THEORETICAL VERIFICATION OF THE SCALING RELATION TO DETERMINE SURFACE FRACTAL DIMENSION OF POROUS MEDIA

A simplified procedure is developed to acquire the surface fractal dimension of porous media utilizing the PSD information from mercury porosimetry or other analyses.The self similarity of the inner surface of Sierpinski sponge is analyzed,the result of which demonstrates that the inner surface of Sierpinski sponge is not scale invariant over the whole range of scale transformations.By applying the simplified procedure to analyze and treat the PSD information of Sierpinski sponge over the scale invariant range,it is found that the surface fractal dimension calculated by the scaling relation is in very good agreement with its theoretical value,which virtually provides a theoretical affirmation of the method.

Implication of Surface Fractal Analysis to Evaluate the Relative Sensitivity of Topography to Active Tectonics,Zagros Mountains, Iran

Fractal geometry is increasingly becoming a useful tool for modeling and quantifying the complex patterns of natural phenomena. The Earth's topography is one of these phenomena that have fractal characteristics. This paper investigates the relative sensitivity of topography to active tectonics using ASTER Global Digital Elevation Model. The covering divider method was used for direct extraction of surface fractal dimension(D surf) to estimate the roughness-surface of topography with aid of geographic information system(GIS)techniques. This evaluation let us highlight the role of the geomorphic and tectonic processes on the spatial variability of fractal properties of natural landforms.Geomorphic zones can be delineated using fractal dimension mapping in which variability of surface fractal dimension reflects the roughness of the landform surface and is a measure of topography texture. Obtained results showed this method can be a quick and easy way to assess the distribution of land surface deformation in different tectonic settings. The loose alluvial deposits and irregularities derived by tectonic activity have high fractal dimensions whereas the competent formations and higher wavelength folded surfaces have lower fractal dimensions.According to the obtained results, the Kazerun Fault Zone has a crucial role in the separation of the Zagros Mountain Ranges into the different lithological,geomorphological and structural zones.

Classification of forearm action surface EMG signals based on fractal dimension

Surface electromyogram （EMG） signals were identified by fractal dimension.Two patterns of surface EMG signals were acquired from 30 healthy volunteers＇ right forearm flexor respectively in the process of forearm supination （FS） and forearm pronation （FP）.After the raw action surface EMG （ASEMG） signal was decomposed into several sub-signals with wavelet packet transform （WPT）,five fractal dimensions were respectively calculated from the raw signal and four sub-signals by the method based on fuzzy self-similarity.The results show that calculated from the sub-signal in the band 0 to 125 Hz,the fractal dimensions of FS ASEMG signals and FP ASEMG signals distributed in two different regions,and its error rate based on Bayes decision was no more than 2.26%.Therefore,the fractal dimension is an appropriate feature by which an FS ASEMG signal is distinguished from an FP ASEMG signal.

Effects of fractal surface on rheological behavior and combustion kinetics of modified brown coal water slurries

The paper reports the effects of surface fractal structures on the rheological behavior and combustion kinetics of raw brown coal and three modified coal water slurries （CWSs）. The results show that the fractal structures and physicochemical properties of samples are dependent on various modification processes. The apparent viscosities of the coal water slurries increase with increasing surface fractal dimensions （D）, especially with decreasing shear rates. Fur- thermore, it has been proved that the ignition temperatures and apparent activation energies of modified CWSs are lower than that of raw coal water slurry. Compared with the traditional qualitative analysis of the effect of pore structures on CWSs properties, D can more efficiently indicate the quantificational effect of pore structures on the rheological behavior and combustion kinetics of CWSs.

Fractal characteristics of surface crack evolution in the process of gas-containing coal extrusion

In this paper, simulated experiment device of coal and gas outburst was employed to perform the experiment on gas-containing coal extrusion. In the experiment, coal surface cracks were observed with a high-speed camera and then the images were processed by sketch. Based on the above description, the paper studied the fractal dimension values from different positions of coal surface as well as their changing laws with time. The results show that there is a growing parabola trend of crack dimension value in the process of coal extrusion. Accordingly, we drew the conclusion that extruded coal crack evolution is a process of fractal dimension value increase. On the basis of fractal dimension values taken from different parts of coal masses, a fractal dimension of the contour map was drawn. Thus, it is clear that the contour map involves different crack fractal dimension values from different positions. To be specific, where there are complicated force and violent movement in coal mass, there are higher fractal dimension values, i.e., the further the middle of observation surface is from the exit of coal mass, and the lower the fractal dimension value is. In line with fractal geometry and energy theory of coal and gas outburst, this study presents the relation between fractal dimension and energy in the process of extruding. In conclusion, the evolution of crack fractal dimension value can signify that of energy, which has laid a solid foundation for the quantification research on the mechanism of gas-containing coal extrusion.

Fractal Surface Synthesis Based on Two Dimensional Discrete Fourier Transform

The discrete Fourier transform（DFT） is used for fractional Brownian motion（FBM） surface synthesis in tribology（i.e., contact, sliding, and sealing, etc）. However, the relationship between fractal parameters（fractal dimension and scale factor） and traditional parameters, the influence of fractal parameters on surface appearance, have not been deeply discussed yet. These lead to some kind of difficulty to ensure the synthesized surfaces with ideal fractal characteristic, required traditional parameters and geometric appearance. A quantitative relationship between fractal parameters and the root mean square deviation of surface （Sq） is derived based on the energy conservation property between the space and frequency domain of DFT. Under the stability assumption, the power spectrum of a FBM surface is composed of concentric circles strictly, a series of FBM surfaces with prescribed Sq could be synthesized with given fractal dimension, scale factor, and sampling numbers, but the ten-point height（Sz）, the skewness（Ssk） and the kurtosis（Sku） are still in random, where the probability distributions of Sz and Ssk are approximately normal distribution. Furthermore, by iterative searching, a surface with desired Abbott-Firestone curve could be obtained among those surfaces. An intuitive explanation for the influence of fractal dimension and scale factor on surface appearance is obtained by discussing the effects on the ratio of energy between high and low frequency components. Based on the relationship between Sq and surface energy, a filtering method of surface with controllable Sq is proposed. The proposed research ensures the synthesized surfaces possess ideal FBM properties with prescribed Sq, offers a method for selecting desired Abbott-Firestone curve of synthesized fractal surfaces, and makes it possible to control the Sq of surfaces after filtering.

The autocorrelation of speckles in deep Fresnel diffraction region and characterizations of random self-affine fractal surfaces

This paper studies the correlation properties of the speckles in the deep Fresnel diffraction region produced by the scattering of rough self-affine fractal surfaces. The autocorrelation function of the speckle intensities is formulated by the combination of the light scattering theory of Kirchhoff approximation and the principles of speckle statistics. We propose a method for extracting the three surface parameters, i.e. the roughness w, the lateral correlation length ξ and the roughness exponent α, from the autocorrelation functions of speckles. This method is verified by simulating the speckle intensities and calculating the speckle autocorrelation function. We also find the phenomenon that for rough surfaces with α= 1, the structure of the speckles resembles that of the surface heights, which results from the effect of the peak and the valley parts of the surface, acting as micro-lenses converging and diverging the light waves.

An ultrathin wide-band planar metamaterial absorber based on a fractal frequency selective surface and resistive film

We propose an ultrathin wide-band metamaterial absorber （MA） based on a Minkowski （MIK） fractal frequency selective surface and resistive film. This absorber consists of a periodic arrangement of dielectric substrates sandwiched with an MIK fractal loop structure electric resonator and a resistive film. The finite element method is used to simulate and analyze the absorption of the MA. Compared with the MA-backed copper film, the designed MA-backed resistive film exhibits an absorption of 90% at a frequency region of 2 GHz-20 GHz. The power loss density distribution of the MA is further illustrated to explain the mechanism of the proposed MA. Simulated absorptions at different incidence cases indicate that this absorber is polarization-insensitive and wide-angled. Finally, further simulated results indicate that the surface resistance of the resistive film and the dielectric constant of the substrate can affect the absorbing property of the MA. This absorber may be used in many military fields.

ON STAR PRODUCTFRACTAL SURFACES AND THEIR DIMENSIONS

In this paper, by using fractal curves, a family of fractal surfaces are defined. Each fractal surface of this family is called Star Product Fractal Surface (SPFS). A theorem of the dimensions of the SPFS is strictly proved. The relationship between the dimensions of the SPFS and the dimensions of the fractal curves constructing the SPFS is obtained.

Universality and Specificity of Fractal Dimension of Fractured Surfaces in Materials

After calculation on the fracture angles under various conditions of specific surface energies with different symmetry operations of rotation, the complicated behavior of dependence of fractal dimension on the structure of crystal is shown. It is found that the crack propagates along the weakest crystal plane no matter what the direction of the maximum stress is if the anisotropy is sufficiently strong; and then, the fractal dimension of the fractured surfaces might be determined by the approximate fractal structure already existed in the material. Specificity of the fractal dimension of fractured surfaces would be easy to appear in this case. Reversely, the crack propagates along the direction of the maximum stress no matter what direction of the weakest crystal plane is if the anisotropy is sufficiently weak. Universality of the fractal dimension of fractured surfaces would be possible to appear in this case. In many real materials, universality and specificity of the materials are associated. The fractal dimension measured may more or less be influenced by the structure of materials and it shows its universality through the specificity of materials.

Polarimetric scattering from a two-dimensional improved sea fractal surface

This paper studies the influence of wind parameters and fractal dimension from an improved two-dimensional sea fractal surface on the polarimetric scattering by using facet integration. A two-dimensional improved sea surface simulated is discretized into three matrices of sea surface facets including a height matrix and two slope matrices on orthogonal directions. Based on the Kirchhoff approximation, the polarimetric scattered field is derived in the Cartesian coordinate system by integration of three matrices mentioned above. Finally, the fully polarised radar cross section is numerically simulated and the dependence of the polarimetric scattering on the sea fractal surface, such as the wind speed, the wind direction, as well as the fractal dimension, is discussed in detail.

A two scale nonlinear fractal sea surface model in a one dimensional deep sea

Using the theory of nonlinear interactions between long and short waves, a nonlinear fractal sea surface model is presented for a one dimensional deep sea. Numerical simulation results show that spectra intensity changes at different locations （in both the wave number domain and temporal-frequency domain）, and the system obeys the energy conservation principle. Finally, a method to limit the fractal parameters is also presented to ensure that the model system does not become ill-posed,

THE THEORY OF FRACTAL INTERPOLATED SURFACE AND ITS APPLICATIONS

In this paper, the principle of construction of a fractal surface is introduced, interpolation functions for a fractal interpolated surface are discussed, the theorem of the uniqueness of an iterated function system of fractal interpolated surface is proved, the theorem of fractal dimension of fractal interpolated surface is derived, and the case that practical data are used to interpolate fractal surface is studied.

The fractal characteristic change in the Huanghe River Estuary due to the hydrodynamic conditions’variation

The study area lies in the subaqueous delta, which came into being in 1964 - 1976. Oneoil-field road has been built for exploring petroleum to form a wave barrier. The hydrodynarnic conditions on the north side of the road are relatively violent, on the contrary the hydrodynarnic conditions on the south side of the road are nearly placid. This makes the study area a natural laboratory for studying the influence of the hydrodynarnic conditions on the fractal characteristics of the tidal flat. Selecting an area is named Case Ⅰ on the side of stronger hydrodynarnic activities and an area is named Case Ⅱ on the other side. Measuring the topography and sampling and analyzing the granulometrical composition, it is found that the hydrodynarnic conditions have fatal influence on the surface fractal dimensions and the granulometrical fractal dimensions of the area. In Case Ⅰ , which has strong hydrodynarnic conditions, the surface fractal dimensions are obviously larger than those of Case Ⅱ , and the granulometrical fractal dimensions are relatively smaller than those of Case Ⅱ , the surface fractal dimensions of Case Ⅰ decrease quickly with the increase of grid size; the granulometrical fractal dimensions are disperse, while the hydrodynarnic conditions of Case Ⅱ are just reverse. A sampling line and a core sampling on each side of the road are selected. It is found that on the south side of the road the granulometrical fractal dimensions vary regularly in the line and with the depth, the farther apart from the road, the smaller the fractal dimensions, and the deeper the sampling position the larger the fractal dimensions, while granulometrical fractal dimensions on the north side of the road have no such regularity pattern. The mechanism of the influence of the hydrodynarnic conditions on the fractal characteristics is discussed.

Fractal analysis of polyferric chloride-humic acid(PFC-HA) flocs in different topological spaces

The fractal dimensions in different topological spaces of polyferric chloride-humic acid （PFC-HA） flocs, formed in flocculating different kinds of humic acids （HA） water at different initial pH （9.0, 7.0, 5.0） and PFC dosages, were calculated by effective densitymaximum diameter, image analysis, and N2 absorption-desorption methods, respectively. The mass fractal dimensions （De） of PFC-HA floes were calculated by bi-logarithm relation of effective density with maximum diameter and Logan empirical equation. The Df value was more than 2.0 at initial pH of 7,0, which was 11% and 13% higher than those at pH 9.0 and 5.0, respecively, indicating the most compact flocs formed in flocculated HA water at initial pH of 7.0. The image analysis for those flocs indicates that after flocculating the HA water at initial pH greater than 7.0 with PFC flocculant, the fractal dimensions of D2 （logA vs. logdL） and D3 （logVsphere vs. logdL） of PFC-HA floes decreased with the increase of PFC dosages, and PFC-HA floes showed a gradually looser structure. At the optimum dosage of PFC, the D2 （logA vs. logdL） values of the flocs show 14%-43% difference with their corresponding Dr, and they even had different tendency with the change of initial pH values. However, the D2 values of the floes formed at three different initial pH in HA solution had a same tendency with the corresponding Df. Based on fractal Frenkel-Halsey-HiU （FHH） adsorption and desorption equations, the pore surface fractal dimensions （Ds） for dried powders of PFC-HA flocs formed in HA water with initial pH 9.0 and 7.0 were all close to 2.9421, and the Ds values of flocs formed at initial pH 5.0 were less than 2.3746. It indicated that the pore surface fractal dimensions of PFC-HA floes dried powder mainly show the irregularity from the mesopore-size distribution and marcopore-size distribution.

Adhesive Contact of Elastic Solids with Self-Affine Fractal Rough Surfaces

The elastic adhesive contact of self-affine fractal rough surfaces against a rigid flat is simulated using the finite element method. An array of nonlinear springs, of which the force-separation law obeys the Lennard–Jones potential, is introduced to account for the interfacial adhesion. For fractal rough surfaces, the interfacial interaction is generally attractive for large mean gaps, but turns repulsive as the gap continuously shrinks. The interfacial interactions at the turning point corresponding to the spontaneous contact are shown for various surfaces. For relatively smooth surfaces, the probability density distributions of repulsion and attraction are nearly symmetric. However, for rougher surfaces, the simulation results suggest a uniform distribution for attraction but a monotonously decreasing distribution with a long tail for repulsion. The pull-off force rises with increasing ratio of the work of adhesion to the equilibrium distance, whereas decreases for solids with a higher elastic modulus and a larger surface roughness. The current study will be helpful for understanding the adhesion of various types of rough solids.

Influence of Drying Methods on Fractal Geometric Characteristics of Mesoporous Silica Aerogels

Chemical modification/ambient drying method and freeze drying method were introduced to research the synthesis of mesoporous silica aerogels. By analyzing N2 gas adsorption/desorption isotherms, the fractal geometric characteristics of gels were focused. The overall surface fractal dimensions were determined by analyzing N2 gas adsorption branch and a Frenkel-Halsey-Hill （FHH） equation was empolyed to determine surface fractal dimension Df It is found that, during ambient drying process, VTMCS/VWetgel ratio plays a crucial role in the changes of geometric feature, the key point is 50%, when the ratio is lower, and surface roughness increases with the ratio, when it exceeds 50%, the surface is almost unaffected by the modification. While freeze drying always tends to get larger Df freeze drying process could cause a rough surface of the gels. Compared with traditional porosity and specific surface area analyses, fractal geometry may be expected to be favorable for mesoporous structural analyses of materials.

Surface characteristics analysis of fractures induced by supercritical CO_(2)and water through three-dimensional scanning and scanning electron micrography

Morphology of hydraulic fracture surface has significant effects on oil and gas flow,proppant migration and fracture closure,which plays an important role in oil and gas fracturing stimulation.In this paper,we analyzed the fracture surface characteristics induced by supercritical carbon dioxide(SC-CO_(2))and water in open-hole and perforation completion conditions under triaxial stresses.A simple calculation method was proposed to quantitatively analyze the fracture surface area and roughness in macro-level based on three-dimensional(3D)scanning data.In micro-level,scanning electron micrograph(SEM)was used to analyze the features of fracture surface.The results showed that the surface area of the induced fracture increases with perforation angle for both SC-CO_(2)and water fracturing,and the surface area of SC-CO_(2)-induced fracture is 6.49%e58.57%larger than that of water-induced fracture.The fractal dimension and surface roughness of water-induced fractures increase with the increase in perforation angle,while those of SC-CO_(2)-induced fractures decrease with the increasing perforation angle.A considerable number of microcracks and particle peeling pits can be observed on SC-CO_(2)-induced fracture surface while there are more flat particle surfaces in water-induced fracture surface through SEM images,indicating that fractures tend to propagate along the boundary of the particle for SC-CO_(2)fracturing while water-induced fractures prefer to cut through particles.These findings are of great significance for analyzing fracture mechanism and evaluating fracturing stimulation performance.

Comments on "Reexamination of Correlations for Nucleate Site Distribution on Boiling Surface by Fractal Theory

In recently published paper by Yang Chunxin[1], I reexamined the paper. On page 128, the paper 'pointed out that the size and spatial distribution density of nucleation sites presented on real boiling surface can be described by the normalized fractal distribution function, and the physical meaning of parameters involved in some experimental correlations proposed by early investigations are identified according to fractal distribution'. However, the definition on fractal dimension given by Yang[1] is highly questionable, and the results obtained by Yang are contradictory to the basic fractal theory. Here are my comments:

Numerical simulation of bistatic scattering from fractal rough surface in the finite element method

By using the Monte Carlo method and numerical finite elementapproach, bistatic scattering from the fractal and Gaussian rough surfaces is studied. The difference between these two surfaces and their functional dependence on the surface parameters are discussed. Angular variation of bistatic scattering from the fractal surface is very significant, even for fairly smooth surface, whilst scattering from the Gaussian rough surface tends to the specular reflection. The slope of angular variation is linearly related with the fractal dimension. If an electrically_large target is placed over the rough surface, the fractal dimension inverted from bistatic scattering would be reduced. As the surfaces become very rough, scattering from different fractal and Gaussian surfaces would be not identified.