Weierstrass-Mandelbrot Function Research Articles

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.

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.

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.

Simulation and modeling of grinding surface topography based on fractional derivatives

The grinding surface’s 3-D topography has self-affinity. To more accurately simulate this characteristic, this paper proposes for the first time to construct a grinding surface simulation model based on fractional derivatives, including: utilizing the Weierstrass-Mandelbrot function to simulate the point cloud of the grinding wheel surface; further deriving the trajectory equation based on the He’s fractal derivative; the semi-inverse method is employed to construct the fractional-order variational principle of the vibration equation. Experiments show that the simulation model designed in this paper can predict the surface height more accurately, and is better than the traditional model in the calculation of roughness parameters Sa and Sq. The model is helpful to thoroughly understand the formation mechanism of grinding surface topography, and provides simulation data guidance for machine vision inspection of grinding surface in micron scale.

Effect of surface roughness on lubrication performance of water-lubricated bearing for energy recovery turbocharger

In this study, the lubrication performance of the water-lubricated bearing of the energy recovery turbocharger was studied considering the surface roughness. By simulating the rough surface of the bearing with the Weierstrass-Mandelbrot function and combining the water film thickness equation, Reynolds average equation, Greenwood-Tripp contact model and motion equation, the theoretical model of water-lubricated bearing is established and its solution is developed. The variational rules for the hydrodynamic pressure and force, asperity contact pressure and force, and other performance parameters of the water lubricated bearings during start-up and stable operation were obtained by solving this model, and the effect of different surface roughness on the start-up performance and steady operation lubrication performance is studied. The results show that the hydrodynamic force increases rapidly and the asperity contact force sharp decreases during the early start-up stage, and the smaller the surface roughness, the faster the hydrodynamic increase and asperity contact force decrease. The thickness distribution and pressure distribution of water film with different surface roughness have slight differences in stable operation. Therefore, the surface roughness has an important effect on the lubrication performance of the water lubricated bearing in the start-up stage, but has little effect on the stable operation.

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.

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.

Description of multi-periodic signals generated by complex systems: NOCFASS - New possibilities of the Fourier analysis

<p style='text-indent:20px;'>Here, we show how to extend the possibilities of the conventional F-analysis and adapt it for quantitative description of multi-periodic signals recorded from different complex systems. The basic idea lies in filtration property of the Dirichlet function that allows finding the leading frequencies (having the predominant amplitudes) and the shortcut frequency band allows to fit the initial random signal with high accuracy (with the value of the relative error less than 5%). This modification defined as NOCFASS-approach (Non-Orthogonal Combined Fourier Analysis of the Smoothed Signals) can be applied to a wide class of different signals having multi-periodic structure. We want to underline here that the shortcut frequency dispersion has linear dependence <inline-formula><tex-math id="M1">\begin{document}$ \Omega_{k} = c.k+d $\end{document}</tex-math></inline-formula> that differs from the conventional dispersion accepted in the conventional Fourier transformation <inline-formula><tex-math id="M2">\begin{document}$ \omega(k) = \frac{2\pi k}{T} $\end{document}</tex-math></inline-formula>. (<i>T</i> is a period of the initial signal). With the help of integration procedure one can extract a low-frequency trend from trendless sequences that allows to applying the NOCFASS approach for calculation of the desired amplitude-frequency response (AFR) from different "noisy" random sequences. In order to underline the multi-periodic structure of random signals under analysis we consider two nontrivial examples. (a) The peculiarities of the AFR associated with Weierstrass-Mandelbrot function. (b) The random behavior of the voltammograms (VAGs) background measured for an electrochemical cell with one active electrode. We do suppose that the proposed NOCFASS-approach having new attractive properties as the simplicity of realization, agility to the problem formulated will find a wide propagation in the modern signal processing area.</p>

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.

A rough surface electrical contact model considering the interaction between asperities

A rough surface electrical contact model considering the asperity interaction is developed to study the electrical contact behavior. The rough surface is generated by using a two-dimensional Weierstrass-Mandelbrot function. First, the mechanical contact area of the rough surface is determined by the boundary element method in the absence of the electric potential. The electrical contact area is then calculated by a fitting function based on the mechanical contact area. The fitting data is provided by the single-asperity electrical contact model developed in our previous study. By using the electrical contact area of each asperity, the contact force, electrical contact resistance, electric current, and temperature rise on the asperities are calculated. Finally, the electrical contact behavior on the rough surface is thus determined by summarizing the contributions of each contact spot. The finite element simulation of electrical contact of rough surfaces is also conducted to validate the proposed model. The proposed model is used to investigate the effects of the electric potential, fractal surface parameters, and material parameters on the electrical contact behavior. The results illustrate that the developed model can accurately predict the multi-physics behavior of electrical contact.

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.

Study on the Movement of Pulverized Coal Particles in Fractal Fracture Network.

Coalbed methane (CBM) exploitation is a complex multiphase flow process. With the development of CBM extraction technology, it is recognized that the migration of reservoir fluid in the fracture network is affected by the matrix fracture structure, fracture width, and pulverized coal particles plugging, therefore the fluid-particle coupling problem has received increasing attention. In this paper, six groups of fracture models with different fractal dimensions are established using the Weierstrass-Mandelbrot function, and the fluid-particle coupling phenomena in fractures with different roughness and aperture are simulated using an immersed boundary-lattice Boltzmann method (IB-LBM), by comparing the numerical simulation results with the theoretical analysis, it is proved that the IB-LBM numerical method has high accuracy and effectiveness. The effects of fractal dimension, particle size, and particle concentration on the plugging effect were investigated. The results show that the plugging effect on the fracture is enhanced with the increase of the pulverized coal particles content. When the concentration and size of pulverized coal particles are the same, the migration path of pulverized coal particles in complex fractures is more tortuous and more likely to be plugged. It may lead to a shift in the main seepage network of the fracture and the formation of new seepage channels. The research in this paper provides a data basis for the formation and plugging pattern of pulverized coal particles in coal seam fractures, which can provide a basis for the control of pulverized coal particles concentration in CBM drainage.

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.

Morphology differences between fractional Brownian motion and the Weierstrass-Mandelbrot function and corresponding Hurst evaluation

Fractional Brownian motion (FBM) and the Weierstrass-Mandelbrot (W-M) function are two important methods for constructing self-affine objects. Accurately characterizing their features, such as the morphology and fractal geometry, is fundamental for their follow-up applications. However, due to the differences between self-similar and self-affine properties, the fractal dimension evaluation is difficult and sometimes unconvincing. In addition, the sampling length and diversity of calculation methods both lead to the non-uniqueness of the fractal dimension. This study compared the morphology differences between FBM and the W-M function and then analyzed the effects of each parameter on their morphology. Comparisons indicated that FBM has fewer control parameters than the W-M function. Finally, from basic fractal geometric properties, we derived the relationship between the measurement scale r and profile lengths L(r) for self-affine profiles and found that it is a complicated form rather than L(r)sim c r^{H-1}, where H is the Hurst exponent. Meanwhile, the equivalent vertical height v(r) has an intuitive and clear power law relationship to the measurement scale v(r)sim sigma r^H, which provides a method to estimate the Hurst exponent.

A novel fractal contact model based on size distribution law

This study develops a new fractal contact model to evaluate contact status on rough surfaces. Specifically, the three-dimensional Weierstrass-Mandelbrot function is used to characterize fractal surfaces, in which the asperity level range is determined by the power spectral density of surface. A novel size distribution law of truncation regions has been proposed based on rough surface truncation in our previous study to overcome the limitation of the law proposed by Mandelbrot. By using the size distribution law, a multi-asperity contact model is established to calculate the contact force and real contact area. The evolution of the asperity level range is considered during the compression process. Furthermore, the elastic or plastic deformation status of each asperity in the level range is determined. In the implementation of the proposed theoretical model, the numerical truncation of fractal surfaces is conducted to obtain the effective asperity level range. Finite element simulation of fractal surface contact and compression experiments of copper specimens are conducted to validate the fractal contact model. The theoretical results of the contact force and total contact area agree well with the simulation and experimental results. The effects of the fractal parameters on the contact behavior are further investigated. The results indicate that the proposed fractal contact model can provide an accurate prediction of rough contact behaviors.

Study on seepage characteristics of cross fractures in adjacent and convective flow modes

The distribution of fractures in the underground rock mass is complex, so it is of great significance to study the parameter characteristics of fracture seepage for geothermal exploitation, land subsidence, and other fields. Based on the Weierstrass-Mandelbrot function, several X-shaped fracture models with different roughness are established, and the fracture seepage law under different roughness and inlet pressure is studied. At the same time, the center (cross-region) of the X-shaped fracture is selected as the research object, and the seepage difference characteristics of the cross-region under the two inlet modes of adjacent flow and convective flow are compared and analyzed. The results show that the outlet velocity of fracture fluid is inversely proportional to the fractal dimension and is directly proportional to the inlet pressure.

Fractal characteristics of surface roughness and their effects on laser shock waves

The Weierstrass–Mandelbrot function model was used to simulate the rough topography of a target surface, and the fractal theory was used to study the fractal characteristics of the target surface topography, which is the self-similarity of surface topography. The 3D fluid dynamics governing equations of the laser-supported detonation plasma flow field in Cartesian coordinates are given and discretized by a high-order difference scheme. The equation of state of plasma under different conditions is considered, and a new method of flux splitting is introduced to carry out numerical simulations. The numerical results of plasma pressure, density, temperature, and velocity at different times and different surface roughnesses are obtained, which show the evolvement of the ablative plasma. The simulated results show that a rough surface can obviously affect the propagation velocity of the detonation wave near the wall. In addition, the waveform is also destroyed, forming a broken waveform.

Characterization of interlayer mechanical performance of double-layer porous asphalt compacted by three methods: Simulations and observations

The compaction method directly governs the interlayer patterns, which significantly influence the performance of asphalt pavement. The objective of this research aimed to investigate the interlayer mechanical performance of double-layer porous asphalt (DLPA) samples compacted by three methods: hot-on-cold (HoC), hot-on-warm (HoW), and hot-on-hot (HoH). The interlayer patterns at the upper-lower layer interface of DLPA samples were measured by the thin aluminum sheet and oil-saturated paper and characterized by the modified Weierstrass-Mandelbrot (W-M) function with two fractal parameters. The virtual direct shear test and pull-off test were conducted on the simulated DLPA samples incorporating interlayer patterns. The interlayer mechanical strengths from simulation were verified by laboratory experiments. The results showed that the fractal dimension and scale parameter in the modified W-M function characterized the interlayer patterns. The DLPA samples using HoH compaction exhibited the highest shear and tensile strengths that were inversely proportional to the fractal dimension, owing to the interlocking interlayer pattern generated. The HoH method was recommended to compact the DLPA pavement if the double-layer asphalt paver was available. The findings are useful to improve construction quality of the DLPA pavement.

On the size distribution of truncation areas for fractal surfaces

The size distribution of contact spot areas is critical for the evaluation of contact behavior of rough surfaces. A size distribution law proposed by Mandelbrot has been used widely in multi-asperity fractal contact models to predict the rough contact behavior. However, the Mandelbrot law has a limitation on the representation of the truncation area distribution for rough surfaces. Therefore, the present study develops a new size distribution law of the truncation areas for fractal surfaces. Specifically, the three-dimensional Weierstrass-Mandelbrot (WM) function is used to characterize rough surfaces. A novel method is proposed to determine the fractal parameters of the WM function. Numerical investigation of rough surface truncation is conducted to obtain size distribution information of the truncation regions for various rough surfaces. Based on the numerical results, the new size distribution law of the truncation areas is proposed and validated by fractal surface and realistic rough surface. It is found that the proposed law can accurately reproduce the size distribution of the truncation areas for rough surfaces with a wide range of fractal parameters.