Koch Fractal Research Articles

In-plane wave propagation analysis for waveguide design of hexagonal lattice with Koch snowflake

In this study, the band structure and directional characteristics of the hexagonal lattice with Koch snowflake are investigated. The finite element method and Bloch theorem are used to analyze the wave propagation in hexagonal lattice with Koch snowflake for different fractal orders. The effects of the geometric parameters on band structures are discussed in detail. The group velocities at given frequencies are calculated for analyzing the directional characteristics of the wave propagation. The dynamic responses of the periodic lattice are investigated via numerical simulation to confirm the directional features. The vibration experiments are conducted to verify the existence of band gap and vibration suppression performances of the structures, which are used to design the waveguide in the lattices. The results demonstrate that the hexagonal lattice with Koch snowflake for different fractal orders can tune the elastic wave behaviors in a specific way. As the fractal order increases, multiple band gaps appear and the band structure has self-similarity in the low-frequency range. The directional characteristics of wave propagation in the hexagonal lattice with Koch snowflake can be used to further expand the vibration attenuation range. The results also show the potential importance and feasibility of using the hexagonal lattice with Koch snowflake for vibration isolation and control using the hexagonal lattice with Koch snowflake.

An extensive survey on fractal structures using iterated function system in patch antennas

SummaryOver the last few years, fractal structure has been used in several engineering fields such as surface physics, telecommunication, fluid mechanics, and so forth. In telecommunication, often there is a demand for miniaturized patch antennas with fractal structures because it is capable of handling very high frequencies with its diminutive structure. The space filling capability of the fractal structures is used to effectively distribute the surface current all over the radiating material. Fractal structures have a better size reduction capability which aids a miniaturized antenna design. This article provides a generalized and detailed survey on fractal structures in a Euclidean space using iterated function system (IFS) in patch antennas. There are various techniques to design a fractal structure, namely, IFSs, escape‐time fractals, strange attractors, L‐systems, random fractals, and so forth. IFSs are widely used technique to construct a finite set of construction mapping on a complete metric space. The fractal geometries designed using the technique are Sierpinski carpet and gasket, Koch snowflake, deterministic tree/binary tree, Gosper island, Minkowski island, Cantor set, T‐square, and some space filling fractals. This paper deals with the above given fractal structures with their design expressions and shapes in recent developments and applications mentioned. The fractal geometries are used in MIMO antennas to abridge the mutually coupled radiators and improve the gain and efficiency of the antenna. This paper also analyzes the effects of fractal structures on MIMO antennas and discusses the ascendancy fractal structures by cross‐referencing the significant antenna parameters.

Network and Field Analysis of Koch Snowflake Fractal Geometry Radiofrequency Coils for Sodium MRI

Sodium is one of the most abundant physiological cations and is a key element in many cellular processes. It has been shown that several pathologies, including degenerative brain disorders, cancers, and brain traumas, express sodium deviations from normal. Therefore, sodium magnetic resonance imaging (MRI) can prove to be valuable for physicians. However, sodium MRI has its limitations, the most significant being a signal-to-noise ratio (SNR) thousands of times lower than a typical proton MRI. Radiofrequency coils are the components of the MRI system directly responsible for signal generation and acquisition. This paper explores the intrinsic properties of a Koch snowflake fractal radiofrequency surface coil compared to that of a standard circular surface coil to investigate a fractal geometry’s role in increasing SNR of sodium MRI scans. By first analyzing the network parameters of the two coils, it was found that the fractal coil had a better impedance match than the circular coil when loaded by various anatomical regions. Although this maximizes signal transfer between the coil and the system, this is at the expense of a lower Q, indicating greater signal loss between the tissue and coil. A second version of each coil was constructed to test the mutual inductance between the coils of the same geometry to see how they would behave as a phased array. It was found that the fractal coils were less sensitive to each other than the two circular coils, which would be beneficial when constructing and using phased array systems. The performance of each coil was then assessed for B1+ field homogeneity and signal. A sodium phantom was imaged using a B1+ mapping sequence, and a 3D radial acquisition was performed to determine SNR and image quality. The results indicated that the circular coil had a more homogeneous field and higher SNR. Overall while the circular coil proved to generate a higher signal-to-noise ratio than the fractal, the Koch coil showed higher versatility when in a multichannel network which could prove to be a benefit when designing, constructing, and using a phased array coil.

SETI strategy with FAST fractality

We applied the Koch snowflake fractal antenna in planning calibration of the Five-hundred-meter Aperture Spherical radio Telescope (FAST), hypothesizing second-order fractal primary reflectors can optimize the orientated sensitivity of the telescope. Meanwhile, on the grounds of NASA Science Working Group Report in 1984, we reexamine the strategy of Search for Extraterrestrial Intelligence (SETI). A mathematical analysis of the radar equation will be performed in the first section, aiming to make it convenient to design a receiver system that can detect activities of an extraterrestrial civilization, according to the observable region of the narrowband. Taking advantage of the inherent potential of FAST, we simulate the theoretical detection of a Kardashev Type I civilization by a snowflake-selected reflecting area (Drake et al.).

GENERALIZED KOCH CURVES AND THUE–MORSE SEQUENCES

Let [Formula: see text] be the well-known [Formula: see text] Thue–Morse sequence [Formula: see text] Since the 1982–1983 work of Coquet and Dekking, it is known that [Formula: see text] is strongly related to the famous Koch curve. As a natural generalization, for [Formula: see text], we use [Formula: see text] to define the generalized Koch curve, where [Formula: see text] is the generalized Thue–Morse sequence defined to be the unique fixed point of the morphism [Formula: see text] [Formula: see text] beginning with [Formula: see text] and [Formula: see text], and we prove that generalized Koch curves are the attractors of the corresponding iterated function systems. For the case that [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], the open set condition holds, and then the corresponding generalized Koch curve has Hausdorff, packing and box dimension [Formula: see text], where taking [Formula: see text] and then [Formula: see text] will recover the result on the classical Koch curve.

Modified Koch Fractal Antenna for Multi and Wideband Wireless Applications

This paper focuses on the design and development of modified Koch fractal antenna. Compared to traditional Koch curve antenna, the presented antenna possesses a greater number of frequency bands and better impedance matching. Furthermore, the bacterial foraging optimization (BFO) approach is implemented to enhance the impedance bandwidth. The developed technique has been verified by employing various numerical simulations. The design parameters generated from the optimization procedure have been utilized to manufacture the antenna and the respective experimental and simulated results compared. The measured results show that the designed antenna exhibits multi and wideband behavior, covering WLAN, WIMAX, and various other wireless applications.

Power Number and Mixing Drag Coefficient of Koch Fractal Impellers used by Koch Curve

Power Number and Mixing Drag Coefficient of Koch Fractal Impellers used by Koch Curve

Peano–Gosper, Koch and Minkowski fractal curves-based novel hybrid antenna using modified partial ground plane for multi-standard wireless applications

This paper presents the unique structure of a hybrid antenna with a blend of Peano–Gosper, Koch and Minkowski curves using a modified partial ground plane for multi-standard wireless applications. Initially, the Minkowski and Koch curves are amalgamated by using a versatile mathematical tool called Iterative Function System (IFS), and furthermore, their resultant structure is superimposed on each segment of Peano–Gosper curve to obtain the novel geometry of hybrid fractal antenna named Gosper–Koch–Minkowski Hybrid Fractal Antenna (GKMHFA). The partial ground plane with 50 Ω transmission line fed is used to analyse the performance of designed structure and found that it exhibits five different frequency bands above 9 GHz in the frequency range 1–20 GHz. Consequently, to attain the frequency bands below 9 GHz with enhanced bandwidth and reflection coefficient, the partial ground plane has been modified. The proposed antenna with the modified ground plane reveals seven distinct frequency points 1.5, 3.4, 8.6, 12.2, 14.1, 15.3 and 18.9 GHz with a maximum bandwidth 5.0 GHz along with improved reflection coefficient i.e. −45.65 dB at 1.5 GHz. The proposed antenna also reports positive gain and stable radiation patterns in the operational frequency bands, which make it a proficient candidate for the multi-standard wireless applications such as GPS, terrestrial communication, PCS, UMTS, advance wireless services, Bluetooth, WLAN, Wi-Fi, Wi-MAX, military satellite communication, television broadcasting, broadcasting satellite, FSS, Ku-band and K-band.

Compact Koch Fractal Dipole RFID Antenna

In this paper, the design of Koch Fractal dipole antenna for first, second and third iteration have been presented. The third iteration of Koch Fractal Dipole Antenna shows a total size reduction of 21.25% compared to the basic dipole antenna optimized at the same frequency of 0.92 GHz. The new antenna is proposed based on the structure of the third iterated Koch fractal is designed using Rogers RO3010 substrate with thickness on 1.27 mm. The proposed antenna shows improvement of return loss to -34.57 dB, gain of 1.92 dB, 99% efficiency. The proposed antenna had reduced 17% of length compared to the basic dipole antenna.

Computation of sharp estimates of the Poincaré constant on planar domains with piecewise self-similar boundary

We establish a strategy for finding sharp upper and lower numerical bounds of the Poincaré constant on a class of planar domains with piecewise self-similar boundary. The approach consists of four main components: W1) tight inner-outer shape interpolation, W2) conformal mapping of the approximate polygonal regions, W3) grad-div system formulation of the spectral problem and W4) computation of the eigenvalue bounds. After describing the method, justifying its validity and determining general convergence estimates, we show concrete evidence of its effectiveness by computing lower and upper bound estimates for the constant on the Koch snowflake.

A Uhf Near Field RFID Reader Antenna Implemented with Fractal Koch Curve

This paper comprises a fractal Koch curve with tapered slot structure subjected to magnetic field strengthanalysis for Ultra High Frequencies (UHF) RFID near field reader antenna. In the design, Variable distance with opposite direction current principle is used. The center frequency of the proposed antenna operates at 897 MHz with bandwidth of 101 MHZ. (853 to 954 MHz). The obtained bandwidth is sufficient to cover the UHF RFID frequency for, Canada (902-928 MHz), Europe andUSA (865-868 MHz). The near field analysis was verified by keeping different liquids and the antenna performance is calculated. During the measurement, the antenna is kept at a distance of 50 mm in z direction and is observed that the interrogation area of proposed antenna is achieved with 243 mm X 190 mm. The result shows that the antenna exhibits the ability to track chemicals in industries and pharmaceutical applicationsmanagement. It is also observed that the proposed antenna is sufficient to operate as commercial tags for near field magnetic induction communication.

Compressive behaviors of fractal-like honeycombs with different array configurations under low velocity impact loading

Fractal-like structures have been found to offer excellent mechanical performance, showing remarkable potential in engineering applications. This study proposed the fractal-like honeycombs with different array configurations. Their intricate patterns were constructed based on a simple fractal cell inspired by Koch curve. Compressive behaviors of the proposed honeycombs were investigated theoretically and numerically. Compared with conventional honeycombs, the specific energy absorption (SEA) of the proposed honeycombs can be improved by 85%, 110% and 95% for triangular, square and hexagonal honeycombs, respectively. For every array configuration, the proposed structures all exhibit higher crushing force efficiency (CFE). Theoretical models of conventional honeycombs and the optimal fractal-like honeycombs were established. Folding mechanism was analyzed to reveal the mechanical improvement by comparing details of folding lobes in the forming process. Analytical solutions were compared in terms of the half-wave length, mean crushing force and SEA by a solution map, reflecting the ratios and trends of increment in each criterion. The influence of the cell number on the half-wave length was studied. The findings of this study provide new directions and guidelines for designing novel energy absorbers with enhanced crashworthiness.

Fractal Interpolation Using Harmonic Functions on the Koch Curve

The Koch curve was first described by the Swedish mathematician Helge von Koch in 1904 as an example of a continuous but nowhere differentiable curve. Such functions are now characterised as fractal since their graphs are in general fractal sets. Furthermore, it can be obtained as the graph of an appropriately chosen iterated function system. On the other hand, a fractal interpolation function can be seen as a special case of an iterated function system thus maintaining all of its characteristics. Fractal interpolation functions are continuous functions that can be used to model continuous signals. An in-depth discussion on the theory of affine fractal interpolation functions generating the Koch Curve by using fractal analysis as well as its recent development including some of the research made by the authors is provided. We ensure that the graph of fractal interpolation functions on the Koch Curve are attractors of an iterated function system constructed by non-constant harmonic functions.

A NOVEL 3D MICROMIXER WITH QUARTIC KOCH CURVE FRACTAL

In this paper, we mainly study the mixing performance of the micromixer with quartic Koch curve fractal (MQKCF) by numerical simulation. Changing the structure of the microchannel based on the fractal principle can significantly improve the fluid flow state in the microchannel and improve the mixing efficiency of the micromixer. This paper discussed the effects of different fractal deflection angles, microchannel heights and different fractal times on the mixing efficiency under four different Reynolds numbers (Re). It is found that changing the deflection angle of the fractal can bring extremely high benefits, which makes the fluid deflect and fold in the microchannel, enhancing the chaotic convection in the microchannel, and improve the mixing efficiency of the fluid. Under the reasonable arrangement of the quartic Koch curve fractal principle, it can give the micro-mixture more than 99% mixing efficiency. Based on the excellent mixing performance of MQKCF, it also has extremely high application value in the biochemical neighborhood.

Traces and extensions of certain weighted Sobolev spaces on $$\mathbb {R}^n$$ and Besov functions on Ahlfors regular compact subsets of $$\mathbb {R}^n$$

The focus of this paper is on Ahlfors Q-regular compact sets $$E\subset \mathbb {R}^n$$ such that, for each $$Q-2<\alpha \le 0$$ , the weighted measure $$\mu _{\alpha }$$ given by integrating the density $$\omega (x)=\text {dist}(x, E)^\alpha $$ yields a Muckenhoupt $$\mathcal {A}_p$$ -weight in a ball B containing E. For such sets E we show the existence of a bounded linear trace operator acting from $$W^{1,p}(B,\mu _\alpha )$$ to $$B^\theta _{p,p}(E, \mathcal {H}^Q\vert _E)$$ when $$0<\theta <1-\tfrac{\alpha +n-Q}{p}$$ , and the existence of a bounded linear extension operator from $$B^\theta _{p,p}(E, \mathcal {H}^Q\vert _E)$$ to $$W^{1,p}(B, \mu _\alpha )$$ when $$1-\tfrac{\alpha +n-Q}{p}\le \theta <1$$ . We illustrate these results with E as the Sierpinski carpet, the Sierpinski gasket, and the von Koch snowflake.

FRACTAL INTERPOLATION FUNCTIONS ON AFFINE FRACTAL INTERPOLATION CURVES

In this paper, we give fractal interpolation functions generated on some special affine fractal interpolation curve by harmonic functions of fractal analysis. In the case of Koch Curve, same statement is regarded as a corollary, that is, we show that it is possible to ensure that graphs of fractal interpolation functions on the Koch Curve are attractors of iterated function systems. In the case of fractal interpolation functions generated on general affine fractal interpolation curves by harmonic functions of fractal analysis, all the discussions are similar to those performed in some special affine fractal interpolation curve if their harmonic structures are given.

Computational analysis of a dual‐port semi‐circular patch antenna combined with Koch curve fractals for ultra‐wideband systems

AbstractThis paper presents a two‐port microstrip‐fed semi‐circular patch antenna integrated with Koch curve fractals and common ground with defected ground structure for the effective implementation in handheld ultra‐wideband (UWB) systems. The proposed array is engraved on a 1.57 mm thick FR‐4 substrate with an overall array size of 30.5 × 47 × 1.64 mm3. The parametric analysis of the proposed array indicates that the two optimized Koch curve fractal (second iteration order) radiators (upper substrate layer) provides an optimum matching performance in a wide operational band (4.395–10.184 GHz, 79.4%) and minimizes the patch area by 41.2% as compared to the conventional circular patch. To realize good port‐to‐port isolation (S21/S12 ≤ −16.8 dB), the reduced ground is slotted and incorporated with a funnel‐shaped decoupling structure at the center. The prototype of the proposed array is fabricated and experimentally tested to justify the simulated S‐parameter results. All diversity metrics like envelope correlation coefficient, diversity gain (DG), mean effective gain, channel capacity loss, and total active reflection coefficient lie within their acceptable limits which affirm a high signal‐to‐noise ratio and the proficiency of the proposed array in the development of high data‐rate UWB communication devices.

PSO AND IFS TECHNIQUES FOR THE DESIGN OF WEARABLE HYBRID FRACTAL ANTENNA

ABSTRACT This paper presents the design of Meander and Koch Curves based Wearable Hybrid Fractal Antenna (MKWHFA) using Particle Swarm Optimisation (PSO) and Iterative Function System (IFS). The proposed antenna design combines two fractal geometries i.e. Meander and Koch curve to enhance its performance characteristics. It uses the Jean material as a substrate having a dielectric constant of 1.7 and height 1 mm. The patch and the ground plane made using the copper tape. ANSYS/Ansoft HFSS V13 software is used for simulating and designing for the proposed MKWHFA. Parametric analysis on this antenna’s ground plane is done to achieve the desired resonant frequency, gain, and bandwidth. A mathematical model is derived using the IFS to generate the fractal geometry in MATLAB software. The feed position of MKWHFA optimised using PSO technique. The MKWHFA with Meander-Koch slot resonates at 2.4508 GHz, 5.4005 GHz, 6.6503 GHz, and 8.0002 GHz with corresponding gains 5.61 dB, 8.55 dB, 5.07 dB and 3.24 dB, bandwidths 291 MHz, 94 MHz, 222 MHz and 310 MHz respectively. The proposed MKWHFA fabricated and tested in the laboratory. There is a good agreement between the simulated and experimentally measured results.

Boundary element methods for acoustic scattering by fractal screens

We study boundary element methods for time-harmonic scattering in {mathbb {R}}^n (n=2,3) by a fractal planar screen, assumed to be a non-empty bounded subset Gamma of the hyperplane Gamma _infty ={mathbb {R}}^{n-1}times {0}. We consider two distinct cases: (i) Gamma is a relatively open subset of Gamma _infty with fractal boundary (e.g. the interior of the Koch snowflake in the case n=3); (ii) Gamma is a compact fractal subset of Gamma _infty with empty interior (e.g. the Sierpinski triangle in the case n=3). In both cases our numerical simulation strategy involves approximating the fractal screen Gamma by a sequence of smoother “prefractal” screens, for which we compute the scattered field using boundary element methods that discretise the associated first kind boundary integral equations. We prove sufficient conditions on the mesh sizes guaranteeing convergence to the limiting fractal solution, using the framework of Mosco convergence. We also provide numerical examples illustrating our theoretical results.

Research of antenna complexes using chiral metamaterials and fractal geometry of radiators for MIMO systems

This article is devoted to the study of the possibilities of increasing spectral efficiency in MIMO systems by using antennas with substrates of biisotropic and bianisotropic chiral metamaterials and various types of fractal emitters, in particular, fractal structures in the form of a Sierpinski triangle, Koch and Gilbert curves, as well as a dipole triangular antenna of complex configuration FRM. The spectral efficiency was calculated by using one of the variations of the Shannon formula, which includes a complete matrix of Z-parameters. In turn, this matrix was determined using the software package of electrodynamic modeling. It is shown that the use of such antennas with the fractal geometry of the emitters located on chiral substrates reduces the mutual influence between the emitters, and, in turn, increases the spectral efficiency in several frequency ranges compared to traditional solutions.