Fractal Dimension Number Research Articles

The Fractal Characteristics of Soft Soil under Cyclic Loading Based on SEM

Cyclic loading always results in great damage to the pore structure and fractal characteristics of soft soil. Scanning electron microscope (SEM) can help collect data to describe the microstructure of soft soil. This paper conducted a series of SEM tests to interpret the effect of consolidation confining pressure, circulating dynamic stress ratios and overconsolidation ratio on soil’s micro-pore structure and fractal characteristics. The results demonstrate that fractal dimension can well represent the complex characteristics of the microstructure of the soil; the larger the consolidation confining pressure, the greater the cyclic dynamic stress ratio, and the greater the overconsolidation ratio, the smaller the fractal dimension number of soil samples. Finally, an empirical fitting formula for cumulative strain considering microstructure parameters is established through data fitting.

Broadband Electromagnetic Absorption Effect of Topological Structure Using Carbon Nanotube Based Hybrid Material.

The development of microwave absorbing technology raises the demands for all-band absorption. The topological structures expand the frequency range of electromagnetic wave absorption and eliminate the differences caused by scattering in different incident directions. The multi-wall carbon nanotube and carbonyl iron particles were mixed with polylactic polymer to fabricate filaments for fused deposition. The distribution characteristics of the structures using carbonyl iron/carbon nanotube hybrid material for the key absorption frequency band are obtained. The reflectivity of the honeycomb structure in X and Ku bands is verified experimentally through the preparation method of fused deposition modeling 3D printing. With the decrease of the fractal dimension number, the electromagnetic loss performance basically increases. Preliminary research results showed that the topological structure could significantly expand the absorbing frequency range, and the effective frequency band less than −10 dB is 2–40 GHz, which has a clear application potential for radar absorption.

Effective stiffness, wave propagation, and yield surface attributes of Menger sponge-like pre-fractal topologies

In the current work, the effective stiffness, long-wavelength attributes, and yield limits of Menger sponge-like pre-fractal topologies are investigated for the first time. Three types of Menger sponge-like structures with circular (MC), rhombic (MR), and square (MS) cavities are considered up to their third-level fractal topology. For each level, the elastic, shear, and bulk moduli, as well as the Poisson's ratio values are computed. Moreover, third-level Menger sponge-like fractal structures are additively manufactured and experimentally tested. A clear difference between the elastic stiffness and yield strength of the different Menger sponge-like structures is numerically and experimentally identified. It is observed that the cavity type considerably affects the scaling of elastic stiffness attributes upon increasing fractal level. The MS and MC topologies relate to lower relative density structures compared to the MR topology for all fractal levels investigated. The percentage relative density reductions induced by the fractal topology are a multiple of the corresponding decrease in the magnitude of the propagating shear and longitudinal phase velocities. Furthermore, all Menger sponge levels and inner cavity types yield analogous wave propagation directional dependencies, though with different phase velocity magnitudes. It is also found that increasing fractal topologies diminish the yield strength limits of the Menger fractal structures, with the factor associating the normal and shear loading yield strength of two successive Menger topology levels to be lower, but comparable in magnitude with the fractal dimension number. The fractal level evolution of the yield surface is well-captured by the extended Hill's criterion both for bi-axial and axisymmetric loads. The corresponding coefficients are identified for first, second, and third-level MS structures.

Analysis of Dendrite Images Formed by Electrochemical Migration in a Semiconductor Sensor

Dendrites were observed in the failure of semiconductor sensor devices. EDX analysis showed that the dendrites grown from bare sensor dice consisted of tin metal. The tin dendrites exhibited massive and dense branches. Dendrites grown from mechanically decapped parts consisted of silver. The silver dendrites exhibited delicate, lace-like structure. Binary and grey scale images of dendrites were analyzed for fractal dimension number and branch density. The tin dendrites had a higher, statistically significant branch density number than silver, due to tin’s more intricate branching pattern. Fractal numbers can be used to differentiate between tin and silver dendrites, even in the absence of EDX analysis equipment.

Morphological Characterization of Graphene Flake Networks Using Minkowski Functionals

Two Minkowski functionals were tested in the capacity of morphological descriptors to quantitatively compare the arrays of vertically-aligned graphene flakes grown on smooth and nanoporous alumina and silica surfaces. Specifically, the Euler-Poincare characteristic and fractal dimension graphs were used to characterize the degree of connectivity and order in the systems, i.e. in the graphene flake patterns of petal-like and tree-like morphologies on solid substrates, and meshlike patterns (networks) grown on nanoporous alumina treated in low-temperature inductivelycoupled plasma. It was found that the Minkowski functionals return higher connectivity and fractal dimension numbers for the graphene flakepatterns with more complex morphologies, and indeed can be used as morphological descriptors to differentiate among various configurations of vertically-aligned graphene flakes grown on surfaces.

The Blemish Identify in Ultrasonic Testing of Friction Welded Joints Based on Fractal Theory

The friction welded joints made by GH4169 heat metal alloys are detected by U1traPAC system of the ultrasonic wave explore instrument. Aimed at the blemish signal characteristics, a method is put forward to identify the blemish signal in ultrasonic testing of friction welded joints based on fractal theory., A new kind of fractal dimension number calculation method—normalized scale box-counting dimension method is used to calculate the box-dimension of the blemish signal, and statistic and analysis the scopes of the box-dimension of the blemish signal and the square differ of the result. The preliminary experimental results show that the blemish signal and have each fractal dimension zones, and don’t hand over to fold. So it can use to judge whether the blemish exist or not. The method is identified to have better covariance characteristics about the blemish identify of the friction welded joints.

Modeling of aggregation kinetics by a new moment method

A new flocculation dynamics model is established based on the proposed Taylor expansion moment method, for the fine particle aggregation and floc formation. Mechanisms of hydrodynamic shear and differential sedimentation dominated flocculation are studied in the framework of Smoluchowski mean field theory. The Taylor expansion technique is applied and the fractal dimension is introduced into the system of particle dynamics equations. The effects of particle diffusion and vertical transport have been included. The influence of collision efficiency and the combination with fractal dimension on the calculation results are studied. The formula for estimating the floc time accounting for the fractal dimension is derived. The proposed model is applicable to all the three types of floc mechanisms: Brownian motion, differential settlement and orthokinetic flocculation. Increase of fractal dimension number results in smaller critical floc size and reduced floc structure coefficient and smaller standard deviation of floc size distribution, which reflects the self-holding. Results of the present research show that particle flocculation and fragmentation can be balanced in a certain scope, to achieve the so-called self-retention.

The dynamic analysis of lake landscape of Wuhan City in recent 40 years

城市湖泊是城市湿地的重要组成部分,其演变受到人类活动的严重干扰,也影响城市的可持续发展.利用19732013年40年间Landsat MSS(3景)、Landsat 5 TM(5景)和Landsat 8(1景)共9景遥感影像,从景观角度分析武汉市主要湖泊的变化.首先借助遥感和地理信息系统技术提取湖泊信息,然后通过主成分分析法选取平均斑块面积(MPS)、斑块面积标准差(PSSD)、边界密度(ED)和斑块平均分维数(MPDF)4个景观指数用于湖泊景观分析.结果表明,武汉市湖泊变化经历了4大阶段:(1) 1970s,由于“围湖造田”政策的推行,湖泊总面积、MPS和PSSD急剧减小,湖泊斑块数量(尤其是小型湖泊)急剧增加,大湖破碎、小湖增加;(2) 1980s,推行“退田还湖”政策,湖泊总面积有较大回升,但小型湖泊消亡现象较为严重;(3) 1990s,由于经济发展和人口增长等原因,湖泊水面面积仍缓慢波动减小;(4) 21世纪后,湖泊斑块数量持续缓慢增加,但湖泊总面积呈减少趋势,故又可能进入一轮大湖萎缩或破碎、小湖增加的阶段.总体而言,武汉城市湖泊受到人为干扰和政策导向影响明显,应大力加强湖泊的科学管理,合理地开发利用,保护好湖泊自然资源.;Inland urban lakes are important components of urban wetland, and they are greatly affected by the disturbance of human activities; while the conditions of urban lakes also affect the sustainable development of cities. This study aimed to analyse the dynamic changes of major lakes in Wuhan City from a landscape perspective using nine Landsat images(including three Landsat MSS images, five Landsat 5 TM images and one Landsat 8 image). Remote sensing and geographic information systems technologies were firstly used to extract lakes from the images, and suitable landscape matrices were then selected by applying principal component analysis. Four landscape matrices were selected for qualifying the dynamic changes, including the mean patch size(MPS), standard deviation of patch size(PSSD), edge density(ED) and mean patch fractal dimension number(MPDF). The results showed that the lake change of Wuhan had four stages:(1) during the 1970s, total area of the lake, MPS and PSSD reduced drastically; while patch number(especially the number of small lakes) increased dramatically, which indicated that big lakes fragmented into small lakes mainly due to the guide of “Reclaim the land” policy;(2) during the 1980s, the implementation of “Returning farming land to lakes” policy caused an increase in the total area of the lakes, but the small lakes were dying out seriously during that period;(3) during the 1990s, due to the development of economy and the growth of population, the problem of the shortage of land arose. The lakes suffered from the overexploitation again, and the surface area of lakes reduced with fluctuation;(4) since 21st century, lakes patches continued to increase slowly in numbers, while the total area of the lake showed a decreasing trend, which indicated that big lakes fragmented or broken into small lakes. We concluded that the lakes of Wuhan city were greatly influenced by the human activities and policies, and that the scientific management of lakes should be carried out for keeping the lakes under rational developments and for protecting these valuable natural resources.

Effects of chemical mechanical planarization slurry additives on the agglomeration of alumina nanoparticles II: Aggregation rate analysis

The aggregation rate and mechanism of 150nm alumina particles in 1mM KNO3 with various additives used in chemical mechanical planarization of copper were investigated. The pH of each suspension was ∼8 such that the aggregation rate was slow enough to be measured and analyzed over ∼120min. In general, an initial exponential growth was observed for most suspensions indicating reaction-limited aggregation. After aggregate sizes increase to >500nm, the rate followed a power law suggesting diffusion-limited aggregation. Stability ratios and fractal dimension numbers were also calculated to further elucidate the aggregation mechanism.

Study of Flocs Morphology of Chitosan Coagulation Aid Treatment and Fractal Dimensions

Flocculation water treatment is one of the important aspects of water treatment .The aim of this work is to use PFS for treating Yellow River surface water and different molecular weights of chitosan as coagulant aid. Electron microscope was used to observe flocs morphology and study CTS coagulation aid morphological change, floc image with microscope camera system analysis, calculation of floc fractal dimension. The results show that the different molecular weights of CTS in dealing with surface raw water to play a good coagulation aid effect. Microscope and image analysis system of flocs morphology show that fractal dimension of morphology of PFS / CTS were significantly higher ,Among them，high molecular weight of CTS as coagulant aid，and fractal dimension of morphology of PFS/ CTS were 1.1026,1.5593 , low molecular weight of CTS as coagulant aid，and fractal dimension of morphology of PFS/ CTS were 1.0904,1.5440 respectively. Floc fractal dimension changes to better reflect the flocculation level and treatment effect for the establishment of floc fractal dimension number of changes in the coagulation control is provided for reference.

Fractal characteristics of two-dimensional images of ‘Fuji’ apple trees trained to two tree configurations after their winter pruning

The structural diagrams of apple trees are the comprehensive reflection of the effects of their training and pruning as well as their physiological and ecological characteristics and yield. However, there have been few research reports on the characteristics of the structural diagrams of apple trees. The study investigated the fractal dimension numbers and fractal characteristics of the two-dimensional images of 5-year-old and 10-year-old ‘Fuji’ apple trees trained to the tall spindle configuration and the open-center configuration employing box-counting in combination with the image processing technology of the Photoshop. The two-dimensional images of apple trees with the different configurations differed and varied with their ages. The fractal dimension numbers of the two-dimensional images of the 10-year-old apple trees with the tall spindle configuration and with the open-center configuration were 1.6625 and 1.6531 respectively while the fractal dimension numbers of the two-dimensional images of the 5-year-old apple trees with the tall spindle configuration and with the open-center configuration were 1.6429 and 1.6377 respectively. As the age of the apple trees increased, the spatial quantities and distributions of trunks and branches got slightly intensified, and the fractal dimension numbers of their two-dimensional images and the apple yield increased correspondingly. The comparison of the fractal characteristics of the apple trees with the same age, which were trained to the different configurations, revealed that under the same age, the branch quantities and the apple yield of the apple trees with the tall spindle configuration were higher than those with the open-center configuration, so that under the same age the fractal dimension numbers with the tall spindle configuration were higher than those with the open-center configuration. These results showed that the fractal dimension number of the two-dimensional images of apple trees depended upon their trunk and bough distribution and at the same time it increased with their apple yield as well. Therefore, the fractal dimension numbers of the two-dimensional images of apple trees could be employed as an indicator for assessing training and pruning effects on apple trees and their fruit yield.

Fractal Dimension of Particle Aggregates in Magnetic Fields

ABSTRACTParticle flocculation plays a major role in water treatment processes. In flocculation kinetics models it is usually assumed that spherical particles collide and form spherical aggregates. Real aggregates, however, are of irregular shapes and can be considered as fractal objects. The structure of fractal objects can be described by a fractal dimension number that plays an important role in aggregation kinetics. Two-dimensional computer simulations of particle aggregation are carried out in this work to directly observe the evolution of floc size and to determine their fractal dimension. The computer program developed in this study simulates random particle motion as well as cluster growth. The simulation results are visualized using Java programming language. The fractal dimension of the simulated clusters is determined based on the linear relationship between log-(mass of clusters) and log-(radius of clusters). Primary forces acting on individual particles, including van der Waals, electrostatic, magnetic dipole, and hydrodynamic interparticle forces, are examined to determine the collision efficiency at different collision angles, as well as the structure of the aggregates. The effect of magnetic dipole forces on the fractal dimension and chain formation is examined. It is shown that when the magnetic dipole force is of the same magnitude as the double-layer force within a narrow range of zeta potential values, one-dimensional or two-dimensional clusters may be obtained.