Random Fractal Structures Research Articles

Fully coupled aero-hydrodynamic analysis of a biomimetic fractal semi-submersible floating offshore wind turbine under wind-wave excitation conditions

With wind energy utilization shifting to the deep sea, floating offshore wind turbines (FOWTs) have huge potential to improve the competitiveness of offshore wind energy generation. However, traditional FOWT systems are often in force imbalance due to the unique swaying characteristics caused by the unfixed foundation and high center of mass of the platform. Therefore, a novel bionic semi-submersible floating platform based on Victoria Amazonia (with random fractal structure) is proposed to increase the hydrodynamic stability of FOWT. A regular fractal structure with similar number of perforations and perforated area is also implemented to verify the effectiveness of the bionic FOWT. In this study, the unsteady computational fluid dynamic approach is adopted to simulate the aerodynamic and hydrodynamic response. More specifically, the dynamic fluid-body interaction module integrated in STAR-CCM+ is employed to establish a FOWT numerical model with rotating blades, tower, and mooring line system. Furthermore, the volume of fluid model in conjunction with the 6-DOF solver was used to efficiently solve the fluid-induced dynamic motion of FOWT in a multi-phase flow composed of air and water. Finally, the fully coupled calculation of the fractal structure FOWT was performed using the established reliable numerical model. The maximum average thrust and power of 736.43 kN and 5294.04 kW are obtained by random fractal FOWT in aerodynamic responses, while in hydrodynamic amplitude responses, the maximum decreases of 19.16% are obtained by both random fractal and regular fractal structures. As for amplitude responses in heave and surge, the random fractal FOWT of 12.91% and 5.05% decreases performs relatively better than the regular one (11.48% and 4.09% decreases). In addition, the visualization of unsteady flow fields and vortices in fractal structures is investigated in detail. Compared with the regular fractal structure, vortices within random fractal structure interact adequately with the pontoon walls, resulting in higher energy absorption effect and stability.

Quantum Gravity Emergence from Entanglement in a Multi-Fold Universe

We start from a hypothetical multi-fold universe U_MF, where the propagation of everything is slower or equal to ‎the speed of light and where entanglement extends the set of paths available to Path Integrals. This multifold ‎mechanism enables EPR (Einstein-Podolsky-Rosen) “spooky actions at distance” to result from local interactions ‎in the resulting folds. It produces gravity-like attractive effective potentials in the spacetime, between entangled ‎entities, that are caused by the curvature of the folds. When quantized, multi-folds correspond to gravitons and ‎they are enablers of EPR entanglement. Gravity emerges non-perturbative and covariant from EPR entanglement ‎between virtual particles surrounding an entity. In U_MF, we encounter mechanisms that predict gravity fluctuations ‎when entanglement is present, including in macroscopic entanglements. Besides providing a new perspective on ‎quantum gravity, when added to the Standard Model as (SMG), with non-negligible affects at its scales, and to the ‎Standard Cosmology, U_MF can contribute explanations of several open questions and challenges. It also clarifies ‎some relationships and challenges met by other quantum gravity models and Theories of Everything. It leads to ‎suggestions for these works. We also reconstruct the spacetime of U_MF, starting from the random walks of particles ‎in an early spacetime. U_MF now appears as a noncommutative, discrete, yet Lorentz symmetric, spacetime that ‎behaves roughly 2-Dimensional at Planck scales, when it is a graph of microscopic Planck size black holes on a ‎random walk fractal structure left by particles that can also appear as microscopic black holes. Of course, at larger ‎scales, spacetime appears 4-D, where we are able to explain curvature and recover Einstein’s General Relativity. We ‎also discover an entanglement gravity-like contributions and massive gravity at very small scales. This is ‎remarkable considering that no Hilbert Einstein action, or variations expressing area invariance, were introduced. ‎Our model also explains why semi classical approaches can work till way smaller scale than usually expected and ‎present a new view on an Ultimate Unification of all forces, at very small scales. We also explore opportunities for ‎falsifiability and validation of our model, as well as ideas for futuristic applications, that may be worth considering, ‎if U_MF was a suitable model for our universe U_real.‎

Diffraction on random fractal structures

In this paper, the formation of diffraction patterns for regular and random fractals of Cantor is numerically studied. The calculation of diffraction on one-dimensional, two-dimensional and radial fractal structures is performed using a fractional Fourier transform. This propagation operator makes it possible to obtain a field distribution (with the accuracy of scale) in any paraxial region, both in the Fresnel diffraction zone and in the far field.

An embarrassingly parallel algorithm for random walk simulations on random fractal structures

Anomalous diffusion is often simulated by random walks on random fractal structures. As existing simulation methods either lack a high degree of parallelism or impose restrictions on the choice of fractal structures, a new approach is proposed here.We present a parallel algorithm for simulating random walks on fractal structures that is suitable for a wide variety of hardware architectures. The degree of parallelism of the algorithm equals the number of random walkers, which is achieved by its communication-avoiding design. In contrast to other approaches, the random fractal structure is not pre-computed at whole. Instead, only the surrounding of each random walker is calculated by the parallel threads while the random walker moves around on the fractal structure.

Strongly enhanced light trapping in a two-dimensional silicon nanowire random fractal array.

We report on the unconventional optical properties exhibited by a two-dimensional array of thin Si nanowires arranged in a random fractal geometry and fabricated using an inexpensive, fast and maskless process compatible with Si technology. The structure allows for a high light-trapping efficiency across the entire visible range, attaining total reflectance values as low as 0.1% when the wavelength in the medium matches the length scale of maximum heterogeneity in the system. We show that the random fractal structure of our nanowire array is responsible for a strong in-plane multiple scattering, which is related to the material refractive index fluctuations and leads to a greatly enhanced Raman scattering and a bright photoluminescence. These strong emissions are correlated on all length scales according to the refractive index fluctuations. The relevance and the perspectives of the reported results are discussed as promising for Si-based photovoltaic and photonic applications.

Real-Time Computational Model of Ball-Milled Fractal Structures

Ball milling (BM) offers a flexible process for nanomanufacturing of reactive bimetallic multiscale particulates (nanoheaters) for self-heated microjoining engineering materials and biomedical tooling. This paper introduces a mechanics-based process model relating the chaotic dynamics of BM with the random fractal structures of the produced particulates, emphasizing its fundamental concepts, underlying assumptions, and computation methods. To represent Apollonian globular and lamellar structures, the simulation employs warped ellipsoidal (WE) primitives of elasto-plastic strain-hardening materials, with Maxwell–Boltzmann distributions of ball kinetics and thermal transformation of hysteretic plastic, frictional, and residual stored energetics. Interparticle collisions are modeled via modified Hertzian contact impact mechanics, with local plastic deformation yielding welded microjoints and resulting in cluster assembly into particulates. The model tracks the size and diversity of such particulate populations as the process evolves via sequential collision and integration events. The simulation was shown to run in real-time computation speeds on modest hardware, and match successfully the fractal dimension and contour shape of experimental ball-milled Al–Ni particulate micrographs. Thus, the model serves as a base for the design of a feedback control system for continuous BM.

Influence of soot aggregate structure on particle sizing using laser-induced incandescence: importance of bridging between primary particles

Soot aggregates formed in combustion processes are often described as clusters of carbonaceous particles in random fractal structures. For theoretical studies of the physical properties of such aggregates, they have often been modelled as spherical primary particles in point contact. However, transmission electron microscopy (TEM) images show that the primary particles are more connected than in a single point; there is a certain amount of bridging between the primary particles. Particle sizing using the diagnostic technique laser-induced incandescence (LII) is crucially dependent on the heat conduction rate from the aggregate to the ambient gas, which depends on the amount of bridging. In this work, aggregates with bridging are modelled using overlapping spheres, and it is shown how such aggregates can be built to fulfil specific fractal parameters. Aggregates with and without bridging are constructed numerically, and it is investigated how the bridging influences the heat conduction rate in the free-molecular regime. The calculated heat conduction rates are then used in an LII model to show how LII particle sizing is influenced by different amounts of bridging. For realistic amounts of bridging (\(C_{\rm{ov}}\leq0.25\)), the primary particle diameters were overestimated by up to 9 % if bridging was not taken into account.

Dynamic conductivity of composites of fractal structure

We have studied the dynamic conductivity of a composite consisting of well- and weak-conducting components with random fractal structure. In order to calculate effective properties of composite medium, we used hierarchic structure model and innovative iterative averaging method based on renormalization group transformations idea. Our results show, that the behavior of a composite over a magnetic field become even more complicated.Unusual peaks and oscillations appear in frequency dependencies of effective conductivity, permittivity and other properties. We discuss the influence of fractal parameters of the composite structure on such unusual behavior of effective properties.

Conductivity in a magnetic field of a three-dimensional composite with a random fractal structure

Conductivity in a magnetic field of a three-dimensional composite with a random fractal structure

The coastline and lake shores on a fractal island

We compute the fractal dimensions of the 'hulls' or external boundary and the boundaries of the internal cavities in several deterministic as well as random fractal structures. Our conclusion is that the two fractal dimensions are in fact identical. The deterministic fractals we study are Sierpinski carpets (SC) in a two-dimensional space and the random fractals are percolation clusters at criticality. As an intermediate case, we present results on some randomized SC. In the random structures, statistics of the area and perimeters of all internal cavities or holes are taken and the fractal dimension of the hull borderline is computed. Two different definitions of the borderline are used, considering nearest neighbours as well as nearest and second nearest neighbours as connected. The conclusion is valid for both cases.

05/00933 Random fractal structures in North American energy markets

05/00933 Random fractal structures in North American energy markets

Mode Selection of Diffusion-limited Aggregation

The diffusion-limited aggregation (DLA) is a pattern formation governed only by the nature of diffusion field. Irreversible sticking process of particles gives rise to isotropic random fractal structure of DLA. Since Witten and Sander proposed such a simulation model, many researchers have been devoted to various structures of offlattice DLA, such as self-similarity, active zone, multifractality, branching distribution, fractal dimension, and so on. However, many authors have indicated complex structures of DLA cluster. Mysterious one among them is near pentagonal symmetry with the outline shape of DLA cluster. A wedge model was proposed from the viewpoint of harmonic measure, which gives the fractal dimension D 1:714 for a symmetric cone of pentagon. The details of mysterious symmetry in DLA cluster are studied by numerical simulations. In order to study the exterior shape of DLA, we measure the coordinates of 10 sticking particles without cluster growth that is the same manner as the study of active zone. The sticking simulations are performed on the 2500 offlattice clusters composed of 3 10 particles. Here, simulation methods with high accuracy are used as reported in ref. 14. Averaged power spectrum of sticking distribution in angle direction is shown in Fig. 1 as a function of spectrum mode. The peak position of mean spectrum can be estimated as the mode of 5:4 by the fitting of parabolic curve near the peak. Thus, the mysterious symmetry of DLA is characterized by the near pentagon as indicated before reports. However, this peak mode is slightly increasing with increasing of cluster size in the present simulation up to 3 10 particles. Above ordinary DLA simulation is started from the initial condition of a seed particle with the radius RS 1⁄4 0. In order to clarify the stability of near pentagonal symmetry, following DLA simulations are performed on the initial conditions of circular seed up to RS 1⁄4 500. As shown in Fig. 2 for RS 1⁄4 500, it is obvious that the near pentagonal mode in angle direction is extremely stable for large size of cluster independent of initial condition. Cluster density as a function of angle is analyzed by power spectrum, and then spectrum peak is obtained from the mean data for 1000 clusters (30000 clusters for RS 1⁄4 0). An example of power spectrum along the cluster growth from the seed of RS 1⁄4 500 is drawn in Fig. 3(a), which shows decreasing of peak mode with increasing of cluster size. As shown in Fig. 3(b), the Fig. 1. Power spectrum of angle distribution for sticking particles onto the surface of 2500 ordinary DLA clusters composed of N 1⁄4 3000000 particles. The peak position is estimated as the mode of 5:4.

Random fractal structures in North American energy markets

This paper uses daily observations on West Texas Intermediate (WTI) crude oil prices at Chicago and Henry Hub natural gas prices at LA (over the deregulated period of the 1990s) and various tests from statistics and dynamical systems theory to support a random fractal structure for North American energy markets. In particular, this evidence is supported by the Vassilicos et al. (1993) multifractal structure test and the Ghashghaie et al. [Nature 381 (1996) 767] turbulent behavior test.

Random fractal structures in North American energy markets

This paper uses daily observations on West Texas Intermediate (WTI) crude oil prices at Chicago and Henry Hub natural gas prices at LA (over the deregulated period of the 1990s) and various tests from statistics and dynamical systems theory to support a random fractal structure for North American energy markets. In particular, this evidence is supported by the Vassilicos et al. (1993) multifractal structure test and the Ghashghaie et al. [Nature 381 (1996) 767] turbulent behavior test.

Spatial gliding, temporal trapping, and anomalous transport

We introduce and study a model of stochastic dynamics, governed by Markovian laws of motion in a phase space filled with random traps. The resulting (non-Markovian) stochastic trapped motion alternates intermittently between periods of spatial gliding, where the motion ‘glides’ the underlying Markovian dynamics, and periods of temporal trapping, where the motion is halted in random traps. We investigate the asymptotics and scaling limits of this model. We prove that when the random trappings are heavy-tailed then Mittag–Leffler functions and probability laws emerge and govern the functional structure and statistics of the system, and that the time flow has a random fractal structure whose fractal exponent (dimension) is determined by the ‘heaviness’ of the trappings. We study the effect of random trapping on general Lévy dynamics. We prove that subjecting Lévy dynamics to heavy-tailed trapping will always result in: (i) sub-diffusive behavior—when the underlying Lévy dynamics are of finite variance; and (ii) space–time fractal behavior—when the underlying Lévy dynamics are scale-invariant. Furthermore, we explore the issue of first exit times. To that end, a general Feynman–Kac framework for trapped processes is developed, and a method of transforming ‘trapped’ Feynman–Kac equations to ‘standard’ ones is established. The study of first exit times enables us to quantitatively connect macroscopic observations to microscopic behavior in general Markovian dynamics subjected to random trapping. In the case of Lévy dynamics, first exit times from balls are computed, and the relationships between their statistics and the statistics of the trapped motion are derived.

Superlocalization of Waves and Electrons in Hopping Conductivity

We have found the two different densities of states d s = 4/3 and d v = 1.05 and two inverse localization lengths for fractons with the scalar and vector interactions, respectively. In this concept the wave functions of electrons or waves on an incipient percolation cluster and fractal dilute structure exhibit superlocalization behavior of the form ψ(r) exp [-r do ] with values of d o1 = 1.73 and d o2 = 2.4 for the former and the latter. Applications of these results for thermally activated hopping conductivity a(t) exp [-(T 0 /T) β ] between impurities on a random fractal structure give the values of β = 2/5 for the scalar and of β = 1/2 for the vector interactions, respectively.

Superlocalization of waves and electrons and hopping conductivity on classical and superconductive fractals

We have found an unexpected paradoxical situation in the percolation transition: the superconductive behavior below and above the threshold. We have found also the two different density of states d s=4/3 and d v=1.05 and the inverse localization lengths for fractons with the scalar and vector interactions, respectively. In this concept the wave functions of electrons or waves on an incipient percolation cluster and fractal dilute structure exhibit superlocalization behavior of the form ψ( r)∝exp[− r d φ ] with values of d φ1 =1.73 and d φ2 =2.4 for the former and the latter. Applications of these results for thermally activated hopping conductivity σ( t)∝exp[−( T 0/ T) β ] between impurities on a random fractal structure give the values of β=2/5 for the scalar and β=1/2 (Mott's law) for the vector interactions, respectively. Band states are localized in classical and superlocalized in superconductive percolations.

Non-linear behaviour of human EEG: fractal exponent versus correlation dimension in awake and sleep stages

The question of whether the finite values of the correlation dimension (D2), used as an index of EEG complexity are due to its chaotic nature or they reflect its behaviour as linearly-correlated noise, remains open. This report aims at clarifying this by measuring D2 and analysing the non-linear nature of EEG through the method of surrogate data as well as by calculating the fractal exponent ( β) via coarse graining spectral analysis (CGSA) in nine adult subjects during waking and sleep states. The results show that even if it is possible to get an estimation of D2 in all states, non-linear structure appears to be present only during slow wave sleep (SWS). EEG exhibits random fractal structure with 1/ f − β spectrum (1< β<3) and a negative linear correlation between D2 and β in all states except during SWS. In consequence, in those states, finite D2 values could be attributed to the fractal nature of EEG and not to the presence of low-dimensional chaos, and therefore, it the use of β would be more appropriate to describe the complexity of EEG, due to its lower computational cost.

Pseudo-scaling and multifractality

Random systems with stretched (and fluctuating) exponential form of probability density functions (PDFs), such as turbulence (distribution of velocity differences) and percolation (visiting frequency in percolative diffusion), are studied. Dimensionless moments are defined as F np ( r) = ƒ p / ƒ n p/ n , where ƒ p ( r) is the standard moments of order p ( p > n). Pseudo-scaling (PS) is defined as existence of power relationships F np ∼ F nq ρ , where exponent ρ npq depends on p, q and n. It is shown that for large enough n, p, q, the PS takes place even if the ordinary scaling is broken. It is shown that there are two kinds of the asymptotical pseudo-scaling: with ρ npq = ( p− n)/( q− n) and with ρ npq = ( p/ q) [ln( p/ n)/(ln( q/ n)]. If ordinary scaling also takes place in the systems then these two kinds of the pseudo-scaling lead to two kinds of corresponding ordinary scaling laws. A speculation concerning genesis of the stretched exponential PDFs is briefly discussed. Phase transition from random fractal structures to homogeneity is studied in the framework of the pseudo-scaling approach and it is shown that multifractality of passive scalar concentration has the pseudo-scaling form of the second kind near the transition (whereas for systems far from the transition the multifractality has the pseudo-scaling form of the first kind). The condition of compatibility of the pseudo-scaling and of the generalized dimension approaches is also investigated. Agreement between the theoretical results and experimental data of different authors is established both for the situations where ordinary scaling takes place and for situations where ordinary scaling does not take place.

Multifractal Properties of Phase Transition from Random Fractal Structures to Homogeneity

The transport properties of particles (of passive scalar concentration) in random media with anomalous diffusion are studied. The generalized multifractal hypothesis of Havlin-Bunde: ∼ τ q (τ q ∼ q γ , γ ∼ R - s q , where \({\mit\Theta}_R = \int_v P(r,t) {\rm d}v/v\,\,(v \sim R^d\) is an arbitrary volume with scale R ). It is shown that, in this limit, s q = A q l n ( q / q o ). The value q 0 = 2 can be obtained from the condition of compatibility between this approach and the generalized dimension description. Agreement between the theoretical results and experimental data is estab...