Fractal Model Research Articles

Fractal Calculus Facilitates Rethinking ‘Hard Problems’: A New Research Paradigm

This paper introduces a non-standard research technique to clarify how complex phenomena, such as those that are abundantly present in human physiology, can be faithfully described using fractal dynamical models with and without stochastic forces. This method for conducting research involves tracing the historical evolution of understanding an empirical medical process facilitated by the fractal-order calculus perspective. Herein, we trace the analysis of the time series for heart rate variability (HRV) developed for diagnosing the cardiovascular health of a patient. This is performed herein by introducing four (one empirical, which entails three theoretical fractal models) distinct but related fractal models, each one introduced to solve a particular problem arising from a fundamental defect in the previous model, but in generalizing a model at one stage to resolve the problem associated with the defect, another is invariably introduced by the replacement model. It is through the utilization of the fractal-order calculus that the necessity for rethinking how to systematically incorporate additional layers of complexity is revealed, ultimately resulting in a ‘complete’ description of its empirical dynamics in fractal terms.

Fractal Strategy for Improving Characterization of N2 Adsorption–Desorption in Mesopores

The current studies primarily analyze the heterogeneity and complexity of mesopore structures based on low-temperature nitrogen (N2) adsorption curves and the Frenkel–Halsey–Hill (FHH) fractal model. However, these studies ignore the fact that the low-temperature N2 desorption curve can also reflect the desorption performance of the mesopore structure. In this research, novel fractal indicators for characterizing the adsorption–desorption performance of mesopores based on the fractal dimension from the N2 adsorption curves and N2 desorption curves are proposed. The novel fractal indicators I1 and I2 are applied to evaluate the adsorption–desorption performance of mesopores with pore size 2–5 nm and pore size 5–50 nm, respectively. The fractal indicator I1 shows an increasing trend with coalification, reflecting that the gas adsorption performance of 2–5 nm mesopores is enhanced with coalification. The fractal indicator I2 exhibits a trend of first increasing and then decreasing with coalification, indicating the gas desorption performance of mesopores with pore size 5–50 nm decreases first and then increases. The proposed indicators provide novel analytical parameters for further understanding the gas adsorption–desorption mechanism of porous coal-based or carbon-based materials.

Fractal Feature Modeling: Multi-Dimensional Attention and Spatial Adaptive Relationship Learning for Person Re-Identification

Fractal Feature Modeling: Multi-Dimensional Attention and Spatial Adaptive Relationship Learning for Person Re-Identification

Effect of pore structure characteristics on the properties of red mud-based alkali-activated cementitious materials

This study used a mercury intrusion porosimeter (MIP) to establish a Menger sponge model and a thermodynamic fractal model to investigate the influence mechanism of changes in the pore structure of red mud-based alkali-activated cementitious materials under different Na2O contents on their mechanical properties and volume stability. The results indicate that the thermodynamic fractal model can better reflect the pore size distribution within the entire pore size measurement range, with a correlation coefficient R 2 maintained between 0.98347 and 0.99854. Moreover, the thermodynamic fractal model can well reflect the relationship between the pore structure characteristics of the matrix and its macroscopic properties, with the increase of Na2O content, the N-(C)-A-S-H gel content continuously increased, and the pore fractal dimension increased from 2.42609 to 2.72440, indicating a denser pore structure, which promoted the improvement of compressive and flexural strength of the matrix at 28 days while deteriorating volume stability.

Modelling nematic liquid crystal in fractal dimensions

In this paper we present a fractal Berreman's model which account for azimuthal surface anchoring of nematic liquid crystals which play a leading role nowadays in biomedical field and pharmaceutical controlled release dosage forms. The model is based on the concept of "product-like fractal measure" in anisotropic media which permits to describe liquid crystals formation in terms of fractal dimensions. It was observed that fractal dimensions affect both the polar azimuthal angle and the Franck excess elastic energy density of the distortion. For fractal dimensions close to unity, our analysis reveals the emergence of fluctuations and large deformations in the dynamical variables of the theory which suggest the presence of non-linear elastic instabilities or emergence of defect structures in liquid crystals. These fluctuations fade away for fractal dimensions much less than unity. We have evaluated the elastic energy per unit of area which is found to depend on the square of the fractal dimension. This leads to a decrease of the saturation voltage which is necessary to reduce the elastic energy of liquid crystals films in order to minimize the elastic energy. This reduction may describe nematic liquid crystals free from deformations, singularities or weak anchoring surfaces.

Surface integrity analysis and inspection for nanochannel sidewalls using the self-affine fractal model-based statistical quality control for the atomic force microscopy (AFM)-based nanomachining process

The atomic force microscopy (AFM) technology is a promising method for nanofabrication due to the high tunability of this affordable platform. The quality inspection and control significantly impact the manufacturing effectiveness for realizing the functionality of the achieved nanochannel. Particularly, the surface characteristics of nanochannel sidewalls, which play a significant role in determining the quality of the nanomachined products, can not be accurately captured using conventional surface integrity metrics (e.g., surface roughness). Therefore, it is necessary to propose a method to quantitatively characterize the surface morphology and detect the abnormal parts/regions of the nanochannel sidewall. This paper presents a statistical process control approach derived from the self-affine fractal model to detect the sidewall surface anomalies. It evaluates changes in the self-affine fractal model parameters (standard deviation, correlation length, and roughness exponent), which can be used to signify the changes on the sidewall surface; the statistical distributions of these parameters are derived and used to develop control charts to allow inspection of the sidewall morphology. The approach was tested on the AFM-based nanomachined samples. The results suggest that the presented approach can effectively reflect the abnormal regions on the machined parts, which opens up a new avenue toward guiding the quality control and rework for process improvement for AFM-based nanomachining.

A coupled fractal model for predicting the relative permeability of rocks considering both irreducible fluid saturation and stress effects

The relative permeability of rocks is an essential parameter for evaluating two-phase flow characteristics and plays an important role in engineering fields such as resource exploitation. To this end, a mathematical model for predicting relative permeability was first developed based on an equivalent capillary model and fractal theory. The proposed model considers the irreducible fluid saturation under stress and quantifies the influence of the pore structure characteristics on the relative permeability. This model was then compared with relevant experimental data and existing theoretical expressions to verify its validity. Finally, the factors affecting the two-phase seepage characteristics were discussed. The results show that the irreducible fluid saturation is intimately connected to the fractal dimensions, pore size, fluid viscosity, pressure drop gradient, and elastic modulus. Fluid properties and pore structure characteristics are the main factors affecting relative permeability. The wetting phase relative permeability is more sensitive to pore structure and irreducible fluids. Increased effective stress increases irreducible fluid saturation, reduces two-phase flow capacity, and significantly decreases the relative permeability of wetting phase fluids. Increased elastic modulus and Poisson's ratio decrease the irreducible fluid content under stress and increase the permeability.

On a Symmetry-Based Structural Deterministic Fractal Fractional Order Mathematical Model to Investigate Conjunctivitis Adenovirus Disease

In the last few years, the conjunctivitis adenovirus disease has been investigated by using the concept of mathematical models. Hence, researchers have presented some mathematical models of the mentioned disease by using classical and fractional order derivatives. A complementary method involves analyzing the system of fractal fractional order equations by considering the set of symmetries of its solutions. By characterizing structures that relate to the fundamental dynamics of biological systems, symmetries offer a potent notion for the creation of mechanistic models. This study investigates a novel mathematical model for conjunctivitis adenovirus disease. Conjunctivitis is an infection in the eye that is caused by adenovirus, also known as pink eye disease. Adenovirus is a common virus that affects the eye’s mucosa. Infectious conjunctivitis is most common eye disease on the planet, impacting individuals across all age groups and demographics. We have formulated a model to investigate the transmission of the aforesaid disease and the impact of vaccination on its dynamics. Also, using mathematical analysis, the percentage of a population which needs vaccination to prevent the spreading of the mentioned disease can be investigated. Fractal fractional derivatives have been widely used in the last few years to study different infectious disease models. Hence, being inspired by the importance of fractal fractional theory to investigate the mentioned human eye-related disease, we derived some adequate results for the above model, including equilibrium points, reproductive number, and sensitivity analysis. Furthermore, by utilizing fixed point theory and numerical techniques, adequate requirements were established for the existence theory, Ulam–Hyers stability, and approximate solutions. We used nonlinear functional analysis and fixed point theory for the qualitative theory. We have graphically simulated the outcomes for several fractal fractional order levels using the numerical method.

Fractional-Order Modeling of Heat and Moisture Transfer in Anisotropic Materials Using a Physics-Informed Neural Network.

Mathematical models of heat and moisture transfer for anisotropic materials, based on the use of the fractional calculus of integro-differentiation, are considered because such two-factor fractal models have not been proposed in the literature so far. The numerical implementation of mathematical models for determining changes in heat exchange and moisture exchange is based on the adaptation of the fractal neural network method, grounded in the physics of processes. A fractal physics-informed neural network architecture with a decoupled structure is proposed, based on loss functions informed by the physical process under study. Fractional differential formulas are applied to the expressions of non-integer operators, and finite difference schemes are developed for all components of the loss functions. A step-by-step method for network training is proposed. An algorithm for the implementation of the fractal physics-informed neural network is developed. The efficiency of the new method is substantiated by comparing the obtained numerical results with numerical approximation by finite differences and experimental data for particular cases.

Analysis of Spatial Association of Mineralization and Faults Through Fry and Fractal Modelling in Anarak Metallogenic Zone, Iran

Analysis of Spatial Association of Mineralization and Faults Through Fry and Fractal Modelling in Anarak Metallogenic Zone, Iran

Hidden deposit exploration using the tectono-geochemistry method in the western Xicheng ore field, China

Geochemical anomalies involve complex geological and geochemical processes. Integrating metallogenic processes into the interpretation of geochemical data can promote mineral exploration. In this study, 3080 subsamples were collected from the western Xicheng ore field and combined into 1312 composite samples using the tectono-geochemistry method. Nineteen elements were analyzed for each composite sample. Factor analysis based on the CLR-transformed data yielded four factors, including the Ag–Sb–Hg–Pb–Au–(B–Ba) association of F1, Zn–Cd–Pb association of F2, Bi–Sn–(Au–As) association of F3, and W–Sn–(Cu) association of F4. Thresholds of each factor were obtained using the concentration–number (C–N) fractal model. Six targets were delineated based on the factor anomaly maps, and one Pb–Zn and two Au deposits were discovered in Targets I and II, respectively. These discoveries and the good spatial correspondence between known deposits and anomalies provide compelling evidence for the effectiveness of the tectono-geochemistry method in the study area. More importantly, a model for the genetic relationship between geochemical anomalies and metallogenesis was constructed. The late tectonic-magmatic-hydrothermal transformation dominated the geochemical pattern in the study area. The degree of interaction between the Au-rich magmatic-hydrothermal fluids and SEDEX-style Pb–Zn mineralizations yielded leakage halos with various elemental assemblages. In addition, W–Sn anomalies may serve as auxiliary exploration indicators for Au mineralizations.

Multi-scale quantitative characterization of three-dimensional pores and fissures in deep coal and study of the evolution laws

Native pores and fissure in deep coal bodies directly influence the deformation and destruction of coal seams. The pore structure characteristics of coal tend to be complicated because of the damage caused by tectonic stress; hence, it is necessary to study the development of pore fracture in coal. In this study, two types of coal samples—primary structural coal and tectonically deformed coal—were selected from the 12403 comprehensive mining face in the Shangwan Coal Mine, part of the Shendong Coal Mine Group. The developmental characteristics, distribution patterns, and morphological and structural differences of pores and fissures in these types of structural coal were examined using X-ray diffraction, scanning electron microscopy, mercury intrusion porosimetry, and computed tomography scanning, combined with digital image processing technology. A fractal model for the evolutionary characteristics of the pore-fracture network in a coal body is proposed. The characteristics of the pore-fracture network under different measurement accuracies are also investigated. The results showed that the primary structural coal tended to develop small fissures of simple morphologies, with poor fissure connectivity and connection strength. In contrast, the morphologies of fissures in tectonically deformed coal were complex, with better fissure connectivity and connection strength, and a higher fissure rate than that of primary structural coal. A fractal model was developed to describe and characterize the geometrical features of the pore-fracture network at various measurement accuracies. The dominant pore-size segments that can be characterized are 50 nm ≤ r ≤ 10 μm and r > 10 μm, respectively. The fractal dimensions of porosity and pore-fracture network volume increase as measurement scales decrease. The accuracy and applicability of the fractal model in predicting the porosity and volume fractal dimension of the pore-fracture network across various scales were confirmed.

Theoretical study on nonlinear seepage mechanism in fractal dendritic fracture network of low permeability coal with water injection

Coal seam water injection technology is adopted by many mines as an effective means of dust reduction in coal mines. There is a threshold pressure gradient phenomenon in the process of water injection in low permeability coal seam, which makes the flow of pressure water in the fracture structure of coal body present nonlinear seepage characteristics. To reveal the theoretical relationship between the structural parameters of coal and the nonlinear seepage characteristics, firstly, the Bingham fluid constitutive equation is used to describe the non-Newtonian behavior in low-permeability coal. Combined with the fractal tree-like bifurcation fracture network model, a mathematical analytical model of threshold pressure gradient is established. Secondly, the model was verified by high-pressure water invasion and radial seepage experiments, and the sensitivity of the model was analyzed. The results show that the error between the theoretical calculation value and the experimental measurement value is between 8.65 % and 42.4 %, which verifies the validity of the model. The above research results can provide a theoretical basis for improving the water injection effect of low permeability coal seam.

Insights into Transferal to Fractal Space Modeling: Delayed Forced Helmholtz-Duffing Oscillator with the Non-Perturbative Approach

Abstract The damped Helmholtz-Duffing oscillator is a topic of great interest in many different fields of study due to its complex dynamics. By transitioning from conventional continuous differential equations to their fractal counterparts, one gains insights into the system's response under new mathematical frameworks. This paper presents a novel method for converting standard continuous differential equations into their fractal equivalents. This conversion process occurs after the nonlinear system is transformed to its linear equivalent. Numerical analyses show that there are several resonance sites in the fractal system, which differ from the one resonance point found in the continuous system. One important finding is that the fractal system loses some of its stabilizing power when decaying behavior is transformed into a diffuse pattern. Interestingly, a decrease in the fractal order in resonance settings shows a stabilizing impact, highlighting the dynamics' complexity inside fractal systems.

A Novel Fractal Model for Contact Resistance Based on Axisymmetric Sinusoidal Asperity

In this paper, a novel fractal model for the contact resistance based on axisymmetric sinusoidal asperity is proposed, which focuses on the resistance characteristics of the rough interface at a microscopic scale. By introducing the unique geometric shape of axisymmetric sinusoidal asperity, and combining it with a three-dimensional fractal theory, the micro-morphology characteristics of the rough interface can be characterized more precisely. Subsequently, by conducting a theoretical analysis and numerically solving the deformation mechanisms of asperities on the rough interface, a refined model for contact resistance is constructed. This research comprehensively employs theoretical analysis, numerical simulation, and experimental testing methods to deeply explore the current transmission mechanisms during the contact process of the rough interface. The findings suggest that the proposed model is capable of precisely capturing the intricate interplay of various factors, including contact area, contact load, and material properties, with the contact resistance. Compared to the existing models, the presented model demonstrates significant advantages in terms of prediction accuracy and practicality. This research provides an important theoretical basis and design guidance for optimizing the electrical performance of the rough interface, which has great significance for engineering applications.

Fractal model of the strength of lightweight concrete based on volcanic tuff taking into account the scale effect

The study proposes a fractal model based on B. Mandelbrot's geometry. Mandelbrot geometry to describe the scale effect in concrete. It has been experimentally established that under-loading concrete undergoes degradation of its structure, which is expressed in the sequential destruction of fractals at different scale levels. It was found that the scale reduction leads to an increase in concrete strength: 20x20x20 mm specimens showed the highest strength among all four compositions (23.3-25.3 MPa). The average density of tuff concrete was also investigated: the smallest specimens had a density of 1961.3 kg/m³, while the largest specimens had a density of 1974.2 kg/m³, which is explained by the heterogeneity of the structure and pore distribution. The water permeability of concrete, evaluated through air permeability, showed that specimens with higher air permeability (30.2-30.5 s/cm³) had better water resistance (W14). These data indicate a correlation between air permeability and water resistance, which may be due to the denser structure and improved pore distribution. The study of air permeability makes it possible to predict the durability of concrete structures when exposed to moisture, as well as to improve the quality of building materials and reduce the cost of their operation.

Stress sensitivity analysis of vuggy porous media based on two-scale fractal theory

Interparticle pore space and vugs are two different scales of pore space in vuggy porous media. Vuggy porous media widely exists in carbonate reservoirs, and the permeability of this porous media plays an important role in many engineering fields. It has been shown that the change of effective stress has important effects on the permeability of vuggy porous media. In this work, a fractal permeability model for vuggy porous media is developed based on the fractal theory and elastic mechanics. Besides, a Monte Carlo simulation is also implemented to obtain feasible values of permeability. The proposed model can predict the elastic deformation of the fractal vuggy porous media under loading stress, which plays a crucial role in the variations of permeability. The predicted permeability data based on the present fractal model are compared with experimental data, which verifies the validity of the present fractal permeability model for vuggy porous media. The parameter sensitivity analysis indicates that the permeability of stress-sensitivity vuggy porous media is related to the capillary fractal dimension, capillary fractal tortuosity dimension, Young’s modulus, and Poisson’s ratio.

A hydraulic-mechanical coupled triple porosity fractal model considering action mechanisms and gas transport mechanisms in carbon-hydrogen reservoirs

The shale formation, a common carbon-hydrogen reservoir, has numerous nanoscale pores that facilitate multiple gas transport mechanisms, including viscous flow, slip flow, Knudsen diffusion, and surface diffusion. Additionally, various action mechanisms involving effective stress, real gas effect, gas adsorption and desorption, and fracture-pore characteristics significantly influence gas transport. In this study, we conceptualize shale as a triple porosity medium comprising organic matrix, inorganic matrix, and microfractures. For each system, its hydraulic-mechanical coupled model with action mechanisms and gas transport mechanisms is derived based on fractal theory. After validating the model with Barnett and Marcellus data, the evolution of apparent permeability and gas pressure is studied, as well as the influences of pore fractal features, the real gas effect, and the number of hydraulic fractures on gas extraction. Simulation results indicate that, in comparison to organic matrix, inorganic pore fractal features exert a more noticeable effect on cumulative production. The real gas effect is critical in gas transport and reservoir reserve estimation; disregarding this effect results in a 17.4% underestimation of cumulative gas production by the 1600th day. Fracture interference attenuates the improvement in cumulative production gains from increased hydraulic fracture numbers.

Fractal permeability model for power-law fluids in embedded tree-like branching networks based on the fractional-derivative theory

The investigation of permeability in tree-like branching networks has attracted widespread attention. However, most studies about fractal models for predicting permeability in tree-like branching networks include empirical constants. This paper investigates the flow characteristics of power-law fluids in the dual porosity model of porous media in embedded tree-like branching networks. Considering the inherent properties of power-law fluids, non-Newtonian behavior effects, and fractal properties of porous media, a power-law fluids rheological equation is introduced based on the fractional-derivative theory and fractal theory. Then, an analytical formula for predicting the effective permeability of power-law fluids in dual porous media is derived. This analytical formula indicates the influences of fractal dimensions and structural parameters on permeability. With increasing length ratio, bifurcation series, and bifurcation angle, as well as decreasing power-law exponent and diameter ratio, the effective permeability decreases to varying degrees. The derived analytical model does not include empirical constants and is consistent with the non-Newtonian properties of power-law fluids, indicating that the model is an effective method for describing the flow process of complex non-Newtonian fluids in porous media in natural systems and engineering. Therefore, this study is of great significance to derive analytical solutions for the permeability of power-law fluids in embedded tree-like bifurcation networks.

The influence of supervised and unsupervised methods in the analysis of low density stream sediments in the regional prospecting of pegmatites in the state of Espírito Santo, Araçuaí Orogen - Eastern Pegmatite Province of Brazil

The state of Espírito Santo is located in Brazil southeastern and it is geologically inserted in the Eastern Brazilian Pegmatite Province in the Araçuaí Orogen. This province is one of the most important pegmatite provinces in the world, known mainly for the occurrence of gem-quality minerals and high grade lithium deposits. Although Espírito Santo has a geological framework with many granite intrusions similar to those hosting mineralized pegmatites in Minas Gerais, it hasn't been acknowledged as a potential pegmatitic area. There are reports of mineralization of beryl (aquamarine), topaz, amethyst, and chrysoberyl in the state's mineral history. I this research we use regional stream sediment geochemical data of the Espirito Santo State, to create prospective geochemical maps to map gemological pegmatites. To do so, primary descriptive statistical analyses as well as Pearson and Spearman correlations and Principal Component Analysis (PCA) were used to select elements associated with pegmatites. The geochemical prospectivity mapping was carried out by supervised and unsupervised approaches. The supervised method was based on integration of the elements by C-A fractal models, used to define the geochemical thresholds, and fuzzy logic methodology to integrate weighted geochemical maps. The unsupervised approach, however, was based on interpolation of the representative principal component scores. All models were validated using the Prediction Area plot (P-A plot). The results of our research showed that unsupervised methodology overperformed the other weighted supervised models, with a prediction rate of 78.14 % and prospectivity of 0.87. The final results also show a pegmatite prospectivity map for the Espirito Santo state, that indicates a great exploration potential for gemological minerals.)