Periodic Lattice Research Articles

Equivariant lattice bases

We study lattices in free abelian groups of infinite rank that are invariant under the action of the infinite symmetric group, with emphasis on finiteness of their equivariant bases. Our framework provides a new method for proving finiteness results in algebraic statistics. As an illustration, we show that every invariant lattice in Z ( N × [ c ] ) \mathbb {Z}^{(\mathbb {N}\times [c])} , where c ∈ N c\in \mathbb {N} , has a finite equivariant Graver basis. This result generalizes and strengthens several finiteness results about Markov bases in the literature.

Mechanical evaluation of elastomeric thermoplastic polyurethane additively manufactured triply periodic minimal surface area lattice structures for adjustable cushioning properties

PurposeCushioning is a useful material property applicable for a range of applications from medical devices to personal protective equipment. The current ability to apply cushioning in a product context is limited by the appropriateness of available materials, with polyurethane foams being the current gold standard material. The purpose of this study is to investigate additively manufactured flexible printing of scaffold structures as an alternative.Design/methodology/approachIn this study, this study investigates triply periodic minimal surface (TPMS) structures, including Gyroid, Diamond and Schwarz P formed in thermoplastic polyurethane (TPU), as a possible alternative. Each TPMS structure was fabricated using material extrusion additive manufacturing and evaluated to ASTM mechanical testing standard for polymers. This study focuses attention to TPMS structures fabricated for a fixed unit cell size of 10 mm and examine the compressive properties for changes in the scaffold porosity for samples fabricated in TPU with a shore hardness of 63A and 90A.FindingsIt was discovered that for increased porosity there was a measured reduction in the load required to deform the scaffold. Additionally, a complex relationship between the shore hardness and the stiffness of a structure. It was highlighted that through the adjustment of porosity, the compressive strength required to deform the scaffolds to a point of densification could be controlled and predicted with high repeatability.Originality/valueThe results indicate the ability to tailor the scaffold design parameters using both 63A and 90A TPU material, to mimic the loading properties of common polyurethane foams. The use of these structures indicates a next generation of tailored cushioning using additive manufacturing techniques by tailoring both geometry and porosity to loading and compressive strengths.

On the data-driven description of lattice materials mechanics

In the emerging field of mechanical metamaterials, using periodic lattice structures as a primary ingredient is relatively frequent. However, the choice of aperiodic lattices in these structures presents unique advantages regarding failure, e.g., buckling or fracture, because avoiding repeated patterns prevents global failures, with local failures occurring in turn that can beneficially delay structural collapse. Therefore, it is expedient to develop models for computing efficiently the effective mechanical properties in lattices from different general features while addressing the challenge of presenting topologies (or graphs) of different sizes. In this paper, we develop a deep learning model to predict energetically-equivalent mechanical properties of linear elastic lattices effectively. Considering the lattice as a graph and defining material and geometrical features on such, we show that Graph Neural Networks provide more accurate predictions than a dense, fully connected strategy, thanks to the geometrically induced bias through graph representation, closer to the underlying equilibrium laws from mechanics solved in the direct problem. Leveraging the efficient forward-evaluation of a vast number of lattices using this surrogate enables the inverse problem, i.e., to obtain a structure having prescribed specific behavior, which is ultimately suitable for multiscale structural optimization problems.

THE ROLE OF ATOMIC SCATTERING FACTORS IN DETERMINING THE MINERAL COMPOSITION OF THE BAZHENOV SUITE'S ROCKS

For the first time, it was demonstrated, that the use of standard atomic-structural parameters of minerals in X-ray diffraction (XRD) analysis to determine the phase composition in the Bazhenov suite's core is incorrect and most often leads to inaccurate information when calculating their quantitative ratio. In particular, by decomposing the XRD — based quantitative data on the mineral composition of the core of the Bazhenov suite into the main rock-forming oxides and further comparing them with reliable X-ray fluorescence spectrometry (XRF) data, an unsatisfactory convergence between them was established. At the same time, significant differences in the results of XRD and XRF have been shown in terms of oxides of silicon, sodium, magnesium, potassium, and iron, both at large and small mass fractions. It has been found that even with perfect XRD, achieved by optimizing the standard mineral structure parameters (lattice period, crystallite size, crystal lattice microdistortion, absorption coefficient, crystallographic texture, background) and achieving a very high degree of convergence (more than 95%) of experimental and theoretical diffractograms, the differences between oxide forms are still significant. It is shown, that the results of XRD of Bazhenov suite are questioned due to detected inconsistencies in oxide forms. Using the analysis of multiple factors and data from diffractograms of many samples, it has been established that the identified differences are primarily caused by the use of incorrect initial atomic-structure parameters of minerals in modeling. It is shown that rock minerals in the Bazhenov suite are primarily characterized by significant changes in the values of atomic scattering factors and significant displacements of atomic positions in the nodes of the crystal lattice, caused by the action of the high pressures and temperatures in the formation. For the first time, the XRD algorithm that has been developed, is presented and implemented in Bazhenov suite rocks, with consideration of the state of atoms in the analyzed mineral. Furthermore, it was found that the authors XRD approach enables the identification and quantification of the Bazhenov suite minerals with a relatively low content, which are challenging to identify during traditional XRD.

Broadband vibration mitigation using a two-dimensional acoustic black hole phononic crystala).

Acoustic black holes (ABHs) are known as efficient structural dampers. Periodic lattices are identified as an efficient way to forbidden wave propagation in targeted frequency bandgaps (BGs). The paper demonstrates the possibility to merge the ABH effect with Bragg BGs. The geometrical layout leading to this double effect consists of a plate of periodically modulated thickness by a combination of cosine functions of the spatial coordinates constituting an ABH-like cell, coated with a thin damping layer. The resulting metamaterial allows the realization of solid, stiff, and nonresonant panels over a wide frequency range, including low frequencies, without increasing the mass. First, the band structure is analyzed in the conservative case (without damping layer) using a plane wave expansion model following Kirchhoff's assumptions. The results show the existence of low-frequency BGs that can be controlled by only three geometric parameters, which are defined on the type of lattice chosen (square or hexagonal). Next, a finite size panel is designed for the hexagonal lattice. Experimental characterization of the demonstrator with and without viscoelastic coating shows very attractive broadband vibration mitigation performances due to the fact that the dissipation produced by the ABH effect does not deter filtering effects produced in the BG.

Exploring edge states in square–octagon moiré lattice

Moiré lattices, achieved by the superposition of two or more twisted identical periodic lattices, are of interest to various fields because they provide additional degrees of freedom. Here, we theoretically and experimentally study the edge states in a square–octagon moiré lattice. This moiré lattice is created by superimposing two identical square sublattices with an antiphase and a special twist angle. Five different edges, named type-I zigzag edge, type-II zigzag edge, type-I bearded edge, type-II bearded edge, and armchair edge, are explored. Through band structure analysis and numerical simulation of edge excitation, we find that all five edges support edge states. The topological property of the type-I edge states is verified by calculating the 2D polarization of the lattice. Furthermore, the edge mode distribution manifests that multiple bands support identical edge states at the armchair and type-II bearded edges. In the experiment, the moiré lattice is generated by the CW-laser-writing technique, thereby observing all the edge states with corresponding edge excitations. This study broadens the understanding of edge states in the coupled moiré photonic lattices and provides a new platform for exploring topological physics.

All-optical 3D blue phase photonic crystal switch with photosensitive dopants

Blue phase (BP) liquid crystals (LC) have lately become the focus of extensive research due to their peculiar properties and structure. BPs exhibit a highly organized 3D structure with a lattice period in the hundreds of nm. Owing to such structure, BPs are regarded as 3D photonic crystals. The unique properties of this complex LC phase are achieved by the self-assembly of the LC molecules into periodic cubic structures, producing bright selective Bragg reflections. Novel applications involving 3D photonic crystals would certainly benefit from enhanced ground-breaking functionalities. However, the use of BPs as 3D has been traditionally curtailed by the BP crystals trend to grow as random polycrystals, making it difficult to develop practical BP-based photonic devices. The possibility of generating mm-sized BP monocrystals was recently demonstrated. However, besides increasing the scarce number of 3D photonic structural materials, their applications as 3D photonic crystals do not show apparent advantages over other solid materials or metamaterials. Having a tunable BP monocrystal, where crystals could be switched, modulating simultaneously some of their properties as 3D photonic crystals, they would constitute a new family of materials with superior performance to other existing materials, opening up a plethora of new applications. In this work, an all-optical switchable 3D photonic crystal based on BPs doped with tailored photoactive molecules is demonstrated. Two switching modes have been achieved, one where the BP reversibly transitions between two BP phases, BPI and BPII, (two different cubic crystal systems) while maintaining the monocrystallinity of the whole system. The second mode, again reversible, switches between BPI and isotropic state. None of these modes are related to the regular thermal transitions between LC phases; switching is triggered by light pulses of different wavelengths. This all-optical approach allows for a seamless fast remotely controlled optical switch between two 3D photonic crystals in different cubic crystal systems and between a photonic crystal and an isotropic matrix. Applications of switchable BPs for adaptive optics systems or photonic integrated circuits would make great advances using 3D photonic crystal switches. All-optical photonic systems such as these hold great promise for the development of tunable and efficient photonic devices such as dynamic optical filters and sensors, as they enable light-driven modulation and sensing applications with unprecedented versatility.

Poissonian pair correlation for directions in multi-dimensional affine lattices and escape of mass estimates for embedded horospheres

Abstract We prove the convergence of moments of the number of directions of affine lattice vectors that fall into a small disc, under natural Diophantine conditions on the shift. Furthermore, we show that the pair correlation function is Poissonian for any irrational shift in dimension 3 and higher, including well-approximable vectors. Convergence in distribution was already proved in the work of Strömbergsson and the second author [The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems. Ann. of Math. (2)172 (2010), 1949–2033], and the principal step in the extension to convergence of moments is an escape of mass estimate for averages over embedded $\operatorname {SL}(d,\mathbb {R})$ -horospheres in the space of affine lattices.

Additive manufactured metal lattice structures used as flow-through electrodes in a filter-press electrochemical reactor

This study deals with the electrochemical and hydraulic performances of additive manufactured metal periodic lattice structure electrodes. These electrodes are fabricated by Laser Powder Bed Fusion (LPBF) from the titanium alloy TA6V and two structures have been studied: diagonal and octet-truss. They are integrated, as flow-through electrodes, in an electrochemical filter-press reactor. Current–potential curves are experimentally obtained by linear sweep voltammetry when feeding the reactor with a ferricyanide solution. From the measurement of the limiting current of ferricyanide ions reduction, the volumetric mass transfer coefficient kAe is determined for several flow rates. Numerical simulations of fluid flow and mass transfer in these structures are also performed. Results show high values of kAe for these structures (competing with carbon felt electrodes) while maintaining a high permeability (low flow resistance). The surface roughness, inherent to the additive manufacturing process, is found to induce a low (≲10%) enhancement of mass transfer despite the roughness size (Ra in the range 10-20μm) is close to the average thickness of the diffusion layer (for the investigated fluid velocities). As a result, these electrode structures show promising potential to be used in intensified electrochemical reactors by being further optimized to find the optimal internal dimensions that maximize mass transfer while limiting flow resistance.

Crack-insensitive fracture of elastomer-based soft network materials under monotonic and cyclic loads

In various applications of stretchable electronics and soft robots, soft materials are subjected to long-term load-bearing conditions, requiring them to possess good extreme mechanical properties such as highly stretchable, strong, fracture-resistant and anti-fatigue. Conventional soft materials with a single polymer network typically exhibit relatively low fracture resistance, while commonly employed toughening method involves introducing mechanical dissipation into original polymer networks often require complicated chemical synthesis and result in significant hysteresis under cyclic loads. Designing soft solid materials into soft network materials with periodic lattice structure paving another effective route to improve the fracture-resistant performance of soft materials, which has not been reported in the literature. This work provides a combined experimental and computational study on the mechanical properties and fracture behaviors of elastomer-based soft network materials with/without a precut crack under monotonic and cyclic loads, aiming to present a structural design strategy for fracture-resistant soft materials. The elastomer-based soft network materials are proven to be crack-insensitive, low-hysteresis and anti-fatigue. The nonlinear finite element analysis (FEA) is employed to reveal the underlying mechanism of crack-insensitivity in elastomer-based soft network materials. Both the mechanical properties and deformed configurations of soft network materials with/without a precut crack are accurately predicted by the FEA method. The effect of microstructure geometry and network topology on the mechanical properties of soft network materials is systematically studied. Under various amplitudes of cyclic applied strain, the elastomer-based soft network materials can survive even after individual microstructures ruptured, and the fatigue fracture process slows down as the applied strain amplitude decreases. In contrast to the behavior observed in monotonic loads where fracture initiates at the microstructure located ahead of the crack tip, fatigue fracture initiation in soft network materials with a precut crack exhibits a random distribution.

The effect of permeability on the flow structure of porous square cylinders

This study experimentally investigates the wake structure of a porous square cylinder in terms of permeability over two decades of $Da$ (i.e. $2.4 \times 10^{-5} < Da < 2.9 \times 10^{-3}$ ). The porous cylinder, featuring a simple cubic lattice structure, was fabricated using an additive manufacturing technique. This unique method, combined with a periodic and scalable lattice structure, effectively isolates permeability from porosity, making it suitable for an in-depth parametric study. The key parameter, permeability, was directly estimated by measuring the pressure drop and superficial velocity for each porous case in an open-loop pipe flow system. The downstream flow fields were obtained using standard planar particle image velocimetry measurements in an open-loop wind tunnel. Based on the experimental data, structural modifications in the near wake were examined in relation to permeability, leading to the identification of four distinct flow regimes depending on $Da$ . Additionally, the downstream flow adjustment length ( $L_i$ ) was assessed by introducing a permeability-based source term into the momentum equation, facilitating the development of an analytical model for $L_i$ . The present experimental data support this analytical model, and our results further confirmed that $L_i$ plays a crucial role as a characteristic length scale in the near wake.

Study on the crashworthiness of multi-cell thin-walled tubes filled with triply periodic minimal surface lattices

To improve energy absorption and efficiency and develop a structure with outstanding energy absorption capability, this study proposed designing multi-cell tubes filled with triply periodic minimal surface (TPMS) lattices. Based on experimental and simulation methods, the crashworthiness of multi-cell tubes filled with three types of TPMS lattices (Diamond, Gyroid, and Primitive) was investigated. Five multi-cell tube structures were fabricated by increasing the thin-walled tube corner units, improving the energy absorption efficiency by 67.98%. The results show that due to the coupling between TPMS lattices and thin-walled tubes, multi-cell tubes filled with TPMS lattices exhibited better crashworthiness performance. The energy absorption of multi-cell tubes filled with Diamond, Gyroid, and Primitive lattices increased sequentially. The energy absorption per unit mass and energy absorption capability were most notably enhanced for the multi-cell tubes filled with Diamond lattice, with energy absorption efficiency 17.24% higher and energy absorption 104.26% greater than empty multi-cell tubes. This inspires crash designs of rail vehicles and automotive fronts.

Strain-Induced Ferromagnetism in Monolayer T″-Phase VTe2: Unveiling Magnetic States and Anisotropy for Spintronics Advancement.

Two-dimensional (2D) ferromagnets have attracted significant interest for their potential in spintronic device miniaturization, especially since the discovery of ferromagnetic ordering in monolayer materials such as CrI3 and Fe3GeTe2 in 2017. This study presents a detailed investigation into the effects of the Hubbard U parameter, biaxial strain, and structural distortions on the magnetic characteristics of T″-phase VTe2. We demonstrate that setting the Hubbard U to 0 eV provides an accurate representation of the observed structural, magnetic, and electronic features for both bulk and monolayer T″-phase VTe2. The application of strain reveals two distinct ferromagnetic states in the monolayer T″-phase VTe2, each characterized by minor structural differences, but notably different magnetic moments. The T″-1 state, with reduced magnetic moments, emerges under compressive strain, while the T″-2 state, featuring increased magnetic moments, develops under tensile strain. Our analysis also compares the magnetic anisotropy between the T and T″ phases of VTe2, highlighting that the periodic lattice distortion in the T″-phase induces an in-plane anisotropy, which makes it a material with an easy-axis of magnetization. Monte Carlo simulations corroborate our findings, indicating a high Curie temperature of approximately 191 K for the T″-phase VTe2. Our research not only sheds light on the critical aspects of the VTe2 system but also suggests new pathways for enhancing low-dimensional magnetism, contributing to the advancement of spintronics and straintronics.

Scalable Nanophotonic Structures Inside Silica Glass Laser‐Machined by Intense Shaped Beams

AbstractAll‐dielectric nanophotonic devices are usually fabricated by engraving arrays of nanoholes at the surface of high‐index materials, to engineer dedicated optical functions. However, their direct 3D integration in the volume of a material is challenging, inaccessible to current planar nanolithography methods. Here is introduced an ultrafast laser‐machining method that opens the possibility to realize scalable arrays of hollow nanochannels directly inside the bulk of silica glass within a single‐step, maskless, and digital approach. Using a custom‐shaped micro‐Bessel beam and by tuning laser pulse durations from femtoseconds to picosecond to boost processing versatility, dense assemblies of nanochannels with adjustable lengths (up to 30 µm), and submicron lattice periodicity (down to 0.7 µm) are achieved. As a proof‐of‐principle demonstration, a gradient‐index metaphotonic structure is realized and its performance is experimentally characterized, demonstrating its relevancy for imprinting phase functions with magnitude up to in the short‐wave infrared spectral range. Results show that the unique flexibility and scalability provided by individual control of each channel opens a new realistic alternative approach for 3D fabrication of monolithic integrated nanophotonic devices inside a wide range of low‐index standard optical materials.

Large deformation response of a novel triply periodic minimal surface skeletal-based lattice metamaterial with high stiffness and energy absorption

Triply periodic minimal surface (TPMS) lattice metamaterials present superior mechanical performance over traditional strut-based lattice metamaterials due to their unique structural characteristics. However, the high stiffness–stable plastic response trade-off dilemma is still the major challenge for skeletal-based metamaterials. Herein, a novel TPMS skeletal lattice metamaterial is proposed. Finite element simulations are conducted to reveal its elastic properties and plastic response under compression. Afterwards, validation experiments are performed on the lattice specimens fabricated by the most common Fused Deposition Modeling (FDM) process. The nominal stress–strain curves and collapse mode of the lattice materials are obtained accordingly. Both the numerical simulations and experiments demonstrate that the proposed TPMS lattice metamaterial exhibits high specific modulus, strength and energy absorption, together with a smooth and elongated stress plateau, thereby overcoming the strength-efficiency trade-off. Different from the rigid nodal region in traditional strut-based lattices, the strut connection parts in the novel TPMS lattices experience shear and twisting, which avoids the catastrophic failure of the struts and enhances the loading efficiency of the structure under compression. Meanwhile, the numerical results indicate that the stable plastic response of the designed architecture is insensitive to the plastic flow behavior of the bulk material, which is also supported by the experimental results. It is also demonstrated that the novel TPMS lattice metamaterial presents several times higher stiffness as well as energy absorption simultaneously than the current strut-based lattice materials, which can be potentially applied as load-bearing components and impact energy absorbers.

Time-dependent properties of run-and-tumble particles. II. Current fluctuations.

We investigate steady-state current fluctuations in two models of hardcore run-and-tumble particles (RTPs) on a periodic one-dimensional lattice of L sites, for arbitrary tumbling rate γ=τ_{p}^{-1} and density ρ; model I consists of standard hardcore RTPs, while model II is an analytically tractable variant of model I, called a long-ranged lattice gas (LLG). We show that, in the limit of L large, the fluctuation of cumulative current Q_{i}(T,L) across the ith bond in a time interval T≫1/D grows first subdiffusively and then diffusively (linearly) with T: 〈Q_{i}^{2}〉∼T^{α} with α=1/2 for 1/D≪T≪L^{2}/D and α=1 for T≫L^{2}/D, where D(ρ,γ) is the collective- or bulk-diffusion coefficient; at small times T≪1/D, exponent α depends on the details. Remarkably, regardless of the model details, the scaled bond-current fluctuations D〈Q_{i}^{2}(T,L)〉/2χL≡W(y) as a function of scaled variable y=DT/L^{2} collapse onto a universal scaling curve W(y), where χ(ρ,γ) is the collective particle mobility. In the limit of small density and tumbling rate, ρ,γ→0, with ψ=ρ/γ fixed, there exists a scaling law: The scaled mobility γ^{a}χ(ρ,γ)/χ^{(0)}≡H(ψ) as a function of ψ collapses onto a scaling curve H(ψ), where a=1 and 2 in models I and II, respectively, and χ^{(0)} is the mobility in the limiting case of a symmetric simple exclusion process; notably, the scaling function H(ψ) is model dependent. For model II (LLG), we calculate exactly, within a truncation scheme, both the scaling functions, W(y) and H(ψ). We also calculate spatial correlation functions for the current and compare our theory with simulation results of model I; for both models, the correlation functions decay exponentially, with correlation length ξ∼τ_{p}^{1/2} diverging with persistence time τ_{p}≫1. Overall, our theory is in excellent agreement with simulations and complements the prior findings [T. Chakraborty and P. Pradhan, Phys. Rev. E 109, 024124 (2024)1539-375510.1103/PhysRevE.109.024124].

Ultra-high Q resonances based on zero group-velocity modes accompanied by bound states in the continuum in 2D photonic crystal slabs

Optical resonators made of 2D photonic crystal (PhC) slabs provide efficient ways to manipulate light at the nanoscale through small group-velocity modes with low radiation losses. The resonant modes in periodic photonic lattices are predominantly limited by nonleaky guided modes at the boundary of the Brillouin zone below the light cone. Here, we propose a mechanism for ultra-high Q resonators based on the bound states in the continuum (BICs) above the light cone that have zero-group velocity (ZGV) at an arbitrary Bloch wavevector. By means of the mode expansion method, the construction and evolution of avoided crossings and Friedrich-Wintgen BICs are theoretically investigated at the same time. By tuning geometric parameters of the PhC slab, the coalescence of eigenfrequencies for a pair of BIC and ZGV modes is achieved, indicating that the waveguide modes are confined longitudinally by small group-velocity propagation and transversely by BICs. Using this mechanism, we engineer ultra-high Q nanoscale resonators that can significantly suppress the radiative losses, despite the operating frequencies above the light cone and the momenta at the generic k point. Our work suggests that the designed devices possess potential applications in low-threshold lasers and enhanced nonlinear effects.

SCATTERING OF ELECTROMAGNETIC WAVES ON FLAT GRID TWO-PERIODIC STRUCTURES

Context. One of the scientific hypotheses for the creation of nonreciprocal optical metasurfaces is based on the use of a wave channel in which rays of the direct and reverse diffraction scenarios are realized on two-periodic flat structures with nonlinear elements. Such processes in the nanometer wavelength range of electronic devices require precise calculations of the interaction of waves and microstructures of devices. It is also important to describe the behavior of antenna devices in mobile communications. Expanding the wavelength range of stable communication is achieved by using prefractal structures in antenna devices in combination with periodic structuring. Similar modeling problems arise when electromagnetic waves penetrate materials with a crystalline structure (radio transparency).
 Objective. To test this hypothesis, it is necessary to carry out mathematical modeling of the process of scattering of electromagnetic waves by metasurfaces under conditions of excitation of several diffraction orders. It is known that among two-periodic flat lattices of different structures there are five types that fill the plane. These are the Bravais grilles. The problem of scattering of an incident monochromatic TE polarized wave on a metal screen with recesses in two-periodic structures filled with silicon was considered.
 Method. The paper builds mathematical models for the study of spatial-amplitude spectra of metasurfaces on Brave lattices and gives some results of their numerical study. The condition for determining the diffraction orders propagating over the grating is proposed. Scattered field amplitudes are from the solution of the boundary value problem for the Helmholtz equation in the COMSOL Multiphysics 5.4 package. Similar problem formulations are possible when studying the penetration of an electromagnetic field into a crystalline substance.
 Results. Obtained relations for diffraction orders of electromagnetic waves scattered by a diffraction grating. The existence of wavelengths incident on a two-periodic lattice for which there is no reflected wave is shown for different shapes (rectangular, square, hexagonal) of periodic elements in the center of which a depression filled with silicon was made. Distributions of reflection coefficients for different geometric sizes of colored elements and recesses are given. The characteristics of the electric field at resonant modes in the form of modulus isolines show the nature of the interaction of the field over the periodic lattice and the scatterersdepressions. At the resonant wavelengths of the incident waves, standing waves appear in the scatterers.
 Conclusions. A mathematical model of the set of diffraction orders propagating from a square and hexagonal lattice into halfspace is proposed z >= 0. It has been shown that flat periodic lattice with square or hexagonal periodicity elements and resonant scatterers in the form of cylindrical recesses filled with silicon can produce a non-mirrored scattered field in metal. The response of the lattices to changes in the wavelength of the incident field by the structure of diffraction orders of the scattered field and high sensitivity to the rotation of the incident plane were revealed. The two-periodic lattices have prospects for creating anti-reflective surfaces of various devices. Two-periodic lattices have prospects for creating anti-reflective surfaces for various devices, laser or sensor electronic devices, antennas in mobile communication elements, and radio transparency elements. They have more advanced manufacturing technologies in relation to spatial crystal structures.

Hyperuniformity in two-dimensional periodic and quasiperiodic point patterns.

We study hyperuniform properties in various two-dimensional periodic and quasiperiodic point patterns. Using the histogram of the two-point distances, we develop an efficient method to calculate the hyperuniformity order metric, which quantifies the regularity of the hyperuniform point patterns. The results are compared with those calculated with the conventional running average method. To discuss how the lattice symmetry affects the order metric, we treat the trellis and Shastry-Sutherland lattices with the same point density as examples of periodic lattices, and Stampfli hexagonal and dodecagonal quasiperiodic tilings with the same point density as examples of quasiperiodic tilings. It is found that the order metric for the Shastry-Sutherland lattice (Stampfli dodecagonal tilings) is smaller than the other in the periodic (quasiperiodic) tiling, meaning that the order metric is deeply related to the lattice symmetry. Namely, the point pattern with higher symmetry is characterized by the smaller order metric when their point densities are identical. Order metrics for several other quasiperiodic tilings are also calculated.

Crashworthiness of 3D Lattice Topologies under Dynamic Loading: A Comprehensive Study.

Periodic truss-based lattice materials, a particular subset of cellular solids that generally have superior specific properties as compared to monolithic materials, offer regularity and predictability that irregular foams do not. Significant advancements in alternative technologies-such as additive manufacturing-have allowed for the fabrication of these uniquely complex materials, thus boosting their research and development within industries and scientific communities. However, there have been limitations in the comparison of results for these materials between different studies reported in the literature due to differences in analysis approaches, parent materials, and boundary and initial conditions considered. Further hindering the comparison ability was that the literature generally only focused on one or a select few topologies. With a particular focus on the crashworthiness of lattice topologies, this paper presents a comprehensive study of the impact performance of 24 topologies under dynamic impact loading. Using steel alloy parent material (manufactured using Selective Laser Melting), a numerical study of the impact performance was conducted with 16 different impact energy-speed pairs. It was possible to observe the overarching trends in crashworthiness parameters, including plateau stress, densification strain, impact efficiency, and absorbed energy for a wide range of 3D lattice topologies at three relative densities. While there was no observed distinct division between the results of bending and stretching topologies, the presence of struts aligned in the impact direction did have a significant effect on the energy absorption efficiency of the lattice; topologies with struts aligned in that direction had lower efficiencies as compared to topologies without.