Fractal Dimension Theory Research Articles

Investigation of the Impact of Moisture in Various States on the Adsorption of CH4, N2, and CO2 by Coal.

To investigate the impact of moisture in various states on the adsorption of CH4, N2, and CO2 by coal, this study conducted isothermal adsorption experiments on dry coal for moisture and on coal with varying moisture content for CH4, N2, and CO2. The results revealed that coal moisture adsorption progresses through four stages: monolayer adsorption, multilayer adsorption, cluster formation, and condensation into free moisture filling pores. The presence of moisture in coal varies across these stages, with its impact on CH4, N2, and CO2 adsorption decreasing as moisture content increases. As the moisture content rises, the adsorption constants (a and b values) progressively decline and eventually level off. Based on differences in coal moisture states, delineated by a relative humidity of 0.50 (corresponding to a moisture content of 2.39% in experimental samples), adsorption potential theory identified low moisture content stages where the monolayer and multilayer adsorption potentials for moisture (E1 = 4.99 kJ/mol, E2 = 39.71 kJ/mol) exceeded those for CH4, N2, and CO2, suggesting competitive adsorption effects. In the high-water content stage, the mesoporous structural fractal dimension D2 of the coal samples was found to be 2.77, nearly reaching 3, as determined through liquid nitrogen adsorption experiments and fractal dimension theory. This suggests that the intricate pore structure leads to water condensation, free water formation, and pore blockage, making it the primary factor influencing the adsorption of CH4, N2, and CO2 in coals, which is inherently linked to the characteristics of coal.

Spontaneous combustion coal gangue-based composite cement: Compressive performance and environmental benefits under grinding kinetics control

This study aims to tackle the environmental pressure of spontaneous combustion coal gangue by using its active silica-alumina content to create eco-friendly composite cement through a low-energy process. The Divas-Aliavden grinding kinetics equation was established, and powder characteristics were analyzed using particle size distribution, the Rosin-Rammler-Bennett model, and fractal dimension theory. The compressive performance of the coal gangue-based composite cement was tested. A comprehensive evaluation model linked grinding time, powder characteristics, compressive performance, and energy consumption, while also analyzing hydration products and environmental impact. Results showed that initial grinding stages had larger particle sizes and higher energy utilization efficiency. As grinding progressed, energy consumption increased. A grinding time of 35 min achieved optimal balance between energy efficiency and compressive performance, improving particle distribution uniformity and complexity. The compressive performance of the composite cement reached 95 % of ordinary Portland cement after 28 days of curing. The evaluation model indicated that specific surface area, d50, and fractal dimension are key factors influencing energy efficiency, determining energy consumption and particle distribution during grinding. Environmental assessment revealed that substituting 30 % of cement with coal gangue could reduce CO2 emissions by 250 kg CO2/t, achieving a reduction rate of 29.5 %. This study supports the large-scale application of spontaneous combustion coal gangue, contributing to efficient resource utilization and environmental protection.

Particle crushing mechanism of recycled aggregate with different component categories: applied to pervious concrete base

This paper focuses on the particle crushing mechanism of construction solid waste recycled aggregate (CWRA), particularly recycled concrete (RC) and recycled bricks (RB), using natural gravel (NG) as a control group. Compression tests, combined with the Weibull distribution, fractal dimension theory, and relative crushing rate, were conducted to investigate the crushing behavior of these materials under different loads. Results indicate that, under the same load, the crushing degree ranks from highest to lowest as RB, RC, and NG. The Weibull distribution effectively describes the crushing state, with “fragmentation” being the primary mode for most materials, while RC also exhibits “grinding.” As the load increases, the relative crushing rate rises for all materials but slows beyond a certain load threshold, indicating greater difficulty in further crushing. The study identifies a load threshold of 400 kN, beyond which changes in void fraction and relative crushing rates become less significant, particularly in stronger aggregates. Highlights Identified that mortar attached to the RC surface is the primary cause of the increased particle crushing below 0.6 mm. Introduced Weibull distribution parameters a and b to accurately characterize the crushing state of RA particles. Revealed the errors between fractal dimension-based evaluations and the actual crushing state of RC particles. Proposed a two-stage crushing mode for RC. Determined the critical threshold value for the variation of crushing degree with load level.

Experimental Determination and Simulation Validation: Johnson–Cook Model Parameters and Grinding Simulation of 06Cr18Ni11Ti Stainless Steel Welds

Hydrogen permeation resistance in the welded region of 06Cr18Ni11Ti steel is relatively weak due to surface defects, which need high integrity surface machining. The parameters of the welding material for 06Cr18Ni11Ti steel are currently unavailable, which causes some inconvenience for simulation studies. To fill the lack of 06Cr18Ni11Ti steel weld material parameters in the relevant literature at the present stage, the quasi-static tensile test at different strain rates and notch specimen tensile tests were conducted in this paper and determined the Johnson–Cook (J-C) constitutive model parameters and Johnson–Cook failure model parameters. Subsequently, a multi-grain grinding simulation model was built based on W-M fractal dimension theory by using the determined material parameters. The influence of processing parameters on grinding heat was analyzed. Grinding experiments were conducted to analyze the influence of processing parameters on grinding heat and grinding force. By comparing the simulation and experimental results, it is revealed that the average error is 9.37%, indicating relatively small discrepancy. It is demonstrated that the grinding simulation model built in this paper could efficiently simulate the grinding process, and the determined weld material parameters of 06Cr18Ni11Ti steel have been verified to possess high accuracy and reliability.

Sustainable Transportation: Exploring the Node Importance Evolution of Rail Transit Networks during Peak Hours

The scientific and rational assessment of the evolution of node importance in rail transit line networks is important for the sustainability of transportation systems. Based on the complex network theory, this study develops a weighted network model using the Space L method. It first considers the network topology, the mutual influence of neighboring nodes of the transportation system, and the land use intensity in the station influence domain to construct a comprehensive index evaluation system of node importance. It then uses the covariance-weighted principal component analysis algorithm to more comprehensively evaluate the node importance evolution mechanism and analyzes the similarity and difference of the sorting set by adopting three different methods. The interaction mechanism between the distribution of important nodes and the evolution of land use intensity is explored in detail based on the fractal dimension theory. The Xi’an rail transit network is considered an example of qualitative and quantitative analysis. The obtained results show that the importance of nodes varies at different times of the day and the complexity of the morning peak is more prominent. Over time, articulated fragments with significance values greater than 0.5 are formed around the station, which are aligned with the direction of urban development, creating a sustainable mechanism of interaction. As the network’s crucial nodes in the center of gravity increase and the southern network expands, along with the increased intensity of the city’s land utilization, the degree of alignment in evolution becomes increasingly substantial. Different strategies for attaching the network, organized based on the size of Si can lead to the rapid damage of the network (reducing it to 0.2). The identification of crucial nodes highlighted in this paper serves as an effective representation of the functional characteristics of the nodes in transportation networks. The results obtained can provide a reference for the operation and management of metro systems and further promote the sustainable development of transportation networks.

Fractal Dimension (Df) Theory of Ismail’s Entropy (IE) with Potential Df Applications to Structural Engineering

As an ultimate generalisation to several kinds of generalised entropy in the literature, a novel entropy measure, namely, Ismail’s entropy, or (IE), is presented. This article spotlights the significance of fractal dimension, through highlighting several possible applications of fractal dimension to structural engineering. In addition to several difficult open problems and the next step of inquiry, the paper ends with some concluding observations

Fractal Dimension(Df )Theory of Ismail's Entropy(IE) with Potential Df Applications to Smart Cities

Ismail entropy (IE), my newly developed entropy measure, is a final generalisation to many generalised entropies found in the literature. This research identifies the fractal dimension of IE. Additionally, this study presents some possible uses of fractal dimension in smart city applications. Concluding thoughts are included at the end of the work along with several difficult outstanding problems and the direction of future research

High sedimentation efficiency and enhanced rare earth recovery in the impurity removal process of rare earth leachate by flocculation system

The main impurity ion in rare earth (RE) leachate are aluminum ions, which can be effectively removed by precipitation in the form of aluminum hydroxides. The flocculent colloidal precipitation formed by aluminum ions has a loose structure and high liquid content, resulting in lower sedimentation efficiency and higher RE loss. The addition of flocculants in the impurity removal process of RE leachate is expected to solve this problem. The cationic polyacrylamide flocculant (CPAM) was screened as the flocculant to accelerate the sedimentation of the impurity removal process as well as to effectively improve the recovery of RE. The effects of CPAM concentration, dosage, stirring rate and stirring time on the flocculation process was investigated. The optimal flocculation effect could be obtained under conditions of dosage of 0.5 mL, concentration of 0.1%, stirring speed of 700 r/min and stirring time of 2 min. Under this condition, RE recovery efficiency was 97%, Al removal efficiency was 96%, and the sedimentation time of the flocs was 20 min. Compared with absence of flocculants in this process, Al removal efficiency remained almost unchanged, while RE recovery efficiency and the sedimentation time of flocs increased by 4% and shortened by 40 min, respectively. The flocculation mechanism was discussed by the theory of two-dimensional fractal dimension, and it was found that the flocculation was mainly based on the combined effect of bridge flocculation and electrical neutralization.

Effects of rock water with different acidity on microstructure and reburning ability of residual coal in goaf

To explore the influences of rock water acidity (PH = 4,5,7) over the microstructure and reburning behaviors of oxidized residual coal in the overlying goaf, the pore morphology, coal surface, functional group content and activity of pre-oxidized coal (O0) and water-immersed oxidized coal (OI) were analyzed by Low-temperature nitrogen adsorption and In-situ diffuse reflection experiments, combining with fractal dimension theory and functional group oxidation kinetics. The oxidation and reburning ability of coal samples has been analyzed by Thermogravimetric-Differential thermal analysis experiment. The results are as follows: Part of the type-III pores of pre-oxidized coal become type-II pores when immersed. The acidic solution has stronger corrosivity to O0 than distilled water, playing a role in smoothing coal and improving the uniformity of pore structure. The order of reburning ability of samples is OI7>OI4>O0>OI5 according to the ignition and stage average mass loss rate (RM). The reactivity of –OH may affect the reburning ability of coal more than its initial content. Soaking is beneficial to the decrease of activation energy of aliphatic groups, and the activity of –C=O- and -C-O- in acid-soaked coal molecules is stronger, so that they can take part in coal-oxygen composite reactions at low temperature.

Identification of 4FGL Uncertain Sources at Higher Resolutions with Inverse Discrete Wavelet Transform

In the forthcoming era of big astronomical data, it is a burden to find target sources from ground-based and space-based telescopes. Although machine-learning methods have been extensively utilized to address this issue, the incorporation of in-depth data analysis can significantly enhance the efficiency of identifying target sources when dealing with massive volumes of astronomical data. In this work, we focused on the task of finding active galactic nucleus (AGN) candidates and identifying BL Lacertae objects (BL Lac) or flat spectrum radio quasar (FSRQ) candidates from the 4FGL_DR3 uncertain sources. We studied the correlations among the attributes of the 4FGL_DR3 catalog and proposed a novel method, named fractal dimension–inverse discrete wavelet transform (FDIDWT), to transform the original data. The transformed data set is characterized as low-dimensional and feature-highlighted, with the estimation of correlation features by fractal dimension theory and the multi-resolution analysis by inverse discrete wavelet transform (IDWT). Combining the FDIDWT method with an improved lightweight MatchboxConv1D model, we accomplished two missions: (1) to distinguish the AGNs from others (non-AGNs) in the 4FGL_DR3 uncertain sources with an accuracy of 96.65% ± 1.32%, namely Mission A; and (2) to classify blazar candidates of uncertain type into BL Lacs or FSRQs with an accuracy of 92.03% ± 2.2%, namely Mission B. There are 1354 AGN candidates in Mission A, and 482 BL Lacs candidates and 128 FSRQ candidates were found in Mission B. The results show a high consistency of greater than 98% with the results in previous works. In addition, our method has the advantage of finding less variable and relatively faint sources than ordinary methods.

Fractal Few-Shot Learning.

Forming deep feature embeddings is an effective method for few-shot learning (FSL). However, in the case of insufficient samples, overcoming the task complexity while improving the accuracy is still a major challenge. To address this problem, this article considers the consistency between similar data from the fractal perspective, introduces a priori knowledge, and proposes a fractal embedding model by combining FSL with fractal dimension theory for the first time. We improve the original fractal dimension algorithm used to describe image texture roughness to suit a neural network. Moreover, in accordance with the improved algorithm, prior knowledge of the quantized image is integrated into the features to reduce the impact of the data distribution on the model. Experimental results obtained on multiple image benchmark datasets show that the performance of the proposed model exceeds or matches that of previous state-of-the-art models. In addition, the proposed model achieves the best performance in cross-domain scenarios, further illustrating its robustness.

Macro-micro crack and damage evolution characteristics of concrete: After the action of acidic drying-saturation cycle

Repeated saturation and drying of acidic mine water have a major influence on the mechanical properties and stability monitoring of artificial dams (concrete walls) in underground reservoirs. A uniaxial compression acoustic emission (AE) monitoring test was conducted to quantify the evolution characteristics of micro cracks in concrete. In addition, the evolution characteristics of macro cracks during sample failure were quantitatively analyzed by fractal dimension theory and Image-Pro. Finally, the damage constitutive model was established and the damage evolution law was studied. The results of the study showed that the percentage of micro shear cracks in dry samples exhibited a decreasing and then increasing trend during the loading stage before the uniaxial compressive strength (UCS). However, after the acidic drying-saturation cycle (ADSC), this percentage of concrete showed an increasing trend. As the number of drying-saturation cycles increases or the pH decreases, the fractal dimension, total length, and area of the macro cracks also increase. The damage constitutive model, based on ADSC and loading conditions, works well. After the ADSC, the concrete sample will have three damage stages. Additionally, the strain at entry into each stage increases. Expansion and shrinkage of hydration products and chemical damage due to ion exchange are the mechanisms that change the mechanical properties and the macro-micro crack evolution pattern of concrete after ADSC.

Обоснование фрактального абсолютного объёма компонентами дробной размерности

For materials science, an urgent task is the task of obtaining composite materials, lightweight and other concretes with the required complex of physical and technical properties. The solution of this problem is associated with the study of the processes of self-organization, stability and decay of various nonequilibrium systems, which include composite materials and concretes in the form of capillary-pore structures. This possibility seems promising if the analysis and quantitative descriptions of dissipative systems and structures of concrete are carried out taking into account the theory of fractals and fractal dimension. The paper investigates the structure of foam concrete with different magnification and scaling down multiplicities. The position is substantiated that the structure of foam concrete consists of pores similar in shape but different in size. Therefore, such a structure is self-similar at various scale levels. It can be assumed that the foam concrete material is formed from pores, i.e. thin walls-bridges, tightly packed.

Vertical vs. Horizontal Fractal Dimensions of Roads in Relation to Relief Characteristics

This paper investigated the surface length of roads from both horizontal and vertical perspectives using the theory of fractal dimension of surfaces and curves. Three progressive experiments were conducted. The first demonstrated the magnitude of the differences between the planar road length and the DTM-derived surface road length and assessed its correlation with the DTM-calculated road slope. The second investigated the road distance complexity through the fractal dimension in both planar and vertical dimensions. The third related the vertical with the horizontal fractal dimension of roads across a range of distinct physiographic regions. The study contributed theoretically by linking the planimetric complexity to vertical complexity, with clear applications for advanced transportation studies and network analyses. The core methodology used geographic information systems (GIS) to integrate a high resolution (1 × 1 m) digital terrain model (DTM) with a road network layer. A novel concept, the vertical fractal dimension of roads was introduced. Both the vertical and horizontal fractal dimensions of the roads were calculated using the box-counting methodology. We conducted an investigation into the relationship between the two fractal dimensions using fourteen study areas within four distinct physiographic regions across Slovenia. We found that the average slope of a three-dimensional (3D) road was directly related to the length difference between 3D and two-dimensional (2D) roads. The calculated values for the vertical fractal dimension in the study areas were only slightly above 1, while the maximum horizontal fractal dimension of 1.1837 reflected the more sinuous properties of the road in plan. Variations in the vertical and horizontal fractal dimensions of the roads varied between the different physiographic regions.

A Study on the Pore Structure and NMR Fractal Characteristics of Continental Shale in the Funing Formation of the Gaoyou Sag, Subei Basin

The continental shale oil resource in China exhibits significant potential and serves as a crucial strategic alternative to the country’s conventional oil and gas reserves. The efficacy of shale oil exploration and production is heavily contingent upon the heterogeneity of the pore structure within the reservoir. However, there remains a scarcity of research pertaining to the pore structure of continental shale and the factors that influence it. The objective of this study is to provide a quantitative characterization of the heterogeneity exhibited by the continental shale of the Funing Formation in the Gaoyou Sag. In this study, the research focus is directed toward the continental shale of the Funing Formation located in the Gaoyou Sag of the Subei Basin. This paper examines the correlation between the fractal dimension of nuclear magnetic resonance (NMR) and various factors including the total organic carbon (TOC), mineral composition, geochemical parameters, and physical properties, utilizing the principles of fractal dimension theory. The findings indicate that the primary pore types observed in the Funing Formation continental shale are inorganic matrix pores, which encompass dissolution pores, clay mineral intergranular pores, and a limited number of pyrite intergranular pores. By employing a relaxation time cutoff, the NMR fractal dimension can be categorized into two distinct dimensions: the bound-fluid-pore fractal dimension (0.5795~1.3813) and the movable-fluid-pore fractal dimension (2.9592~2.9793). The correlation between mineral composition and the fractal dimension indicates a negative relationship between the fractal dimensions of bound-fluid pores and movable-fluid pores and the content of quartz. The correlation between clay minerals and the fractal dimension indicates a significant negative relationship between the fractal dimensions of bound-fluid pores and movable-fluid pores with illite. There exists a negative correlation between the pore fractal dimension of bound fluid and the content of organic matter, whereas a positive correlation is observed between the pore fractal dimension of mobile fluid and the content of organic matter. The range of maturity of organic matter within the Funing Formation exhibits a relatively limited span, as indicated by the vitrinite reflectance (Ro) values falling between 0.8% and 0.9%. This narrow range of maturity does not exert a substantial influence on the two fractal dimensions. The NMR fractal dimension exhibits a negative correlation with permeability in relation to reservoir physical properties, while the bound-fluid-pore fractal dimension demonstrates a negative correlation with the total porosity. The findings suggest that the NMR fractal dimension can serve as a valuable indicator for evaluating the physical characteristics of reservoirs. The present study successfully examined the pore structure of continental shale through the utilization of nuclear magnetic resonance technology. This innovative technique provides a novel avenue for the assessment of continental shale reservoirs and the investigation of pore heterogeneity on a global scale.

Microstructure response to shear strength deterioration in loess after freeze-thaw cycles

Shear strength deterioration owning to freeze-thaw (F-T) cycles affects loess slope stability in seasonal frozen region. The change in loess shear strength is closely related to particle microstructure. In this study, multiple F-t-tests (i.e., 0, 1, 3, 5, 10, 15, 20, and 30 cycles) were performed on natural loess from Jing'yang, China. Unconsolidated-undrained triaxial tests and scanning electron microscopy (SEM) were performed on the specimens after the predefined number of F-T cycles. Particle microstructure was quantitatively analyzed using fractal dimension theory. The relationship between shear strength index and particle microstructure characteristics parameters was evaluated using the grey correlation method. Changes in the loess microstructure involved two main processes: transformation and stabilization, and the particle shape, arrangement, and interparticle contact were transformed into different types. The particle shape changed from complex to round with debris detaching from the particle surface. The orderliness of particle arrangement increased and point-face intercontact was formed. The internal friction angle and cohesion decreased with an increasing number of F-T cycles, and the change pattern was similar to that of the particle microstructure characteristics, both of which changed significantly before 5 F-T cycles and stabilized after 15 F-T cycles. An evaluation of the correlation between shear strength index and particle microstructure characteristic parameters demonstrated that, the change in particle shape contributed the most to deterioration of shear strength during F-T cycles, followed by particle arrangement and particle distribution. These results provide insights into the mechanism of degradation of loess shear strength in seasonally frozen areas at the microstructural scale.

Experimental evaluation of the frost resistance durability of RCC layer subjected to early-age freezing

Freeze–thaw cycle, shear and nuclear magnetic resonance tests were conducted to investigate the effect of early-age freezing on the frost resistance durability of roller compacted concrete (RCC) layers during their service period. The effects of interval time and pre-curing time on the frost durability of early-age frozen RCCs were discussed. The appearance damage, shear failure mode, shear strength and pore structure of the early-age frozen RCC layers after freeze–thaw cycles were evaluated. Results showed that the appearance damage of the early-age frozen RCC layers after the freeze–thaw cycles began with microcracks at the layer joints, which then extended to the upper and lower layers. The shear failure mode was dominated by slip failure. Early-age freezing caused initial damage to the layer and increased the rate of loss of peak shear strength and the volume of harmful pores during the freeze–thaw cycles. The degree of shear strength loss due to interval time was in a descending order as follows: 24, 12, 4 and 0 h. The loss in shear strength was the largest in the specimens pre-cured for 12 and 24 h, followed by those pre-cured for 48 and 72 h. The loss in shear strength was the smallest for the specimens pre-cured for 7 days, similar to the result for the unfrozen control group. In addition, the corresponding relationship between shear strength and pore structure at the layer of the early-age frozen RCCs during the freeze–thaw cycles was quantitatively described based on fractal dimension theory. A freeze–thaw damage model was developed to consider the effects of early-age freezing on the surface of RCC, and the calculated results of the model were in good agreement with the test results.

Tensile failure analysis of laser stake-welded T-joints: X-ray tomography based conformal finite element and fractal theoretical approaches

To promote the application of complex structural materials in large size, the interfacial joint is the key component for sandwich structure to wipe off the sample size limitation and fabricate multi-material structures. However, the laser welding process inevitably induced void defects in the welded joint, influencing the mechanical properties of the whole structure. In this paper, the aluminum alloy laser stake-welded T-joints were fabricated in a laser welding system. The geometric deviations and internal void defects of T-joints were visualized and quantified by micro X-ray computed tomography (μCT). Two image-based finite element models were reconstructed from CT images to discuss the effect of different manufacturing-induced defects on the tensile behaviour of T-joints. The geometric deviations of T-joints make a response for the decrease of strength and the void defects result in the decrease of Young’s modulus of T-joints. The fractal dimension theory, based on the value of void volume and the distance of the void to the interface obtained from CT images, was developed to predict the strength of the T-joint. The results matched well with the experimental results.

High-Temperature-Induced Pore System Evolution of Immature Shale with Different Total Organic Carbon Contents.

The pyrolysis process of source rock, especially organic-rich immature shale, is required for oil and gas extraction, during which the evolution of the pore structure system in the immature shale determines the heat conduction and fluid flow under the heating treatment. Although some sound achievements have been made regarding the pyrolysis of immature shale, the effect of the total organic carbon (TOC) content on the pore structure evolution of immature shale remains unclear. With respect to this issue, in this work, a series of N2 adsorption/desorption and nuclear magnetic resonance (NMR) experiments were conducted, and fractal dimension theory was also introduced to analyze the pore structure evolution of immature shale subjected to heating treatment in a quantitative manner. The results indicate that the adsorption branch of the nitrogen adsorption-desorption isotherm can be divided into three stages. The pore structure of different TOC immature shales does not change significantly, and they are all slit-shaped. In addition, immature shale with a higher organic content has a higher hydrocarbon expulsion strength and a higher pore volume growth rate, which indicate that the pyrolysis of organic matter greatly affects the pore structure of immature shale during heating. This phenomenon shows that the pyrolysis of organic matter greatly influences the pore structure of immature shale during the heating process. The pores of immature shale in the study area have significant fractal characteristics, the fractal dimension is between 2.397 and 2.636, the pore space of the sample is extremely small, the pore structure is extremely complex, and the heterogeneity is strong.

Sound quality prediction of unsteady vehicle interior sound

In this paper, the noise samples of pure electric vehicles under unsteady-state are collected. Then, the subjective evaluation test is carried out after pre-processing such as screening and intercepting. The subjective sound quality scores of the noise samples are obtained. Meanwhile, the noise samples are calculated of four conventional psychoacoustic objective parameters such as loudness, sharpness, roughness, and speech intelligibility. Preprocessing by ensemble empirical mode decomposition (EEMD) is performed. Characteristic parameters of noise samples such as time-frequency domain fractal dimension difference and sample entropy of noise samples are obtained based on fractal dimension and sample entropy theory. Finally, the quality prediction model of vehicle interior sound is established based on six characteristic parameters and BP neural network. The results show that the prediction effect is excellent for the subjective scores of sound quality of unsteady interior sound in the vehicle.