Fractal Fields Research Articles

Numerical and experimental study on fractal flow field for improving the performance of non-aqueous redox flow battery

The flow field design of the non-aqueous redox flow battery (RFB) decides the electrolyte flow and mass transfer of active species inside porous electrode, which has a significant influence on the operation of RFB. In this work, a fractal tree-like flow channel is proposed to enhance the electrolyte convection and optimize the concentration distribution of active species inside the graphite felt electrode, thus boosting the overall battery performance. Using the finite element simulation method, a three-dimensional numerical model of single cell is established to comparatively study the steady-state discharge process characteristics of deep eutectic solvent (DES) electrolyte-based vanadium‑iron RFBs with fractal and serpentine flow fields, respectively. Moreover, by assembling the single cell of this RFB and building the full cell test system, an experimental study is conducted to compare the discharging output voltage and power of the RFBs with serpentine flow field and fractal flow fields of different fractal dimensions. The numerical results show that the fractal tree-like flow field can improve the discharging power of RFB as well as reduce the pumping loss compared with the conventional serpentine flow field. That is mainly attributed to that fractal flow field is conducive to reducing the polarization loss and strengthening the convection in porous electrode simultaneously. What's more, the role of fractal dimension of flow field is also investigated numerically, thus knowing that large fractal dimension promotes lower pumping loss while small fractal dimension is in favor of higher output voltage. In addition, the experimental results also demonstrate that the fractal flow field could improve the output voltage and maximum discharging power of RFB compared to serpentine flow field. This work provides an optimization method for the flow field design of DES electrolyte-based vanadium‑iron RFB, and is expected to be employed in the other non-aqueous RFBs.

Synchronization of Julia Sets in Three-Dimensional Discrete Financial Models

When aiming to achieve consistency in fractal characteristics between different models, it is crucial to consider the synchronization of Julia sets. This paper studies the synchronization of Julia sets in three-dimensional discrete financial models. First, three-dimensional discrete financial models with different model parameters are proposed and their Julia sets are presented. According to the model forms, two kinds of synchronous couplers that can achieve synchronization of Julia sets between different models are designed by changing the synchronization parameters. The proposed synchronization method is theoretically derived and the efficiency of different synchronous couplers are compared. Finally, the effectiveness is verified by Julia sets graphics. This method has reference value for theoretical research into financial models in the field of fractals.

Complex field reversal dynamics in nanomagnetic systems

Nanomagnetic materials, built from thin, patterned films of ferromagnetic materials, began as analogues to frustrated magnetism. Their low energy of operation and emergent properties make them strong candidates for physics-based devices. A recent model of how nanomagnetic domains flip, the Glauber mean-field model, is used here to understand how systems of nanomagnets evolve when opposed by external field. This reversal can be expressed in an analytical form in the case of one-dimensional chains and trees at zero temperature, where the cascade of spin flips gives rise to harmonic power spectra. The same cascades in two and three dimensions form fractal field reversal clusters whose shape depends on the strength of the field and the tuning of interactions between nanomagnets.

Critical properties of the optical field localization in a three-dimensional percolating system: Theory and experiment

We systematically study the optical field localization in an active three-dimensional (3D) disordered percolating system with light nanoemitters incorporated in percolating clusters. An essential feature of such a hybrid medium is that the clusters are combined into a fractal radiation pattern, in which light is simultaneously emitted and scattered by the disordered structures. Theoretical considerations, based on systematic 3D simulations, reveal nontrivial dynamics in the form of propagation of localized field bunches in the percolating material. We obtain the length of the field localization and dynamical properties of such states as functions of the occupation probability of the disordered clusters. A transition between the dynamical states and narrow point-like fields pinned to the emitters is found. The theoretical analysis of the fractal field properties is followed by an experimental study of the light generation by nanoemitters incorporated in the percolating clusters. The experimental results corroborate theoretical predictions.

RESEARCH ON THE FLOW FIELD OF THE PEMFC BIPOLAR PLATE BASED ON THE TREE-LIKE FRACTAL THEORY

Tree-like branching structures occur in both natural and artificial transport systems, which have fascinated multidisciplinary researchers to study and apply the transport mechanisms of tree-like branching structures for decades. In this paper, the flow field of a proton exchange membrane fuel cell (PEMFC) with a Y-shaped tree-like fractal structure was studied utilizing Murray’s law in fractal theory. The polarization curve, gas concentration distribution, water concentration distribution, pressure drop distribution, and current density distribution of the PEMFC are numerically simulated in this research, and the transmission law is analyzed with different channel branching angles [Formula: see text]. The results demonstrate that the optimal branching angle of the designed tree-like fractal flow field is [Formula: see text]. When compared to a parallel flow field under identical conditions, the maximum output power density of the fractal flow field with a branching angle of [Formula: see text] is 26.7% higher. The optimal angle of [Formula: see text] for symmetric branching flow derived from Murray’s law was shown to be applicable to the flow field design of the fuel cell, improving the transport characteristics of the reaction gases and the overall performance of the PEMFC. This research may provide further references for the design of flow fields in fuel cells.

Properties of Fractal Speckle Fields: Theory and Applications

Properties of Fractal Speckle Fields: Theory and Applications

FRW model with two-fluid source in fractal cosmology

Present paper deals with flat Friedmann - Robertson - Walker (FRW) model with two - fluid source in fractal cosmology. In this model one fluid represents matter content of the universe and another fluid is radiation field modeling the cosmic microwave background. To get the deterministic model, we have used the relation between pressure and density for matter through the gamma law equation of state $p_{m} = (\gamma-1)\, \rho_{m}, \,\, 1\leq \gamma\leq2$ . The solutions of fractal field equations are obtained in terms of Kummer's confluent hypergeometric function of the first kind. Some physical parameters of the models are obtained and discussed their behavior with the help of graphs in detail.

Three-dimensional modeling and analysis of fractal characteristics of rupture source combined acoustic emission and fractal theory

Deep mining exploration and the sustainable use of enormous data via acoustic emission (AE) systems is a major experimental approach for fulfilling the perception, characterization, and forecasting for rock engineering catastrophes such mines, tunnels, and water conservation. To carry out the fine numerical characterization of the nonlinear and disordered time-space-energy information in the AE system, we use the fractal theory to analyzing. The key elements such as time, space, and energy of the fracturing source are comprehensively extracted. The fractal dimension statistical framework and its optimal dimension are designed, and the three-dimensional (3D) analysis model that can characterize the actual failure and instability process is constructed. The AE experimental data and numerical simulation are used to obtain the spatial clustering and energy dissipation fractal dimension characteristics of regional AE events in the time domain. The model is optimized and verified on the spot. The results show that the time-space-energy parameters of AE can be characterized by fractal dimension. The fractal dimension analysis model can transform and finely characterize the failure anomaly data. The distribution results of the fractal dimension value field can be consistent with the actual failure. Through iterative optimization and comprehensive verification of the scientificity and applicability of the fractal dimension analysis model, a comprehensive fractal dimension representation method of time-space-energy of fracture source in rock failure process is formed, which can provide guidance for the service and analysis of existing monitoring systems in mines, and provide theoretical support and technical conditions for fine judgment of rock mass failure in mines.

General escape criteria for the generation of fractals in extended Jungck–Noor orbit

The aesthetic patterns play a key role in the field of fractals. Due to self similarities in the nature of fractals, researchers used the fractals in many fields of sciences (i.e. in Mathematics, Computer Science, Physics, Image Encryption, Biology and Chemistry). The most studied fractals types are the Mandelbrot sets (MSs) and Julia sets (JSs). To generate fractals, escape criteria is required. In this work, a general escape criteria is proved via extended Jungck-Noor iteration with s-convexity. These results are used in algorithms to present the generation of fractals in extended Jungck-Noor orbit for general complex polynomial f(x)=∑i=0paixi with p≥2, where ai∈ℂ for i=0,1,2,…,p. The graphics of MSs and JSs are demonstrated in the examples. The variations in MSs and JSs for different values of involved parameters are also shown.

Influence of Conducting Particle on DC Flashover Characteristics and Tracking Property of GIS/GIL Insulator

Flashover at gas/insulator interface acts as a vital challenge to the safe and stable operation of gas-insulated system especially when involved with surface metallic particle. In this paper, flashover characteristics at SF₆/epoxy insulator interface were investigated concerning on the tracking property based on image analysis. DC flashover experiments were operated for basin-type spacer with metallic particle at various locations. Flashover voltage was obtained and tracking on the surface after successive flashover was captured. Improved image analysis method based on fractal theory was carried out to evaluate tracking pattern quantitatively. The relation between the fractal dimension of tracking and field distribution at particle tip was revealed. Obtained results showed that flashover voltage decreased due to severe field distortion with moving particle from ground to HV electrode. Meanwhile, fractal dimension of tracking covering the area from particle to ground electrode indicated centralized tendency. Simulation result confirmed this variation as the consequence of centralized field distribution at the tip of particle. The quantitative relation between fractal dimension and field distribution suggests new sight to estimate the influence of metallic particle on insulation property in the gas-insulated system.

Fractal diffusion patterns of periodic points in the Mandelbrot set

The Mandelbrot set is one of the most profound study objects in the fractal field. However, the characteristics of internal structures of the Mandelbrot set are not fully studied, which is an obstacle to further understanding of its convergence mechanism. Here, we demonstrate that if the threshold is not taken when calculating the periodic number in the iteration process, the internal structures formed by the collection of periodic points will show fractal diffusion patterns from the theoretical area to the target area. Even when the threshold is taken, these diffusion characteristics remain until the threshold is larger than a certain value. This critical value of the threshold can be regarded as a transition gate of the internal structure from an unstable state to a stable state. The findings of this study indicate that the fractal patterns of the Mandelbrot set can not only exist in the outer area but also the interior area of the Mandelbrot set.

The extended generalized radial flow model and effective conductivity for truncated power law variograms

Pumping tests are established for characterizing spatial average properties of aquifers. At the same time, they are promising tools to identify heterogeneity characteristics such as log-conductivity variance and correlation scales. We present the extended Generalized Radial Flow Model (eGRF) which combines the characterization of well flow in fractal geometry with an upscaled conductivity for pumping tests in heterogeneous media. We show that the eGRF is a generalization of previously solutions, such as that of Barker, Butler and Neuman. We derive effective conductivities for uniform and well flow conditions in heterogeneous log-normal media with a truncated power law correlation structure through the upscaling procedure Coarse Graining. The radial-dependent effective conductivity for well flow reflects the gradual change of heterogeneity impact on average pumping test drawdowns. We then combine upscaled conductivities with the eGRF model to determine the effective pumping test solution. We provide a proof of concept by comparing theoretical upscaling results with Monte Carlo well flow simulations in heterogeneous fractal fields. The eGRF and upscaling results are implemented and made freely available as python code for transport simulation as well as pumping test analysis.

Infilling missing data of binary geophysical fields using scale invariant properties through an application to imperviousness in urban areas

ABSTRACT High-resolution modelling is needed to improve the understanding and management of storm water in cities. It requires data, which is not always available; hence the growing importance of handling missing data. Here, we use impervious areas in cities as case study. They are responsible for rapid runoff that can generate surface flooding. A methodology to handle such binary missing data relying on scale-invariant properties is presented. It uses a previous study, which showed in 10 peri-urban areas that imperviousness exhibits scale-invariant features from metres to kilometres, to generate realistic scenarios for the missing impervious data. More precisely, fractal fields are commonly simulated thanks to a simple binary multiplicative cascade process (β-model). Here we condition it to the available data. Numerical simulations are used to confirm theoretical expectations. They are then implemented to infill missing impervious data on a 3 km2 catchment and the corresponding uncertainty is quantified.

Fractal movement in the visualisation processes

The article is dedicated to the transformation of learning processes. The causal reasons for the widespread use of the fractal approach in the use of techniques and methods of visualising educational information are considered. The specifics of the use of fractal movement in the organisation of the educational process are described. The role and place of the teacher in the management of transfer and disclosure processes is noted. The fractal organisation of the learning route algorithm is characterised by a markedly deep determinism of the factors that determine the purpose and result of learning. The text notes the conditions of filling the thesaurus of the learner, improving its intellectual property in the form of transmission, assimilation and rejection of information. The elements of the thesaurus formed in the fractal educational field orient both the teacher and the learner to an extended view of a qualitatively in-depth approach to presenting in a specific unity of elements of visualised educational material. The connection between the individual abilities of the teacher and the potential of the learner is been traced. The conditions of the optimal combination of information flow and abilities of the learner, determining the success of training, are highlighted. Positive experience of the implementation of the proposed provisions of the educational process implies further analysis of the mechanisms and procedures of the fractal educational space.

Numerical study of a three‐dimensional electroosmotic micromixer with Koch fractal curve structure

AbstractBACKGROUNDA new type of three‐dimensional electroosmotic micromixer is designed based on the Koch fractal curve principle. According to the fractal principle, the influences of fractal structure, voltage size, electrode spacing and electrode configuration on the mixing performance were proposed and studied.RESULTSThe simulation results show that under the combined action of fractal and electric field, irregular alternating vortices are generated in the microchannel, which can disturb the flow of fluid and greatly improve the mixing of fluids in the electroosmotic micromixer. In particular, when the direct current voltage is 10 V, the electrode spacing is 0.15 mm and the electroosmotic micromixer has three groups of alternately placed electrode pairs, the mixing efficiency can reach 98%.CONCLUSIONSThis new type of electroosmotic micromixer is suitable for many occasions, including the mixing of biochemical reagents, and has high application value. © 2021 Society of Chemical Industry (SCI).

Some Remarks on Geometry of Fractional Calculus

In recent years, researchers interested in the field of fract als and related subjects have also begun using the concept of fractional calculus in some of their investigations. In thi s paper, some interesting aspects and features of fractiona l connection derivatives in differential manifold were discussed. In particular, tr ansformation of Christoffel symbols for fractional connection, the torsion tensor of a fractional connection, and difference tensor of two fra ctional connection are presented.

A Robust Authentication Algorithm for Medical Images Based on Fractal Brownian Model and Visual Cryptography

In this paper, we aimed to discuss the security authentication requirements of medical images in the medical network, and a security authentication method is designed based on fractal and visual cryptography. Based on the discrete fractal Brownian random field model, the gray-level statistical information and spatial structure information of medical images is fully mined. The gray distribution of medical images is expressed in the form of fractal features. By using the spatial data mining methods, the data of fractal structure space is analyzed, and by using the stability of the energy structure, the authentication features are formed. Using the visual cryptography (VC), the robustness of the authentication method is further enhanced. Through the centralized test of common medical images and the comparison analysis with existing methods, it is further verified that the method is effective against common attacks such as JPEG compression, scaling, rotation operation, clipping, added noise, filtering, and blurring.

ФРАКТАЛЬНИЙ АНАЛІЗ КОСМІЧНИХ ЗНІМКІВ SENTINEL-2 ДЛЯ МОНІТОРИНГУ СІЛЬСЬКОГОСПОДАРСЬКИХ КУЛЬТУР

The subject of the study in the article is using the new approach to the processing of spatial information from satellites for more effective and operational evaluation of crops. This is due to the growing trend of access to remote sensing data, due to the improvement of spatial and temporal resolution, which can be used in the analysis of vegetation cover and other related work. The goal of the article is the capability assessment of processing the Sentinel-2 satellite imagery using fractal dimensions to agricultural plant monitoring at different phases of the vegetative. The tasks: to research the method of constructing fractal dimensions for the Sentinel-2 satellite imagery to assess the state of crops during the vegetative phase; to assess the relationship between changes in FD averages and changes in the NDVI index of different time series remote images, to determine the advantage of calculation method fractal dimensions compared to the NDVI index. The following results were obtained. It was found that the NDVI index is most often used to quantify the state of biomass during different time intervals. But this index becomes ineffective during periods of weakening of the vegetation active phase. Accordingly, it is of practical interest to evaluate the possibility of using fractal analysis of agricultural crop satellite imagery at different vegetation phases. The basis of fractal analysis of digital images is the formation of fractal dimensions fields. The analysis of changes in the FD values on different remote images time series of the grain cornfields from the «sliding window» values is carried out. The dependences of the maximum and minimum values of FD, which are in the images, on the «window» size are investigated. It is shown that the homogeneity of the underlying surface can be estimated from the magnitude of changes in the maximum values of FD with the increasing size of the «window». It is established that the pattern of the change of the FD minimum values when changing the «window» size is due to the large sharpness of the underlying surface in the images, and the anomalous behavior of these values allows determining anomalous areas of different sizes in satellite imagery. The pattern of the change in the range of FD with increasing size of the «window», which can be used to determine the homogeneity of the underlying surface in satellite imagery, as well as during the detection of abnormal areas on them. The change analysis of FD average values with an increase in the sizes of «sliding window» is carried out. It is shown that with the same size of the «window» for different image time series, the average FD will be different, which can be used to characterize the agriculture crop vegetation phase. It is established that the pattern of changes in the FD average values is the same as the NDVI indices for different satellite imagery time series of the corn crop fields and that the magnitudes of the FD average values depend on the size of the «window». The size of the «window» is recommended, which provides accommodation between the speed of image processing and the quality of the assessment state vegetation crop. It is shown that to increase the speed of formation of the FFD during the processing of large images, it is advisable to use a «jumping window» instead of a «sliding window». It is mentioned that the «jump» value can be equal to the «window» size. This «jump» value provides maximum speed and does not affect the crop satellite imagery processing quality. Conclusions. The recommended approach to the processing of spatial data from satellites allows assessing the crops' consistency using FD. The pattern of the change in the FD mean values is identical to the NDVI change in different satellite imagery time series of corn crops. In that event, when forming the FFD, data from only one channel of the Sentinel-2 satellite can be used (for example, from the near-infrared channel – b8), and to calculate the NDVI index it is necessary to obtain data from two channels (from the near-infrared and red channels – channels b8 and b4 of the satellite Sentinel-2, respectively), which will reduce the processing time. The scale of FD average values allows detecting a qualitative change in biomass. During further research, it is advisable to perform fractal analysis of Sentinel-2 satellite imagery for other crops at different phases of the vegetation.

On the fractal structure of soil moisture fields

We study the spatial structure of soil moisture fields within savanna ecosystems, whose persistence is vital because it is the driver of the entire ecological structure and function. These include changes in the physical and biogeochemical conditions of the landscape, affecting vegetation state, soil composition, water fluxes, and solar radiation. We focus on computations of the probabilistic structure of islands of soil moisture, known empirically to be related to that of tree clusters, defined as crossing properties of simulated soil moisture fields. Rainfall is modelled via Cox-Isham space-time fields endowed with characteristic scales. Results show that clusters of soil moisture islands are characterized by robust scale-free structures in the region of a phase transition whose order parameter depends on mean soil moisture. Signatures of this fractal structure are well-defined power laws of size distributions of soil moisture clusters; their perimeters-vs-area relations; variance-vs- area of the fields. These characteristics allow for the estimation of the fractal dimension of the field, and its Hurst coefficient. From the general covariance equation of a fractal field, spatial simulations are possible because its mean and variance are known from the probabilistic structure of soil moisture at a point. Our results identify the statistics of hotspots of microbial activity deduced from proper moisture islands, unattainable otherwise, and thus may guide the design of field and remote observations. The critical order parameter characterizing the phase transition establishes where the fractal structure of soil moisture fields exists as a function of the climatic drivers, and the thresholds reflecting where vegetation survives in the field. An example of application of the phase transition diagram presented here is carried out with reference to the Nylsvley savanna in South Africa.

Scaling Analysis of Two‐Phase Flow in Fractal Permeability Fields

AbstractFluid mixing in permeable media is essential in many practical applications. The mixing process is a consequence of velocity fluctuations owing to geological heterogeneities and mobility contrast of fluids. Heterogeneities in natural rocks are often spatially correlated, and their properties, such as permeability, may be described using fractal distributions. This work models the fractal characteristics of such permeability fields in which the covariance function is expressed as a power‐law function. A generalized scaling relation is derived relating various fractal permeability fields using the magnitude of their fluctuations. This relation reveals the self‐similar behavior of two‐phase flow in such permeable media. To that end, a recently developed, high‐resolution numerical simulator is employed to validate the analytically derived scaling relations. Two flow problems are considered in which flow is governed by (1) a linear and (2) a nonlinear transport equation. Due to the probabilistic representation of the fractal permeability fields, a sensitivity study is conducted for each flow scenario to determine the number of realizations required for statistical convergence. Scaling analysis is performed using ensemble averages of simulated saturation profiles and their mixing lengths. Results support the validity of the developed scaling relation across the range of investigated flow conditions at intermediate times. The dynamics of linear flow in the asymptotic regime is affected by the correlation structure of heterogeneity. In nonlinear flow, scaling behavior appears to be dominated by the degree of nonlinearity.