2011), are inspired by physical processes. The human-based algorithms mimic the human behaviors, and some examples are tabu search (TS) (Glover, 2007), socio evolution and learning optimization (SELO) (Kumar et al., 2018), harmony search algorithm (HS) (Manjarres et al., 2013), and teaching learning based optimization (TLBO) (Rao et al., 2012). The main idea of the SI algorithms is to imitate the social behaviors of animals, and some examples of SI algorithms include PSO, ABC, fruit fly optimization algorithm (FOA) (Pan, 2012), moth-flame optimization (MFO) (Mirjalili, 2015, Yildiz and Yildiz, 2017), Harris hawks optimization algorithm (HHO) (Heidari et al., 2019), and grasshopper optimization algorithm (GOA) (Saremi et al., 2017). These metaheuristic algorithms have been successfully applied to different real world problems, such as structural design optimization (Pholdee et al., 2017, Yildiz et al., 2016a, Yildiz et al., 2016b, Yildiz et al., 2019a, Yildiz and Lekesiz, 2017, Yildiz and Yildiz, 2018, Yildiz and Yildiz, 2019), PID controllers (Azmi et al., 2019, Hamza et al., 2017), automotive industries (Kiani and Yildiz, 2016, Yildiz, 2017), scheduling (Akbari et al., 2017, Ribas et al., 2013), and manufacturing problems (Yildiz et al., 2019b, Yildiz et al., 2019c). However, many metaheuristic algorithms perform inadequately when the complexity of real world problems increases.
The bat algorithm (BA), which is a swarm intelligence method, was developed by Yang (2010). Using its echolocation capability, micro-bat can determine a prey’s distance, shape, and location in the dark. Exploration and exploitation are the crucial components of a metaheuristic algorithm (Topal and Altun, 2016). Exploration is the search for diverse solutions in new and undiscovered regions. Exploitation is to find the best solution among the explored neighbors. The BA has been proven to be effective and robust on low dimensional problems and real world applications because of its great exploration capability, but it still experiences the problem of premature convergence. To address this problem, an increasing number of BA variants have been developed. These BA variants can be mainly classified into two major categories.
The first category emphasizes the combination of the advantages of other optimization methods with the standard BA. Nguyen et al. (2014) presented a hybridization of BA and ABC with a communication strategy. Additionally, Murugan et al. (2018) presented an algorithm hybridizing BA and ABC with a chaotic based self-adaptive search strategy. For solving global numerical optimization problems, Wang and Guo (2013) proposed a hybrid metaheuristic algorithm (HS/BA) adopting a pitch adjustment operation in HS algorithm as a mutation operator during the bat exploration process. To increase the diversity of the BA, Ghanem and Jantan (2018) introduced a hybrid bat harmony (HBH) algorithm by further modifying and fine-tuning the pitch adjustment operation in the HS algorithm. Ge et al., 2018a, Ge et al., 2018b, Ge et al., 2018c combined BA with the Powell algorithm, simplex algorithm, and steepest descent, which are called PBA, SABA, and SD-BA, respectively. Integrating the mutation operator in the DE/rand/1/bin scheme with BA, Meng et al. (2015a) designed a novel hybrid bat algorithm with the DE strategy (BADE) for solving constrained optimization problems. Chen and Xu (2019) investigated a hybrid multiobjective bat algorithm (BA-DE) that combined features of BA with DE. Latif et al. (2018) proposed a new bat genetic algorithm (BGA) in which the mutation step of GA was carried out after new solutions were obtained using BA. Ferdowsi et al. (2019) developed a hybrid algorithm (HA) of BA and PSO that had an ability to escape from the local optima and exhibited a high convergence rate. The main idea of the HA was to replace the weak responses of an algorithm with the strong responses of another algorithm. To reinforce the local and global search characteristics of BA, an enhanced bat algorithm (EBA), in which two modifications and the invasive weed optimization algorithm were embedded into BA, was proposed by Yilmaz and Küçüksille (2015).
The second category focuses on modifying the BA mechanism itself, which includes formula modifications, parameter-tuning methods, and auxiliary search strategies. Chakri et al. (2017) developed a directional bat algorithm (dBA) that integrated the directional echolocation into BA, introduced a local random walk to modify the position equation, and designed new updating formulas of the pulse emission rate and loudness. Wang et al. (2019) adopted eight strategies to modify the velocity and position equation, and designed a probability formula to select the most appropriate strategy. Xie et al. (2013) defined a self-adaptive adjustment strategy for the frequency updating formula and introduced the Lévy flights trajectory to modify the position updating formula. Haji and Monje (2019) developed a dynamic bat algorithm (DBA) that utilized a new mechanism to dynamically choose an optimal combination of the pulse emission rate, the pulse frequency, and the population size, while Pérez et al. (2015) employed a fuzzy system to dynamically adjust BA’s parameters. Introducing three different centroid strategies and the velocity inertia-free updating equation, Cui et al. (2019) presented six different BA variants. Gan et al. (2018) incorporated the iterative local search and stochastic inertia weight with BA (ILSSIWBA). With the purpose of empowering BA’s capability to control its diversity concepts, Al-Betar and Awadallah (2018) adopted the strategy of island model for BA. Gandomi and Yang (2014) embedded different chaotic maps into BA and validated the improvement of different chaotic BA (CBA) variants. Moreover, Liang et al. (2018) presented a hybrid BA (RCBA) that combined a chaotic map and random black hole model for solving economic dispatch problems in power systems. Aiming at the structural design problems, Meng et al. (2015b) proposed a novel bat algorithm (NBA) by creatively introducing the bats’ habitat selection into BA and defining a self-adaptive local search strategy. Additionally, Meng et al. (2019) embedded the reinforcement learning into the NBA to adaptively select between different operators, used the individual difference-based strategies to adaptively tune the algorithm’s parameters, and redefined the local search strategy.
The effectiveness of the abovementioned BA variants for solving their specific cases has been validated. However, how to further improve BA’s performance to make it more suitable for real world problems and applications is still important.
The aim of this study is to propose a novel bat algorithm with double mutation operators (TMBA) to enhance BA’s performance on nonlinear optimization problems and to employ the proposed algorithm and the triangulation method to solve the low-velocity impact localization problem. To avoid BA’s problem of premature convergence, three modifications, which include the modified time factor (M1), the Cauchy mutation operator (M2), and the Gaussian mutation operator (M3), are integrated into the standard BA. By utilizing 29 benchmark functions, the contributions of the three modifications are analyzed, and the significant improvement of TMBA compared with several advanced optimization algorithms and improved BA variants is proven. Furthermore, the proposed algorithm is applied to the low-velocity impact localization system based on fiber Bragg grating (FBG) sensors to determine the locations of the low-velocity impact damages on an aluminum plate.
The organization of this paper is given as follows. In Section 2, the relative theories of the BA and the localization method are provided. The details of the novel bat algorithm (TMBA) are given in Section 3. In Section 4, the results obtained by TMBA and comparative algorithms in the numerical experiments and the low-velocity impact localization problem are discussed. Finally, Section 5 draws the conclusion and provides suggestions for further work.
Section snippets
Principle of FBG sensor
The fiber Bragg grating (FBG) sensor is used as a filter that selects and reflects back a part of the input spectrum of light (Lu et al., 2015). The center wavelength of the reflected light is called the Bragg wavelength and is expressed by Eq. (1). where is the Bragg wavelength of an FBG sensor. Here, is the average refractive index of the optic fiber, and is the Bragg grating spacing.
The refractive index and the Bragg grating spacing vary as the temperature and
A novel bat algorithm with double mutation operators
The standard BA has the problem of premature convergence when solving complex optimization problems (Meng et al., 2015b). In this study, we have modified the time factor to improve BA’s convergence speed and local search capability and introduced two mutation operators into the standard BA to increase its capability of escaping from the local optima. The three modifications are abbreviated as M1, M2, and M3. The pseudocode of the novel bat algorithm with double mutation operators (TMBA) is
Experiments and discussions
In this section, two numerical experiments based on 29 benchmark functions (Pan et al., 2014) were executed to evaluate TMBA’s performance in solving nonlinear optimization problems. Firstly, the contributions of the three modifications were analyzed using 30-dimensional functions. Secondly, the performance of TMBA was compared with that of several well-known optimization algorithms using 50-dimensional functions. Finally, an application based on the low-velocity impact localization system was
Conclusion
In this study, a novel bat algorithm with double mutation operators (TMBA) is proposed to solve the low-velocity impact localization problem on an aluminum plate. To avoid the problem of premature convergence of the standard bat algorithm (BA), three different modifications are integrated into BA: the modified time factor (M1), the Cauchy mutation operator (M2), and the Gaussian mutation operator (M3). The modified time factor is developed to increase the convergence speed during early
CRediT authorship contribution statement
Qi Liu: Conceptualization, Methodology, Validation, Investigation, Writing - original draft, Writing - review & editing. Jindong Li: Software, Validation, Investigation, Writing - review & editing. Lei Wu: Validation, Formal analysis, Writing - review & editing. Fengde Wang: Validation, Visualization, Writing - review & editing. Wensheng Xiao: Validation, Resources, Data curation, Supervision, Project administration, Funding acquisition.
Acknowledgment
The authors would like to acknowledge the High-tech Shipping Research Project from Ministry of Industry and Information Technology of China for supporting this study through the project “Research of full life-cycle reliability guarantee technology system for subsea oil and gas production system” with the grant number of 2018GXB01-02-003.
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2021, Engineering Applications of Artificial IntelligenceCitation Excerpt :Therefore, these kinds of algorithms have a wide range of application (Stojanovic et al., 2020; Tian et al., 2018, 2017; Kaur et al., 2020; Hu et al., 2020, 2021). For example, optimization design of structures (Prendes Gero et al., 2006; Mortazavi and Toğan, 2017; Li et al., 2021; Zhang et al., 2019; Liu et al., 2020b; Hayyolalam and Pourhaji Kazem, 2020; Wang et al., 2021), design of new materials (Li et al., 2019, 2020c, 2018), inversion of key parameters (Li et al., 2020d; Xu et al., 2021), optimal control of robot (Stojanovic et al., 2016; Stojanovic and Nedic, 2016; Pršić et al., 2016) and path planning (Ebadinezhad, 2020; Al-Gaphari et al., 2021). The classical bio-inspired optimization algorithms include particle swarm optimization (PSO) algorithm (Li and Qin, 2002; Stacey et al., 2004; Liu et al., 2020c), genetic algorithm (GA) (Deb, 1999; Holl, 1992; Tang et al., 1996), artificial bee colony (ABC) algorithm (Karaboga and Basturk, 2007; Tereshko and Lee, 2002; Tian et al., 2019; Wang et al., 2020), whale optimization algorithm (WOA) (Mirjalili and Lewis, 2016; Yan et al., 2021), ant colony optimization (ACO) algorithm (Ebadinezhad, 2020; Dorigo and Caro, 1999) and so on.
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No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.engappai.2020.103505.