Koch Fractal Research Articles

Size reduction of the square microstrip patch antenna with Koch fractal for 1.575 GHz applications

This paper discusses a square patch microstrip antenna with a resonant frequency of 1.575 GHz. The initial design of the antenna was carried out using a theoretical approach and then applied fractal Koch with 1-iteration and 2-iteration. The design optimization is done using the AWR simulator. Based on the optimal design results, the use of Koch fractals has been able to reduce the antenna size. In the first iteration, the antenna size can be reduced by 28% with a minimum return loss value of -33.36 dB at a frequency of 1.575 GHz. In the second iteration, the antenna size can be reduced by 42.4% from the design without fractals with a minimum return loss of -37.18 dB at a frequency of 1.575 GHz. The gain of each antenna design has almost the same value, namely 5.115 dB.

Hölder Derivative of the Koch Curve

In this paper, we introduce a K Hölder p-adic derivative that can be applied to fractal curves with different Hölder exponent K. We will show that the Koch curve satisfies the Hölder condition with exponent and has a 4-adic arithmetic-analytic representation. We will prove that the Koch curve has exact -Hölder 4-adic derivative.

Koch Snowflake Fractal Embedded Octagonal Patch Antenna with Hexagonal Split Ring for Ultra-wide Band and 5G Applications

Koch Snowflake Fractal Embedded Octagonal Patch Antenna with Hexagonal Split Ring for Ultra-wide Band and 5G Applications

Double Negative (DNG) Metamaterial-Based Koch Fractal MIMO Antenna Design for Sub-6-GHz V2X Communication

Double Negative (DNG) Metamaterial-Based Koch Fractal MIMO Antenna Design for Sub-6-GHz V2X Communication

Differential and Integral Operations in Hidden Spherical Symmetry and the Dimension of the Koch Curve

Приведены примеры дифференциальных и интегральных операций, размерность которых меняется в результате введения новых радиальных переменных. На основе интегральной меры со слабой особенностью $x^\gamma dx$, $\gamma>-1$, вводится оператор, который интерпретируется как оператор Лапласа в пространстве функций дробного числа переменных. Интегрирование по мере $x^\gamma dx$, $\gamma>-1$, также может интерпретироваться как интегрирование по области дробной размерности. Вводится коэффициент скрытой сферической симметрии $\gamma>-1$. Получена формула, связывающая этот коэффициент с размерностью Хаусдорфа множества в $\mathbb{R}_n$ и евклидовой размерностью $n$. Существование скрытых сферических симметрий проверено вычислением размерности $m$-го поколения кривой Коха для произвольных натуральных значений $m$. Библиография: 17 названий.

Design of Optimized Koch Based Fractal Patch Antenna for Multiband Wireless Applications

An optimized Koch based fractal patch antenna with twin lower leafs has been designed for appropriate wireless communication applications and capable of exhibiting penta-band behavior. The proposed structure has been created by introducing modified Koch curve on a rhombus shaped fractal patch having compact dimensions of 60 mm×50 mm×1.6 mm with defected ground structure. The reflection coefficient (S11) values of the proposed antenna are -11.97 dB, -20.29 dB, - 25.38 dB, -13.91 dB and -23.48 dB at resonating frequencies of 2.8649 GHz, 6.3514 GHz, 7.1622 GHz, 8.2973 GHz, and 8.9459 GHz, respectively. It has values of VSWR at all resonating frequencies within acceptable range of 1 to 2. The prototype of the proposed structure has been designed on easily available and low cost FR4 epoxy substrate material. The bandwidths of 1.39 GHz, 3.30 GHz, 10.19 GHz, 1.93 GHz and 2.23 GHz has been obtained at all resonating frequencies. The measured results has been analyzed and compared with simulated results and bears a close approximation. The developed prototype can be utilized for applications in S, C, and X frequency band which can further be used for various communication applications like radar, satellite communication and wireless computer networks.

Fractal fuzzy logic using Kelly plots

Strictly defined, the concept of self-similarity or self-similarity applies only to mathematical fractals - which arise from the iteration of simple formulae but lead to very complex structures, Cantor Dust, Peano Curve, Koch Snowflake, whereas in natural or physical fractals - those found in nature, such as a fern leaf, an arborisation, capillaries - the concept of self-similarity applies, since their fractality is only statistical and they possess, consequently, an anisotropic scaling,(not having the same properties in all dimensions of analysis), which does not allow an amplified part of a figure to maintain exactly the characteristics of the figure as a whole, is where we find Kelly plots.

Design of 1 × 2 MIMO Palm Tree Coplanar Vivaldi Antenna in the E-Plane with Different Patch Structure

In this paper, 1 × 2 MIMO of Palm Tree Coplanar Vivaldi Antenna is presented that simulated at 0.5–4.5 GHz. Some GPR applications require wideband antennas starting from a frequency below 1 GHz to overcome high material loss and achieve deeper penetration. However, to boost the gain, antennas are set up in MIMO and this is costly due to the large size of the antenna. When configuring MIMO antenna in the E-plane, there is occasionally uncertainty over which antenna model may provide the optimum performance in terms of return loss, mutual coupling, directivity, beam squint, beam width, and surface current using a given substrate size. However, the configuration of E-plane antenna in MIMO has an issue of mutual coupling if the distance between elements is less than 0.5λ. Furthermore, it produces grating lobes at high frequencies.We implement several types of patch structures by incorporating the truncated, tilt shape, Hlbert and Koch Fractal, Exponential slot, Wave slot, the lens with elips, and metamaterial slot to the radiator by keeping the width of the substrate and the shape of the feeder. The return loss, mutual coupling, directivity, beam squint, beamwidth, and surface current of the antenna are compared for 1 × 2 MIMO CVA. A continuous patch MIMO has a spacing of 0.458λ at 0.5 GHz, which is equivalent to its element width. From the simulation, we found that Back Cut Palm Tree (BCPT) and Horizontale Wave Structure Palm Tree (HWSPT) got the best performance of return loss and mutual scattering at low-end frequency respectively. The improvement of directivity got for Metamaterial Lens Palm Tree (MLPT) of 4.453 dBi if compared with Regular Palm Tree-Coplanar Vivaldi Antena (RPT) at 4 GHz. Elips Lens Palm Tree (ELPT) has the best beam squint performance across all frequencies of 0°. It also gots the best beamwidth at 4.5 GHz of 3.320. In addition, we incorporate the MLPT into the radar application.

The Influence of the Aperture Stop Fractal Shape of an Optical System on the Illuminance Distribution

In the article the options for the application of aperture shapes with fractal properties in the design of optical systems are considered. Calculations of mathematical models of point spread functions of a diffraction-limited optical system are performed. The diffraction patterns of the light distribution in these systems are presented, and the point spread functions are considered for various shapes of the aperture stop. Analytical expressions are obtained for the light distribution depending on the pupil shape, which can be used to control the process of image formation. The pupil shape, which has the shape of an equilateral triangle, is chosen as the basic one, and the shape of the pupil as a "Koch snowflake" curve is also considered. Using the Fraunhofer integral, the dependences of the distribution of the spectral density of the complex amplitude on the aperture located on an opaque screen are derived in the Fraunhofer approximation and under the condition of illumination by a plane monochromatic wave. Using the relationship with the complex amplitude, the sought-for intensity distribution in the plane of the diffraction pattern is obtained. Taking into account the simplifications adopted in this article, the solution of the Fraunhofer integral is found, by setting the integration limits, depending on: the selected aperture profile, the coordinate system chosen for it, and the position of nodal points in this system

A novel probe‐fed directional dielectric resonator antenna for mobile communication

AbstractA directional dielectric resonator antenna (DDRA) for 5G mobile communication system is proposed. The antenna is side‐fed with the metal probe to make it work in quasi‐TEδ11 mode, so as to realize the characteristics of directional radiation. The stacked structure is adopted, and the two side walls of the rectangular dielectric resonator parallel to the microstrip line are grooved to improve the frequency bandwidth of the antenna. Furtherly, the Koch snowflake fractal is used for the second iteration of the top cylindrical dielectric resonator, and the parasitic metal columns inside the DRA are also helpful for the enhancement of bandwidth. The prototype was fabricated and tested to verify the design, and the measured results show −10 dB impedance bandwidth of about 23% and an average gain of 5.2 dBi within the operation band.

Numerical investigation of thermal energy storage system loaded with nano-enhanced phase change material with Koch snowflake fractal cross-section

The melting of nano-enhanced phase change material (NEPCM) in a latent heat thermal storage system is investigated for PCM loaded within an annulus between an outer wavy cylinder and an inner Koch snowflake cylinder. The inner Koch snowflake cylinder is used in this study to increase the heating surface area and enhance the NEPCM melting process. The inner cylinder is hot, while the outer one is insulated. The system governing equations and the associated boundary conditions were numerically solved by employing the Galerkin weighted residual finite element method. The impacts of nanoparticle concentration, undulation number of the outer cylinder, annulus eccentricity, and inner cylinder geometry on the melting performance of the storage unit were examined and represented by the liquid fraction, isotherms, Bejan number, velocity, and Nusselt number. The usage of more blades and snowflake surfaces near the hot source's periphery portions disrupts the PCM surface's cohesiveness and increases heat penetration between the PCM layers. The nanoparticle concentration of 6 vol% enhances the melting time by about 32 %, and the undulation number of 8 participates in 25 % of performance enhancement. Putting the hot cylinder closer to the bottom of the storage unit expands the buoyancy effect and enhances the melting process by >3.2 times. The utilization of a Koch snowflake cylinder instead of a triangular one also shows an enhancement of >30 %.

Four variations on a fractal theme

Four variations of the Koch curve are presented. In each case, the similarity dimension, area bounded by the fractal and its initiator, and volume of revolution about the initiator are calculated. A range of classroom exercises are proved to allow students to investigate the fractals further.

A novel modified Kock–Snowflake fractal MIMO antenna for 4G/5G UWB communication gadgets

AbstractA dual‐port, tapered‐fed fractal antenna with a partial ground is proposed for effective implementation in portable ultra‐wideband–multiple‐input multiple‐output devices. The array is modeled on a 41 × 71 × 1.57 mm3 dielectric FR4 substrate. Each circular radiator defected with a modified Koch–Snowflake fractal (4th iteration) provides 46.7% size miniaturization. The array yields 14.7 GHz bandwidth (3.3–18 GHz, 138%) by defecting the partial ground and offsetting the tapered feedlines. By embedding an H‐shaped decoupler and plus‐shaped slots in the ground, significant isolation (S21/S12 ≤ −16 dB) is accomplished. Various diversity metrics are investigated and are found to follow their permissible standards. To justify the simulated results, the fabricated array is tested where a favorable match between the results makes it a suitable candidate for high data‐rate wireless applications.

MHD natural convection of Fe3O4- MWCNT/Water hybrid nanofluid filled in a porous annulus between a circular cylinder and Koch snowflake

In this work, the numerical investigation was conducted for the MHD natural convection and entropy generation characteristics of water-based hybrid nanofluid (Fe3O4-MWCNT) in a porous annulus between a cooled circular cylinder and a heated Koch snowflake subjected to a uniform magnetic force. The novelty of this work is presented by the special shape and different studied positions of the hot barrier. The governing equations are explained by employing the Finite Element Method. The impacts of nanoparticle volume fraction (φ = 2 %, 3%,4%, and 5 %), Rayleigh number (Ra = 103 to 106), Hartman number (Ha = 0,25,50,100), Darcy number (Da = 10-2, 10-3, 10-4, and 10-5), and the position of the Koch snowflake (four cases) on the distributions of isotherms, streamlines, average Nusselt number (Nuavg) as well as on total entropy generation and Bejan number are thoroughly examined. The computational outcomes indicate that increasing the Ra number is possible by changing the temperature between the hot and cold sources. By increasing this parameter, the buoyancy force of the fluid is strengthened. As the Da number increases, the penetration of the flow cross-section in the cavity increases, and the flow circulates in the cavity with less depreciation due to the buoyancy force. Applying Lorentz force, if not in the direction of natural flow, causes the flow velocity to be depleted and fluid flow in the cavity to be facilitated. With increasing Ra number, the application of Lorentz force with different intensities becomes important due to the importance of fluid circulation in the cavity. Applying a Lorentz force with a higher Ha number at high Ra numbers reduces heat transfer due to junction balance and flow separation on hot and cold surfaces. In the lower Ra numbers, the Bejan number tends to 1, indicating a significant increase in the temperature gradient in the cavity and the expansion of the thermal boundary layer at low fluid velocities. At the highest Ra, increasing Ha from 0 to 100 decreases Nuavg by 50 %, while decreasing Da from 10−2 to 10−5 reduces Nuavg by 70 %.

Correction to: A Low Profile Meta-Material Loaded Koch Fractal Antenna for Broadband Public Safety Applications

Correction to: A Low Profile Meta-Material Loaded Koch Fractal Antenna for Broadband Public Safety Applications

A Low Profile Meta-Material Loaded Koch Fractal Antenna for Broadband Public Safety Applications

A Low Profile Meta-Material Loaded Koch Fractal Antenna for Broadband Public Safety Applications

A Passive Micromixer with Koch Snowflakes Fractal Obstacle in Microchannel

The passive micromixer is one of the essential devices that can be integrated into the Lab on Chip (LoC) system. Micromixer is needed to increase mixing efficiency. In this paper, two Koch fractal obstacle-based micromixer models of Secondary Snowflakes Fractal Micromixer (SSFM) and Tertiary Snowflakes Fractal Micromixer (TSFM) were designed. The effect of the Koch fractal resistance angles (15o, 30o, 45o, 315o, 330o, 345o) and the influence of the inlet (T and T-vortex) were studied in this paper using COMSOL Multiphysics numerical simulations. The results showed that the TSFM structure with a 30o angle on the T-vortex inlet is optimal. The deflection phenomena generated by the TSFM obstacle enhance the contact area between the two fluids and chaotic convection can be increased at Reynolds Number (Re) 0.05 and Re 100. This paper examines concentration curves along the channel ranging from 1 mol/L to 5 mol/L. This clearly shows that the fluid flow direction changes within the microchannel. This work provides a new design for the micromixer.

Design of Spiral Resonator Based on Fractal Metamaterials and Its Improvement for MCR-WPT Performance

To further optimize the electromagnetic characteristics of square spiral resonator (SSR) in wireless power transfer (WPT) system, the SSR fractal finite element model is established in the high frequency structure simulator (HFSS) simulation software based on Koch curve and Hilbert curve. The mechanism of the effect of Koch and Hilbert fractal structure on frequency and permeability characteristics of SSR is analyzed in depth. The inductance of the resonant coil is calculated by the Maxwell software and then a magnetically coupling resonant WPT (MCR-WPT) system operating at 6.78 MHz is constructed in HFSS. The influence mechanism of the spiral metamaterial (MM) structure combined with the Koch/Hilbert fractal on the transfer performance of MCR-WPT system is analyzed. The simulation and experimental results show that the power transfer efficiency (PTE) of the system can be improved effectively when the negative permeability/hybrid MM slab (NMS/HMS) based on Koch/Hilbert/initial SSRs is inserted in the middle of the transfer distance. In the experiment, the PTE of the HMS based on Hilbert fractal can be improved by up to 21.62%.

Sophisticated acoustofluidic patterns generated in quasi-Sierpiński-carpet shaped chambers with heterogeneous radiation surface distributions

In this research, an original strategy to generate diversified acoustofluidic fields in quasi-Sierpiński-carpet shaped chambers has been proposed and simulated for patterned manipulation of massive micro-scale particles. All of the structural elements in the fractalized chambers are symmetrical in spatial arrangement, and all radiation surfaces possess the same setting of input frequency point and oscillation amplitude along individual normal direction. Plenty of splendid acoustofluidic patterns can be obtained in the originally-static quasi-Sierpiński-carpet shaped chambers generated at different levels of recursion without complicated parameter modulation. The simulation results of acoustofluidic distributions together with micro-scale particle movement trajectories under different oscillation conditions further demonstrate the manipulation functionality of these artificially-designed devices. In comparison with the existing structural design schemes of miniaturized lab-on-a-chip systems, the introduction of fractal features like Sierpiński carpet/triangle and Koch snowflake can provide extraordinary insights and broaden application prospects of acoustofluidics, which is conducive to ultrasonic micro/nano manipulations from simplification to pluralism. The preliminary work shows the possibility of utilizing Sierpiński-type fractal structures as amenable components to customize acoustofluidic fields for the investigation of topographical manipulation of biological samples and orientational manoeuvre of micro/nano-scale machines in ways that are not achievable via traditional approaches.

Application of Selected Fractal Geometry Resonators in Microstrip Strain Sensors

Until now, strain gauges, fiber optic sensors, magnetic sensors, and piezoelectric sensors have been used for strain measurements. In recent years, the idea of using microstrip antennas in strain measurement applications has emerged. One of the key requirements for deformation sensors is the ability to mount them on small-size elements (therefore the size of these sensors is crucial). The possible way to miniaturize microstrip sensors is the application of appropriate patch geometry, e.g. fractal geometry. In this article application of various fractal patch geometries in microstrip strain sensors were investigated. All sensors were designed for the same operating frequency <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${f}_{\textbf {r}} =2.725$ </tex-math></inline-formula> GHz and on the same laminate. This approach showed how the influence of a specific geometry affects the sensitivity and the transducer size. The Sierpinski carpet, the Sierpinski triangle, the Sierpinski curve, and the combination of the Koch curve and the Sierpinski carpet were used to obtain the fractal geometry and miniaturize the sensor dimensions. They have been compared with the widely used rectangular patch. Numerical and experimental analyzes for the proposed sensors were carried out. The best solution to this problem was to use a combination of the Koch curve and the Sierpinski carpet. This fractal geometry, compared to a rectangular patch, enables the reduction of the patch size by 70% while reducing the sensitivity by 26%.