Distribution Of Points Research Articles

Improvements for the solution of crack evolution using extended finite element method.

It is demonstrated that the eXtended Finite Element Method (XFEM) is of remarkable efficiency in simulating crack evolution by eliminating the need for remeshing and refinement. In this paper, it is shown how to enhance the solution efficiency through a comprehensive mathematical investigation of the solution process using XFEM. A typical example is presented to illustrate the disparities in nodal displacements along the two symmetric faces of the crack resulting from the approximation of XFEM. By analysing the structure and components of the global stiffness matrix, the underlying causes of these discrepancies are identified. Building upon these findings, two improvements of the solution are proposed to gain an acceptable accuracy in computing the nodal displacements. The first improvement consists of the subdivision of the enriched elements depending on the characteristic of the distribution of Gauss points. The second improvement is set by determining the optimal number of Gauss points in each sub-element near the crack tip. To calculate the stress intensity factor of the crack under surface pressure, such improvements are applied in conjunction with the interaction integral method, which significantly reduces computational time and eliminates the influence of surface tractions. The numerical solution is validated by comparing it with the analytical solution and the standard XFEM solution. The proposed improvements can enhance both the accuracy of the solution and the computational efficiency of XFEM.

Temperature Distribution of Frozen Wall Formed by Irregular Hole Arrangement During In Situ Repair of Underground Shield Machine

In order to study the development law of the irregular hole freezing temperature field, combined with the shield solution project of the Nanjing Water Supply Corridor, the distribution characteristics and influencing factors of the irregular freezing temperature field of the river bottom shield machine are studied by numerical simulation. The following conclusions are obtained: (1) The extension length of the outer ring pipe is correlated approximately positively with the thickness and average temperature of the freezing wall at the bottom of the cup. The thickness increases by 0.25 m, and the average temperature decreases by 1.25 °C for every 1 m increase in the extension length. (2) The intersection time decreases logarithmically with the increase in the extension length of the outer ring tube. (3) As the ratio of the axial angle between the two adjacent tubes in the weak area of the outer ring tube becomes larger, the temperature of the weak point in the center of the two tubes increases approximately linearly. The midpoint temperature of the two tubes increases by 3.3 °C for every 1 increase in the angle coefficient. (4) With the increase in the opening angle of the inner ring hole, the thickness and average temperature change, respectively, at 150 d are not more than 0.15 m and 0.6 °C. The results show that under the irregular freezing form, the angle and length of the outer ring pipe have a great influence on the temperature field, and the angle of the inner ring pipe has little influence on the final distribution of the temperature field. The average temperature and the temperature distribution of the weak points show a trend of decreasing first and then increasing along the shield advancing direction, reaching a minimum near the cutterhead.

A Study on the Impact of Overtaking Lane-Changing Behaviour in Expressway Interchange Weaving Areas

This study investigates the overtaking lane-changing (OLC) behaviour in expressway interchange weaving areas, aiming to analyse these behaviours’ causes and potential impacts. Field data are utilised to analyse the statistical characteristics of lane-changing points and spatio-temporal utilisation in weaving areas. A modified NS model, which considers the distribution pattern of vehicle speeds, and a rigid lane-changing rule based on Gaussian distribution are proposed. Additionally, a cellular automaton simulation model is constructed to quantify the influence of OLC behaviour on traffic efficiency and spatio-temporal utilisation based on simulated data. The findings indicate that the imbalanced distribution of lane-changing points and spatio-temporal utilisation in weaving segments, caused by rigid lane-changing behaviour, is an objective factor that triggers OLC behaviour. When the traffic volume in weaving areas ranges from 500 to 1,100 pcu/5 min and the proportion of OLC behaviour is between 0.35 and 0.7, the behaviour will significantly enhance the average vehicle speeds of the outermost lane of the main road and normal rigid lane-changing (NRLC) vehicles, with increases of up to 48% and 51%, respectively. Moreover, OLC behaviour also improves the balance of spatio-temporal utilisation in weaving areas and reduces the average spatio-temporal utilisation. This study clarifies the positive impact of OLC behaviour on expressway interchange weaving areas and provides new research ideas for enhancing the efficiency of these areas.

Analytic mappings of the unit disk which almost preserve hyperbolic area

AbstractIn this paper, we study analytic self‐maps of the unit disk which distort hyperbolic areas of large hyperbolic disks by a bounded amount. We give a number of characterizations involving angular derivatives, Lipschitz extensions, Möbius distortion, the distribution of critical points and Aleksandrov–Clark measures. We also examine the Lyapunov exponents of their Aleksandrov–Clark measures.

Characterisation of the stellar wind in Cyg X-1 via modelling of colour-colour diagrams

Context. Cygnus X-1 (Cyg X-1) is a high-mass X-ray binary where accretion onto the black hole (BH) is mediated by the stellar wind from the blue supergiant companion star HDE 226868. Due to its inclination, the system is a perfect laboratory to study the not yet well-understood stellar wind structure. In fact, depending on the position of the BH along the orbit, X-ray observations can probe different layers of the stellar wind. Deeper wind layers can be investigated at superior conjunction (i.e. null orbital phases). Aims. We aim to characterise the stellar wind in the Cyg X-1/HDE 226868 system, analysing one passage at superior conjunction covered by XMM-Newton during the ‘Cyg X-1 Hard state Observations of a Complete Binary Orbit in X-rays’ (CHOCBOX) campaign. Methods. To analyse the properties of the stellar wind, we computed colour-colour diagrams. Since X-ray absorption is energy-dependent, colour indices provide information on the parameters of the stellar wind, such as the column density, NH, w, and the covering factor, fc. We fitted colour-colour diagrams with models that include both a continuum and a stellar wind component. We used the kernel density estimation method to infer the unknown probability distribution of the data points in the colour-colour diagram, and selected the model corresponding to the highest likelihood. In order to study the temporal evolution of the wind around superior conjunction, we extracted and fitted time-resolved colour-colour diagrams. Results. We found that the model that best describes the shape of the colour-colour diagram of Cyg X-1 at superior conjunction requires the wind to be partially ionised. The shape of the colour-colour diagram strongly varies during the analysed observation, due to concurrent changes of the mean NH, w and the fc of the wind. Our results suggest the existence of a linear scaling between the rapid variability amplitude of NH, w (on timescales between 10 s and 11 ks) and its long-term variations (on timescales > 11 ks). Using the inferred best-fit values, we estimated the stellar mass loss rate to be ∼7 × 10−6 M⊙ yr−1 and the clumps to have a characteristic mass of ∼1017 g.

Investigations on opportunities and challenges of brilliant high-power laser beam welding with 24 kW and adjustable power distribution for different materials

Solid-state laser beam sources offer the possibility of generating high-brilliance laser beams with low expansion and high usable intensity at the focal point. New approaches include beam shaping with the use of core and ring fiber and, therefore, variable power distribution in the laser beam focal point and material interaction area. Particularly, high-power laser beam welding benefits from beam shaping because of the stabilizing effect on the weld pool. Furthermore, the technical progress achieved with regard to beam quality also allows one to achieve high Rayleigh lengths and, therefore, a more uniform beam diameter over the whole material thickness. In this study, investigations on high-power laser beam welding with a 24 kW disk laser beam source are conducted for three different materials (mild steel, aluminum alloy, and copper), which are of high interest for welding in different sectors. The influence of power distribution between the core and the ring as well as welding speed on weld geometry (depth and width), weld pool stability, and the resulting weld seam quality is investigated. It is shown that the welding process cannot just be scaled up in comparison with welding with lower laser beam power but has its own challenges. It is possible that high welding depths (12 mm for copper, more than 12 mm is possible for aluminum, and 25 mm for mild steel) could be achieved in one pass. To achieve this, aluminum needs the lowest energy per unit length per mm of sheet thickness and copper the highest.

High density mapping accuracy and efficiency in atrial fibrillation: real-world outcomes from the SECURE registry

Abstract Background/Introduction A high-density (HD) mapping catheter is valuable for validating pulmonary vein isolation (PVI) and analysing arrhythmic substrates of the left atrium (LA) during atrial fibrillation (AF) ablation. Multiple HD mapping catheters are available for use in clinical practice. A novel octa-spline mapping catheter with a close internal unipolar reference electrode is now available and may improve mapping efficiencies and image resolutions (Figure). Purpose This study aims to report real-world experiences with different commercially available HD mapping catheters in AF ablation and compare mapping performance. Methods SECURE is an ongoing observational, prospective, post-marketing study collecting real-world data on ablation procedures performed per standard-of-care. At the time of data extraction (Nov. 2023), 16 sites across 8 countries had enrolled 2,366 patients (pts), with 12-month follow-up data available for 828 pts. The current analysis, which included consecutive pts undergoing AF ablation, evaluated mapping efficiency and procedural outcomes with 3 HD mapping catheters: circular catheter (CC), penta-spline catheter (PSC), and octa-spline catheter (OSC). Mapping data were analysed with a cloud-based data management and artificial intelligence-powered algorithm. Results Overall, 859 pts with AF (paroxysmal AF [PAF], n=465; persistent AF [PsAF], n=240; long-standing PsAF [LS PsAF], n=154) were included in this analysis. The CC, PSC, and OSC were used in 428, 210, and 221 pts, respectively. Procedures using the OSC had a significantly higher mean mapping area (31,989.9 mm2) than those performed with the CC (23,152.2 mm2; P<0.0001) and PSC (29,804.6 mm2; P=0.0027), as well as a >3-fold higher mean mapping rate (138.4 vs 20.8 and 43.4 points/min, respectively; both P<0.0001). The higher number of mapping points translated to a higher mean mapping density (OSC, 0.34 vs CC, 0.05 and PSC, 0.07 points/mm2; both P<0.0001), with more uniform point distribution and less interpolation. With the CC, PSC, and OSC, respectively, total mean procedure times (incl PVI + other targets) were 118.0, 118.7, and 165.1 min, (P<0.0001 for OSC vs. both CC and PSC) and mean pre-ablation mapping times were 25.2, 25.0, and 21.5 min (P<0.0004 for OSC vs. both CC and PSC). The improved mapping quality and efficiency with the OSC allowed to include specific ablation targets beyond PVI. Fewer PVI-only procedures were performed with the OSC (29.4%) versus the CC (62.1%) and PSC (54.8%), while ablation of the PVI+ other targets was more common with the OSC versus the CC and PSC, including the cavotricuspid isthmus (37.1% vs 17.3% and 16.7%), LA roof (42.5% vs 13.1% and 21.4%), and LA posterior wall (21.7% vs 10.5% and 16.2%). Results were overall consistent per AF type (Table). Conclusions By providing more accurate mapping, the OSC allows for improved analysis of the arrhythmic substrate, which may improve ablation outcomes for pts with PAF and PsAF.Figure 1

Evaluation of Surface Finishing Efficiency of Titanium Alloy Grade 5 (Ti–6Al–4V) After Superfinishing Using Abrasive Films

Ti–6Al–4V is the most commonly used alpha–beta titanium alloy, making it the most prevalent among all titanium alloys. The processed material is widely employed in aerospace, medical, and other industries requiring moderate strength, a good strength-to-weight ratio, and favorable corrosion resistance. A microfinishing process on the titanium alloy surface was conducted using abrasive films with grain sizes of 30, 12, and 9 μm. Superfinishing with abrasive films is a sequential process, where finishing operations are performed with tools of progressively smaller grains. The surface topography measurements of the workpiece were taken after each operation. The experiment was in the direction of developing a new surface smoothness coefficient considering the number and distribution of contact points so as to properly evaluate the quality of the surface finishing. The results showed that the finest-grain films gave the most uniform contact points, thus offering the best tribological characteristics; the 9 LF (micron lapping film) tools gave the smoothest surfaces (Sz = 2 µm), while the biggest-grain films, such as the 30 FF (micron microfinishing film), were less effective since large protrusions formed. This is a suitable study to explore the optimization paths for the superfinishing of titanium alloys, with implications for improving the performance and longevity of components in critical industrial applications.

Riemann–Hilbert problem and soliton solutions with their asymptotic analysis for the focusing nonlocal Hirota equation with step-like initial data

The Riemann–Hilbert (RH) problem is developed to study the Cauchy problem of focusing space symmetric nonlocal Hirota (fsNH) equation with step-like initial data. Firstly, we analyze the characteristics of the Jost solutions and spectral functions, including analyticity, symmetry, and asymptotics as k→∞ and k→0. Then, we discuss three cases of zero point distributions of the spectral functions, and derive the expressions of spectral functions by constructing the special analytic functions and establishing the scalar RH problem. Furthermore, the constraint relations among the zeros of the spectral functions are analyzed. Next, the corresponding RH problem equipped with residual and singularity conditions is successfully derived. These singularity conditions are then transformed into a general residue condition, which reduces the problem of solving the RH problem with reflection-less potential to solving a system of algebraic equations. Ultimately, the corresponding soliton solutions of the fsNH equation and their asymptotics are derived from the solution of the RH problem.

Data-driven effects of human activities and environmental factors on inland aquatic dissolved organic matter in China: Insights from machine learning

Aquatic DOM, a major carbon pool in inland waters, plays a key role in the global carbon cycle and is influenced by human activities and environmental factors. While watershed surveys have identified some drivers of aquatic DOM changes, national-scale knowledge remains limited. Spectral monitoring data from 721 aquatic DOMs in inland China were collected to predict and quantify the spectral properties of aquatic DOMs and the coupled effects of human activities and environmental factors by complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN), random forest (RF) and structural equation modeling (SEM). Our results show that the dataset is relatively evenly distributed in terms of flow regimes, and the data are adequately representative of various types of water in China. Although the performance of CEEMDAN is affected by the skewed distribution of data sampling points (453 and 449 for eastern regions and eutrophic waters, respectively), CEEMDAN effectively processes the spatial sequences of aquatic DOM spectra, which significantly improves the reliability of the machine learning model (by 13%–68%) and reduces the error (by 63%–92%). CEEMDAN-RF and SEM results indicate that longitude and latitude are the most important environmental factors affecting the spectral properties of aquatic DOM through temperature, light and land cover differences, reducing the biogenic properties of aquatic DOM. Unlike observations in small watersheds and estuaries, environmental factors of season, precipitation and salinity have weak effects on aquatic DOM. Furthermore, the biogenic character of aquatic DOM is enhanced by urban human activities, represented by urbanization, population, and impervious land fraction, which have long-lasting and strong impacts on aquatic DOM by driving water eutrophication and enhancing phytoplankton and microbial activity. Our study informs the prediction and quantification of coupled environmental and anthropogenic impacts on aquatic DOM at large scales.

Electric potential energy optimized 3D radial sampling trajectories for MRI

A novel method for creating “golden” 3D center-out radial MRI sampling trajectories was developed and analyzed. This method, called ELECTRO (ELECTRic potential energy Optimized), uses repulsive forces to minimize electric potential energy. An objective function \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\:G\\left(S\\right)$$\\end{document}, the electric potential energies of all subsets of consecutive readouts in a 3D radial trajectory, and its reduced form were minimized using a multi-stage optimization strategy. A metric called normalized mean nearest neighbor angular distance (NMNA) was proposed for describing distributions of points on a sphere. ELECTRO and other relevant golden trajectories were compared in silico using NMNA and point spread function analysis. Consecutive readouts from an ELECTRO trajectory were well spread out, with consistent NMNA values across sphere sizes (σNMNA = 0.005) and between regions on the sphere (NMNA ≈ 1.49). Conversely, the supergolden trajectory had poor consistency in NMNA values (σNMNA = 0.090) and clustering (NMNA = 1.28 at the pole with 40,000 readouts) that lead to artifact in the point spread function. Multi-stage optimization was faster than single-stage and obtained lower values of \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\:G\\left(S\\right)$$\\end{document} (e.g., 0.87 vs. 0.91, for a sphere size of 40). In conclusion, ELECTRO trajectories are more golden than other 3D center-out radial trajectories, making them a suitable candidate for dynamic 3D MR imaging.

The Study on Sentence Difficulty of the Chinese Components

In recent years, reading comprehension has gradually become a proficiency indicator of interest in lexical and grammar. As sentences are the basic units of discourse structure, sentence difficulty is often applied to the study of text difficulty. Although there have been a number of studies on sentence difficulty, the lack of consistency in the indicators chosen or the discussion of specific grammatical issues have limited the research on sentence difficulty. Therefore, this study adopts a corpus-based approach, using a corpus as an objective and scientific data source. The study utilizes the Digital Platform for Chinese Grammar and the 8000 Chinese Words as important reference sources. Additionally, the CRIE 3.0 is employed to validate the texts and establish sentence difficulty indicators. However, due to the incomplete development of certain indicators in the "Chinese Grammar Digital Platform", the study refers to the Chinese Proficiency Grading Standards for International Chinese Language Education and Hanyu Shuiping Kaoshi to establish comprehensive sentence structure and sentence component difficulty indicators. Subsequently, the established difficulty indicators are validated by conducting comparative analyses using corpora as the basis. Native speaker corpora are used as benchmarks, while Mandarin learner corpora are used for comparison, and then validate objectively through the machine learning model. These validation aims to examine the validity and reliability of the selected indicators and establish a calculation method involving " level of grammar * point distribution ratio of grammar " to determine the difficulty indicators for Chinese sentences, Additionally, expert reliability is accessed to ensure the credibility of indicators.

The influence of a coordination training model that uses various manipulative movements on futsal kick accuracy

Achieving precise kick accuracy and coordinated movements in futsal was essential for competitive success. This study aimed to evaluate the differences between various futsal training approaches that incorporated distinct coordination exercises involving manipulative movements, specifically assessing their impact on kick accuracy. A quasi-experimental design with a control group was employed, focusing on shooting skills within the futsal context. The population consisted of 70 members from the Universitas Muhammadiyah Kalimantan Timur (UMKT) futsal team, from which 40 players were purposively selected based on their participation duration of over six months and their tier status within the university's core teams. Participants were divided into two groups: one received static coordination training (control group), and the other underwent dynamic coordination training (experimental group), each comprising 20 players. The intervention spanned 18 sessions, including pre- and post-tests. Data were collected through a modified shooting kick test at the futsal goal, utilizing a point distribution system. Analysis revealed significant differences in kick performance between the two training methods, with dynamic coordination training yielding superior results (t-count = 3.010 t-table = 2.024). These findings indicated that dynamic coordination exercises were more effective than static ones in enhancing futsal kick accuracy. Future research should have considered expanding the population and sample size to include amateur players, introducing additional variables and diverse training methodologies, extending training duration and frequency, conducting long-term evaluations, measuring shooting technique quality, and employing quantitative research methods. By addressing these suggestions, subsequent studies could contribute more comprehensively to the understanding of effective training strategies in futsal.

A mixture of ellipsoidal densities for 3D data modelling

We introduce a novel probability density function for modeling the distribution of points around an ellipsoidal surface. This density is part of the family of elliptical distributions. We establish the theoretical convergence properties of the parameter estimators and validate them using simulated data. Furthermore, we propose a mixture model utilizing this density, and we estimate its parameters using the Expectation-Maximization (EM) algorithm. To assess its performance, we compare the algorithm to a state-of-the-art ellipse fitting method and conduct experiments on 3D real data obtained from depth cameras.

Impacts of land use/cover change on urban rainfall-flooding processes based on numerical simulations

The evolution of surface conditions caused by urbanization has a great impact on urban flooding, whose increasing frequency of occurrence has severely affected urban safety. Most previous studies have focused on land use/cover change (LUCC) but haven’t specifically explored the effects of LUCC on the spatial distribution of flood points and the ponded depth. We combined the storm water management model (SWMM) with multi-year remote sensing data to simulate the flood points and ponded depth in different rainstorm types. Results showed that the ratios of flood points with ponded depths > 0.7 m increased from 2.58% to 17.12%, with the evolution of the urban surface conditions in 1988-2021. And the flood points with ponded depths > 0.7 m were mainly located in the southern and northwestern parts of Bantian Street. These results can provide better-informed decision making for storm water management and early warning of flooding disasters in Shenzhen.

Analysis of the Grid Quantization for the Microwave Radar Coincidence Imaging Based on Basic Correlation Algorithm

In Microwave Radar Coincidence Imaging (MRCI), the imaging region is typically discretized into a fine grid. In other words, it assumes that the equivalent scatterers of the target are precisely located at the centers of these pre-discretized grids. However, this approach usually encounters the off-grid problem, which can significantly degrade the imaging performance. In this paper, to establish a criterion for grid quantization, the performance of the MRCI system related to the grid size and the distribution of imaging points is investigated. First, the discretization of the imaging scene is regarded as a random sampling problem, and the off-grid imaging model for MRCI is established. Then, the probability distribution function (PDF) of the imaging amplitude for a single point target is analyzed, and the mean first-order imaging error (MFE) for multiple point targets is derived based on the Basic Correlation Algorithm (BCA). Finally, the relationship between the grid quantization of the imaging area and the performance of the MRCI system is analyzed, providing a theoretical guidance for enhancing the performance of MRCI. The validity of the analyses is verified through simulation experiments.

Detecting market bubbles: A generalized LPPLS neural network model

To enhance bubble detection capabilities, we introduce two significant improvements to the Log-Periodic Power Law Singularity (LPPLS) model: (1) a novel fitting approach, which yields more accurate predictions of critical price distributions within a single sample window; (2) a restructured neural network approach further enhances the estimations of the probability distributions of the critical points across both time and price dimensions, and it can be fine-tuned with real-world data. The simulation and practical applications to typical asset price bubbles in cryptocurrencies, commodities, and equity indices demonstrate that our refined model, the Generalized-LPPLS Neural Network (G-LPPLS-NN), outperforms all other models we examined in terms of predictive accuracy for critical point distributions.

Point defect distributions in ultrafast laser-induced periodic surface structures on β-Ga2O3

β-Ga2O3 has received widespread attention due to its ultrawide bandgap, which potentially permits applications in extreme conditions. Ultrafast laser irradiation of β-Ga2O3 provides a means for exploring the response of the material under such conditions, which could result in the generation of point defects as well as a localized modification of structural features that could yield properties that differ from the pristine surface. However, an understanding of defects generated by femtosecond laser irradiation in the vicinity of laser-induced periodic surface structures (LIPSS) remains to be explored. We correlate topographic features with optical and electronic properties by combining near-nm scale resolution cathodoluminescence with Kelvin probe force microscopy. Defects are found to correlate with crystalline order and near-surface morphology, as well as changes in work function. They are also suggested to be closely related to the formation of high spatial frequency LIPSS. These results suggest a need for precise tuning of laser irradiation conditions as well as possible post-processing to control defects in future Ga2O3 devices.

Neutrosophic Approach to Water Quality Assessment: A Case Study of Gomati River, the Largest River in Tripura, India

This study addresses the complexity of assessing river water quality, a multifaceted process influenced by numerous water quality parameters (WQPs) characterized by inherent uncertainties and diverse judgment information from decision-makers. These uncertainties and diverse judgment information can be effectively represented and simulated using Neutrosophic sets. In this study, we propose an effective water pollution rating system, the Neutrosophic water quality index (NWQI), to derive a water pollution score (NWQI-score) for rating water pollution levels. We demonstrate the application of our methodology through an assessment of water quality indices for rating pollution in the Gomati River, the largest river in Tripura, India. Using the NWQI, we highlight the NWQI score as a pivotal indicator for evaluating the river's water quality. For assessing water quality, we consider six crucial parameters, namely Total hardness, Biochemical oxygen demand, Total suspended solids, Electrical conductivity, pH, and total dissolved solids, sampled at five strategic sites along the river. Sampling was conducted from January 2024 to July 2024, guided by the distribution of waste discharge points. Finally, we conclude the study, summarizing findings and suggesting future research directions.

Research on the Markov-Chain State Interval Division Based on Predicted Data Correction

By 2022, the total length of roads in Tibet Autonomous Region reached 121,447 kilometers. Due to the unique geological conditions in Tibet, various natural disasters such as earthquakes, mudslides, landslides, avalanches, and strong winds frequently occur. Along the Sichuan-Tibet Highway alone, over 300 disasters happen each year, significantly impacting the region’s economic development. This study focuses on the complexity and randomness of natural disaster mechanisms and combines Markov chain theory to improve the accuracy of prediction data for mudslides, landslides, and earth subsidence etc. The main method is to modify the state interval of the prediction model parameter-Markov chain based on the distribution of discrete points on the number axis. The following state interval division methods are proposed: (1) If the relative error of the predicted value exceeds 50%, adjust the prediction model. (2) Obtain the lower bound of state E1 by taking the floor value downward. (3) The width of each interval does not need to be uniform. (4) Arrange continuous, dense, and close points on the number line in the same side in batches, and represent a state continuously, dividing it into one suitable interval or batches. Using this method, an improved RMSE of 0.28mm and MAPE of 0.87% were obtained for engineering examples, outperforming other models such as GM(1,1), Verhulst, DGM(2,1) with corresponding RMSE values of 0.86 mm, 0.69 mm, and 1.38 mm, and MAPE values of 2.75%, 2.53%, and 5.99%. The combined prediction results for five sets of data yielded an RMSE of 0.14 mm and MAPE of 0.56%, which are quite close to the results obtained using Markov selection correction with an RMSE of 0.37 mm and MAPE of 1.01%. Furthermore, comparing the four sets of case, the average reduction in RMSE and MAPE is 3.56mm and 1.72%, respectively, demonstrating that this method can further improve the performance of Markov chain prediction.