A NOVEL VARIATIONAL PERSPECTIVE TO FRACTAL WAVE EQUATIONS WITH VARIABLE COEFFICIENTS

This paper aims at establishing two different types of wave models with unsmooth boundaries by the fractal calculus, and their fractal variational principles are successfully designed by employing the fractal semi-inverse transform method. A new approximate technology is proposed to solve the two fractal models based on the variational principle and fractal two-scale transform method. Finally, two numerical examples show that the proposed method is efficient and accurate, which can be extended to solve different types of fractal models.

Investigation and Application of Fractal Theory in Cement-Based Materials: A Review

Cement-based materials, including cement and concrete, are the most widely used construction materials in the world. In recent years, the investigation and application of fractal theory in cement-based materials have attracted a large amount of attention worldwide. The microstructures of cement-based materials, such as the pore structures, the mesostructures, such as air voids, and the morphological features of powders, as well as the fracture surfaces and cracks, commonly present extremely complex and irregular characteristics that are difficult to describe in terms of geometry but that can be studied by fractal theory. This paper summarizes the latest progress in the investigation and application of fractal theory in cement-based materials. Firstly, this paper summarizes the principles and classification of the seven fractal dimensions commonly used in cement-based materials. These fractal dimensions have different physical meanings since they are obtained from various testing techniques and fractal models. Then, the testing techniques and fractal models for testing and calculating these fractal dimensions are introduced and analyzed individually, such as the mercury intrusion porosimeter (MIP), nitrogen adsorption/desorption (NAD), and Zhang’s model, Neimark’s model, etc. Finally, the applications of these fractal dimensions in investigating the macroproperties of cement-based materials are summarized and discussed. These properties mainly include the mechanical properties, volumetric stability, durability (e.g., permeability, frost and corrosion resistance), fracture mechanics, as well as the evaluation of the pozzolanic reactivity of the mineral materials and the dispersion state of the powders.

Physical modeling of water-retention capacity of soils

The function of the water-retention capacity of the soil is necessary, for example, when calculating irrigation norms in irrigation agriculture. Various mathematical models are used to approximate the water-retention capacity, which have a number of disadvantages inherent in these models. For example, the absence of physically adequate analytical descriptions for the coefficients of a given function. The use of physical fractal models for predicting and calculating the water-retention capacity of soils seems promising. Application of the fractal model Pore-Solid-Fractal is necessary to perform the calculation of desorption curves of water-retention capacity of some types of alpha-humus and texture-differentiated soils of light particle size distribution has been performed. The calculated data for the drying branches of the WRC are compared with the experimental data. The study of statistical differences between samples (data convergence) was carried out using the Mann-Whitney test (U). The empirical values of the U-test are from 17.5 to 20. The critical value of the U-test for a given number of compared data series at a probability level of 0.99 is 8. The critical value of the U-test for a given sample size is less than the calculated one, respectively, the difference between the series of empirical and the calculated data in the sample are not statistically significant. The fractal model allows calculating the water-retention capacity function of soils with high accuracy.

Predicting Fluid Flow Regime, Permeability, and Diffusivity in Mudrocks from Multiscale Pore Characterisation

In geoenergy applications, mudrocks prevent fluids to leak from temporary (H2, CH4) or permanent (CO2, radioactive waste) storage/disposal sites and serve as a source and reservoir for unconventional oil and gas. Understanding transport properties integrated with dominant fluid flow mechanisms in mudrocks is essential to better predict the performance of mudrocks within these applications. In this study, small-angle neutron scattering (SANS) experiments were conducted on 71 samples from 13 different sets of mudrocks across the globe to capture the pore structure of nearly the full pore size spectrum (2 nm–5 μm). We develop fractal models to predict transport properties (permeability and diffusivity) based on the SANS-derived pore size distributions. The results indicate that transport phenomena in mudrocks are intrinsically pore size-dependent. Depending on hydrostatic pore pressures, transition flow develops in micropores, slip flow in meso- and macropores, and continuum flow in larger macropores. Fluid flow regimes progress towards larger pore sizes during reservoir depletion or smaller pore sizes during fluid storage, so when pressure is decreased or increased, respectively. Capturing the heterogeneity of mudrocks by considering fractal dimension and tortuosity fractal dimension for defined pore size ranges, fractal models integrate apparent permeability with slip flow, Darcy permeability with continuum flow, and gas diffusivity with diffusion flow in the matrix. This new model of pore size-dependent transport and integrated transport properties using fractal models yields a systematic approach that can also inform multiscale multi-physics models to better understand fluid flow and transport phenomena in mudrocks on the reservoir and basin scale.

Application of Traditional Cultural Symbols in Art Design under the Background of Artificial Intelligence

In order to solve the declining influence of traditional cultural symbols, the research on traditional cultural symbols has become more meaningful. This article aims to study the application of traditional cultural symbols in art design under the background of artificial intelligence. In this paper, a fractal model with self-combined nonlinear function changes is constructed. By combining nonlinear transformations and multiparameter adjustments, various types of fractal models can be automatically rendered. The convolutional neural network algorithm is used to extract the characteristics of the style picture, and it is compared with the trained picture many times to avoid the problem of excessive tendency of the image with improper weight. The improved L-BFGS algorithm is also used to optimize the loss of the traditional L-BFGS, which improves the quality of the generated pictures and reduces the noise of the chessboard. The experimental results in this paper show that the improved L-BFGS algorithm has the least loss and the shortest time in the time used for more than 500 s. Compared with the traditional AdaGrad method, its loss is reduced by about 62%; compared with the traditional AdaDelta method, its loss is reduced by 46%. Its loss is reduced by about 8% compared with the newly optimized Adam method, which is a great improvement.

May the Piezoresistivity of GNP-Modified Cement Mortar Be Related to Its Fractal Structure?

High piezoresistivity of cement-based composites tuned by conductible fillers provides a feasible way to develop self-sensing smart structures and buildings. However, the microstructural mechanisms remain to be properly understood. In the present work, the piezoresistivity of cement mortar with different dosages of graphene nanoplatelets (GNPs) was investigated, and the microstructure was assessed by electron scanning microscopy (SEM) and mercury intrusion porosimetry (MIP). Two surface fractal models were introduced to interpret the MIP data to explore the multi-scale fractal structure of the GNP-modified cement mortars. Results show that the incorporation of GNPs into cement mortar can roughen the fracture surfaces due to the GNPs’ agglomeration. Gauge factor (GF) rises and falls as GNP content increases from 0% to 1% with the optimal piezoresistivity observed at GNP = 0.1% and 0.05%. The GF values of the optimum mortar are over 50 times higher than those of the reference mortar. Fractal dimensions in macro and micro fractal regions change with GNP content. Analysis shows that the fractal dimensions in micro region decrease first and then increase with the increase of GF values. GNPs not only impact the fractal structure of cement mortar, but also alter the tunneling and contact effects that govern the piezoresistivity of composite materials.

Machine Learning, Compositional and Fractal Models to Diagnose Soil Quality and Plant Nutrition

Soils, nutrients and other factors support human food production. The loss of high-quality soils and readily minable nutrient sources pose a great challenge to present-day agriculture. A comprehensive scheme is required to make wise decisions on system’s sustainability and minimize the risk of crop failure. Soil quality provides useful indicators of its chemical, physical and biological status. Tools of precision agriculture and high-throughput technologies allow acquiring numerous soil and plant data at affordable costs in the perspective of customizing recommendations. Large and diversified datasets must be acquired uniformly among stakeholders to diagnose soil quality and plant nutrition at local scale, compare side-by-side defective and successful cases, implement trustful practices and reach high resource-use efficiency. Machine learning methods can combine numerous edaphic, managerial and climatic yield-impacting factors to conduct nutrient diagnosis and manage nutrients at local scale where factors interact. Compositional data analysis are tools to run numerical analyses on interacting components. Fractal models can describe aggregate stability tied to soil conservation practices and return site-specific indicators for decomposition rates of organic matter in relation to soil tillage and management. This chapter reports on machine learning, compositional and fractal models to support wise decisions on crop fertilization and soil conservation practices.