Fractal Dimension Distribution Research Articles

Structure-Elasticity Relationships in Hybrid-Carrageenan Hydrogels Studied by Image Dynamic Light Scattering, Ultra-Small-Angle Light Scattering and Dynamic Rheometry.

Hybrid-carrageenan hydrogels are characterized using novel techniques based on high-resolution speckle imaging, namely image dynamic light scattering (IDLS) and ultra-small-angle light scattering (USALS). These techniques, used to probe the microscopic structure of the system in sol-gel phase separation and at different concentrations in the gel phase, give access to a better understanding of the network's topology on the basis of fractals in the dense phase. Observations of the architecture and the spatial and the size distributions of gel phase and fractal dimension were performed by USALS. The pair-distance distribution function, P(r), extracted from USALS patterns, is a new methodology of calculus for determining the network's internal size with precision. All structural features are systematically compared with a linear and non-linear rheological characterization of the gels and structure-elasticity relationships are identified in the framework of fractal colloid gels in the diffusion limit.

Unveiling the dynamics of "scenes changing as steps move" in a Chinese classical garden: a case study of Jingxinzhai Garden

Chinese classical gardens are a significant cultural heritage in the history of world gardening, with their central architectural principle of "scenes changing as steps move" (步移景异) embodying traditional Chinese aesthetic ideals. This study employed the variation of visual complexity during a tour as a representation of this principle, using Jingxinzhai (静心斋) Garden as a case study. The visual environmental characteristics were quantified using the Fraclab box-counting, Canny edge detection, and DeepLab V3 + model, and the spatial distribution of fractal dimension and visual index of landscape elements were analyzed. Through partial correlation analysis, hierarchical regression analysis, and one-way ANOVA, the relevant factors (BVI, RVI, GVI, WVI) and influencing factors (BVI, RVI, GVI) and the differences among landscape element combinations of visual complexity were identified. Furthermore, the distribution patterns and causes of visual complexity in Chinese classical gardens were then discussed. This study proposes an effective method for quantifying the visual environmental characteristics of Chinese classical gardens and provides an explanation of the concept of "scenes changing as steps move" from the perspective of visual environment. It offers important references for a deeper understanding of Chinese classical garden design and planning.

Fractal evolution model of proton spatial distribution in low water/binder cement-based composites (LW/B-CC) during early-age hydration process

This study establishes the fractal evolution model of proton spatial distribution in low water/binder cement-based composites (LW/B-CC) based on fractal theory and low field nuclear magnetic resonance (LF NMR) technology. The results show that the fractal evolution process of gel water, capillary water and surface water distributions in LW/B-CC presents three stages: the first 4–6 h after the beginning of hydration, 4–6 h to 12 h, and 12 h to long-term stage. In the first and second stages, the fractal dimensions of gel and capillary water increase linearly with hydration time, while the fractal dimension of surface water decreases linearly. The boundary is a sudden change of fractal dimensions after about 4–6 h of hydration. The third stage starts after about 12 h of hydration, at which time the fractal dimensions begin to converge to fixed values. Besides, this study reveals the mathematical relationship between fractal dimension (Df) and particle size distribution modulus (q), which is conducive to achieve the full-cycle regulation of LW/B-CC.

Investigation on the rock-breaking performance of specially-shaped PDC cutter under confining pressure conditions

Cutter shape has a significant influence on the rock-breaking performance of drill bit. Compared with conventional Polycrystalline Diamond Compact (PDC) cutter, the emergence of specially-shaped PDC cutters (SSPCs) with different geometric shapes is of great benefit to solve the problems of low rate of penetration (ROP) and rock-breaking efficiency in deep hard and hot dry rock formations. However, the high confining pressure and high in-situ stress (HCHS) have a significant influence on the rock-breaking performance of PDC cutter which cannot be ignored. Given that the above facts, this paper selects five kinds of PDC cutters with different cutter shapes, and implements a series of cutting experiments of granite under confining pressure conditions (CPCs) and atmospheric pressure condition (APC) using a true triaxial rock cutting device. The influence of cutter profile on the rock-breaking performance of PDC cutter was studied from the perspective of cutting force, mechanical specific energy (MSE), cuttings, coarseness index (CI) and fractal dimension. The results show that the rock breaking under CPC is more difficult than that under APC, resulting in the increased cutting force and MSE. Moreover, the cuttings obtained under CPC are smaller in size and the cutting grooves are smoother. Cutter shape has a great influence on the rock-breaking efficiency of PDC cutter, the rock-breaking efficiency of axe-shaped cutter is higher, while that of stinger cutter is lower. Moreover, PDC cutters with different shapes have different sensitivities to confining pressure, and stinger cutter has the lowest sensitivity to confining pressure, in other words, confining pressure has little influence on the rock breaking process of stinger cutter. What is more, the distribution of cuttings is characterized by the fractal dimension and particle size distribution parameter from the perspective of cuttings morphology and size respectively, and they present completely opposite changing trends. The obtained results are of great significance to the optimization of cutter profile, the design of bit cutter arrangement and the drilling of deep formation.

Association of Genetic Risk for Age-Related Macular Degeneration with Morphological Features of the Retinal Microvascular Network.

Age-related macular degeneration (AMD) is a multifactorial disease encompassing a complex interaction between aging, environmental risk factors, and genetic susceptibility. The study aimed to determine whether there is a relationship between the polygenic risk score (PRS) in patients with AMD and the characteristics of the retinal vascular network visualized by optical coherence tomography angiography (OCTA). 235 patients with AMD and 97 healthy controls were included. We used data from a previous AMD PRS study with the same group. The vascular features from different retina layers were compared between the control group and the patients with AMD. The association between features and PRS was then analyzed using univariate and multivariate approaches. Significant differences between the control group and AMD patients were found in the vessel diameter distribution (variance: p = 0.0193, skewness: p = 0.0457) and fractal dimension distribution (mean: p = 0.0024, variance: p = 0.0123). Both univariate and multivariate analyses showed no direct and significant association between the characteristics of the vascular network and AMD PRS. The vascular features of the retina do not constitute a biomarker of the risk of AMD. We have not identified a genotype-phenotype relationship, and the expression of AMD-related genes is perhaps not associated with the characteristics of the retinal vascular network.

The structure-giving role of Rb+ ions for water-ice nanoislands supported on Cu(111).

We characterize the effect of rubidium ions on water-ice nanoislands in terms of area, fractal dimension, and apparent height by low-temperature scanning tunneling microscopy. Water nanoislands on the pristine Cu(111) surface are compared to those at similar coverage on a Rb+ pre-covered Cu(111) surface to reveal the structure-giving effect of Rb+. The presence of Rb+ induces changes in the island shape, and hence, the water network, without affecting the nanoisland volume. The broad area distribution shifts to larger values while the height decreases from three bilayers to one or two bilayers. The nanoislands on the Rb+ pre-covered surface are also more compact, reflected in a shift in the fractal dimension distribution. We relate the changes to a weakening of the hydrogen-bond network by Rb+.

Identifying and differentiating dementias with EEG fractal dimension distributions

AbstractBackgroundAlzheimer’s disease dementia (AD‐dementia) and non‐AD dementia have both been associated with decreases in fractal dimension and spectral slowing in electroencephalography (EEG). Here, we describe a set of EEG features ‐ Fractal Dimension Distributions (FDD) ‐ which quantifies the stability of fractal dimensions over time within specific frequency bands. We compare the performance of models trained using this FDD feature to models using age, spectral power, or overall fractal dimension.MethodWe analyzed resting‐state data from older adults (age M = 71, range = 55‐86) with subjective cognitive impairment (SCI; N = 97, 59 female) or dementia (N = 51, 32 female), including 38 with AD (N = 33). We calculated the Fractal Dimension Distributions (FDD) by first computing the Higuchi fractal dimension within moving windows, then summarizing the distribution of these fractal values using the mean and standard deviation within standard frequency bands. Finally, we trained regularized logistic regression models to classify participants as having SCI or dementia, and as having SCI or AD‐dementia. We compared the performance of models using FDD or standard measures of fractal dimension. Models were evaluated using balanced accuracy, weighted sensitivity, and weighted specificity calculated using 5‐fold cross‐validation.ResultFor SCI vs Dementia, the model using FDD (accuracy = 74%, sensitivity = 67%, specificity = 81%, p < .001) had higher accuracy, sensitivity, and specificity than models trained on age, standard measures of fractal dimension, or spectral power. Similarly, the model trained to predict SCI vs AD‐Dementia was most accurate when using FDD (accuracy = 73%, sensitivity = 59%, specificity = 97%, p < .005). The features useful for discriminating SCI from Dementia partially overlapped with the features useful for discriminating SCI from AD‐Dementia.ConclusionOur novel feature, FDD, outperforms standard measures of fractal dimensions or spectral band power at discriminating between SCI and dementia. Moreover, FDD may help to distinguish between AD and other varieties of dementia.

Spatial Heterogeneity of Urban Road Network Fractal Characteristics and Influencing Factors

Fractal geometry has provided a new perspective for urban road network morphology research. This study systematically verifies and analyzes the spatial heterogeneity of fractal characteristics and influencing factors of urban road networks using spatial analysis. Here, Tokyo Metropolis was selected as a case, and the fractal dimensions of road networks were calculated. To determine the spatial heterogeneity in the relationship between fractal dimensions and influencing factors, we examined the spatial distribution characteristics of fractal dimensions using spatial autocorrelation analysis, selected population, build-up area density, and road network density as the explanatory variables, and established the global regression model and local regression model using ordinary least squares (OLS) and geographically weighted regression (GWR), respectively. The results indicated that the spatial distribution of fractal dimensions of the urban road network exhibited an obvious tendency toward geographical dependency. Considering the spatial heterogeneity in the relationship between the fractal characteristics of the road network and the influencing factors not only improves the reliability of analysis but also helps planners and decision-makers grasp the morphological characteristics of the urban road network and estimate the evolution of the road network, thereby promoting the development of urban road networks in a more orderly, efficient, and sustainable direction.

A polydisperse model for thixotropic elasto‐viscoplastic suspensions of aggregating particles using population balances

AbstractAn improved population balance‐based rheological constitutive framework for polydisperse aggregating suspensions is derived by incorporating detailed models for orthokinetic and perikinetic aggregation and shear breakage processes. The framework accounts for critical properties such as dynamic arrest, viscoelasticity, kinematic hardening, thixotropy, and yield stress to generate a full range of thixotropic elasto‐viscoplastic (TEVP) response. Additionally, the model is thermodynamically consistent because the dynamics and timescales are completely determined by internal structural and kinetic variables. The model connects the rheological response to the structural descriptors such as the size distribution of agglomerates, mean sizes, fractal dimension, and agglomerate volume fraction. Predictions are compared against a wide range of shear rheology measurements data for model thixotropic suspensions of fumed silica and carbon black, including large amplitude oscillatory shear (LAOS), as well as ultra‐small angle neutron scattering under steady shear (Rheo‐uSANS).

Fabrication process independent and robust aggregation of detonation nanodiamonds in aqueous media

In the past detonation nanodiamonds (DNDs), sized 3–5 nm, have been praised for their colloidal stability in aqueous media, thereby attracting vast interest in a wide range of applications including nanomedicine. More recent studies have challenged the consensus that DNDs are monodispersed after their fabrication process, with their aggregate formation dynamics poorly understood. Here we reveal that DNDs in aqueous solution, regardless of their post-synthesis de-agglomeration and purification methods, exhibit hierarchical aggregation structures consisting of chain-like and cluster aggregate morphologies. With a novel characterization approach combining machine learning with direct cryo-transmission electron microscopy and with X-ray scattering and vibrational spectroscopy, we show that their aggregate morphologies of chain and cluster ratios and the corresponding size and fractal dimension distributions vary with the post-synthesis treatment methods. In particular DNDs with positive ζ-potential form to a hierarchical structure that assembles aggregates into large networks. DNDs purified with the gas phase annealing and oxidation tend to have more chain-like aggregates. Our findings provide important contribution in understanding the DND interparticle interactions to control the size, polydispersity and aggregation of DNDs for their desired applications.

Methodological features of the analysis of the fractal dimension of the heart rate

Objectives. The aim of the present work is to determine the fractal dimension parameter calculated for a sequence of R–R intervals in order to identify the boundaries of its change for healthy and sick patients, as well as the possibility of its use as an additional factor in the detection of cardiac pathology.Methods. In order to determine the fractal dimension parameter, the Hurst-, Barrow-, minimum coverage area-, and Higuchi methods are used. For assessing the stationarity of a number of electrocardiography (ECG) intervals, a standard method is used to compare arithmetic averages and variances from samples of the total data array of ECG intervals. To identify differences in fractal dimensions of healthy and sick patients, this parameter was ranked. Using the Kolmogorov–Smirnov two-sample criterion, the difference between the distribution laws in the samples for healthy and sick patients is shown.Results. Among the considered methods for calculating the fractal dimension, the Higuchi method demonstrates the smallest data spread between healthy patients. By ranking the calculated fractional dimension values, it was possible to identify the difference between this parameter for healthy and sick patients. The difference in the distribution of fractal dimension of healthy and sick patients is shown to be statistically significant for the coverage and Higuchi methods. At the same time, when using the traditional Hurst method, there is no reason to reject the null hypothesis that two groups of patients belong to the same general population.Conclusions. Based on the obtained data, the difference between the fractal dimension indicators of the duration of R–R intervals of healthy and sick patients is shown to be statistically significant when using the Higuchi method. The fractal dimensions of healthy and sick patients can be effectively distinguished by ranking samples. The results of the research substantiate prospects for further studies aimed at using fractal characteristics of the heart rhythm to identify abnormalities of the latter, which can serve as an additional factor in determining heart pathologies.

Fractal Informed Generative Adversarial Networks (FI-GAN): Application to the generation of X-ray CT images of a self-similar partially saturated sand

The recent increase in data availability fostered the development of new and more powerful data-driven numerical models, particularly machine learning. With a limited quantity of data in geomechanics applications, machine learning often suffers from unstable training and lack of convergence. Physics informed machine learning models have been proposed in the literature to mitigate the issue as an alternative to data-only approaches. In the present work, a new machine learning method is proposed: Fractal Informed Generative Adversarial Networks (FI-GAN) and applied to a case of X-ray CT images of partially saturated sand. This method consists of training an informed adversarial model using pore fractal dimension. The semi-supervision provided by pore fractal metrics presents two advantages, firstly, it regularises the training; secondly, it improves the distribution of pore fractal dimension, which is correlated with permeability. Results show that the generated images improve the fractal distribution of the informer phase in the partially saturated material (i.e., water) and the other two phases. The approach is validated with physics flow simulations. The generated images can be applied in the calibration of image processing tools, filling missing data in X-ray CT scans, generating microscales in multiscale applications, and data augmentation, among others.

Влияние фрактальной размерности на качество классификации компьютерных атак методами машинного обучения

For building an effective network protection system in computer network against attacks, a promising direction is joint use of fractal analysis and data mining. It is proposed to increase the efficiency of network attacks classification by introducing additional fractal dimension (FD) statistics of attacks along with other attributes. In contrast to the well-known works, it is proposed to further improve the efficiency of classifying network attacks by using not only the average value, but also other statistical characteristics of the DF of attacks and normal traffic as information features. These can be variance, skewness and kurtosis coefficients that characterize the shape and parameters of the distribution of the RF. The effectiveness of the proposed method is evaluated using machine learning algorithms by assessing the quality of the binary classification of network attacks and normal traffic using the UNSW-NB15 database as an example. The following classification algorithms were used to classify the dataset: k-nearest neighbors (k-NN), multiple logistic regression (LR), decision tree (DTC), random forest (RF), ada boost. The following metrics were used to evaluate the effectiveness of the constructed models: accuracy (precision), recall (recall), F-score (F-score), ROC-curves, AUC-ROC. It is shown that the use of mean value, variance, skewness and kurtosis coefficients, which characterize the shape and distribution parameters of the statistical characteristics of the FD distribution as additional information features, makes it possible to increase the efficiency of attack classification by an average of 10%. K-NN and LR classification algorithms. For the DTC and RF algorithms, the greatest effect from the use of additional attributes is in reducing the training and testing time and is about 3.5 times for each of the algorithms.

A Fractal Model of Effective Thermal Conductivity of Porous Materials Considering Tortuosity

Accurate estimation of the thermal conductivity of porous materials is crucial for the modeling of heat transfer and energy consumption calculation in energy, aerospace, biomedicine and chemical engineering, etc. The series-parallel model is a simple and direct method and is usually used in the prediction of the effective thermal conductivity (ETC) of porous materials. In this work, the weighted coefficients of the series and parallel section were obtained based on the tortuosity of the porous materials. Then, the physical model of the ETC of the porous materials was established. Furthermore, the ETC of the porous materials was developed using the fractal model to calculate the pore cross-sectional area of the porous materials. Finally, quantitative analysis of the characteristic parameters, e.g., porosity, tortuosity, tortuous fractal dimension and pore diameter distribution, of the ETC of the porous materials was conducted. The results show that the proposed model can provide an accurate prediction of the ETC of porous materials.

FRACTAL EVIDENCE OF REACTIVE TRANSPORT IN GEOLOGIC SYSTEMS BASED ON MULTIPHASE FLOW HYDRAULIC MODELS PART II: THE BIOGEOCHEMICAL ASPECT

In the petroleum and soil science literature, fractal models of porous systems abound. Consequently, any process, be it physicochemical or geomechanical, that imparts pore structure will equally impact the fractal nature of the system. Several studies have reported the effect of physicochemical processes, such as the water–rock interactions, on the fractal dimension of the porous system using different approaches involving sophisticated instrumentation. Others have also reported the effect of geomechanical processes such as compaction on fractal dimension using similar approaches. To the best of our knowledge, none or little has been reported regarding the use of multiphase flow hydraulic models for revealing the effect of water–rock interactions on the fractal dimension and pore size distribution of the porous system. Consequently, to fill the knowledge gap in this regard, data on two-phase flow of acid gas and brine in porous media have been used to show that the fractal dimension and pore size distribution of the porous system change in response to water–rock interactions. In view of the injection of acid gas, such as anthropogenic carbon dioxide, into saline aquifers being a feasible means for carbon immobilization by mineral carbonation and biogeochemical means, the effect of change in the fractal properties of the porous system on the microbial transport and distribution in porous system has also been reviewed in relationship to the principal findings of this paper. Most importantly, the results of this study show that data on two-phase flow in porous media involving acid gas and brine can be used to prove changes in the fractal nature of the porous system, and this change can also impact subsurface distribution of methanogens that have the potential to trap carbon by methanogenesis.

A new calculation approach of heterogeneous fractal dimensions in complex hydraulic fractures and its application

Fractal dimension is often used to characterize the distribution of complexity fractures. However, the existing fractal dimension calculation method can only obtain one fractal dimension for a specific fracture network, which cannot accurately describe the complexity hydraulic fractures with strong heterogeneity. This work considers the actual distribution of complex hydraulic fractures in stimulated reservoir area, calculate fractal dimensions of complex hydraulic fractures at different regions. A new quantitative model is established to describes the distribution of heterogeneous fractal dimensions (QM-FD), and obtains the relationship between fractal dimensions and fracture locations. Combined with the heterogeneous fractal dimensions for complex hydraulic fractures, the porosity/permeability/compressibility of complex fractures is determined, and a more general distribution of flow capacity parameters of complex fractures is obtained. A conceptual reservoir model with complex hydraulic fractures is given in the work, and the fractal dimension distribution is described with the proposed approach, then the permeability is calculated with the fractal dimensions. The permeability calculated by heterogeneous fractal dimensions and the actual permeability is compared, and results show that the fitting rate between the calculated permeability and the real value is higher than 93%. The proposed approach is used in four typical reservoirs with complex hydraulic fractures, the fractal dimensions and equivalent permeability is calculated and compared with that from conventional fractal model. The conventional fractal model is poor in dealing with the description of complex hydraulic fractures with non-monotonic changes, and the fitting rate is below then 60%. While the fitting rate of the equivalent permeability calculated by the heterogeneous fractal dimensions in this work and the actual permeability is higher than 92%. The calculation approach of heterogeneous fractal dimensions proposed in the work can be applied to parameter characterization of complex fractures, which provides effective help for flow simulation and productivity prediction of unconventional reservoirs.

Revisiting b-value for extended Kopili region of Northeast India and probabilistic estimation of seismic hazard attributes thereof

Kopili fault and its' surroundings of northeast India have remained witness to some devastating earthquakes. The complex tectonic feature of this region incurs a high seismicity rate relative to other regions. Very recently, an earthquake of magnitude 6.4 (MW) has jolted this region. Considering the persistently rising seismicity pattern of this region, the present study intends to integrate the frequency magnitude distribution of earthquakes with other pertinent seismic hazard parameters. This work presents the estimation of the return period and probability of non-exceedance of medium to large earthquakes for the Kopili fault and its' surroundings via the extreme value method. Two statistical methods, namely the Gutenberg-Richter(G-R) power law and Gumbel extreme value method (GEVM), are applied for the correct estimation of the b-value for the Kopili fault and its' surrounding region. The uniform and homogenous magnitude scale dataset consisting of 550 events observed within a limit of 350 km from the Kopili fault (1964 to 2022) are used in the study. The b-value computed using GR relation was found to be unsatisfactory (0.73) due to an inadequate dataset—thus Gumbel's extreme value method is used for the estimation of the b-value (0.97). Gumbel's extreme value method is used for the calculation of the probability of occurrence of earthquakes, return period, and the most probable earthquake magnitude for the study region. The most probable annual earthquake is found to be ∼ MW 4.99. The return period and probability of occurrence of design-based earthquakes are also calculated. The correlation between b-value, seismic moment, and fractal dimension for seismic hazard analysis is also investigated. The antithetical relationship between seismic moment and b-value is observed for the region while the deep-root character of the Kopili fault and the strong, unfractured character of the Shillong plateau is highlighted with the spatial distribution of fractal dimension in the study region.

Experimental study on rockburst fragment characteristic of granite under different loading rates in true triaxial condition

In order to study the effect of loading rate on rockburst, an indoor test of single-face fast unloading-three directions and five-face stress-vertical continuous loading is conducted under different loading rates using a new true-triaxial rockburst test system. The entire process of rockburst inoculation-occurrence-development is completely and accurately reproduced. The fragmentation degree, fractal dimension, and Weibull distribution characteristics of rockburst fragments under different loading rates are compared and analyzed. The results indicate that the rockburst ejection failure process can be summarized as four stages: grain ejection, rock spalling into plates, rock shearing into fragments, and rock fragment ejection. Rockburst fragments are mostly coarse-grained, medium-grained, and fine-grained. The macroscopic block characteristics of rockburst fragments reflect the fragmentation degree of rock, and the fragmentation degree of rockburst fragments increases with the loading rate. The fractal dimension shows the same trend with different loading rates, that is, the fractal dimension grows as the loading rate increases. As the loading rate increases, the crack expansion rate accelerates, which aggravates the damage to the rock specimen by decreasing its compressive strength and causing greater fragmentation when the specimen is failed. The loading rate has a significant effect on the energy consumption of rockburst fragments.

Multifractal Characteristics of China’s Stock Market and Slump’s Fractal Prediction

It is necessary to quantitatively describe or illustrate the characteristics of abnormal stock price fluctuations in order to prevent and control financial risks. This paper studies the fractal structure of China’s stock market by calculating the fractal dimension and scaling behavior on the timeline of its eight big slumps, the results show that the slumps have multifractal characteristics, which are correlated with the policy intervention, institutional arrangements, and investors’ rationality. The empirical findings are a perfect match with the anomalous features of the stock prices. The fractal dimensions of the eight stock collapses are between 0.84 and 0.98. The fractal dimension distribution of the slumps is sensitive to market conditions and the active degree of speculative trading. The more mature market conditions and the more risk-averse investors correspond to the higher fractal dimension and the fall which is less deep. Therefore, the fractal characteristics could reflect the evolution characteristics of the stock market and investment philosophy. The parameter set calculated in this paper could be used as an effective tool to foresee the slumps on the horizon.

An Online Detection Method for Coal Dry Screening Based on Image Processing and Fractal Analysis

In coal dry screening, online detection for screening efficiency is a significant challenge. Notwithstanding, the method of image processing is strenuous to implement in this field due to the complex surface texture of shattered coal. This method identifies the fractal phenomenon before and after coal screening is discovered for the indirect detection of screening efficiency. For better fractal dimension distribution, an image denoising and filter method for wiping off the coal image surface texture is applied. Additionally, an enhanced Kirsch edge-detection algorithm is employed to obtain coal particle edges. Furthermore, the relation between fractal dimension and screening efficiency is presented by using the box-counting method. In this research, we skilfully transform the tough problem of image detection for particle size distribution into the calculation of the fractal dimension of the coal-edge image, and closely associate the fractal dimension with screening efficiency. With this method, it will be easier to predict the screening efficiency in real-time.