Showing papers on "Fractal dimension published in 2016"

Self-Similar Conformations and Dynamics in Entangled Melts and Solutions of Nonconcatenated Ring Polymers

Abstract: A scaling model of self-similar conformations and dynamics of nonconcatenated entangled ring polymers is developed. Topological constraints force these ring polymers into compact conformations with fractal dimension df = 3 that we call fractal loopy globules (FLGs). This result is based on the conjecture that the overlap parameter of subsections of rings on all length scales is the same and equal to the Kavassalis–Noolandi number OKN ≈ 10–20. The dynamics of entangled rings is self-similar and proceeds as loops of increasing sizes are rearranged progressively at their respective diffusion times. The topological constraints associated with smaller rearranged loops affect the dynamics of larger loops through increasing the effective friction coefficient but have no influence on the entanglement tubes confining larger loops. As a result, the tube diameter defined as the average spacing between relevant topological constraints increases with time t, leading to “tube dilation”. Analysis of the primitive paths ...

Fractal cometary dust – a window into the early Solar system

Abstract: The properties of dust in the protoplanetary disk are key to understanding the formation of planets in our Solar System. Many models of dust growth predict the development of fractal structures that evolve into non-fractal, porous dust pebbles representing the main component for planetesimal accretion. In order to understand comets and their origins, the Rosetta orbiter followed comet 67P/Churyumov-Gerasimenko for over two years and carried a dedicated instrument suite for dust analysis. One of these instruments, the MIDAS atomic force microscope, recorded the 3D topography of micro- to nanometre sized dust. All particles analysed to date have been found to be hierarchical agglomerates. Most show compact packing, however, one is extremely porous. This paper contains a structural description of a compact aggregate and the outstanding porous one. Both particles are tens of micrometres in size and show rather narrow subunit size distributions with noticeably similar mean values of 1.48+0.13−0.59 μm for the porous particle and 1.36+0.15−0.59 μm for the compact. ompact. The porous particle allows a fractal analysis, where a density-density correlation function yields a fractal dimension of Df = 1.70 ± 0.1. GIADA, another dust analysis instrument on-board Rosetta, confirms the existence of a dust population with a similar fractal dimension. The fractal particles are interpreted as pristine agglomerates built in the protoplanetary disk and preserved in the comet. The similar subunits of both fractal and compact dust indicate a common origin which is, given the properties of the fractal, dominated by slow agglomeration of equally sized aggregates known as cluster-cluster agglomeration.

Surface Roughness Measurements Using Power Spectrum Density Analysis with Enhanced Spatial Correlation Length

Abstract: Roughness of a surface as characterized by an atomic force microscope (AFM) is typically expressed using conventional statistical measurements including root-mean-square, peak-to-valley ratio, and average roughness. However, in these measurements only the vertical distribution of roughness (z-axis) is considered. Additionally, roughness of a surface as determined by AFM is a function of the scanning scale, sampling interval and/or scanning methods; therefore, the consideration and quantification of the lateral distribution (x and y) is necessary. Power spectral density (PSD) analysis provides both lateral and vertical signals captured from AFM images. By applying one of the commonly adopted models to the PSD data, the fractal model and k-correlation model, the equivalent root mean squared roughness, correlation length, fractal dimension and Hurst exponent are quantified. These parameters describe the spatial distribution of roughness and spatial length scale of the roughness values. Longer correlation len...

Criticality and Connectivity in Macromolecular Charge Complexation

Abstract: We examine the role of molecular connectivity and architecture on the complexation of ionic macromolecules (polyelectrolytes) of finite size. A unified framework is developed and applied to evaluate the electrostatic correlation free energy for point-like, rod-like, and coil-like molecules. That framework is generalized to molecules of variable fractal dimensions, including dendrimers. Analytical expressions for the free energy, correlation length, and osmotic pressure are derived, thereby enabling consideration of the effects of charge connectivity, fractal dimension, and backbone stiffness on the complexation behavior of a wide range of polyelectrolytes. Results are presented for regions in the immediate vicinity of the critical region and far from it. A transparent and explicit expression for the coexistence curve is derived in order to facilitate analysis of experimentally observed phase diagrams.

A fractal network model for fractured porous media

Abstract: The transport properties and mechanisms of fractured porous media are very important for oil and gas reservoir engineering, hydraulics, environmental science, chemical engineering, etc. In this paper, a fractal dual-porosity model is developed to estimate the equivalent hydraulic properties of fractured porous media, where a fractal tree-like network model is used to characterize the fracture system according to its fractal scaling laws and topological structures. The analytical expressions for the effective permeability of fracture system and fractured porous media, tortuosity, fracture density and fraction are derived. The proposed fractal model has been validated by comparisons with available experimental data and numerical simulation. It has been shown that fractal dimensions for fracture length and aperture have significant effect on the equivalent hydraulic properties of fractured porous media. The effective permeability of fracture system can be increased with the increase of fractal dimensions for fracture length and aperture, while it can be remarkably lowered by introducing tortuosity at large branching angle. Also, a scaling law between the fracture density and fractal dimension for fracture length has been found, where the scaling exponent depends on the fracture number. The present fractal dual-porosity model may shed light on the transport physics of fractured porous media and provide theoretical basis for oil and gas exploitation, underground water, nuclear waste disposal and geothermal energy extraction as well as chemical engineering, etc.

Numerical research of turbulent boundary layer based on the fractal dimension of pressure fluctuations

Abstract: The results of numerical and experimental studies of the structure and the frictional resistance of the turbulent flow with the effects obtained by using a modified model of the mixing Prandtl with the fractal dimension of pressure fluctuations. A construction of a cooled turbine blade was designed based on the results. The construction comprises a combined cooling and cylindrical cavity on the blade surface and the inner surface of the cooling channels.

Fluid flow in porous media with rough pore-solid interface

Abstract: Quantifying fluid flow through porous media hinges on the description of permeability, a property of considerable importance in many fields ranging from oil and gas exploration to hydrology. A common building block for modeling porous media permeability is consideration of fluid flow through tubes with circular cross section described by Poiseuille's law in which flow discharge is proportional to the fourth power of the tube's radius. In most natural porous media, pores are neither cylindrical nor smooth; they often have an irregular cross section and rough surfaces. This study presents a theoretical scaling of Poiseuille's approximation for flow in pores with irregular rough cross section quantified by a surface fractal dimension Ds2. The flow rate is a function of the average pore radius to the power 2(3-Ds2) instead of 4 in the original Poiseuille's law. Values of Ds2 range from 1 to 2, hence, the power in the modified Poiseuille's approximation varies between 4 and 2, indicating that flow rate decreases as pore surface roughness (and surface fractal dimension Ds2) increases. We also proposed pore length-radius relations for isotropic and anisotropic fractal porous media. The new theoretical derivations are compared with standard approximations and with experimental values of relative permeability. The new approach results in substantially improved prediction of relative permeability of natural porous media relative to the original Poiseuille equation.

The Fibonacci Hamiltonian

Abstract: We consider the Fibonacci Hamiltonian, the central model in the study of electronic properties of one-dimensional quasicrystals, and establish relations between its spectrum and spectral characteristics (namely, the optimal Holder exponent of the integrated density of states, the dimension of the density of states measure, the dimension of the spectrum, and the upper transport exponent) and the dynamical properties of the Fibonacci trace map (such as dimensional characteristics of the non-wandering hyperbolic set and its measure of maximal entropy as well as other equilibrium measures, topological entropy, multipliers of periodic orbits). We also exhibit a connection between the spectral quantities and the thermodynamic pressure function. As a result, a detailed description of the spectral properties for all values of the coupling constant is obtained (in contrast to all previous quantitative results, which could be established only in the regime of small or large coupling). In particular, we show that the spectrum of this operator is a dynamically defined Cantor set and that the density of states measure is exact-dimensional; this implies that all standard fractal dimensions coincide in each case. We show that all the gaps of the spectrum allowed by the gap labeling theorem are open for all values of the coupling constant. Also, we establish strict inequalities between the four spectral characteristics in question, and provide the exact large coupling asymptotics of the dimension of the density of states measure (for the other three quantities, the large coupling asymptotics were known before).

Fractal Dimension Estimation for Developing Pathological Brain Detection System Based on Minkowski-Bouligand Method

Abstract: It is of enormous significance to detect abnormal brains automatically. This paper develops an efficient pathological brain detection system based on the artificial intelligence method. We first extract brain edges by a Canny edge detector. Next, we estimated the fractal dimension using box counting method with grid sizes of 1, 2, 4, 8, and 16, respectively. Afterward, we employed the single-hidden layer feedforward neural network. Finally, we proposed an improved particle swarm optimization based on three-segment particle representation, time-varying acceleration coefficient, and chaos theory. This three-segment particle representation encodes the weights, biases, and number of hidden neuron. The statistical analysis showed the proposed method achieves the detection accuracies of 100%, 98.19%, and 98.08% over three benchmark data sets. Our method costs merely 0.1984 s to predict one image. Our performance is superior to the 11 state-of-the-art approaches.

The Role of Surface Structure in Normal Contact Stiffness

Abstract: The effects of roughness and fractality on the normal contact stiffness of rough surfaces were investigated by considering samples of isotropically roughened aluminium. Surface features of samples were altered by polishing and by five surface mechanical treatments using different sized particles. Surface topology was characterised by interferometry-based profilometry and electron microscopy. Subsequently, the normal contact stiffness was evaluated through flat-tipped diamond nanoindentation tests employing the partial unloading method to isolate elastic deformation. Three indenter tips of various sizes were utilised in order to gain results across a wide range of stress levels. We focus on establishing relationships between interfacial stiffness and roughness descriptors, combined with the effects of the fractal dimension of surfaces over various length scales. The experimental results show that the observed contact stiffness is a power-law function of the normal force with the exponent of this relationship closely correlated to surfaces’ values of fractal dimension, yielding corresponding correlation coefficients above 90 %. A relatively weak correlation coefficient of 60 % was found between the exponent and surfaces’ RMS roughness values. The RMS roughness mainly contributes to the magnitude of the contact stiffness, when surfaces have similar fractal structures at a given loading, with a correlation coefficient of −95 %. These findings from this work can be served as the experimental basis for modelling contact stiffness on various rough surfaces.

Strength, fragmentation and fractal properties of mixed flaws

Abstract: Experiments on Portland cement samples containing mixed flaws are conducted to investigate the strength, fragmentation and fractal properties. Flaw geometry is a new combination of two edge-notched flaws and an imbedded flaw, which is different from those in the previous studies, where parallel or coplanar flaws are used. The physical implications of the shear-box test applied to result to rock slopes are studied. The physical and analytical fragmentation characteristics of preflawed samples are analyzed through the sieve test and fractal theory, respectively. Three different patterns of tensile cracks and shear cracks are observed. A sliding crack model is presented to elucidate the brittle failure flaws. In all of the cases of the shear-box tests, the coalescence is produced by the linkage of shear cracks, and two types of coalescence (Type C1 and Type C2) have been classified, which tend to confirm the observations from the numerical model and field of jointed rock slopes. The shear strength is a function of the flaw geometry and the shear–normal stress ratio. The result of sieve tests indicates that the fragment size distribution of fragments has the fractal property, providing a physical understanding of the fragmentation mechanism. The fragments under the shear-box test have fractal dimensions between 2.2 and 2.6, which are larger than those under the compression test but similar to those in the fault cores. The fragmentation in the case of Type C2 has a smaller fractal dimension, corresponding to a larger shear strength.