Fractal Model Research Articles

Novel adaptive control approach to fractal fractional order deforestation model and its impact on soil erosion.

Deforestation exerts profound ramifications on soil quality and biodiversity, thereby exerting substantial economic repercussions. The depletion of organic matter and structural integrity of soil following tree removal for agricultural purposes underscores the severity of this issue. In elucidating the soil pollution stemming from deforestation, this research employs a sophisticated five-compartment SDIFR model integrating fractal dimension and fractional order dynamics. The rigorous analysis, including the application of Picard Lindelof's fixed point theorem, establishes the existence and uniqueness of explicit solutions. Furthermore, the examination of local and global stability sheds light on the system's behavior, delineating between pollution-free equilibrium and pollution-extinct equilibrium states. To regulate system behavior, an adaptive control framework grounded in fractal fractional order is proposed, leveraging the Adams-Bashforth numerical approximation scheme for implementation. Through numerical simulations, the study underscores the pivotal role of parameters, thus substantiating the significance of the proposed model in comprehensively addressing the complexities of soil pollution induced by deforestation.

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 occurs after the nonlinear system is transformed into 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. This endeavor to convert to fractals is a revolutionary technique that is being employed for the first time.

On an abrasion-motivated fractal model

Abstract In this paper, we consider a fractal model motivated by the abrasion of convex polyhedra, where the abrasion is realised by chipping small neighbourhoods of vertices. After providing a formal description of the successive chippings, we show that the net of edges converge to a compact limit set under mild assumptions. Furthermore, we study the upper box-counting dimension and the Hausdorff dimension of the limiting object of the net of edges after infinitely many chipping.

Discussion of “Thermal conductivity of porous building materials: An exploration of new challenges in fractal modelling solutions”

In November 2023 this journal published “Thermal conductivity of porous building ma­terials: An exploration of new challenges in fractal modelling solutions”. That paper gauges four fractal building materials’ thermal conductivity models, concluding that fractal-geometry-based approa­ches appear “a promising method” as they “demonstra­te high reliability in repro­ducing experimental data”. This discussion of the pa­per aims to shine a different light on the potential of fractal thermal conductivity models. It shows that good agreement with experimental data usually ori­ginates from calibration of various “physical” factors comprised in these models, with the fitted numbers commonly deviating from physical rea­lity. Moreover, exemplary instances reveal that good agreement with experimental data is obtained des­pite misinterpretation of measured outcomes or critical defects in model development, or, exceptio­nally, due to fabrication of validation information. This discussion does hence not share the pa­per’s positive opinion on the prospects of fractal-geometry-ba­sed ther­­mal conductivity models, and advises caution instead.

A FRACTAL PORE-SCALE MODEL OF STRESS-DEPENDENT THERMAL CONDUCTIVITY IN A RECTANGULAR CROSS-SECTION TREE-LIKE MICROFRACTURE NETWORK

Stress is an important external element influencing effective thermal conductivity (ETC). However, previous theoretical investigations of stress-dependent effective thermal conductivity have used a simplified parallel plate model, overlooking the complex structure of rock fracture networks. Therefore, this paper establishes a new fractal model for the stress-dependent ETC of a rectangular cross-section tree-like microfracture network. The model derives a constitutive equation between the stress-dependent ETC and the microscopic geometry structural parameters of the microfracture network. The model’s validity is confirmed by the error of less than 0.291% between the theoretical predictions and the experimental data. The influence of geometry structural parameters of the microfracture network (length ratio, width ratio, height ratio, angle, total number of branches and ratios of soft part, hard part, etc.) on the ETC under the low-stress state has been analyzed. It was discovered that the soft part of the rock microfracture network has the most impact on ETC. The fractal pore-scale model clarifies heat transfer mechanism in rock microfracture network under effective stress, with each parameter physically defined and independent of empirical constants.

A revised elastic-plastic contact model of cycloidal pinwheel based on length scale

This study explores the relationship between surface roughness and characteristic scale through theoretical analyses and experimental data. Considering the concavity and convexity of cycloidal gear tooth profile curvatures, a curved surface contact coefficient model suitable for both inner and outer contacts, with either equal or unequal curvatures, has been developed. Moreover, a fractal contact model is constructed for cycloidal pinwheels. This model considers scale correlation and analyzes the influence of microscopic parameters on the elastic-plastic critical grade and the behavior of the contact area. Additionally, the influence of the micro-convex body frequency index on surface contact performance is explored to provide theoretical insights for improving the elastic contact ratio and reducing wear on contact surfaces.

PH-controlled-ion-mobility of Laponite-phosphate dispersion for physical hydrogel with improved mechanical properties and sensitivity to CO2

Self-assembled physical nanocomposite hydrogels based on bio-based poly(itaconic acid) and Laponite® nanoclays were prepared and tested as possible CO2 sensors in smart food packaging. The structure and mechanical properties of the fabricated hydrogels were controlled by changing the initial pH of the aqueous Laponite® dispersions stabilised by tetrasodium pyrophosphate. The dispersions were studied by SAXS and analysed using circular disc and fractal model to determine clay agglomeration tendency. Sodium and pyrophosphate ions mobility was investigated by 23Na and 31P 1D as well as spin-spin relaxation times T2 NMR measurements. The rheological behaviour of the produced hydrogels was studied by oscillatory shear measurements. It was found that decreasing the pH of the Laponite® nanodispersions up to 8 increases the strength of the physical interactions between the nanoclays, dispersant, and polyelectrolyte chains, which enables production of dimensionally stable and mechanically robust hydrogels. Their potential application as the CO2-sensitive matrix for construction of environmentally friendly colorimetric sensors was demonstrated showing the CO2 sorption capacity of 0.07 mmol CO2/g. A prototype device of hydrogel sensor was built and tested in a food packaging application to monitor the plum fruit respiration processes.

Advancing characterization of VOC diffusion in indoor fabrics: A dual-porosity modeling approach

Volatile organic compounds (VOCs) are major chemical pollutants in indoor air. Indoor fabrics, such as curtains, carpets, sofas, and clothes, strongly adsorb VOCs due to their high loading rates and large specific surface areas. The desorption of VOCs from these fabrics can act as a secondary source, worsening indoor air pollution and prolonging its effects. The diffusion coefficient is a key parameter that determines the source-sink properties of fabrics. In this study, the VOC diffusion characteristics in fabrics were investigated through microstructural examination, mass transfer analysis, and environmental chamber experiments. Yarn and fiber gaps were identified as dual mass transfer channels within the fabrics and were represented using two distinct fractal models. A dual-porosity medium (DPM) model, based on these fractal representations, was developed to predict the VOC diffusion coefficients in indoor fabrics and was validated via experiments under various environmental conditions and fabric-VOC combinations. The results highlight the significant impact of fabric structure and composition on VOC adsorption and emission dynamics. The validated DPM model provides a comprehensive approach to predicting VOC diffusion in fabrics, providing a more accurate method for assessing indoor air quality and fabric-mediated human exposure.

Biofilm Growth in Porous Media Well Approximated by Fractal Multirate Mass Transfer With Advective‐Diffusive Solute Exchange

AbstractBiofilm growth in porous media changes not only the hydrodynamic properties of the medium (reduction in porosity and permeability, and increase in dispersivity), but also the transport itself (breakthrough curves display increasingly fast first arrivals and long tails). These features are well reproduced by multicontinuum models (Multi‐Rate Mass Transfer, MRMT) which can be used to describe reactive transport in heterogeneous porous media and facilitate the simulation of reactions that are localized within biofilms. Here, we present a conceptual and numerical model of biochemical reactive transport with dynamic biofilm growth based on MRMT formulations. Mass exchange between mobile water and immobile biofilm aggregates is represented by a memory function, which simplifies definition of MRMT parameters. We successfully tested this model on two sets of laboratory data and found that (a) a basic model based on the growth of uniformly sized biofilm aggregates fails to reproduce laboratory tracer tests and rate of biofilm growth, while a fractal growth model, which we obtain by integrating the memory functions of biofilm aggregates with a power law distribution, does; (b) the biofilm memory function evolves as the biofilm grows; and (c) the early time portion of eluted volume tracer breakthrough curves are independent of flow rate, whereas the tail becomes heavier when the flow rate is decreased, which implies that both advection controlled and diffusion controlled mass exchange coexist in biofilms. These findings imply that porous media biofilms are essentially different from those developing in human tissues or open spaces.

Fractal Modelling of Heterogeneous Catalytic Materials and Processes.

This review considers the use of fractal concepts to improve the development, fabrication, and characterisation of catalytic materials and supports. First, the theory of fractals is discussed, as well as how it can be used to better describe often highly complex catalytic materials and enhance structural characterisation via a variety of different methods, including gas sorption, mercury porosimetry, NMR, and several imaging modalities. The review then surveys various synthesis and fabrication methods that can be used to create catalytic materials that are fractals or possess fractal character. It then goes on to consider how the fractal properties of catalysts affect their performance, especially their overall activity, selectivity for desired products, and resistance to deactivation. Finally, this review describes how the optimum fractal catalyst material for a given reaction system can be designed on a computer.

Investigating the compaction and the mechanical behaviors of coal gangue as subgrade filler and constructing highway subgrade in practice

Using coal gangue as subgrade filler (CGSF) can address the accumulated issues of coal mine waste, but also save the constructing costs, which has the important ecological and engineering practical value. The well-graded limit of CGSF was captured based on the fractal model grading equation (FMGE). The large-scale vibration compaction test (LCT) shows that particle grading has a remarkable influence on the compaction of CGSF, with the increase of fine particle content, the dry density of sample first increases and then decreases. On this basis, the optimal range of particle grading was obtained. A series of large-scale triaxial tests (LTT) were conducted to investigate the mechanical behaviors of CGSF. The results shown that (1) the peak deviatoric stress shows a well-quadratic correlation with the fractal dimension. The optimal grading range of CGSF captured by LTT and LCT is basically the same, hence the suggestion of acquiring the optimal grading through LCT was given. (2) Increasing the compaction degree (DC) can significantly improve the strength of CGSF if DC is smaller, while when DC is greater, the strength improved by increasing DC will not be significant. (3) The grading and DC have a significant impact on the cohesive force of CGSF, while have little effect on internal friction angle. In addition, the research results provide good guidance for the constructing of a highway subgrade in practice.

Chlamydia infection with vaccination asymptotic for qualitative and chaotic analysis using the generalized fractal fractional operator

In this work, we solve a system of fractional differential equations utilizing a Mittag-Leffler type kernel through a fractal fractional operator with two fractal and fractional orders. A six-chamber model with a single source of chlamydia is studied using the concept of fractal fractional derivatives with nonsingular and nonlocal fading memory. The fractal fractional model of the Chlamydia system can be solved by using the characteristics of a non-decreasing and compact mapping. A suggested model with the Lipschitz criteria and linear growth is studied both qualitatively and quantitatively, taking into account boundedness, uniqueness, and positive solutions at equilibrium points with Leray-Schauder results under time scale concepts. We examined the framework of local and global stability and insight into Lyapunov function properties for the infectious disease model. Chaos Control will employ the regulate for linear responses approach to stabilize the system following its equilibrium points. This will take into consideration a fractional order framework with a managed design, where solutions are bounded in the feasible domain and have a greater impact at the lower minimum infectious rate. To illustrate the implications of fractional and fractal dimensions with varying interest rate values through simulations with Newton’s polynomial method under the Mittag-Lefller kernel. Additionally, a comparative analysis of results is also derived by employing power and exponential decay kernels at various fractional orders.

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.