In this work, a physical connection between the fractional time derivative and fractal geometry of fractal media is developed and applied to viscoelasticity and thermal diffusion in elastomers. ...Integral to this formulation is the application of both the fractal dimension and the spectral dimension which characterizes diffusion in fractal media. The methodology extends the generalized molecular theory of Rouse and Zimm where generalized Gaussian structures (GGSs) replace the Rouse matrix with the generalized Gaussian Rouse matrix (GRM). Importantly, the Zimm model is extended to fractal media where the new relaxation formulation contains internal state variables that naturally depend on the fractional time derivative of deformation. Through the use of thermodynamic laws in fractal media, we derive the linear fractional model of viscoelasticity based on both spectral and fractal dimensions. This derivation shows how the order of the fractional derivative in the linear fractional model of viscoelasticity is a rate dependent material property that is strongly correlated with fractal and spectral dimensions in fractal media. To validate the correlation between fractional rates and fractal material structure, we measure the viscoelasticity and thermal diffusion of two different dielectric elastomers: Very High Bond (VHB) 4910 and VHB 4949. Using Bayesian uncertainty quantification (UQ) based on uniaxial stress–strain measurements, the fractional order of the derivative in the linear fractional model of viscoelasticity is quantified. Two dimensional fractal dimensions are also independently quantified using the box counting method. Lastly, the diffusion equation in fractal media is inferred from experiments using Bayesian UQ to quantify the spectral dimension by heating the polymer locally with a laser beam and quantifying thermal diffusion. Comparing theory to experiments, a strong correlation is found between the viscoelastic fractional order obtained from stress–strain measurements in comparisons with independent predictions of fractional viscoelasticity based on the fractal structure and fractional thermal diffusion rates.
Image super-resolution aims to reconstruct a high-resolution image from the known low-resolution version. During this process, it should keep the degree of image roughness non-decreasing, which ...reflects various texture features and appearance. However, this point is not well addressed in the current work. This work argues that reducing roughness during image super-resolution is the key reason causing various problems such as artificial texture and/or edge blur. In this work, keeping the image roughness non-decreasing during super-resolution is being well investigated for the first time to our best knowledge. Image super-resolution is cast as an optimization problem to keep image roughness non-decreasing. In order to tackle this problem, the image super-resolution is approached based on the theory of fractal, where adaptive fractal interpolation function is proposed. In this way, the rational fractal interpolation model is adaptive to every local region. Thus, the roughness of every image region can be best maintained while super-resolution is carried out through fractal interpolation. In this work, the image roughness is reflected by the fractal dimension, which is a key element affecting the construction of fractal interpolation model. That is, the image roughness is measurable using fractal dimension. Mathematically, the overall image super-resolution process can be converted into a fractal interpolation optimization problem where the local fractal dimension is maintained. Although adaptive super-resolution on image segments may best maintain image roughness using the proposed method, it still generates unnecessary block artifacts. To tackle this problem, this work proposes a fine-grained pixel-wise fractal function. Our extensive experimental results demonstrate that the proposed method achieves encouraging performance with the state-of-the-art super-resolution algorithms.
Effective detection of low observable moving target at sea is important for remote sensing and radar signal processing. The non-Gaussian property of sea clutter and lack of accurate model make the ...detection difficult for statistics based detectors. Also the fractal techniques in time domain cannot achieve high detection probability in heavy sea clutter. To help solve the problems, fractal characteristics of IPIX datasets in fractional Fourier transform (FRFT) domain are analysed making use of the fluctuation of FRFT amplitudes and moving target detection algorithms are proposed based on the fractal characteristics in FRFT domain. Firstly, fractal model in FRFT domain is established with fractional Brownian motion model and two judgment and extraction methods are employed for calculating the fractal characteristics in FRFT domain. It is found that sea clutter of different polarisations exhibit fractal behaviours in FRFT domain, that is, self-similarity property, within its corresponding scale-invariant interval. Then, we find that four specific fractal statistics in the best FRFT domain can provide valuable information for developing simple and effective detectors. Finally, traditional amplitude detector and Hurst exponent detector in time domain are compared and the results prove the superior detection ability of low observable moving target without complex computations.
Existing penetration models for grouting slurry seldomly consider the relationship between dry density and complexity of soil. A novel penetration model, in which the complexity of the flow path in ...soil was described by fractal theory, has been proposed for quantifying the maximum penetration distance of grout slurry flowing through soils with different dry densities. The relationship among dry density, fractal dimension and fractal tortuosity of flow path was identified by SEM analysis, showing that fractal tortuosity increases but fractal dimension is almost unchanged with increase of dry density. The results calculated by the penetration model were validated against experimental results for different soils under various pump pressures and volume flow rates. The analysis results show that the grout slurry tends to penetrate to a long distance under high pump pressure, low flow rate, low consistency factor and low fluidity index. High dry density and small average particle size of soil reduce the penetration distance. The proposed model is helpful to achieve a better design in soil ground stabilization via grouting.
We investigate the possibilities and properties of the application of Giuseppe Peano fractal geometry for the miniaturization of microstrip patch antennas and compare its performance with those of ...the usual fractals, such as Koch, Tee-Type and Sierpinski. The length of the Giuseppe Peano fractal patch perimeter increases, while its surface area remains constant without any more space occupation. Consequently the antenna miniaturization, maintenance of its gain and increase of its relative frequency bandwidth are achieved. The proposed antenna has circular polarization at one of its resonance frequencies, which is realized by producing a perturbation on its initial structure. Further miniaturization of antenna may be obtained by cutting slots on its structure and its broadbanding may be achieved by placing an air gap under its metallic patch and more effective miniaturization is obtained by placing a Giuseppe Peano fractal strip along the microstrip patch antenna.
•A fractal scaling law for size distribution of fracture aperture is confirmed.•An analytical solution for dimensionless permeability of dual-porosity media is derived.•Validity of proposed multiple ...fractal model is verified by comparing with in-situ measurements.•Relationships between dimensionless permeability and structural parameters are analyzed.
A multiple fractal model that considers the fractal properties of both porous matrices and fracture networks is proposed for the permeability of dual-porosity media embedded with randomly distributed fractures. In this model, the aperture distribution is verified to follow the fractal scaling law, and the porous matrix is assumed to comprise a bundle of tortuous capillaries that also follow the fractal scaling law. Analytical expressions for fractal aperture distribution, total flow rate, total equivalent permeability, and dimensionless permeability are established, where the dimensionless permeability is defined as the ratio of permeability of the porous matrices to that of the fracture networks. The dimensionless permeability is closely correlated to the structural parameters (i.e., α, θ, Dtf, Dtp, De, Dp, emax, λmax) of the dual-porosity media, and it is more sensitive to the fractal dimension for the size distribution of fracture aperture than to that for the size distribution of pore/capillary diameter. The maximum pore/capillary diameter has a greater impact on the dimensionless permeability than that of the maximum fracture aperture. The dimensionless permeability of fracture networks constructed by the fractal aperture distribution has close values with those of models with lognormal aperture distribution. The proposed multiple fractal model does not involve any empirical constants that do not have clear physical meanings, which could serve as a quick estimation method for assessing permeability of dual-porosity media.
Low-heat Portland (LHP) cement is a kind of high-belite cement, which has the characteristic of low hydration heat. Currently, it is extensively used in the temperature control of mass concrete. ...Based on the thermodynamic database of OPC-based materials, the thermodynamic software GEM-Selektor (noted as GEMS) is used for simulating the hydration products of the LHP cement paste. Then, according to the GEMS thermodynamic simulation results, MATLAB is used to visualize the initial and ultimate stages of LHP cement pastes; the effects of curing temperature and water to cement (w/c) ratio on hydration products are addressed; and the porosity, fractal dimension, and tortuosity of different pastes are calculated. It is found that an appropriately high curing temperature is important for reducing porosity, especially in the early hydration stage. Hydration time also has a significant impact on the hydration of LHP cement paste; long hydration time may reduce the impact of temperature on hydration products. The w/c ratio is another important consideration regarding the hydration degree and porosity of LHP paste, and under different curing temperatures, hydration times, and w/c ratios, the porosity varies from 5.91–32.91%. The fractal dimension of this work agrees with the previous findings. From tortuosity analysis, it can be concluded that the high curing temperature may cause significant tortuosity, further affecting the effective diffusivity of LHP cement paste. For cement pastes with low w/c ratio, this high curing temperature effect is mainly reflected in the early hydration stage, for ones with high w/c ratio, it is in turn evident under long-term curing.
Drought is characterized by a moisture deficit that can adversely impact the environment, economy, and society. In North America, like many regions worldwide, predicting the timing of drought events ...is challenging. However, our novel study in climate research explores whether the Drought Monitor database exhibits fractal characteristics, represented by a single scaling exponent. This database categorizes drought areas by intensity, ranging from D0 (abnormally dry) to D4 (exceptional drought). Through vibration analysis using power spectral densities (PSD), we investigate the presence of power-law scaling in various statistical moments across different scales within the database. Our multi-fractal analysis estimates the multi-fractal spectrum for each category, and the Higuchi algorithm assesses the fractal complexity, revealing that D4 follows a multi-fractal pattern with a wide range of exponents, while D0 to D3 exhibit a mono-fractal nature with a narrower range of exponents.
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•Water treated reduced the pore volume and specific surface area.•Compared with water treated, mesopore volume increased in ionic liquid treated coal.•After ionic liquid treated, the ...pore becomes rougher and more complex.•Ionic liquid treated can change pore structures and improve coalbed methane mining.
In order to study the variation of pore characteristics and fractal dimensions, coal samples are treated by ionic liquid with different concentrations. Then the N2 isotherm adsorption/desorption experiment is carried out. The results show that water treatment caused a decrease in total pore volume. While the total pore volume in ionic liquid treated coal samples are larger than that in water treated coal sample. In comparison with water treated coal sample, the volumes of micropores in ionic liquid treated coal samples are reduced and the specific surface areas are increased. However, the volume and specific surface area of the mesopores are opposite to those in the water treated coal sample. The fractal dimensions D1 and D2 are calculated. The results show D1 decreases in water treated coal sample while increases in ionic liquid treated coal samples. D2 decreases obviously in water treated coal sample. Besides, it is lower than that in ionic liquid treated coal samples. Those indicate single water treatment will decrease the surface roughness and structural complexity of pores, while ionic liquid treatment will increase it. This study is of great significance to enhance the CBM exploitation and reduce the water block effect.
•The statistical results for corrosion pitting parameters to quantitatively illustrate the evolution law of corroded steel surface.•Calculated multi-fractal dimension of corroded steel surface and ...established the regression relation between multi-fractal dimension and statistical pitting results.•Optimizing the fractal reconstructed model by using the noise reduction and the continuous treatment.•Fractal reconstruction model to simulate corroded steel surface with high precision.
Corrosion is one of inevitable problems of constructional steel exposed to atmospheric environment in the long time, of which the quantification usually plays a key role in the accurate safety evaluation of corroded steel structure. Aiming to determine the reasonable quantitative method of corrosion, the morphological parameters were thoroughly investigated based on the surface measurements of 6 groups of corroded steel specimens obtained by accelerated corrosion test. By using the statistical analysis, the corrosion pitting parameters and their evolution rules were systematically analyzed and discussed. The distribution laws of pitting depth, width and spacing distance, and their statistical characteristic parameters (i. e., the expectation and the variance) were revealed. And then, their probability density functions were given at different pitting shape parameter. At last, the fractal character of corroded specimen surface was also researched. The results showed that all the pitting parameters obeyed the chi-square distribution (χ2). The corrosion pit evolved from a single corrosion pit to a double-stepped corrosion pit, and finally to be a single-stepped style. With increase of corrosion degree, the multi-fractal dimension (D) of corroded surface generally increased. Besides, the value of D can be well predicted by the combined effect of all characteristic parameters of corrosion pit. Finally, based on the W-M function and considering the relationship between the characteristic parameters and fractal dimension, and noise reduction and continuous treatment in consideration, the fractal reconstruction model of corroded steel surface was proposed, which can be utilized to accurately simulate the corroded surface of constructional steel.
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