Complex Lattice Research Articles

A Note on the Order Lozanovsky Spectrum for Positive Operators

In this note we consider the order Lozanovsky spectrum of a positive operator on a complex Banach lattice and show that it generally contains the order Weyl spectrum, settling a weaker form of a question raised in Alekhno (Positivity 13(1):3–20, 2009).

Designing an anatomical contour titanium 3D-printed oblique lumbar interbody fusion cage with porous structure and embedded fixation screws for patients with osteoporosis.

This study aimed to design an anatomical contour metal three-dimensional (3D)-printed oblique lateral lumbar interbody fusion (OLIF) cage with porous (lattices) structure and embedded screw fixation to enhance bone ingrowth to reduce the risk of cage subsidence and avoid the stress-shielding effect. Finite element (FE) analysis and weight topology optimization (WTO) were used to optimize the structural design of the OLIF cage based on the anatomical contour morphology of patients with osteoporosis. Two oblique embedded fixation screws and lattice design with 65% porosity and average pore size of 750 μm were equipped with the cage structure. The cage was fabricated via metal 3D printing, and static/dynamic compression and compressive-shear tests were performed in accordance with the ASTM F2077-14 standard to evaluate its mechanical resistance. On FE analysis, the OLIF cage with embedded screw model had the most stability, lowest stress values on the endplate, and uniform stress distribution versus standalone cage and fixed with lateral plate under extension, lateral flexion, and rotation. The fatigue test showed that the stiffnesses/endurance limits (pass 5 million dynamic test) were 16,658 N/mm/6000 N for axial load and 19,643 N/mm/2700 N for compression shear. In conclusion, an OLIF cage with embedded fixation screws can be designed by integrating FE and WTO analysis based on the statistical results of endplate morphology. This improves the stability of the OLIF cage to decrease endplate destruction. The complex contour and lattice design of the OLIF cage need to be manufactured via metal 3D printing; the dynamic axial compression and compressive-shear strengths are greater than that of the U.S. Food and Drug Administration (FDA) standard.

Hierarchical lattice: Design strategy and topology characterization

The structure-material integrated design is an art-of-state concept and be enabled by additive manufacturing. The lattice material is classified into structure as well as material because mechanical properties are determined by its topology. However, the lack of a flexible design strategy hinders the lattice achieve the structure-material integrated material candidate. This work suggests the strut-nested based strategies to effectively conduct the hierarchical lattice design. The strut in the larger-scale lattice can be replaced by the smaller-scale lattice structure through the rotation, stretching, and translation operations combining the local and global numbering, thereby complete the multi-scale lattice design. The design skills are well elucidated with custom-developed algorithm; a serious of complex lattices achieve multi-scale design. The influence of hierarchical structures in lattices on a significant parameter, strut length-to-diameter, is identified. Our work offers the alternative strategy to realize the hierarchical lattice design.

Convergence map with action-angle variables based on square matrix for nonlinear lattice optimization

To analyze nonlinear dynamic systems, we developed a new technique based on the square matrix method. We propose this technique called the \convergence map" for generating particle stability diagrams similar to the frequency maps widely used in accelerator physics to estimate dynamic aperture. The convergence map provides similar information as the frequency map but in a much shorter computing time. The dynamic equation can be rewritten in terms of action-angle variables provided by the square matrix derived from the accelerator lattice. The convergence map is obtained by solving the exact nonlinear equation iteratively by the perturbation method using Fourier transform and studying convergence. When the iteration is convergent, the solution is expressed as a quasi-periodic analytical function as a highly accurate approximation, and hence the motion is stable. The border of stable motion determines the dynamical aperture. As an example, we applied the new method to the nonlinear optimization of the NSLS-II storage ring and demonstrated a dynamic aperture comparable to or larger than the nominal one obtained by particle tracking. The computation speed of the convergence map is 30 to 300 times faster than the speed of the particle tracking, depending on the size of the ring lattice (number of superperiods). The computation speed ratio is larger for complex lattices with low symmetry, such as particle colliders.

Tailored elastic properties of beam-based lattice unit structures

AbstractIn this paper a structural optimization framework is developed to design three-dimensional periodic lattice unit cells that meets specific mechanical requirements. The work is motivated by the high design freedom of additive manufacturing technologies, which enable complex multiscale lattice structures to be printed. An optimized lattice unit cell delivers desired orthotropic elastic material properties, providing a tailored metamaterial. The design variables are the coordinates of lattice skeleton nodes defined within the three-dimensional lattice cell space, and the connectivities between them resulting a strut-skeleton. Genetic algorithm (GA) is combined with posterior particle swarm optimization (PSO) algorithm to establish an integrated topology and shape optimization tool. For the calculation of the elastic properties of the individual lattice cells, an effective Timoshenko beam-based finite element calculation method was developed. The novelty of the work stems from its free topology optimization nature, excluding the strut diameters from the optimization variables. The method is demonstrated by four lattice cell optimization cases, where extreme orthotropic elastic properties were targeted and achieved. The tailored lattice cells represent a metamaterial, that can be used to build a structural component on the macroscopic scale, by stacking the cells periodically together, to fill the macroscopic 3D design space. This framework is a strong basis that can be extended to meet further nonlinear metamaterial requirements, such as energy absorption.

Tensor network construction for lattice gas models: Hard-core and triangular lattice models.

The representation of complex lattice models in the form of a tensor network is a promising approach to the analysis of the thermodynamics of such systems. Once the tensor network is built, various methods can be used to calculate the partition function of the corresponding model. However, it is possible to build the initial tensor network in different ways for the same model. In this work, we have proposed two ways of constructing tensor networks and demonstrated that the construction process affects the accuracy of calculations. For demonstration purposes, we have done a brief study of the 4 nearest-neighbor (NN) and 5NN models, where adsorbed particles exclude all sites up to the fourth and fifth nearest neighbors from being occupied by another particle. In addition, we have studied a 4NN model with finite repulsions with a fifth neighbor. In a sense, this model is intermediate between 4NN and 5NN models, so algorithms designed for systems with hard-core interactions may experience difficulties. We have obtained adsorption isotherms, as well as graphs of entropy and heat capacity for all models. The critical values of the chemical potential were determined from the position of the heat capacity peaks. As a result, we were able to improve our previous estimate of the position of the phase transition points for the 4NN and 5NN models. And in the model with finite interactions, we found the presence of two first-order phase transitions and made an estimate of the critical values of the chemical potential for them.

Congruence RFRS towers

We describe a criterion for a real or complex hyperbolic lattice to admit a residually finite rational solvable (RFRS) tower that consists entirely of congruence subgroups. We use this to show that certain Bianchi groups PSL 2 (𝒪 d ) are virtually fibered on congruence subgroups, and also exhibit the first examples of RFRS Kähler groups that are not a subgroup of a product of surface groups and abelian groups.

Layered BiOI single crystals capable of detecting low dose rates of X-rays

Detecting low dose rates of X-rays is critical for making safer radiology instruments, but is limited by the absorber materials available. Here, we develop bismuth oxyiodide (BiOI) single crystals into effective X-ray detectors. BiOI features complex lattice dynamics, owing to the ionic character of the lattice and weak van der Waals interactions between layers. Through use of ultrafast spectroscopy, first-principles computations and detailed optical and structural characterisation, we show that photoexcited charge-carriers in BiOI couple to intralayer breathing phonon modes, forming large polarons, thus enabling longer drift lengths for the photoexcited carriers than would be expected if self-trapping occurred. This, combined with the low and stable dark currents and high linear X-ray attenuation coefficients, leads to strong detector performance. High sensitivities reaching 1.1 × 103 μC Gyair−1 cm−2 are achieved, and the lowest dose rate directly measured by the detectors was 22 nGyair s−1. The photophysical principles discussed herein offer new design avenues for novel materials with heavy elements and low-dimensional electronic structures for (opto)electronic applications.

Synthesis and characterization of a new cyanato-N cadmium(II) Meso-arylporphyrin complex by X-ray diffraction analysis, IR, UV/vis, 1H MNR spectroscopies and TDDFT calculations, optical and electrical properties

Herein, we have presented a new cadmium(II) pentacoordinate porphyrin complex obtained by the reaction of the [(meso‑tetra(para-methoxyphenyl)porphyrinato]cadmium(II) starting material ([Cd(TMPP)]) with an excess of potassium thiocyanate (KSCN) and crystand-222 (cryst-222) in the dichloromethane solvent leading to the coordination compound with the formula: [K(crypt-222)(H2O)][Cd(TMPP)(NCO)]·2(CH2Cl2)·0.5(C6H14)·H2O (symbolized by complex I). This species was characterized by elemental analysis, UV/Vis, FT-IR, and single crystal X-ray diffraction studies. The intermolecular interactions responsible of the cohesion of the crystal lattice of complex I were investigated using the Hirshfeld surface approach. DFT/TDDFT calculations on complex I were carried out including (i) Molecular structure optimization using the DFT/B3LYP-D3/lanl2dz level of theory, (ii) the frontier molecular orbitals (FMOs) calculations, (iii) the Mulliken charges distribution calculations, (iv) the Molecular electronic potential analysis (MEP), (v) an ab initio UV/Vis and IR calculations and (vi) the QTAIM and NCI-RDG analyses. These latter DFT/B3LYP-D3/lanl2dz calculations studies confirmed, inter alia, the strong hydrogen bond linking the NCO− axial ligand and the water molecule coordinated to the potassium counterion. Furthermore, the electrical conductance and dielectric properties of complex I were investigated shown firstly that the frequency dependence of electrical conductivity, follows the Jonscher's universal dynamic law and secondly that the obtained results have been discussed in terms of the OLPT and NSPT models, which is well adapted to our Cd(II)-porphyrin-NCO semiconductor material.

The effect of geometric design and materials on section properties of additively manufactured lattice elements

Additive manufacturing (AM) technologies such as laser-based powder bed fusion (LB-PBF) facilitate the fabrication of complex lattice structures. However, these structures consistently display dimensional variation between the idealised and as-manufactured specimens. This research proposes a method to characterise the impact of common LB-PBF powders (aluminium and titanium alloys) and geometric design parameters (polygon order, effective diameter, and inclination angle) on section properties relevant to stiffness and strength of as-manufactured strut elements. Micro-computed tomography (µCT) has been applied to algorithmically characterise the as-manufactured variation and identify a scale threshold below which additional geometric resolution does not influence the section properties of as-manufactured parts. This methodology provides a robust and algorithmic design for additive manufacturing (DFAM) tool to characterise the effects of manufacturing and design parameters on the functional response of AM strut elements, as is required for certification and optimisation.

Terahertz Silicon Metagratings: High‐Efficiency Dispersive Beam Manipulation above Diffraction Cone

AbstractOptical wavefront engineering is essential for the development of next‐generation integrated photonic devices. It is used for reflecting terahertz waves in a predesigned nonspecular direction with near‐unitary efficiency, which is a longstanding challenge for high‐performance functional devices. Recently, metagratings have offered an efficient solution for beam steering at large angles without the need for a discretization phase or impedance profile. Here, all‐dielectric metagratings fabricated using a silicon cuboid complex lattice are proposed and demonstrated experimentally to achieve anomalous terahertz beam reflections above the diffraction cone with unitary diffraction efficiency. For the bipartite metagrating system, a single dispersive scatterer per unit is effective for achieving broadband beam steering because of Brillouin zone folding, and another perturbative synergetic scatterer is introduced to slightly tailor the array coupling and improve the performance. High‐efficiency beam steering, including both retroreflection under oblique incidence and one‐way diffraction under normal incidence, can be achieved by breaking structural symmetry and coherently suppressing unnecessary radiation channels. Moreover, silicon metagratings with spatially dispersive response features show perfect anomalous reflection operation in the broadband region, which is promising for leveraging terahertz spatially separated devices.

Spin-Resolved Band Structure of Hoffman Clathrate [Fe(pz)2Pt(CN)4] as an Essential Tool to Predict Optical Spectra of Metal–Organic Frameworks

Paramount spin-crossover properties of the 3D-Hoffman metalorganic framework (MOF) [Fe(pz)2Pt(CN)4] are generally described on the basis of the ligand field theory, which provides adequate insight into theoretical and simulation analysis of spintronic complexes. However, the ligand field approximation does not take into account the 3D periodicity of the actual complex lattice and surface effects and therefore cannot predict a full-scale periodic structure without utilizing more advanced methods. Therefore, in this paper, the electronic properties of the exemplar MOF were analyzed from the band structure perspective in low-spin (LS) and high-spin (HS) states. The density-of-states spectra determined for both spin-up and spin-down electrons of Fe d6 orbitals indicate spin-orbital splitting and delocalization for HS due to spin polarization in the iron atom ligand field. Presence of the surface states in the real crystal causes a red shift of the metal-metal charge transfer (MMCT) and metal-ligand charge transfer (MLCT) peaks for both HS and LS states. The addition of residual water molecules and disorder among the pyrazine rings reveal additional influences on the positions of the pyrazine band and, therefore, on the absorption spectra of the crystal. The results show a magnification of the peak correlated with the MLCT in the HS state and a significant red shift of the LS characteristic absorption band. The presented approach involving band structure analysis delivers a more complete image of the electronic properties of the [Fe(pz)2Pt(CN)4] crystalline network and can be a landmark for insightful studies of other MOFs.

Impact Performance of 3D Printed Spatially Varying Elastomeric Lattices

Additive manufacturing is catalyzing a new class of volumetrically varying lattice structures in which the dynamic mechanical response can be tailored for a specific application. Simultaneously, a diversity of materials is now available as feedstock including elastomers, which provide high viscoelasticity and increased durability. The combined benefits of complex lattices coupled with elastomers is particularly appealing for anatomy-specific wearable applications such as in athletic or safety equipment. In this study, Siemens' DARPA TRADES-funded design and geometry-generation software, Mithril, was leveraged to design vertically-graded and uniform lattices, the configurations of which offer varying degrees of stiffness. The designed lattices were fabricated in two elastomers using different additive manufacturing processes: (a) vat photopolymerization (with compliant SIL30 elastomer from Carbon) and (b) thermoplastic material extrusion (with Ultimaker™ TPU filament providing increased stiffness). Both materials provided unique benefits with the SIL30 material offering compliance suitable for lower energy impacts and the Ultimaker™ TPU offering improved protection against higher impact energies. Moreover, a hybrid lattice combination of both materials was evaluated and demonstrated the simultaneous benefits of each, with good performance across a wider range of impact energies. This study explores the design, material, and process space for manufacturing a new class of comfortable, energy-absorbing protective equipment to protect athletes, consumers, soldiers, first responders, and packaged goods.

Evaluation of the Ultimate Collapse Load of a High-Voltage Transmission Tower under Excessive Wind Loads

Several high-voltage transmission towers failed under excessive wind loads in a mountainous and exposed area. This study discusses the efficient and reliable modeling of lattice towers dominantly loaded by wind, a scenario which led to a collapse cascade in a high-voltage transmission line. The ultimate load-bearing capacity had to be estimated and the failure positions identified. Finite Element Analysis was employed through static analyses, Linear Buckling Analyses (LBA) and RIKS analyses (Arc-Length method) in Abaqus 2021. With the purpose of improving the accuracy in the simulation of structural instabilities of complex lattice structures, the model sensitivity to superimposed geometrical imperfections and the joint stiffness of the truss connections were investigated in brace and lattice structure sub-assemblies. Afterwards, linear analyses and non-linear analyses with imperfections were performed on the single tower model. The analysis proved that solely excessive wind can cause such failure on the lattice structures, and the critical structural elements have been correctly identified. The investigation proved that the towers had not been under-designed with respect to the standards valid at the time of erection. However, they were not designed for this exceptional storm event, and evidence was provided that wind alone could bring about the collapse. It is nevertheless not recommended to increase the safety factors in general for the design of such structures, but to base the assumed loading on actual and local wind and service load measurements.

A virtual stylus method for non-destructive roughness profile measurement of additive manufactured lattice structures

Surface roughness is traditionally evaluated with contact profilometry; however, these methods are not compatible with complex additive manufactured lattice structures due to limited physical access. For these scenarios, computed tomography (CT) is often used to provide qualitative insight into surface roughness but does not directly yield roughness profile data. This research describes a hybrid approach for the non-destructive quantification of roughness profile data for lattice structures based on the mathematical reconstruction and interpretation of CT data. Formal analyses are applied to propose the theoretical minimum CT voxel size required to characterise surface roughness for a specified sampling length. The method is verified against optical data for nominally flat metallic specimens and applied to metallic and polymeric cylinders fabricated by powder bed fusion and material extrusion respectively. This research also assesses the influence of CT reconstruction thresholding as a process variable and finds that roughness profile data is only weakly influenced by thresholding settings, due to scattering effects at the surface — a novel finding that provides certainty for the industrial application of this method. The ability of the proposed method to accurately characterise the inherent surface roughness of these processes as well as the effect of specimen orientation is thus demonstrated, enabling full geometric characterisation supporting subsequent certification analysis. The method can be algorithmically implemented in combination with the generative design of complex lattice structures to support structural certification requirements.

Local Molecular Field Theory for Coulomb Interactions in Aqueous Solutions

Coulomb interactions play a crucial role in a wide array of processes in aqueous solutions but present conceptual and computational challenges to both theory and simulations. We review recent developments in an approach addressing these challenges─local molecular field (LMF) theory. LMF theory exploits an exact and physically suggestive separation of intermolecular Coulomb interactions into strong short-range and uniformly slowly varying long-range components. This allows us to accurately determine the averaged effects of the long-range components on the short-range structure using effective single particle fields and analytical corrections, greatly reducing the need for complex lattice summation techniques used in most standard approaches. The simplest use of these ideas in aqueous solutions leads to the short solvent (SS) model, where both solvent-solvent and solute-solvent Coulomb interactions have only short-range components. Here we use the SS model to give a simple description of pairing of nucleobases and biologically relevant ions in water.

Construction of two-dimensional arbitrary shape <sup>87</sup>Rb atomic array based on spatial light modulator

The ultra-cold atomic system is a clean and highly controllable quantum system, which can be used for quantum simulation of important physical problems in many fields such as condensed matter physics, high-energy physics, astrophysics, and chemical reactions. The constructions of optical lattices with different configurations are an important prerequisite for simulating diverse complex quantum systems, especially solid materials. In this work, we use weighted Gerchberg-Saxton algorithm to generate holograms. By using liquid crystal spatial light modulator and high-resolution imaging system, holograms (in momentum space) are transformed into images in real space for constructing various two-dimensional (2D) optical trap arrays, such as simple triangular, hexagonal, square lattice and more complex honeycomb lattice. We load <sup>87</sup>Rb ultra-cold atoms into the 2D optical trap arrays with a minimal spacing of 3 μm in between. This method is versatile and flexible, which is helpful in expanding the application scope of quantum simulation with optical lattices.

Rational design of intergrowth P2/O3 biphasic layered structure with reversible anionic redox chemistry and structural evolution for Na-ions batteries

Layered oxides have attracted unprecedented attention for their outstanding performance in sodium-ion battery cathodes. Among them, the two typical candidates P2 and O3 type materials generally demonstrate large diversities in specific capacity and cycling endurance with their advantages. Thus, composite materials that contain both P2 and O3 have been widely designed and constructed. Nevertheless, the anionic/cationic ions’ behavior and structural evolution in such complex structures remain unclear. In this study, a deep analysis of an advanced Na0.732Ni0.273Mg0.096Mn0.63O2 material that contains 78.39 wt% P2 phase and 21.61 wt% O3 phase is performed based on two typical cathodes P2 Na0.67Ni0.33Mn0.67O2 and O3 NaNi0.5Mn0.5O2 that have the same elemental constitution but different crystal structures. Structural analysis and density functional theory (DFT) calculations suggest that the composite is preferred to form a symbiotic structure at the atomic level, and the complex lattice texture of the biphase structure can block unfavorable ion and oxygen migration in the electrode process. Consequently, the biphase structure has significantly improved the electrochemical performance and kept preferable anionic oxygen redox reversibility. Furthermore, the hetero-epitaxy-like structure of the intergrowth of P2 and O3 structures share multi-phase boundaries, where the inconsistency in electrochemical behavior between P2 and O3 phases leads to an interlocking effect to prevent severe structural collapse and relieves the lattice strain from Na+ de/intercalation. Hence, the symbiotic P2/O3 composite materials exhibited a preferable capacity and cyclability (∼130 mAh g−1 at 0.1 C, 73.1% capacity retention after 200 cycles at 1 C), as well as reversible structural evolution. These findings confirmed the advantages of using the bi/multi-phase cathode for high-energy Na-ion batteries.

Non-reciprocal wave propagation in time-modulated elastic lattices with inerters

Non-reciprocal wave propagation in acoustic and elastic media has received much attention of researchers in recent years. This phenomenon can be achieved by breaking the reciprocity through space- and/or time-dependent constitutive material properties, which is an important step in overcoming the limitations of conventional acoustic- and phononic-like mechanical lattices. A special class of mechanical metamaterials with non-reciprocal wave transmission are latices with time-modulated mass and stiffness properties. Here, we investigate the non-reciprocity in elastic locally resonant and phononic-like one-dimensional lattices with inerter elements where mass and stiffness properties are simultaneously modulated through inerters and springs as harmonic functions of time. By considering the Bloch theorem and Fourier expansions, the frequency-band structures are determined for each configuration while asymmetric band gaps are found by using the weighting and threshold method. The reduction in frequency due to introduced inerters was observed in both phononic and locally resonant metamaterials. Dynamic analysis of finite-length lattices by the finite difference method revealed a uni-directional wave propagation. Special attention is given to phononic-like lattice based on a discrete-continuous system of multiple coupled beams. Moreover, the existence of edge modes in the discrete phononic lattice is confirmed through the bulk-edge correspondence and their time evolution quantified by the topologically invariant Chern number. The proposed methodology used to investigate non-reciprocal wave transmission in one-dimensional inerter-based lattices can be extended to study more complex two-dimensional lattices.

Grid-tree composite support structures for lattice parts in selective laser melting

Metal lattice structure has excellent mechanical properties to develop lightweight parts, and selective laser melting (SLM) makes it possible to construct complex lattice parts without molds. Support structures are required in SLM to remove heat away from the process and reduce geometrical distortions. However, traditional support structure generation methods have some problems, such as low generation efficiency, difficulty in the removal and not meeting the fabrication requirements of lattice parts. Therefore, this article proposes a grid-tree composite support structure generation algorithm for lattice parts. First, optimize the build orientation of the lattice parts considering the surface quality of functional features and the processability of lattice structures. Then, identify the frame’s overhang area and the point overhang of lattice structures. Finally, generate the grid-tree composite support structure for lattice parts. We conducted experiments to verify our approach and compare our method with three supports of Magics RP. Experimental results reveal that our method is effective in reducing support volume and warpage deformation.