Fractals

In this paper, we construct evolving networks from $n$-dimensional Sierpinski cube. Using the self-similarity of Sierpinski cube, we show the evolving networks have scale-free and small-world properties.

The multiphase flow through unsaturated porous media and accurate estimation of relative permeability are significant for oil and gas reservoir, grounder water resource and chemical engineering, etc. A new fractal model is developed for the multiphase flow through unsaturated porous media, where multiscale pore structure is characterized by fractal scaling law and the trapped water in the pores is taken into account. And the analytical expression for relative permeability is derived accordingly. The relationships between the relative permeability and capillary head as well as saturation are determined. The proposed model is validated by comparison with 14 sets of experimental data, which indicates that the fractal model agrees well with experimental data. It has been found that the proposed fractal model shows evident advantages compared with BC-B model and VG-M model, especially for the porous media with fine content and texture. Further calculations show that water permeability decreases as the fractal dimension increases under fixed saturation because the cumulative volume fraction of small pores increases with the increment of the fractal dimension. The present fractal model for the relative permeability may be helpful to understand the multiphase flow through unsaturated porous media.

The time series plot of electricity daily load demand is seasonal as shown by its regular repetitive pattern during the same period each year. Its behavior is determined by phase-space diagrams that are able to identify any of the following states of the series: fixed point, periodic, or chaotic. The first two deal with predictable systems. This paper focuses on presenting a new methodology to analyze the dynamics of the series in reference by using the curve formed by attractors that move in the complex plane over the Mandelbrot set according to the law dictated by the load curve. Because electrical power is a variable, it is also defined in the complex plane with the components of active power on the real axis and reactive power on the imaginary axis. Therefore, electrical power facilitates a new field of analysis in Mandelbrot fractal space. The obtained temporal curve confirms that the profile of the electric power demand is also mapped with the new fractal geometric space of the Mandelbrot set, thus providing a new contribution that extends knowledge about the dynamics of systems in fractal geometry.

The construction of an abstract expressionist artwork is driven by chaotic mechanisms that sculpt multifractal characteristics. Jackson Pollock’s paintings, for example, arise due to the random process of depositing drops and jets of paint on a canvas. However, most of the paintings and drawings try to recreate with fidelity common forms, natural landscapes, and the human figure. Accordingly, in the context of the formation of statistically self-similar objects, a question persists: will it be possible to find some vestige of multifractal structure in drawings or paintings whose elaboration process tries to avoid chaos? In this work, we scrutinize into several artistic drawings in sand to answer this intriguing question. These pieces of art are elaborated using craters, furrows, and sand piles; and some of them are inscribed on the Representative List of the Intangible Cultural Heritage of Humanity. We prove that the sand drawings analyzed here are multifractal objects. This finding suggests that a piece of visual art, which may initially appear ordered, contains many components distributed at different degrees of self-similarity that substantially increase the structural complexity.

In this paper, we make research on Katugampola and Hadamard fractional integral of one-dimensional continuous functions on $[0,1]$. We proved that Katugampola fractional integral of bounded and continuous function still is bounded and continuous. Box dimension of any positive order Hadamard fractional integral of one-dimensional continuous functions is one.

Coal fractography is a powerful tool for interpreting coal fracture behaviors, which is significant for dealing with failure issues encountered in deep coal mining. However, the accuracy of coal fractography highly depends on the method of quantitatively characterizing coal fracture surfaces. In this study, coal fractography under dynamic impact loading was investigated based on a multifractal method, the multifractal spectrum parameters were proposed to quantitatively describe the coal fracture surfaces. The width of the multifractal spectrum $\mathrm{\Delta }\alpha ={\alpha }_{\mathrm{\text{max}}}-{\alpha }_{\mathrm{\text{min}}}$ characterizes the uniformity of the surface asperity distribution, and the spectrum parameter $\mathrm{\Delta }f(\alpha )=f({\alpha }_{\mathrm{\text{max}}})$–$f({\alpha }_{\mathrm{\text{min}}})$ characterizes the proportion of dominant asperities on fracture surface. The coal fractography results indicate that larger loading rate leads to more asperities on the coal fracture surfaces, i.e. rougher fracture surfaces, and the fracture surfaces are dominated by small asperities induced by dynamic impact loading. In addition, significant anisotropy effect was found on the fracture surfaces under dynamic impact loading by the spatial distributions of multifractal spectrum parameter $\mathrm{\Delta }\alpha$. The parameter $\mathrm{\Delta }\alpha$ was further utilized to determine the macrocrack direction and microfracture markings on the coal fracture surfaces, the results transpire that the multifractal method is feasible for coal fractographic analysis under dynamic loading conditions.

In this paper, we consider decomposition of continuous functions in $C([0,1])$ in terms of Hausdorff dimension and lower box dimension. Precisely, we show that, given real numbers $1\le \alpha \le \beta \le 2$, any real-valued continuous function in $C([0,1])$ can be decomposed into a sum of two real-valued continuous functions each having a graph of Hausdorff dimension $\alpha$ and lower box dimension $\beta$. This generalizes a theorem of Wingren, also Wu and the present author. We also consider the arbitrary decomposition of continuous functions in terms of Hausdorff dimension and lower box dimension.

The boundaries of central place models proved to be fractal lines, which compose fractal texture of central place networks. However, the fractal texture cannot be verified by empirical analyses based on observed data. On the other hand, fractal structure of central place systems in the real world can be empirically confirmed by positive studies, but there are no corresponding models. The spatial structure of classic central place models bears Euclidean dimension $(d=2)$ rather than fractal dimensions $(0<D<2)$. This paper is devoted to deriving structural fractals of central place models from the textural fractals. The method is theoretical deduction based on the dimension rules of fractal sets. The main results and findings are as follows. First, the central place fractals were formulated by the $k$ numbers and $g$ numbers. Second, three structural fractal models were constructed for central place systems according to the corresponding fractal dimensions. Third, the classic central place models proved to comprise Koch snowflake curve, Sierpinski space filling curve, and Gosper snowflake curve. Moreover, the traffic principle plays a leading role in urban and rural settlements evolution. A conclusion was reached that the textural fractal dimensions of central place models can be converted into the structural fractal dimensions and vice versa, and the structural dimensions can be directly used to appraise human settlement distributions in reality. Thus, the textural fractals can be indirectly employed to characterize the systems of human settlements.

In this paper, we study topological properties of some level sets and some multifractal sets induced by Rademacher’s series and Takagi’s series, respectively. By using symbolic space, we obtain necessary and sufficient conditions for them to be residual.

In this paper, with the Hausdorff measure, the Hausdorff integral on fractal sets with one or lower dimension is firstly introduced via measure theory. Then the definition of the integral on fractal sets in ${ℝ}^n$ is given. With the variable substitution theorem in the Riemann integral generalized to the integral on fractal sets, the integral on fractal manifolds is defined. As a result, with the generalization of Gauss’ theorem, Stokes’ theorem is generalized to the integral on fractal manifolds in ${ℝ}^n$.

Electrospinning is a complex process, and it can be modeled by a Bratu-type equation with fractal derivatives by taking into account the solvent evaporation. Though there are many analytical methods available for such a problem, e.g. the variational iteration method and the homotopy perturbation method, a straightforward method with a simple solution process and high accurate results is much needed. This paper applies the Taylor series technology to fractal calculus, and an analytical approximate solution is obtained. A fractal variational principle is also discussed. As the Taylor series is accessible to all non-mathematicians, this paper sheds a bright light on practical applications of fractal calculus.

A novel predictive model for calculating relative permeability was derived based on a capillary tube model with fractal theory. Different forms of immovable water including water film (WF) and microcapillary water were incorporated in the new model. Special immovable water called lost dynamic water (LDW) was introduced in the proposed model. The results of verification show that there is agreement between the calculated results and the published experimental data and analytical model. The results indicated that the effect of LDW, WF, and stress dependence had a significant influence on the relative permeability, which cannot be neglected. A larger LDW coefficient, more dead-end pores, and corners in porous media yielded a more complex pore structure. Therefore, more water was trapped in the pore and became connate water, resulting in higher gas relative permeability and lower water relative permeability at a given water saturation. Due to the microcapillary effect, the relative permeability of the water/gas increased/decreased as the drawdown pressure increased at the same water saturation. Higher effective stress was more likely to cause rock deformation, resulting in higher gas relative permeability and lower water relative permeability at a given water saturation. This study provides a significant reference for reservoir engineers conducting water and gas two-phase flow analysis. The theoretical model is beneficial for research into the interpolation of relative permeability via numerical simulation.

This study proposed analytical solutions for permeability of a fractal-like tree network model with fractures having variable widths, which has not been reported before, if any. This model is more realistic with natural fracture networks than the traditional constant width fracture network models. The results show that considering fracture width variations decreases the permeability. Taking the fracture width ratio that equals to 0.6 and the total number of branching levels that equals to 30 as an example, the permeability decreases by more than three orders of magnitude with respect to that of a constant width fracture network model. The fracture length ratio plays a more significant role in permeability when it is larger than 0.8 than that is less than 0.8. The permeability is more sensitive to the fracture aperture ratio that is less than 0.8. When the total number of branching levels is large (i.e. 30), the permeability changes significantly (i.e. more than three orders of magnitude); whereas when the total number of branching levels is small (i.e. 5), the permeability varies in a small range (i.e. less than one order of magnitude). When taking into account the relationships among fracture length ratio, fracture aperture ratio and fracture width ratio, the parameters can be easily obtained and analytical solutions for permeability can also be easily derived. The empirical function for predicting critical hydraulic gradient is proposed, which can be used to estimate whether the fluid flow is within the linear flow regime and whether the proposed analytical solutions are applicable in the present study.

This paper presents a numerical study on the equivalent permeability of a fractured rock. A series of two-dimensional discrete fracture network (DFN) models for the calculation of equivalent permeability are generated based on discrete element method (DEM). A sufficient large “parent” DFN model is generated based on the data obtained from a site investigation result of Three Gorges Project in China. Smaller DFN models are extracted from the large “parent” DFN model to calculate the equivalent permeability with an interval of rotation angle of $15^{\circ }$. Fluid flow through fractures in both horizontal and vertical directions is simulated. The results show that when the side length of DFN models are larger than 40m, the equivalent permeability of both $k_x$ and $k_y$ become stable, indicating that a DFN model size of 40m can be approximated as a representative elementary volume (REV) for those studied rocks. Penetration ellipses are fitted using the least square method on the basis of the calculated equivalent permeability tensor and the main seepage directions of this fractured rock were determined as 63–67${}^{\circ }$. Fractal characteristics of DFN models are analyzed with box-counting method by changing the fracture trace length and fracture density, and the results show that equivalent permeability exhibits a logarithmic increasing trend with the increment of fractal dimension.

Let $D_k=\{0,r_k,\dots ,(q_k-1)r_k\}$ be an arithmetic digit set for each $k\ge 1$, where $1<q_k\in {\mathbb{Z}}^{+},r_k\in {\mathbb{Z}}^{+}$, and let ${\{b_k\}}_{k=1}^{\infty }$ be a sequence of integers larger than 1. In this paper, we prove that the Moran measure ${\mu }_{\{b_k\}\{D_k\}}$ generated by infinite convolution of finite atomic measures

For the self-affine measure ${\mu }_{M,D}$ generated by an expanding matrix $M\in M_2(\mathbb{Z})$ and an integer digit set

A better comprehension of the behavior of shale gas transport in shale gas reservoirs will aid in predicting shale gas production rates. In this paper, an analytical apparent permeability expression for real gas is derived on the basis of the fractal theory and Fick’s law, with adequate consideration of the effects of Knudsen diffusion, surface diffusion and flexible pore shape. The gas apparent permeability model is found to be a function of microstructural parameters of shale reservoirs, gas property, Langmuir pressure, shale reservoir temperature and pressure. The results show that the apparent permeability increases with the increase of pore area fractal dimension and the maximum effective pore radius and decreases with an increase of the tortuosity fractal dimension; the effects of Knudsen diffusion and surface diffusion on the total apparent permeability cannot be ignored under high-temperature and low-pressure circumstances. These findings can contribute to a better understanding of the mechanism of gas transport in shale reservoirs.

Tool wear is an important issue that happens in all machining operations when the tool exerts forces on the workpiece. Therefore, engineers should choose the optimum values for machining parameters and conditions to reduce the amount of tool wear and increase its life. Machine vibration is one of the factors that highly affects tool wear. Since both tool wear and machine vibration signal have complex structures, in this research we employ fractal theory to find out their relation. In this paper, we analyze the relation between tool wear and machine vibration signal in different experiments where the depth of cut, feed rate and spindle speed change. The obtained results showed that tool wear and machine vibration signal are related to each other in case of variations of depth of cut and feed rate in different experiments, where both fractal structures get more complex by the increment of these machining parameters. The obtained method of analysis in this research can be potentially applied to other machining operations in order to link the machine vibration to the structure of tool wear.

Investigating about the machined surface is an important issue in every machining operation. One of the factors that affect the quality of machined surface is the vibration of machine that is caused by different parameters such as the spindle speed. Therefore, the analysis of the relation between the machine vibration and surface quality of machined surface is very important. In this research, we analyze the relation between the machined surface and machine vibration from the complexity point of view. For this purpose, we employ fractal analysis. We investigate the variations of fractal structure of vibration signal collected from lathe machine versus the variations of fractal structure of the machined surface, for different depths of cut, feed rates and spindle speeds. Based on the obtained results, the variations of fractal structures of machined surface and vibration signal show a similar trend with the variations of depth of cut and feed rate, where by the increment of depth of cut and feed rate, vibration signal and machined surface get more complex. However, the obtained results in the case of variation of spindle speed do not show a consistent trend between variations of fractal structures of vibration signal and machined workpiece. The obtained method in this research can be applied to other machining operations in order to analyze the relation between the complexities of machine vibration and machined surface.