Methods Of Fractal Geometry Research Articles

Городская среда и теория детерминированного хаоса

One of the main directions in the application of fractal geometry is to create complex objects using simple algorithms using computer technology. The use of fractal geometry methods and the theory of deterministic chaos to describe real systems, in particular, in the development of the architectural appearance of the city, will allow solving problems of research and forecasting the development of the urban environment.

Automation of thermal processing of quartz glass with the following computer treatment of the quality of siloxane communication

The strength characteristics and isotropic physical properties of quartz glass give broad prospects for its use as a resonator for navigation systems of aircraft, as well as for complex optical systems and automation elements. To obtain a sufficiently hard surface with the desired properties, it is necessary to create conditions for the occurrence of siloxane bonds - silicon and oxygen compounds. During the formation of such bonds, the process of polymerization and annealing of quartz glass, high-strength resonator samples are obtained. The quality of the resulting surface is determined by different methods - fractal geometry, recumbent drop and Vickers. In the presented material, the authors propose to use the recumbent drop method (distilled water was used). To obtain samples of resonators, a resistance furnace has been designed and constructed, which operates in an automated mode using a personal computer control. Also, a computer program “Droplet stacking method” was written in Delphi 7.0, which presents the results of monitoring the annealing process, including the surface annealing temperature and contact angle. The results of measurements and calculations can be recorded as a file and observed on the display. The program interface is intuitive and convenient for controlling the process of obtaining samples. In addition, to form a high-quality connection, a program was also written to determine the radius of the nucleus and the value of the Gibbs free energy, which is necessary for the formation of the nucleus. The experimental results obtained and the calculations performed by the lying drop and fractal geometry methods have demonstrated the agreement between the results within the error limits of the methods.

SOME RESULTS OF STUDIES OF THERMAL FIELDS OF CARBON YARNS

In the present work, a study of the thermal properties of a carbon yarn was carried out. Were obtained different thermal fields at different structures of the carbon yarn. It was found that thermal fields are highly heterogeneous in the presence of defects in the form of nodes. The novelty of the work is the synthesis of the structure of the cross-section of a carbon yarn using fractal geometry methods. The results of the work can be used in the development of non-destructive testing techniques and in the design of woven electric heaters

Research on Key Technical Indexes of Coastal Reclamation Planning and Design

Large-scale coastal reclamation has become a significant land use issue worldwide for urban construction and economic development. The reclamation of coastal wetlands brings substantial economic benefits, however, the structure and function of coastal ecosystem are affected by drivers of human-caused landscape change. This research takes Hangzhou Bay and Zhoushan Islands as the case study to investigate the correlation between the coastal geomorphic complexity and the tidal range reduction rate, and to explore the control technical indexes of the design in reclamation area by a multidisciplinary approach that integrates the basic theories and quantitative methods of fractal geometry with the hydrodynamic mechanism of ocean dynamics. The results show that the coastal tidal range reduction rate is closely related to the coastline fractal dimension and patch shape index (D, S), and reveals the influence of the complexity of the coastal landscape on the tidal energy loss. In addition, based on model predictions, it can be found that the large-scale reclamation in Zhoushan will cause a serious decline in the complexity of the coastal landscape and the reduction of tidal energy, which is extremely detrimental to coastal disaster prevention. In the end, the scientific design theory and quantitative control indexes of reclamation are put forward to provide theoretical basis and design reference for future coastal reclamation and disaster prevention.

Fractal geometry and design of modern structures

One of the main directions in the application of mathematical methods in construction - is to create the complex objects using simple algorithms of fractal geometry with the use of computer technology. The use of fractal geometry methods to describe real systems, in particular, in the development of the architectural appearance of the city and the design of modern structures of the most amazing shape, will allow solving the problems of research and forecasting the urban environments development.

Multifractal analysis of soil particle size distribution to evaluate the effects of gypsum on the quality of sodic soils

AbstractSoil particle size distribution (PSD) essentially determines the soil's physical characteristics, including aggregate stability and porosity. Evaluating the PSD of sodic soils altered by reclamation is beneficial to the development of strategies for addressing land degradation. However, it remains unclear how to extract the features of PSD of sodic soils sensitively under gypsum reclamation. In this study, a total of 108 soil samples collected from a reclaimed sodic field were measured for PSD along with other soil properties, including hydraulic conductivity. Almost all the reclaimed sodic soil was measured to be silt loam, which was not obviously different from their original texture. The concentration, uniformity and symmetry of PSD of sodic soils have been quantified by fractal and multifractal geometry methods, in which the multifractal method provided more parameters than the fractal method to characterize the features of PSD more sensitively. The multifractal method offering predictive parameters (i.e., D1, D2, αmin and α0‐αmin) was able to significantly (p < 0.05) differentiate the features of PSD. The multifractal results indicated that the heterogeneity of PSD at 0–60‐cm depth was increased by reclamation, whereas the major particle size (silt) in these sodic soils decreased, and the combined treatment (i.e., using flue gas desulphurization [FGD] gypsum plus leaching water) created the most heterogeneous PSD. The redistribution of soil particle size caused by reclamation treatments resulted in the improvement of soil physical and chemical properties, showing increased soil aggregate diameter, raised total porosity, enhanced saturated hydraulic conductivity and decreased pH. The findings of this study bring a new opportunity to evaluate the effects of gypsum on the quality of sodic soils from the PSD perspective, which also provides a reference for evaluation of reclamation of other poor soils.Highlights The quantification of soil particle size distribution (PSD) of sodic soils remains unclear. A field evaluation using the multifractal method to characterize the PSD of sodic soil. With the reduction of silt particles, the heterogeneity of PSD was increased by reclamation. This study provides a new perspective for evaluation of the effects of gypsum on quality of sodic soils.

The analysis of Venus’ physical surface using methods of fractal geometry

In this paper, the work on investigating fractal structures on Venus was performed on the basis of observations taken by the “Magellan” spacecraft (NASA). The uncertainties in some data produced by “Magellan” were filled by the information that had been collected before – in “Venera 15”, “Venera 16”, and “Pioneer” missions. During the implementation of the work a digital map of Venus’ surface was built, and its spatial model was created. It is worth noting that the choice of basic level surface on Venus is defined by a certain value of potential or a point on its surface through which the geoid passes. The model of Venus’ physical surface was created use the harmonic expansion into spherical functions of altimetry data the “Magellan” mission. In the present paper, for determining and analyzing fractal dimensions the Minkowski mathematical algorithm, which is a simplified option of Hausdorff-Besicovitch dimension and provides high reliability and accuracy, was used. As a result, fractal correlations of Venus’ geoid anomalies in both longitude and latitude as well as the mean value of fractal dimensions were calculated. The following values of mean fractal dimension for Venus surface are obtained: in latitude – Dβ = 1.003; in longitude – Dλ = 0.98. Based on these values, we may conclude that the topographic model of Venus’ physical surface is close to spherical figure. The comparison between the obtained Venus fractal parameters with the ones of the Earth shows the good agreement.

Microfracture development and chemical damage analysis of limestone based on fractal theory

To explore the influence of microfracture development caused by chemical dissolution on the mechanical properties of limestone, this paper presents a new numerical simulation and quantitative analysis method. First, the dissolution rate was determined by the theory of chemical kinetics, and a differential equation that can be solved for results of the fracture evolution process by COMSOL Multiphysics was established to describe the microfracture's expansion. The fractal dimensions of the microfractures were found to have a linear relationship with the damage variables at different time periods through analysis of the simulation results with the fractal geometry method using fracture width as the index, which proves that the evolution of damage has a fractal nature. After that, a damage evolution equation was fitted to predict the deterioration in rock mechanical properties under hydrochemical actions and the predictive uniaxial compressive strength of limestone is seen to be in agreement with experimental test results. The application of the fractal geometry method has important engineering significance as it relates the development of microscopic fractures to changes in the macroscopic mechanical properties and predicts the mechanical properties of the rock under chemical damage.

Algorithmization for Calculating Fractal Parameters of the Relief of a Rough Surface According to GOST R ISO 25178-2-2014

The application of fractal geometry methods for characterizing the properties of surfaces of materials and products is described. The terminology of the standard GOST R ISO 25178-2-2014, “Geometric characteristics of products (GPS). Surface structure. Areal. Part 2. Terms, definitions and parameters of surface structure.” The functions provided by the standard for calculating fractal dimension are presented; algorithms for their calculation in the Mathcad environment are described. We give an example of calculations based on the results of measurements of the heights of surface irregularities of a gauge block, obtained using an atomic force microscope.

Fractal characteristic of microscopic pore structure of tight sandstone reservoirs in Kalpintag Formation in Shuntuoguole area, Tarim Basin

Using the fractal geometry method, the microscopic pore structures of tight sandstone reservoirs in Kalpintag Formation of Shuntuoguole area in Tarim Basin were conducted fractal characterization on the base of test analysis data such as physical property, cast thin section, scanning electron microscope and mercury injection, and the genetic mechanism of pore structure heterogeneity was investigated. The storage spaces are dominated by intergranular dissolved pore, intragranular dissolved pore and residual intergranular pore, and the throat type consists of the necking throat, lamellar throat, curved lamellar throat and tube-shaped throat. The microscopic structure type includes Type I (fractal dimension≤2.350), Type II (2.350<fractal dimension<2.580), Type III (fractal dimension≥2.580) and fracture type. The most favorable reservoirs with Type-I microscopic pore structure are mainly distributed in the Upper Member of Kalpintag Formation, while the reservoirs with Type-II and Type-III microscopic pore structures are mainly in the Lower Member of Kalpintag Formation. The sedimentation controls the heterogeneity of microscopic pore structure, and the differences on composition and particle size of sandstone lead to differentiation of microscopic pore structures. The Lower Member of the Kalpintag Formation experiences stronger compaction and cementation but weaker dissolution than the Upper Member of the Kalpingtag Formation, and thus the microscopic pore structure of Upper Member of the Kalpintag Formation is significantly worse that of the Lower Member o the Kalpingtag Formation. The Upper Member of the Kalpintag Formation with high content of brittle mineral develops microscopic fractures due to tectonic rupture, thus the permeability is improved and the heterogeneity of microscopic pore structures is enhanced; but the Lower Member of Kalpintag Formation is characterized by attrition crushing of particles and strong compaction.

СВЯЗЬ ФРАКТАЛЬНОЙ РАЗМЕРНОСТИ ПОВЕРХНОСТИ РАЗРУШЕНИЯ БЕТОНА С ХАРАКТЕРИСТИКАМИ МЕХАНИКИ РАЗРУШЕНИЯ

As it was found previously, the concrete fracture surface formed from tensile force is described by fractal geometry methods. It is shown thatthe fractal dimension value is related to the tensile stress gradient φ_i, to the aggregate size and, as shown earlier, does not depend on the strength of concrete. Moreover, the fractal dimension depends on the size of the sample only until its size reaches a value to which linear fracture mechanics is applicable. The stress intensity factor is related to the fractal dimension, and both characteristics are related to the aggregate size. A connection for the critical stress intensity factor K_Ic^f(l,φ_i) characterizing the crack resistance of the material in nonlinear fracture mechanics with the crack size l and the specimenis proposed. The stress intensity factor for a fractal crack K_Ic^f(l,φ_i) can be used to calculate structures using nonlinear fracture mechanics.

Quantitative analysis of null-dimensional (points) multiplicity by methods of fractal geometry

The algorithms of analysis and quantitative estimation of point sets on the basis of fractal geometry methods are proposed. It is shown the possibility of application of the developed method of sputtering and sputter analysis for solving applied technical problems. In particular, the method of fractal diagnostics of fuel spray nozzles with diesel engine nozzles was proposed to evaluate the efficiency of repair of fuel equipment components, as well as the method of quantitative fractal estimation of the quality of applying powder materials to metal surfaces in a tribostatic way. Using methods of fractal geometric analysis of point sets allows to efficiently analyze and optimize technological processes at the enterprises of machine-building and automobile industry.

Quantification of Fractures of Rock Analogue and Rock Analogue Mixed with Chemical Substance (Salt) Using Box-counting Method

The main aim of this paper is to quantify complex flow patterns of fractures of a mixture of raw starch and water and also that of starch, water and salt. The specific objectives are to: reveal the sort of fractures to be developed in the rock analogue, determine the effect of salt inclusion in the complex flow pattern, use the box-counting tool to quantify the complexity of the rock analogue patterns, and also deduce the significance of the quantification to geology. Rock analogue is a structure that is similar in function to a rock but the chemical composition is different. The rock analogue used in this experiment was fine-grained starch. The experiment reveals the fractal nature in flow patterns as different forms of weight % and different acts of viscosities and densities exhibit different forms and sizes of fractures. Different weights of starch were sampled with a specific and constant weight of 100g of water. 10 % weights of starch were sampled for the weight of salt. To quantify this flow experiment, two different methods of fractal geometry are applied, namely the MORFA and Box-counting. It could be observed that the higher and better the linear correlation, the lower the value of the fractal dimension and vice-versa. The average fractal dimension (D f) values for the fractures or cracks with salt (1.60) corresponds better to the D f values of Takayasu, NASA, Hirata, Barton and that of the general D f data for geophysics, than the D f values without salt (1.62).

Fractal characterization of the oxidation structure in ZrB2-SiC composite after thermal shocks

The effect of the oxidation structure on the residual strength of ZrB2-SiC composite after repeated thermal shocks was investigated. The silica-rich glasses on the oxidation surface and pores in the oxidation layer were characterized using the fractal geometry method. Their fractal dimensions were calculated by Box-counting method. The increase of silica-rich glasses and the decrease of pores contributed to the increase of the residual strength of ZrB2-SiC after thermal shocks.

Estimation of models and cycles in time series applying fractal geometry

A time series, also called a time series or chronological series, consists of a set of data, coming from realizations of a random variable that are observed successively in time. Its analysis involves the use of statistical methods to adjust models to explain their behavior and make reliable forecasts. In this article integrated autoregressive models of moving average are adjusted to the studied series, complemented with specific methods of fractal geometry as support for the detection of the existence of random cycles in the series. The present investigation implies the realization of simulations, in a first phase and, later, the analysis of temporal series of economic and social variables of the country and the region.

A Uniaxial Compression Experiment with CO2-Bearing Coal Using a Visualized and Constant-Volume Gas-Solid Coupling Test System.

Injecting carbon dioxide (CO2) into a deep coal seam is of great significance for reducing the concentration of greenhouse gases in the atmosphere and increasing the recovery of coalbed methane. A visualized and constant-volume gas-solid coupling system is introduced here to investigate the influence of CO2 sorption on the physical and mechanical properties of coal. Being able to keep a constant volume and monitor the sample using a camera, this system offers the potential to improve instrument accuracy and analyze fracture evolution with a fractal geometry method. This paper provides all steps to perform a uniaxial compression experiment with a briquette sample in different CO2 pressures with the gas-solid coupling test system. A briquette, cold-pressed by raw coal and sodium humate cement, is loaded in high-pressure CO2, and its surface is monitored in real-time using a camera. However, the similarity between the briquette and the raw coal still needs improvement, and a flammable gas such as methane (CH4) cannot be injected for the test. The results show that CO2 sorption leads to peak strength and elastic modulus reduction of the briquette, and the fracture evolution of the briquette in a failure state indicates fractal characteristics. The strength, elastic modulus, and fractal dimension are all correlated with CO2 pressure but not with a linear correlation. The visualized and constant-volume gas-solid coupling test system can serve as a platform for experimental research about rock mechanics considering the multifield coupling effect.

The Fractal Geometry of the Cosmic Web and Its Formation

The cosmic web structure is studied with the concepts and methods of fractal geometry, employing the adhesion model of cosmological dynamics as a basic reference. The structures of matter clusters and cosmic voids in cosmological N-body simulations or the Sloan Digital Sky Survey are elucidated by means of multifractal geometry. A nonlacunar multifractal geometry can encompass three fundamental descriptions of the cosmic structure, namely, the web structure, hierarchical clustering, and halo distributions. Furthermore, it explains our present knowledge of cosmic voids. In this way, a unified theory of the large-scale structure of the universe seems to emerge. The multifractal spectrum that we obtain significantly differs from the one of the adhesion model and conforms better to the laws of gravity. The formation of the cosmic web is best modeled as a type of turbulent dynamics, generalizing the known methods of Burgers turbulence.

Effect of steel fiber and carbon black on the self-sensing ability of concrete cracks under bending

Abstract This work explores the mechanical and self monitoring performance of concrete beams containing steel fiber (SF), carbon black (CB) and carbon fiber (CF), when subjected to bending. The effect of the utilization of multiphase conductive materials on the compressive strength of the concrete and the load - deflection - fractional change in impedance (FCI) relationship of the concrete beam are investigated. The relationships between FCI and crack opening displacement (COD) and sensitivity (gauge factor) are also studied. Moreover, the quantitative analysis of signal noise in FCI-COD curves has been accomplished by the fractal geometry method. The result shows that a monotonic linearity and bi-linearity increasing relationship between FCI and COD can be observed for beams with deflection-softening (Group A and Group B specimens) and deflection-hardening (Group C specimens) behavior, respectively. The diphasic conductive admixture (SF + CB) improves the FCI-COD curve with higher sensitivity and lower noise signal compared to the monophasic conductive admixture (SF) and the triphasic conductive admixture (SF + CB + CF) specimen.

New Method of Visibility Network and Statistical Pattern Network Recognition Usage in Terrain Surfaces

Abstract Many problems in the analysis of natural terrain surface shapes and the construction of terrain maps to model them remain unsolved. Almost the whole process of thematic interpretation of aerospace information consists of a step-by-step grouping and further data conversion for the purpose of creating a completely definite, problematically oriented picture of the earth’s surface. In this article, we present application of a new method of drawing 3D visibility networks for pattern recognition and its application on terrain surfaces. For the determination of complexity of 3D surface terrain, we use fractal geometry method. We use algorithm for constructing the visibility network to analyse the topological property of networks used in complex terrain surfaces. Terrain models give a fast overview of a landscape and are often fascinating and overwhelmingly beautiful works by artists who invest all their interest and an immense amount of work and know-how, combined with a developed sense of the portrayed landscape, in creating them. At the end, we present modelling of terrain surfaces with topological properties of the visibility network in 3D space.

Effect of wall roughness on performance of microchannel applied in microfluidic device

The influence of wall roughness on flow and heat transfer performance in microchannels at low Reynolds number is investigated in this paper. Two sizes of PMMA microchannels are fabricated by microinjection molding and the width is 20 μm and 800 μm respectively. The surface profile of bottom wall is described by the two-dimensional fractal geometry method and it is found there is error within 5% between surface roughness obtained by the fractal geometry method and actual roughness. Then, the effects of dimensionless relative roughness (5–7.5%), fractal dimension (1.5–1.8), aspect ratio (0.025–4) on the flow resistance and heat transfer performance are analyzed by numerical and experimental method respectively. Reynolds number considered here are 10–60. The results show that the better flow performance and heat transfer performance can be obtained with high aspect ratio of rectangular microchannel. However, increasing surface roughness not only increases the heat transfer performance, but also introduces a large flow resistance, which makes the friction coefficient rise sharply. As a result, surface roughness has great influence on the flow and heat transfer performance, and the most suitable surface morphology should be obtained according to the specific application.