Regular Hexagonal Lattice Research Articles

Closed-form effective out-of-plane elastic properties of regular and non-regular hexagonal lattice plates

This study derives closed-form expressions for the effective out-of-plane elastic properties of regular and non-regular hexagonal lattice plates. The derivation utilizes the homogenization method based on equivalent strain energy and the Kirchhoff plate theory. Rigorous accuracy assessments of the closed forms are conducted through comparisons with the existing literature and direct finite element analysis. A detailed parametric study is also conducted using the obtained closed forms to provide insights into the relationships between the effective elastic properties and the unit-cell shape. Lastly, general forms for some effective properties applicable to out-of-plane and in-plane situations are presented for hexagonal lattices.

Bounds on the Optimal Radius When Covering a Set with Minimum Radius Identical Disks

The problem of covering a two-dimensional bounded set with a fixed number of minimum-radius identical disks is studied in the present work. Bounds on the optimal radius are obtained for a certain class of nonsmooth domains, and an asymptotic expansion of the bounds as the number of disks goes to infinity is provided. The proof is based on the approximation of the set to be covered by hexagonal honeycombs and on the thinnest covering property of the regular hexagonal lattice arrangement in the whole plane. The dependence of the optimal radius on the number of disks is also investigated numerically using a shape-optimization approach, and theoretical and numerical convergence rates are compared. An initial point construction strategy is introduced, which, in the context of a multistart method, finds good-quality solutions to the problem under consideration. Extensive numerical experiments with a variety of polygonal regions and regular polygons illustrate the introduced approach. Funding: This work was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo [Grants 2013/07375-0, 2016/01860-1, 2018/24293-0, and 2019/25258-7] and Conselho Nacional de Desenvolvimento Científico e Tecnológico [Grants 302682/2019-8, 303243/2021-0, 304258/2018-0, and 408175/2018-4].

Hamiltonian cycles on bicolored random planar maps

We study the statistics of Hamiltonian cycles on various families of bicolored random planar maps (with the spherical topology). These families fall into two groups corresponding to two distinct universality classes with respective central charges c=−1 and c=−2. The first group includes generic p-regular maps with vertices of fixed valency p≥3, whereas the second group comprises maps with vertices of mixed valencies, and the so-called rigid case of 2q-regular maps (q≥2) for which, at each vertex, the unvisited edges are equally distributed on both sides of the cycle. We predict for each class its universal configuration exponent γ, as well as a new universal critical exponent ν characterizing the number of long-distance contacts along the Hamiltonian cycle. These exponents are theoretically obtained by using the Knizhnik, Polyakov and Zamolodchikov (KPZ) relations, with the appropriate values of the central charge, applied, in the case of ν, to the corresponding critical exponent on regular (hexagonal or square) lattices. These predictions are numerically confirmed by analyzing exact enumeration results for p-regular maps with p=3,4,…,7, and for maps with mixed valencies (2,3), (2,4) and (3,4).

The mechanics and optimal design of micro-architectured stepped hexagonal lattices

This paper conceptualises and analyses a new class of stepped hexagonal lattice achieved through modifying the cross-section of the constituent beams in a controlled manner. The main idea lies in the redistribution of the mass of the constituent beam to obtain a range of equivalent elastic properties, which is not possible within the scope of regular hexagonal lattices. The mass of the stepped lattices is kept the same as the regular hexagonal lattices with equivalent geometry. The in-plane mechanics of such mass-conserved hexagonal lattices are investigated, considering stepped beams as constituent members. The mechanics of a unit cell is exploited to derive the closed-form analytical expressions of equivalent elastic properties of the lattice. The derivation utilises the stiffness elements of the constituent stepped beam members. The stiffness matrices are obtained using two different approaches. They include a semi-analytical condensed stiffness matrix approach based on static substructuring and an analytical method based Castigliano’s energy formulation. Both of these approaches are general and can handle arbitrary geometry for the constituent beams. New closed-form analytical expressions of equivalent elastic properties of the lattice are derived. The optimum geometric parameters are obtained by formulating an analytical optimisation problem. It is shown that a unique solution is possible by solving two simultaneous nonlinear equations. The general closed-form expressions of the equivalent elastic properties can be considered benchmark solutions. In the particular case of the regular lattice, they reduce to the well-known classical expressions. Numerical results show that the values of equivalent elastic properties of the lattice can change significantly by redistributing the mass of the constituent beams. It is demonstrated that up to a 37% increase in the equivalent elastic modulus can be achieved compared to the regular lattice by optimally choosing a stepped profile of the constituent elements.

The Refractive Index Of Photonic Crystal Fibers As A Function Of Some Parameters And Temperature

Photonic crystal fibers in the late period occupied a wide range of studies and research because of the ease of dealing with them in terms of design and installation, as there is a group of parameters used in them that can affect the refractive index of the pulse propagation through it, including the diameter of the air holes, the distance between the air holes and their number, as this study showed in addition to the aforementioned parameters The effect of temperature on the refractive index was also studied. It has been observed that with an increase in the diameter of the air holes, the refractive index increases and, conversely, the increase in the distance between them leads to a decrease in the refractive index. As for the number of air holes, it has no clear effect. As for the temperature, which is proportional to the frequency and intensity, this increase in temperature leads to an increase in the refractive index of the pulse passing through the photonic crystal fiber. Changing the temperature of the photonic crystal fiber is an interesting for dynamics fine refractive index tuning in active refractive index shift compensation system. this paper present a numerical analysis on the effect of photonic crystal fiber temperature on refractive index and modal features. the research depend on regular hexagonal crystal lattice fibers with specific geometric parameters using finite element method.

Control of the piezoelectric and flexoelectric homogenized properties of architected materials by tuning their inner topology

The present contribution aims to study the effect the inner topology of periodic architected materials on their effective piezoelectric and flexoelectric properties. An analysis is performed to study the sensitivity of the effective flexoelectric moduli to the geometrical variables of the considered unit cells of periodic lattice materials. More specifically, hexagonal, re-entrant auxetic, and rectangular unit cells are investigated in terms of their relative piezoelectric and flexoelectric response. The base material obeys a piezoelectric behavior, and the homogenized electromechanical properties are evaluated using the extended Hill–Mandel macrohomogeneity condition articulated with variational principles in the context of periodic multiphysical homogenization. The developed modeling framework of coupled electromechanical phenomena accounts for the higher gradient effects modeled by flexoelectricity that are induced by the strong gradient of properties that exist at the microlevel of the lattice unit cell. Regular hexagonal, re-entrant hexagonal and rectangular lattice unit cells are mutually compared in terms of their flexoelectric properties. The regular hexagonal UC shows the highest piezoelectric moduli, whereas the flexoelectric moduli are found to be maximum for the re-entrant UC. The coupling between mechanical and electrical fields is attested by the non-uniform distribution of the electric potential within the three studied architected materials subjected to a uniform strain.

Energy-based fracture mechanics of brittle lattice materials

This study focuses on the fracture properties of strut-based lattice materials following a global energy-based approach. We focus on two typical isotropic lattices, the regular triangular and hexagonal lattices, with distinct mechanical behavior owing to their widely different stretching-to-bending ratios of the stresses within their struts. Numerical simulations are used to investigate crack orientations for both lattices and subsequently estimate the displacement along the crack surfaces under uniaxial tension. The numerically measured displacements are then employed to derive closed-form expressions for the corresponding energy release rates. We perform tensile tests on additively manufactured specimens with different crack-lengths to measure the fracture energy for each lattice. It is found that the triangular lattice has a much higher fracture energy, and therefore toughness, than the hexagonal honeycomb. These results are compared with a simple theoretical analysis involving the energy release from a single strut, as well as with numerical simulations of crack propagation that employ a stress-based criterion. Finally, we introduce an effective fracture toughness for lattice-based materials and compare the values with other monolithic and architected solids.

Topology optimization of anisotropy hierarchical honeycomb acoustic metamaterials for extreme multi-broad band gaps

Acoustic metamaterials (AMMs), especially based on hierarchical honeycomb structure and topology-optimized design with suitable mechanical properties and low-frequency band gaps (BGs), have been studied recently. The present work reports a series of anisotropy hierarchical honeycomb structure lattices composed of the cell walls and vertices of regular hexagonal lattice replacement honeycombs of various sizes, as well as improved BG evaluation indicators. The band structures of three different types of unfilled hierarchical honeycomb lattices are calculated and the mechanism of BGs generation is analyzed. To obtain multi-broad BGs in the low-frequency range, the scatterers filling schemes are optimized by using a genetic algorithm (GA). Additionally, the effects of structural/material parameters on the BG characteristics of the optimized structures are discussed. Finally, the transmission loss (TL) experiment verified the optimization of the structures. This paper aims to promote the application of hierarchical honeycomb metamaterials with low-frequency vibration isolation in the industrial field.

The Refractive Index Of Photonic Crystal Fibers As A Function Of Some Parameters And Temperature

Photonic crystal fibers in the late period occupied a wide range of studies and research because of the ease of dealing with them in terms of design and installation, as there is a group of parameters used in them that can affect the refractive index of the pulse propagation through it, including the diameter of the air holes, the distance between the air holes and their number, as this study showed in addition to the aforementioned parameters The effect of temperature on the refractive index was also studied. It has been observed that with an increase in the diameter of the air holes, the refractive index increases and, conversely, the increase in the distance between them leads to a decrease in the refractive index. As for the number of air holes, it has no clear effect. As for the temperature, which is proportional to the frequency and intensity, this increase in temperature leads to an increase in the refractive index of the pulse passing through the photonic crystal fiber. Changing the temperature of the photonic crystal fiber is an interesting for dynamics fine refractive index tuning in active refractive index shift compensation system. this paper present a numerical analysis on the effect of photonic crystal fiber temperature on refractive index and modal features. the research depend on regular hexagonal crystal lattice fibers with specific geometric parameters using finite element method.

Elastic valley Hall edge wave in a hierarchical hexagonal lattice

By exploiting topological phases, a new approach can be taken to design elastic waveguides with robust wave propagation against sharp bends or defects. In particular, the topological phase based on the quantum valley Hall effect, or valley topological phase, is an attractive candidate for elastic waveguide because its mechanism can be realized in a relatively simple way. In this work, we report a waveguide design with in-plane dynamics based on the acoustic valley Hall effect, leveraging a hierarchical hexagonal lattice built by replacing the vertices of a regular hexagonal lattice with smaller hexagons. These are purposely introduced because they provide better structural performance than the regular hexagonal lattice. We modified the size of the adjacent smaller hexagons using two different values to break spatial inversion symmetry, which leads to topologically distinct lattices depending on the sign of the modification parameter and the emergence of the valley-dependent edge modes at the interface of two lattice structures. We numerically demonstrated the robust propagation of the valley-dependent edge states through sharp bends. Notably, we aggressively exploited the ease of tuning the geometric parameters to design frequency-tailorable edge modes, to demonstrate wave splitting and frequency filtering with the waveguide. The proposed structure can be a pathway for practically utilizing the elastic topological phase, and can also widen the scope of multifunctional (elastic wave control and load-bearing) structures to include the topological phase.

Mechanical Properties of Semi-Regular Lattices

The mechanical properties of seven semi-regular lattices were derived analytically for in-plane uniaxial compression and shear. These analytical expressions were then validated using Finite Element simulations. Our analysis showed that one topology is stretching-dominated; two are stretching-dominated in compression but bending-dominated in shear; and four are bending-dominated. To assess their potential, the properties of these seven semi-regular topologies were compared to regular lattices. We found the elastic buckling strength of the stretching-dominated semi-regular tessellation to be 43% higher than a regular triangular lattice. In addition, three of the four bending-dominated semi-regular topologies had a higher elastic modulus than a regular hexagonal lattice. In fact, one of these bending-dominated topologies was 85% stiffer and 11% stronger than a hexagonal lattice. This topology would be ideal for applications requiring a high stiffness and high energy absorption.

Face index of nanotubes and regular hexagonal lattices

AbstractThe interdependency of the molecular structures of drugs and their biomedical characteristics have already been proved by lab experiments. We can approximate these characteristics by computing numerical invariants associated to their molecular structures. These invariants are called topological indices. Among many topological indices, the face index (FI) is defined most recently. The FI may be helpful in approximating the boiling point andπ‐electron energy of benzenoid hydrocarbons and some other drugs with the correlation coefficient. In this article, we compute the FI for Boron triangular nanotubes, the nanotubesand, Also compute the FI for quadrilateral sections cut from regular hexagonal latticeand.

Irregular hexagonal core based surface plasmon resonance sensor in near-infrared region

In this work, a highly sensitive yet simple hexagonal photonic crystal fiber (PCF) based surface plasmon resonance (SPR) refractive index (RI) sensor is proposed. The proposed structure consists of twelve purely circular air-holes to render the fabrication process easily realizable. For the feasibility of operation, the target analyte, and the plasmonic material, Indium Tin Oxide (ITO) are employed around the outer surface of the sensor. The performance of the sensor is numerically analyzed using the full-vectorial finite element method (FEM) for different fiber geometric parameters and RI variations. A step-by-step design procedure has been followed starting from the regular hexagonal lattice PCF structure. The optimal sensor performance is obtained by tuning the geometric parameters over the relevant numeric range. It is shown that the proposed sensor achieves a maximum wavelength sensitivity of 37000 nm/RIU and a maximum amplitude sensitivity of 407.285 RIU-1for a dielectric RI range from 1.33 to 1.40 in the near-infrared wavelength regime (1470 nm − 2220 nm). Apart from this, to ensure high detection accuracy of small refractive index (RI) changes, the sensor achieves the highest wavelength resolution and amplitude resolution of 2.70×10-6 RIU and 2.46×10-5RIU-1, respectively, along with the highest Figure of Merit of 217.65 RIU-1. Hence, the proposed PCF-SPR sensor can potentially be used to detect unknown refractive indices for various applications including biosensing.

Comparative Study of Zagreb Indices for Capped, Semi-Capped, and Uncapped Carbon Nanotubes

A carbon nanotube is made up of a molecular structure of a regular hexagonal lattice. This tube can be extended by closing one- or both end of the tube, denoted by semi-capped and capped nanotube. Due to its different structures, it is categorized into three forms, despite every form or shape, it has a wide range of usage. As a memory device, current-carrying device, regenerative study of medicines, and highly recommended in tissue engineering. Due to its numerous applications, there is immense interest to study its physical and chemical properties. As nanotube and its shape is motivated by a molecular structure and topological descriptors are also tools to study different molecular structures, we use this tool to analyze different nanotube structures like armchair carbon, semi-capped, and capped nanotube by Zagreb indices and its co-indices. Moreover, we also compared capped (semi or full) and uncapped armchair carbon nanotube and provided sufficient results of the comparison.

Topological interpretation of color exchange invariants: hexagonal lattice on a torus

We explain a correspondence between some invariants in the dynamics of color exchange in the coloring problem of a 2d regular hexagonal lattice, which are polynomials of winding numbers, and linking numbers in 3d. One invariant is visualized as linking of lines on a special surface with Arf-Kervaire invariant one, and is interpreted as resulting from an obstruction to transform the surface into its chiral image with special continuous deformations. We also consider additional constraints on the dynamics and see how the surface is modified.

A highly birefringent bend-insensitive porous core PCF for endlessly single-mode operation in THz regime: an analysis with core porosity

We present an endlessly single-mode, bend-insensitive and highly birefringent photonic crystal fibre (PCF) for efficient wave transmission in the terahertz (THz) spectrum. To facilitate flat dispersion and high birefringence, a diamond-shaped porous core is introduced inside the Zeonex (background material)-based hexagonal porous cladding. Moreover, the dense geometrical arrangement of the regular hexagonal lattice in the cladding region holds maximum usable optical power inside the porous core. The major geometrical parameter of the proposed PCF such as core porosity has been optimized to improve the optical guiding parameters. The numerically investigated outcomes have shown an ultrahigh birefringence and numerical aperture of 0.0888 and 0.57, respectively. Apart from this, very low bending and material losses of $$3.01 \times 10^{ - 20}$$ cm−1 and 0.01451 cm−1 with flattened dispersion variation of ± 0.2463 ps/THz/cm have been achieved over the broad THz band (i.e. 0.8–1.3 THz). Additionally, our proposed PCF sustains endlessly single-mode operation with a very high core power fraction of 75.28%. With all these promising results, the proposed PCF would be a potential candidate for polarization preserving as well as for efficient broadband transmission applications in the THz regime.

Anisotropy tailoring in geometrically isotropic multi-material lattices

This article proposes the concept of anisotropy tailoring in multi-material lattices based on a mechanics-based bottom-up framework. It is widely known that isotropy in a mono-material lattice can be obtained when the microstructure has an isotropic geometry. For example, regular hexagonal lattices with a unit cell comprised of six equal members and equal internal angle of 120o each, show isotropy in the elastic properties. Such limited microstructural configuration space for having isotropy severely restricts the scope of many multi-functional applications such as space filling in 3D printing. We first demonstrate that there are multiple structural geometries in mono-material lattices that could lead to isotropy. It is shown that the configuration space for isotropy can be expanded by multiple folds when more than one intrinsic material is introduced in the unit cell of a lattice. We explicitly demonstrate different degrees of anisotropy in regular geometrically isotropic lattices by introducing the multi-material architecture. The contours of achieving minimum anisotropy, maximum anisotropy and a fixed value of anisotropy are presented in the design space consisting of geometric and multi-material parameters. Proposition of such multi-material microstructures could essentially expand the multi-functional design scope significantly, offering a higher degree of flexibility to the designer in terms of choosing (or identifying) the most suitable microstructural geometry. An explicit theoretical characterization of the contours of anisotropy along with physical insights underpinning the configuration space of multi-material and geometric parameters will accelerate the process of its potential exploitation in various engineered multi-functional materials and structural systems across different length-scales with the demand of any specific degree of anisotropy but limitation in the micro-structural geometry.

The optimized layout strategy of ring oscillator network for Trojan detection

The method of Trojan detection based on the RON (Ring Oscillator Network) has been widely proposed and has a good effect. However, the method needs to integrate a RON on the chip to sense the voltage drop generated by the Trojan. Currently, the number of ROs (Ring Oscillator) required for a RON has not been discussed, which has a large impact on the additional hardware overhead. Therefore, based on the RON for Trojan detection, we propose an optimized layout strategy that can effectively control the number of ROs integrated into the circuit. By measuring the sensing range that can be perceived by a single RO, the sensing radius of the RO is obtained; then, ROs are placed in the circuit in a way of regular hexagonal lattice to clarify the number of ROs required in the network. Determining the appropriate number of ROs not only reduces the hardware overhead required for on-chip integration, but also ensures the reliability of Trojan detection, ensuring seamless coverage of the circuit with as few ROs as possible.

Effect of hydrogen adsorption energy on the electronic and optical properties of Si-modified single-layer graphene with an Al decoration

The adsorption energy and changes in the structural, electronic, and optical properties of an Al-modified Si-doped single-layer graphene (SLG) structure after hydrogen adsorption were studied using first-principles calculations. The simulation results revealed that the hydrogen adsorption energy of SLG + Si + Al gradually increased to 0.571 eV with an increase in the number of hydrogen molecules. After structural optimization, the C–C bond and the C–Si–C bond angle of SLG + Si + Al regular hexagonal lattice changed within 1.430–1.440 Å and decreased by 29.997°, respectively. In the SLG + Si + Al structure, the resonance of electrons was strengthened, the absorption spectrum of light was enhanced, the width of the absorption peak increased, and electrons absorbed more energy in the transition process.

Diameter-dependent polygonal cross section for holey phenine nanotubes

The cross-sectional shape of the nanotube is a key factor governing fundamental mechanical properties of the nanotube and the nanotube forest. In contrast to most circular nanotubes, in the present work, we demonstrate that the holey phenine nanotubes have polygonal cross sections with diameter-dependent number of sides. The non-circular cross section is attributed to the high twistability of the continuous C–C chains in the phenine nanotube. Consequently, the phenine nanotube forest has a square lattice structure rather than the regular hexagonal lattice of the carbon nanotube forest, resulting in a smooth buckling process under biaxial compression. The buckling pattern of the phenine nanotube forest is highly ordered with the orientation determined by the initial dislocation that frequently appears in the phenine nanotube forest.