A Successive Shift Box-Counting Method for Calculating Fractal Dimensions and Its Application in Identification of Faults

Fractal dimensions of a terrain quantitatively describe the self-organizedstructure of the terrain geometry. However, the local topographic variation cannot be illustrated bythe conventional box-counting method. This paper proposes a successive shift box-counting method,in which the studied object is divided into small sub-objects that are composed of a series of gridsaccording to its characteristic scaling. The terrain fractal dimensions in the grids are calculatedwith the successive shift box-counting method and the scattered points with values of fractaldimensions are obtained. The present research shows that the planar variation of fractal dimensionsis well consistent with fault traces and geological boundaries.

A Statistical Method for Determining the Fractal Dimension of Time Series

In this paper,we present a new method for determining the fractal dimension of time series and the algorithm of H index(Hurst index).

Relevance between abutment pressure and fractal dimension of crack network induced by mining

Based on the geological conditions of coal mining face No.15-14120 at No.8 mine of Pingdingshan coal mining group,the real-time evolution of coal-roof crack network with working face advancing was collected with the help of intrinsically safe borehole video instrument.And according to the geology of this working face,a discrete element model was calculated by UDEC.Combining in situ experimental data with numerical results,the relationship between the fractal dimension of boreholes'wall and the distribution of advanced abutment pressure was studied under the condition of mining advance.The results show that the variation tendency of fractal dimension and the abutment pressure has the same characteristic value.The distance between working face and the peak value of the abutment pressure has a slight increasing trend with the advancing of mining-face.When the working face is set as the original point,the trend of fractal dimension from the far place to the origin can be divided into three phases:constant,steady increasing and constant.And the turning points of these phases are the max-influencing distance(50 m)and peak value(15 m)of abutment pressure.

A novel box-counting method for quantitative fractal analysis of threedimensional pore characteristics in sandstone

Fractal theory offers a powerful tool for the precise description and quantification of the complex pore structures in reservoir rocks,crucial for understanding the storage and migration characteristics of media within these rocks.Faced with the challenge of calculating the three-dimensional fractal dimensions of rock porosity,this study proposes an innovative computational process that directly calculates the three-dimensional fractal dimensions from a geometric perspective.By employing a composite denoising approach that integrates Fourier transform(FT)and wavelet transform(WT),coupled with multimodal pore extraction techniques such as threshold segmentation,top-hat transformation,and membrane enhancement,we successfully crafted accurate digital rock models.The improved box-counting method was then applied to analyze the voxel data of these digital rocks,accurately calculating the fractal dimensions of the rock pore distribution.Further numerical simulations of permeability experiments were conducted to explore the physical correlations between the rock pore fractal dimensions,porosity,and absolute permeability.The results reveal that rocks with higher fractal dimensions exhibit more complex pore connectivity pathways and a wider,more uneven pore distribution,suggesting that the ideal rock samples should possess lower fractal dimensions and higher effective porosity rates to achieve optimal fluid transmission properties.The methodology and conclusions of this study provide new tools and insights for the quantitative analysis of complex pores in rocks and contribute to the exploration of the fractal transport properties of media within rocks.

Running-in Test and Fractal Methodology for Worn Surface Topography Characterization

Studying and understanding of the surface topography variation are the basis for analyzing tribological problems,and characterization of worn surface is necessary.Fractal geometry offers a more accurate description for surface roughness that topographic surfaces are statistically self-similar and can be quantitatively evaluated by fractal parameters.The change regularity of worn surface topography is one of the most important aspects of running-in study.However,the existing research normally adopts only one friction matching pair to explore the surface topography change,which interrupts the running-in wear process and makes the experimental result lack authenticity and objectivity.In this paper,to investigate the change regularity of surface topography during the real running-in process,a series of running-in tests by changing friction pairs under the same operating conditions are conducted on UMT-II Universal Multifunction Tester.The surface profile data are acquired by MiaoXAM2.5X-50X Ultrahigh Precision Surface 3D Profiler and analyzed using fractal dimension D,scale coefficient C and characteristic roughness Ra ＊based on root mean square（RMS） method.The characterization effects of the three parameters are discussed and compared.The results obtained show that there exists remarkable fractal feature of surface topography during running-in process,both D and Ra ＊increase gradually,while C decreases slowly as the wear-in process goes on,and all parameters tend to be stable when the wear process steps into the normal wear process.Ra ＊illustrates higher sensitivity for rough surface characterization compared with the other two parameters.In addition,the running-in test carried with a set of identical surface properties is more scientific and reasonable than the traditional one.The proposed research further indicates that the fractal method can quantitatively measure the rough surface,which also provides an evidence for running-in process identification and tribology design.

Fractal Method for Statistical Analysis Geological Data

This paper establishes the phase space in the light of spacial series data , discusses the fractal structure of geological data in terms of correlated functions and studies the chaos of these data . In addition , it introduces the R/S analysis for time series analysis into spacial series to calculate the structural fractal dimensions of ranges and standard deviation for spacial series data -and to establish the fractal dimension matrix and the procedures in plotting the fractal dimension anomaly diagram with vector distances of fractal dimension . At last , it has examples of its application .

基于三维体视分析技术对金属断口的分形研究

金属断口包含金属材料的性质、性能及服役过程的重要信息,是金属断裂定性分析、定量表征的关键证据。研究证明分形维数与材料断裂参数密切相关,但传统二维图像分析难以全面揭示断口特征。为了更深入了解断裂机制和材料性能,三维形貌技术逐渐成为研究热点。在分析仪器中三维体视显微镜可以提供更便捷、更准确的断口信息,进而通过计算分形维数推测材料断裂性质。其中计盒维数分析方法具有易于实现和理解、应用范围广、稳定性好、可解释性强等优势。利用三维体视显微镜和MATLAB程序获取断口三维形貌并计算分形维数。实验表明,拉伸断口抗拉强度和屈服强度与分形维数呈负相关,冲击断口的冲击值与分形维数呈正相关。

Evaluation of dimension of fractal time series with the least square method

Properties of fractional Brownian motions(f Bms)have been investigated by researchers in different fields,e.g.statistics,hydrology,biology,finance,and public transportation,which has helped us better understand many complex time series observed in nature[1-4].The Hurst exponent H(0<H<1)is the most important parameter characterizing any given time series F(t),where t represents the time steps,and the

Research on nonlinear R／S method and its application in earthquake prediction

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Realizing the Box-counting Method for Calculating Fractal Dimension of Urban Form Based on Remote Sensing Image

在分数维的城市的研究，分数维的尺寸是很重要的。描述城市的分数维的特性被用来。作者设计了二条途径通过北京的一个例子由数盒子的方法计算分数维的尺寸，它分别地被称为向量方法和格子方法。前者在 ArcView 通过交叉分析计算分数维的尺寸；并且后者被编程在 Matlab 执行。他们从三个方面被比较：精明的过程，在使用的限制，和结果。作为结果，结论被成为那在两个方法上有优点和差错，并且他们应该处于实际状况适当地被选择到使用。

Fractal analysis of granular ore media based on computed tomography image processing

九组矿石样品的代表性的图象被 X 光检查获得计算断层摄影术(CT ) 扫描仪。基于 CT 图象分析，稳固的矩阵的分数维的尺寸，毛孔空格和每件样品的矩阵 / 毛孔接口被使用框数测量方法。有粒子尺寸，孔，和渗出物系数的三种分数维的尺寸的关联被调查。结果证明为这些样品的所有图象，矩阵阶段有最高的尺寸，由矩阵毛孔接口的毛孔阶段，和尺寸列在后面有最小的价值；当毛孔阶段的尺寸与孔断然并且线性地联系时，矩阵阶段和矩阵毛孔接口的尺寸否定地并且线性地与孔被相关；矩阵毛孔接口的分数维的尺寸与渗出物系数否定地并且线性地联系。矩阵 / 毛孔接口的更大的分数维的尺寸为解决方案流动显示更不规则的复杂隧道，导致低渗透。

From Fractal Geometry to Fractal Analysis

Euclidian geometry pertained only to the artificial realities of the first, second and third dimensions. Fractal geometry is a new branch of mathematics that proves useful in representing natural phenomena whose dimensions (fractal dimensions) are non-integer values. Fractal geometry was conceived in the 1970s, and mainly developed by Benoit Mandelbrot. In fractal geometry fractals are normally the results of an iterative or recursive construction using corresponding algorithm. Fractal analysis is a nontraditional mathematical and experimental method derived from Mandelbrot’s Fractal Geometry of Nature, Euclidean geometry and calculus. The main aims of the present study are: 1) to address the dimensional imbalances in some texts on fractal geometry, proving that logarithm of a physical quantity (e.g. length of a segment) is senseless;2) to define the modified capacity dimension, calculate its value for Koch fractal set and show that such definition satisfies basic demands of physics, before all the dimensional balance;and 3) to calculate theoretically the fractal dimension of a circle of unit radius. A quantitative determination of the similarity using the set of Koch fractals is carried out. An important result is the relationship between the modified capacity dimension and fractal dimension obtained using the log-log method. The text includes some important modifications and advances in fractal theory. It is important to notice that these modifications and quantifications do not affect already known facts in fractal geometry and fractal analysis.

R/S method for evaluation of pollutant time series in environmental quality assessment

The significance of the fluctuation and randomness of the time series of each pollutant in environmental quality assessment is described for the first time in this paper. A comparative study was made of three different computing methods： the same starting point method, the striding averaging method, and the stagger phase averaging method. All of them can be used to calculate the Hurst index, which quantifies fluctuation and randomness. This study used real water quality data from Shazhu monitoring station on Taihu Lake in Wuxi, Jiangsu Province. The results show that, of the three methods, the stagger phase averaging method is best for calculating the Hurst index of a pollutant time series from the perspective of statistical regularity.

基于分形理论的雷电冲击土体等离子体通道发展研究

等离子体通道对岩土体破坏过程具有内动力作用,针对雷电冲击岩土体后的等离子体通道发展进行研究,从电击穿的角度构建岩土体击穿通道的随机概率模型,采用拉普拉斯方程计算电场,引入分形理论计算发展概率,得到等离子体通道路径模拟图。采用盒维数法计算分形维数来反映等离子体通道发展情况,分析了土体电阻率、雷电流幅值及内部电压降对分形维数的影响规律。结果表明:随着土体电阻率增大,分形维数逐渐减小;随着雷电流幅值增大,分形维数逐渐增大;随着土体内部电压降增大,分形维数逐渐增大,且在土体电阻率越小时,分形维数变化越显著。研究证明,运用二维图像方式直观表现出岩土体内等离子体通道发展,并利用分形几何定量描述岩土体内等离子体通道的发展规律,有助于预测岩土体破坏过程的发展,便于进一步研究岩土体电击穿的物理本质。

不同量粉煤灰漂珠制备氧化物多孔陶瓷的性能研究

为进一步拓展粉煤灰漂珠在多孔陶瓷领域的应用,以粉煤灰漂珠、镁橄榄石、绢云母粉及铝矾土为主要原料,其中粉煤灰漂珠加入量分别为40%、60%、80%(w),成型后在空气气氛下于1100℃热处理1h制得多孔陶瓷材料,分析其物相组成、显微结构以及常规物理性能。结果表明:1)粉煤灰漂珠加入量由40%(w)增加至80%(w),粉煤灰漂珠与基质的间隔变大,试样的气孔率增大、体积密度和常温耐压强度减小,但其气孔的分形维数未有明显变化;2)当粉煤灰漂珠加入量为60%(w)时,试样综合性能较优,此时其体积密度约为(0.90±0.01)g·cm^(-3),且在气孔率为(66.9±0.5)%时,仍具有约(12.37±0.69)MPa的常温耐压强度,在300、600及900℃下的热导率分别为0.201、0.232和0.262W·m^(-1)·K^(-1),因此在隔热耐火材料领域具有良好应用前景。

DLH油田低渗砂岩孔隙分形定量表征方法研究

针对常规表征方法难以精确表征低渗砂岩储层孔隙空间分布复杂性和不规则性的问题,提出了适用于低渗砂岩储层的分形维数计算方法,实现了低渗砂岩储层孔隙特征的定量表征。基于不同分形维数计算方法差异性的分析,优选采用MIFA方法求解低渗砂岩储层的分形维数(在2.042~2.324),相关性最佳;确定了排驱压力、平均孔喉半径、变异系数以及均值系数作为储层孔喉分布复杂程度和非均质程度的综合表征参数;基于恒速压汞分形维数的求解,发现低渗砂岩储层非均质程度呈中小孔喉大于微小孔喉,喉道分布大于孔隙分布的特点;低渗砂岩储层的分形维数与启动压力梯度和应力敏感性损害率的实验结果均存在一定的相互关系,分形维数越大,孔喉分布的非均质性越强,启动压力梯度越大且应力敏感性的损害程度也将加剧。低渗砂岩储层分形维数的计算可用于室内实验结果的定性预测和判断,也可作为油藏工程中应用相渗曲线时的重要判别标准。

永定河典型护岸植被土壤水分入渗特征及影响因素

为探究永定河岸坡缓冲带不同护岸植被覆盖下土壤水分入渗状况及植被根系对其影响,为区域河岸带生态修复和水资源保护提供参考,选取永定河流域典型护岸植被柳树林、荆条林、狗尾草地为研究对象,通过亮蓝染色法结合Photoshop、Image Pro6.0等图像处理技术,基于分形维数和多指标评价法定量评价了优先流发育程度,并探讨了不同植被根系对优先流的影响。结果表明:①研究区3种植被类型覆盖地土壤均存在水分优先入渗现象,土壤染色面积比随土层深度增加而逐渐减小,柳树林地平均染色面积比(38.32%)最大,为荆条林地和狗尾草地的1.11倍和1.37倍。狗尾草地的长度指数、优先流分数均显著低于柳树林地(P<0.05),3种植被覆盖地基质流深度无显著差异。②以优先流评价指数和分形维数评价优先流发育程度的结果呈一致,为柳树林地>荆条林地>狗尾草地,柳树林地的湿润锋迹线破碎化程度最高,优先流发育最显著。③有机质、非毛管孔隙度和根长密度对水分入渗均具有显著影响,总根长密度、非毛管孔隙度与染色面积比呈显著正相关,≥1 mm根系根长密度、有机质含量与分形维数呈显著正相关。3种植被覆盖地染色面积比和根长密度关系曲线拟合较好。总之,在外界条件相同的情况下,柳树林地水分优先入渗现象最明显;植物根系的径级、密度对优先流的发育具有显著影响,≥1 mm根系越丰富优先流现象越明显,而径级<1 mm的根系主要促进表层土壤水分均匀入渗。

Theoretical and numerical studies of rock breaking mechanism by double disc cutters

A plane mechanical model of rock breaking process by double disc cutter at the center of the cutterhead is established based on contact mechanics to analyze the stress evolution in the rock broken by cutters with different spacings. A continuous-discontinuous coupling numerical method based on zero-thickness cohesive elements is developed to simulate rock breaking using double cutters. The process, mechanism,and characteristics of rock breaking are comprehensively analyzed from five aspects: peak force, breaking form, breaking efficiency, crack mode, and breaking degree. The results show that under the penetrating action of cutters, dense cores are formed due to shear failure under respective cutters. The tensile cracks propagate in the rock, and then rock chips form with increasing penetration depth. When the cutter spacing is increased from 10 to 80 mm, the peak force gradually increases, the rock breaking range increases first and then decreases, the specific energy decreases first and then rises, and the breaking coefficient of intermediate rock decreases from 0.955 to 0.788. The area of rock breaking is positively correlated with the length of the tensile crack. Furthermore, the length of the tensile crack accounts for 14.4%–33.6% of the total crack length.

基于自动化量规法的中国大陆海岸线分形特征研究——与网格法的比较分析

分形维数是描述海岸线复杂程度的重要指标,以手工绘制量规圆计算分形维数的方法在准确性和便捷性上已不适用于当前信息时代大空间尺度的海岸线研究。该文基于开源PyQGIS开发自动化量规法计算软件,实现了中国大陆及不同海区、不同沿海省域海岸线量规维的高效计算,并与网格法计算的盒计维进行对比分析。研究结果表明:(1)与传统手工量规法相比,自动化量规法能在延长计算标度的同时保证运算速度和准确性,得到的量规维总体上高于网格法得到的盒计维;(2)经自动化量规法计算得到的2020年中国大陆海岸线的分形维数为1.1948,杭州湾以北和以南海岸线的分形维数分别为1.1704和1.2175,说明杭州湾以南海岸线更复杂;(3)中国大陆海岸线的分形维数呈现“南北高、中间低”的空间分布特征,在海区层面上,南海海区分形维数最大(1.2091),南黄海海区最小(1.1563),省域层面上,天津的分形维数最大(1.2904),上海最小(1.0875)。自动化量规法计算软件可为海岸线分形研究提供便捷有力的计算工具,同时可为海岸线资源调查和综合管理提供技术支持。

混凝土细观断裂模拟及损伤演化表征

为定量表征混凝土破坏过程中裂纹扩展及损伤演化规律,借助激光扫描技术与骨料投放算法构建混凝土细观结构模型,结合室内试验,基于连续-非连续单元方法(continuous-discontinuous element method,CDEM)对数值混凝土切面细观模型进行单轴压缩数值模拟。结果表明:模拟所得应力-应变曲线及破坏模式与室内试验吻合较好;混凝土细观损伤演化过程表现为裂纹萌生、扩展、贯通、最终破坏的典型阶段特征,细观破裂度可揭示混凝土在破坏过程中的细观损伤演化规律;分形维数能够定量表征混凝土受压破裂过程中裂纹发展的复杂随机性,sigmoid型演化模型适用于描述混凝土单轴压缩裂纹分形维数的非线性演变过程。