Fractals

Hall–Petch strengthening has been widely used in materials science, but its mechanism is not very clear yet, some inverse phenomena were observed. This paper gives a fractal approach to explanation of the Hall–Petch effect, revealing the value of the fractal dimensions is the key factor: when it is larger than one, the Hall–Petch effect is predicted; while when it is smaller than one, an inverse Hall–Petch relationship is obtained. The fractal theory can also explain the nano-effort (size effort) in nanotechnology and spider silk’s strength.

We describe the shrinking target problem for random iterated function systems which are semi-conjugate to random subshifts. We get the Hausdorff dimension of the set based on shrinking target problems with given targets. The main idea is an extension of ubiquity theorem which plays an important role to get the lower bound of the dimension. Our method can be used to deal with the sets with respect to more general targets and the sets based on the quantitative Poincaré recurrence properties.

The polar bear hairs have special hierarchical structure with fractal dimensions of golden ratio, which endows the creature with remarkable cool prevention. Fractal calculus is adopted in this paper to reveal its thermal properties, and a fractal derivative model of one-dimensional heat conduction along the hair is established and solved, the results reveal that there is an optimal hair length for the cool prevention. This paper sheds a new light on bio-inspired fabrics for fire-protection clothing and clothing for extreme environment.

Flow mechanism of water–gas phase is of significant importance to accurately understand and predict the fluid flow behavior in many fields, such as water and oil exploration, hot dry rock development and optimization of fuel cells. In this work, a novel relative permeability (RP) model for two-phase flow in fracture networks is proposed based on the fractal theory. The proposed model is expressed as a nonlinear function of saturation of water and gas with no empirical parameter. A stronger interference between water and gas phases than that of the X-model is found. The proposed model may provide a better understanding of the fundamental mechanism of water–gas phase flow in fracture network.

Friction tests were conducted by sliding a ring against a disc under oil lubricated condition. The running-in attractor was generated from measured friction force, vibration and noise signals. On the basis of correlation dimension, predictability was introduced from chaos theory to characterize the running-in attractor. Two parameters, i.e. enclosing radius and average distance were proposed innovatively. The algorithms for the proposed parameters were given as well. Correlation dimension, predictability, enclosing radius and average distance were used to describe the complexity, randomness, bound and convergence degree of a running-in attractor, respectively. This study provides new parameters to quantify the running-in attractor and is meaningful for revealing the nonlinear property of the attractor.

Analysis of cutting forces in machining operation is an important issue. The cutting force changes randomly in milling operation where it makes a signal by plotting over time span. An important type of analysis belongs to the study of how cutting forces change along different axes. Since cutting force has fractal characteristics, in this paper for the first time we analyze the variations of complexity of cutting force signal along different axes using fractal theory. For this purpose, we consider two cutting depths and do milling operation in dry and wet machining conditions. The obtained cutting force time series was analyzed by computing the fractal dimension. The result showed that in both wet and dry machining conditions, the feed force (along $x$-axis) has greater fractal dimension than radial force (along $y$-axis). In addition, the radial force (along $y$-axis) has greater fractal dimension than thrust force (along $z$-axis). The method of analysis that was used in this research can be applied to other machining operations to study the variations of fractal structure of cutting force signal along different axes.

We studied soybean leaves of the main soybean cultivars and their mutants from Shanxi Province, China, using multifractal spectrum analysis of transmission scanned gray scale images of excised leaves. We demonstrate that the parameters of multifractal spectra correlate with the biological status of the plant. The width $\mathrm{\Delta }\alpha$ of the multifractal spectrum, a parameter of the transmission scanned gray scale image, is a good indicator of the metabolic state and growth rate of a soybean leaf. In general, the larger the value of $\mathrm{\Delta }\alpha$, the more exuberant the metabolism and the faster the growth rate of the leaf. ${\alpha }_{min}$ is a parameter related with the distribution of chlorophyll in leaf, it reflects the concentration degree of chlorophyll; the smaller the value of ${\alpha }_{min}$, the higher the concentration degree of chlorophyll in the leaf. ${\alpha }_{max}$ is a parameter related with the distribution of water in the leaf, it reflects the concentration degree of water; the larger the value of ${\alpha }_{max}$, the higher the concentration degree of water in the leaf. In order to reduce the impact of errors in the calculation of $\mathrm{\Delta }\alpha$, ${\alpha }_{min}$ and ${\alpha }_{max}$, we introduced three new parameters — the characteristic width $\mathrm{\Delta }{\alpha }_C$, the characteristic ${\alpha }_S$ and the characteristic ${\alpha }_L$. We compare the values of $\mathrm{\Delta }\alpha$, ${\alpha }_{min}$ and ${\alpha }_{max}$ among the different leaves qualitatively by replacing $\mathrm{\Delta }\alpha$, ${\alpha }_{min}$ and ${\alpha }_{max}$ with $\mathrm{\Delta }{\alpha }_C$, ${\alpha }_S$ and ${\alpha }_L$, respectively. In research of soybean leaf growth, stress resistance of soybean cultivars and the breeding of excellent varietals, $\mathrm{\Delta }{\alpha }_C$ is shown here to be a new and useful indicator for analysis.

In this paper we study the solutions to nonlinear mean-value formulas on fractal sets. We focus on the mean-value problem

Analysis of the brain response to different types of external stimuli has always been one of the major research areas in behavioral neuroscience. The electroencephalography (EEG) technique combined with different signal analysis approaches has been especially successful in revealing the detailed dynamic properties of the neural response to exogenous stimulation. In this analysis, we evaluated the nonlinear structure of the EEG signal using fractal theory in rest and visual stimulation (checkerboard reversal at 8, 14 and 28Hz). Our analysis showed a significant influence of stimulation on the fractal structure of EEG signal. On comparison between different conditions, 14-Hz steady-state visual evoked potentials (SSVEPs), previously shown to trigger an optimal brain response, exhibited the greatest influence on the complexity of the EEG signal. On the other hand, we observed the lowest complexity of EEG signal in the post-stimulation rest period. Statistical analysis confirmed significant differences in the fractal structure of the EEG signal between rest and different stimulation conditions. These findings demonstrate for the first time a direct relationship between the efficiency of brain processing and the complexity of the measured EEG signal, which could be employed for objective assessment and classification in various experimental paradigms.

In 1984, solid-state materials were discovered that had diffraction pattern symmetries that are forbidden for three-dimensional crystalline structures. The structures of these materials have become known as quasicrystals. A considerable literature has developed for both physical and mathematical quasicrystals. There are essentially three forms of mathematical three-dimensional quasicrystals. In this paper, we will develop computational techniques to construct these three forms of quasicrystals as point sets, and then organize those point sets in order to visualize patches of quasicrystals using parallelepipeds.

The new fractal models of the $(1+1)$-dimensional and $(n+1)$-dimensional nonlinear local fractional Harry Dym equation (HDE) on Cantor sets are derived and the analytical approximate solutions of the above two new models are obtained by coupling the fractional complex transform via local fractional derivative (LFD) and local fractional reduced differential transform method (LFRDTM). Fractional complex transform for functions of $(n+1)$-dimensional variables is generalized and the theorems of $(n+1)$-dimensional LFRDTM are supplementary extended. The travelling wave solutions of the fractal HDE show that the proposed LFRDTM is effective and simple for obtaining approximate solutions of nonlinear local fractional partial differential equations.

Complex networks are omnipresent in science and in our real life, and have been the focus of intense interest. It is vital to research the impact of their characters on the dynamic progress occurring on complex networks for weight-dependent walk. In this paper, we first consider the weight-dependent walk on one kind of transfractal (or fractal) which is named the weighted transfractal $(1,3)$-flowers. And we pay attention to the first return time (FRT). We mainly calculate the mean and variance of FRT for a prescribed hub (i.e. the most concerned nodes) in virtue of exact probability generating function and its properties. Then, we obtain the mean and the secondary moment of the first return time. Finally, using the relationship among the variance, mean and the secondary moment, we obtain the variance of FRT and the scaling properties of the mean and variance of FRT on weighted transfractals $(1,3)$-flowers.

Duo to different transport mechanisms and gas storage in organic and inorganic systems, a new triple-continuum model coupling Discrete Fracture Model (DFM) was established to investigate gas flow in shale gas reservoir. Considering the multi-scale and heterogeneity of shale matrix, fractal theory was used to calculate the apparent permeability of organic and inorganic systems while multiple gas transport mechanisms such as viscous flow, Knudsen diffusion, surface diffusion, gas absorption/desorption effect and real gas effect were incorporated. This coupled mathematical model was solved by Finite Element Method (FEM) and the presented fractal apparent permeability model was validated with the experimental data. The results show that fractal characteristics of shale matrix have great impact on gas reservoir performance. The model without considering the influence of fractal characteristics could lead to underestimate gas production by approximately 17%. Viscous flow is the dominate transport mechanisms of shale gas and Knudsen diffusion has an impact on gas flow when the pressure declines. Surface diffusion should be only considered in organic systems and can be ignored. Then the results of sensitivity analysis show that the characteristic parameters of inorganic matter have a greater impact than those of organic matter and establishing a triple-continuum model with considering comprehensive effect of organic and inorganic matter is necessary. In addition, gas production would decrease as the pore fractal dimension and tortuosity fractal dimension increase, which results from the increasing number of small pores and more tortuous path for gas flow.

It has been proved that effective stimulated reservoir volume (ESRV) is a significant area dominant to the production of the fractured well in unconventional gas reservoirs. Although ESRV properties can be estimated based on the microseismic technology and the analysis of actual fracturing data, the operations are complicated and results are inaccurate. Due to the complex structure of stimulated reservoir volume (SRV) with fractal and chaotic characteristics, a fractal evaluation model for ESRV (ESRV-FEM) of fractured wells in unconventional gas (both tight and shale gas) reservoirs is developed. Multiple gas transport mechanisms, SRV and unstimulated reservoir volume (USRV) are included. According to the pressure transient analysis (PTA), influences of multiple transport mechanisms on gas transport behaviors in ESRV are conducted. Moreover, the fractal index representing the heterogeneity degree is applied to estimate the ESRV under different inter-porosity flow coefficients and storage ratio conditions based on the ESRV-FEM. In addition, the presented ESRV-FEM is validated by an actual field case. The results show that gas adsorption has a significant effect on the radial flow duration time in SRV, and the heterogeneity makes the radial flow on PTA curves no longer show a horizontal line with the value of 0.5. Calculated ESRV sizes are compared with the assumed ones under different fractal indexes. The stronger the heterogeneity, the smaller ESRV is. The ESRV size of a fractured well in shale gas reservoirs is only 52.11% of SRV size when the fractal index equals to 0.6. The ESRV-FEM presented in this paper is expected to provide an effective method for the evaluation of the ESRV of fractured wells in unconventional gas reservoirs.

This paper presents an investigation of the convex hulls of the Sierpiński relatives. These fractals all have the same fractal dimension but different topologies. We prove that the relatives have convex hulls with polygonal boundaries with at most 12 vertices. We provide a method for finding the convex hull of a relative using its scaling and symmetry properties and present examples. We also investigate the connectivity properties of certain classes of relatives with the same convex hulls.

The effective permeability is one of the key parameters for porous fibrous gas diffusion layer (GDL), which plays a crucial role in the proton exchange membrane fuel cells (PEMFCs). However, the effect of the surface morphology of porous fibrous GDL on gas transport behaviors is so far been neglected. In order to take that into consideration, a new analytical model is presented for gas flow in porous fibrous GDL with roughened micro-channels based on the fractal scaling laws. Due to the existence of very small pores in the porous fibrous GDL, gas slippage effects through the small pores are also taken into account. The fractal gas permeability is expressed in terms of the relative roughness, other microstructural parameters as well as gas properties. The proposed fractal model is validated by comparing the model prediction with available experimental data. The effect of relative roughness and other micro-structural parameters of GDL on the permeability is analyzed in detail.