Sierpinski Carpet Research Articles

Corner and edge states in topological Sierpinski Carpet systems

Fractal lattices, with their self-similar and intricate structures, offer potential platforms for engineering physical properties on the nanoscale and also for realizing and manipulating high order topological insulator states in novel ways. Here we present a theoretical study on localized corner and edge states, emerging from topological phases in Sierpinski Carpet within a $\pi$-flux regime. A topological phase diagram is presented correlating the quadrupole moment with different hopping parameters. Particular localized states are identified following spatial signatures in distinct fractal generations. The specific geometry and scaling properties of the fractal systems can guide the supported topological states types and their associated functionalities. A conductive device is proposed by coupling identical Sierpinski Carpet units providing transport response through projected edge states which carry on the details of the system's topology. Our findings suggest that fractal lattices may also work as alternative routes to tune energy channels in different devices.

Design 3D Wallpaper Motifs from Sierpinski Carpet Fractals Using Mathematica Applications

The development of geometric studies is very rapid, including fractal geometry. A fractal is a geometric shape that fulfills the properties of the fractal dimension. If you look at a fractal at first glance, it has an irregular shape, but if you look further, there is a regularity. One of the properties of fractals is self-similarity which occurs when a fractal is enlarged. The fractal form has a pattern obtained from iterating a function with infinite repetition. Among the fractal shapes there is the Sierpinski carpet shape. Fractals in 2D form can be transformed using geometric concepts into 3D fractals. Fractal shapes can be applied for cultural development, namely to form wallpaper motifs. Fractal geometric motifs are obtained from iterating a function with the help of Wolframe Mathematica.

Energy absorption characteristics of fractal multi-cell square tubular structures under axial crushing

In this study, a new fractal multi-cell square tube (FMST) based on the fractal design of the Sierpinski carpet is proposed for energy absorption. The dynamic crushing performance and energy absorption characteristics of the FMST are numerically and theoretically investigated. Extensive numerical simulations are carried out for the FMST structures with different fractal orders and masses of the structures. The findings reveal that the specific energy absorption of the 3rd order FMST is significantly higher, exhibiting a remarkable 100 % increase compared to the 0th order FMST. Furthermore, the undulation of the load-carrying capacity of the 3rd order FMST is reduced by up to 88.5 % compared to a conventional square tube, indicating the substantial potential of the FMST for designing highly efficient energy absorbers. Comparative analysis against other hierarchical multi-cell square tubes reported in the literature confirms that the specific energy absorption of the FMST surpasses existing designs. In addition, a theoretical study is presented for the mean crushing force of the proposed FMST, employing the simplified super folding element theory. The theoretical predictions agree well with the numerical results, further validating the effectiveness of the proposed design. This study provides an innovative design of a multi-cell energy absorber with exceptional energy absorption efficiency. The incorporation of fractal principle in the FMST design holds promise for advancing the field of energy absorption, with potential applications in various industries.

Invasive weed optimization based compact hybridized fractal antenna design for RF energy harvesting and multifunctional wireless applications

This paper describes the optimal design and analysis of compact multiband Hybrid Fractal Antenna using Invasive Weed Optimization (HFAIWO). The configured structure involves the fusion of Minkowski, Sierpinski carpet and Giuseppe Peano in a coherent manner. The two-iterative conglomerated fractal antenna is realized physically by considering FR4 epoxy material (dielectric constant, εr = 4.4 and mass density = 1,900 kg/m3). The volumetric dimensions of the fabricated prototype are 36 x 36 x 1.6 mm3. In the designing process, the geometrical descriptors, namely, feedline width ‘WF’ and ground plane dimension ‘LG’ are optimized specifically using Invasive Weed Optimization (IWO) and Harmony Search Algorithm (HSA) strategies. The optimal solution is searched by changing the descriptor values under a set of practical constraints. The examined values of ‘WF’ and ‘LG’ using IWO and HSA are 3 and 31 mm, and 2.6 and 28.7 mm, respectively. Comparative results reveal that IWO offers superior solution quality and convergence than HSA. Afterwards, experimentation of the Projected HFAIWO is done to justify the recommended fractal hybridization approach. Measurements reveal that for VSWR ≤ 2, the fabricated structure produces nine resonance points (1.84, 3.99, 5.15, 5.55, 6.30, 6.58, 8.00, 9.29, and 12.01 GHz) with appreciable gain values. At the respective resonance points, the −10 dB impedance bandwidth is 3.9 %, 3.1 %, 2.1 %, 2.2 %, 1.4 %, 1.2 %, 6.9 %, 1.2 %, and 12.23 %. The provided radiation patterns are arbitrary bidirectional/omnidirectional. By applying the above-said approaches, the radiator size is reduced by 54 %. Importantly, the constructed structure covers two important bands i.e. 1.84 GHz (GSM 1800 band, downlink) and 5.55 GHz (WLAN (Lower) 5.150–5.725 GHz) associated with energy harvesting applications. The practical outcomes suggest that the introduced prototype is a proficient candidate for energy harvesting systems, Satellite communication for downlink, Long range tracking, Battlefield surveillance, Digital broadcast satellite service, Weather Radar, and Maritime navigation radar.

A Novel Contact Stiffness Model for Grinding Joint Surface Based on the Generalized Ubiquitiformal Sierpinski Carpet Theory

A novel analytical model based on the generalized ubiquitiformal Sierpinski carpet is proposed which can more accurately obtain the normal contact stiffness of the grinding joint surface. Firstly, the profile and the distribution of asperities on the grinding surface are characterized. Then, based on the generalized ubiquitiformal Sierpinski carpet, the contact characterization of the grinding joint surface is realized. Secondly, a contact mechanics analysis of the asperities on the grinding surface is carried out. The analytical expressions for contact stiffness in various deformation stages are derived, culminating in the establishment of a comprehensive analytical model for the grinding joint surface. Subsequently, a comparative analysis is conducted between the outcomes of the presented model, the KE model, and experimental data. The findings reveal that, under identical contact pressure conditions, the results obtained from the presented model exhibit a closer alignment with experimental observations compared to the KE model. With an increase in contact pressure, the relative error of the presented model shows a trend of first increasing and then decreasing, while the KE model has a trend of increasing. For the relative error values of the four surfaces under different contact pressures, the maximum relative error of the presented model is 5.44%, while the KE model is 22.99%. The presented model can lay a solid theoretical foundation for the optimization design of high-precision machine tools and provide a scientific theoretical basis for the performance analysis of machine tool systems.

Interpolation problems for random fields on Sierpinski’s carpet

The prediction of stochastic processes and the estimation of random fields of different natures is becoming an increasingly common field of research among scientists of various specialties. However, an analysis of papers across different estimating problems shows that a dynamic approach over an iterative and recursive interpolation of random fields on fractal is still an open area of investigation. There are many papers related to the interpolation problems of stationary sequences, estimation of random fields, even on the perforated planes, but all of this still provides a place for an investigation of a more complicated structure like a fractal, which might be more beneficial in appliances of certain industry fields. For example, there has been a development of mobile phone and WiFi fractal antennas based on a first few iterations of the Sierpinski carpet. In this paper, we introduce an estimation for random fields on the Sierpinski carpet, based on the usage of the known spectral density, and calculation of the spectral characteristic that allows an estimation of the optimal linear functional of the omitted points in the field. We give coverage of an idea of stationary sequence estimating that is necessary to provide a basic understanding of the approach of the interpolation of one or a set of omitted values. After that, the expansion to random fields allows us to deduce a dynamic approach on the iteration steps of the Sierpinski carpet. We describe the numerical results of the initial iteration steps and demonstrate a recurring pattern in both the matrix of Fourier series coefficients of the spectral density and the result of the optimal linear functional estimation. So that it provides a dependency between formulas of the different initial sizes of the field as well as a possible generalizing of the solution for N-steps in the Sierpinski carpet. We expect that further evaluation of the mean squared error of this estimation can be used to identify the possible iteration step when further estimation becomes irrelevant, hence allowing us to reduce the cost of calculations and make the process viable.

Dirichlet forms on unconstrained Sierpinski carpets

Dirichlet forms on unconstrained Sierpinski carpets

Fractal Inspired Hybrid Microstrip Patch Antenna for Surveillance Drone Applications

Drone based surveillance and communication systems are playing a vital role in modern days for both military and civilian applications. Antennas with low profile, smaller size, and light weight are essential for the realization of these systems. A novel fractal-inspired hybrid microstrip patch antenna realized with Sierpinski gasket fractal slots cut in the radiating patch along with corresponding ground structure simultaneously defected with Sierpinski carpet fractal slots is reported for the first time. The antenna resonates at 2.4 GHz with a measured return loss of 17 dB, gain of 7 dB and measured 3 dB beamwidth is of the order of 70 deg. The current approach leads to the size reduction of the antenna to the extent of 55% in comparison to standard radiating patch antenna resonating at the same frequency. The fractal-inspired hybrid microstrip patch antenna is a promising candidate for drone-based surveillance and communication applications owing to its miniaturization and good directional radiation properties.

Characterization of the effect of fractal micro-protrusion and heat on the wettability behavior on chips in medical MEMS

Interfacial wettability, a fundamental property of solid surfaces in the medical field, is known to be affected by surface microstructure and temperature. This research aims to examine the influence of temperature on wettability by creating two distinct types of fractal single-stage microstructures on medical silicon wafers, utilizing the combined features of single-stage microstructure and fractal micro-nano structure. The experimental findings indicate that the area fraction and roughness of the fractal microstructure exhibit variations based on the specific fractal dimension employed. The hydrophobicity of the Sierpinski triangle fractal microstructure surpasses that of other tested microstructures under the same machining accuracy, which can increase the hydrophobicity of the chip surface by 47.9% on average. Furthermore, the hydrophobicity of the Sierpinski triangle fractal microstructure is found to be 13.1% higher than that of the Sierpinski carpet fractal microstructure, despite the temperature stability of the latter is 35% higher. When compared to a uniform micropillar structure with identical parameters, the Sierpinski carpet fractal microstructure demonstrates superior hydrophobicity, while the Sierpinski triangle fractal microstructure exhibits lower hydrophobicity. These findings suggest that the incorporation of fractal microstructures significantly influences the wettability of medical chips under different temperatures, which holds promise for its application in medical MEMS.

A uniform trace theorem for Dirichlet forms on Sierpinski fractals

We establish trace theorems for the self-similar Dirichlet forms on the Sierpinski gasket and the Sierpinski carpet to their subsets generated by cutting with a straight line. For the Sierpinski gasket, the straight line can be in any direction. For the Sierpinski carpet, we require the straight line parallel to an edge of the carpet. The trace forms are expressed in term of values of functions along the cut, in a uniform way independent of the choices of the line.

A novel linear and planar multiband fractal‐shaped antenna arrays with low sidelobes

SummaryFractal antenna arrays are usually used to tune multiband frequencies. However, these types of iteratively constructed antennas are associated with undesirable high sidelobe levels and low directivities. In this paper, an optimization procedure based on the genetic algorithm is used to find the optimal weights of the array elements such that the corresponding array patterns have low sidelobe levels and good directivity. Moreover, the fractal nature in the proposed arrays is maintained regardless of the optimized weights. Thus, the proposed fractal‐shaped array maintains its capability in performing multiband frequency operation. These good radiation features make the proposed fractal‐shaped array more appropriate for the current and future wireless communication applications. Simulation results confirm the superiority of the presented linear and planar fractal‐shaped array structures with compared to the conventional fractal cantor linear array and the standard Sierpinski carpet planar array. For the proposed fractal cantor linear array, the sidelobe level has been reduced to more than −20 dB at different operating frequencies, and the directivity has been improved by more than 8 dB, while the modified Sierpinski carpet planar array has achieved −30 dB depressions in the sidelobe level and 6 dB improvement in the directivity.

Construction of the Aerogel Thermal Conductivity Model based on k×k Sierpinski Carpet Fractal Units

With the characteristics of low density, high specific surface area, and high porosity, aerogel boasts prominent advantages in the field of thermal protection. The thermal insulation performance of aerogel has a significant relationship with its internal microstructure. In this study, the thermal conduction model of Sierpinski aerogel filled with solid in ga
 With the characteristics of low density, high specific surface area, and high porosity, aerogel boasts prominent advantages in the field of thermal protection. The thermal insulation performance of aerogel has a significant relationship with its internal microstructure. In this study, the thermal conduction model of Sierpinski aerogel filled with solid in gas is established based on the equivalent circuit method. We calculated the optimal fractal unit structure of the aerogel via its porosity, applied it to the thermal conductivity calculation of four types of aerogels, and revealed the average relative error of less than 11.58%, which is lower than the calculation results of the thermal conductivity model of the aerogel with the fractal unit structure of , indicating the effectiveness and reliability of the proposed thermal conductivity prediction model.
 s is established based on the equivalent circuit method. We calculated the optimal fractal unit structure of the aerogel via its porosity, applied it to the thermal conductivity calculation of four types of aerogels, and revealed the average relative error of less than 11.58%, which is lower than the calculation results of the thermal conductivity model of the aerogel with the fractal unit structure of , indicating the effectiveness and reliability of the proposed thermal conductivity prediction model.

Prediction of the thermo-physical properties of FGPM aerogels, regarding the development of the Sierpinski carpet model

The Sierpinski carpet is mainly used to model the structure and properties of homogeneous media because of the unique characteristics of fractal materials (self-similarity) and the reduction of calculations complexity. Later, the unit cell of the Sierpinski carpet was modified once with a binomial distribution function to model the functionally gradient aerogels synthesized by the layering method. In this work, the structure and thermal properties of functionally gradient Novolac aerogels are introduced and synthesized by the thermal gradient device (a monolithic, continuous, and controllable method). The structural parameters of the proposed model are determined due to the changes in the sample concentration and gyration radius of colloid particles caused by the thermodiffusion effect. The appropriate range of structural parameters is 0.1 <C/L< 0.93, t = 0, and n = 1. The accuracy of the proposed model is reviewed by experimental samples, which showed suitable matches and reasonable agreement. The main challenge of this study is to involve the effect of the synthesis method (thermal gradient device) in modeling the structure and properties of the Sierpinski carpet in a functionally graded heterogeneous structure.

SHORTEST PATH DISTANCE AND HAUSDORFF DIMENSION OF SIERPINSKI NETWORKS

In this paper, we will study the geometric structure on the Sierpinski networks which are skeleton networks of a connected self-similar Sierpinski carpet. Under some suitable condition, we figure out that the renormalized shortest path distance is comparable to the planar Manhattan distance, and obtain the Hausdorff dimension of Sierpinski networks.

Numerical evaluation of singular integrals on non-disjoint self-similar fractal sets

We consider the numerical evaluation of a class of double integrals with respect to a pair of self-similar measures over a self-similar fractal set (the attractor of an iterated function system), with a weakly singular integrand of logarithmic or algebraic type. In a recent paper (Gibbs et al. Numer. Algorithms 92, 2071–2124 2023), it was shown that when the fractal set is “disjoint” in a certain sense (an example being the Cantor set), the self-similarity of the measures, combined with the homogeneity properties of the integrand, can be exploited to express the singular integral exactly in terms of regular integrals, which can be readily approximated numerically. In this paper, we present a methodology for extending these results to cases where the fractal is non-disjoint but non-overlapping (in the sense that the open set condition holds). Our approach applies to many well-known examples including the Sierpinski triangle, the Vicsek fractal, the Sierpinski carpet, and the Koch snowflake.

Fractal photonic anomalous Floquet topological insulators to generate multiple quantum chiral edge states

Anomalous Floquet topological insulators with vanishing Chern numbers but supporting chiral edge modes are attracting more and more attention. Since the existing anomalous Floquet topological insulators usually support only one kind of chiral edge mode even at a large lattice size, they are unscalable and unapplicable for multistate topological quantum systems. Recently, fractal topological insulators with self-similarity have been explored to support more nontrivial modes. Here, we demonstrate the first experimental realization of fractal photonic anomalous Floquet topological insulators based on dual Sierpinski carpet consisting of directional couplers using the femtosecond laser direct writing. The fabricated lattices support much more kinds of chiral edge states with fewer waveguides and enable perfect hopping of quantum states with near unit transfer efficiency. Instead of zero-dimensional bound modes for quantum state transport in previous laser direct-written topological insulators, we generate multiple propagating single-photon chiral edge states in the fractal lattice and observe high-visibility quantum interferences. These suggest the successful realization of highly indistinguishable single-photon chiral edge states, which can be applied in various quantum operations. This work provides the potential for enhancing the multi-fold manipulation of quantum states, enlarging the encodable quantum information capacity in a single lattice via high-dimensional encoding and many other fractal applications.

Design and implementation of an efficient sierpinski carpet fractal antenna with defected ground structure for 2.45 GHZ applications

A sierpinski carpet fractal antenna with Defected Ground Structure (DGS) is unveiled. The fractal dimensions had paved the way for numerous applications. Fractal technique is applied to the patch and rectangular slot is made in the ground plane which acts as a DGS. The DGS disturbs the shield current and improves the performance of designed antenna. Dimension of designed antenna is 52 × 71 × 5.1 mm3. Rogers RO4350 substrate and coaxial feed technique are utilized in the design. The projected antenna is simulated and fabricated with the resonant frequency of 2.45 GHz. Commencing from the outcome of the projected antenna, it is evident that 23% wider Bandwidth, 13% increment in Gain, improved Return Loss of 15% and 26% VSWR are obtained compared to existing work. The measurements attained by fabricated antenna endow with excellent conformity to the simulated design.

All silicon MIR super absorber using fractal metasurfaces

Perfect absorbers can be used in photodetectors, thermal imaging, microbolometers, and thermal photovoltaic solar energy conversions. The spectrum of Mid-infrared (MIR) wavelengths offers numerous advantages across a wide range of applications. In this work, we propose a fractal MIR broadband absorber which is composed of three layers: metal, dielectric, and metal (MDM), with the metal being considered as n-type doped silicon (D-Si) and the dielectric is silicon carbide (SiC). The architectural design was derived from the Sierpinski carpet fractal, and different building blocks were simulated to attain optimal absorption. The 3D finite element method (FEM) approach using COMSOL Multiphysics software is used to obtain numerical results. The suggested fractal absorber exhibits high absorption enhancement for MIR in the range between 3 and 9 µm. D-Si exhibits superior performance compared to metals in energy harvesting applications that utilize plasmonics at the mid-infrared range. Typically, semiconductors exhibit rougher surfaces than noble metals, resulting in lower scattering losses. Moreover, silicon presents various advantages, including compatibility with complementary metal–oxide–semiconductor (CMOS) and simple manufacturing through conventional silicon fabrication methods. In addition, the utilization of doped silicon material in the mid-IR region facilitates the development of microscale integrated plasmonic devices.

An Innovative Topologies Based on Hypercube Network Interconnection

A parallel processing system's most crucial part is a network interconnection that links its processors. The hypercube topology has interesting features that make it a great option for parallel processing applications. This paper presents two innovative configurations of interconnection networks based on fractal Sierpinski and a hypercube. These are called the Sierpinski Triangle Topology (STT) and Sierpinski Carpet Topology (SCT). Compared to a hypercube, the Sierpinski Triangle topology (STT) noticed a significant decrease in the number of nodes and links as large networks grew. Hence, it is considered a great way to reduce costs because it uses fewer nodes and links. The average distance is also shorter, which is better. Despite it having a smaller bisection width and a higher degree than a hypercube by one. The Sierpinski Carpet Topology (SCT) has the advantage of having a high bisection width compared to a hypercube. That is preferable because it places a lower restriction on the difficulty of parallel algorithms. While the drawback of this topology is that it has a diameter and average distance more than a hypercube.

Numerical modelling of blast mitigation of pre-fractal obstacles

The mechanics of downstream blast wave attenuation caused by interaction with obstacles arranged into a pre-fractal shape based on the Sierpinski carpet was numerically investigated using a high-fidelity CFD solver. The blast mitigation was qualitatively and quantitatively assessed for four pre-fractal iterations at three different scaled distances ( Z = 1.87, 2.24, 2.99 m/kg1/3). Mitigation was seen to occur in zones associated with the location of destructive wave interference patterns in the downstream region. Crucially, these zones were found to widen spatially with increasing pre-fractal iteration, and strong shock-shock interactions that result in load amplification, commonly encountered in downstream regions of a solitary block-like obstacle, were not observed for the more fractal-like obstacles. The mechanisms of attenuation are explored in terms of wave impedance. It is found that pre-fractals reduce wave transmission in the downstream, increase reflection of the blast wave, and enhance trapping within the confines of the pre-fractal obstacle, dramatically changing the directionality and hence the strength of the transmitted wave. Reductions in peak pressure of up to 60% and peak specific impulse of up to 40% were recorded for the highest iteration pre-fractal, that is, obstacles that most closely represent a true fractal, thereby highlighting the effectiveness of such shapes for protective structure design for improved blast mitigation.