scholarly journals Fractal Modeling and Fractal Dimension Description of Urban Morphology

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 961 ◽  
Author(s):  
Yanguang Chen
Keyword(s):  
Fractal Dimension ◽  
Urban Area ◽  
Fractal Geometry ◽  
Urban Form ◽  
Characteristic Length ◽  
Fractal Theory ◽  
Urban Morphology ◽  
Scaling Analysis ◽  
Mathematical Methods ◽  
Scale Free

The conventional mathematical methods are based on characteristic length, while urban form has no characteristic length in many aspects. Urban area is a scale-dependence measure, which indicates the scale-free distribution of urban patterns. Thus, the urban description based on characteristic lengths should be replaced by urban characterization based on scaling. Fractal geometry is one powerful tool for the scaling analysis of cities. Fractal parameters can be defined by entropy and correlation functions. However, the question of how to understand city fractals is still pending. By means of logic deduction and ideas from fractal theory, this paper is devoted to discussing fractals and fractal dimensions of urban landscape. The main points of this work are as follows. Firstly, urban form can be treated as pre-fractals rather than real fractals, and fractal properties of cities are only valid within certain scaling ranges. Secondly, the topological dimension of city fractals based on the urban area is 0; thus, the minimum fractal dimension value of fractal cities is equal to or greater than 0. Thirdly, the fractal dimension of urban form is used to substitute the urban area, and it is better to define city fractals in a two-dimensional embedding space; thus, the maximum fractal dimension value of urban form is 2. A conclusion can be reached that urban form can be explored as fractals within certain ranges of scales and fractal geometry can be applied to the spatial analysis of the scale-free aspects of urban morphology.

scholarly journals The Solutions to the Uncertainty Problem of Urban Fractal Dimension Calculation

Entropy ◽  
2019 ◽  
Vol 21 (5) ◽  
pp. 453 ◽  
Author(s):  
Chen
Keyword(s):  
Fractal Dimension ◽  
Spatial Analysis ◽  
Fractal Geometry ◽  
Fractal Dimensions ◽  
Dimension Estimation ◽  
Scale Free ◽  
Factors Influencing ◽  
Calculation Results ◽  
Fractal Patterns ◽  
Main Factors

Fractal geometry provides a powerful tool for scale-free spatial analysis of cities, but the fractal dimension calculation results always depend on methods and scopes of the study area. This phenomenon has been puzzling many researchers. This paper is devoted to discussing the problem of uncertainty of fractal dimension estimation and the potential solutions to it. Using regular fractals as archetypes, we can reveal the causes and effects of the diversity of fractal dimension estimation results by analogy. The main factors influencing fractal dimension values of cities include prefractal structure, multi-scaling fractal patterns, and self-affine fractal growth. The solution to the problem is to substitute the real fractal dimension values with comparable fractal dimensions. The main measures are as follows. First, select a proper method for a special fractal study. Second, define a proper study area for a city according to a study aim, or define comparable study areas for different cities. These suggestions may be helpful for the students who take interest in or have already participated in the studies of fractal cities.

Fractal Study on the Complexity of Limestone Surface Pore Structure

2012 ◽  
Vol 548 ◽  
pp. 275-280
Author(s):  
Xin Wu ◽  
Si Long ◽  
Guo Hui Li
Keyword(s):  
Fractal Dimension ◽  
Rock Mass ◽  
Pore Structure ◽  
Fractal Geometry ◽  
Fractal Theory ◽  
Microscopic Images ◽  
Simple Rules ◽  
Complex Phenomena ◽  
Mathematics And Physics

Complex characteristics of pore structure of rock mass, such as limestone, are difficult to describe by means of general mathematics and physics. While, the fractal geometry can describe some simple rules behind complex phenomena; and these simple rules can describe the complex phenomena. Therefore in this paper, the fractal theory is applied to study the complexity of the limestone pore structure. Through calculating the fractal dimension of the limestone pore microscopic images of different zoom scales, the scale-independence is proved to be possessed by complexity of pore, which indicates that the limestone is a good fractal body, and its complexity can be studied by means of fractal dimension.

Microstructure Characterization of Ceramics by Fractal Geometry

1997 ◽  
Vol 492 ◽  
Author(s):  
Z. X. Xiong ◽  
K. Z. Baba-Kishi ◽  
F. G. Shin
Keyword(s):  
Electron Microscopy ◽  
Image Processing ◽  
Fractal Dimension ◽  
Fractal Geometry ◽  
Fractal Theory ◽  
Image Processing Techniques ◽  
Fractal Analyses ◽  
Processing Techniques ◽  
Good Agreement

ABSTRACTFollowing Mandelbrot's fractal theory, the irregular characteristics of the microstructural features of ferroelectric Pb(Sc0.5 Ta0.5)O3 ceramics, including grain boundaries and dislocation networks, were investigated. The microstructural features were imaged by electron microscopy. The fractal analyses were carried out manually and by image processing techniques, which show the value of the fractal dimension, D, varies according to the regularity of the microstructure. The value of, D, close to unity is an indication of an increasing degree of microstructural regularity, which is in good agreement with the simulated results.

scholarly journals LOGISTIC MODELS OF FRACTAL DIMENSION GROWTH OF URBAN MORPHOLOGY

Fractals ◽  
2018 ◽  
Vol 26 (03) ◽  
pp. 1850033 ◽  
Author(s):  
YAN-GUANG CHEN
Keyword(s):  
Fractal Dimension ◽  
Urban Form ◽  
Missing Values ◽  
Logistic Models ◽  
Developed Countries ◽  
Logistic Function ◽  
Urban Morphology ◽  
City Development ◽  
Spatial Form ◽  
The Developed Countries

Urban form can be described with fractal dimension, which is a measurement of space filling of urban evolution. However, how to model and understand the fractal dimension growth of urban morphology are still pending questions. This paper is devoted to the research on the fractal dimension curves of urban growth. The principle of squashing function and empirical evidences are employed to demonstrate the following inference: the fractal dimension time series of a city’s spatial form take on a sigmoid curve. Among various sigmoid functions, the logistic function is the most probable selection. The observational data of fractal dimension of different cities from different sources support this logic judgment. A further discovery is that the fractal dimension curves of cities in the developed countries differ from those in the developing countries. A generalized logistic function is thus proposed to model the fractal dimension curves of different types of cities. The general logistic models can be used to predict the missing values and estimate the growth rates of fractal dimension of city development. Moreover, these models can be utilized to analyze when and where there is a fractal of urban form.

TWO-DIMENSIONAL FRACTAL MODEL FOR ULTIMATE CRUSHING STATE OF COARSE AGGREGATES

Fractals ◽  
2019 ◽  
Vol 27 (07) ◽  
pp. 1950109
Author(s):  
QIANMI YU ◽  
JIANKUN LIU ◽  
UJWALKUMAR D. PATIL ◽  
SURYA S. C. CONGRESS ◽  
ANAND J. PUPPALA
Keyword(s):  
Particle Size ◽  
Fractal Dimension ◽  
Fractal Geometry ◽  
Fractal Theory ◽  
Size Fraction ◽  
Size Variation ◽  
Fractal Model ◽  
Fragmentation Model ◽  
Two Dimensional ◽  
Coarse Aggregates

The research on the ultimate crushing state of coarse aggregates is beneficial to analyze and predict the evolutionary process of crushing. The Growing Path method uses the two-dimensional fractal geometry structure to simulate the size variation of particle size fraction during the particle breakage of coarse aggregates and it serves to investigate the ultimate fractal dimension corresponding to the ultimate crushing state of coarse aggregates. This method manifests the self-growing characteristics of particle size distribution in the process of particle crushing. This study found that the two-dimensional image of ultimate fractal model was precisely similar to that of the Sierpinski gasket of fractal theory when the ultimate crushing state was reached. The results from the model analysis show that the theoretically ultimate fractal dimension is about 2.585, which is consistent with the existing results calculated from the three-dimensional ultimate fragmentation model of cataclastic rock located in the fault zones. The relationship between two fractal models was analyzed. Furthermore, the application of fractal geometry presented in this study will also serve as a reference for the analysis of the other chaos phenomena observed in geotechnical engineering.

Predicting Ecological Connectivity in Urbanizing Landscapes

2005 ◽  
Vol 32 (5) ◽  
pp. 763-776 ◽  
Author(s):  
Britta G Bierwagen
Keyword(s):  
Urban Area ◽  
Urban Form ◽  
Urban Morphology ◽  
Landscape Patterns ◽  
Urban Landscapes ◽  
Urban Centers ◽  
Ecological Connectivity ◽  
The Usa ◽  
Level Of Aggregation

Nearly half the world's population lives in urban centers, and these areas are increasingly important components of regional and global land cover. However, their ecological attributes are often overlooked, despite the presence of species, ecosystem services, and risks associated with the spread of pests or threatening processes such as fire. Movement and dispersal of organisms contribute to species persistence in urban landscapes; however, landscape patterns that promote ecological connectivity may also facilitate the spread of undesirable organisms or processes. I investigate how urban form can be used to predict ecological connectivity and assist in prioritizing urban landscapes for conservation activities and risk management. I examine the value of qualitative and quantitative descriptions of urban morphology as predictors of ecological connectivity by comparing sixty-six cities in the USA. Results show that qualitative categories are not adequate for describing ecological connectivity; multivariate descriptions are much better predictors, with urban area, number of urban patches, urban patch extent, level of aggregation, and perimeter area fractal dimension composing the significant synthetic variables. The dominance of area as a differentiating variable led to the development of a new urban connectivity index using a combination of urban area and state population size. This metric, based on readily available aspatial data, explains 78% of variation in ecological connectivity. These results provide a simple but novel tool for beginning to understand the role of urban morphology in promoting desirable environmental outcomes and managing environmental risks in urbanizing landscapes.

Optimal Density Determination of Bouguer Anomaly Using Fractal Analysis (Case of study: Charak area,IRAN)

2005 ◽  
Vol 1 (1) ◽  
pp. 21-24
Author(s):  
Hamid Reza Samadi
Keyword(s):  
Surface Roughness ◽  
Fractal Dimension ◽  
Fractal Analysis ◽  
Fractal Geometry ◽  
Host Rock ◽  
Bouguer Anomaly ◽  
Research Area ◽  
Density Difference ◽  
Optimal Density

In exploration geophysics the main and initial aim is to determine density of under-research goals which have certain density difference with the host rock. Therefore, we state a method in this paper to determine the density of bouguer plate, the so-called variogram method based on fractal geometry. This method is based on minimizing surface roughness of bouguer anomaly. The fractal dimension of surface has been used as surface roughness of bouguer anomaly. Using this method, the optimal density of Charak area insouth of Hormozgan province can be determined which is 2/7 g/cfor the under-research area. This determined density has been used to correct and investigate its results about the isostasy of the studied area and results well-coincided with the geology of the area and dug exploratory holes in the text area

scholarly journals Dynamic characteristics and fractal representations of crack propagation of rock with different fissures under multiple impact loadings

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Bing Sun ◽  
Shun Liu ◽  
Sheng Zeng ◽  
Shanyong Wang ◽  
Shaoping Wang
Keyword(s):  
Fractal Dimension ◽  
Crack Propagation ◽  
Crack Growth ◽  
Impact Test ◽  
Fractal Theory ◽  
Dynamic Change ◽  
Fractal Dimensions ◽  
Peak Stress ◽  
Dynamic Mechanical Properties ◽  
Gravitational Potential Energy

AbstractTo investigate the influence of the fissure morphology on the dynamic mechanical properties of the rock and the crack propagation, a drop hammer impact test device was used to conduct impact failure tests on sandstones with different fissure numbers and fissure dips, simultaneously recorded the crack growth after each impact. The box fractal dimension is used to quantitatively analyze the dynamic change in the sandstone cracks and a fractal model of crack growth over time is established based on fractal theory. The results demonstrate that under impact test conditions of the same mass and different heights, the energy absorbed by sandstone accounts for about 26.7% of the gravitational potential energy. But at the same height and different mass, the energy absorbed by the sandstone accounts for about 68.6% of the total energy. As the fissure dip increases and the number of fissures increases, the dynamic peak stress and dynamic elastic modulus of the fractured sandstone gradually decrease. The fractal dimensions of crack evolution tend to increase with time as a whole and assume as a parabolic. Except for one fissure, 60° and 90° specimens, with the extension of time, the increase rate of fractal dimension is decreasing correspondingly.

scholarly journals Effect of Hydrophobic Silica Nanochannel Structure on the Running Speed of a Colloidal Damper

2021 ◽  
Vol 11 (15) ◽  
pp. 6808
Author(s):  
Gengbiao Chen ◽  
Zhiwen Liu
Keyword(s):  
Fractal Dimension ◽  
Silica Gel ◽  
Fractal Theory ◽  
Percolation Model ◽  
Silica Gels ◽  
Practical Significance ◽  
Running Speed ◽  
Fractal Percolation ◽  
Micropore Structure ◽  
Hydrophobic Silica

A colloidal damper (CD) can dissipate a significant amount of vibrations and impact energy owing to the interface power that is generated when it is used. It is of great practical significance to study the influence of the nanochannel structure of hydrophobic silica gel in the CD damping medium on the running speed of the CD. The fractal theory was applied to observe the characteristics of the micropore structure of the hydrophobic silica gel by scanning electron microscopy (SEM), the primary particles were selected to carry out fractal analysis, and the two-dimensional fractal dimension of the pore area and the tortuous fractal dimension of the hydrophobic silica gel pore structure were calculated. The fractal percolation model of water in hydrophobic silica nanochannels based on the slip theory could thus be obtained. This model revealed the relationship between the micropore structure parameters of the silica gel and the running speed of the CD. The CD running speed increases with the addition of grafted molecules and the reduction in pore size of the silica gel particles. Continuous loading velocity testing of the CD loaded with hydrophobic silica gels with different pore structures was conducted. By comparing the experimental results with the calculation results of the fractal percolation model, it was determined that the fractal percolation model can better characterize the change trend of the CD running velocity for the first loading, but the fractal dimension was changed from the second loading, caused by the small amount of water retained in the nanochannel, leading to the failure of fractal characterization.

scholarly journals Scale Characteristics and Optimization of Park Green Space in Megacities Based on the Fractal Measurement Model: A Case Study of Beijing, Shanghai, Guangzhou, and Shenzhen

2021 ◽  
Vol 13 (15) ◽  
pp. 8554
Author(s):  
Zhen Li ◽  
Wanmin Zhao ◽  
Miaoyao Nie
Keyword(s):  
Fractal Dimension ◽  
Evaluation System ◽  
Goodness Of Fit ◽  
Green Space ◽  
Fractal Theory ◽  
Measurement Model ◽  
Scale Structure ◽  
Qualitative Description ◽  
Urban Spatial Structure ◽  
Factors Affecting

This paper applies fractal theory to research of green space in megacity parks due to the lack of a sufficient qualitative description of the scale structure of park green space, a quantifiable evaluation system, and operable planning methods in traditional studies. Taking Beijing, Shanghai, Guangzhou, and Shenzhen as examples, GIS spatial analysis technology and the Zipf model are used to calculate the fractal dimension (q), the goodness of fit (R2), and the degree of difference (C) to deeply interpret the connotation of indicators and conduct a comparative analysis between cities to reveal fractal characteristics and laws. The research results show that (1) the fractal dimension is related to the complexity of the park green space system; (2) the fractal dimension characterizes the hierarchical iteration of the park green space to a certain extent and reflects the internal order of the scale distribution; (3) the scale distribution of green space in megacity parks deviates from the ideal pyramid configuration; and (4) there are various factors affecting the scale structure of park green space, such as natural base conditions, urban spatial structure, and the continuation of historical genes working together. On this basis, a series of targeted optimization strategies are proposed.

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