Values Of Fractal Parameters Research Articles

Recent strategy to study fractal-order viscoelastic polymer materials using an ancient Chinese algorithm and He’s formulation

This paper introduces a novel methodology to determine the frequency-amplitude relationship of fractal-order viscoelastic polymer materials using the two-scale fractal dimension transform, the equivalent power-form representation of the conservative restoring forces, and a simple coordinate transformation to eliminate viscoelastic effects. Then, the ancient Chinese algorithm Ying Bu Zu Shu and He’s formulation are used for obtaining the frequency-amplitude relationship. Simulation results obtained from the derived expressions exhibit good agreement when compared to numerical integration solutions. This article elucidates how the molecular structure of polymer chains influences the relaxation oscillations as a function of the fractal parameter values.

AN EFFICIENT ANCIENT CHINESE ALGORITHM TO INVESTIGATE THE DYNAMICS RESPONSE OF A FRACTAL MICROGRAVITY FORCED OSCILLATOR

In this paper, the ancient Chinese algorithm is applied to derive fractal microgravity forced oscillator’s frequency–amplitude response curves using the two-scale fractal transform derivative. Depending on the fractal parameter values, it is shown that the system restoring forces exhibit hardening or softening behavior. Furthermore, the proposed solution approach is simple, efficient, and accurate for obtaining the approximate steady-state solution of a fractal microgravity nonlinear oscillator with a purely nonlinear power-form restoring force.

Exact steady-state solution of fractals damped, and forced systems

In this article, the methodology for deriving the exact steady-state solutions of forced oscillatory systems has been extended to obtain the solution of fractal damped and forced differential equations using the two-scale fractal dimension transform. Finally, a numerical example is given to illustrate the applicability of the proposed approach in obtaining the frequency–amplitude response curves for a forced and damped fractal oscillator. Numerical results show that the fractal parameter values shift the frequency–amplitude curves to the left and right from the resonance curve. The present method sheds new light on solving damped and forced fractal oscillators.

Retinal Vessel Phenotype in Patients with a History of Retinal Vein Occlusion

Introduction: The aim of the study was to estimate the phenotype of retinal vessels using central retinal artery equivalent (CRAE), central retinal vein equivalent (CRVE), tortuosity, and fractal analysis in the unaffected contralateral eye of patients with central or branch retinal vein occlusion (CRVO or BRVO). Methods: Thirty-four patients suffering from CRVO, 15 suffering from BRVO, and 49 controlled matched subjects had a fundus image analyzed using the VAMPIRE software. The intraclass correlation coefficient and a Bland-Altman plot were done for the reproducibility study. Results: There was a lack of evidence of difference between the control group and the CRVO group for CRAE (p = 0.06), CRVE (p = 0.3), and arterio-venule ratio (AVR, p = 0.6). Contralateral eyes of CRVO exhibited a significantly higher arterial and minimum arterial tortuosity values (p = 0.012), as compared with control eyes. Contralateral eyes of patients with a history of BRVO had a significantly higher CRAE (p = 0.02), AVR (p = 0.006), and minimal arterial tortuosity (p = 0.05). Fractal analysis showed that contralateral eyes of BRVO had higher values of fractal parameters (D0a, p = 0.005). Conclusion: This study suggests that CVRO or BRVO is not triggered by the same retinal vascular phenotypes in the contralateral eye. The morphology of retinal vasculature may be associated with the occurrence of RVO, independently of known risk factors.

Coupling Fractal Model for Fretting Wear on Rough Contact Surfaces

Abstract In this paper, a fretting damage model based on fractal theory is proposed. The Weierstrass–Mandelbrot function of fractal theory is used to represent the rough contact surface, and a corresponding contact parameter analysis method is also established. Based on neural network algorithm, the values of fractal parameters are fitted, and the fitting accuracy has been greatly improved compared with traditional methods. According to the fractal parameters of the actual surface, the fretting wear process of the rough contact surface is analyzed based on theory of adhesive and three-body abrasive wear. A generic program for the analysis of three-dimensional fretting wear problems is also proposed. Compared with material tests, the prediction error of the fretting wear simulation model is 13.4% for wear depth and 16.7% and 3.9% for width and length of wear scar in stable wear stage. The prediction results show that the model can be applied to the prediction of the actual three-dimensional fretting wear model.

A fractal model for current generation in porous electrodes

This article aims to investigate the solution of the fractional Troesch's equation that models the mechanism of current generation in porous electrodes by numerical integration of its equivalent power-form representation and by deriving its approximate Taylor series solution. Simulation results compare to experimental data demonstrate that the overpotential distribution and efficiency within a porous electrode, from the bottom to the top of the bed is predicted by using Troesch's fractal overpotential diffusion equation, when the fractal parameter values are α ≥ 1.2618 for a porous electrode that follows the Hausdorff dimension, or α ≥ 1.236 for El Naschie's fractal space-time.

Fractal-Based Computation of Heat Source Depths and Temperatures for the Soutpansberg Basin, South Africa

The fractal-based approach was used for computing magnetic depths and temperatures for the Soutpansberg Basin in South Africa. The average depth to the top Zt and basement depth Zo for the Soutpansberg Basin were 4.36 ± 0.28 km, 10.43 ± 0.65 km, respectively. The average temperature at depth Zt was 184.69 ± 7.66 ºC. Magnetic source depths and basal temperatures that were in the Curie point range were determined, to be within 20.35 km to 21.68 km and 549.34 ºC to 585.24 ºC, respectively. Increasing the value of the fractal parameter β from 0 to 4, had an effect of retaining deeper depths and higher temperatures. The fractal parameter values of β > 3 retained Curie point depths and temperatures that indicated basal rock types with an igneous predisposition. The fractal-based approach proved to be an improved technique as compared to the conventional centroid method.

Fractal continuum model for tracer transport in a porous medium

A model based on the fractal continuum approach is proposed to describe tracer transport in fractal porous media. The original approach has been extended to treat tracer transport and to include systems with radial and uniform flow, which are cases of interest in geoscience. The models involve advection due to the fluid motion in the fractal continuum and dispersion whose mathematical expression is taken from percolation theory. The resulting advective-dispersive equations are numerically solved for continuous and for pulse tracer injection. The tracer profile and the tracer breakthrough curve are evaluated and analyzed in terms of the fractal parameters. It has been found in this work that anomalous transport frequently appears, and a condition on the fractal parameter values to predict when sub- or superdiffusion might be expected has been obtained. The fingerprints of fractality on the tracer breakthrough curve in the explored parameter window consist of an early tracer breakthrough and long tail curves for the spherical and uniform flow cases, and symmetric short tailed curves for the radial flow case.

Fractal Analysis of AFM Data Characterizing Strongly Isotropic and Anisotropic Surface Topography

This study discusses changes in the value of fractal parameters determined based on functions of structure S(t), generated in different directions of anisotropy of the examined surfaces. The analyzed material consisted of AFM calibration standards TGT1, PG and TGZ1 which were used as models of strongly isotropic and anisotropic surfaces. The topography of the examined surfaces was imaged by atomic force microscopy. The obtained results indicate that all surfaces can be described mathematically to identify fractal parameters in any anisotropic direction.

A hybrid particle swarm optimization and multi-layer perceptron algorithm for bivariate fractal analysis of rock fractures roughness

Application of fractal theory to quantify the roughness of rock fractures has been well developed, and various methods have been presented to estimate the fractal dimension of roughness profiles, D. However, determination of D is still a challenging issue, and researchers have attributed different values to fractal dimensions of the standard profiles of rock fracture roughness. Two kinds of errors affect the estimated fractal dimension: stochastic errors and systematic errors. Both errors cannot be modeled by any explicit function, because of the complexity and the uncertainty in the relationship between the fractal dimension and the measurable variables. In this paper, a new method is presented to overcome these deficiencies. First, a large number of fractional Brownian profiles with different values of fractal parameters (D and standard deviation σ) were generated and their statistical features were extracted. Then, a hybrid algorithm consisting of two kinds of algorithms, particle swarm optimization (PSO) algorithm and multi-layer perceptron (MLP) neural network, was developed. The inputs of the system were the features of profile namely root mean square of profile and its first derivatives. In each iteration of the hybrid system, the PSO and MLP algorithms exchanged information with each other to optimize values of the fractal parameters. Finally, the system provided the best set of fractal parameters for a given profile. The effectiveness of this method was shown by estimating the fractal parameters of a testing set of profiles. Finally, the values of D and σ were estimated for Barton's standard roughness profiles and for the profiles extracted from digitized surfaces of natural rock fractures.

Structure morphology problems in the air separation by polymer membranes with magnetic particles

Abstract Structure and morphology of dense polymer membranes with dispersed magnetic powder (magnetic membranes) for air separation were investigated. Qualitative and quantitative description of membrane structures based on fractal analysis, is presented. Box counting method for calculating the generalized fractal dimension was used. Structures with the same value of fractal parameters as for real membranes were simulated, and random walk dimensions were evaluated. Study of transport processes in air separation using polymeric membranes with magnetic powder both experimentally and through the random walk, was performed. The experimental results were compared with theoretical predictions. Finally, the diffusion equation with a spatial dependent diffusion coefficient of self-similar type to describe diffusion processes in the aforementioned membranes, was proposed.

Radar polarimetry of Saturn's rings: Modeling ring particles as fractal aggregates built of small ice monomers

We analyze ground-based radar polarimetric observations of Saturn's rings at a wavelength of 12.6 cm by employing the model of a vertically and horizontally plane-parallel homogeneous slab composed of clumpy particles in the form of fractal aggregates of small ice monomers. Our model takes full account of the effects of polarization, multiple scattering, and coherent backscattering. Using efficient superposition T-matrix and vector radiative transfer codes, we perform computations of the backscattering circular polarization ratio for fractal aggregates generated with a cluster–cluster aggregation model and having the following characteristics: monomer refractive index m=1.78+i0.003; monomer packing density p=0.2; fractal dimensions D f=2.5 and 3; and overall fractal radii R in the range 4⩽ R⩽10 cm. In order to obtain physically realistic values of single-scattering properties of the aggregates we perform averaging over an ensemble of clusters generated for the same values of fractal parameters but having different geometrical configurations of the monomers. We conclude that in the framework of the above morphological model of Saturn's rings and the specific cluster–cluster aggregation procedure, it may be problematic to obtain a satisfactory and realistic agreement between theoretical computations and the observed values of the radar circular polarization ratio.

Contact mechanics and friction of fractal surfaces probed by atomic force microscopy

We investigated the contact mechanics and friction forces between atomic force microscope (AFM) probes and self-affine fractal carbon films. We studied single-asperity contacts by means of conventional nanometric conical tips whilst custom-designed micrometric flat tips were adopted to form multiple junctions between the probe and the sample. By varying the externally applied load we found that the average frictional force follows a power-law behavior in the single-asperity regime and a linear behavior in the multi-asperity regime. The friction coefficient was the same for carbon specimens having different fractality. We also acquired quasi-static load–displacement curves on micrometric scale, revealing a strong dependence of the average indentation depth on the values of fractal parameters. A comparison of experimental data with contact theories for randomly rough surfaces is provided.

Requirements for accurate quantification of self-affine roughness using the variogram method

Both stationary and non-stationary fractional Brownian profiles (self-affine profiles) with known values of fractal dimension, D, input standard deviation, σ, and data density, d, were generated. For different values of the input parameter of the variogram method (lag distance, h), D and another associated fractal parameter Kv were calculated for the aforementioned profiles. It was found that σ has no effect on calculated D. The estimated Kv was found to increase with D, σ and d according to the equation Kv = 2.0 × 10−5d0.35σ1.95D14.5. The parameter Kv seems to have potential to capture the scale effect of roughness profiles. Suitable ranges for h were estimated to obtain computed D within ±10% of the D used for the generation and also to satisfy a power functional relation between the variogram and h. Results indicated the importance of removal of non-stationarity of profiles to obtain accurate estimates for the fractal parameters. It was found that at least two parameters are required to quantify stationary roughness; the parameters D and Kv are suggested for use with the variogram method. In addition, one or more parameters should be used to quantify the non-stationary part of roughness, if it exists; at the basic level, the mean inclination/declination angle of the surface in the direction considered can be used to represent the non-stationarity. A new concept of feature size range of a roughness profile is introduced in this paper. The feature size range depends on d, D and σ. The suitable h range to use with the variogram method to produce accurate fractal parameter values for a roughness profile was found to depend on both d and D. It is shown that the feature size range of a roughness profile plays an important role in obtaining accurate roughness parameter values with both the divider and the variogram methods. The minimum suitable h was found to increase with decreasing d and increasing D. Procedures are given in this paper to estimate a suitable h range for a given natural rock joint profile to use with the variogram method to estimate D and Kv accurately for the profile.

A Fractal Analysis of Adhesion at the Contact between Rough Solids

Analyses of adhesion between rough solids hitherto relied on a random rough surface model which is appropriate for engineering surfaces. Adhesion occurs at the peaks of the asperities and is pronounced when the surface roughness effect is small, down to the nanometre level. It therefore follows that while the microscopic roughness would inhibit the two surfaces to come to close separation, adhesive bonds may still form at the peaks of the small-scale asperities. In order to account for the effect of asperities ranging from the nanometre to the micrometre level, it is necessary to introduce a scale-independent model. A fractal approach essentially gives this. The paper describes analysis of adhesion between rough solids using a fractal model and the results are compared with those using the conventional approach and experimental results. Some of the existing experimental observations such as adhesion-induced plasticity have been confirmed. The present analysis also gives a generalized solution and, depending on the values of fractal parameters D, the dimension, and G, the roughness parameter, specific solutions may be obtained. The solution obtained using the conventional stochastic model is thus a specific case in the generalized fractal solution.

A fractal analysis of the influence of non-specific binding on antigen-antibody binding kinetics for biosensor applications

A fractal analysis of the influence of non-specific binding on the specific binding of antigen in solution to antibody immobilized on a biosensor surface is presented for first-, one and a half-, second, and other-order reactions occurring under external diffusion-limited conditions. Both single-step and dual-step binding of antigen in solution to antibody immobilized on the surface is considered. For a first-order reaction, an increase in the fractal parameter, b, leads to a decrease in the amount of antigen in solution bound specifically to the antibody on the surface when non-specific binding is either absent or present. The presence of non-specific binding leads to a decrease in the amount of antigen bound to the antibody on the surface. For a one and a half- and for second-order reactions and when non-specific binding is either absent or present to a small degree (α = 0·01), an increase in the fractal parameter, b, leads to a decrease in the amount of antigen bound specifically to the antibody immobilized on the biosensor surface is However, for an α value of 0·1, the maximum rate and the amount of antigen bound specifically to the antibody immobilized on the biosensor surface is obtained for fractal parameter values of 0·2 and 0·4, and 0·4 for the one and a half- and for second-order reactions, respectively. Apparently, some amount of heterogeneity is helpful in obtaining the optimum amount and rate of antigen in solution bound specifically to the antibody on the surface for reaction orders higher than one. The applicability of the approach to real antibody surfaces is demonstrated.

Self-similarity in particle production in hadron-nucleus interactions at 350 and 200 GeV/c.

Self-similarity in multiparticle production in ${\ensuremath{\pi}}^{\ensuremath{-}}$-nucleus interactions at 350 and at 200 GeV/c has been studied in terms of factorial moments ${F}_{q}$ as well as fractal moments ${G}_{q}$. Fractal parameters, already developed in the classical chaotic system, have been discussed and those parameters for multiparticle production in ${\ensuremath{\pi}}^{\ensuremath{-}}$-nucleus interactions have been evaluated. While the $F$-moment analysis confirms nonstatistical fluctuations, and in terms of anomalous dimensions, predicts cascading in hadronization, multifractality is indicated through the $G$-moment analysis. Variation in the nature or values of fractal parameters have been studied in different sizes of phase space and in different energies for the same reactants. Such a phenomenological analysis is essential at the present stage of development in the study of fractality in multiparticle production.

On grinding and polishing lenses, for optical purposes

Application of fractal theory to quantify the roughness of rock fractures has been well developed, and various methods have been presented to estimate the fractal dimension of roughness profiles, D. However, determination of D is still a challenging issue, and researchers have attributed different values to fractal dimensions of the standard profiles of rock fracture roughness. Two kinds of errors affect the estimated fractal dimension: stochastic errors and systematic errors. Both errors cannot be modeled by any explicit function, because of the complexity and the uncertainty in the relationship between the fractal dimension and the measurable variables. In this paper, a new method is presented to overcome these deficiencies. First, a large number of fractional Brownian profiles with different values of fractal parameters (D and standard deviation σ) were generated and their statistical features were extracted. Then, a hybrid algorithm consisting of two kinds of algorithms, particle swarm optimization (PSO) algorithm and multi-layer perceptron (MLP) neural network, was developed. The inputs of the system were the features of profile namely root mean square of profile and its first derivatives. In each iteration of the hybrid system, the PSO and MLP algorithms exchanged information with each other to optimize values of the fractal parameters. Finally, the system provided the best set of fractal parameters for a given profile. The effectiveness of this method was shown by estimating the fractal parameters of a testing set of profiles. Finally, the values of D and σ were estimated for Barton's standard roughness profiles and for the profiles extracted from digitized surfaces of natural rock fractures.