Search Results (28)

Stylolites are a common mineral dissolution feature in rocks that develop during compression and form distinct tooth structures. On a tectonic plate scale, mid-ocean ridges (MORs) and transform faults are a significant feature of the Earth’s surface that develop due to accretion of new material in an extensional regime. We present a comparison between the two features and argue that transform faults in MOR are similar to the sides of stylolite teeth, with both features representing kinematic faults (KFs). First, we present a numerical model of both stylolite and MOR growth and show that in both cases, KFs nucleate and grow spontaneously. In addition, we use a well-established technique (Family–Vicsek scaling) of describing fractal self-affine interfaces, which has been used for stylolites, to characterize the pattern of MOR systems in both simulations and natural examples. Our results show that stylolites and MOR have self-affine scaling characteristics with similar scaling regimes. They both show a larger roughness exponent at the small scale, a smaller exponent at the intermediate scale, followed by a flattening of the system at the largest scale. For stylolites, the physical forces behind the scaling are the surface energy at the small mineral scale, the elastic energy at the intermediate scale, followed by the system reaching the correlation length where growth stops. For MORs, the physical forces behind the scaling are not yet clear; however, the self-affine scaling shows that transform faults at MORs do not have a preferred spacing, but that the spacing is fractal. Our study offers a new perspective on the study of natural roughening phenomena on various scales, from minerals to tectonic plates, and a new view on the development of MORs. Full article

Numerous analytical radar-scattering laws have been published through the past decades to interpret planetary radar observations, such as Hagfors’ law, which has been commonly used for the Moon, and the cosine law, which is commonly used in the shape modeling of asteroids. Many of the laws have not been numerically validated in terms of their interpretation and limitations. This paper evaluates radar-scattering laws for self-affine fractal surfaces using a numerical approach. Traditionally, the autocorrelation function and, more recently, the Hurst exponent, which describes the self-affinity, have been used to quantify the height correlation. Here, hundreds of three-dimensional synthetic surfaces parameterized using a root-mean-square (rms) height and a Hurst exponent were generated, and their backscattering coefficient functions were computed to evaluate their consistency with selected analytical models. The numerical results were also compared to empirical models for roughness and radar-scattering measurements of Hawaii lava flows and found consistent. The Gaussian law performed best at predicting the rms slope regardless of the Hurst exponent. Consistent with the literature, it was found to be the most reliable radar-scattering law for the inverse modeling of the rms slopes and the Fresnel reflection coefficient from the quasi-specular backscattering peak, when homogeneous statistical properties and a ray-optics approach can be assumed. The contribution of multiple scattering in the backscattered power increases as a function of rms slope up to about 20% of the backscattered power at normal incidence when the rms slope angle is 46°. Full article

When describing the tribological behaviour of technical surfaces, the need for full-length scale microtopographic characterization often arises. The self-affine of surfaces and the characterisation of self-affine using a fractal dimension and its implantation into tribological models are commonly used. The goal of our present work was to determine the frequency range of fractal behaviour of surfaces by analysing the microtopographic measurements of an anodised aluminium brake plunger. We also wanted to know if bifractal and multifractal behaviour can be detected in real machine parts. As a result, we developed a new methodology for determining the fractal range boundaries to separate the nano- and micro-roughness. To reach our goals, we used an atomic force microscope (AFM) and a stylus instrument to obtain measurements in a wide frequency range (19 nm–3 mm). Power spectral density (PSD)-based fractal evaluation found that the examined surface could not be characterised by a single fractal dimension. A new method capable of separating nano- and micro-roughness has been developed for investigating multifractal behaviour. The presented procedure separates nano- and micro-roughness based on the geometric characteristics of surfaces. In this way, it becomes possible to specifically examine the relationship between the micro-geometry that can be measured in each wavelength range and the effects of cutting technology and the material structure that creates them. Full article

The key issues in fractal geometry concern scale invariance (self-similarity or self-affinity) and the notion of a fractal dimension $D$ which exceeds the topological dimension $d$. In this regard, we point out that the constitutive inequality $D>d$ can have either a geometric or topological origin, or both. The main topological features of fractals are their connectedness, connectivity, ramification, and loopiness. We argue that these features can be specified by six basic dimension numbers which are generally independent from each other. However, for many kinds of fractals, the number of independent dimensions may be reduced due to the peculiarities of specific kinds of fractals. Accordingly, we survey the paradigmatic fractals from a topological perspective. Some challenging points are outlined. Full article

A simplified method of determining lattice Boltzmann boundary conditions based on self-affine microchannels with an inherent roughness in a tight reservoir is presented in this paper to address nonlinear efficiency problems in fluid simulation. This approach effectively combines the influence of rough surfaces in the simulation of the flow field, the description of L-fractal theory applied to rough surfaces, and a generalized lattice Boltzmann method with equivalent composite slip boundary conditions for inherent roughness. The numerical simulations of gas slippage in a two-dimensional plate model and rough surfaces to induce gas vortex reflux flow are also successfully carried out, and the results are in good agreement with the simulation results, which establishes the reliability and flexibility of the proposed simplified method of rough surfaces. The effects of relative average height and fractal dimensions of the rough surfaces under exact boundary conditions and equivalent coarsened ones are investigated from three perspectives, namely those of the average lattice velocity, the lattice velocity at average height position at the outlet, and the coefficient of variation for lattice velocity at average height position. It was found that the roughness effect on gas flow behavior was more obvious when it was associated with the enhanced rarefaction effect. In addition, the area of gas seepage was reduced, and the gas flow resistance was increased. When the fractal dimension of the wall was about 1.20, it has the greatest impact on the fluid flow law. In addition, excessive roughness of the wall surface tends to lead to vortex backflow of the gas in the region adjacent to the wall, which greatly reduces its flow velocity. For gas flow in the nanoscale seepage space, wall roughness hindered gas migration rate by 84.7%. For pores larger than 200 nm, the effects of wall roughness on gas flow are generally negligible. Full article

The solute transport in the fractured rock is dominated by a single fracture. The geometric characteristics of single rough-walled fractures considerably influence their solute transport behavior. According to the self-affinity of the rough fractures, the fractal model of single fractures is established based on the fractional Brownian motion and the successive random accumulation method. The Navier–Stokes equation and solute transport convective-dispersion equation are employed to analyze the effect of fractal dimension and standard deviation of aperture on the solute transport characteristics. The results show that the concentration front and streamline distribution are inhomogeneous, and the residence time distribution (RTD) curves have obvious tailing. For the larger fractal dimension and the standard deviation of aperture, the fracture surface becomes rougher, aperture distribution becomes more scattered, and the average flow velocity becomes slower. As a result, the average time of solute transport is a power function of the fractal dimension, while the time variance and the time skewness present a negative linear correlation with the fractal dimension. For the standard deviation of aperture, the average time exhibits a linearly decreasing trend, the time variance is increased by a power function, and the skewness is increased logarithmically. Full article

Polymers and polymer matrix composites are commonly used materials with applications extending from packaging materials to delicate electronic devices. Epoxy resins and fiber-reinforced epoxy-based composites have been used as adhesives and construction parts. Fractal analysis has been recognized in materials science as a valuable tool for the microstructural characterization of composites by connecting fractal characteristics with composites’ functional properties. In this study, fractal reconstructions of different microstructural shapes in an epoxy-based composite were performed on field emission scanning electron microscopy (FESEM) images. These images were of glass fiber reinforced epoxy as well as a hybrid composite containing both glass and electrospun polystyrene fibers in an epoxy matrix. Fractal reconstruction enables the identification of self-similarity in the fractal structure, which represents a novelty in analyzing the fractal properties of materials. Fractal Real Finder software, based on the mathematical affine fractal regression model, was employed to reconstruct different microstructure shapes and calculate fractal dimensions to develop a method of predicting the optimal structure–property relations in composite materials in the future. Full article

In order to further research the relationship between fractals and complicated networks in terms of self-similarity, the uniform convergence property of the sequence of fractal interpolation functions which can generate self-similar graphics through iterated function system defined by affine transformation is studied in this paper. The result illustrates that it is can be proved that the sequence of fractal interpolation functions uniformly converges to its limit function and its limit function is continuous and integrable over a closed interval under the uniformly convergent condition of the sequence of fractal interpolation functions. The following two conclusions can be indicated. First, both the number sequence limit operation of the sequence of fractal interpolation functions and the function limit operation of its limit function are exchangeable over a closed interval. Second, the two operations of limit and integral between the sequence of fractal interpolation functions and its limit function are exchangeable over a closed interval. Full article

The universal fractality of river networks is very well known, however understanding of their underlying mechanisms is still lacking from a stochastic point of view. In this study, we have described the fractal natures of river networks by introducing a stochastic model where the direction of river flow at a site is determined by the dynamical replication probability which depends on the drainage area at the site rather than at random. We found that the probability induces dynamical persistency in river flows resulting in the self-affine properties shown in real river basins. Full article

The continuity, self-similarity, and self-affinity of a microscopic contact surface can be described by the Weierstrass–Mandelbrot (W–M) function in fractal theory. To address the problems that the existing normal contact load fractal model does not take into account the effect of thermal stress and is not applicable to the temperature variation in the joint surface of the giant magnetostrictive ultrasonic vibration systems, a fractal model of thermal–elastic–plastic contact normal load fractal is established based on fractal theory. The model is an extension of the traditional model in terms of basic theory and application scope, and it takes into account the effects of temperature difference, linear expansion coefficient, fractal dimension, and other parameters. Finally, the effect of the temperature difference at the joint surface on the normal load of the thermoelastic contact is revealed through numerical simulations. The results show that the nonlinearity of the contact stiffness of the thermoelastic joint surface is mainly related to the surface roughness and the fractal dimension, while the effect of the temperature change on the joint surface properties within a certain range is linear. Full article

We study the spectrality of a class of self-affine measures with prime determinant. Spectral measures are connected with fractal geometry that shows some kind of geometrical self-similarity under magnification. To make the self-affine measure becomes a spectral measure with lattice spectrum, we provide two new sufficient conditions related to the elements of digit set and zero set, respectively. The two sufficient conditions are more precise and easier to be verified as compared with the previous research. Moreover, these conditions offer a fresh perspective on a conjecture of Lagarias and Wang. Full article

In the present study, the microstructural and statistical properties of unimplanted in comparison to argon ion-implanted tantalum-based thin film surface structures are investigated for potential application in microelectronic thin film substrates. In the study, the argon ions were implanted at the energy of 30 keV and the doses of 1 × 10¹⁷, 3 × 10¹⁷, and 7 × 10¹⁷ (ion/cm²⁾ at an ambient temperature. Two primary goals have been pursued in this study. First, by using atomic force microscopy (AFM) analysis, the roughness of samples, before and after implantation, has been studied. The corrosion apparatus wear has been used to compare resistance against tantalum corrosion for all samples. The results show an increase in resistance against tantalum corrosion after the argon ion implantation process. After the corrosion test, scanning electron microscopy (SEM) analysis was applied to study the sample morphology. The elemental composition of the samples was characterized by using energy-dispersive X-ray (EDX) analysis. Second, the statisticalcharacteristics of both unimplanted and implanted samples, using the monofractal analysis with correlation function and correlation length of samples, were studied. The results show, however, that all samples are correlated and that the variation of ion doses has a negligible impact on the values of correlation lengths. Moreover, the study of height distribution and higher-order moments show the deviation from Gaussian distribution. The calculations of the roughness exponent and fractal dimension indicates that the implanted samples are the self-affine fractal surfaces. Full article

Porous structures exhibiting randomly sized and distributed pores are required in biomedical applications (producing implants), materials science (developing cermet-based materials with desired properties), engineering applications (objects having controlled mass and energy transfer properties), and smart agriculture (devices for soilless cultivation). In most cases, a scaffold-based method is used to design porous structures. This approach fails to produce randomly sized and distributed pores, which is a pressing need as far as the aforementioned application areas are concerned. Thus, more effective porous structure design methods are required. This article presents how to utilize fractal geometry to model porous structures and then print them using 3D printing technology. A mathematical procedure was developed to create stochastic point clouds using the affine maps of a predefined Iterative Function Systems (IFS)-based fractal. In addition, a method is developed to modify a given IFS fractal-generated point cloud. The modification process controls the self-similarity levels of the fractal and ultimately results in a model of porous structure exhibiting randomly sized and distributed pores. The model can be transformed into a 3D Computer-Aided Design (CAD) model using voxel-based modeling or other means for digitization and 3D printing. The efficacy of the proposed method is demonstrated by transforming the Sierpinski Carpet (an IFS-based fractal) into 3D-printed porous structures with randomly sized and distributed pores. Other IFS-based fractals than the Sierpinski Carpet can be used to model and fabricate porous structures effectively. This issue remains open for further research. Full article

The rapid growth of geographic information technologies in the field of processing and analysis of spatial data has led to a significant increase in the role of geographic information systems in various fields of human activity. However, solving complex problems requires the use of large amounts of spatial data, efficient storage of data on on-board recording media and their transmission via communication channels. This leads to the need to create new effective methods of compression and data transmission of remote sensing of the Earth. The possibility of using fractal functions for image processing, which were transmitted via the satellite radio channel of a spacecraft, is considered. The information obtained by such a system is presented in the form of aerospace images that need to be processed and analyzed in order to obtain information about the objects that are displayed. An algorithm for constructing image encoding–decoding using a class of continuous functions that depend on a finite set of parameters and have fractal properties is investigated. The mathematical model used in fractal image compression is called a system of iterative functions. The encoding process is time consuming because it performs a large number of transformations and mathematical calculations. However, due to this, a high degree of image compression is achieved. This class of functions has an interesting property—knowing the initial sets of numbers, we can easily calculate the value of the function, but when the values of the function are known, it is very difficult to return the initial set of values, because there are a huge number of such combinations. Therefore, in order to de-encode the image, it is necessary to know fractal codes that will help to restore the raster image. Full article

We present results showing the capability of concrete-based information processing substrate in the signal classification task in accordance with in materio computing paradigm. As the Reservoir Computing is a suitable model for describing embedded in materio computation, we propose that this type of presented basic construction unit can be used as a source for “reservoir of states” necessary for simple tuning of the readout layer. We present an electrical characterization of the set of samples with different additive concentrations followed by a dynamical analysis of selected specimens showing fingerprints of memfractive properties. As part of dynamic analysis, several fractal dimensions and entropy parameters for the output signal were analyzed to explore the richness of the reservoir configuration space. In addition, to investigate the chaotic nature and self-affinity of the signal, Lyapunov exponents and Detrended Fluctuation Analysis exponents were calculated. Moreover, on the basis of obtained parameters, classification of the signal waveform shapes can be performed in scenarios explicitly tuned for a given device terminal. Full article