Weierstrass-Mandelbrot Fractal Function Research Articles

Investigation on the heat extraction performance considering a rough fracture network in heterogeneous reservoirs

In this paper, a new three-dimensional numerical model of geothermal extraction is established. The model utilizes spatial frequency to describe the heterogeneity of multiple physical parameters of the reservoir, and rough discrete fractures are constructed based on the Weierstrass-Mandelbrot fractal function. Finally, the study examines the impact of geological-engineering parameters on the heat extraction performance. The results show that after the fracture number increases to 7–9, the influence of fracture number on the low-temperature area gradually diminishes. With the increase in fracture width, the thermal breakthrough phenomenon becomes more pronounced, leading to a reduction in the heat extraction performance. The increase in fracture roughness coefficient can delay thermal breakthrough and improve heat extraction performance. When the fracture roughness coefficient increases from 1.6 to 11.6, the effective electric power increases by 19.44 %. The increase in injection rate intensifies thermal breakthrough, but it also enhances the heat extraction performance. When the injection rate increases from 60 l/s to 160 l/s, the effective electric power increases by 55.7 %. Increasing the injection temperature can delay thermal breakthrough and reduce the heat extraction performance. Abnormal high temperature reservoir (>0.03 K/m) has better heat extraction performance.

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.

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.

Fractal contact analysis for transversely isotropic piezoelectric materials: Theoretical and numerical predictions

This paper aims to investigate the micro frictionless and non-adhesive rough contact for transversely isotropic piezoelectric materials. The surface topography is formulated by the two-variable Weierstrass–Mandelbrot fractal function. The system of contact is subjected to light generalized loads. By virtue of the Hertzian solution of transversely isotropic piezoelectric materials and the size distribution function, the generalized loads-area relation is developed. The smallest truncated contact area in the theoretical model is determined by comparing the theoretical and simulation results. The present numerical solutions via the finite element method are validated in comparison to the numerical results in literature. The influence of various fractal parameters on the contact behavior of transversely isotropic piezoelectric solids are studied.

Investigation Regarding the Influence of Contact Condition on the Thermal Contact Resistance Between Copper and Indium

The thermal contact resistance caused by the rough contact interface accounts for a large proportion of the total thermal resistance of microelectronic devices. In this article, the influencing factors of the contact thermal resistance between copper-indium interface in electronic packaging has been investigated using numerical calculation methods. The Weierstrass-Mandelbrot fractal function was used to establish the copper-indium contact surface morphology with different surface characteristics. The temperature field of the copper-indium contact model with different surface roughness was calculated by the finite element method, and the influence of pressure on the contact thermal resistance was analyzed. The heat transfer behavior of the interface gap medium was simulated by setting the thermal contact conductivity that changed with the contact gap distance. The influence of air and grease in the thermal resistance of copper-indium contact was investigated. It was indicated that 1) pressure will rapidly reduce the contact thermal resistance, but the cost-effectiveness ratio gradually decreases and 2) influence of air on the contact thermal resistance decreases rapidly as the pressure increases, but grease has a significant effect under various pressure conditions.

Exact Spectral Moments and Differentiability of the Weierstrass-Mandelbrot Fractal Function

Abstract Fractal mathematics using the Weierstrass-Mandelbrot (WM) function has spread to many fields of science and engineering. One of these is the fractal characterization of rough surfaces, which has gained ample acceptance in the area of contact mechanics. That is, a single mathematical expression (the WM function) contains characteristics that mimic the appearance of roughness. Moreover, the “roughness” is “similar” across large dimension scales ranging from macro to nano. The field of contact mechanics is largely divided into two schools of thought: (1) the roughness of real surfaces is essentially random, for which stochastic treatment is appropriate, and (2) surface roughness can be reduced to fractal mathematics using fractal parameters. Under certain mathematical constraints, the WM function is either stochastic or deterministic. The latter has the appeal that it contains no randomness, so fractal mathematics may offer closed-form solutions. Spectral moments of rough surfaces still apply to both approaches, as these represent physical metrology properties of the surface standard deviation, slope, and curvature. In essence, spectral moments provide a means of data reduction so that other physical processes can subsequently be applied. It is well known, for example, that the contact model of rough surfaces, by Greenwood and Williamson (GW), depends on parameters that are direct outcomes of these moments. Despite the vast amount of publications on the WM function dedicated to surfaces, two papers stand out as originators, where the others mostly rework their results. These two papers, however, contain some omissions and approximations that may lead to gross errors in the estimation of the spectral moments. The current work revisits these papers and adds information, but departs in the mathematical treatment to derive exact expressions for the said moments. Moreover, it is said that the WM function is nondifferential. That is also revisited herein, as another approach to derive the spectral moments depends on such derivatives. First, the complete mathematical treatment of the WM function is made, then the spectral moments are derived to yield exact forms, and finally, examples are given where the physical meanings of the approximate and exact moments are discussed and their values are compared. Numerical procedures will be introduced for both, and the effectiveness of the computational effort is discussed. One numerical procedure is particularly effective for any digitized signal, whether that originates from analytical functions (e.g., WM) or real surface measurements.

A fractal model for predicting thermal contact conductance considering elasto-plastic deformation and base thermal resistances

A prediction model of thermal contact conductance is developed. Engineering rough surfaces are characterized by three-dimensional fractal Weierstrass-Mandelbrot fractal function. Three deformation modes, including fully plastic deformation, elasto-plastic deformation and elastic deformation, are considered to analyze the contact mechanism. Fractal surface and three deformation modes are incorporated into the calculation of thermal contact conductance. A comprehensive thermal contact conductance computation model considering both base thermal resistance and constricted thermal resistance is established. The results show that thermal contact conductance increases with the increase of normal contact pressure; the relative contribution of constricted resistance component to base resistance component tends to increase with the increase of normal contact pressure; fractal dimension and fractal roughness both have significant influences on thermal contact conductance.

Fractal modeling of normal contact stiffness for rough surface contact considering the elastic–plastic deformation

In this work, a new formulation of elastic–plastic contact model for the normal contact between rough surfaces is proposed based on the fractal theory. The surface topography is described using the modified one-variable Weierstrass–Mandelbrot fractal function. A new elastoplastic asperity contact model is developed based on the contact mechanics combined with the continuity and smoothness of mean contact pressure and contact load across different deformation regimes from elastic to elastoplastic, and from elastoplastic to fully plastic. The contact stiffness of a single asperity in the three deformation regimes of fully plastic, elastoplastic and elastic is derived, which changes smoothly at transit points between different deformation modes and overcomes the shortcoming of discontinuity derived from previous model. The contact stiffness and contact load of the whole surface in the three deformation regimes are formulated by integrating the micro-asperity contact. The difference between contact stiffness calculated from the plastic–elastoplastic–elastic three contact regimes and plastic–elastic two contact regimes is not obvious for surface with rougher topograph; however, for surface with smoother topograph, the two contact regimes predict a much smaller contact stiffness. The relationship of the normal contact stiffness and the normal contact force follows a power law function for the fractal surface contact considering three deformation regimes. The power exponent is nonlinearly dependent on the fractal dimension, which is different from the linear relationship for the purely elastic contact.

Rarefaction cloaking: Influence of the fractal rough surface in gas slider bearings

For ultra-thin gas lubrication, the surface-to-volume ratio increases dramatically when the flow geometry is scaled down to the micro/nano-meter scale, where surface roughness, albeit small, may play an important role in gas slider bearings. However, the effect of surface roughness on the pressure and load capacity (force) in gas slider bearings has been overlooked. In this paper, on the basis of the generalized Reynolds equation, we investigate the behavior of a gas slider bearing, where the roughness of the slider surface is characterized by the Weierstrass-Mandelbrot fractal function, and the mass flow rates of Couette and Poiseuille flows are obtained by deterministic solutions to the linearized Bhatnager-Gross-Krook equation. Our results show that the surface roughness reduces the local mass flow rate as compared to the smooth channel, but the amount of reduction varies for Couette and Poiseuille flows of different Knudsen numbers. As a consequence, the pressure rise and load capacity in the rough bearing become larger than the ones in the smooth bearing in the slip and early transition flow regimes, e.g., a 6% roughness could lead to an increase of 20% more bearing load capacity. However, this situation is reversed in the free-molecular flow regime, as the ratio of the mass flow rates between Couette and Poiseuille flows is smaller than that in the smooth channel. Interestingly, between the two extremes, we have found a novel “rarefaction cloaking” effect, where the load capacity of a rough bearing equals to that of a smooth bearing at a certain range of Knudsen numbers, as if the roughness does not exist.

Study on the optical performance of thin-film light-emitting diodes using fractal micro-roughness surface model

Although LEDs have been widely studied using optical simulations, there is no optical model considering the effect of micro-roughness surface (MRS) on the optical performance for packaged LEDs. In this work, we employ the finite-difference time-domain method and the direction-sensitive bidirectional scattering distribution function to characterize the optical properties of the MRS upon the n-GaN layer. The MRS is generated by the Weierstrass–Mandelbrot fractal function. Furthermore, thin-film LEDs (TFLEDs), blue TFLED devices, and white TFLED devices considering the MRS are investigated using the ray-tracing (RT) method. The results show that the MRS has different optical properties when the light propagates out and in the n-GaN layer. In turn, the difference in the scattering ability of various MRS causes a significant effect on the optical performance of packaged TFLEDs, including radiant efficacy, luminous efficacy, intensity pattern and spectrum, as well as the correlated color temperature.

Fractal model and Lattice Boltzmann Method for Characterization of Non-Darcy Flow in Rough Fractures

The irregular morphology of single rock fracture significantly influences subsurface fluid flow and gives rise to a complex and unsteady flow state that typically cannot be appropriately described using simple laws. Yet the fluid flow in rough fractures of underground rock is poorly understood. Here we present a numerical method and experimental measurements to probe the effect of fracture roughness on the properties of fluid flow in fractured rock. We develop a series of fracture models with various degrees of roughness characterized by fractal dimensions that are based on the Weierstrass–Mandelbrot fractal function. The Lattice Boltzmann Method (LBM), a discrete numerical algorithm, is employed for characterizing the complex unsteady non-Darcy flow through the single rough fractures and validated by experimental observations under the same conditions. Comparison indicates that the LBM effectively characterizes the unsteady non-Darcy flow in single rough fractures. Our LBM model predicts experimental measurements of unsteady fluid flow through single rough fractures with great satisfactory, but significant deviation is obtained from the conventional cubic law, showing the superiority of LBM models of single rough fractures.

Comparison of two fractal interpolation methods

As a tool for studying complex shapes and structures in nature, fractal theory plays a critical role in revealing the organizational structure of the complex phenomenon. Numerous fractal interpolation methods have been proposed over the past few decades, but they differ substantially in the form features and statistical properties. In this study, we simulated one- and two-dimensional fractal surfaces by using the midpoint displacement method and the Weierstrass–Mandelbrot fractal function method, and observed great differences between the two methods in the statistical characteristics and autocorrelation features. From the aspect of form features, the simulations of the midpoint displacement method showed a relatively flat surface which appears to have peaks with different height as the fractal dimension increases. While the simulations of the Weierstrass–Mandelbrot fractal function method showed a rough surface which appears to have dense and highly similar peaks as the fractal dimension increases. From the aspect of statistical properties, the peak heights from the Weierstrass–Mandelbrot simulations are greater than those of the middle point displacement method with the same fractal dimension, and the variances are approximately two times larger. When the fractal dimension equals to 1.2, 1.4, 1.6, and 1.8, the skewness is positive with the midpoint displacement method and the peaks are all convex, but for the Weierstrass–Mandelbrot fractal function method the skewness is both positive and negative with values fluctuating in the vicinity of zero. The kurtosis is less than one with the midpoint displacement method, and generally less than that of the Weierstrass–Mandelbrot fractal function method. The autocorrelation analysis indicated that the simulation of the midpoint displacement method is not periodic with prominent randomness, which is suitable for simulating aperiodic surface. While the simulation of the Weierstrass–Mandelbrot fractal function method has strong periodicity, which is suitable for simulating periodic surface.

A geometrical–mechanical–thermal predictive model for thermal contact conductance in vacuum environment

In this article, a comprehensive geometrical–mechanical–thermal predictive model is developed for thermal contact conductance between two flat metallic rough surfaces. The rough surface was characterized by Weierstrass–Mandelbrot fractal function. The micro-morphology was measured by laser microscope to identify the fractal parameters that were then applied to mechanical and thermal modeling. A new contact mechanics model was then proposed to calculate the contact parameters, and different contact scales between asperities and three modes of deformation, elastic, elastic–plastic and fully plastic, were taken into account. The normal contact pressure, which should be equal to the exterior load, was formulated as a function of the fractal parameters, the maximum contact area and the material physical properties of the given surface. Based on the contact mechanics model, first a single pair of contacting asperities was proposed and then multi-contacting asperities were combined to get total thermal contact conductance. The influences of contact load, surface roughness, asperity top radius and contact area as well as the temperature on the thermal contact conductance were investigated by using the proposed model. The investigation results showed that thermal contact conductance increases with the contact load and contact area. The larger the surface roughness, the smaller is the thermal contact conductance. Finally, the experiments were conducted to validate the effectiveness of the thermal contact conductance modeling. This geometrical–mechanical–thermal predictive model was compared with the two existing predictive models and a series of experimental data. The results showed good agreement, demonstrating the validity of the model and providing certainty for further study on the heat transfer between contact surfaces.

On the contact stiffness and nonlinear vibration of an elastic body with a rough surface in contact with a rigid flat surface

Abstract The surface topography of engineering rough surfaces has an enormous influence on contact mechanics of the interface and further the dynamics of the contact system. In this paper, the force-deflection characteristic and the nonlinear vibration of a rough surface interacting with a rigid flat surface are studied. A three-dimensional rough surface is constructed using a modified two-variable Weierstrass–Mandelbrot fractal function and the force-deflection is determined to be a positive power-law function. The power has the values larger than unity and increases with a rougher surface topography. Approximation of the force-deflection characteristic is also presented. Accordingly, the natural frequency is determined both exactly and approximately from the numerical calculation of the natural period and using the multiple scales method on the approximate equation, respectively. The primary resonance responses under harmonic excitation are also determined as well as the jump-up and jump-down responses. The nonlinear contact stiffness characteristics and the effect on natural frequency and the frequency responses are illustrated for different rough surface topographies. It is shown that the variation of natural frequency with amplitude and the multi-valued region of frequency responses increase with a rougher surface topography; however, the jump-up and jump-down frequencies both decrease, as well as the peak amplitude.

An experimental investigation on the mechanism of fluid flow through single rough fracture of rock

The structure of fractures in nature rock appears irregular and induces complicated seepage flow behavior. The mechanism and quantitative description of fluid flow through rock fractures is a difficult subject that has been greatly concerned in the fields of geotechnical, mining, geological, and petroleum engineering. In order to probe the mechanism of fluid flow and the effects of rough structures, we conducted a few laboratory tests of fluid flow through single rough fractures, in which the Weierstrass-Mandelbrot fractal function and PMMA material were employed to produce the fracture models with various fractal roughnesses. A high-speed video camera was employed to record the fluid flow through the entire single rough fracture with a constant hydraulic pressure. The properties of fluid flow varying with the fracture roughness and the influences of the rough structure were analyzed. The components of flow resistance of a single rough fracture were discussed. A fractal model was proposed to relate the fluid resistance to the fracture roughness. A fractal equivalent permeability coefficient of a single rough fracture was formulated. This study aims to provide an experimental basis and reference for better understanding and quantitatively relating the fluid flow properties to the structures of rock fractures.

Encriptación óptica empleando llaves Weierstrass-Mandelbrot

This paper presents the generation of encryption keys using the local oscillating properties of the partial sums of Weierstrass-Mandelbrot fractal function. In this way, the security key can be replicated if the parameters used to obtain it are known. Therefore, these parameters can be sent instead of sending the key. This procedure reduces the amount of information to be sent and prevents possible interception of the key. Moreover, the key can not be affected by data loss or pollution. The effectiveness of the Weierstrass-Mandelbrot keys were demonstrated by computer simulation in a 4f optical encryption system and the double random phase encoding technique. These keys allow us to encrypt and to retrieve the information, which is the evidence of their effectiveness for data security. Furthermore, it is shown that if the encrypted data is obtained under an attack, it is not possible to access the information without the security key.

Fractal Quality of Microaccelerations

This paper deals with fractal quality of microaccelerations which occur in the indoor environment of space laboratory. Change in size of space laboratory results in the fact that dynamics of microaccelerations possess the quality corresponding to self-affinity of fractal functions in the task statement considered. We suggest forming of mass inertia characteristics of space laboratory with predetermined microaccelerations level with the help of the microaccelerations model based on the real part of Weierstrass–Mandelbrot fractal function.

Synthesis of fractal geometry and CAGD models for multi-scale topography modelling of functional surfaces

In order to support the functional design and simulation and the final fabrication processes for functional surfaces, it is necessary to obtain a multi-scale modelling approach representing both macro geometry and micro details of the surface in one unified model. Based on the fractal geometry theory, a synthesized model is proposed by mathematically combining Weierstrass-Mandelbrot fractal function in micro space and freeform CAGD model in macro space. Key issues of the synthesis, such as algorithms for fractal interpolation of freeform profiles, and visualization optimization for fractal details, are addressed. A prototype of the integration solution is developed based on the platform of AutoCAD’s Object ARX, and a few multi-scale modelling examples are used as case studies. With the consistent mathematic model, multi-scale surface geometries can be represented precisely. Moreover, the visualization result of the functional surfaces shows that the visualization optimization strategies developed are efficient.

A novel method to identify the scaling region for chaotic time series correlation dimension calculation

To obtain more accurate correlation dimension estimations for chaotic time series, a novel scaling region identification method is developed. First, points that obviously do not belong to the scaling region associated with the whole double logarithm correlation integral curve are removed using the K-means algorithm. Second, a point-slope-error algorithm is developed to recognize a possible scaling region. Third, the K-means algorithm is used again to further remove a small interval of interfering points in the possible scaling region to obtain a more precise scaling region. The correlation dimension of four typical chaotic attractors and five curves generated by the Weierstrass-Mandelbrot fractal function were calculated using the proposed method. These calculated values were very close to the respective theoretical fractal dimensions. Moreover, the effectiveness of our method in identifying the scaling region was compared with existing methods. Results show that our method can distinguish the scaling region objectively, accurately, automatically and quickly, making estimations of the correlation dimension more precise and affording significant improvements in nonlinear analysis.

An analytical model of thermal contact resistance based on the Weierstrass—Mandelbrot fractal function

A fractal model for analysing the thermal contact resistance (TCR) of rough surfaces is presented; it is based on the classical heat conduction theory and fractal geometry for the surface topography description, elastic—plastic deformation of contacting asperities, and size-dependent constriction resistance. Relations for the TCR in terms of contact load are obtained for heat conductive surfaces with known material properties and surface topography. With the real contact area being approximately 1 per cent of the apparent contact area or less, the microcontact area distribution has a dominant influence on the TCR. Useful design guidelines for heat contacts are extracted from the numerical results. The analytical results agree well with previous experiments.