Graph-directed random fractal interpolation function

"Barnsley introduced in [1] the notion of fractal interpolation function (FIF). He said that a fractal function is a (FIF) if it possess some interpolation properties. It has the advantage that it can be also combined with the classical methods or real data interpolation. Hutchinson and Ruschendorf [7] gave the stochastic version of fractal interpolation function. In order to obtain fractal interpolation functions with more exibility, Wang and Yu [9] used instead of a constant scaling parameter a variable vertical scaling factor. Also the notion of fractal interpolation can be generalized to the graph-directed case introduced by Deniz and  Ozdemir in [5]. In this paper we study the case of a stochastic fractal interpolation function with graph-directed fractal function."

Fractal dimension of coastline of Australia

Coastlines are irregular in nature having (random) fractal geometry and are formed by various natural activities. Fractal dimension is a measure of degree of geometric irregularity present in the coastline. A novel multicore parallel processing algorithm is presented to calculate the fractal dimension of coastline of Australia. The reliability of the coastline length of Australia is addressed by recovering the power law from our computational results. For simulations, the algorithm is implemented on a parallel computer for multi-core processing using the QGIS software, R-programming language and Python codes.

Numerical Study of Thermal Properties of a Silica Aerogel Fractal Material

Silica Aerogel is material with a special structure consisting of interconnected solid particles of SiO2 forming the skeleton that enclose nano-pores filled with confined air and occupying more than 90% of the volume. It is characterised by a thermal conductivity that can reach lower values than any other material. Our aim was to explain the causes of the super-insulation of this material with the use of a calculating method of conductive heat transfer flux in an aerogel structure and determine the equivalent thermal conductivity. For this purpose, numerical specific software was developed to generate random fractal structure of silica aerogel with pre-defined concentration of solid particles and properties of both skeleton and confined air. Calculation of the conductivity at any point in the confined gas domain shows variable values as a function of the pore size and the location of the point in the pore. Heat transfer through the aerogel in the unsteady state as well as in the steady state was simulated by imposing a Dirichlet-type boundary conditions for each side of the domain of the aerogel structure. A new developed numerical method was used to calculate the equivalent thermal conductivity of the whole fractal structure. Concentration of solid particles has proved not to be the only parameter in which depend on the thermal conductivity of silica aerogel and the influence of tortuosity has been demonstrated. A correlation linking thermal conductivity to both concentration of solid particles and tortuosity of the material was suggested.

Numerical Simulation of the Non-Darcy Flow Based on Random Fractal Micronetwork Model for Low Permeability Sandstone Gas Reservoirs

Darcy’s law is not suit for describing high velocity flow in the near wellbore region of gas reservoirs. The non-Darcy coefficient β of the Forchheimer’s equation is a main parameter for the evaluation of seepage capacity in gas reservoirs. The paper presented a new method to calculate β by performing gas and con-water flow simulations with random 3D micropore network model. Firstly, a network model is established by random fractal method. Secondly, based on the network simulation method of non-Darcy flow in the literature of Thauvin and Mohanty, a modified model is developed to describe gas non-Darcy flow with irreducible water in the porous medium. The model was verified by our experimental measurements. Then, we investigated the influence of different factors on the non-Darcy coefficient, including micropore structure (pore radius and fractal dimension), irreducible water saturation ( S wi ), tortuosity, and other reservoir characteristics. The simulation results showed that the value of the non-Darcy coefficient decreases with the increase in all: the average pore radius, fractal dimension, irreducible water saturation, and tortuosity. The non-Darcy coefficients obtained by the fractal method of microparameters are estimated more precisely than the conventional methods. The method provides theoretical support for the productivity prediction of non-Darcy flow in gas reservoirs.

Effect of the Concrete Mesostructure Geometric Form on Its Elastic Modulus

A comprehensive understanding of the geometrical form of the concrete mesostructure is important because it is associated with the complex and random mechanical behavior of the concrete. There is still uncertainty about how to characterize the geometrical form of the concrete mesostructure and its effect on the observed macroscopic behavior. The coarse aggregate content and fractal dimension were used in this study to indicate the geometrical form of concrete at the mesolevel. Taking the mechanical parameters measured in experiments, a group of mesomodels with various fractal dimensions and coarse aggregate contents was simulated by using the random fractal modeling method and then was studied and discussed. The analytical solution, simulation, and experimental data all suggested that the elastic modulus increased with the increasing coarse aggregate content. Meanwhile, the fractal dimension can cause the elastic modulus to decline slightly. The comprehensive consideration of both fractal geometry and classical Euclidean geometry can aid in predicting the macroscopic behavior of concrete.

Assouad Spectrum Thresholds for Some Random Constructions

The Assouad dimension of a metric space determines its extremal scaling properties. The derived notion of the Assouad spectrum fixes relative scales by a scaling function to obtain interpolation behaviour between the quasi-Assouad and the box-counting dimensions. While the quasi-Assouad and Assouad dimensions often coincide, they generally differ in random constructions. In this paper we consider a generalised Assouad spectrum that interpolates between the quasi-Assouad and the Assouad dimension. For common models of random fractal sets, we obtain a dichotomy of its behaviour by finding a threshold function where the quasi-Assouad behaviour transitions to the Assouad dimension. This threshold can be considered a phase transition, and we compute the threshold for the Gromov boundary of Galton–Watson trees and one-variable random self-similar and self-affine constructions. We describe how the stochastically self-similar model can be derived from the Galton–Watson tree result.