Fractal Model Research Articles

Study on coal pulverization characteristics and gas desorption mechanism based on impact crushing experiment

The coal's particle size distribution properties after pulverization and the gas desorption behavior driven by pulverization are of profound meaning to the study of coal and gas outburst mechanism. In this paper, based on the impact crushing experiment, the tectonic coal and primary coal are crushed under different impact energy conditions. After screening the broken coal, the particle size distribution law is analyzed, and the characterization function suitable for the particle size distribution of coal particles after crushing is determined. The relationship between crushing work and new surface area and fractal dimension of coal body is discussed. The consequences indicated that the mass proportion of tectonic coal below 0.074 mm particle size is much huger than that of raw coal. G-S, R–R, and fractal distribution model describe the best particle size distribution of the two coals in the scope of 0.074∼4 mm. The new surface area added increases with the crushing work, and the tectonic coal is 1.34–1.96 times that of the raw coal. The fractal dimension diminishes first and then increases with the crushing work ratio. In addition, the gas desorption amount of coal particles with different particle sizes after coal pulverization was measured, and a dynamic model suitable for coal pulverization-driven gas desorption was established, and the experimental results were verified. The research results of this paper can provide experimental and theoretical basis for the analysis of energy dissipation in coal and gas outburst.

Dynamic Simulation Analysis of Aviation Gear Transmission Based on Rough Tooth Surface

In order to explore the influence of tooth surface microstructure on the dynamics of aviation gear transmission, this paper describes the profile of aviation gear considering rough surface based on fractal theory. With the help of the established gear fractal contact model, the dynamic parameters such as time-varying friction coefficient, time-varying meshing stiffness and transmission error of rough tooth surface are characterized. The nonlinear dynamic model of aviation gear is established to solve the dynamic response of smooth and rough tooth surface. The dynamic meshing force, dynamic friction force, vibration time domain and frequency domain diagram are obtained, and the corresponding experiments are carried out. The simulation and experimental results show that the fluctuation range of the dynamic simulation results of aviation gear transmission based on rough tooth surface will be larger, and it has a certain degree of disorder, which is closer to the experimental results.

Quantitative Description of Pore and Fracture Distribution Heterogeneity Using Mercury Removal Curve and Applicability of Fractal Models

Many studies have used fractal theory to characterize pore structure distribution heterogeneity through mercury intake curves. However, there is relatively little research on the fractal model calculation of mercury removal curves. In this study, a high-pressure mercury intrusion test is used to describe the pore and fracture distribution heterogeneity (PFDH). The fractal physical meaning of the mercury removal curve was determined by calculating the change in the curve’s fractal dimension value. The results are as follows. (1) According to mercury removal efficiency and porosity, samples can be divided into types A (mercury removal efficiency above 35%) and B (mercury removal efficiency below 35%). In general, type A sample belongs to micro-pore-developed types, and type B samples belong to the macro-pore-developed type. (2) The Menger model (M) represents the complexity of a specific surface area, while the Sierpinski model (S) represents the roughness of the pore volume. Among all the samples, the lower-pore-volume region controls PFDH. (3) According to the calculation results of the single fractal model, it can be seen that the PFDH of type B is stronger than that of type A, which is similar to the results of mercury intrusion. According to the calculation structure of the multifractal model, it can be seen that the volume distribution heterogeneity of type B under various pores is significantly stronger than that of type A. This is opposite to the result of mercury injection. (4) DM has a relationship with the pore volume percentage at different stages, so the M model at the mercury inlet stage can better characterize PFDH at the mercury inlet stage.

МОДЕЛИРОВАНИЕ МИКРОРЕЛЬЕФА ПОВЕРХНОСТИ ПОЧВЕННОГО СЛОЯ ДЛЯ УТОЧНЕННЫХ РАСЧЕТОВ РАБОЧИХ ОРГАНОВ НА ПРОЧНОСТЬ

Issues of increasing the reliability of working parts of agricultural machines interacting with the soil surface are discussed in the article. A loading model based on taking into account microroughness of the surface soil layer using a fractal surface model constructed using the diamondsquare algorithm is proposed. It is shown that the parameters of microroughness of the soil surface, calculated using the proposed model, correspond to experimental data obtained by profiling the soil surface during pre-sowing treatment and during the preparation of flax. A comparative calculation of the loading of the needle harrow tooth and the tiller tooth was carried out in accordance with the model of a flat soil surface and a soil surface with a microrelief, modeled according to the proposed model. It is shown that the mechanical stresses in the working bodies interacting with the soil surface, loaded according to the proposed model, are 14% greater during pre-sowing treatment, and 25% greater when preparing flax trust, than when calculated using a model of a flat soil surface. The use of a microrelief model is optimal both when calculating soil tillage machines and when calculating other flax harvesting machines (pulling machines, flax sliver fluffers, balers and other units) that are not intended for direct tillage of the soil, but have contact with it during the technological process.

A Review of Cross-Scale Theoretical Contact Models for Bolted Joints Interfaces

Bolted joints structures are critical fastening components widely used in mechanical equipment. Under long-term loading conditions, the bolted joints interface generates strong nonlinearities within the system. The nonlinear stiffness inside the bolt leads to changes in the stiffness of the whole system. This affects the dynamic characteristics of the whole system. It brings challenges and difficulties to the performance prediction and reliability assessment of the equipment. A cross-scale theoretical model study based on the microscopic contact mechanism can provide a more comprehensive understanding and cognition of the degradation behavior of bolted joints interfaces. The current development status and deformation process of asperity models are summarized. The research progress of statistical summation model and contact fractal model based on microscopic contact mechanism is analyzed. The experimental methods for parameter identification of connection interfaces are reviewed. The study of numerical modelling of bolted joints structures from the surface contact mechanism is briefly described. Future research directions for cross-scale modelling of bolted joints structures are outlined.

FRACTAL FRACTIONAL MODEL OF RADIATIVE HEAT AND MASS TRANSFER CHARACTERISTICS OF TIME DEPENDENT FLOW WITH VARIABLE VISCOSITY, INDUCED MAGNETIC FIELD, DUFOUR AND SORET EFFECTS: AN ENTROPY GENERATION

: In this article, the Caputo Fabrizio fractal fractional order derivatives operator with an exponential kernel was employed to examine the radiative heat and mass transfer characteristics of time dependent flow with variable fluid viscosity, induced magnetic field, Soret and Dufour effects. The local mathematical model for the flow problem is formulated by take into account the impacts of thermal radiation, heat source, and viscous dissipation. The governing equations in-terms of fractal fractional model with an exponential kernel were solved numerically using finite difference method. The influence of flow variables such as induced magnetic field, concentration field, entropy rate, thermal field, and velocity field profiles against the pertinent parameters are discussed through graphs. Increase the values of magnetic Prandtl number results to rises of induced magnetic field. Higher Dufour number significantly grows the thermal field. The fractal fractional parameters enhance the velocity field, thermal field, Bejan number, entropy rate, concentration field and induced magnetic field profiles. The velocity field profiles recede with higher values of fluid variable viscosity parameter whereas the thermal field and induced magnetic field has an opposite effect. Larger Soret number amplifies the concentration field. Increase of Brinkman number, thermal radiation parameter, and magnetic Prandtl number intensifies the entropy generation rate. Increases of Brinkman number, magnetic Prandtl number, Soret number and Dufour number leads to a decrease of Bejan number whereas Bejan number rises with an increase of thermal radiation parameter and heat source

Scaling theory of fractal complex networks

We show that fractality in complex networks arises from the geometric self-similarity of their built-in hierarchical community-like structure, which is mathematically described by the scale-invariant equation for the masses of the boxes with which we cover the network when determining its box dimension. This approach—grounded in both scaling theory of phase transitions and renormalization group theory—leads to the consistent scaling theory of fractal complex networks, which complements the collection of scaling exponents with several new ones and reveals various relationships between them. We propose the introduction of two classes of exponents: microscopic and macroscopic, characterizing the local structure of fractal complex networks and their global properties, respectively. Interestingly, exponents from both classes are related to each other and only a few of them (three out of seven) are independent, thus bridging the local self-similarity and global scale-invariance in fractal networks. We successfully verify our findings in real networks situated in various fields (information—the World Wide Web, biological—the human brain, and social—scientific collaboration networks) and in several fractal network models.

Fractal evolution model of proton spatial distribution in low water/binder cement-based composites (LW/B-CC) during early-age hydration process

This study establishes the fractal evolution model of proton spatial distribution in low water/binder cement-based composites (LW/B-CC) based on fractal theory and low field nuclear magnetic resonance (LF NMR) technology. The results show that the fractal evolution process of gel water, capillary water and surface water distributions in LW/B-CC presents three stages: the first 4–6 h after the beginning of hydration, 4–6 h to 12 h, and 12 h to long-term stage. In the first and second stages, the fractal dimensions of gel and capillary water increase linearly with hydration time, while the fractal dimension of surface water decreases linearly. The boundary is a sudden change of fractal dimensions after about 4–6 h of hydration. The third stage starts after about 12 h of hydration, at which time the fractal dimensions begin to converge to fixed values. Besides, this study reveals the mathematical relationship between fractal dimension (Df) and particle size distribution modulus (q), which is conducive to achieve the full-cycle regulation of LW/B-CC.

Metallogenic prediction based on fractal theory and machine learning in Duobaoshan Area, Heilongjiang Province

The extraction and integrated analysis of multi-source geological data are key steps in the prediction of mineralization. Current studies are focusing on the extraction and integration of the deep-level mineralization information. In the era of big data, mathematical models and computer algorithms for data mining of multi-source prospecting information have emerged as a leading research area in mineral prediction. In this study, we quantitatively analyzed the structure and remote sensing alteration information using the concentration–area (C–A) fractal model and the box-counting method for the Duobaoshan mineralization area, Heilongjiang Province, China. Results indicate that areas of high fractal dimension of remote sensing alteration correspond to abundant alteration anomalies. Fractal characterization of geological structure is consistent with the spatial distribution. Therefore, fractal characterization provides predictive factors of structure and remote sensing alteration in the development of a predictive model of mineralization. Soil geochemical data were analyzed using the component data analysis (CDA) method and the spectrum–area (S–A) fractal model. The analyses identified anomalous and background signals represented by the PC1 and PC2 principal component combinations. These combinations show a strong correlation between geochemical anomaly data and known deposits in the study area, suggesting that the S–A model effectively identifies geochemical anomalies that can be used as a predictive factor of a mineralization prediction model. The mineralization prediction model was developed using random forest (RF) and support vector machine (SVM) algorithms. The model incorporates predictive factors from multiple sources, including the ore-forming geological background, fractal-characterized geological structure, fractal-characterized remote sensing alteration, and geochemical characteristics. The models incorporated the C–A fractal model to evaluate the probability of mineral prediction. By integrating the characteristics of multi-source mineral prospecting information with the predictive results of machine-learning models, we delineated eight prospective mineralization areas. This approach validates the effectiveness of a combined method involving fractal theory and machine-learning in mineral exploration, offering new insights and theoretical guidance for further mineral prospecting in the study area.

Attenuation analysis in coal based on a fractal viscoelastic model

Attenuation analysis in coal based on a fractal viscoelastic model

Crossover effects and dynamic scaling properties from Eden growth to diffusion-limited aggregation

The diffusion-limited aggregation (DLA) model is a classical fractal aggregation model that describes numerous fractal phenomena, and the Eden model is a typical discrete growth model for investigating cell clusters, such as bacteria or tissue cultures. There is a significant difference between these two discrete models in their growth rules and dynamic scaling properties. In this paper, we introduce a generalized growth theory to describe effectively these two distinct growth classes based on the competitive growth rules. By controlling the adhesion coefficient, a continuous transition exists from dense polymerization to finger growth, i.e., the evolution from Eden growth to DLA in both two-dimensional and three-dimensional cases. To simulate effectively the generalized growth system in the most physically relevant case of three-dimensions, we employ a proper technique named the Marsaglia method for correctly obtaining distributed points to avoid the inappropriate simulation schemes commonly used. Furthermore, the crossover of these competitive growth exhibits non-trivial scaling behavior, and the quantitative relation between the corresponding critical exponents and the adhesion parameter is also investigated. Concurrently, the porosities of the generalized growth system are calculated, which exhibit obviously nontrivial dynamic properties depending on the growth dimensions.

Friedmann equations of the fractal apparent horizon

From a fractal perspective, the entropy bound of gravitational systems undergoes changes. Furthermore, in the cosmological setting, the conservation law of a perfect fluid is also altered in such systems, affecting spatial elements like volume, area, and radius. By applying the first law of thermodynamics and deriving the Friedmann equations, we can gain insight into the evolution of such a fractal cosmos. However, observations continue to necessitate the existence of a dark energy source. To address this, in this article, we have created a novel fractal ΛCDM cosmological model and determined the fractal cosmological observables. We show that the spatial fractal dimension is two, and the age of the Universe is 13.91 Gyr, by fitting the model’s parameters to cosmological data.

Fractal-Based Multi-Criteria Feature Selection to Enhance Predictive Capability of AI-Driven Mineral Prospectivity Mapping

AI-driven mineral prospectivity mapping (MPM) is a valid and increasingly accepted tool for delineating the targets of mineral exploration, but it suffers from noisy and unrepresentative input features. In this study, a set of fractal and multifractal methods, including box-counting calculation, concentration–area fractal modeling, and multifractal analyses, were employed to excavate the underlying nonlinear mineralization-related information from geological features. Based on these methods, multiple feature selection criteria, namely prediction–area plot, K-means clustering, information gain, chi-square, and the Pearson correlation coefficient, were jointly applied to rank the relative importance of ore-related features and their fractal representations, so as to choose the optimal input feature dataset readily used for training predictive AI models. The results indicate that fault density, the multifractal spectrum width (∆α) of the Yanshanian intrusions, information dimension (D1) of magnetic anomalies, correlation dimension (D2) of iron-oxide alteration, and the D2 of argillic alteration serve as the most effective predictor features representative of the corresponding ore-controlling elements. The comparative results of the model assessment suggest that all the AI models trained by the fractal datasets outperform their counterparts trained by raw datasets, demonstrating a significant improvement in the predictive capability of fractal-trained AI models in terms of both classification accuracy and predictive efficiency. A Shapley additive explanation was employed to trace the contributions of these features and to explain the modeling results, which imply that fractal representations provide more discriminative and definitive feature values that enhance the cognitive capability of AI models trained by these data, thereby improving their predictive performance, especially for those indirect predictor features that show subtle correlations with mineralization in the raw dataset. In addition, fractal-trained models can benefit practical mineral exploration by outputting low-risk exploration targets that achieve higher capturing efficiency and by providing new mineralization clues extracted from remote sensing data. This study demonstrates that the fractal representations of geological features filtered by multi-criteria feature selection can provide a feasible and promising means of improving the predictive capability of AI-driven MPM.

Spectral analysis for weighted extended Vicsek polygons

Because of the application of fractal networks and their spectral properties in various fields of science and engineering, they have become a hot topic in network science. Moreover, deterministic weighted graphs are widely used to model complex real-world systems. This paper studys weighted extended Vicsek polygons W(G m,t ), which are based on the Vicsek fractal model and the extended fractal cactus model. The structure of these polygons is controlled by the positive integer coefficient m and the number of iterations t. From the construction of the graph, we derive recursive relations of all eigenvalues and their multiplicities of normalized Laplacian matrices from the two successive generations of the weighted extended Vicsek polygons. Then, we use the spectra of the normalized Laplacian matrices to study Kemeny’s constant, the multiplicative Kirchhoff index, and the number of weighted spanning trees and derive their exact closed-form expressions for the weighted extended Vicsek polygons. The above results help to analyze the topology and dynamic properties of the network model, so it has potential application prospects.

Molybdenum and gold distribution variances within Iranian copper porphyry deposits

This study aims to differentiate between Cu-Mo and Cu-Au porphyry deposits within Iran through fractal modeling, utilizing the Mo/Au geochemical index as a primary determinant. Employing the Concentration-Number (C-N) fractal model, the Mo and Au values within Iranian copper porphyry deposits were comprehensively classified. The findings from the fractal modeling reveal distinct criteria for separating these deposits. Cu-Mo porphyry deposits are identified by Mo concentrations ≥141 ppm, while Cu-Au porphyry deposits are characterized by Au concentrations ≥398 ppb. The pivotal threshold for the Mo/Au ratio stands at 1585, facilitating the division between these distinct types of porphyry deposits. Within the Urumieh-Dokhtar magmatic arc, numerous Cu-Mo porphyry deposits, exemplified by globally recognized sites like Sarcheshmeh and Sungun, are prevalent. Conversely, Cu-Au porphyry deposits are found in the NW and SW regions of the Urumieh-Dokhtar magmatic arc and eastern Iran, typified by occurrences like Dalli. The study's findings reveal a correlation between geological characteristics and the ages of porphyry deposits, indicating that those formed during the Miocene period and later are predominantly of the Cu-Mo (Copper-Molybdenum) porphyry type.

Parametric simulations of fractal-fractional non-linear viscoelastic fluid model with finite difference scheme

Fractal-fractional derivatives are more general than the fractional derivative and classical derivative in terms of order. Fractal-fractional derivative is used in those models where the classical continuum hypothesis theory fails. More precisely, these derivative operators are used where the surface or space is discontinuous, e.g., porous medium. Fractal-fractional derivative is considered advance tool to analyze the fluid dynamic model more than fractional and classical model. Given the extensive applicability of fractal-fractional derivatives, the current analysis focuses on investigating the behavior of a non-linear Walter’s-B fluid model under the influence of time-varying temperature and concentration During the simulation process, we have also taken into account the effects of first-order chemical reactions, Soret numbers, thermal radiation, Joule heating, and viscous dissipation of energy. A magnetic field with a strength of B0 was applied to the left plate in the transverse direction. The classical mathematical model was first developed using relative constitutive equations and later generalized with the fractal-fractional derivative operator. Numerical solutions to the generalized model have been obtained using the finite difference method. Various graphs are drawn from the obtained numerical solutions to study the influence of physical parameters on the rheology of Walter’s-B fluid. It has been observed that by varying the fractional and fractal order of the generalized model, one can easily derive fractal, fractional, and classical models.

Fragmentation energy calculation model of coal crushed by impact load

ABSTRACT Rock fragmentation is a common physical and mechanical phenomenon that exists in a variety of ranges from the mining and processing. Communication is an important operation in coal processing whereby the coal block coming from coalmines is crushed into fragmentations with reduced size, which accounts for 80% of the electricity consumption of mineral processing circuits. Fragmentation energy, which is related to the coal particle size distribution, is an essential parameter for improving the efficiency of shearer, road-header, and crusher. Coal particles are generally simplified into fine or near spherical shapes for analytical and numerical simulation purposes. This research aims to establish the fragmentation energy calculation model for coal fragmentations with coarse and wide size distribution based on the analysis of size distribution and energy-size relationships of coal particles. It has been demonstrated by the results that the fragmentation energy of impact load crushed coal can be determined based on Rittinger’s theory with fractal or RR model particle size distribution.

The Influence of Environmental Temperature on the Passive Oxidation Process in the C/SiC Composite

The C/SiC composite plays a crucial role in providing thermal protection for hypersonic vehicles. The SiO2 oxide layer formed via passive oxidation during ablation constitutes a typical porous medium with self-similarity. Given its significant impact on the thermal protection of the material, accurately predicting the variation in the SiO2 oxide layer thickness is of paramount importance. The growth of the oxide layer impedes the diffusion of oxygen within the material. This study considered microstructural parameters of the oxide layer based on high-temperature gas oxidation tests of the C/SiC composite. Fractal theory was utilized to construct a fractal diffusion-reaction kinetics model describing oxygen diffusion within the oxide layer and the evolution of the oxide layer under varying environmental conditions. The finding demonstrated that the existence of the oxide layer significantly influences the passive oxidation of the composite. This study underscored the significance of predicting the impact of environmental parameters on passive oxidation in the practical application of the C/SiC composite and the study result offers a valuable reference for evaluating the thermal resistance of the C/SiC composite.

Pore structure characterization and effective thermal conductivity modeling of fibrous building thermal insulation materials based on X-ray tomography

The existing structural characterization methods of building materials as well as thermal conductivity calculation models can not be directly applied when considering fibrous building thermal insulation materials with intricate skeletons. In this study, rock wool is considered the research object, and the X-ray tomography method is used to characterize the pore structure and extract structural parameters. Based on the actual pore morphology of the material, the pores in the representative elementary volume (REV) are reasonably assumed to be cubic. Additionally, considering the contact thermal resistance of fibers and the randomness of pore distribution, the curved pattern of heat conduction pathway with the ever-changing diameter is reasonably simplified into a series-parallel form of the curved heat chain with the same diameter. On this basis, the fractal models of effective thermal conductivity are established in horizontal and vertical directions. Finally, a modified formula for the effective thermal conductivity calculation of fiber thermal insulation materials is proposed, taking into account the influence of fiber orientation on the material thermal conductivity. It was shown that the relative deviation between the calculated correction value and the experimental value of the effective thermal conductivity is all less than 16.13%.

Nonempirical fractal permeability model and experimental verification of hydrate-bearing sediments based on heterogeneous distribution of particles

The one of the factors limiting production of gas-water is affected by sediments' permeability. Evaluating the normalized permeability (Kn) of sediments containing hydrates is necessary for analyzing the saturation (decomposition level) and flow capacity. The mistakes in Kn prediction are frequently brought about by the assumption such as homogenous particles distribution and ignoring microscopic pore structure. To address the issue, a fractal-based nonempirical prediction model of Kn was developed by modeling, parameter acquisition using SEM and Analyzer, experiment validation, error comparison with published data, and sensitivity analysis for evolution, respectively. Assuming that sediment skeleton particles were misaligned spherical particles of equal diameter. The heterogeneity of sediments was described by the vertical and horizontal distance ratio (m) and offset angle (θ) of the particles. This led to a novel approach for determining average tortuosity (τav,0) and tortuosity fractal dimension (Dτ,0). The viability of the model was confirmed by permeability measurement experiments. Additionally, using the measuring function of SEM, the maximum pore diameter of sediments was obtained. Furthermore, a comparison with earlier data demonstrated the effect. Considering heterogeneity, the average accuracy of Geometric Mean Variance in Kn has increased to 9.66%. Finally, each parameter's impact on prediction was examined. At m = 1.357, Dτ,0 grows symmetrically. Kn,GC varies 15 times more with an increase in Dτ,0 than Kn,PF. All findings indicate that the new model is more likely to be forecasted Kn for a variety of systems, including clayey-silt, sand, and silty-sand system, and especially in reflecting the heterogeneity.