Beam Lattice Research Articles

Elastic calibration of a discrete domain using a proper generalized decomposition

Current discrete/lattice methods can simulate a continuous mechanical behavior thanks to a network of bonds. The main drawback of these approaches is the need of a calibration process to link the emerging behavior of the structure and the parameters of the local mechanical bond. It is proposed in this work to use a fast and recent reduction model technique to build once and for all an exhaustive data chart and thus to avoid the calibration process. The proper generalized decomposition technique was used to build a parametric analysis in the case of a lattice beam structure. The results were in the range of the current calibration values found in the literature and extend it by giving a global calibration curve. They also allowed to discuss about the influence, in this specific case of lattice-beam structure, of the density of beams in terms of number of discrete elements and connectivity.

GEOMETRIC POSITIONING RELIABILITY OF SPACECRAFT CANTILEVER STRUCTURES

The paper describes the results of assessing reliability of spacecraft cantilever structures (SCS) that serve as supports of radio reflectors. The specifics of SCS reliability is that they should be capable of serving 12-15 years in space without changing their 3D geometry. Results of reliability calculation of two types of lattice cantilever carbon polymer beams are presented. Further needed research is suggested.

Load Distribution Coefficient of Node of Lattice Beam Based on Two-Parameter Elastic Foundation Model

According to the force characteristic of lattice beam, used the Two-parameter elastic foundation model to deduce the answer of internal forces of concentrated force on the finite-length beam, and calculate load distribution coefficient of node, then get the internal forces of lattice beam, used the analytical solution combined with practical engineering, and compared with numerical solution of simulation of ABAQUS finite element software, to verify the correctness of theoretical solution, and analyze its deficiency.

Computational design of innovative mechanical metafilters via adaptive surrogate-based optimization

Architected materials and metamaterials are a challenging frontier for the development of optimal design strategies targeted at the active and passive control of elastic wave propagation. Within this research field, the microstructural optimization of mechanical metamaterials for achieving desired spectral functionalities may require considerable computational resources. Based on this motivating framework, the present paper illustrates a machine learning methodology to attack the inverse design problem concerning the optimization of the dispersion properties characterizing a novel layered mechanical metamaterial, conceived starting from the bi-tetrachiral periodic topology. Specifically, an adaptive technique is adopted to surrogate and maximize the objective function purposely defined to determine the optimal beam lattice microstructure characterized by the largest stop bandwidth at the lowest centerfrequency (low-cutting mechanical metafilter). The technique is computationally efficient in identifying the existing optimal solution in the physically admissible parameter space. The designed bi-tetrachiral metamaterial provides satisfying broadband low-frequency filtering performances, not achievable by the component tetrachiral layers.

Study on vibration characteristics of sandwich beam with BCC lattice core

In this study, the vibration characteristics of a sandwich lattice beam have been studied using an analytical approach and finite element(FE) analyses. In the analytical approach, a sixth-order partial differential equation for expressing the motion in terms of the transverse displacement has been solved, which is assumed that the sandwich core is deformed mainly by the shear load. Here, a new theoretical approach for predicting the equivalent shear modulus GC∗ of the lattice core is proposed. The analytical prediction of the natural frequency, accounting for transverse shear, agrees well with the FE results. Also, the vibration response of the first-order natural frequency f1 exhibits a upwardly convex curve as the strut diameter increases, suggesting that there is a maximum value of the frequency f1 for a given facesheet thickness tf. Moreover, it is observed that the maximum frequency of the sandwich lattice beam is always greater than that for a continuum beam, which ensures the effectiveness of the lattice core in enhancing the vibration response. Therefore, vibration suppression in aircraft structures can be achieved by introducing lattice sandwich panels.

Generating a 64 × 64 beam lattice by geometric phase modulation from arbitrary incident polarizations.

A laser beam lattice from tailoring spatial dimensions of lights is a kind of structured optical field, which already have found many applications in lots of domains. Here we propose a geometric phase element made from polymerized liquid crystals to transform Gaussian beams into a 64×64 beam lattice with high performance. Different from other geometric phase elements, the proposed element can introduce identical phase modulations for any polarizations, indicating that the beam lattice could be well generated with arbitrary incident homogeneous polarizations but not limited to specific circular polarizations. In the experiment, a 64×64 beam lattice is well generated. It is estimated that the uniformity of the obtained lattice fluctuates about 60% among various incident polarizations, which is very close to the prediction. This work opens a new site for producing high-dimensional beam lattices and will inspire more advanced applications.

Lightweight design and static analysis of lattice compressor impeller

Taking the compressor impeller as the research object and the lightweight design as the research goal, a lattice filled lattice cell suitable for the application of rotating periodic symmetric structure is designed. Its purpose is to make the rigidity and strength of impeller adjustable and reduce the mass of impeller on the premise of meeting the design requirements. The analysis and comparison of unfilled impeller, solid impeller and lattice filled impeller with different diameters were carried out under the limit condition of 80,000 r/min. The results showed that the average circumferential deformation of lattice impeller tip with beam diameter of 0.2 mm, 0.4 mm and 1 mm was 4.84%, 3.49% and 3.71% lower than that of solid impeller. For the impeller with a lattice beam diameter of 0.4 mm, its weight can be reduced by 22.68% compared with the solid impeller. The average circumferential deformation of the tip of the lattice impeller lies between the unfilled impeller and the solid impeller. The results show that the impeller with lattice filling hub can not only reduce the weight effectively, but also improve the efficiency of the compressor.

Architectured Lattice Materials with Tunable Anisotropy: Design and Analysis of the Material Property Space with the Aid of Machine Learning

Architectured beam lattice materials whose anisotropy can be tuned by varying the composition of their elementary cell are investigated. As an exemplary prototype of such material architecture, a regular triangular lattice with an elementary cell composed of 12 beams is considered. One out of three possible values of the elastic modulus is assigned to each beam. The structure is fully defined by a vector in the 12D composition‐structure space whose components are given by the elastic modulus values of the beams comprising the elementary cell. The elastic properties of this 2D material are represented by the compliance elasticity tensor with six independent compliance coefficients. Aiming at a specific set of properties thus involves finding the point in the 12D composition‐structure space that corresponds to a given point in the 6D property space. This is a problem of large dimensionality. To solve it, the neural network approach is used. This enables creation of architectured materials with tunable elastic anisotropy. A chiral element combining large twist with additional anisotropy requirements is presented as an example of successful machine‐learning‐based optimization of beam lattices proposed.

Symmetry of the phononic landscape of twisted kagome lattices across the duality boundary

In this work, we investigate the symmetry of the phonon landscape of twisted kagome lattices across their duality boundary. The study is inspired by recent work by Fruchart et al. [Nature, 577, 2020], who specialized the notion of duality to the mechanistic problem of kagome lattices and linked it to the existence of duality transformations between configurations that are symmetrically located across a critical transition point in configuration space. Our first goal is to elucidate how the existence of matching phonon spectra between dual configurations manifest in terms of observable wavefield characteristics. To this end, we explore the possibility to aggregate dual kagome configurations into bi-domain lattices that are geometrically heterogeneous but retain a mechanically homogeneous response. Our second objective is to extend the analysis to structural lattices of beams, which are representative of realistic cellular metamaterials. We show that, in this case, the symmetry of the phonon landscape across the duality boundary is broken, implying that the conditions for dual behavior do not merely depend on the geometry, but also on the dominant mechanisms of the cell, suggesting a dichotomy between geometric and functional duality.

Free and forced wave propagation in beam lattice metamaterials with viscoelastic resonators

Beam lattice materials are characterized by a periodic microstructure realizing a geometrically regular pattern of elementary cells. The dispersion properties governing the free dynamic propagation of elastic waves can be studied by formulating parametric discrete models of the cellular microstructure and applying the Floquet-Bloch theory. Within this framework, governing the wave propagation by means of spectral design techniques and/or energy dissipation mechanisms is a major issue of theoretical and applied interest. Specifically, the wave propagation can be inhibited by purposely designing the microstructural parameters to open stop bands in the material spectrum at target center frequencies. Based on these motivations, a general dynamic formulation is presented for determining the dispersion properties of mechanical metamaterials, modeled as locally resonant beam lattices with generic coordination number. The mechanism of local resonance is realized by tuning periodic auxiliary oscillators, viscoelastically coupled with the beam lattice microstructure. As peculiar aspect, the viscoelastic coupling is derived by a mechanical formulation based on the Boltzmann superposition integral, whose kernel is approximated by a Prony series. Consequently, the free propagation of damped waves is governed by a linear homogeneous system of integro-differential equations of motion. Therefore, differential equations of motion with frequency-dependent viscoelastic coefficients are obtained by applying the in-space Z-transform and in-time bilateral Laplace transform. The complex-valued branches characterizing the dispersion spectrum are determined and parametrically analyzed for the beam lattice characterized by quadrilateral periodic cell. The spectral branches may exceed the model dimension, due to the occurrence of pure-damping spectral components. Particularly, the spectra corresponding to Laurent series approximations of the viscoelastic coefficients are investigated and the solution admissibility and convergence for increasing order series is analyzed. The standard dynamic equations with linear viscous damping are recovered at the first-order approximation. Low-order approximations are found to underestimate the real and imaginary parts of the spectrum, as well as the low-frequency stop bandwidth. Finally, the forced response to a harmonic mono-frequent external point excitation is investigated. The metamaterial responses to non-resonant, resonant and quasi-resonant external forces are compared and discussed from a qualitative and quantitative viewpoint.

Weak Solutions within the Gradient-Incomplete Strain-Gradient Elasticity

In this paper we consider existence and uniqueness of the three-dimensional static boundary-value problems in the framework of so-called gradient-incomplete strain-gradient elasticity. We call the strain-gradient elasticity model gradient-incomplete such model where the considered strain energy density depends on displacements and only on some specific partial derivatives of displacements of first- and second-order. Such models appear as a result of homogenization of pantographic beam lattices and in some physical models. Using anisotropic Sobolev spaces we analyze the mathematical properties of weak solutions. Null-energy solutions are discussed.

Hierarchy of beam models for lattice core sandwich structures

A discrete-to-continuum transformation to model 2-D discrete lattices as energetically equivalent 1-D continuum beams is developed. The study is initiated in a classical setting but results in a non-classical two-scale micropolar beam model via a novel link within a unit cell between the second-order macrorotation-gradient and the micropolar antisymmetric shear deformation. The shear deformable micropolar beam is reduced to a couple-stress and two classical lattice beam models by successive approximations. The stiffness parameters for all models are given by the micropolar constitutive matrix. The four models are compared by studying stretching- and bending-dominated lattice core sandwich beams under various loads and boundary conditions. A classical 4th-order Timoshenko beam is an apt first choice for stretching-dominated beams, whereas the 6th-order micropolar model works for bending-dominated beams as well. The 6th-order couple-stress beam is often too stiff near point loads and boundaries. It is shown that the 1-D micropolar model leads to the exact 2-D lattice response in the absence of boundary effects even when the length of the 1-D beam (macrostructure) equals that of the 2-D unit cell (microstructure), that is, when L=l.

Highly anisotropic hexagonal lattice material for low frequency water sound insulation

Insulation of water sound through impedance mismatch has the advantage of broadband effectiveness compared to using materials with bandgaps induced either by local resonance or Bragg scattering. In general, acoustic impedance of an isotropic solid under normal incidence condition is the product of mass density and longitudinal wave velocity. It is derived here, the acoustic impedance of an anisotropic solid depends additionally on a new parameter, and a carefully designed anisotropic solid can achieve a very small impedance along a specific direction. Honeycomb beam lattice is proposed as an example to achieve a much smaller effective impedance than water based on the above principle. Numerical simulation shows, a thin slab, with an overall thickness being two orders of magnitude smaller than the water wavelength, designed from the highly anisotropic lattice can reflect almost 97.7% of incident acoustic energy. A deep subwavelength sample with a thickness 21 mm is then fabricated and measured in a water tube. The experiment shows, the sample can reduce sound transmission by nearly 18.7 dB over the low frequency range 1.5 kHz ∼3.5 kHz. This study demonstrates the potential of anisotropic lattices in engineering effective impedance for insulating water sound at low frequency.

Trapping Yb171 atoms into a one-dimensional optical lattice with a small waist

In most experiments with atoms trapped in optical lattices, the transverse size of the optical lattice beams is on the order of tens of micrometers, and loading many atoms into smaller optical lattices has not been carefully investigated. We report trapping 1500 $^{171}$Yb atoms in a one-dimensional optical lattice generated by a narrow cavity mode at a distance of 0.14 mm from a mirror surface. The simplest approach of loading atoms from a mirror magneto-optical trap overlapped with the cavity mode allows the adjustment of the loading position by tuning a uniform bias magnetic field. The number of atoms trapped in the optical lattice exhibits two local maxima for different lattice depths, with a global maximum in the deeper lattice. These results open a way to quantum mechanical manipulation of atoms based on strong interaction with a tightly focused light field.

In-situ synchrotron X-ray tomography investigation of micro lattice manufactured with the projection micro-stereolithography (PμSL) 3D printing technique: Defects characterization and in-situ shear test

The robust mechanical properties of architected micro lattice mechanical metamaterials are determined by their precisely designed artificial microstructures, and various types of mechanical metamaterials have been proposed for acoustic manipulation, impact energy absorption, vibration attenuation, etc. In this paper, 3D pyramid and hourglass micro lattice samples are fabricated with projection micro-stereolithography (PμSL) 3D printing technique. Synchrotron X-ray tomography 3D imaging at the sub-micrometer precision level is employed for characterizing surface roughness and internal void defects distribution induced by the manufacturing process. Afterwards, synchrotron X-ray tomography 3D imaging of the in-situ shearing deformation process is performed for studying the failure process and deformation modes of the as-fabricated micro-lattice samples. It is found that manufacturing surface roughness and internal defects within the lattice beams and nodes can significantly influence the deformation behaviors and failure modes of these shearing micro lattice samples. Finally, three types of finite elementanalysis (FEA) models based on synchrotron X-ray tomography 3D image reconstruction are proposed for understanding the relation between manufacturing defects and mechanical behaviors of micro lattices, where effects of statistical distributions of geometry defects of micro lattice structures are focused. Based on comparison between experimental and numerical results, it can be concluded that manufacturing defects can change the deformation behaviors and mechanical integrity of micro-lattice remarkably.

Lattice design for a hybrid multi-bend achromat light source

We present a lattice design and beam dynamics analysis for a 4 GeV low-emittance, high-brilliance synchrotron light source. The lattice consisted of 26 six-bend achromats with 26 long and 26 short straight sections. We present the design results for a hybrid multi-bend achromat (HMBA) lattice with a natural emittance of 168 pm and circumference of 729.3 m. Each cell had 4 longitudinal gradient bendings, 2 combined bendings, 14 quadrupoles, and 2 families of sextupoles. The lattice was designed to be flexible to provide both nonzero and zero dispersion functions in the long straight section. The two straight sections in each cell had lengths of 5.6 and 1.2 m. To ensure sufficient injection, we optimized the design lattice and dynamic aperture. We investigated the dynamic aperture in the lattice with machine errors, and the results showed that the lattice design provides sufficient dynamic aperture after COD correction for beam injection. We present the results of an injection scheme that demonstrates the space in the injection section and the particle motions of the injected beam. We also present the results of beam tracking after the beam injection to examine the characteristics of the beam injection. The designed HMBA lattice was well optimized in terms of the beam parameters and brilliance as an intermediate light source at 4 GeV and a circumference of 729.3 m.

Study on Dynamic Response of Pressure Type Anchor and Lattice Beam System Under Seismic Loadings

In order to investigate dynamic response of anchoring landslide under seismic loadings with different loading magnitudes, a large scale shaking table model test was carried out, and the acceleration response of anchoring landslide, mechanical characteristics of anchors and lattice beams were studied respectively. The test results indicate that (1) under lower loading magnitude 0.05–0.2 g, the PGA amplification factors of different soil layers in landslide, and the dynamic response in different layers of anchors and lattice beams all have little differences. (2) Under 0.2–0.4 g, the PGA amplification factors of different soil layers and the anchoring force of different layers of anchors continue to increase. The strain of the top anchors and the bending strains of the top crossbeams are slightly greater than that in other layers. (3) Under high loading magnitude 0.4–0.6 g, the strain of the upper anchors continues to increase and it is much greater than that in other layers, and the bending strain of the top crossbeam increases sharply and generates obvious tension-bend deformation.

A Globally Conforming Lattice Structure for 2D Stress Tensor Visualization

AbstractWe present a visualization technique for 2D stress tensor fields based on the construction of a globally conforming lattice. Conformity ensures that the lattice edges follow the principal stress directions and the aspect ratio of lattice elements represents the stress anisotropy. Since such a lattice structure cannot be space‐filling in general, it is constructed from multiple intersecting lattice beams. Conformity at beam intersections is ensured via a constrained optimization problem, by computing the aspect ratio of elements at intersections so that their edges meet when continued along the principal stress lines. In combination with a coloring scheme that encodes relative stress magnitudes, a global visualization is achieved. By introducing additional constraints on the positional variation of the beam intersections, coherent visualizations are achieved when external loads or material parameters are changed. In a number of experiments using non‐trivial scenarios, we demonstrate the capability of the proposed visualization technique to show the global and local structure of a given stress field.

Shaking table test of seismic responses of anchor cable and lattice beam reinforced slope

As a combined supporting structure, the anchor cable and lattice beam have a complex interaction with the slope body. In order to investigate the seismic behaviors of the slope reinforced by anchor cable and lattice beam, a large-scale shaking table test was carried out on a slope model (geometric scale of 1:20) by applying recorded and artificial seismic waves with different amplitudes. The acceleration and displacement of the slope, the displacement of lattice beam and the axial force of anchor cable were obtained to study the interaction between the slope and the supporting structure. The test results show that: (1) the acceleration responses of the slope at different relative elevations display obvious nonlinear characteristics with increasing of the peak ground acceleration (PGA) of the inputted seismic waves, and the weak intercalated layer has a stronger effect on acceleration amplification at the upper part of the slope than that at the lower part of the slope; (2) the frequency component near the second dominant frequency is significantly magnified by the interaction between the slope and the supporting structure; (3) the anchor cables at the upper part of the slope have larger peak and residual axial forces than that at the lower part of the slope, and the prestress loss of the anchor cable first occurs at the top of the slope and then passes down; (4) the peak and residual displacements inside the slope and on the lattice beam increase with the increase of relative elevation. When the inputted PGA is not greater than 0.5 g, the combined effect of anchor cable and lattice beam is remarkable for stabilizing the middle and lower parts of the potential sliding body. The research results can provide a reference for the seismic design of such slope and the optimization of supporting structure.

Multiple Step Interlaced Beam Scan to Minimize the Deviation of Radar Detection Performance

For detection of targets, a radar conventionally scans a specified search volume with a fixed beam lattice of a specified beam spacing. With the fixed beam lattice, the detection performance of the radar within a unit beam lattice changes depending on the line-of-sight angle to a target. In this paper, multiple step interlaced scan is proposed to minimize the deviation of the detection performance due to the change of the target line-of-sight angle. As a figure of merit for the detection performance, the cumulative probability of detection is analyzed for various values of the interlaced scan step with different beam overlap ratios, and the optimal values of the steps to minimize the deviation of the cumulative probability of detection are derived.