Showing papers on "Fractal dimension published in 2021"
TL;DR: Various network covering algorithms, which form the basis for obtaining fractal dimension, are being reviewed and the different dimensions used to describe the fractal property of networks and their applications are discussed.
95 citations
TL;DR: In this article, a 3D reconstruction of CT images was used for the establishment of fluid-solid conjugate heat transfer model and coal thermal deformation model based on the microstructures of coal.
Abstract: To study the seepage and deformation characteristics of coal at high temperatures, coal samples from six different regions were selected and subjected to computed tomography (CT) scanning studies. In conjunction with ANSYS software, 3D reconstruction of CT images was used for the establishment of fluid-solid conjugate heat transfer model and coal thermal deformation model based on the microstructures of coal. In addition, the structure of coal was studied in 2D and 3D perspectives, followed by the analysis of seepage and deformation characteristics of coal at high temperatures. The results of this study indicated that porosity positively correlated with the fractal dimension, and the connectivity and seepage performances were roughly identical from 2D and 3D perspectives. As the porosity increased, the fractal dimension of coal samples became larger and the pore-fracture structures became more complex. As a result, the permeability of coal samples decreased. In the meantime, fluid was fully heated, generating high-temperature water at outlet. However, when the porosity was low, the outlet temperature was very high. The average deformation of coal skeleton with different pore-fracture structures at high temperatures showed a trend of initial increase and subsequent decrease with the increase of porosity and fractal dimension. The maximum deformation of coal skeleton positively correlated with connectivity but negatively correlated with the fractal dimension.
91 citations
TL;DR: In this paper, the peak of a solitary wave is weakly affected by the unsmooth boundary, and a fractal variational principle is established to obtain the wave solution in fractal space.
Abstract: It is well-known that the boundary conditions will greatly affect the wave shape of a nonlinear wave equation. This paper reveals that the peak of a solitary wave is weakly affected by the unsmooth boundary. A fractal Korteweg-de Vries (KdV) equation is used as an example to show the solution properties of a solitary wave travelling along an unsmooth boundary. A fractal variational principle is established in a fractal space and its solitary wave solution is obtained, and its wave shape is discussed for different fractal dimensions of the boundary.
83 citations
DOI•
01 Dec 2021
TL;DR: In this article, the authors summarized the latest progress in the investigation and application of fractal theory in cement-based materials and summarized the applications of these fractal dimensions in investigating the macro properties of these materials.
Abstract: Cement-based materials, including cement and concrete, are the most widely used construction materials in the world. In recent years, the investigation and application of fractal theory in cement-based materials have attracted a large amount of attention worldwide. The microstructures of cement-based materials, such as the pore structures, the mesostructures, such as air voids, and the morphological features of powders, as well as the fracture surfaces and cracks, commonly present extremely complex and irregular characteristics that are difficult to describe in terms of geometry but that can be studied by fractal theory. This paper summarizes the latest progress in the investigation and application of fractal theory in cement-based materials. Firstly, this paper summarizes the principles and classification of the seven fractal dimensions commonly used in cement-based materials. These fractal dimensions have different physical meanings since they are obtained from various testing techniques and fractal models. Then, the testing techniques and fractal models for testing and calculating these fractal dimensions are introduced and analyzed individually, such as the mercury intrusion porosimeter (MIP), nitrogen adsorption/desorption (NAD), and Zhang’s model, Neimark’s model, etc. Finally, the applications of these fractal dimensions in investigating the macroproperties of cement-based materials are summarized and discussed. These properties mainly include the mechanical properties, volumetric stability, durability (e.g., permeability, frost and corrosion resistance), fracture mechanics, as well as the evaluation of the pozzolanic reactivity of the mineral materials and the dispersion state of the powders.
56 citations
TL;DR: In this article, a fractal contact model suited for gear pair contact has been established, and the asperity contact stiffness is calculated, thus the fractal compliance compliance can be obtained.
55 citations
TL;DR: The proposed Fractal-based DKPCA (FDKPCA) integrates the two strategies to overcome SPCA/D PCA/DKPCA shortcomings, FDim allows verifying the degree of fitting and ensures optimal dimensionality reduction.
48 citations
TL;DR: In this paper, a qualitative analysis of the mathematical model of novel coronavirus ( COVID -19) involving anew devised fractal-fractional operator in the Caputo sense having the fractional-order q and the fractal dimension p is considered.
Abstract: In this manuscript, a qualitative analysis of the mathematical model of novel coronavirus ( COVID -19) involving anew devised fractal-fractional operator in the Caputo sense having the fractional-order q and the fractal dimension p is considered. The concerned model is composed of eight compartments: susceptible, exposed, infected, super-spreaders, asymptomatic, hospitalized, recovery and fatality. When, choosing the fractal order one we obtain fractional order, and when choosing the fractional order one a fractal system is obtained. Considering both the operators together we present a model with fractal-fractional. Under the new derivative the existence and uniqueness of the solution for considered model are proved using Schaefer’s and Banach type fixed point approaches. Additionally, with the help of nonlinear functional analysis, the condition for Ulam’s type of stability of the solution to the considered model is established. For numerical simulation of proposed model, a fractional type of two-step Lagrange polynomial known as fractional Adams-Bashforth ( AB ) method is applied to simulate the results. At last, the results are tested with real data from COVID -19 outbreak in Wuhan City, Hubei Province of China from 4 January to 9 March 2020, taken from a source (Ndairou, 2020). The Numerical results are presented in terms of graphs for different fractional-order q and fractal dimensions p to describe the transmission dynamics of disease infection.
45 citations
TL;DR: In this article, the authors investigate quantum transport in fractal networks by performing continuous-time quantum walks in a fractal photonic lattices and reveal the transport properties through the photon evolution patterns, mean square displacement and the Polya number.
Abstract: Fractals are fascinating, not only for their aesthetic appeal but also for allowing the investigation of physical properties in non-integer dimensions. In these unconventional systems, many intrinsic features might come into play, including the fractal dimension and the fractal geometry. Despite abundant theoretical studies, experiments in fractal networks remain elusive. Here we experimentally investigate quantum transport in fractal networks by performing continuous-time quantum walks in fractal photonic lattices. We unveil the transport properties through the photon evolution patterns, the mean square displacement and the Polya number. Contrarily to classical fractals, we observe anomalous transport governed solely by the fractal dimension. In addition, the critical point at which there is a transition from normal to anomalous transport depends on the fractal geometry. Our experiment allows the verification of physical laws in a quantitative manner and reveals the transport dynamics in great detail, thus opening a path to the understanding of more complex quantum phenomena governed by fractality. Quantum transport in fractal networks is experimentally investigated by performing continuous-time quantum walks in fractal photonic lattices. Contrarily to classical fractals, anomalous transport governed solely by the fractal dimension is observed.
44 citations
42 citations
TL;DR: In this article, a one-dimensional SHPB impact test was carried out to test the dynamic compressive strength, damage morphology, fracture energy dissipation density, and other parameters of the rocks under different strain rates.
Abstract: In order to study the fractal characteristics of the pomegranate biotite schist under the effect of blasting loads, a one-dimensional SHPB impact test was carried out to test the dynamic compressive strength, damage morphology, fracture energy dissipation density, and other parameters of the rocks under different strain rates; besides, sieve tests were conducted to count the mass fractal characteristics of the crushed masses under different strain rates to calculate the fractal dimension of the crushed rock . Finally, the relationships between fractal dimension and dynamic compressive strength, crushing characteristics, and energy dissipation characteristics were analysed. The results show that under different impact loads, the strain rate effect of the rock is significant and the dynamic compressive strength increases with the increasing strain rate, and they show a multiplicative power relationship. The higher the strain rate of the rock, the deeper the fragmentation and the higher the fractal dimension, and the fractal dimension and rock crushing energy density are multiplied by a power relationship. By performing the comparative analysis of the pomegranate biotite schist, a reasonable strain rate range of 78.75 s-1~82.51 s-1 and a reasonable crushing energy consumption density range of 0.78 J·cm-3~0.92 J·cm-3 were determined. This research provides a great reference for the analysis of dynamic crushing mechanism, crushing block size distribution, and crushing energy consumption of the roadway surrounding rock.
42 citations
TL;DR: In this article, the fractal characteristics of crack propagation in hydraulic fracturing and the effect of fractal dimension of induced fractures on the rock breakdown pressure were investigated, and the results showed that fractal dimensions of induced fracture induced by crack propagation can affect the fracture breakdown pressure.
Abstract: This paper mainly investigated the fractal characteristics of crack propagation in hydraulic fracturing and the effect of fractal dimension of induced fractures on the rock breakdown pressure. A se...
TL;DR: In this paper, various methods including MIP test, gas emission index experiments, and gas permeability analysis were employed in order to gain clearer insights into the changing laws of pore structure and gas percolation for samples before and after liquid CO2 phase change fracturing (LCPCF).
TL;DR: In this paper, a hybrid intelligent fuzzy fractal approach for classification of countries based on a mixture of fractal theoretical concepts and fuzzy logic mathematical constructs is presented, which is based on the COVID-19 data of confirmed and death cases.
TL;DR: In this article, a combination of core analysis, MIP analysis and images analysis was used to study the pore structure and petrophysical properties of tight sandstones and the results of pore structures were used to check whether the fractal dimension values from five popular MIP fractal models can represent the heterogeneity of thin sandstones.
TL;DR: In this paper, a novel fractal dimension estimation method based on VMD is proposed to decompose the multi-component signal into several components, and two fractal dimensions corresponding to the small and large time scales are extracted as feature parameters of vibration signals.
TL;DR: Clinker ash is a byproduct obtained from coal-fired electrical generators and is a type of recycled granular material with a complex particle shape and large surface roughness.
Abstract: Clinker ash is a byproduct obtained from coal-fired electrical generators. It is a type of recycled granular material with a complex particle shape and large surface roughness. Particle sh...
TL;DR: In this article, a model of normal stiffness between curved fractal surfaces considering friction factor is proposed based on the continuity of length scale for asperities, and contact stiffness of the whole rough surface is derived by double integral.
01 Jan 2021
TL;DR: In this article, the exact solution of some important ordinary differential equations where the differential operators are the fractal-fractional was derived and a new numerical scheme to obtain solution in the nonlinear case was presented.
Abstract: New class of differential and integral operators with fractional order and fractal dimension have been introduced very recently and gave birth to new class of differential and integral equations. In this paper, we derive exact solution of some important ordinary differential equations where the differential operators are the fractal-fractional. We presented a new numerical scheme to obtain solution in the nonlinear case. We presented the numerical simulation for different values of fractional orders and fractal dimension.
25 Jun 2021
TL;DR: Wood et al. as mentioned in this paper compare three fractal dimension calculation methods applied to shales and reveal the uncertainties that should be taken into account when applying the methods and the appropriate curve-tting optimization configurations.
Abstract: Surface roughness of shales has a key influence on the petroleum resources they are able to store and the fraction of them that can be recovered. The fractal dimension quantifies the degree of roughness and is influenced primarily by the pore surfaces within the shale that typically include micro-, meso-and macro-pores. Isotherms generated by gas adsorption experiments are the common data source used to derive estimates of fractal dimension. The Frenkel-Halsey-Hill fractal technique is the most widely applied fractal dimension estimation method. Other methods can derive fractal dimension from isotherm data but typically the values they generate are different from the Frenkel-Halsey-Hill derived fractal dimension values. Moreover, those differences can vary significantly depending on the type of shales involved. Those shales displaying more complex pore-scale distributions including extensive micro-porosity components tend to be associated with the greatest discrepancies. A comparison of three fractal dimension calculation methods applied to shales reveals aspects of their calculation and interpretation methods that explain the differences in the fractal dimension values they generate. This study identifies the uncertainties that should be taken into account when applying the methods and the appropriate curve fitting optimization configurations that should be evaluated. Taking these factors into account leads to more realistic selections of appropriate fractal dimension values from gas adsorption isotherms of organic-rich shales. Cited as : Wood, D. A. Techniques used to calculate shale fractal dimensions involve uncertainties and imprecisions that require more careful consideration. Advances in Geo-Energy Research, 2021, 5(2): 153-165, doi: 10.46690/ager.2021.02.05
TL;DR: Fracture surfaces after biaxial fatigue tests were compared using fractal dimension for three types of metallic materials in smooth and notched specimens made of S355J2 and 10HNAP steels and 2017-T4 aluminium alloy, considering both proportional and nonproportional cyclic loading as discussed by the authors.
TL;DR: In this paper, a multicore parallel processing algorithm is presented to calculate the fractal dimension of coastline of Australia, which is a measure of degree of geometric irregularity present in the coastline.
Abstract: Coastlines are irregular in nature having (random) fractal geometry and are formed by various natural activities. Fractal dimension is a measure of degree of geometric irregularity present in the coastline. A novel multicore parallel processing algorithm is presented to calculate the fractal dimension of coastline of Australia. The reliability of the coastline length of Australia is addressed by recovering the power law from our computational results. For simulations, the algorithm is implemented on a parallel computer for multi-core processing using the QGIS software, R-programming language and Python codes.
TL;DR: The chaotic phase of the Bose-Hubbard Hamiltonian is identified by the energy-resolved correlation between spectral features and structural changes of the associated eigenstates as exposed by their generalized fractal dimensions.
Abstract: We identify the chaotic phase of the Bose-Hubbard Hamiltonian by the energy-resolved correlation between spectral features and structural changes of the associated eigenstates as exposed by their generalized fractal dimensions. The eigenvectors are shown to become ergodic in the thermodynamic limit, in the configuration space Fock basis, in which random matrix theory offers a remarkable description of their typical structure. The distributions of the generalized fractal dimensions, however, are ever more distinguishable from random matrix theory as the Hilbert space dimension grows.
TL;DR: The primary focus of this article is to provide a compressed overview of the developments in fractal-shaped antennas as well as arrays over the last few decades where the most prominent contributions mostly from IEEE journals have been highlighted.
Abstract: In mathematical definition, a fractal is a self-similar subset of Euclidean space whose fractal dimension strictly exceeds its topological dimension which in turn involves a recursive generating methodology that results in contours with infinitely intricate fine structures. Fractal geometry has been used to model complex natural objects such as clouds coastlines, etc., that has space-filling properties. In the past years, several groups of scientists around the globe tried to implement the structure of fractal geometry for applications in the field of electromagnetism, which led to the development of new innovative antenna configurations called “fractal antennas” which is primarily focused in fractal antenna elements, and fractal antenna arrays. It has been demonstrated that by exploiting the recursive nature of fractals, several marvellous kinds of properties can be observed in antennas and arrays. The primary focus of this article is to provide a compressed overview of the developments in fractal-shaped antennas as well as arrays over the last few decades where the most prominent contributions mostly from IEEE journals have been highlighted. The open intention of this review work is to show an encouraging path to antenna researchers for its advancement using fractal geometries.
TL;DR: In this paper, the stable cubic covering method (SCCM) and stable differential cubic cover method (SDCCM), based on the box-counting method, were proposed to measure the fractal dimension of a rough surface.
TL;DR: In this paper, the effect of static stress on the fractal characteristics and propagation behavior of blast-induced cracking are analyzed using a laboratory model experimental method combined with fractal theory, a dynamic caustics method and high-speed camera technology.
TL;DR: Based on the Weirstrass-Mandelbrot (WM) function model of fractal theory, the fractal parameters of IPMC interface are calculated, and the modified WM model is used to simulate the two-dimensional and three-dimensional interface profiles.
TL;DR: In this article, the fractal Schrodinger equation with position-dependent mass in fractal dimensions is constructed from fractal anisotropy and product-like fractal measure introduced by Li and Ostoja-Starzewski in their formulation of fractal continuum media and elasticity.
Abstract: In this study, the Schrodinger equation with position-dependent mass in fractal dimensions is constructed from fractal anisotropy and product-like fractal measure introduced by Li and Ostoja-Starzewski in their formulation of fractal continuum media and elasticity. The theory is characterized by a fractal uncertainty relation and a generalized fractal momentum operator. The fractal Schrodinger equation is exactly solved for different position-dependent masses and effective potentials. In particular, we discuss the problems of quantum dots and nanocrystals. Modifications in their energies levels are detected which are in agreement with recent studies and experiments.
TL;DR: In this article, the kinetics of heat-induced gelation and the microscopic dynamics of a hen egg white gel are probed using x-ray photon correlation spectroscopy along with ultrasmall-angle xray scattering, revealing an exponential growth of the characteristic relaxation times followed by an intriguing steady state in combination with a compressed exponential correlation function and a temporal heterogeneity.
Abstract: The kinetics of heat-induced gelation and the microscopic dynamics of a hen egg white gel are probed using x-ray photon correlation spectroscopy along with ultrasmall-angle x-ray scattering. The kinetics of structural growth reveals a reaction-limited aggregation process with a gel fractal dimension of ≈2 and an average network mesh size of ca. 400 nm. The dynamics probed at these length scales reveals an exponential growth of the characteristic relaxation times followed by an intriguing steady state in combination with a compressed exponential correlation function and a temporal heterogeneity. The degree of heterogeneity increases with decreasing length scale. We discuss our results in the broader context of experiments and models describing attractive colloidal gels.
01 Jan 2021
TL;DR: In this paper, a new approach to the use of kernel operators derived from fractional order differential equations is proposed, and three different types of kernels are used, power law, exponential decay and Mittag-Leffler kernels.
Abstract: In this paper a new approach to the use of kernel operators derived from fractional order differential equations is proposed. Three different types of kernels are used, power law, exponential decay and Mittag-Leffler kernels. The kernel's fractional order and fractal dimension are the key parameters for these operators. The main objective of this paper is to study the effect of the fractal-fractional derivative order and the order of the nonlinear term, $ 1