Unit Cell Topology Research Articles

Dynamic crushing behavior of closed-cell aluminum foams based on different space-filling unit cells

Closed-cell metal foams are cellular solids that show unique properties such as high strength to weight ratio, high energy absorption capacity, and low thermal conductivity. Due to being computation and cost effective, modeling the behavior of closed-cell foams using regular unit cells has attracted a lot of attention in this regard. Recent developments in additive manufacturing techniques which have made the production of rationally designed porous structures feasible has also contributed to recent increasing interest in studying the mechanical behavior of regular lattice structures. In this study, five different topologies namely Kelvin, Weaire–Phelan, rhombicuboctahedron, octahedral, and truncated cube are considered for constructing lattice structures. The effects of foam density and impact velocity on the stress–strain curves, first peak stress, and energy absorption capacity are investigated. The results showed that unit cell topology has a very significant effect on the stiffness, first peak stress, failure mode, and energy absorption capacity. Among all the unit cell types, the Kelvin unit cell demonstrated the most similar behavior to experimental test results. The Weaire–Phelan unit cell, while showing promising results in low and medium densities, demonstrated unstable behavior at high impact velocity. The lattice structures with high fractions of vertical walls (truncated cube and rhombicuboctahedron) showed higher stiffness and first peak stress values as compared to lattice structures with high ratio of oblique walls (Weaire–Phelan and Kelvin). However, as for the energy absorption capacity, other factors were important. The lattice structures with high cell wall surface area had higher energy absorption capacities as compared to lattice structures with low surface area. The results of this study are not only beneficial in determining the proper unit cell type in numerical modeling of dynamic behavior of closed-cell foams, but they are also advantageous in studying the dynamic behavior of additively manufactured lattice structures with different topologies.

Photonic Band Gap Material Topological Design at Specified Target Frequency

AbstractTo achieve any specified band gap structure around target frequency, a new photonic crystals (PhC) topological design strategy is proposed. In the design optimization model, a band gap index that measures the minimal signed distances between the specified frequency and any two adjacent orders of frequencies is proposed, and the topology of PhC unit cell is represented with a low number of design variables through the material‐field series‐expansion. Then the sequential Kriging‐based algorithm is adopted for solving the multi‐peak and nondifferentiable optimization problems. Based on the numerical implementation of the proposed optimization method, the PhC unit cells are designed to provide band gaps precisely matching any specified central frequency value. The optimized unit cell designs are easily assembled to achieve a PhC device with the large relative band gap.

Additively Manufactured Gradient Porous Ti-6Al-4V Hip Replacement Implants Embedded with Cell-Laden Gelatin Methacryloyl Hydrogels.

Laser additive manufacturing has led to a paradigm shift in the design of next-generation customized porous implants aiming to integrate better with the surrounding bone. However, conflicting design criteria have limited the development of fully functional porous implants; increasing porosity improves body fluid/cell-laden prepolymer permeability at the expense of compromising mechanical stability. Here, functionally gradient porosity implants and scaffolds designed based on interconnected triply periodic minimal surfaces (TPMS) are demonstrated. High local porosity is defined at the implant/tissue interface aiming to improve the biological response. Gradually decreasing porosity from the surface to the center of the porous constructs provides mechanical strength in selective laser melted Ti-6Al-4V implants. The effect of unit cell size is studied to discover the printability limit where the specific surface area is maximized. Furthermore, mechanical studies on the unit cell topology effects suggest that the bending-dominated architectures can provide significantly enhanced strength and deformability, compared to stretching-dominated architectures. A finite element (FE) model developed also showed great predictability (within ∼13%) of the mechanical responses of implants to physical activities. Finally, in vitro biocompatibility studies were conducted for two-dimensional (2D) and three-dimensional (3D) cases. The results of the 2D in conjunction with surface roughness show favored physical cell attachment on the implant surface. Also, the results of the 3D biocompatibility study for the scaffolds incorporated with a cell-laden gelatin methacryloyl (GelMA) hydrogel show excellent viability. The design procedure proposed here provides new insights into the development of porous hip implants with simultaneous high mechanical and biological responses.

Numerical study on bending response of auxetic 2D-lattice plates

In this study, the bending response of 2D-lattice plates having auxetic unit cells is determined by using the homogenization method based on equivalent strain energy. The auxetic unit cells in this study are different topologies of re-entrant hexagonal unit cells. In the homogenization method, the effective out-of-plane elastic properties of an auxetic 2D-lattice plate are obtained from the strain energy values of its unit cell under different curvature modes, determined by the finite element method. In the analysis, the auxetic unit cells are considered as frames whose struts are modeled as Euler beams. The effective elastic properties, i.e., the effective bending moduli, Poisson’s ratios, and shear modulus, are described as the bending response of a 2D-lattice plate subjected to out-of-plane bending. In the validation, the effective elastic properties of some auxetic 2D-lattice plates obtained from their unit cells by the homogenization method are numerically compared with those obtained from direct structural analysis of the plates. Besides, the obtained results show how the bending response of the auxetic 2D-lattice plates can be adjusted by varying their unit-cell geometries, especially the internal cell angle.

Signal and Noise Analysis of an Open-Circuit Voltage Pixel for Uncooled Infrared Image Sensors

An imaging pixel unit-cell topology leveraging a photodetector in the forward-bias region is proposed. Connecting the anode of the photodiode to the gate of a NMOS device operating in the subthreshold region provides the basis for a new open-circuit voltage pixel (VocP) architecture. Theoretical analysis is presented to show the response and performance benefits of the VocP in comparison to a conventional pixel. Based on this analysis, the signal and noise relationships for both pixels are derived and leveraged to construct an end-to-end readout system model. The model results highlight potential performance benefits of the VocP over a conventional direct-injection pixel topology. To verify the analysis, the proposed VocP readout architecture is fabricated along with a conventional direct-injection pixel readout in a $0.18~\mathrm {\mu }\text{m}$ CMOS technology. The VocP performance is compared to a traditional reverse-bias current-mode photodetector configuration. Simulation, modeling, and measurements align with the proposed analytical model. Benefits in system sensitivity and dynamic range are demonstrated showing more than a $2\times $ improvement in noise-equivalent temperature difference and a 4 dB improvement in dynamic range.

Effect of unit cell topology on the tensile loading responses of additive manufactured CoCrMo triply periodic minimal surface sheet lattices

This study involves the fabrication of triply periodic minimal surface sheet lattices possessing two different topologies—Neovius and Schoen’s IWP. The structural characteristics of these materials, which were fabricated by laser powder bed fusion using CoCrMo alloy powder, were investigated. The tensile properties and deformation behavior of the materials were evaluated by increasing the unit cell size (1, 2.5, and 5 mm) for each topology. The results of the tensile tests showed that the yield strengths and Young’s moduli of both topologies decreased, and that their elongations increased with decreasing unit cell size. We compared the mechanical properties of the Neovius and IWP lattices, which had the same unit cell size, and found that the former exhibited higher yield strength, tensile strength, and elongation. Further, the tensile deformation behavior of the specimens was analyzed by applying the Gibson–Ashby analytic model. The Neovius lattice accommodated more uniform deformations in a greater number of cell layers. These experimental observations and the use of a simplified model of the lattice structure can enhance the understanding of the mechanism of the room-temperature tensile deformation of Neovius and IWP sheet-based lattice structures.

Spectroscopic investigations of gamma-ray irradiation effects on metal organic framework

The chemical and physical features of metal organic frameworks (MOFs) at a molecular level are of great merit to understand structure–property relationships and, consequently, to aid in the design of improved high value products. This paper reports on the processing of one MOF-type material, namely MIL-101(Cr), by γ-ray irradiation at low doses up to 30 kGy. The optimum dose is determined to be 15 kGy, at which an increase in the adsorption of iodine of more than 40% is achieved in comparison with thermally activated un-irradiated MIL-101(Cr). Scanning electron microscopy, X-ray diffraction and the Brunauer−Emmett−Teller (BET) isotherms indicate a significant particle agglomeration and a preservation of both the unit cell topology as well as the mesoporous structure of MOF. Two-dimensional correlation Fourier transformation infrared and X-ray diffraction reveals that orientation is the driving force for structural alterations including water type, bonding and contents. These effects account for the increasing adsorption capacity of MOF after irradiation and suggest further exploitation of irradiation as a promising method for MOFs activation.

Theoretical and experimental research on anisotropic and nonlinear mechanics of periodic network materials

Bio-inspired network materials composed of regularly aligned repeating filamentary microstructures exhibit promising application prospects in tissue engineering and bio-integrated devices due to the typical J-shaped mechanics curve precisely matched to the biological tissues. During the tensile process, an evident nonlinear mechanical anisotropy is induced by the drastic changes of microstructure geometries, which is of great concern in some particular requirements such as identifying the direction with the lowest or highest tensile stiffness and inducing alternate positive and negative or nearly zero Poisson's ratio. An efficient theoretical model for nonlinear anisotropic mechanics is developed in this paper by introduction of equilibrium and deformation compatibility conditions of presentative unit cell under finite uniaxial stretching along arbitrary directions, which is then verified by finite element analyses (FEA) and experiments both graphically and quantitatively. Through this model, the anisotropic mechanical responses of the periodic network materials are studied thoroughly and systematically based on the precise prediction of stress-strain relation, transverse-longitudinal strain relation and the deformed configuration. The effects of both the geometric parameters and the unit cell topology on the anisotropic mechanical responses are analyzed thereafter, which implies that the proposed model can play an instructive role in the design, optimization, and application of the periodic network materials.

Mechanical properties of a hybrid auxetic metamaterial and metastructure system

We propose in this work an innovative hybrid auxetic metamaterial with a centersymmetric unit cell and tessellation topology similar to the one provided by the missing rib configuration. The tessellation proposed is applied to different core unit cells (star shape, cross-chiral shape with same dimensions, and reentrant). The effects of the geometric parameters of the cells on the in-plane mechanical properties of this hybrid auxetic metamaterial system are investigated via finite elements (FEMs). Representative unit cells (RUCs) with optimal mechanical behaviors are identified; those configurations exhibit the larger negative Poisson’s ratios and enhanced specific moduli. Designs related to two groups of auxetic metastructures with cylindric and cubic shapes are then developed based on the optimized RUCs along x and y directions. The equivalent mechanical performance of these metastructures under internal pressure is evaluated from a numerical standpoint. Auxetic cylindrical metastructures can be tailored by adjusting the number of the optimized RUCs along the circumferential and longitudinal directions, together with the geometric parameters of the optimized RUC itself. These hybrid auxetic metamaterials and metastructures provide the potential for multifunctional applications in biomechanics, flexible electronics, and aerospace.

Optimization of sound transmission loss of open acoustic barriers with respect to unit cell topology

Open acoustic barriers exhibit excellent sound transmission reduction property at a certain frequency/frequencies which highly depends on the configuration of its unit cell. Design of unit cell configuration for minimum sound transmission at predefined objective frequency remains an open question. This paper aims at providing an automatic design method for open acoustic barriers with multi-material unit cell. Firstly, a wave finite element method is developed to calculate the sound transmission through an infinite array of periodic scatterers. As the unit cell contains infinite fluid domain, the application of Floquet-Bloch theorem to the boundaries of perfectly match layers (PML) is necessary and has been resolved in this paper. This wave finite element method with the implementation of PML is validated by comparing to analytical solution of sound transmission through an array of steel cylinders. Then a genetic algorithm is employed to optimize the sound transmission loss with respect to material distribution of a bi-material unit cell. Finally, the effectiveness of this inverse design is demonstrated by examples with different predefined frequencies. Corresponding unit cell typologies are obtained and the dips of sound power transmission coefficient curve are successfully tuned to objective frequencies.

Shape transformers for phononic band gaps tuning in two-dimensional Bloch-periodic lattice structures

In this paper, shape transformers, a set of non-dimensional geometrical parameters that define beam cross-section efficiencies based on their size and shape, are employed in the wave propagation analysis of two-dimensional periodic lattice structures. Here, shape transformers of lattice unit-cell beams are used as design variables to tailor the frequency band gaps of the lattice structure, in the form of their mid frequencies and aspect ratios, by means of numerical Bloch-periodic simulation. Case studies are presented for lattices with square and hexagon Bravais symmetries where a total of six different unit-cell topologies are studied, considering 12 classes of different cross-section shapes. It is observed that a significant shift of the waveband branches towards lower frequencies are obtained with the variation of shape transformers from a void to a filled envelope configuration. Moreover, while changing the cross-section geometric variables, it is possible to determine associated values of maximum aspect ratios of band gap frequencies, thus identifying cross-section shapes that have better performance with respect to lattice dynamic characteristic. It is also found that the “H” and “I” cross-section shapes, in comparison to other shape families presented, resulted in the maximization of the band gap aspect ratio while maintaining a low frequency range, hence being ideal candidates for designing lattice metamaterials with lower frequency ranges of elastic wave attenuation. Furthermore, among all lattice topologies investigated, it is noted that the employment of diamond cross-section and its derivatives resulted in the greatest sensitivity of the mid-frequency shift of the band gap with respect to variations in cross-section geometric efficiency. It is also observed, as a general behavior for all unit-cells studied, that when scaling up the in-plane height of the cross-section, the aspect ratio of the band gap was maximized. Finally, using the Modal Assurance Criterion, we demonstrated that the variation in the shape transformers conserves the consistency between the eigenwaves of unit-cells at different characteristic points of the Irreducible Brillouin Zone. The presented study paves the way for the development and the systematic implementation of a novel wave attenuation tuning technique.

Silicon nitride programmable photonic processor with folded heaters.

General-purpose programmable photonic processors rely on the large-scale integration of beamsplitters and reconfigurable phase shifters, distributed within unit cells or photonic gates. With their future evolution threatened by several hardware constrains, including the integration density that can be achieved with current mesh topologies, in this work, we present a unit cell topology design to increase the integration density of waveguide mesh arrangements based on folded Mach-Zehnder Interferometers. We report the design details of a 40-unit cell waveguide mesh integrated in a 11mm x 5.5 mm silicon nitride chip achieving, to the best of our knowledge, the highest integration density reported to date for a general-purpose photonic processor. The chip is electrically interfaced to a PCB and we report examples of reconfigurable optical beamsplitters, basic tunable microwave photonic filters with high peak rejection (40 dB approx.), as well as the dynamic interconnection and routing of 5G digitally modulated signals within the photonic mesh.

Topology optimization design of quasi-periodic cellular structures based on erode–dilate operators

In this paper, a novel density-based topology optimization method for cellular structures with quasi-periodic microstructures is proposed. Here, ‘quasi-periodic’ is developed from periodic microstructures composed with a base unit cell, through gradually changing one or multiple alterable microstructural sizing/shape parameters. However, it is a challenge and key issue to identify the explicit alterable parameters in the density-based topology frame, since the microstructural topology is described by voxel densities. For solving this problem, an erode–dilate based method for describing the quasi-periodic microstructures is proposed. A family of quasi-periodic microstructures is formed by executing the erode–dilate operators on the base unit cell with different operator parameters. The topology optimization of quasi-periodic cellular structures is formulated as simultaneously optimizing the topology of the base unit cell and the volume fractions of macro-elements that correspond to the candidate microstructures. Furthermore, to avoid the small structural members being eliminated in the eroded process, the structural skeleton is extracted to modify the eroded process. Sensitivities of the structural compliance with respect to the two types of design variables are derived, and the gradient-based optimization method is applied to update the design variables. In the obtained results, the neighboring microstructures have similar topologies and varying volume fractions. The design space has been expanded compared with the periodic cellular structures, and the connectivity issue of the conventional heterogeneous microstructures has been resolved. The numerical examples validate the effectiveness of the proposed method, and the results show that the quasi-periodic cellular structures have better performances than single-scale structures and periodic cellular structures.

Double Dirac cones and topologically nontrivial phonons for continuous square symmetric C4(v) and C2(v) unit cells

Because phononic topological insulators have primarily been studied in discrete, graphene-like structures with C$_{6}$ or C$_{3}$ hexagonal symmetry, an open question is how to systematically achieve double Dirac cones and topologically non-trivial structures using continuous, non-hexagonal unit cells. Here, we address this challenge by presenting a novel computational methodology for the inverse design of continuous two-dimensional square phononic metamaterials exhibiting C$_{4v}$ and C$_{2v}$ symmetry. This leads to the systematic design of square unit cell topologies exhibiting a double Dirac degeneracy, which enables topologically-protected interface propagation based on the quantum spin Hall effect (QSHE). Numerical simulations prove that helical edge states emerge at the interface between two topologically distinct square phononic metamaterials, which opens the possibility of QSHE-based pseudospin-dependent transport beyond hexagonal lattices.

Anisotropic Metasurface With Asymmetric Propagation of Electromagnetic Waves and Enhancements of Antenna Gain

An anisotropic metasurface that performs a reflectarray and transmitarray on each side is proposed. The proposed configuration comprises 15×15 unit cells that consist of polarization converters and a polarization filter. Two topologies are developed for the unit cells to meet five design goals simultaneously, including reflection magnitude, transmission magnitude, reflection phase difference, transmission phase difference, and polarization conversion ratio. By arranging the two topologies of unit cells according to a required phase distribution, in-phase reflection and transmission are achieved on each side. A prototype is designed, fabricated, and analyzed at 28.0 GHz. The feed is implemented as a quasi-Yagi antenna with gain of 8.0 dBi. When linearly-polarized electromagnetic waves impinge on the top, the proposed metasurface functions as a reflectarray and offers gain of 17.9 dBi. When the feed antenna illuminates the bottom, the proposed metasurface performs a transmitarray and depicts gain of 19.1 dBi. Moreover, when the both sides are arranged with feeds, a unidirectional pattern with power gain of 20.3 dBi is obtained. Such a “twin-feed mode” demonstrates a highly-directed main beam with orthogonal polarization components, which is useful in dual-polarized communications. Thus, the proposed metasurface provides antenna applications with low-cost, low-complexity, and multifunctional wavefront manipulation.

Novel Tubular Structures with Negative Poisson's Ratio and High Stiffness

A stiff tubular cellular structure topology is designed by alternatively indenting rectangular contour patterns on the cell ribs. Modified topologies can be obtained by tailoring the geometric parameters. The Poisson's ratios and elastic moduli of the proposed tubular structures with various geometric parameters are determined using finite element method (FEM). Results show that the minimum Poisson's ratio of the proposed structures can reach −0.28. The unit cell topology and orientation can significantly affect the structures deformation behaviors. The structures exhibit mixed mode of bending‐ and stretching‐dominated deformation responses, and show improved specific elastic moduli compared with traditional open‐cell stochastic foams with positive Poisson's ratios. 3D‐printed samples with identical geometric variables to those of the FE models are fabricated, and their Poisson's ratios and stress–strain relationships are determined experimentally, and compared with simulation results. Excellent agreement is achieved between measurements and simulations. The design concept proposed here can be optimized for specific applications via geometric parameters manipulation.

Compact Double-Layer FR4-Based Focusing Lens Using High-Efficiency Huygens' Metasurface Unit Cells.

High transmission efficiency metasurface unit cells have been designed based on surface electric and magnetic impedances derived from Huygens’ principle. However, unit cells for low transmission loss (<1 dB) over a wide transmission phase range require at least three metallic layers, which complicates the unit cell design process. In this paper, we introduce high-efficiency Huygens’ metasurface unit cell topologies in double-layer FR4 printed circuit board (PCB) by implementing surface electric and magnetic current using the top and bottom metallic patterns and via drills. Eleven unit cells were optimized for wide phase coverage (−150° to 150°) with a low average transmission loss of −0.82 dB at 10 GHz. To demonstrate the high-efficiency of the designed unit cells, we designed and fabricated two focusing lenses with dimensions of near 150 × 150 mm (5λ × 5λ) to focus a spherical beam radiated from short focal distances (f = 100 and 60 mm). The fabricated focusing lens showed 12.87 and 13.58 dB focusing gain for f = 100 and 60 mm at 10 GHz, respectively, with a 1 dB fractional gain bandwidth of near 10%. We expect that the proposed focusing lens based on high-efficiency double-layer metasurface unit cells can help realize compact and high-gain focusing lens-integrated antenna systems.

Elastic and elasto-plastic analysis of Ti6Al4V micro-lattice structures under compressive loads

This article aims to study the modeling and simulation of Ti6Al4V micro-lattice structures under the same compressive loads. Initially, five distinct unit cell topologies (Grid, X, Star, Cross, and Tesseract) were used to design lattice structures. For the modeling of these lattice structures, both three-dimensional wireframe and solid (homogeneous and heterogeneous gradient) meshing were employed. However, the geometric design parameters, such as strut length, strut diameter, and, in particular its configuration (5 × 5 × 5 array), have been assessed as the same for all types of lattice structures. Here, the finite element (FE) modeling approach is used to evaluate the elastic and elasto-plastic behaviors of these structures due to subjected uniaxial compressive loading. The FE results have been validated in previously reported experimental data and have been well comprehended. The static linear and nonlinear FE method formulations are based on the Timoshenko beam theory and Johnson–Cook damaged model, which describe the elasto-plastic behavior. The FE simulations results (linear and nonlinear) were studied and it was shown that Star lattice structure has high stiffness value at low porosity, compared to the other lattice structures. Hence, to further determine the optimum variables (strut diameter –Sd and pore size –PS) for the Star unit cell-based lattice structures, the Taguchi-based optimization technique with L9 orthogonal array was used. In addition, a linear regression model was developed to estimate the maximum initial yield stress for the optimum variables. Finally, the same configuration of Star lattice structures was designed by the heterogeneous solid gradient mesh approach with two swapping conditions, ‘A’ and ‘B.’ These models were employed to perform linear static FE simulations. Therefore, the swapping condition and the type of gradient-based lattice structures, possessing high and low compressive stresses, were identified.

A Filament Level Analysis on 3-D Orthogonal Weave Micro-geometry Modeling under Different Yarn Tension

3-D woven textile is generated through the weaving process. Yarn is a non-continuum material domain, the structure of which is determined by inter fiber movement. In this study, the micro-geometry of 3-D orthogonal weave is generated at filament level predicting through the weaving process implementing the digital element approach (DEA). First, the basic concepts and explicit algorithm of DEA is introduced. The method of calculating adjusted digital fiber material property in terms of discretization resolution is proposed. Second, the unit-cell topology of 3-D orthogonal weave is defined by a position matrix. The calculation of potential energy of the cell is derived. At last, a dynamic weaving process is designed to investigate the effect of yarn tension on fabric micro-geometry and cell energy. 4 unit-cells are generated under 4 sets of tension combination. Results show that as the fabric thickness decrease, the filaments of the weft yarns move towards the center. The weft yarns at the top and bottom deform into their final shape first. The applied tension on weaver plays a dominant role in determining fabric thickness and convergence speed. By comparing the numerical results with the microscope pictures taken from the actual specimen, it is concluded that the fabric micro-geometry produced by tension combination 4 closely matches the experimental results.

Characterisation and modeling of additively-manufactured polymeric hybrid lattice structures for energy absorption

Lightweight cellular materials and structures are widely used in load-bearing and energy absorption applications, because of favorable mechanical properties such as high compressibility and low relative density. Recent progress in additive manufacturing techniques has enabled specific architectural geometries of unit cells in cellular structures to be tailored for particular needs. Numerous metallic and polymeric cellular lattices comprising different unit cell topologies have been manufactured and examined in terms of their energy absorption performance. In this study, two designs of hybrid three-dimensional cubic lattices which combine the advantages of an octet and a bending-dominated structure were established, and fabricated via the Fused Deposition Modeling technique. To validate the energy absorption capability of these new hybrid lattices, quasi-static uniaxial compression tests were conducted on samples made from Polylactic Acid. Numerical simulations were also performed to facilitate analysis of the deformation modes of the specimens tested in experiments. Tensile tests on solid dog-bone samples printed at various angles with respect to the build plate reveal fabrication-angle-dependent anisotropy. Consequently, material properties that depend on cell strut inclination were incorporated into the simulations. Compared with a conventional octet that possesses a high stiffness but a fluctuating post-yield response, the experimental results show that the new designs are capable of producing a relatively stable post-yield stress plateau without sacrificing stiffness and strength significantly. The present study indicates the potential for further enhancement of the energy absorption performance of lattice structures by tuning their topological architectures appropriately.