RESEARCH AND MATHEMATICAL MODELING OF FRACTIONAL-DIFFERENTIAL RHEOLOGICAL MODELS

Deformation processes in media with fractal structure have been studied. At present, research on the construction of mathematical methods and models of interconnected deformation-relaxation and heat-mass transfer processes in environments with a fractal structure is at an early stage. There are a number of unsolved problems, in particular, the problem of correct and physically meaningful setting of initial and boundary conditions for nonlocal mathematical models of nonequilibrium processes in environments with fractal structure remains unsolved. To develop adequate mathematical models of heat and mass transfer and viscoelastic deformation in environments with fractal structure, which are characterized by the effects of memory, self-organization and spatial nonlocality, deterministic chaos and variability of rheological properties of the material, it is necessary to use non-traditional approaches. -differential operators. The presence of a fractional derivative in differential equations over time characterizes the effects of memory (eridity) or non-marking of modeling processes. The implementation of mathematical models can be carried out by both analytical and numerical methods. In particular, in this paper the integral form of fractional-differential rheological models is obtained on the basis of using the properties of the non-integer integral-differentiation operator and the Laplace transform method. The obtained analytical solutions of mathematical models of deformation in viscoelastic fractal media made it possible to obtain thermodynamic functions, creep nuclei and fractal-type relaxation. Developed software to study the effect of fractional differentiation parameters on the rheological properties of viscoelastic media.

Влияние фрактальной структуры на электрическое поле в грозовых облаках

The paper considers the degree of fractal structure impact on the electric field intensity inside thunderstorm clouds using the apparatus of fractional integrodifferentiation. We propose a mathematical model of intensity dynamics of a static electric field in the thunderstorm clouds, taking into account media with fractal dimension. The results obtained confirm the close connection of electrophysical processes in thunderclouds with the fractal medium itself. В статье с помощью аппарата дробного интегро-дифференцирования рассматривается степень влияния фрактальной структуры на напряженность электрического поля внутри грозовых облаков. Предлагается математическая модель динамики напряженности статического электрического поля в грозовых облаках с учетом сред с фрактальной размерностью. Полученные результаты подтверждают тесную связь электрофизических процессов в грозовых облаках с самой фрактальной средой В статье с помощью аппарата дробного интегро-дифференцирования рассматривается степень влияния фрактальной структуры на напряженность электрического поля внутри грозовых облаков. Предлагается математическая модель динамики напряженности статического электрического поля в грозовых облаках с учетом сред с фрактальной размерностью. Полученные результаты подтверждают тесную связь электрофизических процессов в грозовых облаках с самой фрактальной средой.

RECONFIGURABLE MONOPOLE ANTENNA DESIGN BASED ON FRACTAL STRUCTURE FOR 5G APPLICATIONS

a reconfigurable antenna design for 5G applications is presented. It is based on monopole antenna and fractal structure. The design structure is consisted of (monopole) feedline, ground plane, L-shape reflector, fractal structure and PIN diodes. The antenna is printed on (25×29×1.6 mm3) FR-4 substrate of εr=4.3 and tanδ =0.001. The antenna shows a resonant frequency at 4.1 GHz with S11=-11.4 dB and Omni-direction pattern of 1.21 dB gain. The L-shaped reflector is used to maintain the radiation pattern in a specific direction. Moreover, the proposed fractal structure is found to operate as a circuit to give another resonant frequency and enhance the antenna performance. Where it is used to give more manipulation in the antenna performance including: frequency resonance and radiation patterns. The PIN-diodes are used to give many cases for more current manipulation. moreover, the authors used RF (50 SMA port) between monopole antenna and right side of ground plane to optimize directing radiation pattern and to eliminate the problems of interference between AC and DC current that produced from using PIN diode. This manipulation leads to change the resonant frequency and radiation pattern to the desired direction.So all parts are printed on a single side of FR4 substrate

Fractal structure and hydration-driven shape memory of duck down in the dry–wet state

Duck down, as a natural keratin material, has been widely used as a filling material. The multilevel bifurcation structure of down has been observed and characterized through scanning electron microscopy. The structure is a complex fractal structure composed of four-level self-similar structures including five units, that is, the calamus, main barb, barb, barbule, and node or prong. The differential friction effect of the dynamic friction coefficients of the barb was reduced from 0.4 (dry state) to 0.23 (wet state), namely a decrease of 42.5%. The friction locking effect decreases due to the swelling of the fiber diameter. The down is zero gravity in water, and under the action of vibration and internal stress, down that has been subjected to friction or heat setting treatment can quickly return to its original shape in water. This shape memory mechanism was further confirmed, in which down after heat setting can restore its shape to the natural state by shaking it quickly and vigorously. This research provides inspiration to investigate more complicated functions of natural materials and encourages the creation of very intelligent synthetic polymers.

Finite Element Calculation of the Linear Elasticity Problem for Biomaterials with Fractal Structure

Aims: The aim of this study was to develop the mathematical models of the linear elasticity theory of biomaterials by taking into account their fractal structure. This study further aimed to construct a variational formulation of the problem, obtain the main relationships of the finite element method to calculate the rheological characteristics of a biomaterial with a fractal structure, and develop application software for calculating the components of the stress-strain state of biomaterials while considering their fractal structure. The obtained results were analyzed. Background: The development of adequate mathematical models of the linear elasticity theory for biomaterials with a fractal structure is an urgent scientific task. Finding its solution will make it possible to analyze the rheological behavior of biomaterials exposed to external loads by taking into account the existing effects of memory, spatial non-locality, self-organization, and deterministic chaos in the material. Objective: The objective of this study was the deformation process of biomaterials with a fractal structure under external load. Methods: The equations of the linear elasticity theory for the construction of the mathematical models of the deformation process of biomaterials under external load were used. Mathematical apparatus of integro-differentiation of fractional order to take into account the fractal structure of the biomaterial was used. A variational formulation of the linear elasticity problem while taking into account the fractal structure of the biomaterial was formulated. The finite element method with a piecewise linear basis for finding an approximate solution to the problem was used. Results: The main relations of the linear elasticity problem, which takes into account the fractal structure of the biomaterial, were obtained. A variational formulation of the problem was constructed. The main relations of the finite-element calculation of the linear elasticity problem of a biomaterial with a fractal structure using a piecewise-linear basis are found. The main components of the stress-strain state of the biomaterial exposed to external loads are found. Conclusion: Using the mathematical apparatus of integro-differentiation of fractional order in the construction of the mathematical models of the deformation process of biomaterials with a fractal structure makes it possible to take into account the existing effects of memory, spatial non-locality, self-organization, and deterministic chaos in the material. Also, this approach makes it possible to determine the residual stresses in the biomaterial, which play an important role in the appearance of stresses during repeated loads.

Fractal Characteristic Analysis of Urban Land-Cover Spatial Patterns with Spatiotemporal Remote Sensing Images in Shenzhen City (1988–2015)

Understanding the urban land-cover spatial patterns is of particular significance for sustainable development planning. Due to the nonlinear characteristics related to the spatial pattern for land cover, it is essential to provide a new analysis method to analyze them across remote sensing imagery. This paper is devoted to exploring the fractals and fractal dimension properties of land-cover spatial patterns in Shenzhen city, China. Land-cover information was extracted using a supervised classification method with ArcGIS technology from cloud-free Landsat TM/ETM+/OLI imagery, covering 1988–2015. The box-counting method and the least squares regression method are combined to estimate fractal dimensions of the land-cover spatial pattern. The information entropy was used to verify our fractal dimension results. The results show the fractal dimension changes for each land cover type from 1988 to 2015: (1) the land-cover spatial form of Shenzhen city has a clear fractal structure, but fractal dimension values vary in different land cover types; (2) the fractal dimension of build-up land increases and reaches a stable value, while grassland and cultivated land decrease; The fractal structure of grassland and bare land showed a bifractals trend increasing year by year; (3) the information entropy dimension growth is approaching its maximum capacity before 2011. We integrated the information entropy index and fractal dimension to analyze the complexity in land-cover spatial evolution from space-filling, space balance, and space complexity. It can be concluded that driven by policies, the land-cover spatial form in Shenzhen experienced a process from a hierarchical spatial structure with a low evolution intensity to a higher evolution intensity with multiscale differential development. The fractal dimension has been becoming better through self-organization, and its land resources are reaching the growth limits.

Dual-band metamaterial absorber with stable absorption performance based on fractal structure

In this paper, two kinds of dual-band metamaterial absorbers (MMAs) with stable absorption performance based on fractal structures are proposed. As the key feature, with the increase in fractal order, the fractal MMAs can reduce the weight while keeping the absorption performance. The multi-band absorption property is analyzed by multiple L-C resonances generated by the fractal structure. By virtue of good impedance matching characteristics and the synergy of the circuit and electromagnetic resonance, effective and stable microwave absorption is readily achieved. Finally, two prototypes are fabricated for demonstration, and the measurement result is consistent well with the simulation one. As expected, the proposed fractal MMAs have the advantage of low-cost, light-weight, and dual-effective absorption bands, and have great potential in the application of multi-band radar stealth.