Fractal Design Research Articles

The influence mechanism of fractal design method on mechanical properties and corrosion behavior of sheet Gyroid porous structures formed by LPBF

In order to meet the various performance requirements of bone implants, it is necessary to design a porous structure with multi-scale pores. In this paper, a set of 2-level fractal porous structures and 1-level porous structures P80 and P85 were designed based on the sheet-Gyroid porous structure. The influence of porosity and fractal design on the forming quality, mechanical properties and corrosion resistance of the porous structures was investigated. The results showed that the porosity deviation of the 2-level porous structures was in the range of 17.88% to 20.79%, which was about twice that of the 1-level porous structures. The elastic gradient of 2-level fractal porous structures varied from 1.654GPa to 3.636GPa, the compressive offset strength ranged from 50.9MPa to 111.5MPa, and the first maximum compressive strength was in the range of 57.8MPa to 136.6MPa, all of which were lower than those of 1-level porous structures with similar porosity, which extended the design range of mechanical properties of porous structures to a certain extent, and was conducive to avoiding the stress shielding effect of implants. Based on the study of mechanical properties of 2-level fractal porous structures, the mechanical properties prediction models were constructed based on the improved G-A model. In addition, it was found that the corrosion resistance of 2-level fractal porous structures was much higher than that of 1-level porous structures, and the corrosion resistance decreased with the increase of porosity. The fractal design of TPMS structure can effectively expand the design space of porous bone implants.

Exploring the Complex Dynamics of Julia and Mandelbrot Sets for the Complex-valued Mapping sin(z<sup>k</sup>) + az + c Using Four-Step Iterative Scheme with s-Convexity

In this paper, we investigate unique variations of the Julia and Mandelbrot sets, establishing criteria for the escape of a function Tcz = sin(zk) + az + c for all z ∈ C, where k ∈ N \ {1} and a, c ∈ C with a̸ = 0 by utilizing a four-step iterative scheme extended with s-convexity, we analyze the dynamics of these fractals by formulating and implementing the algorithms. Our exploration focuses on understanding how distinct parameters impact these fractals’ color, dynamics, and overall visual characteristics. Our findings reveal that specific fractal patterns resemble exquisite natural objects like flowers and spiders. These fractal designs are also employed as visually striking artistic motifs on garments and textile materials. The captivating and complex properties of fractal patterns within dynamic systems make them an enticing and visually appealing area of research.

Design of a Tri-band Minkowski Fractal MIMO Antenna for FR2 5G Applications

The growing importance of fifth-generation (5G) communication technologies for the Internet-of-Things have made it necessary to design multi-element antennas that operate within the millimetre-wave spectrum. However, simultaneously achieving multiple operating frequencies, high gain and good isolation between the elements of such small-sized antennas is a challenging task. In this research, a composite right/left hand millimetre-wave fractal antenna was designed for 5G mobile terminals, in order to simultaneously satisfy these key design goals. This aim was accomplished by designing a 2 x 2 multiple-input, multiple output (MIMO) antenna based on the second-order Minkowski fractal using an electromagnetics simulation software, Computer Simulation Technology (CST) Studio Suite ®. Two composite-right/left hand structures were included on the reverse side of the antenna, to reduce the coupling between the MIMO antenna elements. Simulation results show that the designed antenna operates at three frequency bands within the millimetre-wave spectrum, namely 25.52 GHz – 28.23 GHz, 36.5 GHz – 37.5 GHz, and 47.7 GHz – 49.7 GHz. The maximum antenna gains at these three bands are 6.9 dBi, 7.5 dBi, and 6.0 dBi, respectively. Also, good isolation is realized between the antenna elements at the operating frequencies, with a minimum value of 10.8 dB. With a spacing of 0.03λ between the MIMO antenna elements, the overall size of the designed antenna was 30 mm x 12 mm x 0.508 mm, which compares positively with other related fractal antenna designs. The designed antenna shows good prospects for integration into various mobile terminal devices to aid 5G-based millimetre-wave communications.

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.

Fractal Circuit Architectures for Spatially Controlled Heating and Multi-Modal Shape Programming of Electro-Active Shape Memory Polymers

Shape memory polymers (SMPs) are commonly activated through external heating or rigid embedded heaters, which lack precise temperature control and restrict programmable shape transformations. This study demonstrates integrating thin films of hard electronic materials patterned in fractal designs into SMPs yields novel thermomechanical responses, enabling flexible implementation of stretchable electronic functionality. Space-filling Hilbert, Moore and Peano fractal curves generate distributed electronic circuitry patterns within the SMP matrices. To investigate the coupled electro-thermal-mechanical behavior, a multi-physics modeling approach is adopted. A nonlinear thermo-visco-hyperelastic constitutive model captures the SMP’s shape memory characteristics. Governing equations for the multi-physics problem are formulated. Experimental data validates the model’s predictions of the SMP’s nonlinear material response under uniaxial loading. This validated framework analyzes SMP composites with integrated fractal circuits. Findings reveal composites with higher-order fractal circuits’ exhibit faster, more uniform resistive heating and smaller electrical resistance changes during stretching, enabling consistent and predictable shape recovery under various deformations. Notably, these advantageous heating and resistance characteristics enhance the force recovery rate during the shape memory cycle. Furthermore, the composites demonstrate excellent shape recovery under bending. Remarkably, uniaxial stretching can induce controlled out-of-plane shape transformations, enabling novel actuation modalities. This work provides insights into leveraging fractal circuit integration to enhance SMP performance and expand capabilities, enabling multi-functional actuators, sensors and programmable soft grippers.

Accessing versatile tensile ductility of amorphous materials by fractal nanoarchitecture design

Amorphous materials have been known to be intrinsically brittle under tensile stress. This is especially true in silicon-based and metallic-based glasses. In particular, for the latter, they quite often show promising mechanical properties superior to their crystalline counterparts. However, lacking tensile ductility strongly prohibits them from servicing the societies. Although with decades of immense research efforts, we are still waiting for a sophisticated solution. To march in this direction, in this work, we are dedicated to finding a practical method by smart fractal nano-architecture design through advanced computational modeling. This mainly aims to circumvent the intrinsic strain localization at the nano-scale to avoid catastrophic failure. By distributing the external strain to numerous self-confined nano-branches, we successfully achieve astonishing and tunable tensile ductility. This strategy proves to be very effective for different classes of amorphous materials. We found that the tensile properties of the nano-architectured glasses depend on both the constituent elements and the nanostructure. This demonstrates our flexible capability to design desired mechanical properties for a specific material at any spatial length scale. The current findings will inspire experimental realization by cutting-edge 3D-printing techniques and call for optimal design by top-notch graph neural network deep learning algorithms. This work also proposes a new ’structure’-property relationship to efficiently bridge experimental fabrication and computational design.

Duality of a coiled phononic crystal enables reflectionless interfaces

Recently, it has been demonstrated that one-dimensional bar-based phononic crystals can exhibit subwavelength Bragg bandgaps by coiling the bars and locking the nodal rotational degrees of freedom to create what is termed a coiled phononic crystal (CPnC) [C. L. Willey et al., Phys. Rev. Appl. 18, 014035 (2022)]. Here, it is shown that the CPnC exhibits duality of its dispersion curves relative to its coiling/twist angle, meaning that the dispersion curves are symmetric about a particular coiling/twist angle defined configuration. An exciting implication of this finding is that under a certain set of constraints, segments of dual unit cells with perpendicular wave propagation directions can be connected such that their wave transmission is equivalent to a finite CPnC entirely composed of identical unit cells with parallel wave propagation directions. The ability to link unit cells with different wave propagation orientations, but the same dispersion/dynamic stiffness, is used to create an elastic hierarchically coiled phononic crystal based on a fractal space-filling curve design. The novelty of this work is that it numerically demonstrates reflectionless wave propagation in large fractal architectures (created from specific combinations of dual unit cells) such that regular phononic properties (i.e., passbands and bandgaps) are preserved, allowing for propagation of broadband signals and filtering.

Crashworthiness study of hexagonal honeycomb based on fractal design

Crashworthiness study of hexagonal honeycomb based on fractal design

UWB-MIMO 4-Port Fractal Antenna with Reduced Mutual Coupling

Abstract: UWB MIMO antenna loaded structure Mutual coupling between the four radiating elements is obtained to be less than - 25 db. The patterns of radiation are stable over the operating range and a peak gain of 4.5 dB is obtained at 7.5GHz and Return loss is obtained less than 10dB. Simulation is conducted using HFSS simulation tool and printed on a 30 X 30 X 1.6 mm3 FR4 material. 4 linear SRR structures are used between adjacent elements to ensure good isolation. MIMO antenna with grounded stubs and fractal geometry are used to improve isolation and to achieve wide band phenomena. UWB-MIMO antenna using to achieve better isolation. The UWB MIMO 4-port fractal antenna combines the characteristics of ultra-wideband operation, multiple input multiple output capability, four-port configuration for simultaneous data streams, and a fractal antenna design for wide bandwidth and compact size. This makes it suitable for high-speed, reliable communication in applications requiring UWB technology, such as high-speed data transfer, localization, and radar systems. By integrating a copper rectangular plane between ports, interference and crosstalk are minimized, enhancing the antenna's capability to handle multiple input and output signals efficiently. This innovative design approach not only improves isolation but also contributes to the antenna's overall effectiveness in various communication applications.

Energy Harvesting Fractal Antenna for Charging Low Power Devices

Radio frequency energy harvesting (RFEH) is a prospective technique that uses electromagnetic waves which waste in the air to create energy. This cutting-edge technology offers the ability to wirelessly charge battery-free gadgets, making it a potential alternative energy source for use in the future. Fractal antenna systems are a type of antenna that is characterized by their self-similarity and ability to operate over a wide range of frequencies. The proposed design combined two popular fractal shapes in one design. This one design combines the advantages of the other two designs. This research presents a novel multi-band fractal slot antenna design and optimization method for RFEH systems. The fractal antenna designs that were simulated using the CAD design suite, as well as the final design and performance that captured multiple bands are described. The rectifying and matching networks are next evaluated, and the ADS (Advanced Construct System) software is used to construct and simulate each circuit. Additionally, the voltage regulator, energy storage, and application are addressed along with the required voltage to completely function a wireless sensor node, which may be utilized in many applications including wearable technology and intelligent traffic systems (ITS) that control roads to improve environmental quality. Finally, experimental results demonstrated good agreement with simulations that had a high radiation efficiency and gain.

Energy absorption characteristics of asymmetric tree-like fractal gradient hierarchical structures

Bionic gradient hierarchy design as a design method that can effectively improve the structural crashworthiness, but its current research only focuses on the symmetric fractal method related to the discussion. In view of this, this article carries out the innovative design and research of asymmetric tree-like fractal gradient hierarchy structure (ATFGHS). The results show that the crashworthiness of the structure can be greatly improved by using the asymmetric tree-like fractal gradient hierarchy design method. Specifically, the ATFGHS-P3 showed a 98% increase in energy absorption (EA), a 98% increase in specific energy absorption (SEA), and an 84% increase in crush force efficiency (CFE) compared to a conventional hexagonal multicellular tube. The completion of this study provides valuable insights into the design and optimization of novel lightweight thin-walled energy-absorbing structures.

3D printed bio-inspired self-similar carbon fiber reinforced composite sandwich structures for energy absorption

Fractals widely found in nature possess amazing simple shape cell yet infinitely complex macrostructures. Importantly, all fractals manifest a degree of self-similarity that provides new structural design strategy for composite materials. At present, enhancing energy absorption capacity of lightweight carbon fiber reinforced composite structures is critical for their further engineering applications, and meanwhile biomimetic structures have demonstrated excellent mechanical properties compared to conventional engineering structures. In this work, we constructed bio-inspired fractal structures through three fundamental shape units (curve, circle, and hexagon), and investigated nonlinear mechanical responses of hierarchical self-similar structures inspired by snake-like serpentine, bamboo, and honeycomb, having distinct structure ratios related to cell sizes at different geometry levels. Firstly, such self-similar structures were fabricated into homogenous ones by 3D printing for assessing the energy dissipation contribution of polymer matrix. Moreover, the selected self-similar carbon fiber reinforced composites were additively manufactured and their energy absorption mechanisms of composite sandwich structures were studied via experiment and numerical methods. Additionally, the influences of the self-similar types, structure ratios, and cellular scales on nonlinear compressive behaviors of self-similar composite structures were investigated in detail. The results indicated that the fractal bamboo composite components with the structure ratio of 0.2 exhibit the highest energy absorption rates, with a satisfactory level of energy absorption capacity in potential engineering applications. This study can provide a useful reference in the field of biomimetic fractal design of fiber reinforced composite structures by 3D printing technique.

Semi-self-similar fractal cellular structures with broadband sound absorption

Honeycomb core materials, recognized for their excellent sound absorption and load-bearing capabilities, are widely employed in engineering applications. The back cavity shape of honeycomb structure absorbers plays a crucial role in their sound absorption performance. In this study, the honeycomb structure absorbers are re-designed using fractal design principles, enabling a modification of the acoustic properties of traditional honeycomb-type absorbers and a significant enhancement in the sound absorption frequency band. The theoretical predictions of the semi-self-similar fractal honeycomb absorber structure are validated through finite element simulations and experimental studies, exhibiting remarkable consistency. The findings indicate that compared to single-layer first-order structures, the performance of dual-layer second-order fractal element Helmholtz resonators is significantly improved. Relative to conventional honeycomb panel structures, the fractal design approach markedly enhances the semi-absorption bandwidth of the absorbers, with an increase in the fractal order further broadening the sound absorption frequency band. The innovation of achieving broadband absorption through fractal resonators not only represents a new direction in the design of advanced honeycomb core materials but also holds substantial promise in the field of noise control.

Mechanical behavior of a novel lattice structure with two-step deformation

Lattice structures are increasingly attracting attention due to their excellent mechanical properties and broad application prospects. However, most developed lattices feature single-step deformation or single plateau stress, which confine its multi-task applications. Herein, a novel body-centered cubic (NBCC) with two-step deformation based on body-centered cubic (BCC) and bionic fractal design is introduced. NBCC exhibits bi-plateau stress in the stress–strain curves. The underlying mechanism is caused by the bending and buckling deformation of the struts. The mechanical behaviors of NBCC are investigated by finite element simulation which verified by experiment. Compared with traditional BCC, NBCC has improved modulus of elasticity by 88 %, yield strength by 21.2 %, and specific energy absorption by 108 % when ρ′ and α are 0.07 and 0.5, respectively. Moreover, the elastic modulus as well as yield strength increase with geometrical ratio α. The specific energy absorption tends to maximum at the geometric ratio α = 0.6 ∼ 0.7. From the Ashby map, the proposed NBCC lattice possesses high energy absorption, exceeding most of the existing architected materials at the same density. Furthermore, the results show that NBCC has better isotropy of elastic modulus and tends to be more isotropic material compared to BCC. Finally, theoretical model of two-step plateau stress is established based on hinge dissipation principle. This work opens up new insights into the use of element replacement design to create multi-step pathways that can be applied to design engineering structures with multiple tasks and application for impact protection.

Public Art Design Practice under Visual Communication Design

Abstract The expansion of visual communication design in public art makes the scope of visual communication wide. This new art form reflects the cross and integration of disciplines but also makes the form of public art in our life richer, and people can get more beautiful enjoyment. Research on the fractal algorithm in the escape time algorithm studied the Julia set in different function conditions of the fractal graph of the change and the function of different indices of the joint. The fractal graph obtained a colorful, more tense structure. Subsequently, it is applied to public art design, and after testing its performance, the fractal design of visual communication is combined with the IPA model to explore the practical effect of the fractal design in public art design. The results show that the improved fractal algorithm proposed in this paper increases the pattern generation rate from 61.5% to 92.6%. The fractal dimension measurement of 15 typical batik patterns shows that more than 85% of the batik patterns have an average value of more than 15000.

FRACTAL GEOMETRY-BASED RESOURCE ALLOCATION FOR MIMO RADAR

The netted Collocated Multiple-Input Multiple-Output (C-MIMO) radar systems have attracted widespread attention due to their high resolution and excellent target detection capabilities. The self-similarity and spatial filling properties of fractal geometry enable antenna layouts to function effectively across multiple scales and frequency bands, which helps improve the overall performance and resource utilization of systems. In this paper, a radar array element model based on the Fudgeflake fractal is used to establish a target localization performance model directly related to power bandwidth allocation. Based on the model, a three-step solving strategy was designed with the Sequence Quadratic Program (SQP) algorithm as the foundation to achieve optimized allocation of power and bandwidth while maximizing overall performance. Experimental results demonstrate that under this algorithm, the use of power bandwidth resources is optimized through fractal geometry-based design.

O-Shape Fractal Antenna Optimized Design with Broad Bandwidth and High Gain for 6G Mobile Communication Devices

Optimization of antenna parameters is important for achieving the best design that has higher results for gain and bandwidth while also having a smaller size. One such antenna design is numerically investigated and presented in this research. The antenna is optimized to an O-shape fractal design from a square patch design. The antenna is created by etching a slot of a square patch and making an O-shape fractal metamaterial patch antenna that operates on the THz band. The THz patch antenna is also investigated for its metamaterial properties. The optimization of the THz patch antenna is carried out for substrate height, slot length, and slot width. The optimized design has a size of 65 × 65 µm2. The highest bandwidth of 31.4 THz (138%) and the highest gain of 11.1 dBi is achieved. The optimized design is then investigated for multiple elements. The two-element MIMO antenna design using an O-shape patch is investigated to observe its performance and compare it with an O-shape single-element design. The two-element MIMO antenna design gives two bands with a bandwidth of 18 THz (113%) and 21 THz (56%). The gain of this design is 5.18 dBi and the size is 130 × 65 µm2. A comparison between the O-shape single-element fractal design, two-element fractal MIMO design, and other published designs is carried out. The compact, broadband, and high gain design presented can be used for 6G high-speed mobile communication devices.

Fractal design of 3D-printing mechanical metamaterial undergoing tailorable zero Poisson’s ratio

With the development of mechanical metamaterials, structures with large and nonlinear deformations have attracted great research passions for their increasingly applications. Although it has been studied for decades, the mechanics of conflict between large deformation with non-buckling has not been well understood. In this study, a new type of mechanical metamaterial, of which the large deformation is achieved without buckling, has been proposed to achieve the zero Poisson’s ratio using fractal design. Effects of fractal iteration () and connection angle () on the Poisson’s ratio and mechanical stress-strain behaviors of 3D printing mechanical metamaterial were studied by finite element analysis and experimental tests. Finally, constitutive relationships among loading force, Poisson’s ratio, iteration time and connection angle have been explored and discussed to achieve a large and non-buckling deformation.

Flexible liquid metal-based microfluidic strain sensors with fractal-designed microchannels for monitoring human motion and physiological signals

With the rapid advancement of wearable electronics, there is an increasing demand for high-performance flexible strain sensors. In this work, a flexible strain sensor based on liquid metal (LM)-integrated into a microfluidic device is developed with Peano-type fractal structure design. Compared with the microfluidic sensors with straight and wavy microchannels, the sensor with Peano-shaped channels shows lower hysteresis and improved stretchability. Furthermore, the increase of the fractal order can further improve the sensing performances. The third-order Peano sensor exhibits excellent mechanical and electrical properties, including high tensile capability (490.3%), minimal hysteresis (DH = 0.86%), ultra-low detection limit (0.1%), low overshoot, rapid response time (117 ms), as well as good stability and durability. By adding two independent and perpendicular straight channels to the Peano sensing unit, the feasibility of multi-directional strain recognition is demonstrated. To further improve the sensitivity of the Peano-shaped sensor, a multi-layer Peano sensor is developed, exhibiting remarkably enhanced sensitivity while maintaining low hysteresis. Overall, the developed LM-based microfluidic strain sensors enrolling Peano fractal geometry hold high potential for various wearable electronics applications.