Weierstrass Mandelbrot Research Articles

A Comparative Study of Fractal Models Applied to Artificial and Natural Data

This paper presents an original and comprehensive comparative analysis of eight fractal analysis methods, including Box Counting, Compass, Detrended Fluctuation Analysis, Dynamical Fractal Approach, Hurst, Mass, Modified Mass, and Persistence. These methods are applied to artificially generated fractal data, such as Weierstrass–Mandelbrot functions and fractal Brownian motion, as well as natural datasets related to environmental and geophysical domains. The objectives of this research are to evaluate the methods’ capabilities in capturing fractal properties, their computational efficiency, and their sensitivity to data fluctuations. Main findings indicate that the Dynamical Fractal Approach consistently demonstrated the highest accuracy across different datasets, particularly for artificial data. Conversely, methods like Mass and Modified Mass showed limitations in complex fractal structures. For natural datasets, including meteorological and geological data, the fractal dimensions varied significantly across methods, reflecting their differing sensitivities to structural complexities. Computational efficiency analysis revealed that methods with linear or logarithmic complexity, such as Persistence and Compass, are most suited for larger datasets, while methods like DFA and Dynamic Fractal Approaches required higher computational resources. This study provides an original comparative study for researchers to select appropriate fractal analysis techniques based on dataset characteristics and computational limitations.

FRACTAL-BASED MODELING OF RAIL ROUGHNESS

This paper uses fractal modeling techniques to effectively depict the roughness characteristics of the rails examined in this study by means of both the structure function and the Weierstrass–Mandelbrot function to capture the intricate nature of the rail roughness. Experimental analysis of the roughness of the railway rails from Faurei Railway Testing Center (RTC Faurei) in Romania proves that the roughness height exhibits distinct mathematical fractal characteristics. The study evaluated and compared 41 classical statistical parameters derived from roughness measurements with simulated fractal parameters. The classical roughness parameters, including the Autocorrelation function, Amplitude Density Function, as well as Bearing Area Curves, and rail acoustic roughness, were determined from a profile obtained using the Weierstrass function and compared to the measured ones, revealing a noticeable congruence between the generated charts. The research findings indicate a strong convergence between experimental measurements and simulated profiles, with most parameters falling within a 10% relative error range. This aspect highlights the approach of fractal potent in assessing rail roughness behavior. Consequently, the simulated parameters could potentially analyze rail roughness quality to maintain and track grinding and mitigate the rolling noise.

A Novel Normal Contact Stiffness Model of Bi-Fractal Surface Joints

The contact stiffness of the mechanical joint usually becomes the weakest part of the stiffness for the whole machinery equipment, which is one of the important parameters affecting the dynamic characteristics of the engineering machinery. Based on the three-dimensional Weierstrass–Mandelbrot (WM) function, the novel normal contact stiffness model of the joint with the bi-fractal surface is proposed, which comprehensively considers the effects of elastoplastic deformation of asperity and friction factor. The effect of various parameters (fractal dimension, scaling parameter, material parameter, friction factor) on the normal contact stiffness of the joint is analyzed by numerical simulation. The normal contact stiffness of the joint increases with an increase in the fractal dimension, normal load, and material properties and decreases with an increase in the scaling parameter. Meanwhile, the fractal parameters of the equivalent rough surface of the joint are calculated by the structural function method. The experimental results show that when the load is between 14 and 38 N∙m, the error of the model is within 20%. The normal contact stiffness model of the bi-fractal surface joint can provide a theoretical basis for the analysis of the dynamic characteristics of the whole machine at the design stage.

Influence of 3D joint roughness on fracture behaviours of rock mass subjected to compression

Jointed rock mass with internal rough structural planes have a significant influence on the mechanical behaviour and damage mode of the rock mass. Based on the Weierstrass-Mandelbrot (W-M) fractal function, a numerical model containing a single rough fracture was established. Multi-parameters sensitivity analysis was carried out to determine the key parameters affecting the mechanical properties. By conducting experimental research on the single fracture inclination, fractal dimension, and surface precision of numerical model uniaxial compression, the influence laws of different parameters and the model cracking process were revealed. The results show that the cracks go through the process of sprouting from the single fracture and expanding to both sides, and the expansion of shear cracks in the structural plane reflects the roughness pattern of the structural plane. The higher the roughness was, the lower the peak uniaxial compression strength of the model was. The change of the parameters influenced the macroscopic cracking morphology of the model, and the model finally showed the destruction along the diagonal, the V-shaped destruction, the X-like destruction, and the X-type destruction. The results have an outstanding reference value for further researches on the application of three-dimensional rough discrete fracture networks.

Prediction of formation abrasiveness under the action of a PDC bit based on the fractal dimension of a rock surface

During rock drilling, a drill bit will wear as it breaks the rock. However, there is no uniform grading standard for rock abrasiveness. To solve this problem, the wear mechanisms of a polycrystalline diamond compact (PDC) bit and the formation it is drilling into are analyzed in depth, and an abrasiveness evaluation method based on the fractal dimension of the rock surface topography is established. Initially, a three-dimensional digital model is generated from a scanning electron microscope image of the rock after drilling; next, an evaluation of the irregularities on the rock surface is performed using an adapted Weierstrass–Mandelbrot (W-M) function to ascertain the fractal dimensionality. Then, the microcontact characteristics of the contact surface between the formation and the PDC bit are analyzed, and the distribution of the microconvex contact points of the two-body friction pair in a region is obtained. Because the sliding friction between the drill bit and the rock produces a large amount of heat, according to the contact area formula of the friction surface and heat conduction theory, the temperature rise and overall temperature distribution of the formation and PDC bit under the condition of sliding friction are revealed, and the real contact area between the formation and the drill bit within a certain temperature range is obtained. Finally, the evaluation index of rock abrasiveness under sliding conditions is established by adopting the wear weight loss of the rock cutting tool per unit volume as the index of rock abrasiveness, and the model is verified by a microdrilling experiment. The research in this paper is highly important for improving the rock-breaking efficiency and bit service life during drilling.

Exploration of heat transfer and entropy generation in an irregular wall cavity filled with Al2O3-water nanofluid

This study investigates 2-D natural convection heat transfer within an irregular wall cavity filled with nanofluid, employing a computational approach. The rough profile of the cavity surface is generated using the Weierstrass-Mandelbrot (W-M) function, and various parameters such as Rayleigh number ( 10 3 ≤ Ra ≤ 10 6 ), Knudsen number ( 0 ≤ Kn ≤ 1.5 ), solid volume fraction ( 0 ≤ ϕ ≤ 0.15 ), and amplitude of the irregular rough wall ( 0 ≤ A 1 ≤ 0.1 ) are varied. The nanofluid consists of water as the base fluid and aluminum oxide (Al2O3) nanoparticles. Heat transfer characteristics are analyzed using the Nusselt number (Nu), while entropy generation is quantified to optimize thermal system efficiency. Grid sensitivity analysis and validation against existing literature are conducted to ensure the accuracy of the numerical methodology. Notably, the study demonstrates that increasing the slip velocity at the cavity wall enhances convection within the cavity, leading to a substantial increase in Nu. At Ra = 106, the Nu increases by 53.52% when Kn changes from 0 to 1. Furthermore, the study highlights the significant influence of nanoparticle volume fraction on Nu, showing that as the volume fraction increases, Nu also increases, indicating improved convective heat transfer performance. The Nu for a cavity with an amplitude of 0.1 is roughly 2.5 times higher compared to a smooth-walled cavity at Ra = 106. Increased surface irregularity enhanced turbulence intensity and mixing, leading to stronger convective currents. The novelty of this study lies in examining natural convection in irregular shaped cavities with wall-slip conditions, which are common in practical applications such as solar collectors, microelectronic devices, energy storage units, industrial furnaces, etc. This study offers valuable insights into the thermal behavior of rough enclosures filled with nanofluids, which will facilitate the optimization of heat transfer systems to minimize energy losses.

A fractal model of rough surfaces based on ellipsoidal asperities

PurposeThis study aims to use fractal theory to investigate the contact mechanics of two bidirectional anisotropic surfaces, taking into account the friction coefficient of the contact interface. This study introduces a model that centers on normal contact load and contact stiffness, providing an extensive framework to elucidate the interactions between these surfaces.Design/methodology/approachThe research adopts the Weierstrass–Mandelbrot (W-M) function for simulating bidirectional surface profiles. Through the application of elastic-plastic contact theory, it evaluates the contact area and load of a singular asperity across elasticity, elastoplasticity and plasticity phases. The contact load and stiffness of the rough surface are determined using a refined asperity density distribution function, and these findings are juxtaposed with extant models to validate their precision and rationality.FindingsThe study delineates the influence of fractal dimension (D), surface roughness (G), ellipse eccentricity (e) and friction coefficient (µ) on the contact area, load and stiffness. It reveals that the contact area enlarges with the fractal dimension (D) yet diminishes with increased eccentricity (e), roughness (G) and friction coefficient (µ). These elements considerably affect the contact load and stiffness, underscoring their significance in comprehending surface interactions.Originality/valueThis study applies fractal theory to analyze the contact mechanics of bidirectional anisotropic surfaces, considering the geometry and mechanics of ellipsoidal asperities on rough surfaces to develop a contact mechanics model. This model clarifies the deformation of an asperity in normal contact, presenting a more rational alternative to current models.

The effect of pore size distribution on the fractal evaporative interface in porous media

This article aims to comprehensively analyze the heat and mass transfer and fractal evaporative interface in porous media with different pore size distribution through experiments and simulations of water. Weierstrass-Mandelbrot(W-M) function is used to establish a fractal evaporative interface based on fractal dimension and scaling parameter, which are determined by parameters of pore. The evaporative interface at the pore level is observed in the experiment with aluminum and nickel porous media, which shows reliability of model. The simulation results reveal that when the mass flow rate enlarges, the average Nusselt number, RMS and STD gradually grow, and the average temperature, vapor volume fraction, and velocity reduce, which indicates better thermal protection. As the initial temperature of the bottom improves, the trends in these variables are opposite except for STD, which means more irregular of evaporative interface. These findings have potential applications in transpiration cooling with high temperature in aerospace field and technologies requiring a stable evaporative interface, such as seawater desalination.

A permeability model for the fractal tree-like fracture network with self-affine surface roughness in shale gas reservoirs

The complex natural fracture network with self-affine rough surface and branching characteristics significantly impacts the gas transport in shale gas reservoirs. However, its effects on the permeability have not been studied so far. This study proposes an analytical permeability model for the fractal tree-like fracture network with self-affine surface roughness and branching characteristics. Firstly, the self-affine rough profiles of fracture surface are generated at different fractal dimensions by the Weierstrass–Mandelbrot function and a rough fractal tree-like fracture network is constructed with these surface profiles and branching characteristics. Then, an analytical permeability model is proposed to consider the effects of fracture surface roughness and tree-like branching characteristics on gas flow. This analytical model is verified by numerical simulations. Finally, the velocity distribution of the fracture network and the sensitivity of its structure parameters are analyzed. It is found that eddy flow is more easily formed on rougher fracture surfaces with larger fractal dimension when their fracture aperture is at millimeter scale. The eddy flow disappears when the fracture aperture is at micron scale. Bigger gas flow resistance and more energy loss are observed for smaller fracture aperture and rougher fracture surface. The gas velocity in rough fractures decreases by 60% at micron scale, but decreases by 50% at millimeter scale. Gas flow resistance also increases with the increase of branch angle, branch level and length ratio, but decreases with aperture ratio. As a result, permeability decreases with fractal dimension, branch angle, branch level and length ratio, but increases with aperture ratio.

Computational investigation of aspect ratio and inclination effects on natural convection heat transfer in nanofluid-filled rough wall cavities

Numerical analysis is performed to deduce the effect of aspect ratio (AR) and inclination of a rough-walled cavity on heat transfer, fluid flow, and entropy generation due to natural convection for Rayleigh number (Ra = 105). The cavity is filled with Al2O3-water nanofluid of 0.05 solid volume fraction. The natural convection process is caused by the temperature difference between the left hot wall and the right cold wall of the cavity. The walls are subjected to slip boundary conditions and the slip velocity is characterized by the Knudsen number (Kn = 0.5). The profile of the rough surface is generated by Weierstrass-Mandelbrot (W-M) function. Numerical simulations have been performed using Finite Element based software COMSOL Multiphysics 6.0. The AR is varied in the range of 0.5 to 4, and the inclination of the cavity is kept at 0°, 30°, and 60°. The numerical results are exhibited in terms of isotherms, streamlines, entropy generation due to heat transfer, and fluid friction irreversibilities. It is observed that the slip velocity on the walls has a strong influence on the fluid flow. Entropy generation due to fluid friction is found to increase with the rise in AR up to 2 and then it starts declining. The entropy generation due to fluid friction irreversibility at the center of the cavity is minimal, so the Bejan number is close to unity. Cavity with an inclination of 30° gives the maximum Nusselt number at any value of AR. Notably, the total entropy generation is observed to reach its maximum at α = 60° and minimum at α = 30°, offering comprehensive insights into the interplay of geometric and flow parameters within the cavity.

PREFACE — SPECIAL ISSUE ON FRACTALS AND LOCAL FRACTIONAL CALCULUS: RECENT ADVANCES AND FUTURE CHALLENGES

Fractal geometry plays an important role in the description of the characteristics of nature. Local fractional calculus, a new branch of mathematics, is used to handle the non-differentiable problems in mathematical physics and engineering sciences. The local fractional inequalities, local fractional ODEs and local fractional PDEs via local fractional calculus are studied. Fractional calculus is also considered to express the fractal behaviors of the functions, which have fractal dimensions. The interesting problems from fractional calculus and fractals are reported. With the scaling law, the scaling-law vector calculus via scaling-law calculus is suggested in detail. Some special functions related to the classical, fractional, and power-law calculus are also presented to express the Kohlrausch–Williams–Watts function, Mittag-Leffler function and Weierstrass–Mandelbrot function. They have a relation to the ODEs, PDEs, fractional ODEs and fractional PDEs in real-world problems. Theory of the scaling-law series via Kohlrausch–Williams–Watts function is suggested to handle real-world problems. The hypothesis for the tempered Xi function is proposed as the Fractals Challenge, which is a new challenge in the field of mathematics. The typical applications of fractal geometry are proposed in real-world problems.

Influence of manufacturing surface topography on torque and load bearing capacity of hydro-viscous drive clutch under mixed lubrication stage

PurposeHydro-viscous drive (HVD) clutches are widely used in equipment requiring soft start, such as fans and pumps, to transmit torque and adjust speed by changing the gap distance between friction pairs. This paper aims to propose a novel two-parameter evaluation method for HVD during the mixed lubrication stage. The objective is to develop an effective model that establishes the relationship between these parameters and the actual surface topography.Design/methodology/approachIn the presented methods, the fractal features of the real manufacturing surface are calculated based on the power spectrum function by the ultra-depth three-dimensional microscope. After that, the hybrid friction model of the friction plate is established based on mixed elasto-hydrodynamic lubrication theory, boundary friction model and fractal theory. Then the torque and load bearing characteristics of the clutch are obtained, and the influences of the surface fractal features are investigated and discussed. Finally, the Weierstrass–Mandelbrot function is adopted for the surface topography characterization and evaluation.FindingsThe results indicate that the proposed method exhibits good accuracy, while the speed difference between the friction pair exceeds 2,500 rpm. It is concluded that this paper proposed a way to evaluate the torque and loading capacity of HVD considering the real manufacturing surface topography and is helpful for surface optimization.Originality/valueThe originality and value of this study lie in its development of a novel torque and load bearing capacity evaluation method for HVD in mixed lubrication stage, considering manufacturing surface topography and describing the real manufacturing surface.

NEW SPECIAL FUNCTIONS APPLIED TO REPRESENT THE WEIERSTRASS–MANDELBROT FUNCTION

This work is devoted to the subtrigonometric and subhyperbolic functions in terms of theWiman class for the first time. The conjectures for the subsine and subcosine functions are considered in detail. The Weierstrass–Mandelbrot function is represented as the hyperbolic subsine, and hyperbolic subcosine functions to get new results for the nondifferentiable functions.

Development of an improved three-dimensional rough discrete fracture network model: Method and application

Structure plane is one of the important factors affecting the stability and failure mode of rock mass engineering. Rock mass structure characterization is the basic work of rock mechanics research and the important content of numerical simulation. A new 3-dimensional rough discrete fracture network (RDFN3D) model and its modeling method based on the Weierstrass-Mandelbrot (W-M) function were presented in this paper. The RDFN3D model, which improves and unifies the modelling methods for the complex structural planes, has been realized. The influence of fractal dimension, amplitude, and surface precision on the modeling parameters of RDFN3D was discussed. The reasonable W-M parameters suitable for the roughness coefficient of JRC were proposed, and the relationship between the mathematical model and the joint characterization was established. The RDFN3D together with the smooth 3-dimensional discrete fracture network (DFN3D) models were successfully exported to the drawing exchange format, which will provide a wide application in numerous numerical simulation codes including both the continuous and discontinuous methods. The numerical models were discussed using the COMSOL Multiphysics code and the 3-dimensional particle flow code, respectively. The reliability of the RDFN3D model was preliminarily discussed and analyzed. The roughness and spatial connectivity of the fracture networks have a dominant effect on the fluid flow patterns. The research results can provide a new geological model and analysis model for numerical simulation and engineering analysis of jointed rock mass.

Simulation of electrical contact wear on the rough surfaces of ultra-high-voltage transmission line fittings

A novel electrical contact wear simulation method was proposed to characterize the electrical contact wear of fittings in ultra-high-voltage transmission lines by combining line fittings with the Archard wear model and oxidation loss theory. In this method, a three-dimensional (3D) rough body was generated by using the Weierstrass–Mandelbrot fractal function to simulate the contact surface. An electrical contact wear subroutine was developed, and the wear state was updated using arbitrary Lagrangian–Eulerian adaptive grid technology. Finally, finite element software was used to perform thermal stress wear coupled analysis. The results show that the wear volume, wear depth and friction temperature obtained by the rough electric contact model were 2.71 times, 4.21 times and 2.18 times of the common ideal plane model, respectively. In the rough model, the wear depth of the nodes initially accelerated, subsequently slowed down, and again accelerated with time. The friction high temperature region was distributed in a point pattern, and the temperature difference between the contact region and the non-contact region became obvious.

Stiffness analysis of nutation face gear considering roughness

PurposeThe purpose of this paper is to calculate the meshing stiffness of nutation face gear considering the roughness, establish the calculation method of time-varying meshing stiffness of rough tooth surface and analyze the influence of roughness, load and other factors on the meshing stiffness of tooth surface.Design/methodology/approachThe Weierstrass–Mandelbrot (W-M) function in the Majumdar–Bhushan model is used to characterize the rough contact line of the tooth surface, the normal height and radius of the micro convex body are calculated and the contact flexibility of the contact point of the tooth surface is obtained. The contact flexibility and the bending shear deformation flexibility obtained previously are substituted into the improved deformation compatibility equation for iterative calculation, and the time-varying meshing stiffness of the nutation face gear considering the roughness is obtained.FindingsCompared with ABAQUS finite element simulation results, it is found that the meshing stiffness curve of rough tooth surface is more gentle than that of smooth tooth surface, the meshing stiffness value is smaller and the meshing stiffness change is smaller at the position where the number of gear teeth coincide changes.Originality/valueIn the process of calculating contact deformation, the fractal theory W-M function is used to characterize the contact line of the rough nutation face gear, and the deformation coordination condition considering roughness is improved. Therefore, the method of time-varying meshing stiffness considering roughness can obtain more accurate results, which provides theory and data for the subsequent dynamics analysis of the nutation face gear transmission.

A statistics view of contact pressure distribution for normal contact of fractal surfaces

Due to inherent nonlinear stiffness and damping characteristics, dry friction interfaces have a significant impact on the dynamics of jointed structures. When two surfaces are brought into purely normal contact, contact pressure distribution is of high concern. This work focused on the statistics of the contact pressure distribution between a rough surface and a smooth rigid plane, which provides new insight into the interface contact behaviour. First, the fractal rough surface is generated using the Weierstrass–Mandelbrot function with measured roughness parameters. With meshed rough surfaces, an elastic–plastic finite element contact analysis is performed to determine the contact pressure distribution. Then, the results of contact pressure are statistically analysed. The effects of roughness and contact load on contact area, contact stiffness and mean contact pressure are thoroughly investigated. The probability distribution of contact pressure is determined by fitting a continuous function using a twofold Weibull mixture model. The proposed probability distribution function is found to be capable of describing contact pressure. The contact pressure distribution is affected by the surface fractal characteristics and evolves with the contact load.

Atomistic understanding of rough surface on the interfacial friction behavior during the chemical mechanical polishing process of diamond

The roughness of the contact surface exerts a vital role in rubbing. It is still a significant challenge to understand the microscopic contact of the rough surface at the atomic level. Herein, the rough surface with a special root mean square (RMS) value is constructed by multivariate Weierstrass–Mandelbrot (W–M) function and the rubbing process during that the chemical mechanical polishing (CMP) process of diamond is mimicked utilizing the reactive force field molecular dynamics (ReaxFF MD) simulation. It is found that the contact area A/A0 is positively related with the load, and the friction force F depends on the number of interfacial bridge bonds. Increasing the surface roughness will increase the friction force and friction coefficient. The model with low roughness and high lubrication has less friction force, and the presence of polishing liquid molecules can decrease the friction force and friction coefficient. The RMS value and the degree of damage show a functional relationship with the applied load and lubrication, i.e., the RMS value decreases more under larger load and higher lubrication, and the diamond substrate occurs severer damage under larger load and lower lubrication. This work will generate fresh insight into the understanding of the microscopic contact of the rough surface at the atomic level.

A two-phase density-based solver for simulating wet steam flows with non-equilibrium condensation. II. Effects of 3D surface topography through nozzles

In Paper I, we developed a new two-phase flow solver and assessed its reliability and effectiveness through a series of numerical simulations. In Paper II, we utilized this solver to study the steam flow through nozzles characterized by three-dimensional surface topography. To generate random rough surfaces, we employed a multivariate Weierstrass–Mandelbrot function. Our numerical simulation results exhibited good agreement with the reported experimental data for pressure distribution. Moreover, we examined the changes in pressure, temperature, humidity, Mach number, average Nusselt number, and thrust as the wall surface morphology varied. We observed that the pressure field was prone to fluctuations, the boundary layer structure thickened and changed, the degree of condensation decreased with reduced humidity, three-dimensional Mach waves occurred, and the averaged Nusselt number decreased while thrust deviation increased. These findings contribute to a better understanding of the effects of fractal dimensions and condensation on random roughness in steam flow through nozzles and highlight the significance of incorporating surface topography into numerical simulations to improve accuracy and predictability in engineering applications.

Analysis and Experimental Verification of the Sealing Performance of PEM Fuel Cell Based on Fractal Theory

Aiming to accurately predict the leakage rate of the sealing interface, this work proposes a two-dimensional finite element model of a proton exchange membrane fuel cell, which includes the microscopic surface morphology and the asperity contact process of the components. First of all, we constructed the surface morphology of the seal by the two-dimensional W-M (Weierstrass–Mandelbrot) fractal function and explored the influence of fractal dimension (D) and scale parameter (G) on the surface profile. Furthermore, the finite element method and Poiseuille fluid theory were adopted to obtain the deformation variables of the asperity under different clamping pressures and leakage rates. Moreover, we quantitatively analyzed the impact of surface roughness on the clamping pressure and leakage rate. It was found that both the surface amplitude and surface roughness are positively correlated with G and negatively correlated with D. Surface morphology is proportional to D but has no relationship with G. Additionally, the deformation asperity decreases exponentially with growing clamping pressure, and the leakage rate is consistent with the experimental values at a clamping pressure of 0.54 MPa. With the same leakage rate, when the seal surface roughness value is less than 1 μm, a doubled roughness value leads to an increase of 31% in the clamping pressure. In contrast, when the surface roughness of the seal is greater than 1 μm, a doubled roughness value induces an increase of 50% in the corresponding clamping pressure.