Methods Of Fractal Geometry Research Articles

Short-process leaching and kinetic behaviour of aluminium and silver from waste photovoltaic modules

Efficient recycling and reuse of resources are essential for global resource transformation. Researchers worldwide have focused on the proper disposal of waste photovoltaic (PV) modules produced by the rapidly developing PV industry. The utilisation of waste PV modules has high economic value and helps prevent the pollution of harmful elements in the environment to achieve sustainable development in the PV industry. This work proposes short 'rapid H2O2-H2SO4 leaching' for the efficient recovery of aluminium and silver from waste PV modules. The extraction for aluminium and silver were 99.20 % and 99.01 %, respectively. Fractal geometry method was used to modify and optimise the leaching kinetics. The fitting calculation indicated that the leaching of aluminium and silver conformed to internal diffusion-controlled process.

FRACTAL NANOPARTICLES OF PHASE-SEPARATING SOLID SOLUTIONS: MORPHOLOGY-DEPENDENT PHASE EQUILIBRIA IN TUNGSTEN HEAVY PSEUDO-ALLOYS

In this paper, we simulate thermodynamically the morphology-dependent phase equilibria in core-shell nanoparticles of a phase-separating solid solution using the example of W-Cr heavy alloy. The morphology of a nanoparticle is described in the framework of the fractal geometry methods. Itis shown that there are two possible heterogeneous states in a nanoparticle while the compositions of phases in both states differ from each other. The dependences of mutual solubilities of components on the temperature are obtained while the behavior of these dependences significantly differs depending on the particular state and the morphology of nanoparticle under consideration. In nanoparticles of a very complicated morphology, the phase separation itself gets suppressed and the nanoparticle remains inthe homogeneous state at the temperatures significantly below the macroscopic value of the upper critical dissolution temperature. The demonstrated regularities are explained based on three mechanisms of reducing the free energy of the system and the “competition” between them. In the final section, a method for calculating the equlibrium size distributions and average characteristics of nanoparticle ensembles is described along with a technique of measuring nanoparticle fractal dimensions based on the microscopy data.

Objects Camouflage Possibilities Analysis in the Modern Military Conflicts Conditions

The article analyses the methods of camouflage used in modern military conflicts. The article gives a historical overview of camouflage of military equipment and personnel since the beginning of the twentieth century. The mathematical and physical principles that should be taken into account in the manufacture of camouflage nets are discussed, the requirement for which is the difficulty in recognising and differentiating between an artificial surface and real terrain. The mathematical model of a camouflage net pattern is a fractal, a self-similar scale-invariant object of noninteger topological dimension. From the point of view of physics, the fractal dimension is a statistical value that demonstrates how densely a fractal fills a space. This means, in particular, that by dividing a photo of a real surface into cells and then counting the number of black cells in relation to all of them (in the simplest case of a photo of a winter forest, for example), you can find out the average indicator of the landscape's scale similarity, which can be reproduced on a camouflage grid. In the context of active military operations caused by russian aggression, for effective camouflage, it is proposed to use the concepts and methods of fractal geometry in the manufacture of camouflage means, in particular camouflage nets, to maximise the imitation of natural landscapes and structures that will not be easy to recognise.

Thermal-hydraulic performance of flat-plate microchannel with fractal tree-like structure and self-affine rough wall

ABSTRACTInspired by the natural bifurcating structures, tree-like microchannels have been applied for microelectronics cooling. In order to understand the thermal-hydraulic performance of a flat-plat tree-like microchannel, successive branching ratios of tree-like structure are optimized based on minimization of flow resistance. It is shown that the optimal successive diameter ratio of symmetrical and dichotomous structures under volume constraint follows Murray’s law, while the optimal successive length ratio under the constraint of fixed channel area follows the power law 2−2/3. A mathematical model of convection in disc-shaped heat sink composed of a tree-like microchannel with self-affine rough surface is developed by the fractal geometry and finite element method. The flat-plate tree-like micro-channel with optimal successive diameter and length ratio shows enhanced thermal-hydraulic performance. The Nusselt number of the flat-plat tree-like micro-channel increases with the inlet Reynolds number and the self-affine fractal dimension of the rough wall. The present optimization method and mathematical model for the flat-plate tree-like microchannel shed light on the design of flat-plate micro-channel heat sinks and flow channel in fuel cell among other potential cooling applications.

Морфометрический анализ водосбора реки Серлиг-Хем (Тува) с использованием ГИС и ДДЗ

The Geographic Information System (GIS) serves as an effective tool for determining the structure of the catchment area, water resources management and planning. An important aspect of watershed characteristics is quantitative analysis performed using geoinformation technologies. In this study, GIS and fractal geometry methods were used to identify morphological features and analyze the properties of the Serlig-Khem River basin (Todzhinsky district), Republic of Tyva. The analysis of morphometric parameters includes both linear and areal aspects of the basin. The area of the studied catchment basin, determined in the GIS environment, is 3032.79 km2, mainly with a tree-like asymmetric type of river network pattern. The average bifurcation coefficient of 3.13 indicates that the basin is largely controlled by the structure. The density of the river network is 0.34 km/km2, the basin is pear-shaped. The territory under study has a predominantly mountainous character, with a slope of up to 40-30 degrees within the Academician Obruchev ridge and on the meridional segment. The fractal analysis carried out in the work confirms the assumption about the relationship between the value of the fractal dimension indicator and the degree of coverage of the catchment area by the lines of the river network.

Interplay of fragmentation, flow, mingling, and mixing in composite pseudotachylyte-ultracataclasite veins – An example from the Main Central Thrust (central Nepal)

The occurrence of pseudotachylyte-ultracataclasite veins at the Main Central Thrust (central Nepal) is reported here for the first time. The veins are planar, mm-thin layers concordant to the main foliation and layering of the wallrock. They contain elongate domains with variable amount of glasses and µm to mm-sized angular to sub-angular fragments. The glasses indicate the presence of pseudotachylyte and its origin by frictional fusion and subsequent rapid cooling. Amoeboid and partly cuspate domain geometries with diffuse contacts result from simultaneous movement and mingling of pseudotachylyte and ultracataclasite materials during injection into opening foliation-parallel cracks. Frequency distributions of fragment sizes, distribution patterns of fragments, and lobate boundaries of mingled materials are quantified by fractal geometry methods. They follow various power laws on various scales and indicate repeated processes of fragmentation, injection of mobile material, flow, mingling, mixing, and solidification. The formation of composite pseudotachylyte-ultracataclasite veins correlates with thrusting along the Main Central Thrust, specifically with the reactivation of the thrust in the Late Miocene to Early Pliocene.

РАЗРАБОТКА ПОДХОДА К ПОВЫШЕНИЮ КАЧЕСТВА РЕМОНТА ОСЕЙ КОЛЕСНЫХ ПАР

One of the main requirements, which is set to executive surfaces of wheelset axles, is an accuracy of a form in transverse direction (deviation from roundness). Since methods of final processing (centerless grinding and rolling) don’t correct inaccuracies of the form, for the provision of the required accuracy of the form turn machining operations are used, including operations on the numerically controlled machines. At the application of such machines it is necessary to reveal processes that have a major impact on processing inaccuracy and to develop an approach that will decrease the impact. The paper considers features of creating a texture of the transverse profile of rotation body parts at turning. With the use of the random processes theory and methods of fractal geometry the authors studied systematic and random components of the profile. On the basis of analysis of gained results the authors show that at finishing processing vibrations of elements of the processing system have a major impact. As a result, they have formulated an approach to automatic provision of required parameters of accuracy of the form of wheelset axles at processing with the use of numerically controlled machinery.

Railways and urban expansion: how does rail transport affect urban expansion in metropolitan areas? (Warsaw and Copenhagen case)

ABSTRACT Railway transportation plays a key role in the formation of the spatial structure of metropolitan areas. The metropolitan areas with coherent and regular railways seem to have more disciplined spatial development. The present study aims to express the relationship between the railways and the discipline of spatial structure using a developmental-applied approach. To this end, two metropolitan areas with railway coverage are selected as case studies, and four methods of minimum-distance estimation, spatial regression, fractal geometry, and cellular automata are used to analyze the data. The results indicate that the railway plays a direct role in urban expansion so that, in Copenhagen with an orderly railway network, urban development is channelized around the railway but in Warsaw with a more complicated railway network, urban expansion is more dispersed and located around rail intersections.

Micro-pore structure and fractal characteristics of deep shale from Wufeng Formation to Longmaxi Formation in Jingmen exploration area, Hubei Province, China

To explore the pore structure and fractal characteristics of shale reservoirs in the Jingmen exploration area and quantitatively evaluate the heterogeneity and complexity of pores, taking Wufeng-Longmaxi formations of Well YT3 as the research object, the pore structure characteristics of shale reservoirs are analyzed by low-pressure nitrogen adsorption experiments, total organic carbon (TOC) content testing, X-ray diffraction analysis, etc., and the FHH fractal model is established based on the low-temperature nitrogen adsorption fractal geometry method. The relationship between fractal dimension and shale composition, pore structure, physical property, gas-bearing property, and burial depth is discussed. The results show that (1) The lower section of Longmaxi Formation and Wufeng Formation have significantly higher TOC concentrations than the upper section of Longmaxi Formation. With the increase of burial depth, the silicate minerals increase gradually, while clay minerals decrease. (2) The on-site desorption gas content of shale shows that the gas content of the upper section of Longmaxi Formation with low TOC abundance is lower than the lower section of Longmaxi Formation and Wufeng Formation. In terms of shale gas composition, nitrogen dominates the upper section of Longmaxi Formation, whereas methane dominates the Wufeng Formation and lower section of Longmaxi Formation. (3) The isothermal curve of shale under low-pressure nitrogen is identical to IUPAC type IV, while the adsorption hysteresis loop is similar to types H3 and H4. The pores are mainly micro medium pores distributed below 50 nm, and they are flat and slit. The lower section of Longmaxi Formation and Wufeng Formation have significantly larger adsorption volumes than the upper section of Longmaxi Formation, and the abundance of organic matter provides a large amount of organic pore storage space. (4) The BET specific surface area and BJH total pore volume in the lower section of Longmaxi Formation and Wufeng Formation with rich organic matter are significantly larger than in the upper section of Longmaxi Formation with low organic matter, whereas the average pore size is significantly smaller. (5) Shale pores have obvious fractal characteristics, with D2 being higher than D1, indicating that the pore structure is more complicated than that of the pore surface. Fractal dimension has a significant positive correlation with TOC content, BET-specific surface area, and burial depth, a weak positive correlation with quartz content and BJH total pore volume, a significant negative correlation with clay mineral content and average pore diameter, and almost no correlation with porosity and permeability. Many factors that affect the fractal dimension. Correlation analysis reveals that the mass fraction of TOC and clay mineral, specific surface area, average pore diameter, and burial depth are the main controlling factors. The fractal dimension can be utilized to quantify the complexity and heterogeneity of pore structure, which provides a concept for studying the distribution characteristics of shale pore structure and reservoir evaluation.

FRACTAL DIMENSION IN MORPHOLOGY AND MEDICINE: THEORETICAL BASES AND PRACTICAL APPLICATION: review

Morphometry is an integral part of most modern morphological studies and the classic morphological morphometric methods and techniques are often borrowed for research in other fields of medicine. The majority of morphometric techniques are derived from Euclidean geometry. In the past decades, the principles, parameters and methods of fractal geometry are increasingly used in morphological studies. The basic parameter of fractal geometry is fractal dimension. Fractal dimension allows you to quantify the degree of filling of space with a certain geometric object and to characterize the complexity of its spatial configuration. There are many anatomical structures with complex irregular shapes that cannot be unambiguously and comprehensively characterized by methods and techniques of traditional geometry and traditional morphometry: irregular linear structures, irregular surfaces of various structures and pathological foci, structures with complex branched, tree-like, reticulated, cellular or porous structure, etc. Fractal dimension is a useful and informative morphometric parameter that can complement existing quantitative parameters to quantify objective characteristics of various anatomical structures and pathological foci. Fractal analysis can qualitatively complement existing morphometric methods and techniques and allow a comprehensive assessment of the spatial configuration complexity degree of irregular anatomical structures. The review describes the basic principles of Euclidean and fractal geometry and their application in morphology and medicine, importance and application of sizes and their derivatives, topological, metric and fractal dimensions, regular and irregular figures in morphology, and practical application of fractal dimension and fractal analysis in the morphological studies and clinical practice.

Modeling the Chemical Pollution of the Area by the Random-Addition Method

Currently, a significant group of industrial facilities can be classified as chemically hazardous facilities (CHFs). To predict the spread of harmful impurities in the programs being developed, Gaussian and Lagrangian models are actively used, on the basis of which the complexes used both in the EMERCOM of Russia and in research organizations are being implemented. These complexes require the introduction of a large amount of information, including the characteristics of the wind field in the distribution of an emergency chemically hazardous substance, which limits their use. In systems, the formation of which is influenced by a large number of different random factors, spatial scaling (similarity) is often found, and one or another parameter can be described using the methods of fractal geometry, which in the past few decades has been actively and successfully applied to the description of various physical objects. The purpose of this study is to analyze the possibility of using the random-addition method for early prediction of the distribution of harmful impurities in the surface air layer during the short-term release of a substance on the surface as a result of an emergency.

The neuroglia of the optic nerve. Part II. Fractal morphometry of astrocytes

The astrocytes of the optic nerve form a network of cells around the nerve fibers and - in particular - establish contacts with the meningeal envelope of the nerve. The morphology of these cells has been analyzed by applying fractal methods (FracLac software) to camera lucida (two-dimensional) drawings of astrocytes, taking the following three parameters into consideration: fractal dimension (DB), lacunarity (lambda) and density. These parameters provide information on the morphological complexity, heterogeneity and size of the cells respectively. All three parameters demonstrate that differences exist between astrocytes located in the central region of the optic nerve and those in a marginal position. The former are larger than the latter, which are closely associated with the meningeal envelope of the nerve. Our results suggest that peculiarities in the morphology of astrocytes are related to the anatomical region in which they are located, where they adapt to the gaps that remain between other neural elements, as is the case of the myelinated nerve fibers of the white matter. This theory is supported by the results of a study in which the morphology of the astrocytes of the corpus callosum were analyzed by applying the same fractal geometry methods.

Application of fractal and multifractal analysis on Blue Nile drainage patterns in the morphostructural analysis of the Ethiopian highlands, Ethiopia

Fractal geometry methods provide a quantitative description of complex landscape networks, such as drainage patterns. This article focuses on the determination of the fractal dimension ( F D) and Renyi multifractal spectrum ( F Dα) of drainage patterns of the Blue Nile Basin in the Ethiopian Highlands. The Ethiopian Highlands are one of the most tectonically influenced areas in the world. The spatial distribution of drainage patterns reflects a decreasing influence of tectonics with increasing distance from the Main Ethiopian Rift (MER) to the center of the Ethiopian Highlands. The novelty of the paper is given by the calculation of F D and F Dα of drainage patterns as a part of morphostructural analysis. We analyzed how the values of F D and F Dα in each of 145 local drainage patterns changed depending on the changes of tectonic influences on landscape formation. Our methodologies provided a way of distinguishing the morphologies of tectonic and non-tectonic drainage patterns using the fractal/multifractal technique. The results of the study show that old dendritic drainage patterns located in the western part with low tectonic activity and an absence of faults had the lowest F D and F Dαvalues. The opening up of the MER and the neotectonic uplift of the Ethiopian Highlands accelerated the development of drainage patterns by activating river erosion and river piracy. Dendritic drainage patterns of the volcanic platforms have changed to young deep cut trellis patterns, and their F D and F Dα values are increased. Accelerating Quaternary uplift created the recent high landscape in the eastern part of the Blue Nile Basin, where the tectonic activities correspond to the MER. This young Quaternary landscape is characterized by prevailing deep and narrow valleys, and the presence of morphodynamic phenomena (erosion, landslides, earthquakes), where the trellis drainage patterns have the highest F D and F Dα.values.

Investigation of the relief geometry of the membrane of human buccal epithelium cells using atomic force microscopy methods

The analysis of the shape of the Brownian relief of the surface of human buccal epithelium cells was carried out using the method of fractal geometry. The scaling coefficients eta, zeta and the fractal dimension Df were used to estimate their actual areas Sc(fact) and linear characteristics lf. Keywords: atomic force microscopy, buccal epithelium, fractal number.

Исследование геометрии рельефа оболочки клеток буккального эпителия человека методами атомно-силовой микроскопии

The analysis of the shape of the Brownian relief of the surface of human buccal epithelium cells was carried out using the method of fractal geometry. The scaling coefficients ,  and the fractal dimension Df were used to estimate their actual areas Scfact and linear characteristics lf.

Damage characteristics of coal under different loading modes based on CT three-dimensional reconstruction

Due to the heterogeneous characteristics of coal materials, it is difficult to carry out repetitive tests and quantitative analysis of internal damage of coal. Therefore, the damage characteristics of coal under uniaxial and true triaxial loading was studied by numerical simulation based on 3D reconstruction modeling, and the corresponding mechanics experiment was used for auxiliary parameter validation. Firstly, the CT scanning system was used to scan the coal specimen, and the Otsu method was used to segment the original CT images to extract the three-dimensional fracture network and the distribution characteristics of mineral components. Secondly, the internal fracture and mineral distribution network model of the coal was reconstructed and imported into FLAC3D for modeling, and the numerical model was validated by fractal geometry method. The average error of fractal dimension between the original CT image and the corresponding numerical model is 1.422%, showing that the numerical model is basically consistent with the CT image of the coal, and the numerical model could reflect the distribution characteristics of the internal fractures and mineral components of the coal. Then, the basic mechanical parameters and acoustic emission (AE) response characteristics of CT reconstructed coal samples were obtained. And the mechanical parameters of the numerical model were calibrated. It can be concluded that the stress–strain curve of the coal is basically consistent with that of the numerical model, and the AE response characteristics of the coal are basically consistent with the evolution of the plastic zone of the numerical simulation, indicating that the numerical model and parameters are reasonable in this work. Moreover, the damage variable evolution of coal specimen in uniaxial compression and true triaxial loading was studied. Finally, the damage equation of coal under uniaxial compression and the relationship between damage variable and body strain energy under true triaxial loading were established.

ON THE TRANSITION BETWEEN FERROMAGNETIC AND PARAMAGNETIC STATES IN MESOPOROUS MATERIALS WITH FRACTAL MORPHOLOGY

In this paper, we have shown how the presence of pores and pore morphology influence on magnetic phase transition temperatures in mesoporous ferromagnetic materials. Model calculations have demonstrated the possibility to obtain macroscopic mesoporous samples with notably reduced Curie temperatures which is also further depressed in the case the pore morphology is more complicated. The results have been obtained on the basis of the experimentally verified correlation between the Curie temperature and cohesive energy of the material and illustrated using the examples of pure mesoporous iron, nickel and cobalt while pore morphology has been determined by the methods of fractal geometry. Several practical applications of mesoporous materials with tuned values of the Curie temperature have also been discussed in the final section.

Combined tillage with elements of Strip-till technology for maize in the Ciscaucasian zone

Relevance. The combined method of the main tillage, using certain methods of influence on the cultivated layer, forms a certain soil density. During the growing season of corn for grain, this indicator changes depending on the seasonal soil moisture and the intensity of the development of the root system of the plant, which is ultimately related to the yield of the crop. Material and method. The objects of research are two systems of basic tillage for corn for grain according to the predecessor winter wheat in the zone of unstable moisture of the Stavropol Territory using a dump and a combined method of basic tillage with elements of Strip-till technology. Soil: southern calcareous chernozem, slightly humus. The functional dependence of soil density on the supply of productive moisture and the intensity of development of the root system of grain corn was established by the method of the theory of dimensions. The method of fractal geometry was used to determine the degree of filling the soil space with root systems of grain corn under various systems of basic tillage.Results and Conclusions. It was found that the density of the soil is in direct functional dependence on the supply of productive moisture in the cultivated soil layer and the intensity of development of plant roots. The soil density during the seeding and flowering periods is higher by the Strip-till technology in comparison with the traditional one on average over the years by 0.02 g / cm3 and 0.03 g / cm3, respectively, and the moisture reserve in the spring period with Strip-till is 12 mm. The intensity of development of the root system according to the indicator D with the Strip-till system (1.58) by 0.31 units, more than with recommended processing (D = 1.27). The yield of corn for grain using the Strip-till technology is on average 0.22 t / g higher than with the recommended one, at a lower cost by 2395 rubles / ha.

Fractal derivative fractional grey Riccati model and its application

Fractal geometry methods are widely used to describe the geometric characteristics of complex systems, statistical behaviors, and power-law characteristics of data results. In this study, the fractal derivative and fractional cumulative generating operators are introduced into the grey Riccati model to establish the fractal derivative and fractional grey Riccati model (FDFGRM), and the analytical solution of the model is obtained. At the same time, multi-objective quantum particle swarm optimization algorithm is used for parameter optimization. We consider China's cement production, natural gas production, and primary aluminum production as examples to verify the validity of the model. The results show that the fitting and testing effects of the FDFGRM are better than those of other models. Finally, FDFGRM is used to predict the future trends of these three cases.

Characterization of discontinuity surface morphology based on 3D fractal dimension by integrating laser scanning with ArcGIS

The fractal geometry method has been employed to quantitatively characterize the roughness of a rock discontinuity, which is one of the key factors affecting its shear strength and the seepage characteristics of a rock mass. However, the current fractal methods involving the three-dimensional discontinuity morphology suffer from one or more problems, such as a complicated calculation procedure, an inaccurate calculation result and an inability to characterize the undulation and anisotropy of a discontinuity. To cope with these problems, the discontinuities in artificial granite samples with irregular and undulating surfaces were taken as examples, and a quantitative study on the discontinuity morphology was conducted based on the method of three-dimensional laser scanning in combination with ArcGIS data processing, geographical research, theoretical calculations and regression analysis. After performing systematic research, we proposed an extensive 3D fractal dimension including three discontinuity morphological parameters, i.e. the fractal dimension of discontinuity morphology, the ratio between the maximal undulating amplitude and the discontinuity length, and the average value of all the apparent dip angles of the discontinuity surfaces dipping opposite the shear direction. The extensive 3D fractal dimension can comprehensively characterize the roughness, undulation and anisotropy of the discontinuity morphology. A set of theoretical calculation methods were then developed to determine the three discontinuity morphological parameters of the extensive 3D fractal dimension based on ArcGIS. We finally established a mathematical expression of the extensive 3D fractal dimension. Compared with the current fractal methods, the extensive 3D fractal dimension can effectively compensate for the inability to characterize the undulation and anisotropy of the discontinuity morphology. Its calculation methods have the advantages of simplification, low-time consumption and high precision.