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1
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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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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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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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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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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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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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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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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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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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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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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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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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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Alexandru, Andrei, and Gabriel Ciobanu. "Fuzzy Results for Finitely Supported Structures." Mathematics 9, no. 14 (July 13, 2021): 1651. http://dx.doi.org/10.3390/math9141651.
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We present a survey of some results published recently by the authors regarding the fuzzy aspects of finitely supported structures. Considering the notion of finite support, we introduce a new degree of membership association between a crisp set and a finitely supported function modelling a degree of membership for each element in the crisp set. We define and study the notions of invariant set, invariant complete lattices, invariant monoids and invariant strong inductive sets. The finitely supported (fuzzy) subgroups of an invariant group, as well as the L-fuzzy sets on an invariant set (with L being an invariant complete lattice) form invariant complete lattices. We present some fixed point results (particularly some extensions of the classical Tarski theorem, Bourbaki–Witt theorem or Tarski–Kantorovitch theorem) for finitely supported self-functions defined on invariant complete lattices and on invariant strong inductive sets; these results also provide new finiteness properties of infinite fuzzy sets. We show that apparently, large sets do not contain uniformly supported, infinite subsets, and so they are invariant strong inductive sets satisfying finiteness and fixed-point properties.
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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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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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Alkhader, Maen, Mohammad Nazzal, and Karim Louca. "Design of bending dominated lattice architectures with improved stiffness using hierarchy." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 233, no. 11 (October 31, 2018): 3976–93. http://dx.doi.org/10.1177/0954406218810298.
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Micro-architectured lattices are a promising subclass of cellular solids whose inner topologies can be tailored to enhance their stiffness. Generally, enhancing lattices' stiffness is achieved by increasing their connectivity. This strategy gives rise to a stiffer response by forcing lattices' ligaments to deform mainly in an axial manner. Conversely, this work is interested in developing micro-architectured lattices with enhanced stiffness, but whose cell walls deform in a flexural manner. Such structures can be more ductile and exhibit better energy mitigation abilities than their stretching dominated counterparts. Enhancing the stiffness of bending dominated lattices without increasing their connectivity can be realized by transforming them to hierarchical ones. This work explores, using experimentally verified finite element simulations, the effect of fractal-inspired hierarchy and customized nonfractal-based hierarchy on stiffness, anisotropy, and deformation mechanisms of an anisotropic bending dominated diamond lattice. Results show that fractal-inspired hierarchy can significantly enhance the stiffness of bending dominated lattices without affecting their deformation mechanisms or anisotropy level; ill-designed hierarchy can have a detrimental effect on lattice's stiffness; and customized hierarchy are more effective than fractal-inspired hierarchy in enhancing lattices' stiffness as well as can be more compatible with traditional, reliable, mass-producing manufacturing processes.
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Chen, Yizhi, and Xianzhong Zhao. "On Decompositions of Matrices over Distributive Lattices." Journal of Applied Mathematics 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/202075.
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LetLbe a distributive lattice andMn,q(L)(Mn(L), resp.) the semigroup (semiring, resp.) ofn×q(n×n, resp.) matrices overL. In this paper, we show that if there is a subdirect embedding from distributive latticeLto the direct product∏i=1mLiof distributive latticesL1,L2, …,Lm, then there will be a corresponding subdirect embedding from the matrix semigroupMn,q(L)(semiringMn(L), resp.) to semigroup∏i=1mMn,q(Li)(semiring∏i=1mMn(Li), resp.). Further, it is proved that a matrix over a distributive lattice can be decomposed into the sum of matrices over some of its special subchains. This generalizes and extends the decomposition theorems of matrices over finite distributive lattices, chain semirings, fuzzy semirings, and so forth. Finally, as some applications, we present a method to calculate the indices and periods of the matrices over a distributive lattice and characterize the structures of idempotent and nilpotent matrices over it. We translate the characterizations of idempotent and nilpotent matrices over a distributive lattice into the corresponding ones of the binary Boolean cases, which also generalize the corresponding structures of idempotent and nilpotent matrices over general Boolean algebras, chain semirings, fuzzy semirings, and so forth.
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Awaga, Kunio. "Solid-State Electrochemistry on Supramolecular Assemblies with Strong Isotropic Property." ECS Meeting Abstracts MA2023-01, no. 15 (August 28, 2023): 1397. http://dx.doi.org/10.1149/ma2023-01151397mtgabs.
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It has been mathematically proven that only honeycomb, diamond, and K4 lattices have a special symmetry called "strong isotropy". Note that crystallographic symmetry is determined only by the position of atoms, while strong isotropy is governed by both the position of atoms and bonds. The K4 lattice, called by various names such as gyroid lattice and srs network, is characterized by the fact that it exhibits chirality. It is recognized that their mathematically-defined “line graphs” correspond to kagome, hyper-kagome, and pyrochlore lattices, respectively, which are well known as spin frustration lattices. This relation suggests that the materials with the strong isotropic lattices possess “hidden” frustration. It is also noteworthy that the band structure of the three lattices contains exotic band dispersions such as Dirac cones due to their lattice symmetry, and that they possess porous structures. From this perspective, we proposed to form the supramolecular assemblies with the with strong isotropic property, and to carry out electrochemical valence control for realizing exotic physical properties such as Dirac electron, spin frustration, etc. In this presentation, we will explain our previous works on the solid-state electrochemistry of LiPc as an introduction. Then, we will report the rational synthesis of the molecule-based honeycomb and K4 structures, using MOF/COF- and supramolecular-chemistry, the nanohybridization using their porous structures, the physical properties such as spin frustration and circularly polarized luminescence, and the solid-state electrochemical redox control on them without destroying the original frameworks.
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Zhu, Lei, Xiaoyang Wang, Liao Sun, Quandong Hu, and Nan Li. "Optimisation of Selective Laser Melted Ti6Al4V Functionally Graded Lattice Structures Accounting for Structural Safety." Materials 15, no. 24 (December 19, 2022): 9072. http://dx.doi.org/10.3390/ma15249072.
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This paper presents a new framework for lightweight optimisation of functionally graded lattice structures (FGLSs) with a particular focus on enhancing and guaranteeing structural safety through three main contributions. Firstly, a design strategy of adding fillets to the joints of body-centred cubic (BCC) type lattice cells was proposed to improve the effective yield stress of the lattices. Secondly, effective properties of lattice metamaterials were experimentally characterised by conducting quasi-static uniaxial compression tests on selective laser melted specimens of both Ti6Al4V BCC and filleted BCC (BCC-F) lattices with different relative densities. Thirdly, a yield stress constraint for optimising FGLSs was developed based on surrogate models quantifying the relationships between the relative density and the effective properties of BCC and BCC-F lattices developed using experimental results assisted by numerical homogenisation. This framework was tested with two case studies. Results showed that structural safety with respect to avoiding yield failure of the optimised FGLSs can be ensured and the introduction of fillets can effectively improve the strength-to-weight ratio of the optimised FGLSs composed of BCC type lattices. The BCC-F FGLS achieved 14.5% improvement in weight reduction compared with BCC FGLS for the Messerschmitt-Bölkow-Blohm beam optimisation case study.
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CASSI, D., and S. REGINA. "RANDOM WALKS ON KEBAB LATTICES: LOGARITHMIC DIFFUSION ON ORDERED STRUCTURES." Modern Physics Letters B 09, no. 10 (April 30, 1995): 601–6. http://dx.doi.org/10.1142/s0217984995000553.
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Kebab lattices are ordered lattices obtained matching an infinite two-dimensional lattice to each point of a linear chain. Discrete time random walks on these structures are studied by analytical techniques. The exact asymptotic expressions of the mean square displacement and of the RW Green functions show an unexpected logarithmic behavior that is the first example of such kind of law on an ordered structure. Moreover the probability of returning to the origin shows the fastest long time decay ever found for recursive random walks.
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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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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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Dong, Wei, Yang Li, Kehao Xin, Dezheng Yin, Longlong Song, and Tong Gao. "A method of designing plate structure consisting of lattices and stiffeners based on topology optimization." Xibei Gongye Daxue Xuebao/Journal of Northwestern Polytechnical University 39, no. 6 (December 2021): 1233–39. http://dx.doi.org/10.1051/jnwpu/20213961233.
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In order to satisfy the lightweight design requirements of the equipment mounting plate in the hypersonic vehicle instrument cabin and improve its static/dynamic performance, a novel structure consisting of both lattices and stiffeners are studied and topology optimization method is proposed in this paper. This structure combines the advantages of lattice structures and conventional stiffened structures. First, the lattice structure is regarded as a kind of virtual material, and its equivalent mechanical properties are calculated by the homogenization method. Then, a marerial interpolation model of the virtual and solid materials are established. A topology optimization problem to minimize the mean compliance under the mass constraint is proposed to realize the layout optimization design of stiffened structure with lattices. Taking an equipment mounting plate as an example, the optimization design of the traditional stiffened structure and the novel stiffened structure with lattices is completed, respectively. Numerical analysis indicates that the lattice stiffened plate structure provides advantageous mechanical performation in the condition of the same weight. The maximum deformation under inertial force is reduced by 11.17% and the peak displacement response under harmonic excitation is reduced by 73.81% by using the stiffened structure with lattices.
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Aichinger, Erhard. "Congruence lattices forcing nilpotency." Journal of Algebra and Its Applications 17, no. 02 (January 23, 2018): 1850033. http://dx.doi.org/10.1142/s0219498818500330.
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Given a lattice [Formula: see text] and a class [Formula: see text] of algebraic structures, we say that [Formula: see text] forces nilpotency in [Formula: see text] if every algebra [Formula: see text] whose congruence lattice [Formula: see text] is isomorphic to [Formula: see text] is nilpotent. We describe congruence lattices that force nilpotency, supernilpotency or solvability for some classes of algebras. For this purpose, we investigate which commutator operations can exist on a given congruence lattice.
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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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Tkachenko, Olga, and Vitaliy Tkachenko. "Sensitivity To Disorder Of Graphene-Like Lattices Of Quantum Dots And Antidots In Two-Dimensional Electron Gas." Siberian Journal of Physics 11, no. 1 (March 1, 2016): 80–87. http://dx.doi.org/10.54362/1818-7919-2016-11-1-80-87.
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We compare three-dimensional electrostatics of semiconductor structures with graphene-like lattices of quantum dots and antidots formed in the plane of the two dimensional electron gas. With lattice constant fixed, the shape of the potential may be tuned so that both lattices have minband spectrum where the second Dirac feature is pronounced and not overlaid by the other states. We show that the lattice of quantum dots is more sensitive to fabrication imperfections, because sources of the disorder are located directly above the electronic channels. Thus the lattices of antidots should be preferred semiconductor artificial graphene candidates.
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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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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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FELSNER, STEFAN, and KOLJA B. KNAUER. "ULD-Lattices and Δ-Bonds." Combinatorics, Probability and Computing 18, no. 5 (September 2009): 707–24. http://dx.doi.org/10.1017/s0963548309010001.
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We provide a characterization of upper locally distributive lattices (ULD-lattices) in terms of edge colourings of their cover graphs. In many instances where a set of combinatorial objects carries the order structure of a lattice, this characterization yields a slick proof of distributivity or UL-distributivity. This is exemplified by proving a distributive lattice structure on Δ-bonds with invariant circular flow-difference. This instance generalizes several previously studied lattice structures, in particular,c-orientations (Propp), α-orientations of planar graphs (Felsner, resp. de Mendez) and planar flows (Khuller, Naor and Klein). The characterization also applies to other instances,e.g., to chip-firing games.
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Frtunić Gligorijević, Milena, Miloš Bogdanović, Nataša Veljković, and Leonid Stoimenov. "TOOL FOR INTERACTIVE VISUAL ANALYSIS OF LARGE HIERARCHICAL DATA STRUCTURES." Facta Universitatis, Series: Automatic Control and Robotics 20, no. 2 (November 24, 2021): 111. http://dx.doi.org/10.22190/fuacr210715009f.
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In the Big Data era data visualization and exploration systems, as means for data perception and manipulation are facing major challenges. One of the challenges for modern visualization systems is to ensure adequate visual presentation and interaction. Therefore, within this paper, we present a tool for interactive visualization of data with a hierarchical structure. It is a general-purpose tool that uses a graph-based approach. However, its main focus is on the visual analysis of concept lattices generated as the output of the Formal Concept Analysis algorithm. As the data grow, concept lattice can become complex and hard for visualization and analysis. In order to address this issue, functionalities important for the exploration of the large concept lattices are applied within this tool. The usage of the tool is presented in the example of visualization of concept lattices generated based on the available data on the Canadas open data portal and can be used for exploring the usage of tags within datasets.
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Papazoglou, Dimitri P., Amy T. Neidhard-Doll, Margaret F. Pinnell, Dathan S. Erdahl, and Timothy H. Osborn. "Compression and Tensile Testing of L-PBF Ti-6Al-4V Lattice Structures with Biomimetic Porosities and Strut Geometries for Orthopedic Implants." Metals 14, no. 2 (February 14, 2024): 232. http://dx.doi.org/10.3390/met14020232.
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In an effort to contribute to the ongoing development of ASTM standards for additively manufactured metal lattice specimens, particularly within the field of medicine, the compressive and tensile mechanical properties of biomimetic lattice structures produced by laser powder bed fusion (L-PBF) using Ti-6Al-4V feedstock powder were investigated in this research. The geometries and porosities of the lattice structures were designed to facilitate internal bone growth and prevent stress shielding. A thin strut thickness of 200 µm is utilized for these lattices to mimic human cancellous bone. In addition to a thin strut size, two different strut geometries were utilized (cubic and body-centered cubic), along with four different pore sizes (400, 500, 600, and 900 µm, representing 40–90% porosity in a 10 mm cube). A 10 mm3 cube was used for compression testing and an experimental pin-loaded design was implemented for tensile testing. The failure mode for each specimen was examined using scanning electron microscopy (SEM). Lattice structures were compared to the mechanical properties of human cancellous bone. It was found that the elastic modulus of human cancellous bone (10–900 MPa) could be matched for both the tensile (92.7–129.6 MPa) and compressive (185.2–996.1 MPa) elastic modulus of cubic and body-centered cubic lattices. Body-centered cubic lattices exhibited higher compressive properties over cubic, whereas cubic lattices exhibited superior tensile properties. The experimental tensile specimen showed reacquiring failures close to the grips, indicating that a different tensile design may be required for consistent data acquisition in the future.
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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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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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Koca, Nazife O., Mehmet Koca, and Ramazan Koc. "Twelvefold symmetric quasicrystallography from the latticesF4,B6andE6." Acta Crystallographica Section A Foundations and Advances 70, no. 6 (September 26, 2014): 605–15. http://dx.doi.org/10.1107/s2053273314015812.
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One possible way to obtain the quasicrystallographic structure is the projection of the higher-dimensional lattice into two- or three-dimensional subspaces. Here a general technique applicable to any higher-dimensional lattice is introduced. The Coxeter number and the integers of the Coxeter exponents of a Coxeter–Weyl group play a crucial role in determining the plane onto which the lattice is to be projected. The quasicrystal structures display the dihedral symmetry of order twice that of the Coxeter number. The eigenvectors and the corresponding eigenvalues of the Cartan matrix are used to determine the set of orthonormal vectors inn-dimensional Euclidean space which lead to suitable choices for the projection subspaces. The maximal dihedral subgroup of the Coxeter–Weyl group is identified to determine the symmetry of the quasicrystal structure. Examples are given for 12-fold symmetric quasicrystal structures obtained by projecting the higher-dimensional lattices determined by the affine Coxeter–Weyl groupsWa(F4),Wa(B6) andWa(E6). These groups share the same Coxeter numberh= 12 with different Coxeter exponents. The dihedral subgroupD12of the Coxeter groups can be obtained by defining two generatorsR1andR2as the products of generators of the Coxeter–Weyl groups. The reflection generatorsR1andR2operate in the Coxeter planes where the Coxeter elementR1R2of the Coxeter–Weyl group represents the rotation of order 12. The canonical (strip, equivalently, cut-and-project technique) projections of the lattices determine the nature of the quasicrystallographic structures with 12-fold symmetry as well as the crystallographic structures with fourfold and sixfold symmetry. It is noted that the quasicrystal structures obtained from the latticesWa(F4) andWa(B6) are compatible with some experimental results.
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Taylor, William R. "Exploring Protein Fold Space." Biomolecules 10, no. 2 (January 27, 2020): 193. http://dx.doi.org/10.3390/biom10020193.
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The model of protein folding proposed by Ptitsyn and colleagues involves the accretion of secondary structures around a nucleus. As developed by Efimov, this model also provides a useful way to view the relationships among structures. Although somewhat eclipsed by later databases based on the pairwise comparison of structures, Efimov’s approach provides a guide for the more automatic comparison of proteins based on an encoding of their topology as a string. Being restricted to layers of secondary structures based on beta sheets, this too has limitations which are partly overcome by moving to a more generalised secondary structure lattice that can encompass both open and closed (barrel) sheets as well as helical packing of the type encoded by Murzin and Finkelstein on small polyhedra. Regular (crystalline) lattices, such as close-packed hexagonals, were found to be too limited so pseudo-latticses were investigated including those found in quasicrystals and the Bernal tetrahedron-based lattice that he used to represent liquid water. The Bernal lattice was considered best and used to generate model protein structures. These were much more numerous than those seen in Nature, posing the open question of why this might be.
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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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Haney, Christopher W., and Hector R. Siller. "Properties of Hyper-Elastic-Graded Triply Periodic Minimal Surfaces." Polymers 15, no. 23 (November 21, 2023): 4475. http://dx.doi.org/10.3390/polym15234475.
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The mechanical behaviors of three distinct lattice structures—Diamond, Gyroid, and Schwarz—synthesized through vat polymerization, were meticulously analyzed. This study aimed to elucidate the intricacies of these structures in terms of their stress–strain responses, energy absorption, and recovery characteristics. Utilizing the described experiments and analytical approaches, it was discerned, via the described experimental and analytical procedure, that the AM lattices showcased mechanical properties and stress–strain behaviors that notably surpassed theoretical predictions, pointing to substantial disparities between conventional models and experimental outcomes. The Diamond lattice displayed superior stiffness with higher average loading and unloading moduli and heightened energy absorption and dissipation rates, followed by the Gyroid and Schwarz lattices. The Schwarz lattice showed the most consistent mechanical response, while the Diamond and Gyroid showed capabilities of reaching larger strains and stresses. All uniaxial cyclic compressive tests were performed at room temperature with no dwell times. The efficacy of hyper-elastic-graded models significantly outperformed projections offered by traditional Ashby–Gibson models, emphasizing the need for more refined models to accurately delineate the behaviors of additively manufactured lattices in advanced engineering applications.
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Egan, Paul F., Nava Raj Khatri, Manasi Anil Parab, and Amit M. E. Arefin. "Mechanics of 3D-Printed Polymer Lattices with Varied Design and Processing Strategies." Polymers 14, no. 24 (December 16, 2022): 5515. http://dx.doi.org/10.3390/polym14245515.
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Emerging polymer 3D-printing technologies are enabling the design and fabrication of mechanically efficient lattice structures with intricate microscale structures. During fabrication, manufacturing inconsistencies can affect mechanical efficiency, thereby driving a need to investigate how design and processing strategies influence outcomes. Here, mechanical testing is conducted for 3D-printed lattice structures while altering topology, relative density, and exposure time per layer using digital light processing (DLP). Experiments compared a Cube topology with 800 µm beams and Body-Centered Cube (BCC) topologies with 500 or 800 µm beams, all designed with 40% relative density. Cube lattices had the lowest mean measured relative density of ~42%, while the 500 µm BCC lattice had the highest relative density of ~55%. Elastic modulus, yield strength, and ultimate strength had a positive correlation with measured relative density when considering measurement distributions for thirty samples of each design. BCC lattices designed with 50%, 40%, and 30% relative densities were then fabricated with exposure-per-layer times of 1500 and 1750 ms. Increasing exposure time per layer resulted in higher scaling of mechanical properties to relative density compared to design alteration strategies. These results reveal how design and fabrication strategies affect mechanical performance of lattices suitable for diverse engineering applications.
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Wood, Katherine M., and Corinne J. Smith. "Clathrin: the molecular shape shifter." Biochemical Journal 478, no. 16 (August 26, 2021): 3099–123. http://dx.doi.org/10.1042/bcj20200740.
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Clathrin is best known for its contribution to clathrin-mediated endocytosis yet it also participates to a diverse range of cellular functions. Key to this is clathrin's ability to assemble into polyhedral lattices that include curved football or basket shapes, flat lattices or even tubular structures. In this review, we discuss clathrin structure and coated vesicle formation, how clathrin is utilised within different cellular processes including synaptic vesicle recycling, hormone desensitisation, spermiogenesis, cell migration and mitosis, and how clathrin's remarkable ‘shapeshifting’ ability to form diverse lattice structures might contribute to its multiple cellular functions.
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Razmjooei, Nasrin, and Robert Magnusson. "Band Dynamics of Multimode Resonant Nanophotonic Lattices with Adjustable Liquid Interfaces." Nanomaterials 13, no. 16 (August 16, 2023): 2350. http://dx.doi.org/10.3390/nano13162350.
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Subwavelength resonant lattices offer a wide range of fascinating spectral phenomena under broadside illumination. The resonance mechanism relies on the generation of lateral Bloch modes that are phase matched to evanescent diffraction orders. The spectral properties and the total number of resonance states are governed by the structure of leaky modes and the mode count. This study investigates the effect of interface modifications on the band dynamics and bound-state transitions in guided-mode resonant lattices. We provide photonic lattices comprising rectangular Si3N4 rods with a liquid film with an adjustable boundary. The band structures and band flips are examined through numerical simulations using the rigorous coupled-wave analysis (RCWA) method and analyzing the zero-order spectral reflectance as a function of the incident angle. The band structures and band flips are examined through numerical simulations, and the influences of the refractive index and the thickness of the oil layer on the band dynamics are investigated. The results reveal distinct resonance linewidths corresponding to different refractive indices of the oil layer. Furthermore, the effect of the oil thickness on the band dynamics is explored, demonstrating precise control over the number of propagating modes within the lattice structure. Theoretical simulations and experimental results are presented for a subwavelength silicon-nitride lattice combined with a liquid film featuring an adjustable boundary. The presence of a relatively thick liquid waveguiding region enables the emergence of additional modes, including the first four transverse-electric (TE) leaky modes, which produce observable resonance signatures. Through experimental manipulation of the basic lattice’s duty cycle, the four bands undergo quantifiable band transitions and closures. The experimental results obtained within the 1400–1600 nm spectral range exhibit reasonable agreement with the numerical analysis. These findings underscore the significant role played by the interface in shaping the band dynamics of the lattice structure, providing valuable insights into the design and optimization of photonic lattices with adjustable interfaces.
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Фомин, Д. В. "APPLICABILITY OF THE COMPACT MATRIX DESCRIPTION METHOD TO ZEOLITE STRUCTURES." Южно-Сибирский научный вестник, no. 4(56) (August 31, 2024): 69–75. http://dx.doi.org/10.25699/sssb.2024.56.4.011.
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Описываются метод компактного векторно-матричного описания регулярных пространственных решёток, а также его модификации. Рассматриваются их сильные и слабые стороны. Отмечается наличие эффективных методов расчёта ряда структурных и энергетических параметров решёток, основанных на использовании компактных описаний в качестве исходных данных. Рассмотрена решётка цеолита кубического типа ACO. Выделен её базовый фрагмент. Выполнен расчёт целочисленных координат частиц этого фрагмента. Показана принципиальная возможность применения модифицированной версии метода компактного матричного описания к сложным пространственным решёткам на примере цеолита ACO. The method of compact vector-matrix description of regular spatial lattices is described, as well as its modification. Their advantages and disadvantages are considered. It is noted that there are effective methods for calculating a number of structural and energy parameters of lattices based on the use of compact descriptions as initial data. A cubic type ACO zeolite lattice is considered. Its basic fragment is selected. The integer coordinates of the particles of this fragment are calculated. The principal possibility of applying a modified version of the compact matrix method to complex spatial lattices using as the example ACOzeolite is shown.
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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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ADARICHEVA, KIRA, and J. B. NATION. "LATTICES OF QUASI-EQUATIONAL THEORIES AS CONGRUENCE LATTICES OF SEMILATTICES WITH OPERATORS: PART I." International Journal of Algebra and Computation 22, no. 07 (November 2012): 1250065. http://dx.doi.org/10.1142/s0218196712500658.
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We show that for every quasivariety 𝒦 of structures (where both functions and relations are allowed) there is a semilattice S with operators such that the lattice of quasi-equational theories of 𝒦 (the dual of the lattice of sub-quasivarieties of 𝒦) is isomorphic to Con(S, +, 0, . As a consequence, new restrictions on the natural quasi-interior operator on lattices of quasi-equational theories are found.
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López, Antonio Fernández, and Eulalia García Rus. "Algebraic lattices and nonassociative structures." Proceedings of the American Mathematical Society 126, no. 11 (1998): 3211–21. http://dx.doi.org/10.1090/s0002-9939-98-04443-8.
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Kalmár, I. G. "$\ast$-structures and orthomodular lattices." Publicationes Mathematicae Debrecen 32, no. 1-2 (July 1, 2022): 1–5. http://dx.doi.org/10.5486/pmd.1985.32.1-2.01.
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Amritkar, R. E. "Structures in coupled map lattices." Physica A: Statistical Mechanics and its Applications 224, no. 1-2 (February 1996): 382–89. http://dx.doi.org/10.1016/0378-4371(95)00351-7.
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Gopal, E. S. R., and S. Baranidharan. "Average lattices and aperiodic structures." Bulletin of Materials Science 17, no. 6 (November 1994): 783–94. http://dx.doi.org/10.1007/bf02757557.
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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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Çaylı, Gül Deniz. "Constructing Uninorms Based on Closure and Interior Operators in Bounded Lattices." Afyon Kocatepe University Journal of Sciences and Engineering 24, no. 6 (December 2, 2024): 1323–32. https://doi.org/10.35414/akufemubid.1439061.
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Uninorms generalizing triangular norms and triangular conorms on bounded lattices have attracted considerable attention recently. In this article, two new approaches are suggested to generate uninorms with an identity element on a bounded lattice. These approaches exploit the existences of a triangular norm (triangular conorm) and a closure operator (interior operator) on a bounded lattice. Meanwhile, two structures of idempotent uninorms on bounded lattices are obtained. In addition, the relationship between the proposed approaches and the existing constructions is investigated.