Definition Of Fractal Dimension Research Articles

Change in the Microstructure and Fractal Characteristics of Intact and Compacted Loess Due to Its Collapsibility

This work aimed to examine the fractal dimension and the difference in the law between intact and compacted loess before and after collapse. Uniaxial compression tests were performed to obtain specimens under various vertical stresses, and mercury intrusion porosimetry (MIP) tests were conducted to determine the pore size distribution (PSD). Three models were selected to determine the fractal dimensions based on PSD. As a result, the pores were classified into ultra-micropores (d < 0.1 μm), micropores (0.1 μm < d < 2 μm), small pores (2 μm < d < 10 μm), and large pores (d > 10 μm). When the fractal dimensions were determined using the capillary pressure model, there were three fractal intervals (Ds1, Ds2, and Ds3), with only Ds1 and Ds2 meeting the definition of fractal dimension. Ds1 increased considerably after the collapse, but Ds2 declined. The thermodynamic law-based model presented the best linear fit, and there was only one fractal interval. The fractal dimension Dn increased dramatically after the specimen underwent wet collapse. In conjunction with fractal theory, it revealed that collapse changed the uniformity of the pore system, making the microscopic pores coarser and more intricate after collapse.

EXPLORING INSECTS FREE FLIGHT: ENHANCING THE DIPTERAN FLIGHT MODEL TO INCLUDE FRACTAL EFFECTS

This paper advances fundamental knowledge of how environmental conditions and physical phenomena at different scales can be included in the differential equation that models the flight dynamics of dipteran insects. The insect’s anatomical capability of modifying their mass inertia and flapping-wing damping properties during flight are included by modeling inertia and damping forces with fractal derivatives. An expression for calculating fractal dimension linked to the temporal distribution of non-geometric quantities related to atmospheric processes such as turbulence flow is introduced using, for the first time ever, the two-scale fractal dimension definition and adopting the flow energy spectrum of eddies that occur at large and small scales. The applicability of the derived expression is illustrated with the prediction of the fractal dimension observed in turbulent flows. Then, the two-scale fractal dimension transform is used to re-write the dipteran flight equation of motion in equivalent form to derive its approximate solution using harmonic balance and homotopy perturbation methods. Numerical predictions computed from the derived approximate solutions allow to elucidate how insects and animals could adapt to flight under different environmental conditions.

Fractal dimension and its variation of intact and compacted loess

To investigate the fractal dimension and its variation of intact and compacted loess, consolidated-drained (CD) triaxial tests were performed to obtain intact and compacted loess specimens at different stress-strain states, their pore-size distribution (PSD) curves were measured by using the mercury intrusion porosimetry (MIP) technique. Based on the PSDs, three models (namely, Menger sponge model, Neimark model and the model based on the law of thermodynamics) were used to determine the fractal dimensions of specimens. The results show that there have 3 or 4 fractal intervals (or fractal dimensions, i.e., Ds1, Ds2, Ds3, Ds4,) when using Menger sponge model, only Ds3 satisfies the definition of fractal dimension. Neimark model can be used to determine the fractal dimensions of inter-aggregate pores and intra-aggregate pores, respectively (i.e., DNmacro and DNmicro). Most DNmacro values are meaningless physically, DNmicro increases with the increase of axial strain during shearing, suggesting that the surfaces of pores become complex with the increase of axial strain. The model based on the law of thermodynamics is the most appropriate if only one fractal dimension (i.e., Df) is expected in the whole measurable range of pore diameter. Df increases linearly with the growth of confining stress during consolidation or axial strain during shearing for both loess soils, indicating that the roughness of the pore surfaces increases with the increase in stress or strain. Besides, the fractal dimensions of intact and compacted loess were compared. According to the variations of PSD and Df, the mechanism for the evolution of the loess soil pore structure is proposed. These findings are expected to provide new insight into establishment of the connection between microstructure and macro stress-strain state of loess.

A Novel Method for a COVID-19 Classification of Countries Based on an Intelligent Fuzzy Fractal Approach.

We outline in this article a hybrid intelligent fuzzy fractal approach for classification of countries based on a mixture of fractal theoretical concepts and fuzzy logic mathematical constructs. The mathematical definition of the fractal dimension provides a way to estimate the complexity of the non-linear dynamic behavior exhibited by the time series of the countries. Fuzzy logic offers a way to represent and handle the inherent uncertainty of the classification problem. The hybrid intelligent approach is composed of a fuzzy system formed by a set of fuzzy rules that uses the fractal dimensions of the data as inputs and produce as a final output the classification of countries. The hybrid approach calculations are based on the COVID-19 data of confirmed and death cases. The main contribution is the proposed hybrid approach composed of the fractal dimension definition and fuzzy logic concepts for achieving an accurate classification of countries based on the complexity of the COVID-19 time series data. Publicly available datasets of 11 countries have been the basis to construct the fuzzy system and 15 different countries were considered in the validation of the proposed classification approach. Simulation results show that a classification accuracy over 93% can be achieved, which can be considered good for this complex problem.

Modeling Urban Growth and Form with Spatial Entropy

Entropy is one of the physical bases for the fractal dimension definition, and the generalized fractal dimension was defined by Renyi entropy. Using the fractal dimension, we can describe urban growth and form and characterize spatial complexity. A number of fractal models and measurements have been proposed for urban studies. However, the precondition for fractal dimension application is to find scaling relations in cities. In the absence of the scaling property, we can make use of the entropy function and measurements. This paper is devoted to researching how to describe urban growth by using spatial entropy. By analogy with fractal dimension growth models of cities, a pair of entropy increase models can be derived, and a set of entropy-based measurements can be constructed to describe urban growing process and patterns. First, logistic function and Boltzmann equation are utilized to model the entropy increase curves of urban growth. Second, a series of indexes based on spatial entropy are used to characterize urban form. Furthermore, multifractal dimension spectra are generalized to spatial entropy spectra. Conclusions are drawn as follows. Entropy and fractal dimension have both intersection and different spheres of application to urban research. Thus, for a given spatial measurement scale, fractal dimension can often be replaced by spatial entropy for simplicity. The models and measurements presented in this work are significant for integrating entropy and fractal dimension into the same framework of urban spatial analysis and understanding spatial complexity of cities.

A new fractal dimension definition based on intuitionistic fuzzy logic

A new fractal dimension definition based on intuitionistic fuzzy logic

Transfinite fractal dimension of trees and hierarchical scale-free graphs

Abstract In this article, we introduce a new concept: the transfinite fractal dimension of graph sequences motivated by the notion of fractality of complex networks proposed by Song et al. We show that the definition of fractality cannot be applied to networks with ‘tree-like’ structure and exponential growth rate of neighbourhoods. However, we show that the definition of fractal dimension could be modified in a way that takes into account the exponential growth, and with the modified definition, the fractal dimension becomes a proper parameter of graph sequences. We find that this parameter is related to the growth rate of trees. We also generalize the concept of box dimension further and introduce the transfinite Cesaro fractal dimension. Using rigorous proofs, we determine the optimal box-covering and transfinite fractal dimension of various models: the hierarchical graph sequence model introduced by Komjáthy and Simon, Song–Havlin–Makse model, spherically symmetric trees and supercritical Galton–Watson trees.

An integrated approach for landslide susceptibility mapping by considering spatial correlation and fractal distribution of clustered landslide data

Natural disasters often show highly heterogeneous character due to complex geo-environmental settings. The spatial distribution of landslides is generally clustered at different scales. In this paper, we proposed a methodology for landslide susceptibility mapping (LSM) with consideration of spatial correlation and distribution of clustered landslide data. To quantify the spatial correlation of landslides, a normalized spatial-correlated scale index (NSCI) was introduced. Based on the definition of landslide frequency ratio, calibrated landslide potential index (CLPI) was proposed to account for the effect of landslide clustering. Considering the fractal distribution of landslides, the variable fractal dimension model (VFDM) was introduced to measure the spatial association between clustered landslides and conditional factors. Based on the definition of fractal dimension (D), the weights of the factors were obtained from fractal perspective. We proposed a weighted calibrated landslide potential model (WCLPM), obtained by the combination of CLPI values and weights of the factors. The proposed method is illustrated by example in Xinjiang, NW China, where landslide points are clustered at regional scale. In the example, the landslides were randomly split into two groups: one for building landslide model (training dataset) and the other for validating the model (validating dataset). Five landslide conditional factors (lithology, tectonic faults, elevation, slope, aspect) were selected, processed, and analyzed in a geographic information system (GIS) environment. Predictive accuracy of the WCLPM was evaluated and compared based on the calculation of area under the prediction-rate curve (AUPRC). The example shows that the proposed WCLPM provides good prediction for the study area (AUPRC = 0.8700). This study provided a novel and practical method for LSM.

FRACTAL CHARACTERISTICS ANALYSIS OF MINE WORKINGS GEOMETRY FOR THE ESTIMATION OF SMOOTH WALL BLASTING QUALITY

The work examines the ways of estimating the smooth wall blasting quality when excavating mine workings. One of the main criteria is to achieve “smooth” mine working contour. It is shown that the mine working contour line represents a fractal object; its geometry is properly estimated by fractional fractal dimension. The definition of fractal dimension is based on the Richardson’s law, which connects the length of infinitely broken line and measurement interval. The calculation algorithm of fractal dimension in computer program is introduced. The analysis of real mine working contours which are formed in the process of drilling and blasting operations, allowed to determine statistically reliable connection of roughness coefficient of the mine working surface and the value of its fractal dimension. Quality criterion for smooth wall blasting is suggested – fractal coefficient of mine working shape. Its value represents the relation of equivalent cross-section area of a mine working to its perimeter, which is determined by fractal dimension of a contour. Obtained ratios allow to reliably estimate the quality of smooth wall blasting when excavating mine workings.

Spatial Measures of Urban Systems: from Entropy to Fractal Dimension.

One type of fractal dimension definition is based on the generalized entropy function. Both entropy and fractal dimensions can be employed to characterize complex spatial systems such as cities and regions. Despite the inherent connection between entropy and fractal dimensions, they have different application scopes and directions in urban studies. This paper focuses on exploring how to convert entropy measurements into fractal dimensions for the spatial analysis of scale-free urban phenomena using the ideas from scaling. Urban systems proved to be random prefractal and multifractal systems. The spatial entropy of fractal cities bears two properties. One is the scale dependence: the entropy values of urban systems always depend on the linear scales of spatial measurement. The other is entropy conservation: different fractal parts bear the same entropy value. Thus, entropy cannot reflect the simple rules of urban processes and the spatial heterogeneity of urban patterns. If we convert the generalized entropies into multifractal spectrums, the problems of scale dependence and entropy homogeneity can be solved to a degree for urban spatial analysis. Especially, the geographical analyses of urban evolution can be simplified. This study may be helpful for students in describing and explaining the spatial complexity of urban evolution.

Fractional Perimeters from a Fractal Perspective

Abstract The purpose of this paper consists in a better understanding of the fractional nature of the nonlocal perimeters introduced in [L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 2010, 9, 1111–1144]. Following [A. Visintin, Generalized coarea formula and fractal sets, Japan J. Indust. Appl. Math. 8 1991, 2, 175–201], we exploit these fractional perimeters to introduce a definition of fractal dimension for the measure theoretic boundary of a set. We calculate the fractal dimension of sets which can be defined in a recursive way, and we give some examples of this kind of sets, explaining how to construct them starting from well-known self-similar fractals. In particular, we show that in the case of the von Koch snowflake S ⊆ ℝ 2 {S\subseteq\mathbb{R}^{2}} this fractal dimension coincides with the Minkowski dimension. We also obtain an optimal result for the asymptotics as s → 1 - {s\to 1^{-}} of the fractional perimeter of a set having locally finite (classical) perimeter.

Application of Fractal Theory in Architectural Design

Firstly, this paper introduces the research background of fractal theory. Secondly, the article gives the definition of fractal dimension, and introduces the calculation method of fractal dimension through examples. At last, the paper discusses the fractal structure of several kinds of typical buildings. And then it is proved that the fractal geometry has a deep influence on the development of architecture in the aspects of epistemology and methodology.

Understanding the Fractal Dimensions of Urban Forms through Spatial Entropy

The spatial patterns and processes of cities can be described with various entropy functions. However, spatial entropy always depends on the scale of measurement, and it is difficult to find a characteristic value for it. In contrast, fractal parameters can be employed to characterize scale-free phenomena and reflect the local features of random multi-scaling structure. This paper is devoted to exploring the similarities and differences between spatial entropy and fractal dimension in urban description. Drawing an analogy between cities and growing fractals, we illustrate the definitions of fractal dimension based on different entropy concepts. Three representative fractal dimensions in the multifractal dimension set, capacity dimension, information dimension, and correlation dimension, are utilized to make empirical analyses of the urban form of two Chinese cities, Beijing and Hangzhou. The results show that the entropy values vary with the measurement scale, but the fractal dimension value is stable is method and study area are fixed; if the linear size of boxes is small enough (e.g., <1/25), the linear correlation between entropy and fractal dimension is significant (based on the confidence level of 99%). Further empirical analysis indicates that fractal dimension is close to the characteristic values of spatial entropy. This suggests that the physical meaning of fractal dimension can be interpreted by the ideas from entropy and scaling and the conclusion is revealing for future spatial analysis of cities.

After notes on self-similarity exponent for fractal structures

AbstractPrevious works have highlighted the suitability of the concept of fractal structure, which derives from asymmetric topology, to propound generalized definitions of fractal dimension. The aim of the present article is to collect some results and approaches allowing to connect the self-similarity index and the fractal dimension of a broad spectrum of random processes. To tackle with, we shall use the concept of induced fractal structure on the image set of a sample curve. The main result in this paper states that given a sample function of a random process endowed with the induced fractal structure on its image, it holds that the self-similarity index of that function equals the inverse of its fractal dimension.

Definition of fractal topography to essential understanding of scale-invariance

Fractal behavior is scale-invariant and widely characterized by fractal dimension. However, the cor-respondence between them is that fractal behavior uniquely determines a fractal dimension while a fractal dimension can be related to many possible fractal behaviors. Therefore, fractal behavior is independent of the fractal generator and its geometries, spatial pattern, and statistical properties in addition to scale. To mathematically describe fractal behavior, we propose a novel concept of fractal topography defined by two scale-invariant parameters, scaling lacunarity (P) and scaling coverage (F). The scaling lacunarity is defined as the scale ratio between two successive fractal generators, whereas the scaling coverage is defined as the number ratio between them. Consequently, a strictly scale-invariant definition for self-similar fractals can be derived as D = log F /log P. To reflect the direction-dependence of fractal behaviors, we introduce another parameter Hxy, a general Hurst exponent, which is analytically expressed by Hxy = log Px/log Py where Px and Py are the scaling lacunarities in the x and y directions, respectively. Thus, a unified definition of fractal dimension is proposed for arbitrary self-similar and self-affine fractals by averaging the fractal dimensions of all directions in a d-dimensional space, which . Our definitions provide a theoretical, mechanistic basis for understanding the essentials of the scale-invariant property that reduces the complexity of modeling fractals.

A survey on fractal dimension for fractal structures

Abstract Along the years, the foundations of Fractal Geometry have received contributions starting from mathematicians like Cantor, Peano, Hilbert, Hausdorff, Carathéodory, Sierpiński, and Besicovitch, to quote some of them. They were some of the pioneers exploring objects having self-similar patterns or showing anomalous properties with respect to standard analytic attributes. Among the new tools developed to deal with this kind of objects, fractal dimension has become one of the most applied since it constitutes a single quantity which throws useful information concerning fractal patterns on sets. Several years later, fractal structures were introduced from Asymmetric Topology to characterize self-similar symbolic spaces. Our aim in this survey is to collect several results involving distinct definitions of fractal dimension we proved jointly with Prof.M.A. Sánchez-Granero in the context of fractal structures.

On discrete models of fractal dimension to explore the complexity of discrete dynamical systems

A fractal structure is a countable family of coverings which displays accurate information about the irregularities that a set presents when being explored with enough level of detail. It is worth noting that fractal structures become especially appropriate to provide new definitions of fractal dimension, which constitutes a valuable measure to test for chaos in dynamical systems. In this paper, we explore several approaches to calculate the fractal dimension of a subset with respect to a fractal structure. These models generalize the classical box dimension in the context of Euclidean subspaces from a discrete viewpoint. To illustrate the flexibility of the new models, we calculate the fractal dimension of a family of self-affine sets associated with certain discrete dynamical systems.

Fractal detector design and analysis for multi‐input multi‐output radar system

The fractal dimension has been used in the monostatic radar to improve the target detection performance, especially in non-homogeneous clutter environments. In this study, the application of fractal dimension is extended to multi-input multi-output (MIMO) radar. A target detection method for MIMO radar using the fractal dimension measurements is developed. Two types of fractal dimension definitions, i.e. Hausdorf dimension and box-counting dimension are explained and the characteristics of the box-counting dimension are analysed in detail. Then, a fractal dimension estimation-based detector is presented and its performance is analysed for colocated MIMO and statistical MIMO. The simulations results verify the theoretical founding of the proposed target detector for MIMO radar based on the fractal dimension estimation.

A new fractal dimension for curves based on fractal structures

In this paper, we introduce a new theoretical model to calculate the fractal dimension especially appropriate for curves. This is based on the novel concept of induced fractal structure on the image set of any curve. Some theoretical properties of this new definition of fractal dimension are provided as well as a result which allows to construct space-filling curves. We explore and analyze the behavior of this new fractal dimension compared to classical models for fractal dimension, namely, both the Hausdorff dimension and the box-counting dimension. This analytical study is illustrated through some examples of space-filling curves, including the classical Hilbert's curve. Finally, we contribute some results linking this fractal dimension approach with the self-similarity exponent for random processes.

Fractal Dimension Analysis of the Julia Sets of Controlled Brusselator Model

Fractal theory is a branch of nonlinear scientific research, and its research object is the irregular geometric form in nature. On account of the complexity of the fractal set, the traditional Euclidean dimension is no longer applicable and the measurement method of fractal dimension is required. In the numerous fractal dimension definitions, box-counting dimension is taken to characterize the complexity of Julia set since the calculation of box-counting dimension is relatively achievable. In this paper, the Julia set of Brusselator model which is a class of reaction diffusion equations from the viewpoint of fractal dynamics is discussed, and the control of the Julia set is researched by feedback control method, optimal control method, and gradient control method, respectively. Meanwhile, we calculate the box-counting dimension of the Julia set of controlled Brusselator model in each control method, which is used to describe the complexity of the controlled Julia set and the system. Ultimately we demonstrate the effectiveness of each control method.