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1
Pan, Chen, Yafeng Han, and Jiping Lu. "Design and Optimization of Lattice Structures: A Review." Applied Sciences 10, no. 18 (September 13, 2020): 6374. http://dx.doi.org/10.3390/app10186374.
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Cellular structures consist of foams, honeycombs, and lattices. Lattices have many outstanding properties over foams and honeycombs, such as lightweight, high strength, absorbing energy, and reducing vibration, which has been extensively studied and concerned. Because of excellent properties, lattice structures have been widely used in aviation, bio-engineering, automation, and other industrial fields. In particular, the application of additive manufacturing (AM) technology used for fabricating lattice structures has pushed the development of designing lattice structures to a new stage and made a breakthrough progress. By searching a large number of research literature, the primary work of this paper reviews the lattice structures. First, based on the introductions about lattices of literature, the definition and classification of lattice structures are concluded. Lattice structures are divided into two general categories in this paper: uniform and non-uniform. Second, the performance and application of lattice structures are introduced in detail. In addition, the fabricating methods of lattice structures, i.e., traditional processing and additive manufacturing, are evaluated. Third, for uniform lattice structures, the main concern during design is to develop highly functional unit cells, which in this paper is summarized as three different methods, i.e., geometric unit cell based, mathematical algorithm generated, and topology optimization. Forth, non-uniform lattice structures are reviewed from two aspects of gradient and topology optimization. These methods include Voronoi-tessellation, size gradient method (SGM), size matching and scaling (SMS), and homogenization, optimization, and construction (HOC). Finally, the future development of lattice structures is prospected from different aspects.
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Horváth, Eszter K., Sándor Radeleczki, Branimir Šešelja, and Andreja Tepavčević. "A Note on Cuts of Lattice-Valued Functions and Concept Lattices." Mathematica Slovaca 73, no. 3 (June 1, 2023): 583–94. http://dx.doi.org/10.1515/ms-2023-0043.
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ABSTRACT Motivated by applications of lattice-valued functions (lattice-valued fuzzy sets) in the theory of ordered structures, we investigate a special kind of posets and lattices induced by these mappings. As a framework, we use the Formal Concept Analysis in which these ordered structures can be naturally observed. We characterize the lattice of cut sets and the Dedekind-MacNeille completion of the set of images of a lattice valued function by suitable concept lattices and we give necessary and sufficient conditions under which these lattices coincide. In addition, we give conditions under which the lattice of cuts is completely distributive.
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Maskery, Ian, Alexandra Hussey, Ajit Panesar, Adedeji Aremu, Christopher Tuck, Ian Ashcroft, and Richard Hague. "An investigation into reinforced and functionally graded lattice structures." Journal of Cellular Plastics 53, no. 2 (July 28, 2016): 151–65. http://dx.doi.org/10.1177/0021955x16639035.
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Lattice structures are regarded as excellent candidates for use in lightweight energy-absorbing applications, such as crash protection. In this paper we investigate the crushing behaviour, mechanical properties and energy absorption of lattices made by an additive manufacturing process. Two types of lattice were examined: body-centred-cubic (BCC) and a reinforced variant called BCC z. The lattices were subject to compressive loads in two orthogonal directions, allowing an assessment of their mechanical anisotropy to be made. We also examined functionally graded versions of these lattices, which featured a density gradient along one direction. The graded structures exhibited distinct crushing behaviour, with a sequential collapse of cellular layers preceding full densification. For the BCC z lattice, the graded structures were able to absorb around 114% more energy per unit volume than their non-graded counterparts before full densification, 1371 ± 9 kJ/m3 versus 640 ± 10 kJ/m3. This highlights the strong potential for functionally graded lattices to be used in energy-absorbing applications. Finally, we determined several of the Gibson–Ashby coefficients relating the mechanical properties of lattice structures to their density; these are crucial in establishing the constitutive models required for effective lattice design. These results improve the current understanding of additively manufactured lattices and will enable the design of sophisticated, functional, lightweight components in the future.
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Guerra Silva, Rafael, María Josefina Torres, Jorge Zahr Viñuela, and Arístides González Zamora. "Manufacturing and Characterization of 3D Miniature Polymer Lattice Structures Using Fused Filament Fabrication." Polymers 13, no. 4 (February 20, 2021): 635. http://dx.doi.org/10.3390/polym13040635.
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The potential of additive manufacturing to produce architected lattice structures is remarkable, but restrictions imposed by manufacturing processes lead to practical limits on the form and dimension of structures that can be produced. In the present work, the capabilities of fused filament fabrication (FFF) to produce miniature lattices were explored, as they represent an inexpensive option for the production of polymer custom-made lattice structures. First, fused filament fabrication design guidelines were tested to assess their validity for miniature unit cells and lattice structures. The predictions were contrasted with the results of printing tests, showing some discrepancies between expected outcomes and resulting printed structures. It was possible to print functional 3D miniature open cell polymer lattice structures without support, even when some FFF guidelines were infringed, i.e., recommended minimum strut thickness and maximum overhang angle. Hence, a broad range of lattice structures with complex topologies are possible, beyond the cubic-type cell arrangements. Nevertheless, there are hard limits in 3D printing of miniature lattice structures. Strut thickness, length and orientation were identified as critical parameters in miniature lattice structures. Printed lattices that did not fully comply with FFF guidelines were capable of bearing compressive loads, even if surface quality and accuracy issues could not be fully resolved. Nevertheless, 3D printed FFF lattice structures could represent an improvement compared to other additive manufacturing processes, as they offer good control of cell geometry, and does not require additional post-processing.
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Abusabir, Ahmed, Muhammad A. Khan, Muhammad Asif, and Kamran A. Khan. "Effect of Architected Structural Members on the Viscoelastic Response of 3D Printed Simple Cubic Lattice Structures." Polymers 14, no. 3 (February 5, 2022): 618. http://dx.doi.org/10.3390/polym14030618.
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Three-dimensional printed polymeric lattice structures have recently gained interests in several engineering applications owing to their excellent properties such as low-density, energy absorption, strength-to-weight ratio, and damping performance. Three-dimensional (3D) lattice structure properties are governed by the topology of the microstructure and the base material that can be tailored to meet the application requirement. In this study, the effect of architected structural member geometry and base material on the viscoelastic response of 3D printed lattice structure has been investigated. The simple cubic lattice structures based on plate-, truss-, and shell-type structural members were used to describe the topology of the cellular solid. The proposed lattice structures were fabricated with two materials, i.e., PLA and ABS using the material extrusion (MEX) process. The quasi-static compression response of lattice structures was investigated, and mechanical properties were obtained. Then, the creep, relaxation and cyclic viscoelastic response of the lattice structure were characterized. Both material and topologies were observed to affect the mechanical properties and time-dependent behavior of lattice structure. Plate-based lattices were found to possess highest stiffness, while the highest viscoelastic behavior belongs to shell-based lattices. Among the studied lattice structures, we found that the plate-lattice is the best candidate to use as a creep-resistant LS and shell-based lattice is ideal for damping applications under quasi-static loading conditions. The proposed analysis approach is a step forward toward understanding the viscoelastic tolerance design of lattice structures.
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Majari, Parisa, Daniel Olvera-Trejo, Jorge A. Estrada-Díaz, Alex Elías-Zúñiga, Oscar Martinez-Romero, Claudia A. Ramírez-Herrera, and Imperio Anel Perales-Martínez. "Enhanced Lightweight Structures Through Brachistochrone-Inspired Lattice Design." Polymers 17, no. 5 (February 28, 2025): 654. https://doi.org/10.3390/polym17050654.
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Lattice structures offer unique mechanical properties and versatility in engineering applications, yet existing designs often struggle to balance performance and material efficiency. This study introduces the brachistochrone curve as a novel framework for optimizing lattice geometries, enhancing mechanical behavior while minimizing material usage. Using finite element simulations and compressive testing of 3D-printed samples, we analyzed the mechanical response of brachistochrone-based (B-) and standard lattice structures (diamond, IWP, gyroid, and BCC). We investigated the scaling behavior of the volume-to-surface area ratio, incorporated fractal dimension analysis, and compared experimental and numerical results to evaluate the performance of B-lattices versus standard designs (S-). Our findings indicate that brachistochrone-inspired lattices enhance mechanical efficiency, enabling the design of lightweight, high-strength components with sustainable material use. Experimental results suggest that B-gyroid lattices exhibit lower stiffness than S-gyroid lattices under small displacements, highlighting their potential for energy absorption applications.
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Echeta, Ifeanyichukwu, Xiaobing Feng, Ben Dutton, Richard Leach, and Samanta Piano. "Review of defects in lattice structures manufactured by powder bed fusion." International Journal of Advanced Manufacturing Technology 106, no. 5-6 (December 30, 2019): 2649–68. http://dx.doi.org/10.1007/s00170-019-04753-4.
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AbstractAdditively manufactured lattice structures are popular due to their desirable properties, such as high specific stiffness and high surface area, and are being explored for several applications including aerospace components, heat exchangers and biomedical implants. The complexity of lattices challenges the fabrication limits of additive manufacturing processes and thus, lattices are particularly prone to manufacturing defects. This paper presents a review of defects in lattice structures produced by powder bed fusion processes. The review focuses on the effects of lattice design on dimensional inaccuracies, surface texture and porosity. The design constraints on lattice structures are also reviewed, as these can help to discourage defect formation. Appropriate process parameters, post-processing techniques and measurement methods are also discussed. The information presented in this paper contributes towards a deeper understanding of defects in lattice structures, aiming to improve the quality and performance of future designs.
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Liu, Tinghao, and Guangbo Hao. "Design of Deployable Structures by Using Bistable Compliant Mechanisms." Micromachines 13, no. 5 (April 19, 2022): 651. http://dx.doi.org/10.3390/mi13050651.
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A deployable structure can significantly change its geometric shape by switching lattice configurations. Using compliant mechanisms as the lattice units can prevent wear and friction among multi-part mechanisms. This work presents two distinctive deployable structures based on a programmable compliant bistable lattice. Several novel parameters are introduced into the bistable mechanism to better control the behaviour of bistable mechanisms. By adjusting the defined geometry parameters, the programmable bistable lattices can be optimized for specific targets such as a larger deformation range or higher stability. The first structure is designed to perform 1D deployable movement. This structure consists of multi-series-connected bistable lattices. In order to explore the 3D bistable characteristic, a cylindrical deployable mechanism is designed based on the curved double tensural bistable lattice. The investigation of bistable lattices mainly involves four types of bistable mechanisms. These bistable mechanisms are obtained by dividing the long segment of traditional compliant bistable mechanisms into two equal parts and setting a series of angle data to them, respectively. The experiment and FEA simulation results confirm the feasibility of the compliant deployable structures.
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Guo, Zhengjie, Yuting Ma, Tayyeb Ali, Yi Yang, Juan Hou, Shujun Li, and Hao Wang. "Enhanced Compressive Properties of Additively Manufactured Ti-6Al-4V Gradient Lattice Structures." Metals 15, no. 3 (February 21, 2025): 230. https://doi.org/10.3390/met15030230.
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Lattice structures are widely used in the aerospace and biomedical fields, due to their lightweight, high specific strength, large specific surface area, good biocompatibility, etc. However, the balancing of the weight and the mechanical properties remains a challenge in designing lattice structures. Combining experiments and simulations, the present work first designs and evaluates the mechanical properties of uniform and gradient topology-optimized Ti-6Al-4V lattices with the same overall porosity of 84.27%, employing finite element simulations. Then, laser powder bed fusion technology is used to fabricate the uniform and gradient Ti-6Al-4V lattices, and their compressive performance is tested. The results indicate that, under longitudinal compression, the gradient lattice structure exhibits good layer-by-layer collapse deformation behavior, achieving better comprehensive performance than the uniform lattice structure. While under horizontal compression, the deformation behavior of the gradient lattice structure is similar to that of the uniform lattice structure, and the deformation is mostly randomly distributed. The cumulative energy absorption of the gradient lattice structure increased by approximately 20% compared with that of the uniform lattice structure. The results provide a technical basis for the integrated design of structural and functional components for aerospace applications.
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Lan, Tian, Chenxi Peng, Kate Fox, Truong Do, and Phuong Tran. "Triply periodic minimal surfaces lattice structures: Functional graded and hybrid designs for engineering applications." Materials Science in Additive Manufacturing 2, no. 3 (September 27, 2023): 1753. http://dx.doi.org/10.36922/msam.1753.
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In this work, we propose the strategies for designing radial graded sheet-based gyroid lattice and the approach to hybridizing solid-network-based gyroid lattice and primitive lattice. The elastic property of triply periodic minimal surfaces (TPMS) sheet-based gyroid lattice structures was explored. We also conducted numerical analysis to investigate the effect of functionally graded sheet-based gyroid lattices on the implant application, and explored the elastic properties of the uniform gyroid lattice parametrically with different relative densities based on the representative volume element model. Analytical equations based on the Gibson-Ashby model were generated to predict the elastic properties. Compressive tests on the samples fabricated by the Stratasys J750 were conducted to validate the feasibility of applying hybridization of different types of lattices. A comparison between radial hybrid primitive-gyroid and gyroid-primitive lattices revealed that the compressive behavior of gyroid-primitive was strengthened. We also found that the gyroid-primitive lattice could achieve auxetic compressive behavior. In conclusion, the numerical analysis illustrates that the application of the functional graded gyroid lattices can relieve the stress shielding effect as well as protects the bone from damage. The hybridization of different lattices can not only strengthen the mechanical properties of TPMS structures but also create a counter-intuitive deformation response.
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Air, A., and A. Wodehouse. "Deformation Taxonomy of Additively Manufactured Lattice Structures." Proceedings of the Design Society 2 (May 2022): 1361–70. http://dx.doi.org/10.1017/pds.2022.138.
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AbstractAdditive manufacturing offers opportunities for designed mechanical deformation within parts by integrating lattice structures into their designs. This work re-analyses and translates data on lattice structure deformation behaviours into a novel taxonomy, enabling their actions to be understood and controlled. Parallels between these actions and the four basic types of mechanical motion are identified. Creating a taxonomy method using these motions enables the future development of a DfAM framework that assimilates controlled anisotropy via lattices and aids the design of compliant mechanisms.
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AKBAY, Özgün Ceren, Burak Özdemir, Erkan Bahçe, and Ender Emir. "Deformation Behaviors Investigation of CoCr Alloy Lattice Structures under Compression Test." Journal of Manufacturing Engineering 18, no. 1 (March 1, 2023): 001–10. http://dx.doi.org/10.37255/jme.v18i1pp001-010.
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ith the development of the additive manufacturing method, the production of lattice structures with complex geometries attracts increasing attention. These lattice structures can be designed with the desired properties, and they are encountered in many areas such as automotive, aerospace and aviation, and manufacturing industries, as they offer the freedom to control their physical, mechanical and geometric properties. The high strength characteristic of lattice structures that can be designed at any scale makes these structures useful for producing different designs. Since the mechanical responses of the lattice structures depend on the lattice design parameters, such as the large number of independent struts forming the lattice, cell size and cell geometry, the mechanical behaviour of these structures should be examined. In this study, a porous lattice structure with four different cell models, namely Dode Medium, Diamond, Rhombic Dodecahedron, and Dode Thin, was produced by Selective Laser Melting (SLM) method. In order to reveal the mechanical properties and deformation responses of the porous lattice structures, they were analyzed under compression test and by the finite element method, and experimental and numerical procedures were compared. The effect of the compression test on the lattice properties and how the deformation is distributed throughout the lattice structure were investigated. The finite Element Analysis and Digital Image Processing (DIP) method was used to determine how the lattices deform. The results obtained will be useful for designing new metallic lattice structures with more excellent deformation resistance in future studies.
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Badard, R. "Extending lattice structures." Journal of Mathematical Analysis and Applications 136, no. 1 (November 1988): 314–24. http://dx.doi.org/10.1016/0022-247x(88)90134-5.
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Karamoozian, Aminreza, Chin An Tan, and Liangmo Wang. "Homogenized modeling and micromechanics analysis of thin-walled lattice plate structures for brake discs." Journal of Sandwich Structures & Materials 22, no. 2 (February 22, 2018): 423–60. http://dx.doi.org/10.1177/1099636218757670.
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Periodic cellular structures, especially lattice designs, have potential to improve the cooling performance of brake disk system. In this paper, the method of two scales asymptotic homogenization was used to indicate the effective elastic stiffnesses of lattice plates structures. The arbitrary topology of lattice core cells connected to the back and front plates which are made of generally orthotropic materials, due to use in brake disc design. This starts with the derivation of general shell model with consideration of the set of unit cell problems and then making use of the model to determine the analytical equations and calculate the effective elastic properties of lattice plate concerning the connected back and front plates. The analytical and numerical method allows determining the stiffness properties and the internal forces in the trusses and plates of the lattice. Three types of core-based lattice plates, which are pyramidal, X-type and I-type lattices, have been studied. The I-type lattice is characterized here for the first time with particular attention on the role that the cell trusses and plates plays on the stiffness and strength properties. The lattice designs are finite element characterized and compared with the numerical and experimentally validated pyramidal and X-type lattices under identical conditions. The I-type lattice provides 4% higher strength more than the other lattice types with 9% higher truss fraction coefficient. Results show that the stiffness and yield strength of the lattices depend upon the stress–strain response of the parent alloy of trusses and plates, the truss mass fraction coefficient, the face carriers thickness and the core elements parameters. The study described here is limited to a linear analysis of lattice properties. Geometric nonlinearities, however, have a considerable impact on the effective behavior of a lattice and will be the subject of future studies.
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Li, Yuhua, Deyu Jiang, Rong Zhao, Xin Wang, Liqiang Wang, and Lai-Chang Zhang. "High Mechanical Performance of Lattice Structures Fabricated by Additive Manufacturing." Metals 14, no. 10 (October 12, 2024): 1165. http://dx.doi.org/10.3390/met14101165.
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Lattice structures show advantages in mechanical properties and energy absorption efficiency owing to their lightweight, high strength and adjustable geometry. This article reviews lattice structure classification, design and applications, especially those based on additive manufacturing (AM) technology. This article first introduces the basic concepts and classification of lattice structures, including the classification based on topological shapes, such as strut, surface, shell, hollow-strut, and so on, and the classification based on the deformation mechanism. Then, the design methods of lattice structure are analyzed in detail, including the design based on basic unit, mathematical algorithm and gradient structure. Next, the effects of different lattice elements, relative density, material system, load direction and fabrication methods on the mechanical performance of AM-produced lattice structures are discussed. Finally, the advantages of lattice structures in energy absorption performance are summarized, aiming at providing theoretical guidance for further optimizing and expanding the engineering application potential of lattices.
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Castle, Toen, Daniel M. Sussman, Michael Tanis, and Randall D. Kamien. "Additive lattice kirigami." Science Advances 2, no. 9 (September 2016): e1601258. http://dx.doi.org/10.1126/sciadv.1601258.
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Kirigami uses bending, folding, cutting, and pasting to create complex three-dimensional (3D) structures from a flat sheet. In the case of lattice kirigami, this cutting and rejoining introduces defects into an underlying 2D lattice in the form of points of nonzero Gaussian curvature. A set of simple rules was previously used to generate a wide variety of stepped structures; we now pare back these rules to their minimum. This allows us to describe a set of techniques that unify a wide variety of cut-and-paste actions under the rubric of lattice kirigami, including adding new material and rejoining material across arbitrary cuts in the sheet. We also explore the use of more complex lattices and the different structures that consequently arise. Regardless of the choice of lattice, creating complex structures may require multiple overlapping kirigami cuts, where subsequent cuts are not performed on a locally flat lattice. Our additive kirigami method describes such cuts, providing a simple methodology and a set of techniques to build a huge variety of complex 3D shapes.
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Khan, Numan, Valerio Acanfora, and Aniello Riccio. "Non-Conventional Wing Structure Design with Lattice Infilled through Design for Additive Manufacturing." Materials 17, no. 7 (March 23, 2024): 1470. http://dx.doi.org/10.3390/ma17071470.
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Lightweight structures with a high stiffness-to-weight ratio always play a significant role in weight reduction in the aerospace sector. The exploration of non-conventional structures for aerospace applications has been a point of interest over the past few decades. The adaptation of lattice structure and additive manufacturing in the design can lead to improvement in mechanical properties and significant weight reduction. The practicality of the non-conventional wing structure with lattices infilled as a replacement for the conventional spar–ribs wing is determined through finite element analysis. The optimal lattice-infilled wing structures are obtained via an automated iterative method using the commercial implicit modeling tool nTop and an ANSYS workbench. Among five different types of optimized lattice-infilled structures, the Kelvin lattice structure is considered the best choice for current applications, with comparatively minimal wing-tip deflection, weight, and stress. Furthermore, the stress distribution dependency on the lattice-unit cell type and arrangement is also established. Conclusively, the lattice-infilled structures have shown an alternative innovative design approach for lightweight wing structures.
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Mechderso, Alachew Amaneh, and Tilahun Mekonnen Munie. "Li-ideals of implicative almost distributive lattices." F1000Research 14 (February 10, 2025): 182. https://doi.org/10.12688/f1000research.159175.1.
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Background In this paper, we introduce the concept of H-implicative almost distributive lattices, which are a special class of lattices with both implicative and almost distributive properties. This study is aimed at extending the understanding of lattice structures and their ideals, particularly focusing on LI-ideals, a novel concept within H-implicative almost distributive lattices. Methods We define an LI-ideal of an implicative almost distributive lattice L and investigate its properties. The paper demonstrates that every LI-ideal in L is an almost distributive lattice ideal of L. Additionally, we explore the relationship between filters and LI-ideals, and we study the process of generating an LI-ideal from a given set. Lastly, we examine the construction of quotient structures via LI-ideals. Results We present several examples showing that every almost distributive lattice ideal is also an LI-ideal in an H-implicative almost distributive lattice. The study establishes key relationships between the concepts of filters and LI-ideals. Furthermore, we provide a method for generating an LI-ideal from a set and construct a quotient structure using an LI-ideal. Conclusions The paper introduces new concepts and relationships within the study of H-implicative almost distributive lattices. Our findings demonstrate the interconnection between almost distributive lattice ideals and LI-ideals and offer insights into how these ideals can be generated and used to construct quotient structures. This work provides a deeper understanding of lattice theory and opens new avenues for further research in the area of lattice ideals.
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McCaw, John C. S., and Enrique Cuan-Urquizo. "Mechanical characterization of 3D printed, non-planar lattice structures under quasi-static cyclic loading." Rapid Prototyping Journal 26, no. 4 (January 27, 2020): 707–17. http://dx.doi.org/10.1108/rpj-06-2019-0163.
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Purpose While additive manufacturing via melt-extrusion of plastics has been around for more than several decades, its application to complex geometries has been hampered by the discretization of parts into planar layers. This requires wasted support material and introduces anisotropic weaknesses due to poor layer-to-layer adhesion. Curved-layer manufacturing has been gaining attention recently, with increasing potential to fabricate complex, low-weight structures, such as mechanical metamaterials. This paper aims to study the fabrication and mechanical characterization of non-planar lattice structures under cyclic loading. Design/methodology/approach A mathematical approach to parametrize lattices onto Bèzier surfaces is validated and applied here to fabricate non-planar lattice samples via curved-layer fused deposition modeling. The lattice chirality, amplitude and unit cell size were varied, and the properties of the samples under cyclic-loading were studied experimentally. Findings Overall, lattices with higher auxeticity showed less energy dissipation, attributed to their bending-deformation mechanism. Additionally, bistability was eliminated with increasing auxeticity, reinforcing the conclusion of bending-dominated behavior. The analysis presented here demonstrates that mechanical metamaterial lattices such as auxetics can be explored experimentally for complex geometries where traditional methods of comparing simple geometry to end-use designs are not applicable. Research limitations/implications The mechanics of non-planar lattice structures fabricated using curved-layer additive manufacturing have not been studied thoroughly. Furthermore, traditional approaches do not apply due to parameterization deformations, requiring novel approaches to their study. Here the properties of such structures under cyclic-loading are studied experimentally for the first time. Applications for this type of structures can be found in areas like biomedical scaffolds and stents, sandwich-panel packaging, aerospace structures and architecture of lattice domes. Originality/value This work presents an experimental approach to study the mechanical properties of non-planar lattice structures via quasi-static cyclic loading, comparing variations across several lattice patterns including auxetic sinusoids, disrupted sinusoids and their equivalent-density quadratic patterns.
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Kitano, Yasuyuki, and Masaki Takata. "Coincidence-Site-Lattice Pattern (Csl Pattern) of 70.5°/[110] Boundary of the 6H-Type Layer Structure." Proceedings, annual meeting, Electron Microscopy Society of America 48, no. 1 (August 12, 1990): 576–77. http://dx.doi.org/10.1017/s0424820100181646.
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The most useful and intuitive model may be a CSL-model to analyze boundary structures. In order to apply the CSL-model to layer structures, we have proposed to use ‘lattice point’ in a wide sence and to add extra lattice points to the Bravais lattice points when interpenetrating(IP)-lattices are drawn. These lattice points will be called ‘extended lattice points‘. It is well known that a layer structure is built up with (almost) identical layers stacking on the top of the others with a cirtain amount of shift in a direction perpendicular to the stacking. Each layer consists of one or more atomic planes and has almost (or exactly) the same atomic configuration. The extended lattice points can be defined as the origins in each layer in crystal. The number of such points depends upon the number of layers in a unitcell.To draw IP-lattices we have adopted all the extended lattice points in addition to the Bravais lattice points. There are three important advantages of doing this extension[ 1,2,3]. First is that the Coincidence-Sites in the IP-lattices drawn do not scatter homogeneously, but gather in a region and make a cluster. They exibit a characteristic pattern of Coincidence-Sites. This pattern is called a CSL-pattern. Second is that the DSC-lattice (Displacement Shift Complete Lattice) provides a set of basic vectors smaller than predicted before extention. Third is that reasonable models of boundaries are able to be made between two different layer structures. This is because the crystal volume attributed to one extended lattice point is exactly the same for the both adjacent crystals.
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Ngapasare, A., G. Theocharis, O. Richoux, Ch Skokos, and V. Achilleos. "Wave-packet spreading in disordered soft architected structures." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 5 (May 2022): 053116. http://dx.doi.org/10.1063/5.0089055.
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We study the dynamical and chaotic behavior of a disordered one-dimensional elastic mechanical lattice, which supports translational and rotational waves. The model used in this work is motivated by the recent experimental results of Deng et al. [Nat. Commun. 9, 1 (2018)]. This lattice is characterized by strong geometrical nonlinearities and the coupling of two degrees-of-freedom (DoFs) per site. Although the linear limit of the structure consists of a linear Fermi–Pasta–Ulam–Tsingou lattice and a linear Klein–Gordon (KG) lattice whose DoFs are uncoupled, by using single site initial excitations on the rotational DoF, we evoke the nonlinear coupling between the system’s translational and rotational DoFs. Our results reveal that such coupling induces rich wave-packet spreading behavior in the presence of strong disorder. In the weakly nonlinear regime, we observe energy spreading only due to the coupling of the two DoFs (per site), which is in contrast to what is known for KG lattices with a single DoF per lattice site, where the spreading occurs due to chaoticity. Additionally, for strong nonlinearities, we show that initially localized wave-packets attain near ballistic behavior in contrast to other known models. We also reveal persistent chaos during energy spreading, although its strength decreases in time as quantified by the evolution of the system’s finite-time maximum Lyapunov exponent. Our results show that flexible, disordered, and strongly nonlinear lattices are a viable platform to study energy transport in combination with multiple DoFs (per site), also present an alternative way to control energy spreading in heterogeneous media.
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Shatabda, Swakkhar, M. A. Hakim Newton, Mahmood A. Rashid, Duc Nghia Pham, and Abdul Sattar. "How Good Are Simplified Models for Protein Structure Prediction?" Advances in Bioinformatics 2014 (April 29, 2014): 1–9. http://dx.doi.org/10.1155/2014/867179.
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Protein structure prediction (PSP) has been one of the most challenging problems in computational biology for several decades. The challenge is largely due to the complexity of the all-atomic details and the unknown nature of the energy function. Researchers have therefore used simplified energy models that consider interaction potentials only between the amino acid monomers in contact on discrete lattices. The restricted nature of the lattices and the energy models poses a twofold concern regarding the assessment of the models. Can a native or a very close structure be obtained when structures are mapped to lattices? Can the contact based energy models on discrete lattices guide the search towards the native structures? In this paper, we use the protein chain lattice fitting (PCLF) problem to address the first concern; we developed a constraint-based local search algorithm for the PCLF problem for cubic and face-centered cubic lattices and found very close lattice fits for the native structures. For the second concern, we use a number of techniques to sample the conformation space and find correlations between energy functions and root mean square deviation (RMSD) distance of the lattice-based structures with the native structures. Our analysis reveals weakness of several contact based energy models used that are popular in PSP.
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Chen, Yu, Jinguo You, Benyuan Zou, Guoyu Gan, Ting Zhang, and Lianyin Jia. "Exploring Structural Characteristics of Lattices in Real World." Complexity 2020 (January 21, 2020): 1–11. http://dx.doi.org/10.1155/2020/1250106.
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There are two important models for data analysis and knowledge system: data cube lattices and concept lattices. They both essentially have lattice structures, which are actually irregular in our real world. However, their structural characteristics and relationship are not yet clear. To the best of our knowledge, no work has paid enough attention to this challenging issue from the perspective of graph data, in spite of the importance of structures in lattice data. In this paper, we first tackle the structural statistics of lattice data from three aspects: the degree distribution, clustering coefficient, and average path length. We demonstrated by various datasets that data cube lattices and concept lattices share similarities underlying their topology, which are, in general, different from random networks and complex networks. Specifically, lattice data follow the Poisson distribution and have smaller clustering coefficient and greater average path length. We further discuss and explain these characteristics intrinsically by building the analytical model and the generating mechanism.
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Bathla, Pranjal, and John Kennedy. "3D Printed Structured Porous Treatments for Flow Control around a Circular Cylinder." Fluids 5, no. 3 (August 14, 2020): 136. http://dx.doi.org/10.3390/fluids5030136.
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The use of porous coatings is one of the passive flow control methods used to reduce turbulence, noise and vibrations generated due to fluid flow. Porous coatings for flow stabilization have potential for a light-weight, cost-effective, and customizable solution. The design and performance of a structured porous coating depend on multiple control parameters like lattice size, strut thickness, lattice structure/geometry, etc. This study investigated the suitability of MSLA 3D printers to manufacture porous coatings based on unit cell designs to optimize porous lattices for flow control behind a cylinder. The Reynolds number used was 6.1×104–1.5×105 and the flow measurements were taken using a hotwire probe. Different experiment sets were conducted for single cylinder with varying control parameters to achieve best performing lattice designs. It was found that lattice structures with higher porosity produced lower turbulence intensity in the wake of the cylinder. However, for constant porosity lattice structures, there was negligible difference in turbulence and mean wake velocity levels. Coating thickness did not have a linear relationship with turbulence reduction, suggesting an optimal thickness value. For constant porosity coatings, cell count in coating thickness did not influence the turbulence or mean wake velocity. Partial coating designs like helical and spaced coatings had comparable performance to that of a full coating. MSLA printers were found capable of manufacturing thin and complex porous lattices.
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Flaut, Cristina, Dana Piciu, and Bianca Liana Bercea. "Some Applications of Fuzzy Sets in Residuated Lattices." Axioms 13, no. 4 (April 18, 2024): 267. http://dx.doi.org/10.3390/axioms13040267.
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Many papers have been devoted to applying fuzzy sets to algebraic structures. In this paper, based on ideals, we investigate residuated lattices from fuzzy set theory, lattice theory, and coding theory points of view, and some applications of fuzzy sets in residuated lattices are presented. Since ideals are important concepts in the theory of algebraic structures used for formal fuzzy logic, first, we investigate the lattice of fuzzy ideals in residuated lattices and study some connections between fuzzy sets associated to ideals and Hadamard codes. Finally, we present applications of fuzzy sets in coding theory.
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Lee, Jun Hak, Seong Je Park, Jeongho Yang, Seung Ki Moon, and Jiyong Park. "Crack Control in Additive Manufacturing by Leveraging Process Parameters and Lattice Design." Micromachines 15, no. 11 (November 10, 2024): 1361. http://dx.doi.org/10.3390/mi15111361.
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This study investigates the design of additive manufacturing for controlled crack propagation using process parameters and lattice structures. We examine two lattice types—octet-truss (OT) and diamond (DM)—fabricated via powder bed fusion with Ti-6Al-4V. Lattice structures are designed with varying densities (10%, 30%, and 50%) and process using two different laser energies. Using additive-manufactured specimens, Charpy impact tests are conducted to evaluate the fracture behavior and impact energy levels of the specimens. Results show that the type of the lattice structures, the density of the lattice structures, and laser energy significantly influence crack propagation patterns and impact energy. OT exhibits straighter crack paths, while DM demonstrates more random fracture patterns. Higher-density lattices and increased laser energy generally improve the impact energy. DM consistently outperformed OT in the impact energy for angle specimens, while OT showed superior performance in stair specimens. Finally, a case study demonstrates the potential for combining OT and DM structures to guide crack propagation along predetermined paths, offering a novel approach to protect critical components during product failure.
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Obadimu, Solomon O., and Kyriakos I. Kourousis. "Compressive Behaviour of Additively Manufactured Lattice Structures: A Review." Aerospace 8, no. 8 (July 30, 2021): 207. http://dx.doi.org/10.3390/aerospace8080207.
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Additive manufacturing (AM) technology has undergone an evolutionary process from fabricating test products and prototypes to fabricating end-user products—a major contributing factor to this is the continuing research and development in this area. AM offers the unique opportunity to fabricate complex structures with intricate geometry such as the lattice structures. These structures are made up of struts, unit cells, and nodes, and are being used not only in the aerospace industry, but also in the sports technology industry, owing to their superior mechanical properties and performance. This paper provides a comprehensive review of the mechanical properties and performance of both metallic and non-metallic lattice structures, focusing on compressive behaviour. In particular, optimisation techniques utilised to optimise their mechanical performance are examined, as well the primary factors influencing mechanical properties of lattices, and their failure mechanisms/modes. Important AM limitations regarding lattice structure fabrication are identified from this review, while the paucity of literature regarding material extruded metal-based lattice structures is discussed.
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Jiang, Cho-Pei, Alvian Toto Wibisono, and Tim Pasang. "Selective Laser Melting of Stainless Steel 316L with Face-Centered-Cubic-Based Lattice Structures to Produce Rib Implants." Materials 14, no. 20 (October 11, 2021): 5962. http://dx.doi.org/10.3390/ma14205962.
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Selective laser melting has a great potential to manufacture biocompatible metal alloy scaffolds or implants with a regulated porosity structure. This study uses five face-centered cubic (FCC) lattice structures, including FCC, FCC-Z, S-FCC, S-FCC-Z, and FCC-XYZ. Specimens with different lattice structures are fabricated using two laser energy densities, 71 J/mm3 and 125 J/mm3. Density, tensile, compressive and flexural test results exhibit the effect of laser parameters and lattice structure geometries on mechanical properties. The higher laser energy density of 125 J/mm3 results in higher properties such as density, strength, and Young’s modulus than the laser energy density of 71 J/mm3. The S-FCC lattice has the lowest density among all lattices. The mechanical tests result show specimen with FCC-XYZ lattice structures fabricated using a laser energy density of 125 J/mm3 meet the tensile properties requirement for human ribs. This structure also meets the requirement in flexural strength performance, but its stiffness is over that of human ribs. The compression test results of lattices are still incomparable due to unavailable compression data of the human ribs. In short, The FCC-XYZ lattice design fabricated by the 125 J/mm3 laser energy density parameter can be used to manufacture customized rib implants.
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Huang, Aiping, and William Zhu. "Geometric Lattice Structure of Covering-Based Rough Sets through Matroids." Journal of Applied Mathematics 2012 (2012): 1–25. http://dx.doi.org/10.1155/2012/236307.
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Covering-based rough set theory is a useful tool to deal with inexact, uncertain, or vague knowledge in information systems. Geometric lattice has been widely used in diverse fields, especially search algorithm design, which plays an important role in covering reductions. In this paper, we construct three geometric lattice structures of covering-based rough sets through matroids and study the relationship among them. First, a geometric lattice structure of covering-based rough sets is established through the transversal matroid induced by a covering. Then its characteristics, such as atoms, modular elements, and modular pairs, are studied. We also construct a one-to-one correspondence between this type of geometric lattices and transversal matroids in the context of covering-based rough sets. Second, we present three sufficient and necessary conditions for two types of covering upper approximation operators to be closure operators of matroids. We also represent two types of matroids through closure axioms and then obtain two geometric lattice structures of covering-based rough sets. Third, we study the relationship among these three geometric lattice structures. Some core concepts such as reducible elements in covering-based rough sets are investigated with geometric lattices. In a word, this work points out an interesting view, namely, geometric lattice, to study covering-based rough sets.
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Kitano, Yasuyuki, and Masaki Takata. "Coincidence-site-lattice-pattern (CSL-pattern) of 70.5°/[110] boundary of the 6H-type layer structure." Proceedings, annual meeting, Electron Microscopy Society of America 48, no. 4 (August 1990): 356–57. http://dx.doi.org/10.1017/s0424820100174916.
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The most useful and intuitive model may be a CSL-model to analyze boundary structures. In order to apply the CSL-model to layer structures, we have proposed to use ‘lattice point’ in a wide sence and to add extra lattice points to the Bravais lattice points when interpenetrating(IP)-lattices are drawn. These lattice points will be called ‘extended lattice points’. It is well known that a layer structure is built up with (almost) identical layers stacking on the top of the others with a cirtain amount of shift in a direction perpendicular to the stacking. Each layer consists of one or more atomic planes and has almost (or exactly) the same atomic configuration. The extended lattice points can be defined as the origins in each layer in crystal. The number of such points depends upon the number of layers in a unit cell.To draw IP-lattices we have adopted all the extended lattice points in addition to the Bravais lattice points. There are three important advantages of doing this extension. First is that the Coincidence-Sites in the IP-lattices drawn do not scatter homogeneously, but gather in a region and make a cluster. They exibit a characteristic pattern of Coincidence-Sites. This pattern is called a CSL-pattern. Second is that the DSC-lattice (Displacement Shift Complete Lattice) provides a set of basic vectors smaller than predicted before extention.
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Grabowski, Adam. "Stone Lattices." Formalized Mathematics 23, no. 4 (December 1, 2015): 387–96. http://dx.doi.org/10.1515/forma-2015-0031.
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Summary The article continues the formalization of the lattice theory (as structures with two binary operations, not in terms of ordering relations). In the paper, the notion of a pseudocomplement in a lattice is formally introduced in Mizar, and based on this we define the notion of the skeleton and the set of dense elements in a pseudocomplemented lattice, giving the meet-decomposition of arbitrary element of a lattice as the infimum of two elements: one belonging to the skeleton, and the other which is dense. The core of the paper is of course the idea of Stone identity $$a^* \sqcup a^{**} = {\rm{T}},$$ which is fundamental for us: Stone lattices are those lattices L, which are distributive, bounded, and satisfy Stone identity for all elements a ∈ L. Stone algebras were introduced by Grätzer and Schmidt in [18]. Of course, the pseudocomplement is unique (if exists), so in a pseudcomplemented lattice we defined a * as the Mizar functor (unary operation mapping every element to its pseudocomplement). In Section 2 we prove formally a collection of ordinary properties of pseudocomplemented lattices. All Boolean lattices are Stone, and a natural example of the lattice which is Stone, but not Boolean, is the lattice of all natural divisors of p 2 for arbitrary prime number p (Section 6). At the end we formalize the notion of the Stone lattice B [2] (of pairs of elements a, b of B such that a ⩽ b) constructed as a sublattice of B 2, where B is arbitrary Boolean algebra (and we describe skeleton and the set of dense elements in such lattices). In a natural way, we deal with Cartesian product of pseudocomplemented lattices. Our formalization was inspired by [17], and is an important step in formalizing Jouni Järvinen Lattice theory for rough sets [19], so it follows rather the latter paper. We deal essentially with Section 4.3, pages 423–426. The description of handling complemented structures in Mizar [6] can be found in [12]. The current article together with [15] establishes the formal background for algebraic structures which are important for [10], [16] by means of mechanisms of merging theories as described in [11].
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Rahman, Hafizur, Ebrahim Yarali, Ali Zolfagharian, Ahmad Serjouei, and Mahdi Bodaghi. "Energy Absorption and Mechanical Performance of Functionally Graded Soft–Hard Lattice Structures." Materials 14, no. 6 (March 11, 2021): 1366. http://dx.doi.org/10.3390/ma14061366.
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Today, the rational combination of materials and design has enabled the development of bio-inspired lattice structures with unprecedented properties to mimic biological features. The present study aims to investigate the mechanical performance and energy absorption capacity of such sophisticated hybrid soft–hard structures with gradient lattices. The structures are designed based on the diversity of materials and graded size of the unit cells. By changing the unit cell size and arrangement, five different graded lattice structures with various relative densities made of soft and hard materials are numerically investigated. The simulations are implemented using ANSYS finite element modeling (FEM) (2020 R1, 2020, ANSYS Inc., Canonsburg, PA, USA) considering elastic-plastic and the hardening behavior of the materials and geometrical non-linearity. The numerical results are validated against experimental data on three-dimensional (3D)-printed lattices revealing the high accuracy of the FEM. Then, by combination of the dissimilar soft and hard polymeric materials in a homogenous hexagonal lattice structure, two dual-material mechanical lattice statures are designed, and their mechanical performance and energy absorption are studied. The results reveal that not only gradual changes in the unit cell size provide more energy absorption and improve mechanical performance, but also the rational combination of soft and hard materials make the lattice structure with the maximum energy absorption and stiffness, in comparison to those structures with a single material, interesting for multi-functional applications.
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Wang, Xinglong, Cheng Wang, Xin Zhou, Di Wang, Mingkang Zhang, Yun Gao, Lei Wang, and Peiyu Zhang. "Evaluating Lattice Mechanical Properties for Lightweight Heat-Resistant Load-Bearing Structure Design." Materials 13, no. 21 (October 27, 2020): 4786. http://dx.doi.org/10.3390/ma13214786.
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Heat-resistant, load-bearing components are common in aircraft, and they have high requirements for lightweight and mechanical performance. Lattice topology optimization can achieve high mechanical properties and obtain lightweight designs. Appropriate lattice selection is crucial when employing the lattice topology optimization method. The mechanical properties of a structure can be optimized by choosing lattice structures suitable for the specific stress environment being endured by the structural components. Metal lattice structures exhibit excellent unidirectional load-bearing performance and the triply periodic minimal surface (TPMS) porous structure can satisfy multi-scale free designs. Both lattice types can provide unique advantages; therefore, we designed three types of metal lattices (body-centered cubic (BCC), BCC with Z-struts (BCCZ), and honeycomb) and three types of TPMS lattices (gyroid, primitive, and I-Wrapped Package (I-WP)) combined with the solid shell. Each was designed with high level of relative density (40%, 50%, 60%, 70%, and 80%), which can be directly used in engineering practice. All test specimens were manufactured by selective laser melting (SLM) technology using Inconel 718 superalloy as the material and underwent static tensile testing. We found that the honeycomb test specimen exhibits the best strength, toughness, and stiffness properties among all structures evaluated, which is especially suitable for the lattice topology optimization design of heat-resistant, unidirectional load-bearing structures within aircraft. Furthermore, we also found an interesting phenomenon that the toughness of the primitive and honeycomb porous test specimens exhibited sudden increases from 70% to 80% and from 50% to 60% relative density, respectively, due to their structural characteristics. According to the range of the exponent value n and the deformation laws of porous structures, we also concluded that a porous structure would exhibit a stretching-dominated deformation behavior when exponent value n < 0.3, a bending-dominated deformation behavior when n > 0.55, and a stretching-bending-dominated deformation behavior when 0.3 < n < 0.55. This study can provide a design basis for selecting an appropriate lattice in lattice topology optimization design.
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Machala, Frantisek, and Vladimir Slezak. "Lattice-inadmissible incidence structures." Discussiones Mathematicae - General Algebra and Applications 24, no. 2 (2004): 199. http://dx.doi.org/10.7151/dmgaa.1085.
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Agterberg, D. F. "Vortex Lattice Structures ofSr2RuO4." Physical Review Letters 80, no. 23 (June 8, 1998): 5184–87. http://dx.doi.org/10.1103/physrevlett.80.5184.
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Losert, B., H. Boustique, and G. Richardson. "Modifications: Lattice-valued structures." Fuzzy Sets and Systems 210 (January 2013): 54–62. http://dx.doi.org/10.1016/j.fss.2012.07.003.
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Darsana, C., and P. Sini. "Lattice of c-structures." Gulf Journal of Mathematics 17, no. 1 (July 1, 2024): 167–78. http://dx.doi.org/10.56947/gjom.v17i1.2078.
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In this paper, we study the properties of the lattice of c-structures. We define the concept of simple expansion and examine some of its properties. Additionally, we determine the automorphism group of the lattice of c-structures and explore the fixed points of the group of automorphisms.
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Wang, Yongchao, Bin Pang, and Fu-Gui Shi. "Lattice-valued coarse structures." Fuzzy Sets and Systems 499 (January 2025): 109175. http://dx.doi.org/10.1016/j.fss.2024.109175.
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Jin, Xin, Guo Xi Li, and Meng Zhang. "Optimal design of three-dimensional non-uniform nylon lattice structures for selective laser sintering manufacturing." Advances in Mechanical Engineering 10, no. 7 (July 2018): 168781401879083. http://dx.doi.org/10.1177/1687814018790833.
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As a kind of novel multifunctional structure with three-dimensional pores characterized by low relative density, lattice structures can attain a lightweight design while maintaining high specific mechanical properties in three-dimensional solid structures. Focusing on the challenge of finding the optimal design of lattice structures in the design object, a design and modeling method of non-uniform three-dimensional lattice structures is proposed while ensuring the selective laser sintering manufacturability. Optimization for cell type, cell size, and strut size distribution of lattices is specified with the mechanical properties analyzed and the material model calculated beforehand. The manufacturing constraints are analyzed and expressed in topology optimization and the optimal distribution of topology optimization results is mapped to the strut size distribution of lattice cells. The rapid and automatic computer-aided design modeling of optimized structures is realized by the parametric definition and assembling of lattice components. Finally, the non-uniform structures are successfully manufactured by selective laser sintering and it is shown by means of finite element analysis and experiments that the proposed design approach can improve the mechanical performance compared to the uniform lattice structure under the same weight reduction. And for the design object in this study, body-centered structure with cell size [Formula: see text]mm is chosen as the optimal cell type and cell size under the given selective laser sintering manufacturing constraints.
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Guerra Silva, Rafael, María Josefina Torres, and Jorge Zahr Viñuela. "A Comparison of Miniature Lattice Structures Produced by Material Extrusion and Vat Photopolymerization Additive Manufacturing." Polymers 13, no. 13 (June 30, 2021): 2163. http://dx.doi.org/10.3390/polym13132163.
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In this paper, we study the capabilities of two additive manufacturing technologies for the production of lattice structures, namely material extrusion and vat photopolymerization additive manufacturing. A set of polymer lattice structures with diverse unit cell types were built using these additive manufacturing methods and tested under compression. Lattice structures built using material extrusion had lower accuracy and a lower relative density caused by the air gaps between layers, but had higher elastic moduli and larger energy absorption capacities, as a consequence of both the thicker struts and the relatively larger strength of the feedstock material. Additionally, the deformation process in lattices was analyzed using sequential photographs taken during the compression tests, evidencing larger differences according to the manufacturing process and unit-cell type. Both additive manufacturing methods produced miniature lattice structures with similar mechanical properties, but vat polymerization should be the preferred option when high geometrical accuracy is required. Nevertheless, as the solid material determines the compressive response of the lattice structure, the broader availability of feedstock materials gives an advantage to material extrusion in applications requiring stiffer structures or with higher energy absorption capabilities.
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McMillan, Matthew Leslie, Marten Jurg, Martin Leary, and Milan Brandt. "Programmatic generation of computationally efficient lattice structures for additive manufacture." Rapid Prototyping Journal 23, no. 3 (April 18, 2017): 486–94. http://dx.doi.org/10.1108/rpj-01-2016-0014.
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Purpose Additive manufacturing (AM) enables the fabrication of complex geometries beyond the capability of traditional manufacturing methods. Complex lattice structures have enabled engineering innovation; however, the use of traditional computer-aided design (CAD) methods for the generation of lattice structures is inefficient, time-consuming and can present challenges to process integration. In an effort to improve the implementation of lattice structures into engineering applications, this paper aims to develop a programmatic lattice generator (PLG). Design/methodology/approach The PLG method is computationally efficient; has direct control over the quality of the stereolithographic (STL) file produced; enables the generation of more complex lattice than traditional methods; is fully programmatic, allowing batch generation and interfacing with process integration and design optimization tools; capable of generating a lattice STL file from a generic input file of node and connectivity data; and can export a beam model for numerical analysis. Findings This method has been successfully implemented in the generation of uniform, radial and space filling lattices. Case studies were developed which showed a reduction in processing time greater than 60 per cent for a 3,375 cell lattice over traditional CAD software. Originality/value The PLG method is a novel design for additive manufacture (DFAM) tool with unique advantages, including full control over the number of facets that represent a lattice strut, allowing optimization of STL data to minimize file size, while maintaining suitable resolution for the implemented AM process; programmatic DFAM capability that overcomes the learning curve of traditional CAD when producing complex lattice structures, therefore is independent of designer proficiency and compatible with process integration; and the capability to output both STL files and associated data for numerical analysis, a unique DFAM capability not previously reported.
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Al Nashar, Mohamad, and Alok Sutradhar. "Design of Hierarchical Architected Lattices for Enhanced Energy Absorption." Materials 14, no. 18 (September 17, 2021): 5384. http://dx.doi.org/10.3390/ma14185384.
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Hierarchical lattices are structures composed of self-similar or dissimilar architected metamaterials that span multiple length scales. Hierarchical lattices have superior and tunable properties when compared to conventional lattices, and thus, open the door for a wide range of material property manipulation and optimization. Using finite element analysis, we investigate the energy absorption capabilities of 3D hierarchical lattices for various unit cells under low strain rates and loads. In this study, we use fused deposition modeling (FDM) 3D printing to fabricate a dog bone specimen and extract the mechanical properties of thermoplastic polyurethane (TPU) 85A with a hundred percent infill printed along the direction of tensile loading. With the numerical results, we observed that the energy absorption performance of the octet lattice can be enhanced four to five times by introducing a hierarchy in the structure. Conventional energy absorption structures such as foams and lattices have demonstrated their effectiveness and strengths; this research aims at expanding the design domain of energy absorption structures by exploiting 3D hierarchical lattices. The result of introducing a hierarchy to a lattice on the energy absorption performance is investigated by varying the hierarchical order from a first-order octet to a second-order octet. In addition, the effect of relative density on the energy absorption is isolated by creating a comparison between a first-order octet lattice with an equivalent relative density as a second-order octet lattice. The compression behaviors for the second order octet, dodecahedron, and truncated octahedron are studied. The effect of changing the cross-sectional geometry of the lattice members with respect to the energy absorption performance is investigated. Changing the orientation of the second-order cells from 0 to 45 degrees has a considerable impact on the force–displacement curve, providing a 20% increase in energy absorption for the second-order octet. Analytical solutions of the effective elasticity modulus for the first- and second-order octet lattices are compared to validate the simulations. The findings of this paper and the provided understanding will aid future works in lattice design optimization for energy absorption.
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Xavier, Jolly, Sunil Vyas, Paramasivam Senthilkumaran, and Joby Joseph. "Complex 3D Vortex Lattice Formation by Phase-Engineered Multiple Beam Interference." International Journal of Optics 2012 (2012): 1–9. http://dx.doi.org/10.1155/2012/863875.
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We present the computational results on the formation of diverse complex 3D vortex lattices by a designed superposition of multiple plane waves. Special combinations of multiples of three noncoplanar plane waves with a designed relative phase shift between one another are perturbed by a nonsingular beam to generate various complex 3D vortex lattice structures. The formation of complex gyrating lattice structures carrying designed vortices by means of relatively phase-engineered plane waves is also computationally investigated. The generated structures are configured with both periodic as well as transversely quasicrystallographic basis, while these whirling complex lattices possess a long-range order of designed symmetry in a given plane. Various computational analytical tools are used to verify the presence of engineered geometry of vortices in these complex 3D vortex lattices.
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Guo, Zhengmiao, Fan Yang, Lingbo Li, and Jiacheng Wu. "Bio-Inspired Curved-Elliptical Lattice Structures for Enhanced Mechanical Performance and Deformation Stability." Materials 17, no. 17 (August 24, 2024): 4191. http://dx.doi.org/10.3390/ma17174191.
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Lattice structures, characterized by their lightweight nature, high specific mechanical properties, and high design flexibility, have found widespread applications in fields such as aerospace and automotive engineering. However, the lightweight design of lattice structures often presents a trade-off between strength and stiffness. To tackle this issue, a bio-inspired curved-elliptical (BCE) lattice is proposed to enhance the mechanical performance and deformation stability of three-dimensional lattice structures. BCE lattice specimens with different parameters were fabricated using selective laser melting (SLM) technology, followed by quasi-static compression tests. Finite element (FE) numerical simulations were also carried out for validation. The results demonstrate that the proposed BCE lattice structures exhibit stronger mechanical performance and more stable deformation modes that can be adjusted through parameter tuning. Specifically, by adjusting the design parameters, the BCE lattice structure can exhibit a bending-dominated delocalized deformation mode, avoiding catastrophic collapse during deformation. The specific energy absorption (SEA) can reach 24.6 J/g at a relative density of only 8%, with enhancements of 48.5% and 297.6% compared with the traditional energy-absorbing lattices Octet and body-center cubic (BCC), respectively. Moreover, the crushing force efficiency (CFE) of the BCE lattice structure surpasses those of Octet and BCC by 34.9% and 15.8%, respectively. Through a parametric study of the influence of the number of peaks N and the curve amplitude A on the compression performance of the BCE lattice structure, the compression deformation mechanism is further analyzed. The results indicate that the curve amplitude A and the number of peaks N have significant impacts on the deformation mode of the BCE lattice. By adjusting the parameters N and A, a structure with a combination of high energy absorption, high stiffness, and strong fracture resistance can be obtained, integrating the advantages of tensile-dominated and bending-dominated lattice structures.
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El-Gayar, Mostafa A., and Radwan Abu-Gdairi. "Extension of topological structures using lattices and rough sets." AIMS Mathematics 9, no. 3 (2024): 7552–69. http://dx.doi.org/10.3934/math.2024366.
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<abstract><p>This paper explores the application of rough set theory in analyzing ambiguous data within complete information systems. The study extends topological structures using equivalence relations, establishing an extension of topological lattice within lattices. Various relations on topological spaces generate different forms of exact and rough lattices. Building on Zhou's work, the research investigates rough sets within the extension topological lattice and explores the isomorphism between topology and its extension. Additionally, the paper investigates the integration of lattices and rough sets, essential mathematical tools widely used in problem-solving. Focusing on computer science's prominent lattices and Pawlak's rough sets, the study introduces extension lattices, emphasizing lower and upper extension approximations' adaptability for practical applications. These approximations enhance pattern recognition and model uncertain data with finer granularity. While acknowledging the benefits, the paper stresses the importance of empirical validations for domain-specific efficacy. It also highlights the isomorphism between topology and its extension, revealing implications for data representation, decision-making, and computational efficiency. This isomorphism facilitates accurate data representations and streamlines computations, contributing to improved efficiency. The study enhances the understanding of integrating lattices and rough sets, offering potential applications in data analysis, decision support systems, and computational modeling.</p></abstract>
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Bari, Klaudio. "Design, Simulation, and Mechanical Testing of 3D-Printed Titanium Lattice Structures." Journal of Composites Science 7, no. 1 (January 11, 2023): 32. http://dx.doi.org/10.3390/jcs7010032.
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Lattice structure topology is a rapidly growing area of research facilitated by developments in additive manufacturing. These low-density structures are particularly promising for their medical applications. However, predicting their performance becomes a challenging factor in their use. In this article, four lattice topologies are explored for their suitability as implants for the replacement of segmental bone defects. The study introduces a unit-cell concept for designing and manufacturing four lattice structures, BCC, FCC, AUX, and ORG, using direct melt laser sintering (DMLS). The elastic modulus was assessed using an axial compression strength test and validated using linear static FEA simulation. The outcomes of the simulation revealed the disparity between the unit cell and the entire lattice in the cases of BCC, FCC, and AUX, while the unit-cell concept of the full lattice structure was successful in ORG. Measurements of energy absorption obtained from the compression testing revealed that the ORG lattice had the highest absorbed energy (350 J) compared with the others. The observed failure modes indicated a sudden collapsing pattern during the compression test in the cases of BCC and FCC designs, while our inspired ORG and AUX lattices outperformed the others in terms of their structural integrity under identical loading conditions.
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Vivek, S., and Sunil C. Mathew. "Some lattices associated with LM-fuzzy topological spaces." Journal of Intelligent & Fuzzy Systems 40, no. 6 (June 21, 2021): 12101–9. http://dx.doi.org/10.3233/jifs-210195.
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This paper studies the closure and interior operators in LM-fuzzy topological spaces. The algebraic structures associated with various collections of closed sets and open sets are identified. Further, certain lattices formed by these algebraic structures are obtained and some lattice theoretic properties of the same are investigated. Corresponding to every element in M, the study associates a lattice of monoids which is determined by various types of closed sets and open sets.
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NURAKUNOV, A. M. "UNREASONABLE LATTICES OF QUASIVARIETIES." International Journal of Algebra and Computation 22, no. 03 (May 2012): 1250006. http://dx.doi.org/10.1142/s0218196711006728.
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A quasivariety is a universal Horn class of algebraic structures containing the trivial structure. The set [Formula: see text] of all subquasivarieties of a quasivariety [Formula: see text] forms a complete lattice under inclusion. A lattice isomorphic to [Formula: see text] for some quasivariety [Formula: see text] is called a lattice of quasivarieties or a quasivariety lattice. The Birkhoff–Maltsev Problem asks which lattices are isomorphic to lattices of quasivarieties. A lattice L is called unreasonable if the set of all finite sublattices of L is not computable, that is, there is no algorithm for deciding whether a finite lattice is a sublattice of L. The main result of this paper states that for any signature σ containing at least one non-constant operation, there is a quasivariety [Formula: see text] of signature σ such that the quasivariety lattice [Formula: see text] is unreasonable. Moreover, there are uncountable unreasonable lattices of quasivarieties. We also present some corollaries of the main result.
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Dal Fabbro, Pierandrea, Stefano Rosso, Alessandro Ceruti, Diego Boscolo Bozza, Roberto Meneghello, Gianmaria Concheri, and Gianpaolo Savio. "Analysis of a Preliminary Design Approach for Conformal Lattice Structures." Applied Sciences 11, no. 23 (December 2, 2021): 11449. http://dx.doi.org/10.3390/app112311449.
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An important issue when designing conformal lattice structures is the geometric modeling and prediction of mechanical properties. This paper presents suitable methods for obtaining optimized conformal lattice structures and validating them without the need for high computational power and time, enabling the designer to have quick feedback in the first design phases. A wireframe modeling method based on non-uniform rational basis spline (NURBS) free-form deformation (FFD) that allows conforming a regular lattice structure inside a design space is presented. Next, a previously proposed size optimization method is adopted for optimizing the cross-sections of lattice structures. Finally, two different commercial finite element software are involved for the validation of the results, based on Euler–Bernoulli and Timoshenko beam theories. The findings highlight the adaptability of the NURBS-FFD modeling approach and the reliability of the size optimization method, especially in stretching-dominated cell topologies and load conditions. At the same time, the limitation of the structural beam analysis when dealing with thick beams is noted. Moreover, the behavior of different kinds of lattices was investigated.
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Zhang, Hongyi, Yang Wang, Shuyou Zhang, Xiaojian Liu, and Xuewei Zhang. "Thermomechanical coupled topology optimization of parameterized lattice structures." Mechanical Sciences 15, no. 2 (October 10, 2024): 555–66. http://dx.doi.org/10.5194/ms-15-555-2024.
Full textAbstract:
Abstract. This paper presents a topology optimization approach for parameterized lattice structures subjected to thermomechanical coupled loads. The proposed approach aims to minimize the compliance of lattice structures while satisfying volume fraction constraints and accurate temperature constraints. A thermomechanical coupled optimization model containing a heat transfer model and a thermoelastic model is utilized for accurate modeling, and the distribution of the temperature field is related to design variables. Numerical homogenization is employed to calculate the effective properties of parameterized lattices, and polynomial interpolation models are used to replace numerical homogenization methods during optimization iterations to reduce computational costs. The proposed method is demonstrated through examples involving battery packs, L-brackets, and machine tool headstocks. Numerical verification results show that the proposed method significantly reduces the compliance of the designed structures compared to traditional solid designs and precisely meets temperature constraints.