Approximation aspects of set-valued fractal surfaces

The application of the spectral representation method in generating Gaussian and non-Gaussian fractal rough surfaces is studied in this work. The characteristics of fractal rough surfaces simulated by the spectral representation method and the conventional Fast Fourier transform filtering method are compared. Furthermore, the fractal rough surfaces simulated by these two methods are compared in the simulation of contact and lubrication problems. Next, the influence of low and high cutoff frequencies on the normality of the simulated Gaussian fractal rough surfaces is investigated with roll-off power spectral density and single power-law power spectral density. Finally, a simple approximation method to generate non-Gaussian fractal rough surfaces is proposed by combining the spectral representation method and the Johnson translator system. Based on the simulation results, the current work gives recommendations on using the spectral representation method and the Fast Fourier transform filtering method to generate fractal surfaces and suggestions on selecting the low cutoff frequency of the power-law power spectral density. Furthermore, the results show that the proposed approximation method can be a choice to generate non-Gaussian fractal surfaces when the accuracy requirements are not high. The MATLAB codes for generating Gaussian and non-Gaussian fractal rough surfaces are provided.

In many in vitro cultures, cells may change their morphology, probably caused by adherence to the surface of the culture dish. Since a fractal alkyl ketene dimer (AKD) surface provides super water-repellency with a contact angle of 174° , we considered that it might provide an improved surface environment for the growth and differentiation of cells by preventing intimate adhesion. C6 glioma cells which were selected to test the effects of the fractal surface, were cultured on a conventional surface, a smooth AKD surface or a fractal AKD surface. On the conventional and smooth AKD surfaces, cells developed bipolar or multipolar shapes with enlarged cell bodies and neurite-like processes. In contrast, cells cultured on the fractal AKD surface presented fine filopodium-like processes like protoplasmic astrocytes in vivo, and higher morphological complexity was revealed by fractal analysis. Reconstruction of three-dimensional shape indicated that cells on the fractal surface were globular, whereas those on the conventional surface were rather flat. Our results suggest that C6 glioma cells on a fractal AKD surface show features of natural astrocytes with their elaborate morphology. The fractal surface thus may provide a new and natural culture environment for experimental assessment of glial structure and function.

We report an experimental and theoretical investigation of the wetting behavior of different model polar and nonpolar liquids and their mixtures on superhydrophobic fractal surfaces made of polymer- or silane-coated "core-shell" particles. We compared the experimental results with the theoretical predictions made according to the theories of Onda-Shibuichi (describes wetting on fractal surfaces) and Cassie-Baxter (describes wetting on generic rough composite surfaces). We found that the experimental findings deviate from the behavior predicted by the Onda-Shibuichi model. On the other hand, the wetting properties were found to be close to the predictions made by the Cassie-Baxter model in the hydrophobic region (the intrinsic contact angle on the flat surface is larger than 90 degrees). However, the wetting behavior in the hydrophilic region (the intrinsic contact angle is less than 90 degrees) could not be described by the Onda-Shibuichi or Cassie-Baxter model. The observed inconsistency between the experimental results and theoretical predictions was explained by the formation of metastable states of a liquid droplet on a fabricated fractal surface according to the theory developed by Johnson and Dettre for generic rough surfaces. The entrapments of the liquid droplets in metastable states resulted in superhydrophobic behavior on fractal surfaces as well, made from nonfluorinated material such as polystyrene with a surface free energy of about 30 mJ/m2. This finding is very promising for real industrial applications where fluorinated compounds are willing to be reduced. It can be concluded that employing a texture with fractal geometry is necessary for the design of superhydrophobic coatings. Thereby, extremely lowering the surface free energy of materials by fluorination is not an obligatory factor for the generation of liquid-repellent superhydrophobic materials. We believe that the results we presented in the paper give new insight into the understanding of wetting not only on general superhydrophobic rough surfaces but also on fractal surfaces.

The research problem is about to generate artificial fractal landscape surfaces from the Digital Elevation Model (DEM) using a stochastic algorithm by Geographic Resources Analysis Support System Geographic Information System (GRASS GIS) software. Fractal surfaces resemble appearance of natural topographic terrain and its structure using random surface modelling. Study area covers Kuril-Kamchatka region, Sea of Okhotsk, North Pacific Ocean. Techniques were included into GRASS GIS modules (r.relief, d.rast, r.slope.aspect, r.mapcalc) for raster calculation, processing and visualization. Module 'r.surf.fractal' was applied for generating synthetic fractal surface from ETOPO1 DEM GeoTIFF using algorithm of fractal analysis. Three tested dimensions of the fractal surfaces were automatically mapped and visualized. Algorithm of the automated fractal DEM modelling visualized variations in steepness and aspect of the artificially generated slopes in the mountains. Controllable topographic variation of the fractal surfaces was applied for three dimensions: dim=2.0001, 2.0050, 2.0100. Auxiliary modules were used for the visualization of DEMs (d.rast, r.colors, d.vect, r.contour, d.redraw, d.mon). Modules 'r.surf.gauss' and 'r.surf.random' were applied for artificial modelling as Gauss and random based mathematical surfaces, respectively. Univariate statistics for fractal surfaces were computed for comparative analysis of maps representing continuous fields by module 'r.univar': number of cells, min/max, range, mean, variance, standard deviation, variation coefficient and sum. The paper includes 9 maps and GRASS GIS codes used for visualization.

Scaling approach to study diffusion and reaction processes on fractal catalysts

A machined surface has observable fractal characteristics, with infinite local and overall self-similar consistency. Therefore, the fractal theory is considered to provide a better description of the morphological characteristics of rough surfaces, which accurately reflects the randomness and multi-scale characteristics of rough surfaces and it is not comparable with the surface characteristics obtained based on statistical parameters limited by sampling length and device resolution. In this study, the Weierstrass-Mandelbrot (W-M) function was applied to construct a fractal reconstruction surface, and the mixed elastohydrodynamic lubrication model was used to investigate the lubrication characteristics of real and reconstructed surfaces under the same fractal parameters. The effects of the fractal parameters on the fractal surface lubrication characteristics were further analyzed. The results demonstrate that the lateral roughness fractal surface provides greater resistance to the entrained flow of lubricant, which leads to a larger average film thickness, than the longitudinal roughness and isotropic fractal surface. With the increase in fractal dimension, the surface roughness peak density increases, which reduces the surface film thickness by 47%, and the friction coefficient increases by 46%. The lubrication parameter fluctuates slightly with the change in the number of overlapping ridges M of the fractal surface. Generally, M has little effect on the surface lubrication characteristics.

Investigation of the Heterogeneous Nucleation on Fractal Surfaces

We study the slippage on hierarchical fractal superhydrophobic surfaces and find an unexpected rich behavior for hydrodynamic friction on these surfaces. We develop a scaling law approach for the effective slip length, which is validated by numerical resolution of the hydrodynamic equations. Our results demonstrate that slippage does strongly depend on the fractal dimension and is found to be always smaller on fractal surfaces as compared with surfaces with regular patterns. This shows that in contrast to naive expectations, the value of effective contact angle is not sufficient to infer the amount of slippage on a fractal surface: depending on the underlying geometry of the roughness, strongly superhydrophobic surfaces may, in some cases, be fully inefficient in terms of drag reduction. Finally, our scaling analysis can be directly extended to the study of heat transfer at fractal surfaces, in order to estimate the Kapitsa surface resistance on patterned surfaces, as well as to the question of trapping of diffusing particles by patchy hierarchical surfaces, in the context of chemoreception.

This paper discusses the scattering of electromagnetic (EM) waves from three – dimensional (3D) rough fractal surfaces using the Kirchhoff approximation. In particular, it introduces a novel method to characterize 3D rough fractal surfaces from spectral information of the backscattered EM wave in Synthetic Aperture Radar (SAR) applications. It represents an important extension of a previous recent paper by the same research group from 2D fractal surfaces to 3D fractal surfaces (the latter representing real – life SAR radar scenes). More specifically, in the present simulation scenarios it is assumed that the radar emits a burst of radar pulses of increasing carrier frequency [therefore a ‘stepped – frequency’ (SF) SAR radar]. By calculating the backscattered EM wave from 3D fractal surfaces as a function of the above radar frequencies (therefore, a ‘spectral method’), it is found here that the slope between the main backscattered lobe and its adjacent side lobes increases with increasing surface fractal dimension (i.e. with increasing surface roughness). In this way a characterization of 3D fractal rough surfaces from backscattered SAR radar data is achieved, as explained in detail in this paper.DOI: http://dx.doi.org/10.5755/j01.eie.23.4.18721

A contact problem is considered in which an elastic half–plane is pressed against a rigid fractally rough surface, whose profile is defined by a Weierstrass series. It is shown that no applied mean pressure is sufficiently large to ensure full contact and indeed there are not even any contact areas of finite dimension — the contact area consists of a set of fractal character for all values of the geometric and loading parameters. A solution for the partial contact of a sinusoidal surface is used to develop a relation between the contact pressure distribution at scale n − 1 and that at scale n . Recursive numerical integration of this relation yields the contact area as a function of scale. An analytical solution to the same problem appropriate at large n is constructed following a technique due to Archard. This is found to give a very good approximation to the numerical results even at small n , except for cases where the dimensionless applied load is large. The contact area is found to decrease continuously with n , tending to a power–law behaviour at large n which corresponds to a limiting fractal dimension of (2 − D ), where D is the fractal dimension of the surface profile. However, it is not a ‘simple’ fractal, in the sense that it deviates from the power–law form at low n , at which there is also a dependence on the applied load. Contact segment lengths become smaller at small scales, but an appropriately normalized size distribution tends to a limiting function at large n . † The authors dedicate this paper to the memory of Dr J. F. Archard, 1918–1989.

We investigate a stochastic lattice model describing a predator-prey system in a fractal scale-free landscape, mimicked by the fractal Sierpinski carpet. We determine the threshold of species coexistence, that is, the critical phase boundary related to the transition between an active state, where both species coexist and an absorbing state where one of the species is extinct. We show that the predators must live longer in order to persist in a fractal habitat. We further performed a finite-size scaling analysis in the vicinity of the absorbing-state phase transition to compute a set of stationary and dynamical critical exponents. Our results indicate that the transition belongs to the directed percolation universality class exhibited by the usual contact process model on the same fractal landscape.

QuestionHow are heterogeneity–diversity relationships (HDRs) influenced by spatial structure in environmental variables, sampling grain and the extent of niche differentiation?MethodsWe developed a spatially explicit simulation model incorporating variable dispersal distances and competition strength on fractal landscapes. By varying the grain used to sample these models, we examined scaling patterns in HDR metrics at fine scales (sampling grain from 100 to 10 000 individuals, sampling extent ca. 260 000 individuals).ResultsEnvironmental geometry exerts an important influence on the ecological processes responsible for HDRs. Unique geometric characteristics of individual landscapes can greatly influence emergent community properties; field studies frequently use inadequate sample sizes to account for this phenomenon. Two opposing processes influence spatial scaling of HDRs: variance partitioning, which favours smaller‐grained samples, and mass effects, which favour larger‐grained samples. In assessing HDRs, diversity is more sensitive than species richness, and should be the preferred measure in field studies. The environmental geometry and age of a community interact: compared to high fractal dimension landscapes, low fractal dimension landscapes are slower to develop HDRs, but in the long term their HDRs will be higher.ConclusionsOur study demonstrates that, despite the superficial simplicity of the concept, HDRs vary in complex and non‐intuitive ways, and warrant further theoretical and empirical study. More generally, environmental geometry is likely to exert a strong influence on many emergent community processes, but we do not yet have a firm understanding of this relationship.

Neutral landscape models predict that habitat loss will abruptly disrupt landscape connectivity. We performed a series of simulation experiments to explore whether thresholds in landscape connectivity affect movement attributes (path length, net displacement, and fractal dimension of pathway) within fractal neutral landscapes. We then tested these assumptions by generating fractal landscape patterns in the field across a range of habitat abundances (0%, 20%, 50%, and 80% grass) and patchiness (clumped vs. patchy) and quantified how patch structure affected movement behavior in a generic organism, the common cricket Acheta domestica (Orthoptera: Gryllidae). In the simulation experiment, individuals constrained to move only through adjacent grass cells (neighborhood size = 4 cells) exhibited abrupt thresholds between 50% and 80% grass cover for all movement parameters in clumped fractal landscapes but exhibited a linear decline in movement with decreasing habitat in patchy landscapes. Individuals constrained to move in sand within these same landscapes did not exhibit thresholds in movement with decreasing sand habitat. The exception is for the fractal dimension of pathways (a measure of tortuosity) in which a threshold occurred between 50% and 80% grass (50% and 20% sand) in patchy landscapes. Increasing the scale of movement by allowing individuals to move through unsuitable habitat (neighborhood size = 12 cells) reduced or eliminated any effects of patch structure on movement. Live crickets can traverse both grass and sand, and thus threshold effects in movement behavior were generally not evident in the field experiment. Only small crickets (15–25 mm) exhibited a threshold response in net displacement (straight-line distance traversed) between 50% and 80% grass cover (50% and 20% sand). Crickets did exhibit significant responses to patch structure, however. Crickets moved faster and with less tortuosity in the control (0% grass) and less-vegetated (20% grass) plots than in plots with greater habitat coverage. Crickets used grass cells significantly more (73%) than expected in the 20% patchy fractal microlandscape; crickets were reluctant to leave isolated cells of grass. Grass provided cover, but sand facilitated movement. While experimentation at the landscape scale is generally intractable or impossible, computer simulation and field experiments founded on neutral landscape models permit initial assessment of how disrupting landscape connectivity affects movement behavior.

By using the Monte Carlo method and numerical finite element approach, bistatic scattering from the fractal and Gaussian rough surfaces is studied. The difference between these two surfaces and their functional dependence on the surface parameters are discussed. Angular variation of bistatic scattering from the fractal surface is very significant, even for fairly smooth surface, whilst scattering from the Gaussian rough surface tends to the specular reflection. The slope of angular variation is linearly related with the fractal dimension. If an electrically-large target is placed over the rough surface. the fractal dimension inverted from bistatic scattering would be reduced. As the surfaces become very rough, scattering from different fractal and Gaussian surfaces would be not identified.

A simple method for estimating the size of nuclei on fractal surfaces