Fractal Geometry Theory Research Articles

Exploring the Role of Fractal Geometry in Engineering Image Processing Based on Similarity and Symmetry: A Review

Fractal geometry theory has been widely used in engineering image processing. In this work, the basic principles and features of fractal geometry are first introduced and its importance in image processing is explained. The features of the symmetry and asymmetry of images are represented in fractal geometry and symmetry scaling is utilized to deal with image processing problems in engineering applications. Subsequently, specific applications of fractal geometry in engineering image processing are discussed in detail in terms of image compression, edge detection, texture analysis, and image reconstruction and restoration. The exploration of these applications reveals the advantages and usefulness of fractal geometry theory in image processing, and it is found that the image has certain symmetry and self-similarity, which is conducive to the establishment of mathematical models for the statistics of graphic contours and shapes. Finally, the unique value of fractal geometry in engineering image processing is further emphasized by comparing the innovations of fractal geometry with traditional image processing methods, which prompts the in-depth consideration of its potential value in this field. This paper provides new insights and directions for the research of engineering image processing, which is of positive significance for future research.

Research on the Impact of Coastal Land Reclamation on Geomorphic Complexity and Tidal Energy Dissipation—A Case Study of Zhoushan Islands, Hangzhou Bay

ABSTRACTCoastal mudflats are vital land resources. In recent years, coastal reclamation has emerged as a vital strategy for alleviating coastal land supply shortages and for developing marine resources in coastal areas. However, large‐scale coastal reclamation changes the natural landscape properties of sea areas, triggering alterations in the surrounding hydrodynamic and ecological environments. Despite the significant impact of coastal geomorphic features on tidal energy loss and disaster prevention, few studies have explored the intrinsic connection between the complexity of coastal landscapes and the rate of tidal reduction. This relationship is crucial for predicting potential tidal energy loss responses to large‐scale changes in geomorphologic features caused by accelerating coastal land reclamation. This study focuses on Hangzhou Bay as the research area and develops a model that combines the correlation between the rate of tidal reduction and the complexity of coastal landscapes. The model is constructed using fractal geometry theory and quantitative methods, while also incorporating principles from ocean dynamics theory. The model is applied to assess the impacts of reclaimed land on the geomorphic complexity and tidal energy dissipation in the Zhoushan Islands. Results indicate a high correlation between differential tide reduction rates and coastal geomorphic complexity. Application of this model reveals that traditional reclamation planning and design substantially alter coastal morphology. Consequently, the significant reduction in the fractal dimension (D) and shape index (S) leads to a tidal reduction rate decrease of over 88%, posing substantial threats to coastal and estuarine bay disaster prevention and ecosystem services. The study proposes effective control indicators and scientific reclamation planning measures, providing a theoretical basis and novel methods for the rational utilization and sustainable development of coastal mudflat resources.

Numerical Simulation on Flow Coupling in a Triple-Fracture Network Model Based on Fractal Geometry Theory

Numerical Simulation on Flow Coupling in a Triple-Fracture Network Model Based on Fractal Geometry Theory

Study on the evolution of fractures in overlying strata during repeated mining of coal seams at extremely close distances

In particular, the secondary development of overlying strata fractures can easily lead to the upper goaf, resulting in gas and water gathered in the goaf entering the working face of the lower coal seam through the overlying strata fractures, threatening the safety of coal mine production. Security risks may arise. To further understand the caving and evolution law of overlying strata during repeated mining in extremely close distance coal seam down mining, 9# coal and 10# coal in the Nanyaotou Coal Industry were used as the engineering background. The caving characteristics and fracture evolution law of overlying strata during single and repeated mining were analyzed through similar material simulation tests. Based on fractal geometry theory, the relationship between the advancing distance of the working face and the fractal dimension of the overlying strata fracture is established to reflect the changing trend of fracture development. The calculation formula is derived from the tensile rate of rock strata to predict the development height of water-conducting fractures. The results show that the overlying strata failure structure is mainly a “hinged structure” and a “step structure,” which respectively promotes and inhibits the development of overlying strata fractures. Repeated mining causes mining-induced fractures in the lower coal seam to pass through the goaf of the upper coal seam and develop more vigorously in the upper coal seam, and the fractal dimension can effectively reflect the development of overlying strata fractures. The height of the water-conducting fracture zone increases in four stages: incubation, gradual increase, further gradual increase, and stability, eventually stopping development under the influence of the key layer (thick mudstone) bearing the load above. The development height of water-conducting fractures predicted by on-site water injection measurement is similar to that predicted by simulation experiments and theoretical calculations, verifying the feasibility of predicting the development height of water-conducting fractures through simulation tests and theoretical analysis. This study provides a reference for coal seam mining under similar conditions.

Using Fractal Geometry Theory to Quantify Pore Structure Evolution and Particle Morphology of Stabilized Kaolinite

Using Fractal Geometry Theory to Quantify Pore Structure Evolution and Particle Morphology of Stabilized Kaolinite

Selective Laser Melting: A Novel Method for Surface Roughness Analysis

The present study introduces a novel approach to analyse the surface roughness of metal parts made by 3D selective laser melting (SLM). This technology, known for its ability to efficiently produce functional prototypes and limited-run series, is particularly effective when surface conditions directly meet usage requir ements. Thus, the suitability of surfaces is a critical factor, emphasizing the importance of new methods for predicting their quality. Here fractal geometry and network theory are integrated to delve into the complexities of SLM-produced surfaces, while machine learning and pattern recognition concepts are employed to evaluate the surface roughness. Specifically, genetic programming, artificial neural networks, support vector machine, random forest, k-nearest neighbors are compared in terms of accuracy demonstrating that only the first method provided valid estimation due to the presence of very little training data. Experimental work with EOS Maraging Steel MS1 and an EOS M 290 3D printer validates the method’s practicality and effectiveness. Then, the research offers a fresh perspective in surface analysis and has significant implications for quality control in additive manufacturing, potentially enhancing the precision and efficiency of 3D metal printing.

Research on the capillary absorption capacity of concrete of different particle sizes after high temperature based on nuclear magnetic resonance technology

Concrete is subjected to thermal stress and thermal loading in a high temperature environment, resulting in the expansion and contraction of internal pores, which adversely affects the stability and durability of concrete structures. To explore the effects of temperature and aggregate particle size on the capillary absorption capacity of concrete, the capillary absorption tests were carried out on the specimens with aggregate particle sizes of 1.25–2.5 mm, 2.5–5.0 mm and 5.0–10 mm after high temperature treatment at 300 °C, 400 °C and 500 °C.In this paper, The capillary absorption coefficient was measured by mass change method, and the migration change law and spatial distribution of water during capillary absorption were monitored by NMR technology, the changes in water content in different pores during water absorption were calculated by nuclear magnetic resonance T2 spectroscopy, and a method for calculating the capillary absorption coefficient based on T2 spectra was proposed. The results showed that the capillary absorption coefficient of the samples without high-temperature treatment decreased with increasing of particle size. The capillary absorption coefficient increases at 300 °C and decreases slightly at 400 °C. At 500 °C, the capillary absorption coefficient increases dramatically. This shows that at 500 °C, the durability of concrete is drastically affected. The capillary absorption coefficient calculated based on the T2 spectrum is linearly related to the results of the traditional weighing method. In this paper, the fractal geometry theory and regarding methods are used to deduce the relationship between the average capillary rise height and t12DT, and the rise rate is related to the pore structure. And the correctness of the model was verified by MRI. This study can provide theoretical guidance for the safety assessment of concrete structures after high temperature exposure.

Highly Non-contractive Iterated Function Systems on Euclidean Space Can Have an Attractor

AbstractIterated function systems (IFSs) and their attractors have been central to the theory of fractal geometry almost from its inception. Moreover, contractivity of the functions in the IFS has been central to the theory of iterated functions systems. If the functions in the IFS are contractions, then the IFS is guaranteed to have a unique attractor. The converse question, does the existence of an attractor imply that the IFS is contractive, originates in a 1959 work by Bessaga which proves a converse to the contraction mapping theorem. Although a converse is true in that case, it is known that it does not always hold for an IFS. In general, there do exist IFSs with attractors and which are not contractive. However, in the context of IFSs in Euclidean space, this question has been open. In this paper we show that a highly non-contractive iterated function system in Euclidean space can have an attractor. In order to do that, we introduce the concept of an L-expansive map, i.e., a map that has Lipschitz constant strictly greater than one under any remetrization. This is necessitated by the absence of positively expansive maps on the interval.

Additive and geometric transversality of fractal sets in the integers

AbstractBy juxtaposing ideas from fractal geometry and dynamical systems, Furstenberg proposed a series of conjectures in the late 1960's that explore the relationship between digit expansions with respect to multiplicatively independent bases. In this work, we introduce and study — in the discrete context of the integers — analogs of some of the notions and results surrounding Furstenberg's work. In particular, we define a new class of fractal sets of integers that parallels the notion of ‐invariant sets on the 1‐torus and investigate the additive and geometric independence between two such fractal sets when they are structured with respect to multiplicatively independent bases. Our main results in this direction parallel the works of Furstenberg, Hochman–Shmerkin, Shmerkin, Wu, and Lindenstrauss–Meiri–Peres and include: a classification of all subsets of the positive integers that are simultaneously ‐ and ‐invariant; integer analogs of two of Furstenberg's transversality conjectures pertaining to the dimensions of the intersection and the sumset of ‐ and ‐invariant sets and when and are multiplicatively independent; and a description of the dimension of iterated sumsets for any ‐invariant set . We achieve these results by combining ideas from fractal geometry and ergodic theory to build a bridge between the continuous and discrete regimes. For the transversality results, we rely heavily on quantitative bounds on the ‐dimensions of projections of restricted digit Cantor measures obtained recently by Shmerkin. We end by outlining a number of open questions and directions regarding fractal subsets of the integers.

The origin, development, and application of fractal geometry theory in architecture

Fractal geometry plays a crucial role in the study of fractals, with wide-ranging applications and it is closely related to architectural design creation, notably in architecture. Fractals are becoming more prevalent in todays society and can be used to describe a variety of extremely irregular items in the actual world. This article aims to provide an overview and analysis of the significant implications of fractal geometry in architecture, exploring its relationship with elements in nature, such as trees. By searching, organizing data, and elucidating examples, the article delves into the origin and basic background of fractal geometry. The use of fractal geometry in architecture demonstrates the perfect symbiosis between fractal and nature through analysis of TOD, the ancient gardens in Suzhou, and other practical applications. One notable fractal algorithm is the L-system, a modeling approach expressing fractal concepts, enabling parameterization and architectural design. Fractal geometry successfully expands possibilities for the harmonious coexistence of architecture and nature, embodying numerous natural features and human-centric ideas.

Simulation and analysis of micro-textured rough curved surface based on fractal characterization method

PurposeMicro-texture is processed on the surface to reduce the friction of the contact surface, and its application is more and more extensive. The purpose of this paper is to create a texture function model to study the influence of surface parameters on the accuracy of the simulated surface so that it can more accurately reflect the characteristics of the real micro-textured surface.Design/methodology/approachThe microstructure function model of rough surfaces is established based on fractal geometry and polar coordinate theory. The offset angle θ is introduced into the fractal geometry function to make the surface asperity normal perpendicular to the tangent of the surface. The 2D and 3D contour surfaces of the surface groove texture are analyzed by MATLAB simulation. The effects of fractal parameters (D and G) and texture parameter h on the curvature of the surface micro-texture model were studied.FindingsThis paper more accurately characterizes the textured 3D curved surface, especially the surface curvature. The scale coefficient G significantly affects curvature, and the influence of fractal dimension D and texture parameters on curvature can be ignored.Originality/valueThe micro-texture model of the rough surface was successfully established, and the range of fractal parameters was determined. It provides a new method for the study of surface micro-texture tribology.Peer reviewThe peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-09-2023-0298/

Ligeti and Artistic Research

AbstractGyörgy Ligeti was very interested in many artistic and scientific fields and drew inspiration for his compositional work from them (his engagement with mathematics – particularly fractal geometry and chaos theory – is perhaps the best known). In this chapter I compare the concept of artistic research with Ligeti's practice and oeuvre. While the notion of artistic research was only appearing in embryonic form during the latter stages of Ligeti's career, many – though not all – of his statements seem to be suitable for describing his artistic processes. The benefit of this investigation is expected to be twofold: applying the concept of artistic research to Ligeti's approaches and practices should yield new insights on the relationship between his work and his interest in other humanities and sciences. Yet this look at Ligeti may also help to refine the concept of artistic research as discussed and applied to the artistic output of today.

A study on the cross-application of fractal geometry and garden design aesthetics

Abstract Currently there is a lack of research on garden design combined with fractal geometry, and most of the research results remain in qualitative analysis and subjective analysis. In this paper, we quantitatively analyze garden design based on fractal geometry to promote the development and innovation of aesthetics in garden design. Based on the reading of literature and other materials, the theory of fractal geometry and its application in the field of garden design are summarized. Based on Gabriel Lame’s design, we constructed the basic unit shape and layout representation using AutoCAD software to realize mathematical modeling design. A good fit of plant fractal dimension was indicated by the coefficient of determination of plant fractal dimension in the study subject garden being 0.9583. Based on the known logN(r) and log(1/r) of 0.1563 and 0.1013, respectively, the value of the fractal dimension of the garden site was derived as 1.54294 by applying the corresponding formula.The regression equation for the contribution of the fractal dimension of each element of the landscape of the mountains and rocks = 0.118-0.0407*Wigwam + 0.277*Wigwam and building combination+0.438*Wigwam and plant combination-0.019*Building+0.306*Building and plant combination-0.022*Plant, and the model R2 is 0.863. This study not only provides reference for future design and quantitative analysis of the antique gardens, but also provides new ideas for the construction of a better human environment and landscaping works.

A STUDY OF THE THERMAL EVOLUTION OF PERMEABILITY AND POROSITY OF POROUS ROCKS BASED ON FRACTAL GEOMETRY THEORY

The temperature effect on the permeability of porous rocks continues to be a considerable controversy in the area of reservoirs since the thermal expansion of mineral grains exhibits complicated influence on pore geometries in them. To investigate the degree of effect of pore structures on the hydro-thermal coupling process, a study of the thermal evolution of permeability and porosity of porous rocks is performed based on fractal theory and on thermal as well as stress effects. This work can provide a general physical explanation on some arguments in this area. The proposed models for permeability and porosity can be associated with temperature and the pore-structural parameters as well as physical parameters of porous rocks, such as the initial porosity ([Formula: see text], the initial fractal dimension ([Formula: see text], the fractal dimension for tortuosity ([Formula: see text] and the thermal expansion coefficient of pore volume ([Formula: see text]. The validity of the proposed models for temperature-dependent permeability and temperature-dependent porosity is validated by comparing them with the available experimental results. The investigations are performed in detail considering the essential effects of pore-structural parameters and physical parameters of porous rock on the dimensionless temperature-dependent permeability and temperature-dependent porosity as well as the fractal dimensions for pore areas and tortuosity. It is found that the pore distribution scale range ratio ([Formula: see text], and pore thermal expansion coefficient ([Formula: see text] have significant effects on the dimensionless temperature-dependent permeability and temperature-dependent porosity of porous rock as well as the fractal dimensions for pore areas and tortuosity. The proposed models may provide a fundamental understanding of the coupled hydro-thermal process of rocks.

A new model for coal gas seepage based on fracture-pore fractal structure characteristics

Coal reservoir is a typical double pore medium, and the magnitude of its permeability is controlled by the parameters of the fracture pore structure. However, the relationship between fracture-pore parameters and permeability is not yet clear. Based on the fractal geometry theory, the fracture-pore morphology of coal body is characterized by tree bifurcation network and tortuous capillary bundle. This study also established a coal gas seepage flow model based on the fracture-pore fractal structure characteristics. The model includes the influence of coal structure parameters, different levels of original coal seam stress, and gas pressure on the gas seepage characteristics. Each parameter in the model has definite physical meaning and does not contain any empirical constants. Finally, the applicability of the model is evaluated by gas radial seepage experiment. It is shown that the gas permeability in the model is a function of fractal dimensions and structural parameters, and based on the sensitivity analysis of each parameter in the model, it is found that the tree diameter ratio β and length ratio γ of the bifurcation network have a significant impact on the permeability. When β increases from 0.1 to 0.9, the model permeability increases by 12 orders of magnitude. However, changes in the characteristic parameters of the pore structure have little effect on gas permeability. For dual porous media such as coal, the permeability of the fracture network dominates the overall permeability of the model, and small changes in the geometric structure of the fracture bifurcation network can lead to significant changes in permeability. The construction of this theoretical model provides a reliable basis for further improving coal-rock porous medium structure seepage models, as well as to better characterize the effect of gas drainage.

Heat Extraction Evaluation of CO2 and Water Flow through Different Fracture Networks for Enhanced Geothermal Systems

Enhanced geothermal system (EGS) technologies have been developed to improve geothermal energy production from hot dry rock (HDR). In this study, discrete fracture network models for geometric topological networks that consider different parameters (the fracture density and the fracture length index) were built on the basis of fractal geometry theory. The heat extraction processes of CO2 and water as the working fluid through different discrete fracture networks were simulated with the application of the thermal–hydraulic–mechanical (THM) coupled method. A series of sensitivity analyses were carried out to reveal the influences of fracture parameters on heat transfer processes. Based on the simulation results, heat extraction efficiencies and temperature distributions in the reservoir of CO2 and water as the working fluid were compared, which showed that CO2 as the working fluid can bring a faster thermal breakthrough. It was found that the fracture length index a = 2.5 and the fracture density I = 5.0 can provide the highest heat extraction rate compared with other cases. This study provides a detailed analysis of fracture parameters and working fluids, which will contribute to the optimized management of geothermal energy production.

A GENERALIZED FRACTAL-BASED APPROACH FOR STRESS-DEPENDENT PERMEABILITY OF POROUS ROCKS

The pore geometry of porous rocks is fundamental for accurate description of stress dependence of effective permeability, which is an important parameter of mass transfer in porous rocks. An important physical assumption that porous rocks contain numerous elliptical or spherical pores has been shown to be successfully applied to many aspects of hydromechanical coupling properties of porous rocks. To investigate the detailed description of pore structures on the degree of effect on the coupled hydromechanical process, in this work, a generalized stress-dependent model for permeability of porous rocks has been proposed based on fractal geometry theory and mechanics of porous rock. The proposed model is expressed as a nonlinear function of pore structure parameters, such as aspect ratio ([Formula: see text]), the fractal dimensions ([Formula: see text] and [Formula: see text]) for tortuosity, the initial fractal dimension ([Formula: see text]), and initial porosity ([Formula: see text]) as well as matrix elastic constants ([Formula: see text] and [Formula: see text]) of porous rocks without any empirical parameter. The validity of the proposed models is verified by the good agreements between available experimental data and theoretical predictions of stress-dependent permeabilities of porous rocks. Detailed discussions of the essential effects of pore structures parameters and material elastic constants of porous rocks on the dimensionless stress-dependent permeabilities are performed. It is found that the stress parameters ([Formula: see text] and [Formula: see text]) have remarkable effects on the dimensionless stress-dependent permeabilities compared with other parameters ([Formula: see text] and [Formula: see text]). The proposed model may contribute to a better quantitative understanding of the coupled hydromechanical properties of porous rocks.

Fractal Characteristics of the Low-Gas Permeability Area of a Fully Mechanized Up-Dip Working Face under Different Dip Angles of Rock Strata

The low-gas permeability area of a fully mechanized up-dip working face was quantitatively studied using a physical similarity simulation test and theoretical analysis under varying dip angles of rock strata. Based on the theory of fractal geometry, this study obtained the fractal dimensions of the low-gas permeability area, the boundary area of the low-gas permeability region, and various layer areas of the low-gas permeability area by increasing the dip angle of rock strata. The findings reveal that the goaf’s high penetration area moved from a symmetrical shape to an asymmetrical one as the dip angle of rock strata increased. The high penetration area on the open-off cut side is notably larger than that on the working face side, due to the effects of advancement at the working face. In the goaf, the lateral length of the cavity decreases as the rock strata’s dip angle increases, while the longitudinal width expands and then contracts until it vanishes because of sliding. In the goaf, the lateral length of the cavity decreases as the rock strata’s dip angle increases, while the longitudinal width expands and then contracts until it vanishes because of sliding. In the goaf, the lateral length of the cavity decreases as the rock strata’s dip angle increases, while the longitudinal width expands and then contracts until it vanishes because of sliding. Moreover, the low-gas permeability area has a larger fractal dimension. The fractal dimension of the area with low gas permeability steadily decreased as periodic weighting emerged, ultimately reaching values of 1.24, 1.27, and 1.34. Moreover, the area’s fractal dimension was greater on the open-off cut side in comparison to the working face side. As the distance from the rock strata floor decreased, the fractal dimension of the area with low gas permeability increased. According to the gradient evolution law, the low-gas permeability area may be divided from bottom to top into three areas: strongly disturbed, moderately disturbed, and lowly disturbed. Based on the theory of mining fissure elliptic paraboloid zones and experimental findings, a mathematical model has been developed to analyze the fractal characteristics of low-gas permeability areas that are influenced by the rock strata’s dip angle. Finally, this study established a dependable theoretical foundation for precisely examining the development of cracks in the area of low gas permeability and identifying the storage and transportation region of pressure relief gas, which is affected by various dip angles of rock strata. It also offered assistance in constructing a precise gas extraction mechanism for pressure relief.

Fractal Study of the Development Law of Mining Cracks

Studying mining fracture development is vital for geotechnical and mining engineering and geological disaster prevention. This research assesses crack effects on rock mass stress equilibrium during coal mining, potentially causing geological disasters such as land subsidence and landslides. Using fractal geometry theory, the present study investigates the development of horizontal and vertical mining cracks, revealing their propagation patterns. The fractal dimension generally increases as the propulsion distance increases; however, fluctuations vary from 250 to 287.5 m, forming a wavering line chart. The proportion of mining fracture area relative to mining space area increases with greater propulsion distance, indicating expanded upward mining space due to separation layers. The horizontal distribution of mining cracks persists, while the vertical distribution decreases, suggesting ground subsidence results from upward transmission. The fastest increase in fractal dimension occurs at 87.5–100 m. At 250 m, it peaks at 1.4136, indicating complex crack structures. During propulsion, the fractal dimension decreases due to upward mining space expansion through overlying rock layer collapse, forming new cracks. The proportion of mining crack area to mining space area increases gradually throughout the mining process. The present study presents a simulation model for crack identification, noting limitations in identifying tiny cracks.

FRACTAL DIMENSIONS OF k-AUTOMATIC SETS

Abstract This paper seeks to build on the extensive connections that have arisen between automata theory, combinatorics on words, fractal geometry, and model theory. Results in this paper establish a characterization for the behavior of the fractal geometry of “k-automatic” sets, subsets of $[0,1]^d$ that are recognized by Büchi automata. The primary tools for building this characterization include the entropy of a regular language and the digraph structure of an automaton. Via an analysis of the strongly connected components of such a structure, we give an algorithmic description of the box-counting dimension, Hausdorff dimension, and Hausdorff measure of the corresponding subset of the unit box. Applications to definability in model-theoretic expansions of the real additive group are laid out as well.