The depletion of shallow and easily exploitable ore reserves has compelled humanity to extract mineral resources at increasingly greater depths (Ranjith et al. 2017; Zhu et al. 2018; Xie et al. 2019). Deep open-pit mining, the transition from open pit to underground mining, and deep underground mining have emerged as the prevailing trends in the mining industry. However, these trends have introduced more frequent geological hazards, e.g., landslides, collapses, and rockbursts, making mineral exploitation an inherently challenging endeavor (Lou et al. 2021; Kang et al. 2023). Fundamentally, the occurrence of any geological hazard can be traced back to rock damage and failure. Consequently, the study of rock failure mechanisms and the identification of precursors preceding rock failure hold immense significance for safety production.

Acoustic Emission (AE) localization technology, as a non-destructive detection method, has garnered increasing attention for its capacity to accurately identify the location, extent, and progression of damage in rock materials (Xu et al. 2012; Zhao et al. 2023; Wei et al. 2020). AE source localization is crucial for assessing the dynamic status of the rock. The estimation of AE source location is fundamentally an inverse problem, wherein the target model is the location coordinates of passive AE sources (Haldorsen et al. 2013; Wang and Alkhalifah 2018). The studies of AE location began in the 1960s. Mogi (1968) conducted line location and plane location based on the difference in P-wave arrival time detected by four sensors. Scholz (1968) utilized the difference in S-wave arrival time and a least square algorithm to calculate the location of AE events. To date, numerous AE location algorithms have been developed (Dong et al. 2022). Based on differences in each element of an inverse problem, AE localization can be categorized into various methods (Brantut 2018). Commonly used AE location algorithms encompass the least square algorithm, relative location algorithm, simultaneous inversion algorithm, Geiger location algorithm, and simplex location algorithm (Yu et al. 2022; Shang et al. 2023). Building upon these AE location algorithms, numerous research results regarding the temporal-spatial evolution of microcracks within rocks have been obtained (Aggelis 2011; Amitrano et al. 2012; Aker et al. 2014).

The swarm intelligence optimization algorithm is a relatively novel algorithm among rock AE localization algorithms, and many scholars have conducted studies on swarm intelligence optimization algorithms. Rawa et al. (2021) introduced a modified grey wolf optimizer based on the analytic hierarchy process for optimizing power generation costs, minimizing losses, and reducing emissions. Subsequent experimental validation demonstrated the improved GWO’s superiority over alternative algorithms in the quest for optimal solutions to these challenges. Motivated by the migratory and foraging patterns of red cranes, Wang and Liu (2021) presented an innovative swarm intelligence optimization algorithm known as the Flamingo Search Algorithm (FSA). This algorithm is rooted in a mathematical model of flamingo behavior, with the attributes of global exploration and local exploitation necessary for an optimization algorithm. Experiments have substantiated FSA’s superior performance in addressing a variety of optimization challenges when compared to alternative algorithms. Teng et al. (2019) proposed a novel approach, combining the Grey Wolf Optimization algorithm with the Particle Swarm Optimization algorithm (PSO-GWO), to address the common issues of slow convergence and low accuracy observed in conventional GWO algorithms. By incorporating Tent chaotic sequences for initializing individual positions, this approach enhances diversity within wolf packs, better preserves optimal individual position information, and reduces the risk of the algorithm converging to local optima. The results of benchmark simulations demonstrate that the proposed algorithm excels in searching for global optimal solutions and exhibits superior robustness compared to other algorithms. Lei et al. (2022) compared the accuracy of backpropagation neural networks and swarm intelligence optimization algorithms for the rock uniaxial compressive strength prediction. The results indicated that the neural network model based on the GWO had the best prediction performance. Wang et al. (2022) introduced the grey wolf algorithm (GWO) into AE localization research to address the limitations of traditional AE localization algorithms in rock mechanics experiments. A modified GWO based on population memory elimination mechanism was proposed to address the deficiency of poor local search ability in the original algorithm. Through experiments, it has been verified that this algorithm outperformed other algorithms in terms of AE localization search efficiency and accuracy.

In summary, the swarm intelligence optimization algorithm is currently a popular choice for optimizing rock AE localization. While the GWO algorithm stands out as a promising approach, the conventional GWO algorithm still exhibits limitations in terms of convergence speed and solution accuracy (Xu et al. 2019; Zhang et al. 2021, 2023). In light of these challenges, the present study attempts to establish a method known as Simplex Improved Grey Wolf Optimizer (SMIGWO) and conducts comparative experiments involving relevant algorithms to confirm the efficacy of this novel approach. This paper is structured into four sections. The first section offers an overview of the current research landscape in rock AE localization technology and optimization algorithms related to rock AE localization. The second section is the specific design of the GWO algorithm for rock AE localization based on iterative chaotic mapping, inverse incomplete Г function, and simplex method improvement. The third section analyzes the experimental results of the SMIGWO algorithm and introduces other relevant algorithms for comparative analysis. The fourth section is the conclusion of the study.

In response to the problems of poor local search ability of the general GWO algorithm, resulting in low solution accuracy and slow convergence speed, the study will introduce iterative chaotic sequences instead of randomly generating initial populations. The search process will be optimized using the convergence factor optimization algorithm based on the inverse incomplete Г function. Additionally, the simplex method will be utilized to address issues related to poorly positioned grey wolves, thereby enhancing the performance of the GWO algorithm.

2.1 Iterative chaotic sequence method for initial population generation

Mirjalili et al. (2014) proposed the Grey Wolf Optimizer (GWO), a population intelligence optimization algorithm inspired by the behavior of grey wolves. The GWO algorithm mimics the hierarchical leadership structure and hunting strategies observed in grey wolves in the wild. It employs four types of grey wolves to simulate the leadership hierarchy, and the entire process from prey detection to capture involves three primary phases: prey search, prey encirclement, and prey attack (Uzlu 2021; Waziri and Yakasai 2022). The GWO algorithm has obvious advantages in search speed, accuracy, stability, and other aspects for rock AE location. Therefore, the study will use the GWO algorithm to analyze the AE localization of rocks.

The grey wolf hierarchical system and possible updated positions of a grey wolf are shown in Fig. 1. The grey wolf population is categorized into four hierarchical systems, with social levels ranging from high to low denoted as α wolf, β wolf, δ wolf, and ω wolf (Choudhuri et al. 2023). Among them, α wolf is the highest leader of the grey wolf group, followed by β wolf. δ wolf is on the third level of the hierarchy and can command individuals at lower levels while following commands from α and β. The bottom layer is the ω wolf, responsible for balancing relationships between different levels of the population and caring for the young wolves. The primary processes of the GWO algorithm involve population initialization, social hierarchy, surrounding prey, hunting, attacking prey, and searching for prey. Among these, the concept of social hierarchy is illustrated in Fig. 1a, while the processes of surrounding prey, hunting, attacking prey, and searching for prey are depicted in Fig. 1b. The initial step in hunting is to chase and surround the prey, and its mathematical model is represented by

where D is the distance between the prey and the grey wolf; t indicates the current iteration step; C denotes the synergy coefficient; X_p(t) represents the position of the prey; X(t) expresses the position of the grey wolf, as presented in Eq. (2):

where X(t+1) denotes the position of the grey wolf at t+1 step and A represents the synergy coefficient. The mathematical expressions for the synergy coefficients A and C are given out:

where a denotes the convergence factor, decreasing linearly from 2 to 0 over iteration times, r₁ and r₂ are random values within the range of [0, 1]. Figure 1b presents the potential updated positions of a grey wolf in 3D space. A grey wolf at the position of (X, Y, Z) can update its position based on the prey's location (X*, Y*, Z*). The random synergy coefficients enable wolves to reach positions anywhere between the points illustrated in Fig. 1b. Thus, a grey wolf can update its position randomly within the space around the prey by employing Eqs. (1)-(3).

Principle of the GWO algorithm (Mirjalili et al. 2014)

Grey wolves have the capability to identify the location of prey and encircle them. Typically, the hunt is directed by α wolf, with occasional participation from the β and δ wolves. However, in an abstract search space, we lack information about the optimum location (prey). To mathematically simulate the hunting behavior of grey wolves, it is assumed that α, β, and δ wolves possess a good understanding of prey positions. The positions of ω and other grey wolves are updated based on the positions of α, β, and δ wolves. Figure 2 shows how a search agent updates its position according to α, β, and δ wolves. The expressions for updating the locations of α, β, and δ wolves are given as

where D_α, D_β, D_δ denote the distance between the current candidate grey wolf and the optimal 3 wolves; X_α, X_β, X_δ express the positions of the optimal 3 wolves in the current population; A₁, A₂, A₃ and C₁, C₂, C₃ represent the synergy coefficients; X represents the position vectors of other candidate grey wolves, which is defined as

where X₁, X₂, and X₃ indicate the updated positions of α, β, and δ wolves.

Location updading in GWO

Due to the randomness of the initial population, which may impact the diversity of the population and result in low solution accuracy, the study incorporates chaotic mapping instead of random generation to filter the initial population of GWO. Chaotic systems, characterized by periodicity, randomness, and regularity, enhance the diversity and computational efficiency of the initial population (Zhang et al. 2019). This study utilizes iterative chaotic mapping sequences to generate the initial position of the grey wolf in the GWO algorithm. The mathematical equation for iterative chaotic mapping is expressed as

where \(x_{k + 1}\) is the grey wolf’s update position; a represents the control parameters of the algorithm; x_k denotes the initial position of the grey wolf in the current GWO algorithm. Meanwhile, owing to the slow convergence speed of the GWO algorithm, this study introduces the inverse incomplete Г function to update the convergence factor of the GWO algorithm. The expression for updating the convergence factor of the inverse incomplete Г function is formulated as

where a_max and a_min represent the max and the mini value of a, respectively; t and t_max indicate the current and the total number of iterations, respectively; λ is a random variable greater than 0. The parameter values selected for this study are a_max = 2, a_min = 0, λ = 0.01.

2.2 Design of SIMGWO algorithm

The GWO algorithm has the issue of neglecting the selection of poor-positioned grey wolves, making it prone to falling into local optima. This study introduces four operators of the simplex method to handle poorly positioned grey wolves, reducing the possibility of the algorithm falling into local optima (Dong et al. 2023). The fundamental idea of the simplex algorithm is to start from an initial basic feasible solution and iteratively find the best path to reach that solution. The principle of the simplex algorithm and the schematic diagram of the four operators selected for research are depicted in Fig. 3.

Sketch of simplex location principle

Initially, an initial basic feasible solution X₀(x₀, y₀, z₀) is chosen, and maximum error and step size are manually set, as depicted in Fig. 3a. The selection of initial points significantly influences the convergence and localization performance of the simplex method. Subsequently, the chosen basic feasible solution undergoes testing to determine whether the error between the actual read arrival time difference and the arrival time difference calculated via coordinates falls within the pre-defined allowable error range. If within this range, the coordinates of the initial point represent the required source position coordinates. Conversely, if the deviation exceeds the allowable limit, the coordinates of the other three points X₁, X₂, and X₃ are constructed based on the preset step size.

After comparing and analyzing the four indicators, the smallest indicator is the most optimal outcome. This particular point is utilized to reconstruct a simplified shape, and subsequently, an iterative search ensues to discover a newer and improved point. As the search progresses and no superior point is found, the preset step size is reduced, effectively diminishing the simplex size, thus approaching closer to the minimum point. If a point is found within a simple form that meets the predetermined error tolerance, it is identified as the localized source location. Within the GWO algorithm, the new vertex of the underperforming grey wolf position is generated through four operators of the simplex method: reflection (X_r), expansion (X_e), outer contraction (X_t), and inner contraction (X_w). The schematic diagram depicting the new vertex formation by these four operations is illustrated in Fig. 3b. Here, X_g represents the most advantageous point, X_b denotes the secondary advantageous point, X_c stands for the center between X_g and X_b, and X_s refers to the inferior point. The expressions for the four operations are delineated in Eqs. (8)-(11):

where α indicates the reflection coefficient, usually taken as 1, γ represents the expansion coefficient, usually taken as 2, β stands for the outer shrinkage coefficient, usually taken as 0.5, β’ is the internal shrinkage coefficient, usually taken as 0.5. That is the method of generating new vertices through the simplex method, wherein these vertices substitute the poorest vertex to optimize the inferior grey wolf positions within the GWO algorithm. The study investigates the use of iterative chaotic sequences as a replacement for random initial population generation. Additionally, the convergence factor based on the inverse incomplete Г function is employed to more accurately represent the algorithm's search process. Utilizing the simplex algorithm, enhancements are made to elevate the performance of grey wolves in disadvantaged positions.

AE source localization methods are developed based on the time difference method, which achieves source localization by establishing a travel time equation. In a homogeneous isotropic medium, the travel time equation between the AE source and the i-th sensor reads:

where L_i denotes the distance from the AE source to the i-th sensor, x_i, y_i, and z_i represent the coordinates of the i-th sensor, i = 1, 2, 3…, N, x₀, y₀, and z₀ represent the coordinates of the AE source, V_p stands for the P-wave velocity, t_i is the time when the i-th sensor receives the P-wave, and t₀ is the time when the AE source emits the signal. The termination criterion in the acoustic emission localization process aims to minimize the time difference between the calculated arrival time of the acoustic emission signal and the actual signal received by the sensor. For each iteration result, a set of calculation times for the signal to reach the sensor can be obtained. By comparing this set of calculation times with the actual signal time detected by the system, an error value can be determined to assess whether the positioning result meets the accuracy requirements. The absolute deviation method is used to estimate the error, as follows:

where t_ci is the calculated P-wave arrival time of the i-th sensor. A visual representation of the SMIGWO algorithm flow is provided in Fig. 4. The specific procedural steps of the SMIGWO algorithm are as follows: Initially, the algorithm defines the population size, maximum iteration count, reflection coefficient, expansion coefficient, as well as inner and outer contraction coefficients within the GWO algorithm. Additionally, parameters such as 'a', 'A', and 'C' are randomly generated. Subsequently, chaotic iterative mapping is employed to generate the initial grey wolf population, and the convergence factor is updated using the inverse incomplete Г Function. Following this, fitness values for all grey wolf individuals are computed and ranked, with the top 3 best grey wolf positions recorded. Then, utilizing four operations from the simplex method, adjustments are made to the position of the suboptimal grey wolf, alongside updates to other individual positions and related parameters. Finally, the algorithm verifies if the predetermined maximum iteration count has been reached. Upon completion, it outputs the optimal position, thereby concluding the SMIGWO optimization (Aldino and Ulfa 2021; Fang et al. 2022).

SMIGWO algorithm flowchart

3.1 Experimental setup

To assess the efficacy of the AE localization method based on the SMIGWO algorithm, pencil lead break (PLB) tests were conducted on three different materials: aluminum, granite, and sandstone. Additionally, the sandstone specimen was subjected to uniaxial compression testing. The specimens were cubic, measuring 100 mm × 50 mm × 100 mm. Prior to the experiment, each specimen underwent three ultrasonic tests. The results of the P-wave velocity tests are presented in Table 1. The AE monitoring was carried out using the PCI-2 AE system from Physical Acoustics Corp. The preamplifiers were configured to 40 dB. Full-waveform AE data were recorded, employing a threshold value of 45 dB at a sampling rate of 10 MHz. The AE system utilized an array of 8 Nano-30 sensors, and the specific sensor layout coordinates were detailed in Table 2. To attach the sensors to rock faces, gum bands were utilized, and Vaseline was applied between the interfaces of the sandstone and AE sensors to ensure optimal coupling.

3.2 Results of PLB test

A typical PLB event location (25, 25, 100) mm was chosen for testing, and the corresponding localization results are presented in Table 3. The SMIGWO algorithm, theoretically independent of material type, relies solely on the assumption of sample homogeneity and uniform wave velocity. This point is supported by the experimental results, revealing good convergence of the SMIGWO algorithm for PLB events. The location errors for aluminum, granite, and sandstone specimens were measured at 2.03 mm, 3.27 mm, and 3.71 mm, respectively. In comparison, employing the GWO algorithm resulted in larger location errors at 4.72 mm, 5.23 mm, and 5.41 mm for the same specimens. This illustrates the superior solution accuracy of the proposed SMIGWO algorithm over the GWO algorithm.

To compare population initialization methods with different strategies, the experimental results of the PLB test on different materials were examined. The convergence curves under various initialization strategies are presented in Fig. 5. GWO represents the use of random mapping, and SMIGWO represents the utilization of iterative mapping. As illustrated in Fig. 5, both the results of the GWO and SMIGWO algorithms converged. However, there was almost no fluctuation in the results of the SMIGWO algorithm. The curve of the SMIGWO algorithm for the aluminum was the most stable, with the minimum residual value stabilizing at approximately 2.2. The iterative mapping employed in the SMIGWO algorithm resulted in faster convergence, shorter runtimes and lower error, which is beneficial for improving calculation accuracy.

Convergence curves of different algorithms

To further analyze the performance of the proposed algorithm, this study examined the convergence efficiency of both the GWO and SMIGWO algorithms. The convergence efficiency of different algorithms is presented in Fig. 6. Based on the experimental findings, the aluminum PLB result exhibited the fewest convergence steps, reaching a stable state after 5 iterations, whereas the sandstone PLB result had the highest number of stable convergence steps. Notably, the SMIGWO algorithm demonstrated significantly faster convergence steps compared to the GWO algorithm, indicating that the proposed algorithm exhibits higher convergence efficiency.

Analysis of convergence efficiency of different algorithms

The GWO and SMIGWO algorithms were run 30 times to analyze each PLB test data. The statistical analysis of the GWO and SMIGWO algorithms is listed in Table 4. As depicted, the average values of the SMIGWO algorithm consistently surpassed those of the GWO algorithm. This suggests that the SMIGWO algorithm exhibited relatively higher solution accuracy. Additionally, the standard deviation of the SMIGWO algorithm was the smallest among all tests, indicating superior stability for the research algorithm. The experimental results demonstrate that, in comparison to traditional GWO, the proposed algorithm exhibited higher solving accuracy, improved stability, and enhanced optimization performance.

3.3 Results of uniaxial compression test on sandstone

In this section, a uniaxial compression test was conducted on a standard cubic sandstone specimen. Figure 7 presents the experimental results of the sandstone specimen under uniaxial compression. Figure 7a illustrates the rock fracture morphology. The uniaxial compression specimen generated a macro-scale fracture, shearing along a single plane, eventually breaking into two halves. Figure 7b, c depict the spatial distributions of AE events based on the GWO and SMIGWO algorithms, respectively. The circle size and color are proportional to the magnitude of the source energy. The larger the source energy, the bigger the circle size and the darker the circle color. It can be observed that both the AE localization results based on the GWO and SMIGWO algorithms are primarily consistent with the actual rock fracture pattern. The majority of AE events are clustered at the locations where fractures eventually occurred. However, the AE event localization results based on the GWO algorithm are distributed in a wide range around the macro fracture, exhibiting relatively scattered patterns with noticeable errors. In contrast, the AE event localization results based on the SMIGWO algorithm are more accurate and better aligned with the actual crack location, highlighting the higher accuracy of the SMIGWO algorithm in AE localization.

Experimental results of the sandstone specimen under uniaxial compression

The study addressed the issue of the non-uniform distribution of grey wolf positions by replacing random initial population generation with iterative chaotic sequences. Subsequently, the search process of the convergence factor optimization algorithm, based on the inverse incomplete Г function, was fine-tuned to strike a balance between global and local search. By incorporating the simplex method to address grey wolves in disadvantageous positions, the enhanced GWO algorithm exhibited superior performance, resulting in the establishment of an improved GWO algorithm, termed SIMGWO, based on iterative and simplex methods.

Experimental results indicated that the improved SMIGWO algorithm achieved higher average values and enhanced solution accuracy in comparison to the traditional SMIGWO algorithm. Moreover, it demonstrated the smallest standard deviation among all events, indicating superior stability. The SMIGWO algorithm also showcased a faster convergence rate, evident from fewer convergence steps. Notably, its advantages were applicable to various materials such as aluminum, granite, and sandstone, affirming consistent effectiveness across material types. The findings highlighted that, compared to conventional GWO optimization techniques, the algorithm proposed in this study exhibited superior solving accuracy, increased stability, and remarkable optimization performance.