Two-dimensional Unit Cell Research Articles

Design and experimental validation of a finite-size labyrinthine metamaterial for vibro-acoustics: enabling upscaling towards large-scale structures.

In this article, we present the design and experimental validation of a labyrinthine metamaterial for vibro-acoustic applications. Based on a two-dimensional unit cell, different designs of finite-size metamaterial specimens in a sandwich configuration including two plates are proposed. The design phase includes an optimization based on Bloch-Floquet analysis with the aims of maximizing the band gap and extruding the specimens in the third dimension while keeping the absorption properties almost unaffected. By manufacturing and experimentally testing finite-sized specimens, we assess their capacity to mitigate vibrations in vibro-impact tests. The experiments confirm a band gap in the low- to mid-frequency range. Numerical models are employed to validate the experiments and to examine additional vibro-acoustic load cases. The metamaterial's performances are compared with benchmark solutions, usually employed for noise and vibration mitigation, showing a comparable efficacy in the band gap region. To eventually improve the metamaterial's performance, we optimize its interaction with the air and test different types of connections between the metamaterial and the homogeneous plates. This finally leads to metamaterial samples largely exceeding the benchmark performances in the band gap region and reveals the potential of interfaces for performance optimization of composed structures.This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 1)'.

Investigation of additively manufactured lattice structures for refractive optical systems based on auxetic properties

This paper reports on a novel concept of thermally stable lattice structures for next-generation refractive optical systems. The aim is to develop a metallic mount for optical lenses based on auxetic structures fabricated by additive manufacturing. Auxetic structures exhibit an effective negative Poisson ratio at the macroscopic level. This property is used to develop a mounting structure that compensates for thermal defocus between refractive optical elements. Numerical and experimental studies of the mechanical behavior of additively manufactured body-centered cubic cells provide the basis for the development of this application-specific mounting structure. Several static load cases are analyzed and deviations between numerical and experimental data are examined to quantify differences in dependence of element type, element size, and manufacturing-related geometrical inaccuracies. These results are used to compare the numerical analysis of the Poisson ratio of an auxetic unit cell with the analytical descriptions. This comparison is then applied to two-dimensional auxetic unit cell compounds. These studies provide the basis for the further development process of an auxetic mounting structure.

Parametric optimization of selected auxetic structures

Auxetic materials exhibit an interesting, counterintuitive behavior—when subjected to uniaxial tension, they stretch laterally, and when uniaxially compressed, they shrink laterally. In contrast to conventional materials, in auxetics, the value of Poisson’s ratio is negative. Behavior of auxetic materials is an effect of their internal structures. The auxetic effect depends mostly on the geometry of their internal unit cells and not on the properties of the bulk material. This paper presents the results of parametric optimization of selected two-dimensional auxetic unit cells with the aim to identify the geometrical parameters which exhibit the strongest influence on the value of Poisson’s ratio in each unit cell, and to identify geometries which exhibit the strongest auxetic effect. The optimization was conducted through numerical simulation with the use of the finite element method in commercial software. Response surface optimization and multi-objective genetic algorithm (MOGA) were applied. Obtained candidate geometries were verified via additional FEM analyses and confirmed to have improved auxetic effect and reduced equivalent stress. 5 × 5 structures composed of reference and optimized geometries of analyzed unit cells were subjected to similar analyses and it was confirmed that the optimization of singular unit cells caused an improvement of auxetic effect and reduction in equivalent stress in regular structures composed of multiple unit cells.

On the realization of periodic boundary conditions for hexagonal unit cells

In the context of homogenization of micro-heterogeneous materials, the choice of the Representative Volume Element (RV E) plays a crucial role. For periodic microstructures, an RV E is an underlying unit cell with periodic boundary conditions. Nevertheless, the question of the implementation of periodic boundary conditions may arise here; for example, some of the applications of periodic boundary conditions in the literature for hexagonal cells are incorrect or they are not given in detail. In this paper, we analyze periodic boundary conditions for two-dimensional hexagonal unit cells. Periodic boundary conditions are characterized by the periodic fluctuations in the displacement fields and anti-periodic traction vectors at associated points of the boundary of the unit cell. From comparative calculations with an ensemble of unit cells, it is evident that the natural choice for vanishing fluctuations is to be set on the midpoints of the six perimeter lines of the cell. The partially applied choice of vanishing fluctuations in the six corner points of the outer edge of the unit cell leads to wrong results. The boundary conditions proposed here are analyzed on the basis of representative examples and compared to the results with the incorrect boundary conditions.

Inverse Design of Two-Dimensional Shape-Morphing Structures

Abstract This study proposes an inverse method for synthesizing shape-morphing structures in the lateral direction by integrating two-dimensional hexagonal unit cell with curved beams. Analytical expressions are derived to formulate the effective Young’s modulus and Poisson’s ratio for the base unit-cell as a function of its geometric parameters. The effective lateral Poisson’s ratio can be controlled by manipulating a set of geometric parameters, resulting in a dataset of over 6000 data points with Poisson’s ratio values ranging from −1.2 to 10.4. Furthermore, we utilize the established dataset to train an inverse design framework that utilizes a physics-guided neural network algorithm, and the framework can predict design parameters for a targeted shape-morphing structure. The proposed approach enables the generation of structures with tailored Poisson’s ratio ranging from −1.2 to 3.4 while ensuring flexibility and reduced stress concentration within the predicted structure. The generated shape-morphing structures’ performance is validated through numerical simulation and physical tensile testing. The finite element analysis (FEA) simulation results confirm agreement with the designed values for the shape-morphing structure, and the tensile testing results reveal the same trend in shape-morphing behavior. The proposed design automation framework demonstrates the feasibility of creating intricate and practical shape-morphing structures with high accuracy and computational efficiency.

Physics-informed machine learning of redox flow battery based on a two-dimensional unit cell model

In this paper, we present a physics-informed neural network (PINN) approach for predicting the performance of an all-vanadium redox flow battery, with its physics constraints enforced by a two-dimensional (2D) mathematical model. The 2D model, which includes 6 governing equations and 24 boundary conditions, provides a detailed representation of the electrochemical reactions, mass transport and hydrodynamics occurring inside the redox flow battery. To solve the 2D model with the PINN approach, a composite neural network is employed to approximate species concentration and potentials; the input and output are normalized according to prior knowledge of the battery system; the governing equations and boundary conditions are first scaled to an order of magnitude around 1, and then further balanced with a self-weighting method. Our numerical results show that the PINN is able to predict cell voltage correctly, but the prediction of potentials shows a constant-like shift. To fix the shift, the PINN is enhanced by further constrains derived from the current collector boundary. Finally, we show that the enhanced PINN can be even further improved if a small number of labeled data is available.

Design of innovative self-expandable femoral stents using inverse homogenization topology optimization

In this study, we propose a novel computational framework for designing innovative self-expandable femoral stents. First, a two-dimensional stent unit cell is designed by inverse homogenization topology optimization. In particular, the unit cell is optimized in terms of contact area with the target of matching prescribed mechanical properties. The topology optimization is enriched by an anisotropic mesh adaptation strategy, enabling a time- and cost-effective procedure that promotes original layouts. Successively, the optimized stent unit cell is periodically repeated on a hollow cylindrical surface to construct the corresponding three-dimensional device. Finally, structural mechanics and computational fluid dynamics simulations are carried out to verify the performance of the newly-designed stent. The proposed workflow is being tested through the design of five proof-of-concept stents. These devices are compared through specific performance evaluations, which include the assessments of the minimum requirement for usability, namely the ability to be crimped into a catheter, and the quantification of the radial force, the foreshortening, the structural integrity and the induced blood flow disturbances.

A semi-analytic method for the computation of the effective properties of composites of two isotropic constituent materials

We develop a computational framework for the semi-analytic representation of the effective transport properties of two-component composite materials, in the spirit of Bergman’s spectral representation. The components of the effective permittivity (or conductivity) tensor are expanded as power series in a contrast parameter between the two phases. The coefficients (the moments of a spectral measure) of these expansions are determined recursively through the numerical evaluations of integrals defined solely on the boundary between the constituent phases, and discretized by high-order (possibly exponentially converging) quadrature schemes. The high precision reached by these coefficients allows the series representation to be recast into Padé approximants (or continued fractions) to provide analytical expressions valid over large (possibly infinite) regions of the parameter complex plane. Examples, thus far limited to two-dimensional rhombic unit cells, demonstrate that the method is reliable for microgeometries and material parameters for which other methods are far less dependable. The analytical representations are particularly useful to study the optical or electrostatic resonant behaviour of some composite materials.

Energy absorption and vibration mitigation performances of novel 2D auxetic metamaterials

This paper proposes novel unit cells that can be arranged periodically constructing disparate auxetic metamaterials with negative Poisson’s ratio for engineering applications. The two-dimensional auxetic unit cells are designed utilizing the mathematical model of the modified Solid Isotropic Material with Penalization (SIMP) topological optimization method with the objective of maximizing the magnitude of the negative Poisson’s ratio. The deformation mechanisms of the auxetic structures are controlled effectively by straight and oblique struts attached at rotation joints since the optimum designs are a combination of chiral and re-entrant unit cells. Three structures with optimum auxetic designs are chosen to verify the modified SIMP method using numerical simulation. Additionally, the energy absorption and vibration mitigation of auxetic metamaterials have been extensively studied. The results showed a significant impact of negative Poisson’s ratio on the considered dynamic properties of the auxetic structures.

Printable assemblies of perovskite nanocubes on meter-scale panel.

Hierarchical assemblies of functional nanoparticles can have applications exceeding those of individual constituents. Arranging components in a certain order, even at the atomic scale, can result in emergent effects. We demonstrate that printed atomic ordering is achieved in multiscale hierarchical structures, including nanoparticles, superlattices, and macroarrays. The CsPbBr3 perovskite nanocubes self-assemble into superlattices in ordered arrays controlled across 10 scales. These structures behave as single nanoparticles, with diffraction patterns similar to those of single crystals. The assemblies repeat as two-dimensional planar unit cells, forming crystalline superlattice arrays. The fluorescence intensity of these arrays is 5.2 times higher than those of random aggregate arrays. The multiscale coherent states can be printed on a meter-scale panel as a micropixel light-producing layer of primary-color photon emitters. These hierarchical assemblies can boost the performance of optoelectronic devices and enable the development of high-efficiency, directional quantum light sources.

Two-Dimensional Distributed Transmission-Line Models for Broadband Full-Tensor Anisotropic Acoustic Metamaterials Based on Transformation Acoustics

We propose the design method for broadband acoustic metamaterials based on the concept of transformation acoustics. Two-dimensional distributed transmission-line (TL) models for full-tensor anisotropic electromagnetic metamaterials are applied to full-tensor anisotropic acoustic metamaterials and the design formulas are shown to uniquely determine the structural parameters of the unit cells. Two-dimensional acoustic waveguide unit cell structures for realizing the TL models are proposed and an acoustic carpet cloak and an acoustic illusion medium are designed according to the introduced theory. The complex sound pressure distributions in the acoustic waveguides of the unit cells are calculated by full-wave simulations to verify the validity of the proposed method, and the broadband operations of the designed carpet cloak and illusion medium are confirmed from the results.

Investigating the stochastic dispersion of 2D engineered frame structures under symmetry of variability

Additive manufacturing has enabled the construction of increasingly complex mechanical structures. However, the variability of mechanical properties may be higher than that of conventionally manufactured structures. Typically, the computational cost of the numerical modeling of such structures considerably increases when variability is considered. In deterministic analyses of periodic structures, the dispersion diagrams obtained for the first Brillouin zone (FBZ) can be used to predict attenuation bands for any direction of propagation. This can be further simplified considering only the contour of the irreducible Brillouin zone (IBZ) if the unit cell presents symmetries. The objective of the current investigation is to present evidence that, similarly to what occurs in deterministic cases, the stochastic results obtained by scanning only the IBZ contour of the proposed two-dimensional unit cell under 4-fold rotational symmetry of statistical variability coincides with statistical results obtained scanning the FBZ. This is not a direct result, because each individual unit cell sample is asymmetric. We show that, under symmetry of variability statistics, the stochastic results computed for supercells and for finite metastructures consisting of a finite number of cells also coincide with the results for the IBZ contour of the unit cell. This result is important, as it dramatically reduces the computation cost of the stochastic analysis of such structures.

Defect-mode-induced energy localization/harvesting of a locally resonant phononic crystal plate: Analysis of line defects

Phononic crystals that are artificially engineered structures have recently been introduced for vibration energy harvesting and sensing applications due to their unique features of band gaps and wave propagation control. Conventional energy harvesters made of phononic crystals are mainly designed for acoustic energy harvesting at a high-frequency vibration source (i.e., kHz levels). In this work, a defect-mode-induced energy harvester is designed for low-frequency excitations in the range of 0–300 ​Hz. The entire system that is a locally resonant phononic crystal (LRPC) plate with line defect patterns is consisted of elastic-wrapped core scatterers periodically embedded in epoxy resin. A two-dimensional (2D) three-component unit cell structure is arranged on the plate and the band gap property is analyzed to optimize the geometric parameters. Defects are then introduced to the LRPC plate with a 7 ​× ​7 point array for analysis. In addition, numerical and experimental studies are conducted to investigate the performance of energy harvesting when attaching a piezoelectric patch on the defect points. The results demonstrate that the proposed LRPC vibration energy harvester having a line defect mode (with continuous or alternate points) shows good performance in energy harvesting, in which a peak power output of 42.72 ​mV can be achieved under 10 ​m/s2 and 252 ​Hz. The performance is almost 6 times more than that of the single-point defect model under the same excitation conditions. The present LRPC-type energy harvester with a line defect mode is more suitable for energy harvesting for low-frequency and broadband conditions.

Double Fourier Integral Analysis Based Convolutional Neural Network Regression for High-Frequency Energy Disaggregation

Non-Intrusive Load Monitoring aims to extract the energy consumption of individual electrical appliances through disaggregation of the total power load measured by a single smart-meter. In this article we introduce Double Fourier Integral Analysis in the Non-Intrusive Load Monitoring task in order to provide more distinct feature descriptions compared to current or voltage spectrograms. Specifically, the high-frequency aggregated current and voltage signals are transformed into two-dimensional unit cells as calculated by Double Fourier Integral Analysis and used as input to a Convolutional Neural Network for regression. The performance of the proposed methodology was evaluated in the publicly available U.K.-DALE dataset. The proposed approach improves the estimation accuracy by 7.2% when compared to the baseline energy disaggregation setup using current and voltage spectrograms.

Equivalent 3D properties of thin-walled composite structures using mechanics of structure genome

Equivalent 3D properties are obtained for composite thin-walled 3D structures using the concept of mechanics of structure genome. The adopted homogenization technique interprets the unit cell associated with the thin-walled 3D structures as an assembly of plates, and the overall strain energy density of the unit cell as a summation of the plate strain energies of these individual plates. The concept of mechanics of structure genome is then applied to drop all higher-order terms and the remaining energy is minimized with respect to the unknown fluctuating functions. This has been done by discretizing the two-dimensional unit cell into one-dimensional frame elements in a finite element description. This allows the handling of structures with different levels of complexities and internal geometry within a general framework. Comparisons have been made with existing work to show the advantage the proposed model offers over existing ones.

Correlation between two- and three-dimensional crystallographic lattices for epitaxial analysis. II. Experimental results.

While the crystal structure of the polymorph phase can be studied in three dimensions conveniently by X-ray methods like grazing-incidence X-ray diffraction (GIXD), the first monolayer is only accessible by surface-sensitive methods that allow the determination of a two-dimensional lattice. Here, GIXD measurements with sample rotation are compared with distortion-corrected low-energy electron diffraction (LEED) experiments on conjugated molecules: 3,4;9,10-perylenetetracarboxylic dianhydride (PTCDA), 6,13-pentacenequinone (P2O), 1,2;8,9-dibenzopentacene (trans-DBPen) and dicyanovinyl-quaterthiophene (DCV4T-Et2) grown by physical vapor deposition on Ag(111) and Cu(111) single crystals. For these molecular crystals, which exhibit different crystallographic lattices and crystal orientations as well as epitaxial properties, the geometric parameters of the three-dimensional lattice are compared with the corresponding geometry of the first monolayer. A comparison of the monolayer lattice from LEED investigations with the multilayer lattices determined by rotated GIXD experiments reveals a correlation between the first monolayer and the epitaxial growth of three-dimensional crystals together with lattice distortions and re-alignment of molecules. The selected examples show three possible scenarios of crystal growth on top of an ordered monolayer: (i) growth of a single polymorph, (ii) growth of three different polymorphs; in both cases the first monolayer serves as template. In the third case (iii) strong lattice distortion and distinct molecular re-alignments from the monolayer to epitaxially grown crystals are observed. This is the second part of our work concerning the correlation between two- and three-dimensional crystallographic lattices for epitaxial analysis. In the first part, the theoretical basis has been derived which provides a mathematical relationship between the six lattice parameters of the three-dimensional case and the three parameters obtained for the two-dimensional surface unit cell, together with their orientation to the single-crystalline substrate. In this work, a combined experimental approach of GIXD and LEED is introduced which can be used to investigate the effect of the epitaxial monolayer on the structural properties of molecular crystals grown on top.

Correlation between two- and three-dimensional crystallographic lattices for epitaxial analysis. I. Theory.

The epitaxial growth of molecular crystals at single-crystalline surfaces is often strongly related to the first monolayer at the substrate surface. The present work presents a theoretical approach to compare three-dimensional lattices of epitaxially grown crystals with two-dimensional lattices of the molecules formed within the first monolayer. Real-space and reciprocal-space representations are considered. Depending on the crystallographic orientation relative to the substrate surface, proper linear combinations of the lattice vectors of the three-dimensional unit cell result in a rhomboid in the xy plane, representing a two-dimensional projection. Mathematical expressions are derived which provide a relationship between the six lattice parameters of the three-dimensional case and the three parameters obtained for the two-dimensional surface unit cell. It is found that rotational symmetries of the monolayers are reflected by the epitaxial order. Positive and negative orientations of the crystallographic contact planes are correlated with the mirror symmetry of the surface unit cells, and the corresponding mathematical expressions are derived. The method is exemplarily applied to data obtained in previous grazing-incidence X-ray diffraction (GIXD) measurements with sample rotation on thin films of the conjugated molecules 3,4;9,10-perylenetetracarboxylic dianhydride (PTCDA), 6,13-pentacenequinone (P2O), 1,2;8,9-dibenzopentacene (trans-DBPen) and dicyanovinyl-quaterthiophene (DCV4T-Et2) grown by physical vapor deposition on Ag(111) and Cu(111) single crystals. This work introduces the possibility to study three-dimensional crystal growth nucleated by an ordered monolayer by combining two different experimental techniques, GIXD and low-energy electron diffraction, which has been implemented in the second part of this work.

A Computational Simulation on Size-dependent Plastic Behavior in Polycrystalline Austenitic Steel based on the Notion of Microforce

Steel containing the metastable austenitic phase shows excellent mechanical properties such as strength, ductility and toughness. The transformation-induced plasticity (TRIP) in the steel can optimize these mechanical properties. Since strain-induced martensitic transformation (SIMT) causes TRIP, deep understanding of the continuous development of SIMT is very important to control the mechanical properties by TRIP. In addition, a dislocation attributes to the strain hardening so it is very important to clearly explain its complexity. It is necessary to construct a model on the dislocation deeply related to size effects. In this study, a hardening model and yield conditions for monocrystalline austenitic steel are established based on the notion of microforce. By incorporating the finite element crystal plasticity method, the two-dimensional unit cells of polycrystalline austenitic steel are obtained by Voronoi tessellation, and the deformation behavior and microstructure are simulated computationally.

Thermal debonding in compacted graphite iron: effect of interaction of graphite inclusions

This paper focuses on investigation of the interaction effect of graphite inclusions on failure of compacted graphite iron (CGI) subjected to thermal load. Although CGI has been used extensively in industrial applications and studied for many years, its failure under pure thermal load has not yet been fully understood, nor has the influence of interacting graphite particles on it. To analyse this phenomenon, a finite-element two-dimensional unit cell is constructed, consisting of two graphite particles presented as ellipses embedded in a metallic matrix. Each inclusion is surrounded by a thin layer that accounts for its interface in order to study its decohesion from the surrounding matrix. An elastoplastic behaviour is assumed for both graphite and matrix material, described by classical J2 flow theory of plasticity, whereas a traction-separation law is applied to the interface. The model is analysed numerically, with an emphasis on the influence of different parameters such as the inclusion size, the distance between the particles, and the combination of their shapes on thermal debonding, identifying critical combinations that either deteriorate or relieve the effect. The obtained results can provide significant knowledge on the response of CGI under thermal load at the microscale, contributing also to understanding of its macroscopic behaviour at elevated temperatures.

Tunable and reflective polarization converter based on single-layer vanadium dioxide-integrated metasurface in terahertz region

A single-layer and actively tunable reflective polarization converter is proposed, which is composed of periodical two-dimensional unit cell of inclined split circular ring filled into by vanadium oxide (VO2) and central rod. Through numerical simulation obtained by the CST Microwave Studio, we find that the broad bandwidth of reflective cross-polarization conversion above 90% is normally obtained from 2.03 to 5 THz at the temperature about 25 °C for both the linearly and circularly polarized wave incidence. At the same time, the polarization conversion ratio (PCR) and energy conversion ratio (ECR) have been furtherly investigated, which reflect high cross-polarization conversion efficiency with nearly no energy loss at room temperature of 25 °C. Thereafter, namely, the cross-polarization conversion can be achieved by theoretical analysis from the views on qualitative variables of polarization azimuth angle (θ), ellipticity (η) and symmetrical polarization decomposition. What is more, the device is sensitive to the incident angles which result into the split reflected frequency spectrum. Afterwards, the actively dynamic responses for reflective frequency spectrum depending on the insulator-to-metal phase transition of VO2 have been investigated. To be demonstrated, the physical mechanism of changeable polarization conversion has been discussed by the distributions of current densities. Lastly, also, the influences of structural parameter variation can be further studied. According to the interesting results, the method provides a strategy that manipulate active and passive metadevices for polarization converter, wireless communication in THz wavelength domain.