Fractal Region Research Articles

Nano-micro pore structure characteristics of carbon black and recycled carbon fiber reinforced alkali-activated materials

Microscopic scrutiny aids in alkali-activated materials’ (AAM) application in construction industry. This study delves into the pore structure and properties of one-part alkali-activated slag (AAS) mortar modified by carbon black (CB) and recycled carbon fiber (rCF). The additives enhanced flexural strength by 51.82% (12.16 MPa) with lower water absorption (10.24%). Refinement of pore size and reduction of connectivity are key factors in improving properties. The densification effect of CB and the strong interface between rCF and gel were observed. Furthermore, AAS mortars exhibited multifractal characteristics within the range of micropores and capillary pores. Despite altering fractal regions, the additives did not affect its size dependence. The backbone fractal dimension increases with the addition of CB and rCF, exhibiting strong correlations with various macro properties, thus serving as a comprehensive parameter to characterize pore shape and distribution. This study deepens understanding of AAM composites, facilitating their adoption of low-carbon building materials.

FractalRG: Advanced fractal region growing using Gaussian mixture models for left atrium segmentation

This paper presents an advanced region growing method for precise left atrium (LA) segmentation and estimation of atrial wall thickness in CT/MRI scans. The method leverages a Gaussian mixture model (GMM) and fractal dimension (FD) analysis in a three-step procedure to enhance segmentation accuracy. The first step employs GMM for seed initialization based on the probability distribution of image intensities. The second step utilizes fractal-based texture analysis to capture image self-similarity and texture complexity. An enhanced approach for generating 3D fractal maps is proposed, providing valuable texture information for region growing. In the last step, fractal-guided 3D region growing is applied for segmentation. This process expands seed points iteratively by adding neighboring voxels meeting specific similarity criteria. GMM estimations and fractal maps are used to restrict the region growing process, reducing the search space for global segmentation and enhancing computational efficiency. Experiments on a dataset of 10 CT scans with 3,947 images resulted in a Dice score of 0.85, demonstrating superiority over traditional techniques. In a dataset of 30 MRI scans with 3,600 images, the proposed method achieved a competitive Dice score of 0.89±0.02, comparable to Deep Learning-based models. These results highlight the effectiveness of our approach in accurately delineating the LA region across diverse imaging modalities.

Variational approaches to Dirichlet gradient-type systems on the Sierpiński gasket

In this paper, we study the multiple solutions of parametric quasi-linear systems of the gradient-type on the Sierpiński gasket arising in physical problems leading to nonlinear models involving reaction-diffusion equations, in problems on elastic fractal media or fluid flow through fractal regions. By using some critical point theorems, we give some new results to establish one, two and three weak solutions for our system. Finally, we give four examples to illustrate the main results.

Static and dynamic nonlinear behavior of a multistable structural system consisting of two coupled von Mises trusses

Recent years have witnessed a growing interest in multistable structures. In most cases it is attained by a sequence of bistable elements. However, little is known on their nonlinear static and dynamic responses. In this work a detailed static and dynamic nonlinear analysis of a multistable structural system consisting of two coupled bistable von Mises trusses is conducted to enlighten the complex behavior of these systems. For this, the exact nonlinear equilibrium equations and equations of motion in their dimensionless forms are obtained through the principle of stationary potential energy and Hamilton’s principle, respectively, considering a linear elastic material. Using continuation algorithms, the nonlinear equilibrium paths are obtained. The stability of each configuration is analyzed using the principle of minimum potential energy. Multiple equilibrium paths of the coupled system are identified, leading to several coexisting stable and unstable solutions, bifurcations and potential wells which are closely connected to the symmetries of the system. It is shown that the number of coexisting solutions increases exponentially with the number of bistable units. The effect of unavoidable imperfections is also clarified. The nonlinear dynamics and bifurcations of the system under harmonic forcing and static pre-load are then studied. Bifurcation diagrams, Poincaré maps and cross-sections of the basins of attraction are used to study the influence of coexisting attractors due to multiple potential wells, resonances and bifurcations. The coexisting responses can merge giving rise to several types of cross-well motions, the threshold value being closely connected with the limit point load of the coupled system. The increase in coexisting solutions leads to increasingly complex basins of attraction with broad fractal regions. On the one hand, the complexity of the bifurcation scenarios seems valuable in several applications to encode information, to respond to external forcing and to switch between different states. On the other hand, multiple attractors and their fractal basins can lead to the loss of global stability and dynamic integrity due to the small uncorrupted basins surrounding each attractor. Thus, the knowledge of the static and dynamic behavior of multistable systems is of great significance to either inducing or avoiding this phenomenon or making use of it.

Impacts of mineralogy and pore structure on spontaneous imbibition in tight glutenite reservoirs

Spontaneous imbibition plays a critical role in the two-phase flow in hydraulic fractures of tight reservoirs. Although massive contributions have been devoted to addressing this issue, very few studies focus on tight glutenite reservoirs, which is an important member of tight reservoirs. As a result, the imbibition mechanism and its influencing factors are still unclear, several key factors fail to be captured, and in-depth analyses are lacking in spontaneous imbibition for tight glutenite reservoirs. In light of the current knowledge gaps, this study conducted a series of water imbibition experiments, combined with X-ray diffraction tests, thin section and scanning electron microscopy observations, high-pressure mercury intrusion experiments, and corrected contact angle experiments to reveal the impacts of rock mineralogy and pore structure on spontaneous imbibition, and significance levels of main influential factors were determined by grey relation analysis. Results show that: 1) the hydrophilic mineral that exceeds 94% provides a strong driving force for water imbibition. Intergranular pore and gravel boost imbibition recovery through enhancing imbibition and drainage ability, respectively. However, intergranular dissolved pore is unfavorable for water imbibition due to extremely rough surface and small radius. Medium-size sand also has a negative impact because of incremental flow tortuosity and pore coordination number reduction; 2) A large percentage of micropore and macropore can significantly enhance imbibition rate and recovery. Three fractal characteristics in micropore, identified for the first time, complicates water imbibition process in tight glutenites. A critical total fractal dimension of micropore is observed, above which the negative influence magnitude of complex pore structure on spontaneous imbibition gradually decreases; 3) Water imbibition sequence and diversities of imbibition phenomena are significantly affected by the combination of multi-fractal dimensions and heterogeneous distributions of pore volume in different fractal regions. The superposition of smallest fractal dimension and the largest pore volume leads to the best imbibition recovery in the transition stage of imbibition, which is an extremely time-consuming process; 4) Upon grey relation analysis, oil drainage ability is the most critical factor for water imbibition in tight glutenites. These observations deepen our understanding of the distinctive imbibition mechanisms in tight glutenite reservoirs.

Effects of solvent vapor annealing on the optical properties and surface adhesion of conjugated D : A thin films.

A diketopyrrolopyrrole (DPP) and perylene diimide (PDI)-based molecule, denoted as PDI-DPP-PDI, was investigated as an electron acceptor material in bulk heterojunction (BHJ) solar cells, with poly[[4,8-bis [5-(2-ethylhexyl)-2-thienyl]benzo[1,2-b:4,5-b']dithiophene-2,6-diyl] [2-(2-ethyl-1-oxohexyl)thieno[3,4-b]thiophenediyl]] (PBDTTT-CT) as an electron donor. The donor polymer and the acceptor molecule have complementary absorption spectra, which is an essential feature for energy collection in organic solar cells. However, AFM images indicated the presence of isolated and microsized PDI-DPP-PDI domains along the surface of the films, which reduced the power conversion efficiency. Therefore, to improve the homogenization of the acceptor along the film, a post-deposition treatment, denoted as solvent vapor annealing (SVA), was performed in a saturated atmosphere containing the vapour of an organic solvent for 3-10 minutes. This procedure changed the optical and morphological properties of the PBDTTT-CT : PDI-DPP-PDI active layer, resulting in increased power conversion efficiency values by more than 2.5 times (reaching 5.1%). Theoretical simulation pointed out that the experimental absorbance band localized at 580 nm, which appeared after SVA treatment, is possibly related to an intense simulated band with a maximum at 572 nm, resulting from a pair of transitions starting in the copolymer and ending in PDI-DPP-PDI, in regions where both are stacked at about 3 Å. The most significant natural transition orbitals (NTOs) related to these transitions indicated charge transfer character. Moreover, analyses carried out by power spectrum density (PDS) of images acquired from the SVA-treated film indicated that in the region of larger frequencies, across the length scale at around 30-70 nm, an additional fractal region appeared with a Ds of 0.95, indicating a flattened region, possibly related to changes in the overall conformation after SVA treatment. This indicates an improvement in the molecular packing, a feature not observed in the as-cast film. The analyses by force curve spectroscopy pointed out increased adhesion forces and adhesion energy in the PBDTTT-CT : PDI-DPP-PDI film after SVA treatment; this feature enhanced the interfacial interaction with the top electrodes, reflecting improved charge extraction in the photovoltaic device.

Volume fractal and surface fractal analysis of the pore structure of natural and heat-treated spruce wood using the mercury intrusion porosimetry test

Abstract Fractal geometry describes the complex pore structure in natural and heat-treated wood and the relationship between pore structure and wood properties, such as strength, heat conductivity, and transport properties. However, the fractal types and the scale-dependent properties of natural and heat-treated wood remain unclear. In this study, comprehensive fractal analysis of the pore structure of natural and heat-treated spruce wood was carried out based on mercury intrusion porosimetry data. Both the volume fractal and surface fractal of natural and heat-treated wood were determined. The results showed that the two fractal types had different scale-dependent fractal properties. Four regions were identified in the pore structures. A volume fractal region was identified for pores in the region of 2–90 μm, while a surface fractal region was identified for pores in the region of 90 nm–7 μm. The pore structure in the region of 2–90 μm that corresponded to the large pore (the lumina in the cell) range showed strong volume fractal properties, and the fractal dimensions were 2.645–2.884. The pore structure in the region of 90 nm–7 μm that corresponded to the small pore (voids on or in cell walls) range showed strong surface fractal properties, and the fractal dimensions were 2.323–2.999. The range of fractal regions was hardly affected by the heat treatment atmospheres. These results showed that fractal geometry can be used to characterize the pore structures of natural and heat-treated wood. These findings are expected to explain the differences in properties between natural and heat-treated wood in the future.

Transport through quantum anomalous Hall bilayers with lattice mismatch

We theoretically investigate quantum transport properties of quantum anomalous Hall bilayers, with arbitrary ratio of lattice constants, i.e., with lattice mismatch. In the simplest case of ratio 1 (but with different model parameters in two layers), the inter-layer coupling results in resonant traversing between forward propagating waves in two layers. In the case of generic ratios, there is a quantized conductance plateau originated from two Chern numbers associated with two layers. However, the phase boundary of this quantization plateau consists of a fractal transitional region (instead of a clear transition line) of interpenetrating edge states (with quantized conductance) and bulk states (with unquantized conductance). We attribute these bulk states as mismatch induced in-gap bulk states. Different from in-gap localized states induced by random disorder, these in-gap bulk states are extended in the limit of vanishing random disorder. However, the detailed fine structure of this transitional region is sensitive to disorder, lattice structure, sample size, and even the configuration of leads connecting to it, due to the bulk and topologically trivial nature of these in-gap bulk states.

RETRACTED: Efficient intelligent compression and recognition system-based vision computing for computed tomography COVID images

Computed tomography (CT) image-based medical recognition is extensively used for COVID recognition as it improves recognition and scanning rate. A method for intelligent compression and recognition system-based vision computing for CT COVID (ICRS-VC-COVID) was developed. The proposed system first preprocesses lung CT COVID images. Segmentation is then used to split the image into two regions: nonregion of interest (NROI) with fractal lossy compression and region of interest with context tree weighting lossless. Subsequently, a fast discrete curvelet transform (FDCT) is applied. Finally, vector quantization is implemented through the encoder, channel, and decoder. Two experiments were conducted to test the proposed ICRS-VC-COVID. The first evaluated the segmentation compression, FDCT, wavelet transform, and discrete curvelet transform (DCT). The second evaluated the FDCT, wavelet transform, and DCT with segmentation. It demonstrates a significant improvement in performance parameters, such as mean square error, peak signal-to-noise ratio, and compression ratio. At similar computational complexity, the proposed ICRS-VC-COVID is superior to some existing techniques. Moreover, at the same bit rate, it significantly improves the quality of the image. Thus, the proposed method can enable lung CT COVID images to be applied for disease recognition with low computational power and space.

Relationship between fractal feature and compressive strength of concrete based on MIP

Concrete is a heterogeneous, multiphase, and multilayer composite system with extremely intricate microscopic pore structure. The structural characteristics of microscopic pore of concrete are closely related to its mechanical properties and durability. The pore structure of concrete with various strength grades (C10-C150) was analyzed by Mercury intrusion porosimetry (MIP) from the aspects of pore diameter distribution, pore volume, pore specific surface area and fractal feature. The Menger sponge model was used to calculate the fractal dimension of different pore regions in each test group, and the most suitable pore fractal region was selected to establish the mathematical model between fractal dimension and compressive strength. The results showed that all fractal curves are divided into four segments according multi-fractal characteristic of concrete, namely Region I, Region II, Region III and Region IV. Region I is macropores fractal region (d = 10 μm ∼ 950 μm), representing the packing patterns of hydrated cement grains, while Region II - Region Ⅳ is micropores fractal region (d = 0.0064 μm ∼ 10 μm), reflecting mainly the microstructure of C-S-H hydrates. Region I and Region Ⅳ exhibit obvious fractal features and the fractal dimension D is between 2 and 3. However, the fractal dimension D > 3 in Region II and Region III, which are nonphysical from the point of view of surface geometry. The fractal dimension can clearly describe the internal pore characteristic, and it is a real good correlation with the most probable pore diameter and the pore specific surface area. Moreover, the mathematical model between the fractal dimension and the compressive strength both accords with the positive power exponential function in Region I and Region IV. The fractal dimension can be viewed as a comprehensive parameter of pore shape and spatial distribution of pore structure, which can more accurately represent the relationship between the microscopic pore structure features and compressive strength of concrete. In practical engineering, it serves to realize the damage detection of concrete on site.

Strange attractors and nontrivial solutions in games with three players

Two-player games have been sufficiently researched, but three-player games have a much greater wealth of solutions, in particular, nonstationary solutions. However, unsteady solutions of three-player games have not been studied in the literature. On the basis of numerical modeling, the dynamic behavior of a 2 × 2 × 2 game with three players is investigated in the range of payoff matrices from -1 to 1. The system of dynamic equations includes three variables (probabilities of pure strategies) and twelve parameters associated with cubic payoff matrices of players. The phase space of a three-player game is a cube (probabilities vary from 0 to 1). It was found that for some payoff matrices, there are no stable solutions in the system. In addition, the structure of the phase space is complex: the volume of the cube is a fractal repulsive region from which the phase trajectories leave, a strange attractor is located near the edge corresponding to the solution of the "node-center" type, and another strange attractor is a chaotic motion between adjacent vertices of the cube. The obtained dynamic properties of the game can be used for making decisions in economics, negotiations between three persons, and modeling relationships between biological species.

Dynamical integrity assessment of stable equilibria: a new rapid iterative procedure

A new algorithm for the estimation of the robustness of a dynamical system’s equilibrium is presented. Unlike standard approaches, the algorithm does not aim to identify the entire basin of attraction of the solution. Instead, it iteratively estimates the so-called local integrity measure, that is, the radius of the largest hypersphere entirely included in the basin of attraction of a solution and centred in the solution. The procedure completely overlooks intermingled and fractal regions of the basin of attraction, enabling it to provide a significant engineering quantity in a very short time. The algorithm is tested on four different mechanical systems of increasing dimension, from 2 to 8. For each system, the variation of the integrity measure with respect to a system parameter is evaluated, proving the engineering relevance of the results provided. Despite some limitations, the algorithm proved to be a viable alternative to more complex and computationally demanding methods, making it a potentially appealing tool for industrial applications.

May the Piezoresistivity of GNP-Modified Cement Mortar Be Related to Its Fractal Structure?

High piezoresistivity of cement-based composites tuned by conductible fillers provides a feasible way to develop self-sensing smart structures and buildings. However, the microstructural mechanisms remain to be properly understood. In the present work, the piezoresistivity of cement mortar with different dosages of graphene nanoplatelets (GNPs) was investigated, and the microstructure was assessed by electron scanning microscopy (SEM) and mercury intrusion porosimetry (MIP). Two surface fractal models were introduced to interpret the MIP data to explore the multi-scale fractal structure of the GNP-modified cement mortars. Results show that the incorporation of GNPs into cement mortar can roughen the fracture surfaces due to the GNPs’ agglomeration. Gauge factor (GF) rises and falls as GNP content increases from 0% to 1% with the optimal piezoresistivity observed at GNP = 0.1% and 0.05%. The GF values of the optimum mortar are over 50 times higher than those of the reference mortar. Fractal dimensions in macro and micro fractal regions change with GNP content. Analysis shows that the fractal dimensions in micro region decrease first and then increase with the increase of GF values. GNPs not only impact the fractal structure of cement mortar, but also alter the tunneling and contact effects that govern the piezoresistivity of composite materials.

A CONTINUOUS VARIATION OF ROUGHNESS SCALING CHARACTERISTICS ACROSS FRACTAL AND NON-FRACTAL PROFILES

In this study, the scaling characteristics of root-mean-squared roughness ([Formula: see text]) was investigated for both fractal and non-fractal profiles by using roughness scaling extraction (RSE) method proposed in our previous work. The artificial profiles generated through Weierstrass–Mandelbrot (W–M) function and the actual profiles, including surface contours of silver thin films and electroencephalography signals, were analyzed. Based on the relationship curves between [Formula: see text] and scale, it was found that there was a continuous variation of the dimension value calculated with RSE method ([Formula: see text]) across the fractal and non-fractal profiles. In the range of fractal region, [Formula: see text] could accurately match with the ideal fractal dimension ([Formula: see text]) input for W–M function. In the non-fractal region, [Formula: see text] values could characterize the complexity of the profiles, similar to the functionality of [Formula: see text] value for fractal profiles, thus enabling the detection of certain incidents in signals such as an epileptic seizure. Moreover, the traditional methods (Box-Counting and Higuchi) of [Formula: see text] calculation failed to reflect the complexity variation of non-fractal profiles, because their [Formula: see text] was generally 1. The feasibility of abnormal implementation of W–M function and the capability of RSE method were discussed according to the analysis on the properties of W–M function, which would be promising to make more understandings of the nonlinear behaviors of both theoretical and practical features.

Stereometric characterization of Dinizia excelsa Ducke wood from Amazon rainforest using atomic force microscopy.

Dinizia excelsa Ducke under three different cut conditions were carefully analyzed. The morphology and stereometry of different wood cutting surfaces (longitudinal radial, longitudinal tangential, and transversal) were studied by SEM and AFM. The results obtained in this study suggest that both the height parameters and the advanced stereometric parameters of the surfaces did not reveal a significant difference, indicating that the spatial patterns do not change according to the type of cut. In this way, the surface microtexture does not vary depending on the cut type. Similarly, the Hurst's coefficients did not show any significant difference in the spectrum of the PSD fractal region. On the other hand, Minkowski functionals presented a morphological difference between the samples. These results showed that the microtexture of the wood surface does not change as a function of the type of cut submitted to the same polishing process.

Pore structure, fractal characteristics and permeability prediction of tight sandstones: A case study from Yanchang Formation, Ordos Basin, China

Many fractal models have been proposed based on mercury intrusion porosimetry (MIP) for investigating the fractal property of porous materials. However, results calculated from different fractal models may be not always consistent according to previous works. In view of this, a combination of techniques including routine core analysis, MIP analysis and images analysis (thin sections and BSE) was used to study the pore structure and petrophysical properties of tight sandstones. The results of pore structure and petrophysical properties were used to check whether the fractal dimension values from five popular MIP fractal models can represent the heterogeneity of tight sandstones.It turned out that different MIP fractal models usually give different results. A single fractal dimension value representative of the entire range of pore sizes can be obtained by either Zhang and Li's model or Shen and Li's model. As fractal dimension increases, pore structure becomes more complex and permeability decreases. However, permeability or pore structure shows no significant correlation with fractal dimensions obtained from other models (Neimark's, Li's and Friensen and Mikula's model) by zone analysis. Nevertheless, two fractal regions can be identified by Neimark's or Li's model and clear inflection points could be identified on the fractal curves. The inflection points are found to be almost equal to the apex determined by Pittman's method, and could be regarded as good indicators for permeability prediction. The results obtained from Friensen and Mikula's model is more complex, and not very satisfactory to describe the heterogeneity for tight sandstones.

Fine structure of heating in a quasiperiodically driven critical quantum system

We study the heating dynamics of a generic one dimensional critical system when driven quasiperiodically. Specifically, we consider a Fibonacci drive sequence comprising the Hamiltonian of uniform conformal field theory (CFT) describing such critical systems and its sine-square deformed counterpart. The asymptotic dynamics is dictated by the Lyapunov exponent which has a fractal structure embedding Cantor lines where the exponent is exactly zero. Away from these Cantor lines, the system typically heats up fast to infinite energy in a non-ergodic manner where the quasiparticle excitations congregate at a small number of select spatial locations resulting in a build up of energy at these points. Periodic dynamics with no heating for physically relevant timescales is seen in the high frequency regime. As we traverse the fractal region and approach the Cantor lines, the heating slows enormously and the quasiparticles completely delocalise at stroboscopic times. Our setup allows us to tune between fast and ultra-slow heating regimes in integrable systems.

New approach for evaluating chloride ion penetration of mortar in cold environments

The aim of this paper was to use the surface fractal dimension (Ds) to evaluate the chloride ion penetration of cement mortar in extremely cold regions. Pore structure is a critical factor that influences the permeability of cement mortar. This study investigated surface fractal dimensions of the pore structure of cement mortars, which were subject to one-time freezing for 7 days after casting. A thermodynamic model was applied to calculate Ds on the basis of mercury intrusion porosimetry data. The results show that the pore structures of all specimens exhibit the scale-dependent fractal property, and three distinct fractal regions (I, II, III) are obtained. According to the regression analysis results of fractal dimension and pore structure parameters, it was found that the fractal dimension can characterise the complexity and chaos of pore structure and predict pore size distribution. Based on this, a linear regression between chloride diffusion coefficient (Dnssm) and Ds in fractal regions was carried out. The results show that Ds of micro pores (region I) have good correlation with Dnssm, up to 0.92908, which is more accurate than the porosity. An index of surface fractal dimension is proposed to evaluate the durability of concrete buildings in extremely cold regions.

Small-Angle Scattering and Multifractal Analysis of DNA Sequences.

The arrangement of A, C, G and T nucleotides in large DNA sequences of many prokaryotic and eukaryotic cells exhibit long-range correlations with fractal properties. Chaos game representation (CGR) of such DNA sequences, followed by a multifractal analysis, is a useful way to analyze the corresponding scaling properties. This approach provides a powerful visualization method to characterize their spatial inhomogeneity, and allows discrimination between mono- and multifractal distributions. However, in some cases, two different arbitrary point distributions, may generate indistinguishable multifractal spectra. By using a new model based on multiplicative deterministic cascades, here it is shown that small-angle scattering (SAS) formalism can be used to address such issue, and to extract additional structural information. It is shown that the box-counting dimension given by multifractal spectra can be recovered from the scattering exponent of SAS intensity in the fractal region. This approach is illustrated for point distributions of CGR data corresponding to Escherichia coli, Phospholamban and Mouse mitochondrial DNA, and it is shown that for the latter two cases, SAS allows extraction of the fractal iteration number and the scaling factor corresponding to “ACGT” square, or to recover the number of bases. The results are compared with a model based on multiplicative deterministic cascades, and respectively with one which takes into account the existence of forbidden sequences in DNA. This allows a classification of the DNA sequences in terms of random and deterministic fractals structures emerging in CGR.

The unpredictability of the basins of attraction in photogravitational Chermnykh’s problem

This paper deals with the photogravitational circular restricted three-body problem, including the effect of circular cluster of material points. The parametric evolution of the position of equilibrium points and the existence of a different number of equilibrium points are illustrated using graphs. A detailed investigation to find out the influence of the radiation pressure $q$ , mass ratio $\mu $ and the circular cluster of material points $M_{b}$ on the geometry of basins of attraction is carried out. Further, the number of iterations required to obtain the desired level of accuracy is recorded and presented using probability distribution diagram. The correlations between the domain of convergence of equilibrium points and the corresponding number of the iterations required to obtain the desired level of accuracy are explained. We monitor the parametric evolution of the basin entropy to reveal the unpredictability in the basins of attractions. It is found that unpredictable (fractal) regions exist in the vicinity of boundaries of the basins of attraction.