Fractal Surface Research Articles

Evaluating Effects of Travel-resonance Method with Color Soy Source and Clear Water on Substance using Fractal Dimension Analysis of Awareness Outward-form

Soy source has a useful effect on the human body and is the important ingredients in various foods and is an important factor of food engineer that tastes food ingredients. The forming model of the soy source was configured asymmetrical on the fractal surface and mixed with water using travel-resonance method to change a certain amount of viscosity into the shape of the surface. Travel mutational-status technology is structured the resonance status for rainbow dot pattern of the glitter awareness rate (GAR) and inequality awareness rate (IAR) on soy source travel awareness outward-form. Awareness rate condition by soy source travel awareness outward-form is structured with the circulate resonance system. Rainbow dot pattern of the happen mutational-status is to check up to structure of soy source travel value with travel layer location by the circulate-down structure. Concept of awareness rate is check up the reference of glitter rate and inequality rate for mutational-status signal by soy source travel resonance outward-form. Happen mutational-status of the GAR-IAR is presented position value of maximum-minimum on soy source travel-resonance outward-form. Far mutational-status of the Tr-aof-FA-?MAX-MIN presented the travel value with 14.67±7.32 units. Convenient mutational-status of Tr-aof-CO-?MAX-MIN presented the travel value with 4.48±0.46 units. Flank mutational-status of the Tr-aof-FL-?MAX-MIN presented the travel value with 1.29±0.32 units. Vicinage mutational-status of the Tr-aof-VI-?MAX-MIN presented the travel value with 0.23±0.07 units. Circulate resonance will be to inquire into at the ability of soy source travel-resonance outward-form for the restrain degree awareness rate on the GAR-IAR that is stick-out the happen glitter and inequality outward-form by the awareness rate system. Awareness outward-form LED will be to inquire into of outward-form by the inequality signal and to figure a travel data of circulate resonance rate by the circulate awareness system. due to the effect of image resolution.

Sealability Analyses of Premium Connections Characterized by a Surface Fractal Function

With the development of high-pressure, high-temperature wells, the sealability of premium connections is being threatened by heavy loads. For the sake of researching the sealability of premium connections at the microlevel on the sealing surface, a rough profile was characterized according to a fractal function, and the influence of the fractal parameters of contact behavior on the rough surface was analyzed. A full-size premium connection model with a fractal surface FEM was established using ABAQUS software to analyze the Von Mises stress and contact pressure, and the sealability was then analyzed using contact strength theory. It was found that utilizing fractal functions to describe the rough morphology on the sealing surface of premium connections could reveal the contact behavior of the sealing surface more realistically. The premium connection’s fractal FEM was closer to the actual situation, considering the effect of the asperities’ elastic–plastic deformation on the sealability. The fractal dimension D had a greater influence on the contact area and contact pressure than the scale coefficient G. When the fractal dimension D was less than 2.5, the maximum Von Mises stress and maximum contact pressure on the sealing surface were 8.81 × 108 Pa and 1.20 × 109 Pa, respectively, making the surface prone to gluing. The sealability of the premium connection was primarily affected by the axial tension. When the axial tension was 1.2 × 106 N, the sealing surface experienced significant displacement along the axial direction, the effective contact length reduced from 2.72 × 10−3 m to 2.24 × 10−3 m, and the maximum contact pressure reduced from 8.10 × 108 Pa to 6.39 × 108 Pa, which led to a 30% decrease in sealing strength, which may pose a high risk of sealing failure. When the internal pressure reached 1 × 108 Pa, the effective contact length increased from 9.08 × 10−3 m to 1.06 × 10−2 m, the maximum contact pressure increased from 8.67 × 108 Pa to 1.37 × 109 Pa, and the sealing strength increased by 23%.

Analytically decoupling of friction coefficient between mixed lubricated fractal surfaces

Herein, we present a generalized close-formed model of the friction coefficient between fractal surfaces in mixed lubrication. A mathematical transformation from asperity area to oil film thickness is conducted, which changes the fractal method’s traditional basis of the area distribution to establish an analytical fractal model of oil film thickness. A dynamic film thickness model with external load under the determination of the surface roughness is further proposed. Based on the fractal-based oil film thickness, the contact force for each part is solved under the external normal force, and decoupling is achieved presented by contact ratio. Through the contact ratio, the constitutive model of friction coefficient is elucidated. The effects of fractal parameters and important working parameters on friction coefficient are then analyzed, and comparison with experimental data is conducted. This work provides a general analytical expression for friction coefficient in mixed lubrication with explicit form and wide applications.

Surface morphology in high-speed grinding of TMCs fabricated by selective laser melting

TiC-reinforced titanium matrix composites (TMCs) are typically difficult-to-machine materials. It is prospective to combine the advantages of selective laser melting (SLM) and high-speed grinding to manufacture TMCs parts. This study investigates the surface morphology in high-speed grinding of TMCs fabricated by SLM. Grinding experiments with 28.75–232.66 m/s are conducted. Two surface features are found in the ground surface of TMCs, i.e. smearing at low grinding speeds and scratch at high grinding speeds. Surface morphologies are evaluated based on fractal dimension FD and surface roughness Ra. The results show that the correlations between FD and Ra indicate the ground surface formation mechanisms of TMCs. It is found that submicron TiC generated in SLM cooperating with high grinding speed achieves a good surface quality in terms of reduced surface roughness and subsurface damages for TMCs, which could be beneficial to the fabrication of other metal matrix composites.

Dual-band and dual-polarized reflective Fresnel zone plate design based on Fractal shape

In this study, a dual-polarized and dual band reflective Fresnel zone plate with reconfigurable beam is proposed on the basis of fractal frequency selective surface (FSS) unit with nearly 360° phase tunability. Firstly, a new phase distribution calculation strategy based on Fresnel diffraction theory is proposed to improve the performance under certain scenarios like sparse arrays. Then, a novel fractal shape is put forward and applied to the design of the Fresnel zone plate. The introduction of the fractal structure makes the unit cell perform dual band, dual polarization and 309° phase tunability characteristics. Due to the self-symmetry of the unit cell, the proposed fractal Fresnel zone plate (FFZP) is capable of beam steering in ± 45° in both TE and TM incident waves. Besides, the proposed structure shows small performance degradation when it comes to oblique incidence up to 45°, which decreases the focal diameter ratio and profile of the proposed FFZP. The operating bandwidth of the FFZP can reach up to 700 MHz at X and Ku bands. It is applicable in a wide range of RF and microwave settings such as satellite and base station.

Hierarchical structured surfaces enhance the contact angle of the hydrophobic (meta-stable) state.

The relation between wetting properties and geometric parameters of fractal surfaces are widely discussed on the literature and, however, there are still divergences on this topic. Here we propose a simple theoretical model to describe the wetting properties of a droplet of water placed on a hierarchical structured surface and test the predictions of the model and the dependence of the droplet wetting state on the initial conditions using simulation of the 3-spin Potts model. We show that increasing the auto-similarity level of the hierarchy - called n - does not affect considerably the stable wetting state of the droplet but increases its contact angle. Simulations also explicit the existence of metastable states on this type of surfaces and shows that, when n increases, the metastability becomes more pronounced. Finally we show that the fractal dimension of the surface is not a good predictor of the contact angle of the droplet.

A Study of Chaos in Nonlinear Electronic Circuits and Applications in Game Theory

Chaos is a ubiquitous part of daily life. The weather, fractal landscapes, and the double pendulum all embody the notion of chaos. The general concepts and the key characteristics of chaos are explored and manifested in a nonlinear electronic circuit known as the Chua’s circuit. The Chua’s Circuit was built and characterized to illustrate chaos behavior in the form of Lorenz attractors. The circuit chaotic behavior was further demonstrated in a numerical model built in MATLAB. More practical chaos applications in secure communication and game theory are discussed. A case study is presented at the end to apply the chaos concept into the model of the strength of poker hands, which displays properties of a chaotic system and resembles Elementary Cellular Automata (ECA) in the chaos application.

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.

Normal Contact Analysis Between Two Self-affine Fractal Surfaces at the Nanoscale by Molecular Dynamics Simulations

Normal Contact Analysis Between Two Self-affine Fractal Surfaces at the Nanoscale by Molecular Dynamics Simulations

Humic Substances: From Supramolecular Aggregation to Fractal Conformation—Is There Time for a New Paradigm?

Natural organic matter, including humic substances (HS), comprises complex secondary structures with no defined covalent chemical bonds and stabilized by inter- and intra-molecular interactions, such as hydrogen bonding, Van der Waal’s forces, and pi-pi interactions. The latest view describes HS aggregates as a hydrogel-like structure comprised by a hydrophobic core of aromatic residues surrounded by polar and amphiphilic molecules akin a self-assembled soft material. A different view is based on the classification of this material as either mass or surface fractals. The former is intended as made by the clustering of macromolecules generating dendritic networks, while the latter have been modelled in terms of a solvent-impenetrable core surrounded by a layer of lyophilic material. This study reviews the evolution of the increasingly refined models that appeared in the literature, all capable to describing the physicochemical properties of HS. All the models are critically examined and revisited in terms of their ability to provide key information on the structural organization of HS. Understanding how the molecular association pathway influences aggregation of HS also provides a key acknowledgment of their role in the environment.

Design and experimental validation of customised fractal FSS

ABSTRACT The present work deals with the combination of Artificial Neural Network (ANN) and Particle Swarm Optimization (PSO) for the development of a flexible, time-efficient CAD module for design of customised Fractal Frequency Selective Surfaces (FSS). The design task of FSS is approached as an optimization problem and solved using PSO whose fitness function is evaluated using a previously trained ANN to estimate the design parameters of the unit cell of fractal FSS for multiple frequencies as per the user’s demand. The results of the developed approach are cross-verified with experimental results. The developed approach can easily be extended for the design of other multifrequency structures including metasurfaces.

Study on nonlinear dynamic characteristics of gear system with 3D anisotropic rough tooth surface based on fractal theory

Different from the mainstream research, this study fully considers the anisotropic micro-topography of the tooth surface when studying the dynamic characteristics of gears. An anisotropic 3D fractal rough tooth surface model is proposed based on fractal geometry theory. The major advantage of the novel model is that it contains more surface micro details, especially on the tooth width, which are ignored in general studies. The backlash and time-varying meshing stiffness (TVMS) are then revised on the basis of this model. Finally, a 6-degree-of-freedom nonlinear dynamic model considering surface topography is established and the influence of fractal parameters on gear dynamics is investigated. The results show that the fractal parameters change the backlash and TVMS and have an obvious impact on the dynamics and vibration responses of the transmission system. The rough gear system enters chaos earlier and has better stability at high speeds in comparison with the smooth system. As the fractal dimension increases, the dynamic transmission error amplitude gradually decreases and the system tends to stable periodic motion. The characteristic scale coefficient yields a more moderate but opposite effect on the dynamic response of the system compared with the fractal dimension. The finding of this study can provide theoretical guidance for tooth surface machining from the perspective of tooth surface morphology.

Time-Varying Wear Calculation Method for Fractal Rough Surfaces of Friction Pairs

For the wear problem of the real rough surface during sliding friction, based on fractal theory and Hertz contact theory, a 3-D fractal rough surface with random characteristics is constructed, and the relationship between the wear deformation depth of the rough peak and its real contact area during the wear process is derived. Furthermore, considering the peak wear and pit scratch phenomena of rough surfaces in different contact states, the time-varying wear calculation model of the worn surface and the compensation wear calculation model of the unworn surface are established, respectively, and the relationship between the instantaneous wear amount and the dynamic change in the rough surface topography is comprehensively characterized. Combined with image digitization technology, the 3-D rough surface is converted into a 2-D discrete plane with 3-D information. According to the dynamic real-time update of the graph data, the iterative calculation of the wear cycle is completed, the time-varying wear calculation method for fractal rough surfaces of friction pairs is proposed, and the dynamic change in the wear amount and surface topography of the rough surface is simulated. The simulation results are experimentally verified and the influence of friction parameters on the surface topography is analyzed. The results show that after the wear simulation, the profile height of the rough surface is reduced, and the average wear depth is 0.02 mm. Increases in rotational speeds and external loads can exacerbate surface wear, surface topography tends to be flattened, and surface carrying capacity increases. This provides theoretical guidance for the development and manufacture of friction pairs.

Universal Langmuir and Fractal Analysis of High-Resolution Adsorption Isotherms of Argon and Nitrogen on Macroporous Silica.

High-resolution isotherms of argon and nitrogen adsorption on macroporous silica have been simulated with universal Langmuir and fractal models. A four-parameter, fractal universal Langmuir equation is a good fit to the data at low pressures. Standard Gibbs energy changes calculated from equilibrium adsorption coefficients show a series of broad peaks that indicate adsorbate structural transformations as a function of pressure and coverage. The Freundlich equation or mean fractal model is also a good fit to isotherms at low pressures. Pressure-varying fractals are accurate fits to the data. Fractal exponents provide information on adsorbate coverage and surface access. Broad peaks in pressure-varying exponents are indicators of adsorbate structure. From adsorptive gas amounts, mean and pressure-varying fractal exponents provide details of adsorbate fractal dimensions and surface roughness. Both Ar and N2 adsorption cause increases in mean surface roughness when compared with pure silica. Surface roughness fluctuations from pressure-dependent adsorptive gas fractal dimensions are associated with adsorbate structure. At one trough, the surface is smooth and is linked to close-packed Ar or N2. For Ar adsorption at 87 K, this structure is a complete monolayer (1.00(4)), while for Ar (77 K), 1.15(4) layers and for N2 (87 K), 2.02(10) layers. The universal Langmuir specific area of the silica is 10.1(4) m2 g-1. Pressure- and coverage-dependent adsorbate structures range from filling defects and holes on the surface to cluster formation to adsorbed Ar or N2 evenly distributed or packed across the surface. The Ar (87 K) isotherm is most sensitive to adsorbate structural transformations.

Structural and Mechanical Properties of CrN Thin Films Deposited on Si Substrate by Using Magnetron Techniques

Chromium nitride thin films are known for their good mechanical properties. We present the characteristics of ultrathin chromium nitride films under 400 nm thickness deposited on silicon substrates by direct current and high-power impulse magnetron sputtering techniques. The methods of investigation of the CrN films were scanning electron microscopy, atomic force microscopy, and nanoindentation. Qualitative and quantitative analyses were performed using AFM and SEM images by fractal dimension, surface roughness and gray-level co-occurrence matrix methods. Our results show that using magnetron techniques, ultrathin CrN films with excellent mechanical properties were obtained, characterized by values of Young’s modulus between 140 GPa and 250 GPa for the samples obtained using high-power impulse magneton sputtering (HiPIMS) and between 240 GPa and 370 GPa for the samples obtained using direct current sputtering (DC). Stiffness measurements also reveal the excellent mechanical properties of the investigated samples, where the samples obtained using HiPIMS sputtering had stiffness values between 125 N/m and 132 N/m and the samples obtained using DC sputtering had stiffness values between 110 N/m and 119 N/m.

Tactile perception of fractal surfaces: An EEG-fNIRS study

In telesurgery and other virtual reality techniques, most virtual surfaces for tactile interactive contact are lacking fractal features which are common in organic surfaces. This paper aims to study the relationship between fractal surfaces and tactile perception. The ‘Hurst exponent’ is applicable for quantitative analysis of fractal textures. A multimodal tactile experiment was carried out using EEG-fNIRS simultaneous joint imaging platform. The Mel-frequency cepstral coefficient (MFCC) was extracted from the sound signal generated by friction vibration. The sample entropy and permutation entropy were calculated from the EEG signal. The results show that the fractal surfaces can be characterized by MFCC, sample entropy and permutation entropy. These three samples also have different brain functional reflections from fNIRS results.

Role of Ir Decoration in Activating a Multiscale Fractal Surface in Porous Ni for the Oxygen Evolution Reaction

Interfacial engineering of electrocatalysts is a pivotal approach for promoting the energy conversion efficiency of water electrolysis systems. Anchoring clusters or even single atoms of a foreign element on the surface of a support, usually 3D structured, can alter its local electronic structure leading to superior electrocatalytic activity. Herein, we report a method to decorate a 3D fractal Ni electrode with Ir atoms via a galvanic displacement reaction. The resulting electrode has a very low Ir loading of 1.6 μmole cmgeo–2, with iridium atoms present as 4–5 nm diameter nanoclusters on the electrode surface. The activity for the electrochemical oxygen evolution reaction (OER) of the 3D fractal Ni electrode was improved by decoration with Ir, resulting in an overpotential of 195 mV at 10 mA cm–2 and a Tafel slope of 44 mV dec–1. Apart from increasing the electrocatalytic activity for the OER due to the very presence of more active iridium atoms, the galvanic displacement reaction resulted in a factor 8–10 increase of the electrochemically active surface area due to the creation/activation of a secondary pore structure that contributes also to the better electrocatalytic performance of the resulting electrode. Using operando acoustic emission, it is demonstrated that the galvanic displacement reaction and the presence of Ir nanoclusters have the additional effect of reducing the average O2 bubble size formed during the OER. As a result, the blocking effect of O2 bubbles at high current density is less drastic than on the 3D fractal Ni electrode, resulting in a less severe decrease of the electrochemically active area at large current density. All three effects contribute toward improving the OER performance of the Ir-decorated 3D fractal Ni electrode.

Experimental observations of fractal landscape dynamics in a dense emulsion.

Many soft and biological materials display so-called 'soft glassy' dynamics; their constituents undergo anomalous random motions and complex cooperative rearrangements. A recent simulation model of one soft glassy material, a coarsening foam, suggested that the random motions of its bubbles are due to the system configuration moving over a fractal energy landscape in high-dimensional space. Here we show that the salient geometrical features of such high-dimensional fractal landscapes can be explored and reliably quantified, using empirical trajectory data from many degrees of freedom, in a model-free manner. For a mayonnaise-like dense emulsion, analysis of the observed trajectories of oil droplets quantitatively reproduces the high-dimensional fractal geometry of the configuration path and its associated local energy minima generated using a computational model. That geometry in turn drives the droplets' complex random motion observed in real space. Our results indicate that experimental studies can elucidate whether the similar dynamics in different soft and biological materials may also be due to fractal landscape dynamics.

Simulation of Lunar Comprehensive Substructure With Fracture and Imaging of Later LPR Data From Chang’e-4 Mission

As one of the most important geophysical methods for detecting the lunar underground structure of the Moon, the lunar penetrating radar (LPR) is applied to the Chang’e-4 (CE-4) mission to explore the Von Kármán crater on the far side of the Moon. With the Yutu-2 rover continues to move towards the Zhinyu crater at the west, radar will likely detect the fractures created by the impact that formed the Zhinyu Crater. Therefore, the establishment of a comprehensive lunar subsurface structure model with fracture, random media and fractal terrain is crucial for the LPR data processing and the understanding of the lunar geological impact process. In addition, the method that can image the radar data with fractures well has great significance to the interpretation of the LPR data. In this paper, we establish a comprehensive lunar subsurface structure model firstly, the considering factors including fractures, random media, fractal terrain, and the radar response are calculated through forward modeling. Secondly, according to the high resolution and the sensitive to inhomogeneities of LPR data, we design a set of processing flow including plane-wave destruction, velocity analysis based on the focusing analysis method and migration imaging based on the velocity continuation. Finally, the results including simulation data and Antarctic fracture data are used to verify the effectiveness of imaging method.

FRACTAL INTERPOLATION ALGORITHMS FOR COMPLEX CURVES AND SURFACES

This paper provides a new idea for interpolating complex curves and surfaces by fractal interpolation. A type of bivariate fractal interpolation functions with function vertical scaling factors for the special interpolation data set is presented. The interpolation algorithm is proposed for complex curves by contour lines of two types of fractal interpolation functions which are proposed by this paper and Navascués et al. [Construction of fractal surfaces, Fractals 28(2) (2020) 2050033], respectively. Meanwhile, some properties of the interpolation algorithm are introduced. Then, the two classes of fractal interpolation functions are extended to the cases of trivariate functions. We also introduce the interpolation algorithm for complex surfaces based on iso-surfaces of fractal interpolation functions constructed. Finally, several examples are provided.