Fractal Dimension Values Research Articles

Assessment of mandibular trabecular structure and cervical vertebral maturation using fractal analysis.

Changes in bone trabeculae occur during skeletal maturation. Fractal analysis is a technique for assessing changes in the internal structure of the bone. The aim of this study was to investigate correlation of bone trabeculation measured on panoramic and lateral cephalometric radiographs with cervical vertebral maturation stages (CVMS). CVMS was evaluated on lateral cephalometric radiographs of 120 individuals aged 6 to 18 years. The subjects were divided into 6 groups (n= 20 each) according to CVMS. On panoramic and lateral cephalometric radiographs, fractal dimension (FD) analyses were performed in mandibular corpus, gonial and condylar regions and cervical vertebral bodies (C2, C3, C4). FD measurements from the mandibular corpus region bilaterally and C2-C4 vertebral bodies showed significant positive correlations with CVMS (r= 0.418, r= 0.412, r= 0.324, r= 0.304, r= 0.263, respectively). Significant differences were found in the FD values of the right/left mandibular corpus and C2 and C3 vertebrae compared to CVMS (P< 0.001, P< 0.001, P= 0.005, P= 0.019, respectively). Significant positive moderate correlations between the FD values of the right and left mandibular corpus and CVMS suggest that fractal analysis may be useful in determining skeletal maturity stage. The significant FD values obtained from the right/left mandibular corpus, and C2 and C3 vertebra indicate that the complexity of the internal trabecular structure increases with maturation.

Anatomical assessment of the trabecular structure of the alveolar bone in periodontal disease by fractal analysis method.

Early diagnosis and treatment of periodontitis, which can cause loss of bone support of the teeth, is of great importance. The use of fractal analysis method is being investigated in order to differentiate periodontal disease radiographically. Fractal analysis presents the degree of complexity in the structure of fractal objects as a numerical data, and has been used to measure changes in trabecular bone. The aim of this study was to compare the trabecular bone fractal dimension (FD) values of patients with periodontitis and gingivitis using panoramic radiographs, and to evaluate the possible relationship between age and gender with fractal dimension. Panoramic radiographs of 64 patients with gingivitis and 64 patients with periodontitis were evaluated retrospectively in the study. Using the radiographs of the patients, FD values measured from the trabecular bone were calculated with the box-counting method in the Image J programme. The FD values of both groups were compared. In addition, the relationship between age and gender parameters and FD values was evaluated within the groups. According to the results of the study, the calculated average FD value of the patients in the gingivitis group was 1.195, while the calculated average FD value of the patients in the periodontitis group was 1.196. No statistically significant difference was observed between the FD values of the gingivitis group and the periodontitis group (p > 0.05). No statistically significant correlation was observed between FD values and age and gender (p > 0.05). No statistically significant results were obtained for the calculated mean FD values of the patients in the gingivitis and periodontitis groups.

Manipulating the McGinty Equation to Create Stable Micro-Wormholes

Fractal wormholes represent a novel theoretical concept at the intersection of classical physics and quantum mechanics. This article introduces the ΨFractal equation, a theoretical construct that seeks to describe these hypothetical entities. The equation integrates fundamental constants with parameters like mass, charge, and fractal dimension, suggesting intriguing properties and interactions with the cosmos. The McGinty equation, Ψ(x,t) = ΨQFT(x,t) + ΨFractal(x,t,D,m,q,s), can be used to help explain the fractal structure of space-time. The ΨFractal(x,t,D,m,q,s) term in the equation represents the fractal properties of space-time, where D represents the fractal dimension, m represents the mass of the system, q represents the charge, and s represents the spin. By manipulating the values of these variables, scientists can create a stable micro-wormhole in a controlled environment. The fractal dimension D, for example, can be used to control the size and stability of the wormhole. A higher fractal dimension value would result in a larger and more stable wormhole, while a lower value would result in a smaller and less stable wormhole. The mass of the system, represented by the variable m, can also play a role in the stability of the wormhole. A higher mass would result in a more stable wormhole, while a lower mass would result in a less stable wormhole. The charge and spin of the system, represented by the variables q and s, respectively, can also have an effect on the stability of the wormhole. A higher charge would result in a more stable wormhole, while a lower charge would result in a less stable wormhole. Similarly, a higher spin would result in a more stable wormhole, while a lower spin would result in a less stable wormhole. By manipulating the values of these variables, scientists can create a stable micro-wormhole in a controlled environment. This leads to advancements in fractal engine technology which in turn leads to the development of practical applications for faster-than-light travel, such as faster communication and interstellar exploration. For example, if we consider the equation for a fractal wormhole, ΨFractal(x,t,D,m,q,s) = [(G * m^2 * D) / (h * q * s)] * e^-(D * m * x^2) Where G is the gravitational constant, h is the Planck constant, and x is distance. By manipulating the value of D, m, q and s, scientists can control the size and stability of the wormhole, which in turn can be used for faster than light communication and interstellar travel.

Dynamic evolution of crack fractal of polypropylene fiber reinforced geopolymer during flexural process

Crack mouth opening displacement (CMOD) is the most widely used parameter to evaluate the crack width changing process in concrete fracture research. However, it is difficult to accurately and comprehensively describe the multi-crack cracking characteristics of high-toughness concrete with only crack width. This paper investigates the dynamic evolution of the fractal dimension (FD) of surface cracks on polypropylene fiber-reinforced geopolymer (PFRG) during flexural process. The box counting method was used to calculate the FD of cracks in the segmentation images that were generated by deep learning. Crack FD at the peak load and the fracture failure load could be used as indicators of deflection hardening behavior, with threshold values of 0.83 and 0.86, respectively. Crack development can be divided into four stages based on load, FD, CMOD, and energy capacity with time: elastic segment, internal crack development segment, surface crack extension segment, and crack destabilization segment. FD shows an obvious exponential increase in the surface crack extension segment and crack destabilization segment, and the changing trend is basically parallel with the load in the surface crack extension segment. The FD of peak load and fracture failure load showed a trend of increasing and then decreasing as the fiber factor was increased. The FD value peaks at 585 when W/B = 0.35 and at 252 when W/B = 0.38. To achieve the best stress diffusion and crack propagation process and achieve the toughening objective, it is necessary to reasonably match the fiber factor and matrix strength.

Global dynamics and computational modeling approach for analyzing and controlling of alcohol addiction using a novel fractional and fractal–fractional modeling approach

In recent years, alcohol addiction has become a major public health concern and a global threat due to its potential negative health and social impacts. Beyond the health consequences, the detrimental consumption of alcohol results in substantial social and economic burdens on both individuals and society as a whole. Therefore, a proper understanding and effective control of the spread of alcohol addictive behavior has become an appealing global issue to be solved. In this study, we develop a new mathematical model of alcohol addiction with treatment class. We analyze the dynamics of the alcohol addiction model for the first time using advanced operators known as fractal–fractional operators, which incorporate two distinct fractal and fractional orders with the well-known Caputo derivative based on power law kernels. The existence and uniqueness of the newly developed fractal–fractional alcohol addiction model are shown using the Picard–Lindelöf and fixed point theories. Initially, a comprehensive qualitative analysis of the alcohol addiction fractional model is presented. The possible equilibria of the model and the threshold parameter called the reproduction number are evaluated theoretically and numerically. The boundedness and biologically feasible region for the model are derived. To assess the stability of the proposed model, the Ulam–Hyers coupled with the Ulam–Hyers–Rassias stability criteria are employed. Moreover, utilizing effecting numerical schemes, the models are solved numerically and a detailed simulation and discussion are presented. The model global dynamics are shown graphically for various values of fractional and fractal dimensions. The present study aims to provide valuable insights for the understanding the dynamics and control of alcohol addiction within a community.

Determination of local fractal dimension of the catalyst support MgAl2O4 surface

In this paper, a procedure for estimating local fractal dimensionality values for greyscale images was developed. This procedure was tested on synthesised fractal and multifractal Brownian surfaces. The use of the method of detrended fluctuation analysis to determine the Hurst index for samples with a size not exceeding 2000 points was analysed. The optimal values of the sample size and the scale parameter have been obtained, allowing the Hurst index to be calculated with an error of less than 20%. The procedure was also tested on greyscale images, which are 8-bit representations of a synthesised fractal surface. The peculiarities of the application of the procedure for calculating the local fractal dimension for images obtained by scanning electron microscopy have been analysed. The values of the local fractal dimension of the oxide support MgAl2O4 are calculated using SEM images at different magnifications. It has been shown that, taking into account the fractal dimensionality calculation, different morphologies of the particle surface are observed at different scales of image magnification. At the highest magnification, the particle surface is found to consist of embedded smooth micron particles. The surface morphology of the MgAl2O4 particle depends on the micro- and mesopores between the embedded particles, this surface can be characterised by a fractal surface with FD2 = 2.3–2.4.

Measurement of 2D and 3D Fractal Features of Urban Morphology from an Architectural View and Its Influencing Factors

Urban morphology has been empirically demonstrated to be self-organized and can be quantified by fractal dimension. However, the spatial variation rule of fractal features at the sub-zone scale has yet to be uncovered, as well as the relationship between fractal dimension values and road network or land-use patterns. In this study, the urban area is partitioned into 158 grid units, with subsequent calculations conducted to determine the fractal dimensions (using 2D box-counting and 3D voxel-counting methods), road network characteristics, and land-use patterns within each individual unit. The pattern of how architectures fill into the 2D or 3D embedding space at the grid level is revealed. Moreover, the spatial relationship between the road network, land-use, and their impacts on the local architectural layout is elucidated by employing MGWR, a model that incorporates the principles of fitting localized spatial regression. The results are as follows: (1) urban morphology follows fractal laws at a sub-zone scale, both in a 2D plane and 3D volume; (2) the filling degree of architecture is high in the urban center but low in the periphery areas; (3) the selected variables fit well with the regression models; (4) there is spatial heterogeneity regarding the influence of each factor. The research findings provide valuable insights into the theoretical relationship between urban morphology and the composite structure of road networks and land use. This facilitates identifying crucial areas and priority directions for urban renewal construction, as well as optimizing architectural design to improve efficiency and functionality.

On the fractal dimension of carbon black particles in pyrolysis flow reactors

This study presents a numerical approach for simulating the internal morphology of carbon black particles formed in a pyrolysis reactor. The simulation process involves a two-step approach using population balance models (PBM) and detailed population balance models (DPBM), solved via sectional and stochastic methods, to simulate the arrangement of primary particles within aggregates to allow determination of their fractal dimension (FD). The outcome is a novel introduction of simulating real flow reactors that for the first time provides the fractal dimension of particles as an output, rather than an assumed input. The results of this study have practical implications for optimizing the synthesis processes in carbon black production. The effects of various production parameters, including aggregation efficiency, temperature, pressure, and acetylene concentrations, on the fractal dimension values of carbon black particles are examined. It is observed that higher temperatures lead to the formation of larger fractal shapes with lower fractal dimensions and larger primary particle diameters. Moreover, increased reactor pressure and higher aggregation efficiency enhance the formation of carbon black aggregates, but also have a time-based effect with higher compactness at longer residence times. The time-based effect reveals the importance of sintering, where high loads of small particles enhance the overall sintering of the aggregates. These findings provide insights into the interplay between temperature, pressure, and particle morphology, highlighting the dynamic nature of carbon black nanoparticles and their response to synthesis process conditions.

Comparison of periapical radiography, panoramic radiography, and CBCT in the evaluation of trabecular bone structure using fractal analysis.

The aim of this study is to compare imaging techniques to evaluate trabecular bone structure using Fractal Analysis (FA). Fifteen sheep hemimandibles were used for this study. Digital images were obtained using periapical radiography, panoramic radiography, and cone-beam computed tomography (CBCT). CBCT imaging was performed in standard (STD) and high-resolution (HR) modes. FA was conducted using ImageJ 1.3 software with the box-counting method on the images. The fractal dimension (FD) values were analyzed by the statistical software Jamovi 1.6.23. Statistical significance was accepted as p < 0.05. The highest mean FD value was the FD on digital periapical radiographs (PaFD) (1.28 ± 0.04), and the lowest mean FD value was the FD on standard resolution cone-beam computed tomography images (STD-CBCTFD) (1.12 ± 0.10). Although there was no statistically significant difference between the PaFD and the FD on digital panoramic radiographs (PanFD) (p = 0.485), the PaFD was found to be significantly higher than STD-CBCTFD (p < 0.001), and the FD on high-resolution cone-beam computed tomography images (HR-CBCTFD) (p = 0.007). The PanFD was found to be significantly higher than the STD-CBCTFD (p = 0.004). According to our results, in the evaluation of trabecular bone structure using FA, periapical radiographs and panoramic radiographs have similar image quality for assessment of the FD. On the other hand, CBCT results did not correlate with results from any of the other techniques in this study.

Bone Trabecular Pattern Analysis in Odontogenic Cysts Using Cone Beam Computed Tomography: A Clinical Retrospective Study.

Introduction The cysts of the maxillofacial region account for one of the most common pathologies of the head and neck region after the mucosal pathologies. Radiography provides an essential clue in early diagnosis and triaging, but it continues further as it is used to evaluate the post-treatment outcome. However, manual analysis is prone to errors. In this scenario, fractal analysis (FA) in radiographs uses mathematical methods to analyse the changes in grey scales in a given radiographic image. FA in odontogenic cysts is used to characterise their complexity, uncover hidden patterns, monitor treatment response, and potentially provide prognostic information. This paper aimed to assess the fractal characteristics of the radicular cyst (RC), dentigerous cyst (DC), and odontogenic keratocyst (OKC) using cone beam computed tomography (CBCT). The objective was to calculate fractal dimension (FD) values expressed in each of these cysts, which could prove to be a radiological adjunct in diagnosing the above cysts. Materials and methods As this is a retrospective study, the archives of CBCT images from June 2021 to December 2023 were obtained from patients diagnosed and confirmed with a histopathological diagnosis with RC, DC, and OKC. The FA was performed using Image J Software(Ver 1.51, National Institute ofHealth Bethesda, Fiji). The cortical and cancellous bones were segmented using thresholding techniques and converted to binary images. The mean FD of the three planes was then compared to establish the distinctive fractal characteristic for the specific odontogenic cysts. A one-way ANOVA was performed using the Statistical Product and Service Solutions (SPSS) (version 23.0; IBM SPSS Statistics for Windows, Armonk, NY) to determine the difference between FD values of RC, DC, and OKC with a significance level less than 0.05. Results The FD values of DC, RC, and OKC were 1.33 ± 0.17, 1.08 ± 0.16, and 1.65 ± 0.12, respectively. The results indicated that OKC had higher FD values than DC and RC, which means that OKC had lesser bone destruction compared to DC and RC. Inferential statistics showed that the one-way ANOVA was used to compare the means of the three groups of FD data. When calculated for the three groups, the F-statistic value was at 7.29, which yielded a P value of 0.03, making it statistically significant for a 95% confidence interval (p<0.05). Conclusion Our CBCT study on bone trabecular pattern analysis using FD and FAin odontogenic cysts reveals distinct alterations in bone parameters among different cyst types. The probability of higher FD values in OKC is because of lesser cortical bone destruction in OKC compared to the other cyst types. These findings have potential implications for diagnosing, treating, and prognosticating odontogenic cysts.

Fractal Changes of the Retinal Microvasculature in Syphilitic Uveitis

ABSTRACT Purpose To quantify chorioretinal microvascular damage and recovery post-treatment in patients with acute syphilitic posterior placoid chorioretinitis (ASPPC) using fractal dimension (FD). Methods Retrospective cohort study of patients with serologically confirmed syphilitic uveitis. We obtained optical coherence tomography angiography (OCTA) scans at baseline and follow-up after intravenous penicillin treatment and computed FD of the superficial capillary plexus (SCP), deep capillary plexus (DCP), and choriocapillaris (CC) using ImageJ. Results We enrolled seven patients with ASPPC (11 eyes), and 17 control subjects (34 eyes). Pre-treatment averages of FD-SCP, FD-DCP, and FD-CC were: 1.672 (±0.115), 1.638 (±0.097), and 1.72 (±0.137); post-treatment: 1.760 (±0.071), 1.764 (±0.043), and 1.898 (±0.047). After treatment FD-CC increased in all 11 eyes with an average of 0.163 (p = 0.003); FD-DCP increased in 10 (91%) eyes with an average of 0.126 (p = 0.003); and FD-SCP increased in seven (64%) eyes with an average of 0.089 (p = 0.059). Compared to the post-treatment FD values in the syphilitic group, controls had similar FD-SCP (p = 0.266), FD-DCP (p = 0.078), and FD-CC (p = 0.449). Conclusions CC and DCP are mostly affected in ASPPC with minimal changes in the SCP. All vascular layers FD recovered after completing antibiotic treatment.

Evaluation of Jaw Bone Change in Children and Adolescents with Rheumatic Heart Disease by Fractal Analysis.

Recognizing the oral needs of people living with rheumatic heart disease (RHD), particularly in relation to their jaws, is of paramount importance. This study aimed to evaluate potential jaw bone changes in children and adolescents diagnosed with RHD and compare them with a group of healthy children and adolescents. The dental panoramic radiographs (DPRs) of 32 children and adolescents diagnosed with RHD comprised the case group. In contrast, the control group consisted of DPRs from 32 children and adolescents without any known diseases. Two regions of interest (ROI1 and ROI2) of trabecular bone and a separate region of interest (ROI3) of cortical bone were selected for fractal analysis. The mandibular cortical index (MCI) was evaluated. The ROI1 fractal dimension (FD) value of the individuals in the case group was statistically higher in the right jaw than in the left jaw (P < 0.01). Significant differences were determined between the case and control groups regarding the left ROI1, and the left and the right ROI2 (P < 0.05). No significant difference was found between the patient and control groups regarding the MCI value (P > 0.05). The results of this study showed that bone metabolism was affected unilaterally in the condyle region and bilaterally in the gonial region of patients with RHD. It also revealed that the cortical layer was not affected.

Optical coherence tomography angiography macular biomarkers of peripheral retinal ischemia in diabetic macular edema: secondary endpoints from the clinical study "FOVEA".

To investigate the relationship between the macular values of fractal dimension (FD) and lacunarity (LAC) on optical coherence tomography angiography (OCTA) images and the presence of peripheral retina non-perfusion areas (NPAs) on fluorescein angiography (FA) in patients with treatment-naïve diabetic macular edema (DME). Fifty patients with treatment-naïve DME underwent a full ophthalmic examination, including best-corrected visual acuity measurement, FA, spectral-domain optical coherence tomography, and OCTA. Specifically, FA was performed to detect the presence of retinal NPAs, whereas fractal OCTA analysis was used to determine macular FD and LAC values at the level of thesuperficial and deep capillary plexus (SCP and DCP). FA montage frames of the posterior pole and peripheral retina, as well as macular OCTA slabs of the SCP and DCP, were obtained. Thirty (60%) eyes with FA evidence of peripheral retinal NPAs in at least one quadrant showed significantly lower FD and higher LAC in both SCP and DCP, when compared with eyes presenting a well-perfused peripheral retina. Furthermore, macular FD and LAC values were found to be significantly associated with the extent of retinal NPAs. Macular FD and LAC of both SCP and DCP seem to be strongly associated with the extent of peripheral retinal NPAs, thus suggesting that may be useful predictive biomarkers of peripheral ischemia in treatment-naïve DME eyes.

Kinematics, intensity, and spatial arrangement of extensional fractures in the Tuxtla-Malpaso fault system: Chiapas strike-slip fault province, southern Mexico

We describe the orientation, number, size, distribution, and fracture spatial arrangement in the central part of the kilometric Tuxtla-Malpaso fault system, one of the largest in the south of Mexico which cuts Cretaceous calcareous rocks. In this area, there are two sub-vertical extensional fracture sets (T1 and T2). The T1 fractures have ∼ N–S orientation and recorded 4.48 % of extension. The T2 fractures cut the T1 fractures, have NE-SW orientation, and accommodate an amount of extension of 3.87 %. These two fracture sets are parallel to two normal fault groups. The youngest normal faults (F2) have NE-SW strike and cut the N–S normal faults (F1). The F1 faults record a reactivation as strike-slip faults contemporaneous with F2.The T1 and T2 fractures have a fracture intensity between 9.08 and 9.39 (P10) on mesoscopic scale. In the case of the spacing, the value of this parameter (0.28–1) is related to fracture clustering. This behavior agrees with the values of fractal box dimension (0.72–0.81), correlation dimension (0.24–0.6), Lyapunov exponent (1.31–2.42), and coefficient of variation (0.94–1.85) on mesoscopic scale. The cumulative aperture vs. distance shows that T1 and T2 fractures have heterogeneous strain distribution. The fracture spatial arrangement is verified with the normalized correlation count analysis (NCC). The NCC graphs show a clustering behavior for T1 and T2 fractures, which are related to a regulary-spaced fractal clusters pattern. We suggest that this pattern is related to the mechanical stratigraphy and the strain location in the fault zones, while the fracture intensity depends on the degree of fracture clustering and rock type more than the amount of strain.Finally, the T1 fractures were associated with the right strike-slip kinematic for the NW-SE Tuxtla-Malpaso fault. This activity recorded a transtension behavior during the middle Miocene. The T2 fractures were formed during the reactivation (Pliocene) of the Tuxtla-Malpaso fault as the left strike-slip fault behind wrenching-transpression kinematics.

Prognostic value of right ventricular trabecular complexity in patients with arrhythmogenic cardiomyopathy.

The present study aimed to investigate the incremental prognostic value of the right ventricular fractal dimension (FD), a novel marker of myocardial trabecular complexity by cardiac magnetic resonance (CMR) in patients with arrhythmogenic cardiomyopathy (ACM). Consecutive patients with ACM undergoing CMR were followed up for major cardiac events, including sudden cardiac death, aborted cardiac arrest, and appropriate implantable cardioverter defibrillator intervention. Prognosis prediction was compared by Cox regression analysis. We established a multivariable model supplemented with RV FD and evaluated its discrimination by Harrell's C-statistic. We compared the category-free, continuous net reclassification improvement (cNRI) and integrated discrimination index (IDI) before and after the addition of FD. A total of 105 patients were prospectively included from three centers and followed up for a median of 60 (48, 66) months; experienced 36 major cardiac events were recorded. Trabecular FD displayed a strong unadjusted association with major cardiac events (p < 0.05). In the multivariable Cox regression analysis, RV maximal apical FD maintained an independent association with major cardiac events (hazard ratio, 1.31 (1.11-1.55), p < 0.002). The Hosmer-Lemeshow goodness of fit test displayed good fit (X2 = 0.68, p = 0.99). Diagnostic performance was significantly improved after the addition of RV maximal apical FD to the multivariable baseline model, and the continuous net reclassification improvement increased 21% (p = 0.001), and the integrated discrimination index improved 16% (p = 0.045). In patients with ACM, CMR-assessed myocardial trabecular complexity was independently correlated with adverse cardiovascular events and provided incremental prognostic value. The application of FD values for assessing RV myocardial trabeculae may become an accessible and promising parameter in monitoring and early diagnosis of risk factors for adverse cardiovascular events in patients with ACM. • Ventricular trabecular morphology, a novel quantitative marker by CMR, has been explored for the first time to determine the severity of ACM. • Patients with higher maximal apical fractal dimension of RV displayed significantly higher cumulative incidence of major cardiac events. • RV maximal apical FD was independently associated with major cardiac events and provided incremental prognostic value in patients with ACM.

Schramm-Loewner Evolution in 2D Rigidity Percolation.

Amorphous solids may resist external deformation such as shear or compression, while they do not present any long-range translational order or symmetry at the microscopic scale. Yet, it was recently discovered that, when they become rigid, such materials acquire a high degree of symmetry hidden in the disorder fluctuations: their microstructure becomes statistically conformally invariant. In this Letter, we exploit this finding to characterize the universality class of central-force rigidity percolation (RP), using Schramm-Loewner evolution (SLE) theory. We provide numerical evidence that the interfaces of the mechanically stable structures (rigid clusters), at the rigidification transition, are consistently described by SLE_{κ}, showing that this powerful framework can be applied to a mechanical percolation transition. Using well-known relations between different SLE observables and the universal diffusion constant κ, we obtain the estimation κ∼2.9 for central-force RP. This value is consistent, through relations coming from conformal field theory, with previously measured values for the clusters' fractal dimension D_{f} and correlation length exponent ν, providing new, nontrivial relations between critical exponents for RP. These findings open the way to a fine understanding of the microstructure in other important classes of rigidity and jamming transitions.

STUDY ON QUANTIFICATION OF ROCK FRACTURE NETWORK TO PROMOTE SHALE GAS DEVELOPMENT

Shale gas is produced by fracturing the reservoir to increase production. Quantifying the complex structure of the rock fracture network has a very important role in shale gas development. In order to quantify the complex structure of rock fracture network, in this paper, combined with the hierarchical structure of fractal figure, long tail distribution and fractal self-similarity, the current calculation method of fractal dimension is improved, and the basic fractal dimension calculation method is proposed based on the principle. By removing the limit calculation method which causes the deviation of numerical results, a reasonable and convenient calculation model for engineering application is proposed. A set of two-dimensional fracture pictures are used to test the model. In the 10 pictures, the more complex the figure is, the larger the new fractal dimension value is, which is in line with the characteristics of fractal dimension describing the complexity of graphics. Furthermore, lattice Boltzmann method (LBM) is used to simulate the velocity field, and the permeability of the fracture network is calculated according to Darcy’s law. The permeability and fractal dimension of the fracture network are compared and analyzed. The conclusion that the permeability is positively related to the fractal dimension of the fracture network is obtained, which provides theoretical support for the further study of shale mining problems such as hydraulic fracturing.

Procena primenljivosti fraktalne analize za opisivanje površine aluminijumskih kompozita nakon bušenja

The paper presents the results of the drilling of an aluminum matrix composite reinforced with ceramic fibers. The drilling process was carried out dry and with assistance from oil mist. The 3D surface roughness was measured using the contact method. A number of roughness parameters were analyzed: average, height, and statistical parameters. The box-counting fractal dimension of the hole surfaces was also determined. Correlation coefficients between the fractal dimension and roughness parameters were calculated. The fractal dimension was found to describe the irregularity of the surface. Fractal dimension values are dependent only on the drilling parameters if the process is carried out wet with the processing fluid. The fractal dimension does not correlate with surface roughness parameters such as Sa, Sq, and Sz. Some correlation was observed between it and the Sku parameter, but only after drilling with oil mist. The geometric structure of the surface after drilling does not show clear features of directionality and periodicity.

Determination of cutting modes when turning railway wheelsets used in heavy traffic

Increasing the durability of the cutting tool when restoring the profile of the rolling surface of railway wheels by controlling the vibrations of the machine-tool-part system. Vibrations arising during the turning of the rolling surface of wagon wheels under various cutting modes were studied. To do this, a matrix of experimental planning was built using the Box-Wilson method, according to which all factors change in turn (cutting depth, cutting speed, feed). After the experiment, a mathematical model is selected and the numerical values of the coefficients of this equation are estimated. In accordance with the power dependences accepted in the theory of cutting, a mathematical model was constructed for the experiments in the form of a logarithmic polynomial of the first degree series. Mathematical processing of measurement results using fractal dimension was carried out. The fractal dimension was determined by the normalized span method. A correspondence has been established between the durability of the he cutter and the intensity of its vibrations when turning railway wheels. A mathematical dependence of the fractal dimension on the cutting modes is obtained, which makes it possible to adjust the cutting speed or feed during the restoration of the rolling surface of the wheel in the presence of defects. Optimal cutting modes, which allow to achieve a given quality of the rolling surface of the wheel in the shortest possible time, give the highest value of fractal dimension. Defects in railway wheels such as sliders, bumps, and gouges lead to a decrease in fractal dimension. The determination of the fractal dimension in the process of turning wheelsets makes it possible to automatically adjust the cutting modes to ensure a given quality of the rolling surface of the wheel and the required durability of the cutting tool.

Prognostic value of left ventricular trabeculae fractal analysis in patients with dilated cardiomyopathy

The prognostic value of left ventricular (LV) myocardial trabecular complexity on cardiovascular magnetic resonance (CMR) in dilated cardiomyopathy (DCM) remains unknown. This study aimed to evaluate the prognostic value of LV myocardial trabecular complexity using fractal analysis in patients with DCM. Consecutive patients with DCM who underwent CMR between March 2017 and November 2021 at two hospitals were prospectively enrolled. The primary endpoints were defined as the combination of all-cause death and heart failure hospitalization. The events of cardiac death alone were defined as the secondary endpoints.LV trabeculae complexity was quantified by measuring the fractal dimension (FD) of the endocardial border based on fractal geometry on CMR. Cox proportional hazards regression and Kaplan-Meier survival analysis were used to examine the association between variables and outcomes. The incremental prognostic value of FD was assessed in nested models. A total of 403 patients with DCM (49.31±14.68years, 69% male) were recruited. After a median follow-up of 43months (interquartile range, 28-55months), 87 and 24 patients reached the primary and secondary endpoints, respectively. Age, heart rate, New York Heart Association functional class >II, N-terminal pro-B-type natriuretic peptide, LV ejection fraction, LV end-diastolic volume index, LV end-systolic volume index, LV mass index, presence of late gadolinium enhancement, global FD, LV mean apical FD, and LV maximal apical FD were univariably associated with the outcomes (all P<0.05). After multivariate adjustment, LV maximal apical FD remained a significant independent predictor of outcome [hazard ratio=1.179 (1.116, 1.246), P<0.001]. The addition of LV maximal apical FD in the nested models added incremental prognostic value to other common clinical and imaging risk factors (all <0.001; C-statistic: 0.84-0.88, P<0.001). LV maximal apical FD was an independent predictor of the adverse clinical outcomes in patients with DCM and provided incremental prognostic value over conventional clinical and imaging risk factors.