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A new proof of Honeycomb Conjecture by fractal geometry methods

Tong ZHANG, Kai DING

《机械工程前沿（英文）》 2013年第8卷第4期页码 367-370 doi: 10.1007/s11465-013-0273-7

摘要：

Based on fractal geometry, we put forward a concise and straightforward method to prove Honeycomb Conjecture—a classical mathematic problem. Hexagon wins the most efficient covering unit in the two- dimensional space, compared with the other two covering units—triangle and square. From this point of view, honeycomb is treated as a hierarchical fractal structure that fully fills the plane. Therefore, the total side length and area are easily calculated and from the results, the covering efficiency of each possible unit is provided quantitatively.

关键词： Honeycomb Conjecture fractal geometry hierarchical fractal structure

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基于分形和小波的几何型面加工误差综合分析

廖小云,唐倩,赵英,郑世泽

《中国工程科学》 2002年第4卷第5期页码 75-78

摘要：

对几何型面加工误差提出并建立了基于分形和小波的综合分析方法，可以精细分析几何型面加工误差的微细成分，并重构出型面轮廓加工误差曲线，以便进行公差与性能关系分析以及加工过程质量监控等相关工作。应用实例表明，该方法是非常有效的。

关键词：机加工加工误差分析分形几何小波分析

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Hierarchical fractal structure of perfect single-layer graphene

T. Zhang, K. Ding

《机械工程前沿（英文）》 2013年第8卷第4期页码 371-382 doi: 10.1007/s11465-013-0279-1

摘要：

The atomic lattice structure of perfect single-layer graphene that can actually be regarded as a kind of hierarchical fractal structure from the perspective of fractal geometry was studied for the first time. Three novel and special discoveries on hierarchical fractal structure and sets were unveiled upon examination of the regular crystal lattices of the single-layer graphene. The interior fractal-type structure was discovered to be the fifth space-filling curve from physical realm. Two efficient methods for calculating the fractal dimension of this fresh member was also provided. The outer boundary curve had a fractal dimension equal to one, and a multi-fractal structure from a naturally existing material was found for the first time. A series of strict self-similar hexagons comprised a rotating fractal set. These hexagons slewed at a constant counterclockwise angle of 19.1° when observed from one level to the next higher level. From the perspective of fractal geometry, these pioneering discoveries added three new members to the existing regular fractal structures and sets. A fundamental example of a multi-fractal structure was also presented.

关键词： hierarchical fractal structure fractal dimension the fifth space-filling curve multi-fractal structure

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Theoretical study on the confine-stiffening effect and fractal cracking of square concrete filled steel

《结构与土木工程前沿（英文）》 2021年第15卷第6期页码 1317-1336 doi: 10.1007/s11709-021-0763-3

摘要： Tension stress in steel-concrete composite is widely observed in engineering design. Based on an experimental program on tension performance of three square concrete-filled tubes (SCFT), the tension theory of SCFT is proposed using a mechanics-based approach. The tension stiffening effect, the confining strengthening effect and the confining stiffening effect, observed in tests of SCFTs are included in the developed tension theory model. Subsequently, simplified constitutive models of steel and concrete are proposed for the axial tension performance of SCFT. Based on the MSC.MARC software, a special fiber beam-column element is proposed to include the confining effect of SCFTs under tension and verified. The proposed analytical theory, effective formulas, and equivalent constitutive laws are extensively verified against three available tests reported in the literature on both global level (e.g., load-displacement curves) and strain level. The experimental verification proves the accuracy of the proposed theory and formulations in simulating the performance of SCFT members under tension with the capability to accurately predict the tensile strength and stiffness enhancements and realistically simulate the fractal cracking phenomenon.

关键词： square concrete filled tubes confine-stiffening confine-strengthening fractal cracking fracture

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Copper fractal growth during recycling from waste printed circuit boards by slurry electrolysis

《环境科学与工程前沿（英文）》 2021年第15卷第6期 doi: 10.1007/s11783-021-1405-7

摘要：

• Copper fractal growth was observed during WPCBs recycling by slurry electrolysis.

关键词： Dendritic copper Fractal growth WPCBs Slurry electrolysis

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Fractal characteristic evaluation and interpolation reconstruction for surface topography of drilled

《机械工程前沿（英文）》 2021年第16卷第4期页码 840-854 doi: 10.1007/s11465-021-0643-5

摘要： In this paper, an improved fractal interpolation model is proposed to reconstruct the surface topography of composite hole wall. This model adopts the maximum positive deviations and maximum negative deviations between the measured values and trend values to determine the contraction factors. Hole profiles in 24 directions are measured. Fractal parameters are calculated to evaluate the measured surface profiles. The maximum and minimum fractal dimension of the hole wall are 1.36 and 1.07, whereas the maximum and minimum fractal roughness are 4.05 × 10 ⁻⁵ and 4.36 × 10 ⁻¹⁰ m, respectively. Based on the two-dimensional evaluation results, three-dimensional surface topographies in five typical angles (0°, 45°, 90°, 135°, and 165°) are reconstructed using the improved model. Fractal parameter D _s and statistical parameters Sa, Sq, and Sz are used to evaluate the reconstructed surfaces. Average error of D _s, Sa, Sq, and Sz between the measured surfaces and the reconstructed surfaces are 1.53%, 3.60%, 5.60%, and 9.47%, respectively. Compared with the model in published literature, the proposed model has equal reconstruction effect in relatively smooth surface and is more advanced in relatively rough surface. Comparative results prove that the proposed model for calculating contraction factors is more reasonable.

关键词： surface topography fractal evaluation fractal interpolation reconstruction composite

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Russian school of the theory and geometry of gearing: Its origin and golden period (1935 --1975)

Dmitry BABICHEV,Sergey LAGUTIN,Natalya BARMINA

《机械工程前沿（英文）》 2016年第11卷第1期页码 44-59 doi: 10.1007/s11465-015-0360-z

摘要：

During the second half of the twentieth century, Soviet scientists were among the world leaders in the theory and geometry of gearing or the theoretical fundamentals of designing gears and gear-cutting tools. The authors of the current paper review works related to the development of the theory of gearing (TG) written in Russian from the mid-1930s to the present. Overall, the review covers more than 400 publications of nearly 200 authors (among more than 1000 studied works). This review involves some works on gear strength and accuracy to attract more experts.

This paper presents a short list of works from the seventeenth century to the nineteenth century and discusses the basic concepts of the TG. This paper also includes a review of more than 160 papers in Russian for the period of 1935–1975, which is the “golden age” of the TG. The main features and achievements during this period are also considered.

The distinctive feature of this review is the use of diagrams that visually present TG experts and the dates and themes of their published works. The diagrams also show the following aspects: 1) The most popular and important themes in a specific period, 2) the most important publications on the themes of interest to readers, and 3) the direction of the works and scope of interests of every researcher.

关键词： theory of gearing (TG) gear geometry generating surface gear synthesis gear analysis

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Permeability analysis and seepage process study on crystal layer in melt crystallization with fractal

Xiaobin JIANG, Baohong HOU, Yongli WANG, Jingkang WANG

《化学科学与工程前沿（英文）》 2011年第5卷第4期页码 435-441 doi: 10.1007/s11705-011-1139-3

摘要： In this paper a porous media seepage model was applied to analyze the permeability and study the seepage process of crystal pillar formed in the preparation of electronic grade phosphoric acid (EGPA). By inspecting the seeping process, the structure parameter of crystal pillar could be obtained. Two basic ideal models (perfectly separated model and perfectly connected model) were presented and a characterized factor was introduced to modify the model. A good simulation result was obtained which met the experiment result well. The relationship between and permeability were also discussed. The characterized factor showed potential application on optimizing process.

关键词： melt crystallization porous media fractal permeability hyperpure material simulation

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Aggregating structure in coal water slurry studied by eDLVO theory and fractal dimension

Qiang LI, Qian WANG, Jian HOU, Jiansheng ZHANG, Yang ZHANG

《能源前沿（英文）》 2023年第17卷第2期页码 306-316 doi: 10.1007/s11708-021-0736-1

摘要： Coal water slurry gasification is a main source of hydrogen in the developing hydrogen economy. Moreover, biomass and waste can be added, making gasification process greener. To expand the application of coal water slurry and gasification process, it is necessary to understand the micro-structure in this large particle suspension system. In this paper, the micro-structure in coal water slurry was studied by extended DLVO (eDLVO) theory and fractal dimension, which is used to explain the mechanism of stability in large particle suspension systems. The interaction between two coal particles was characterized from the interparticle potential and energy barrier based on the eDLVO theory. The rheology and stability between different types of coals are measured and explained by the aggregating structure and fractal dimension in coal water slurry. The results indicated that there would be an aggregating structure in high rank coals, due to the interparticle potential caused by the surface properties, but probably not in low rank coals. This aggregating structure can be described and characterized by fractal dimension. The aggregation of particles is the source of the stability for high rank coals, as the close-packed 3D network structure in large particle suspension can support coal particles from settling down. The results have demonstrated that the combination of the eDLVO theory and rheological measurement is an effective way to investigate the stability of large particle suspension systems.

关键词： coal water slurry extended DLVO (eDLVO) fractal dimension stability

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Application of fractal theory to unsaturated soil mechanics

XU Yongfu, TONG Lixin

《结构与土木工程前沿（英文）》 2007年第1卷第4期页码 411-421 doi: 10.1007/s11709-007-0056-5

摘要： The mechanical properties of unsaturated soils are a function of the saturation degree or matric suction, and can be obtained based on currently available procedures. However, each procedure has its limitations and consequently, care should be taken in the selection of a proper procedure. The fractal approach seems to be a potentially useful tool to describe hierarchical systems and is suitable to model the structure and hydraulic properties of unsaturated soils. In this paper, the soil-water characteristics, unsaturated hydraulic conductivity function, unsaturated shear strength, swelling deformation and compression were derived from the fractal model for the pore-size distribution, and were expressed by only two independent physical parameters, the fractal dimension and the air entry value. The predictions of the proposed soil-water characteristics, unsaturated hydraulic conductivity, unsaturated shear strength, swelling deformation and compression were in good agreement with published experimental data. Comparisons between the experimental results of unsaturated hydraulic conductivity and the predictions of the both fractal model and the van Genuchten-Mualem model were also performed, and it was found that the predictions of the fractal model were better than that of the van Genuchten-Mualem model.

关键词： selection soil-water independent physical unsaturated strength

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Transmission probability method based on triangle meshes for solving the unstructured geometry neutron

WU Hongchun, LIU Pingping, ZHOU Yongqiang, CAO Liangzhi

《能源前沿（英文）》 2007年第1卷第2期页码 158-166 doi: 10.1007/s11708-007-0020-z

摘要： The fuel assembly or core with unstructured geometry is frequently used in the advanced reactor. To calculate the fuel assembly, the transmission probability method (TPM) is widely used. However, the rectangular or hexagonal meshes are mainly used in the TPM codes for the normal core structure. The triangle meshes are most useful for expressing the complicated unstructured geometry. Even though the finite element method and Monte-Carlo method are well suited for solving the unstructured geometry problem, they are very time-consuming. Therefore, a TPM code based on the triangle meshes is developed here. This code was applied to the hybrid fuel geometry, and compared with the results of the MCNP code and other codes. The results of the comparison were consistent with each other. The TPM with triangle meshes can thus be applied to the two-dimensional arbitrary fuel assembly.

关键词： unstructured geometry assembly transmission probability TPM rectangular

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分形模型参数与粗糙度参数R_a关系的研究

潘玉良,吴立群,张云电

《中国工程科学》 2004年第6卷第5期页码 49-51

摘要：

介绍了W－M分形模型参数（D,G,γ）的物理含义，研究了分形参数（D,G,γ）与粗糙度参数R_a之间的联系，导出了分形参数（D,G,γ）与粗糙度参数R_a关系式，用数值分析拟合方法验证了关系式的正确性。

关键词： W－M分形模型分形参数粗糙度R_a

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一种分形图像编码的新方法

王秀妮,姜威,王利村

《中国工程科学》 2006年第8卷第1期页码 54-57

摘要：

由于分形图像编码过程非常费时，因此在前人的基础上，通过分析影响分形图像编码速度的相关因素，提出了使用图像块的改进方差来提高分形压缩性能的思想。在证明迭代函数系统不会改变图像块的改进方差的基础上，给出了基于改进方差的分形图像压缩方法。实验结果说明该方法在保证快速性的同时，重建图像的质量又得到了进一步的提高。

关键词：分形编码图像压缩方差

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PD pattern recognition based on multi-fractal dimension in GIS

ZHANG Xiaoxing, YAO Yao, TANG Ju, ZHOU Qian, XU Zhongrong

《机械工程前沿（英文）》 2008年第3卷第3期页码 270-275 doi: 10.1007/s11465-008-0042-1

摘要： This paper designs four types of gas insulated substation (GIS) defect models based on partial discharge (PD) characteristics and its defections. The GIS gray intensity images are constructed based on the mass specimens gathered by the ultra-high frequency and high-speed sampling systems. The multi-fractal dimension is founded on the box-counting dimension and multi-fractal theories. The GIS gray intensity images distillation methods, based on multi-fractal characteristics, is put forward. The box-counting dimension, multi-fractal dimension, and discharge centrobaric characteristics of the PD images are also extracted. The characteristic variables are then classified by the radial basis function (RBF) network. Identified results show that the methods can effectively elevate the discrimination of the four types of defects in GIS.

关键词： substation high-speed discharge centrobaric network ultra-high frequency