2025-08-15 2025, Volume 23 Issue 4

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  • research-article
    Hehua Zhu
  • research-article
    Raymond L. Sterling

    For thousands of years, humans have used the underground for many purposes and we are now in an era when such uses are becoming more important to support our living patterns, our material needs and to improve the sustainability of our way of life. Many underground facilities serve their intended function well and have proven to have long lifetimes. Some have not been so successful for a variety of reasons or have been retired as no longer meeting the original purpose and not being suitable for conversion to another purpose. While the difference between success and failure is often tied to the specifics of a particular project, this paper seeks to extract some of the general principles that underlie the benefits or drawbacks of different types of underground space uses and how to maximize “success”. The paper is a mixture of the general and the specific because both play a role in success. The paper draws significantly from a recent study of the “lessons learned” from 42 worldwide underground facilities with an average of over 37 years of service mixed with other observations by the author from a career of studying underground space use and underground construction technologies.

  • research-article
    Zixin Zhao, Guolong Zhu, Xingliang Peng, Fawu Wang

    Bimrocks have a block matrix structure that affects their characteristic complex geometric features and widely varying engineering properties. Volumetric block proportion (VBP) significantly affects the strength, extent of the plastic zone, and failure mode of bimrocks. VBP is an essential parameter for evaluating bimrock mechanical properties. However, current unadjusted VBP estimations may have large errors that can exceed 100%. Obtaining more accurate VBP estimates at engineering sites is an ongoing problem that needs more attention. In this study, a tunneling-based bimrock model was constructed. The blocks were ellipsoidal, and their geometric features were designed to be changeable. Based on this model, the influence of bimrock block geometry features on the estimation of the VBP was explored. To reduce the estimation error of the VBP in the tunnel, correction factors corresponding to different geometrical features were proposed. The results of this study can help in estimating VBP more accurately in the field. The findings can also provide useful information for the tunnel construction.

  • research-article
    Takeshi Sato, Fumiharu Nakahara, Kazuo Sakai, Weiren Lin, Kiyoshi Kishida

    Displacement monitoring provides essential information for safe and rational tunnel excavation. The data obtained allow engineers to analyze and predict tunnel behavior, facilitating the selection of appropriate supports and the evaluation of their effectiveness. In a recent tunnelling project in the Akaishi Mountains of central Japan, displacement monitoring was intensively implemented to ensure the stability of the 4.2-km-long Hirogawara adit, excavated to a maximum depth of 832 m. Analysis revealed a strong correlation between initial and final displacements. However, the tunnel experienced occasional support deformations. To address this, the trend of 3D absolute displacements was analyzed to predict and evaluate asymmetric deformation. The effective use of 1-cycle displacement monitoring proved critical for predicting final displacements and optimizing rock supports, particularly in cases with high overburden and limited geotechnical information.

  • research-article
    Hao Zhang, Xueyang Xing, Yiteng Du, Tingchun Li, Jianxin Yu

    Various blasting methods in underground engineering involve rock-breaking processes in enclosed spaces. A whole process of rock blasting is completed by the combination of blasting waves and explosion gas. The two actions exhibit different blasting fracturing characteristics in different time and spatial stages. In this study, an approach of rock blasting simulation of equivalent blasting dynamic-static action is proposed. A set of model blasting experiments under plane strain conditions are carried out to verify from the aspects of feasibility and reliability. The results show that the new method realizes the effective coupling of blast waves and explosion gas in terms of rock-breaking characteristics and pressure wave characteristics. The blasting effects have a high similarity between the simulation result and experimental result, and the maximum error on the damage range and the peak stress is 4.02% and 8.90%. The rock breaking mechanisms of three blasting methods in underground engineering that affect the blasting waves and explosion gas are discussed. The superiority of the new method is evaluated. When the decoupling coefficient is increased, an optimal decoupling coefficient is discovered, which reflects the consistency between the blasting results and the actual situation. When the confining pressure is increased, the inhibition ability on quasi-static action is obviously stronger than that of blasting dynamic action. In slotting blasting, the quasi-static action is the main contributor in the formation of holes penetration. The simulation results identify the rock breaking contributions between blasting waves and explosion gas well. The new simulation method can provide a reliable tool for understanding of the rock-blasting mechanism and restoring the whole blasting process.

  • research-article
    Giulia Guida, Arianna Pucci, Eliano Romani, Giulia M.B. Viggiani, Francesca Casini

    This paper describes a field trial of artificial ground freezing (AGF) carried out in connection with the construction of Line C of Roma underground. AGF was one of the options considered for the temporary stabilisation of the ground during the excavation of Colosseo-Fori Imperiali Station. The field trial aimed at assessing the feasibility of AGF in the complex soil profile and groundwater regime of the subsoil of the historical centre of Roma by establishing the response of the subsoil to the imposed freezing loads, the ability to create a continuous frozen wall, and the associated coolant consumption. The extensive monitoring data were exploited to conduct a detailed analysis of the transient freezing process in the stratified subsoil and used to develop and validate a three-dimensional thermo-hydraulic numerical model. Special attention was given to defining and applying appropriate boundary conditions at the freezing pipes. The paper discusses the main factors affecting the time-dependent freezing process and explores the applicability of simplified two-dimensional models for the Fori Imperiali AGF field trial.

  • research-article
    Yongzhi Zhao, Zhenming Shi, Zhiyong Ai

    This paper presents a solution for the time-dependent behaviors of energy piles embedded in transversely isotropic soils, which considers the mechanical and thermal consolidation. By using the transformed differential quadrature method, kernel functions of coupled thermal-hydro-mechanical solution on the soil-energy pile interface are obtained and the boundary integration is conducted. Then, the energy pile is discretized into finite elements. After introducing the displacement coordination and boundary conditions, matrix equations to reflect the interaction between the surrounding soils and energy piles are formulated and solved. Since the consolidation is considered, the solution for energy pile behaviors with time including displacements and thermal stresses are achieved. Computational results are compared with data of existed literatures and field tests to validate the theory in this study. Finally, numerical examples are conducted to discuss the effects of transverse isotropy of soils, consolidation process and the length-diameter ratio of the energy pile.

  • research-article
    Cheng Chen, Guan-Nian Chen, Song Feng, Xiao-Zhen Fan, Liang-Tong Zhan, Yun-Min Chen

    Monitoring lateral displacement in deep excavation projects is crucial for structural stability and safety. Traditional methods, like manual inclinometers, are accurate but costly and labor-intensive. Automated systems provide real-time data but face challenges with dense sensor placement and high costs. This study presents a novel prediction method using an extreme learning machine (ELM) optimized by an improved particle swarm optimization (IPSO) algorithm. The IPSO-ELM approach utilizes sparse automated measurements to accurately predict lateral displacement profiles, minimizing the need for dense sensor deployment. A case study of a 30.2-m-deep excavation project in Hangzhou, China, demonstrates the method’s effectiveness. The results demonstrate that the IPSO-ELM model maintains high prediction accuracy, with low root mean square error (RMSE) and mean absolute error (MAE) values, even under conditions of sparse sensor placement. Across the entire test dataset, with a sensor spacing of 5.0 m, the model achieved maximum RMSE values ranging from 0.94 to 2.79 mm and maximum MAE values ranging from 0.77 to 2.18 mm, thereby showcasing its robustness and reliability in predicting lateral displacement. A detailed discussion was conducted on the errors associated with various sensor spacing intervals when implementing the proposed method. This study underscores the potential of IPSO-ELM as a cost-effective and reliable tool for automatic monitoring in increasingly complex urban excavation projects.

  • research-article
    Jianghao Ji, Hong Zhu, Zhiqiang Dong, Yijie Pan, Shitong Hou, Liping Cui

    This paper proposes an integrated system for monitoring and strengthening the prestressed concrete cylinder pipe (PCCP) with broken wires, which is based on distributed acoustic sensing (DAS) and self-prestressing iron-based shape memory alloy (Fe-SMA). This system was evaluated in a full-scale study on a PCCP with an inner diameter of 1400 mm and a length of 6000 mm. Firstly, the wire breakage signals were monitored by the DAS system. After that, the PCCP with broken wires were strengthened by Fe-SMA bars, and the mechanical properties were tested. The parameters such as different wire breakage ratios and self-prestressing degrees of Fe-SMA bars were also studied. The results show that the DAS system can identify the time and location of wire breakage; the wire breakage signal is characterized by high amplitude and short duration. After being prestressed with Fe-SMA bars, both the width and length of the main crack, as well as the strains in the concrete, mortar coating, and prestressed steel wires, significantly decreased. Additionally, the higher activation temperature of Fe-SMA bars can effectively offset the negative impact caused by the wire breakage development of PCCP. Combined Fe-SMA with the DAS monitoring system, it enables precise positioning and efficient strengthening of in-service PCCP with broken wires.

  • research-article
    Jiawei Xie, Baolin Chen, Shui-Hua Jiang, Hongyu Guo, Si Xie, Jinsong Huang

    Vast amounts of valuable historical tunnelling site investigation data remain underutilized due to inefficient content-based archiving and searching tools. This study introduces a novel data-driven framework that integrates transfer learning with reverse image search to revolutionize the utilization of historical data in tunnelling projects. The method indexes excavated tunnel sections with corresponding tunnel face images and identifies similarities between projects based on geological features. Transfer learning with pre-trained deep learning models is employed to compress tunnel face images into compact, lower-dimensional vectors, enabling efficient similarity searches. This transformation converts geological information into comparable vectors, enhancing the efficiency and speed of data searches. An online cloud service is developed to allow engineers to access similar historical projects in real-time. To enhance the quality of the compressed vectors, this study developed a multi-level feature extraction method. This method markedly improves the deep learning models’ ability to accurately identify major features from rock images. When applied to a diverse range of tunnel excavation projects in China, the model exhibited an impressive accuracy of over 90% in retrieving projects with similar geological features. This underscores the model’s potential as a robust tool for enhancing data management and decision-making in tunnelling engineering.

  • research-article
    Honggan Yu, Yin Bo, Quansheng Liu, Xuhui Yang, Shuzhan Xu, Xing Huang

    Accurately estimating the monthly advance rate of hard rock tunnel boring machine is of great significance for construction method selection, machine type determination, and project planning. However, current researches mainly focus on estimating the advance rate during construction, and few studies can estimate the advance rate from the entire tunnel scale. To overcome above shortcomings, a monthly advance rate estimation method based on rock mass classification and data augmentation is proposed. Firstly, 56 cases of tunnel boring machine are collected, and proportions of all rock mass grades in basic quality system of the entire tunnel are selected as main inputs of the model. Then, a two-stage data augmentation method based on synthetic minority over-sampling technique and modified auxiliary classifier generative adversarial network is developed. Finally, monthly advance rate estimation models based on machine learning and augmented datasets are established. The results show that the proposed method can accurately estimate the monthly advance rate and the data augmentation method can significantly augment the dataset. The average accuracy of the models is improved by 44.82% after data augmentation. Extreme gradient boosting model performs the best, with an accuracy of 90.31%. Therefore, the proposed method can accurately estimate the monthly advance rate of tunnel boring machine from the tunnel scale and has essential academic and engineering value.

  • research-article
    Xu Zhou, Xiaoling Cao, Ziyu Leng, Chao Zeng, Yanping Yuan, Shady Attia

    In the field of design and application of the energy diaphragm wall (EDW), plenty of research was focused on thermal performances and induced mechanical behaviors. The coupled heat and moisture transfer process and the induced impact on the adjacent underground space were lack of attention, which is inevitable due to the high humidity of the surroundings. Therefore, in this paper, a numerical model taking the gradient of the temperature and relative humidity as the driving potential was established to investigate the characteristics of the coupled heat and moisture transfer in the EDW. Firstly, the behavior of the coupled heat and moisture transfer in the summer and winter was investigated separately, and it was compared with the pure thermal model. Results show that the colder the wall surface, the more humid it is. The heat flux is enlarged by the operation of the EDW. Moreover, the heat flux will be underestimated by more than 3.43% in the heat extraction season and by more than 3.90% in the heat injection case if the moisture transfer is not considered. The following long-running investigations have revealed that the latent flux reaches its maximum and minimum value in transition seasons, with a value that is ten times smaller than that of the sensible heat flux. The sensible heat flux reaches 18.7 W/m2 in summer, while in winter it is −27.4 W/m2. The peak latent heat flux is reduced by 14.7% as a result of the combined effect of changes in surface temperature and humidity, due to the operation of the EDW. Additionally, the magnitude of these fluxes is affected by the indoor conditions (temperature and relative humidity of the indoor air) and the operating temperature of EDW. Therefore, an orthogonal test is performed to evaluate how much the discrepancies are induced by variations in those parameters. The impact of each parameter varies across the seasons (summer, transition season, and winter). However, the indoor relative humidity has a more significant influence on the water vapor flux in all the seasons. This paper provided details about the coupled heat and moisture transfer process in the EDW. Moreover, it attempts to raise an issue about the impact on the hygrothermal load induced by the heat and moisture flux through the wall surface when applying EDW in underground engineering.

  • research-article
    Hongwei Guo, Chao Zhang, Hongyuan Fang, Timon Rabczuk, Xiaoying Zhuang

    Soil liquefaction assessment remains a crucial and complex challenge in seismic geotechnical engineering due to various liquefaction records and limited information, which entails a more generalized off-the-shelf model that can achieve favourable performance on different data sources. In this work, a deep learning model is built and investigated on the soil liquefaction prediction and a modified transfer learning scheme between different data sources is presented. Various datasets, including shear wave velocity-based, CPT-based, SPT-based, and real cases, are collected and utilized to verify the effectiveness and accuracy of the proposed model. Because different data sources in soil liquefaction generally share several geotechnical and mechanical parameters, we work to combine model prior information, feature mapping and data reconstruction in transfer learning models to tackle multi-source domain adaption, which can be further applied to other predictive analysis and facilitate online learning models in geotechnical engineering. Also, the deep learning model is compared with several classical machine learning and ensemble learning models and the modified transfer learning model is formulated by comparing different feature transformation techniques integrated with various feature-based and instance-based transfer learning methods. The accuracy and effectiveness of the deep learning and modified transfer learning models have been validated in the numerical study.

  • research-article
    Zhongxian Liu, Jiaqiao Liu, Haitao Yu, Weiguo He

    This paper introduces a novel two-step multi-scale coupled method for simulating the nonlinear dynamic behavior of a mountain tunnel subjected to fault movement. In the first step, the broadband seismic responses within a large-scale mountain-fault model can be accurately solved by the indirect boundary element method, converting them into effective input forces around the specified region of interest within the mountain. The second step involves finely simulating the nonlinear dynamic response of the tunnel cross-section in the designated region using the finite element method, with the implementation of a viscoelastic artificial boundary to absorb the reflection of scattered waves at truncated boundaries. Two verification processes are employed to validate the accuracy of the multi-scale coupled method. Furthermore, we illustrate the applicability and efficacy of the new method with an example involving the elastoplastic dynamic analysis of a mountain tunnel under the influence of normal fault movement. The presented example highlights the impact of fault motion parameters, including fault dislocation value and dip angle, on the responses of the mountain tunnel. The results demonstrate that the proposed multi-scale coupled method can achieve full-process seismic simulation, ranging from kilometer-scale fault rupture to centimeter-scale mountain tunnel section damage, with a considerably reduced computational expense.

  • research-article
    Zhiwang Lu, Youlin Ye, Pengpeng Ni, Zijie Qian, Ben Niu, Shijian Shang

    Stability of tunnel face is crucial, but previous studies often overlooked the effect of longitudinal tunnel inclination, leading to inaccurate stability assessments. In this study, nine groups of 1g model tests were conducted to study the influence of longitudinal tunnel inclination on passive limit support pressure and passive failure mode of soil in front of the tunnel face under shallow burial conditions (i.e., cover depth ratio of 0.25, 0.50 and 0.75) in a sand stratum. In addition, discrete element method (DEM) analyses at the same scale were established and calibrated against the model test results. Accordingly, the micromechanical information of soil was derived from a microscopic perspective. The results indicate that upon the passive instability of tunnel face, the soil in front of the tunnel face firstly moved approximately perpendicular to the tunnel face, and then it deflected. The instability area of soil in front of the tunnel face increased with the decrease of longitudinal inclination, when the tunnel cover depth was fixed. Furthermore, microscopic analyses indicate that the longitudinal inclination could significantly affect the soil contact orientation in front of the tunnel face. This was more likely to cause the failure zone to rotate.

  • research-article
    Chong Wei, Derek B. Apel, Huawei Xu, Jun Wang, Krzysztof Skrzypkowski

    This study presents uniaxial and triaxial compression tests on large-scale cemented rockfill (CRF) core samples from a Canadian hard-rock mine. Stress-strain curves indicate heterogeneity in strength and deformation properties at various depths. Segregation causes uneven cement and aggregate distribution, affecting uniaxial compressive strength, which decreases with proximity to the discharge point. Findings confirm CRF column strength variability, aiding stability assessment and optimization.

  • research-article
    Mingjun Liu, Jianqin Liu, Wei Guo, Hongxu Liu, Xiao Guo

    To address the research gap in multivariable long-term time series forecasting in the field of tunnel boring machine (TBM) and provide long-term insights for decision-making in TBM construction, this paper studies a novel Transformer-based forecasting model. Leveraging a multi-patch attention mechanism, the newly developed multi-patch attention Transformer (MPAT) model is designed to predict long-term trends of multiple TBM operation parameters. The innovation lies in finding the most relevant time delay series of the input series through autocorrelation calculation, and designing a multi-patch attention mechanism to replace the traditional attention mechanism of Transformer, so that the model can capture local and global information of the series and improve the accuracy of long-term prediction of high-frequency and weakly periodic TBM data. Experimental results have shown that MPAT model has a significant effect on capturing TBM data in terms of temporal dependencies. In a case study, we applied MPAT to the Rongjiang Guanbu Water Diversion Project in Guangdong Province and predicted four excavation parameters. The experimental results show that MPAT exhibits accurate predictive ability when the input length is 36 and the outputs are 12, 24, 48, and 72, respectively. In comparison with some state-of-the-art models, MPAT outperforms MSE by 19.1%, 23.6%, 36.4%, and 48.3%, respectively. We also discussed the impact of input length and the number of patches on performance, and found that each prediction length has the best input length corresponding to it, and longer inputs don’t represent more accurate predictions. The determination of the number of patches should also depend on the input length, as too many or too few patches can affect the capture of local information in the sequence.

  • research-article
    Yichao Rui, Jie Chen, Junsheng Du, Xiang Peng, Zelin Zhou, Chun Zhu

    The layout of a sensor network is a critical determinant of the precision and reliability of microseismic source localization. Addressing the impact of sensor network configuration on positioning accuracy, this paper introduces an innovative approach to sensor network optimization in underground space. It utilizes the Cramér-Rao Lower Bound principle to formulate an optimization function for the sensor network layout, followed by the deployment of an enhanced genetic encoding to solve this function and determine the optimal layout. The efficacy of proposed method is rigorously tested through simulation experiments and pencil-lead break experiments, substantiating its superiority. Its practical utility is further demonstrated through its application in a mining process within underground spaces, where the optimized sensor network solved by the proposed method achieves remarkable localization accuracy of 15 m with an accuracy rate of 4.22% in on-site blasting experiments. Moreover, the study elucidates general principles for sensor network layout that can inform the strategic placement of sensors in standard monitoring systems.

  • research-article
    Hui Li, Weizhong Chen, Xiaoyun Shu, Xianjun Tan, Qun Sui

    The layout of underground engineering objects significantly influences the stability of the surrounding rock mass and construction safety. Despite advancements toward intellectualization and informatization in design optimization and safety assessments, mechanical analysis-based engineering computations still face certain impediments. Consequently, this paper proposes a comprehensive framework integrating tunnel information modelling (TIM), finite element method (FEM) and machine learning (ML) technology to optimize the tunnel longitudinal orientation. It also delves into the specifics of addressing the challenges associated with each technology. The framework encompasses three phases: parametric modelling based on TIM, automatic numerical simulation based on FEM, and intelligent optimization leveraging ML. Initially, geometric models of the geological formations and engineering structures are constructed on the TIM platform. Subsequently, data conversion is facilitated through the proposed transformation interface. Python codes are programmed to realize automatic processing of numerical simulation and results are extracted to the ML algorithm for the prediction model. An optimization algorithm is implanted in the numerical stream file to retrieve the optimal relative intersection angle between the tunnel axis and the trend of rocks. A case study is conducted to evaluate the feasibility of the proposed framework. Results demonstrate a substantial improvement in design and optimization accuracy and efficiency. This framework holds immense potential to propel the intellectualization and informatization of underground engineering.

  • research-article
    Qing Xu, Pengfei Li, Chongbang Xu, Siqing Wang, Sulei Zhang

    Effective control of the tunnel seepage field is crucial in water-abundant regions to ensure the safety and stability of underground structures. Therefore, it is imperative to investigate the effects of the geological factors and tunnel permeability parameters on the drainage capacities of such structures. The Tongzi Tunnel was subjected to model tests using a self-developed testing apparatus to investigate the spatial distribution of tunnel seepage under varying conditions of sand permeability, number of primary support layers, and number of primary support openings. Subsequently, numerical models were developed to validate the observed tunnel seepage field based on experimental conditions. On this basis, an effective water pressure ratio is proposed to evaluate the hydraulic safety of the tunnel spatial distribution. The results indicated a positive correlation between the tunnel water discharge and sand permeability, primary support layers, and primary support openings. Among these factors, the primary support openings exhibited the highest sensitivity to tunnel water discharge, whereas the impact of the primary support layers was relatively low. Additionally, the external water pressure in the tunnel exhibited a negative correlation with sand permeability, primary support layers, and primary support openings. The sensitivity ranking of the structural water pressure fluctuations to the parameters is as follows: primary support openings > sand permeability > primary support layers. Furthermore, the longitudinal water pressure values in the tunnel gradually increase from Section A (circular drainage section) to Section B (middle circular drainage section). Model tests and numerical simulations were performed to demonstrate the data reliability. Finally, with the increase of sand permeability and the number of primary support openings, the effective drainage area (η < 0.6) around the tunnel spatial gradually expands. Besides, the tunnel longitudinal effective drainage interval progressively degrades from the vault (A1 area) to the tunnel bottom (A7 area), and even the tunnel bottom areas are not effectively drained (η > 0.6).

  • research-article
    Chukwuemeka Daniel, Shouye Cheng, Xin Yin, Zakaria Mohamed Barrie, Yucong Pan, Quansheng Liu, Feng Gao, Minsheng Li, Xing Huang

    Rockbursts pose severe risks to underground engineering projects, including mining and tunnelling, where sudden rock failures can lead to substantial infrastructure damage and loss of human lives. An accurate assessment of rockburst damage is essential for safety and effective risk mitigation. This study investigates the effectiveness of ensemble machine learning models optimized through Bayesian optimization (BO) in predicting rockburst damage scales. Nine classifier algorithms, including random forest (RF), were evaluated using a dataset of 254 samples. The research considered factors such as stress conditions, support system capacity, excavation span, geological characteristics, seismic magnitude, peak particle velocity, and rock density as input variables. The rockburst damage scale, categorized into four severity levels based on displaced rock mass, served as the target variable. Among the models evaluated, BO-RF model demonstrated the highest predictive accuracy and generalization capability, achieving 92% testing accuracy. BO-RF model also ranked top in a multi-criteria evaluation framework. This devised ranking system underscores the importance of evaluating model performance on both training and unseen testing data to ensure robust generalization. The findings underscore the effectiveness of BO-RF in enhancing rockburst risk assessment and providing reliable predictive insights for underground engineering applications.

  • research-article
    Youlin Qin, Li Yu, Mingnian Wang, Zhaohui Chen, Hong Jin, Mingyang Yu, Songshen Wang

    Cutter spacing is a key factor influencing the efficiency of TBM operations. Meanwhile, rock brittleness, as a critical indicator of rock fracture, significantly influences fragmentation behavior and rock-breaking efficiency. This study investigates the influence of rock brittleness on rock-breaking through numerical penetration experiments based on the hybrid finite-discrete element method (FDEM) and proposes four intelligent hybrid models to optimize cutter spacing. The results show that as the rock brittleness index (BI) increases from 4.731 to 32.588, the count, depth, width, and proportion of tensile cracks increase, and crack propagation shifts from horizontal to oblique orientations. Moderate cutter spacing (90-110 mm) is optimal for generating tensile cracks. The rock-breaking force increases significantly with higher BI; for instance, at 80 mm spacing, the maximum force for rock with a BI of 13.134 is 5.51 times that for rock with a BI of 4.731. The influence of BI on cutter work and specific energy (SE) is more substantial than the effect of cutter spacing. As BI increases, both cutter work and SE rise considerably. Among the proposed models, the particle swarm optimization and extreme gradient boosting (PSO-XGBoost) model demonstrates the highest performance, achieving an R2 of 0.994, VAF of 99.418%, RMSE of 0.987, and MAPE of 5.217% on the test datasets. An optimization method for cutter spacing is proposed based on this model.