Regular Lattice Research Articles

Impact of Surface Finishing on Ti6Al4V Voronoi Additively Manufactured Structures: Morphology, Dimensional Deviation, and Mechanical Behavior

Additively manufactured medical devices require proper surface finishing before their use to remove partially adhered particles and provide adequate surface roughness. The literature widely investigates regular lattice structures—mainly scaffolds with small pores to enhance osseointegration; however, only a few studies have addressed the impact of surface finishing on the dimensional deviation and the global and local mechanical responses of lattice samples. Therefore, the current research investigates the impact of biomedical surface finishing (i.e., corundum sandblasting and zirconia sandblasting) on Voronoi lattice structures produced by laser powder bed fusion (LPBF) with large pores and different thicknesses on the surface morphology and global and local mechanical behaviors. MicroCT and SEM are performed for the assessment of dimensional mismatch and surface evaluation. The mechanical properties are investigated with 2D digital image correlation (DIC) in quasi-static compression tests to estimate the impact of surface finishes on local maps of strain. In the quasi-static tests, both the global mechanical performances, as expected, and local 2D DIC strain maps were mainly affected by the strut thickness, and the impact of different surface finishings was irrelevant; on the contrary, different surface finishing processes led to differences in the dimensional deviation depending on the strut thickness. These results are relevant for designing lattice structures with thin struts that are integrated into medical prostheses that undergo AM.

Evolution of cooperation with asymmetric rewards

Rewards, as a form of positive reinforcement, effectively encourage cooperation. In this paper, we study a multi-population prisoner's dilemma game with asymmetric rewards, where agents in the same population play prisoner's dilemma game, and agents from the giver population can reward agents from the recipient population only if they make the same choice. In well-mixed populations, asymmetric rewards can facilitate cooperation. Similarly, asymmetric rewards on the regular square lattice can effectively prevent complete defection. In both well-mixed and structured populations, seemingly disadvantageous cooperative givers play an important role in the maintenance and spread of cooperation. Especially on lattice, cooperative givers with asymmetric rewards can be active in the system through diverse cases of cyclic dominance. Our findings provide deeper insights into the impact of asymmetry on cooperative behavior and the development of altruistic behavior in real-world scenarios.

Pore-scale study of coupled heat and mass transfer during primary freeze-drying using an irregular pore network model

This study presents a pore network (PN) model with transient heat transfer and quasi-steady transitional vapor transport that is for the first time applied to irregular porous structures that are obtained by reconstruction of X-ray tomography image data. In contrast to previous studies, the irregular pores are not approximated by spheres but implemented in their original shape. Secondly, instead of assuming cylindrical throats as pore connections, the actual distance between pore centers as well as the pore cross sections are used for the computation of the vapor transport coefficient. The control volume elements of the computational model are matched with the cells obtained by Voronoi tessellation. The improvements have clear advantages over former approaches where the reconstructed void space is usually strongly simplified by balls and sticks confined in a regular lattice structure. A freeze-dried sample of maltodextrin DE12 with 20% (w/w) solid content is used for benchmarking the new methodology. Its morphological and thermal properties are determined by the novel PN model. The simulation results of primary freeze-drying (FD) are compared to reference cases in two ways. First, the differences in heat and mass transfer kinetics as compared to regular PNs are emphasized. Secondly, the PN simulation results are confronted with a simple literature model that neglects pore size distribution (PSD) and transient heat transfer. It is shown that already in small domains with relatively narrow PSD, the variation of the mass transfer coefficient affects the computed sublimation fluxes and yields a significant deviation from the simpler literature model. Moreover, it is revealed in this study that the structure has a significant impact on the sublimation front temperature. This is demonstrated by the comparison of FD in the irregular PN to a regular PN with almost identical PSD but different porosity. The development, verification, and benchmarking of the new PN model can be seen as an important step for studies of the structure dependence of FD.

Laplacian operator and its square lattice discretization: Green function vs. Lattice Green function for the flat 2-torus and other related 2D manifolds

Abstract The paper investigates the truncation error between the Green function and the lattice Green function (LGF) for the Laplacian operator defined on the 2-torus and its discretization on a regular square lattice. Extensions to the cylinder and the rectangular domain with free (or Neumann) boundary conditions are also proposed. In each of these instances, the Green function and its discrete analog are given in exact analytical closed-form allowing to infer accurate estimates as the lattice spacing tends to zero. As expected, it is shown that the continuum limit of the LGF coincides well with the Green function in every case. In particular, the issue of logarithmic singularity regularization of the Green function by the lattice discretization is addressed through two related application examples regarding the rectangular domain, and devoted to the computation of corner-to-corner resistance of an electrical conducting square and the mean first-passage time between the diagonally opposite vertices of a square for a standard Brownian motion, both derived considering the continuum limit.

Development and Analysis of novel Integrable Nonlinear Dynamical Systems on Quasi-One-Dimensional Lattices. Parametrically Driven Nonlinear System of Pseudo-Excitations on a Two-Leg Ladder Lattice

Following the main principles of developing the evolutionary nonlinear integrable systems on quasi-one-dimensional lattices, we suggest a novel nonlinear integrable system of parametrically driven pseudo-excitations on a regular two-leg ladder lattice. The initial (prototype) form of the system is derivable in the framework of semi-discrete zero-curvature equation with the spectral and evolution operators specified by the properly organized 3 × 3 square matrices. Although the lowest conserved local densities found via the direct recursive method do not prompt us the algebraic structure of system’s Hamiltonian function, but the heuristically substantiated search for the suitable two-stage transformation of prototype field functions to the physically motivated ones has allowed to disclose the physically meaningful nonlinear integrable system with time-dependent longitudinal and transverse inter-site coupling parameters. The time dependencies of inter-site coupling parameters in the transformed system are consistently defined in terms of the accompanying parametric driver formalized by the set of four homogeneous ordinary linear differential equations with the time-dependent coefficients. The physically meaningful parametrically driven nonlinear system permits its concise Hamiltonian formulation with the two pairs of field functions serving as the two pairs of canonically conjugated field amplitudes. The explicit example of oscillatory parametric drive is described in full mathematical details.

Measurement of seasonal variations of the NSLS-II circumference and energy

Routine beam-based characterization of the NSLS-II storage ring indicated probable seasonal variations of the ring circumference and energy. We arranged regular direct measurements of the ring circumference during 2023 and compared the results with the archived data of the magnet and undulator settings. A clear correlation between the ring circumference and energy was observed. These seasonal variations of the ring energy are found to be the main reason for regular lattice corrections required to keep the machine performance stable for user operations. The measurement techniques and results are discussed in this article.

Percolation in a three-dimensional nonsymmetric multicolor loop model.

We conduct Monte Carlo simulations to analyze the percolation transition of a nonsymmetric loop model on a regular three-dimensional lattice. We calculate the critical exponents for the percolation transition of this model. The percolation transition occurs at a temperature that is close to, but not exactly, the thermal critical temperature. Our finite-size study on this model yields a correlation length exponent that agrees with that of the three-dimensional XY model with an error margin of 6%.

Fitness noise in the Bak-Sneppen evolution model in high dimensions.

We study the Bak-Sneppen evolution model on a regular hypercubic lattice in high dimensions. Recent work [Phys. Rev. E 108, 044109 (2023)2470-004510.1103/PhysRevE.108.044109] showed the emergence of the 1/f^{α} noise for the fitness observable with α≈1.2 in one-dimension (1D) and α≈2 for the random neighbor (mean-field) version of the model. We examine the temporal correlation of fitness in 2, 3, 4, and 5 dimensions. As obtained by finite-size scaling, the spectral exponent tends to take the mean-field value at the upper critical dimension D_{u}=4, which is consistent with previous studies. Our approach provides an alternative way to understand the upper critical dimension of the model. We also show the local activity power spectra, which offer insight into the return time statistics and the avalanche dimension.

Validity of an approximate electrostatic interaction model

We explain the reasons why a simple approximate expression for the interaction energy between two parallel uniformly charged wires works so well for the calculation of the stored electrostatic energy of a uniformly charged square and/or rectangular plate. The expression under consideration was also found to be very accurate when calculating the total energy of a model consisting of an arbitrary number of uniformly charged wires placed parallel to each other along a one-dimensional regular lattice when the number of wires is not very small. One can use the derived approximate expression for the interaction energy instead of the more elaborate exact one in all circumstances where there is no need for an extremely high accuracy.

HYPERTILING — a high performance Python library for the generation and visualization of hyperbolic lattices

HYPERTILING is a high-performance Python library for the generation and visualization of regular hyperbolic lattices embedded in the Poincaré disk model. Using highly optimized, efficient algorithms, hyperbolic tilings with millions of vertices can be created in a matter of minutes on a single workstation computer. Facilities including computation of adjacent vertices, dynamic lattice manipulation, refinements, as well as powerful plotting and animation capabilities are provided to support advanced uses of hyperbolic graphs. In this manuscript, we present a comprehensive exploration of the package, encompassing its mathematical foundations, usage examples, applications, and a detailed description of its implementation.

Codebase release 1.3 for HYPERTILING

HYPERTILING is a high-performance Python library for the generation and visualization of regular hyperbolic lattices embedded in the Poincaré disk model. Using highly optimized, efficient algorithms, hyperbolic tilings with millions of vertices can be created in a matter of minutes on a single workstation computer. Facilities including computation of adjacent vertices, dynamic lattice manipulation, refinements, as well as powerful plotting and animation capabilities are provided to support advanced uses of hyperbolic graphs. In this manuscript, we present a comprehensive exploration of the package, encompassing its mathematical foundations, usage examples, applications, and a detailed description of its implementation.

Forecasting Lattice and Point Spatial Data: Comparison of Unilateral and Multilateral SAR Models

Spatial auto-regressive (SAR) models are widely used in geosciences for data analysis; their main feature is the presence of weight (W) matrices, which define the neighboring relationships between the spatial units. The statistical properties of parameter and forecast estimates strongly depend on the structure of such matrices. The least squares (LS) method is the most flexible and can estimate systems of large dimensions; however, it is biased in the presence of multilateral (sparse) matrices. Instead, the unilateral specification of SAR models provides triangular weight matrices that allow consistent LS estimates and sequential prediction functions. These two properties are strictly related and depend on the linear and recursive nature of the system. In this paper, we show the better performance in out-of-sample forecasting of unilateral SAR (estimated with LS), compared to multilateral SAR (estimated with maximum likelihood, ML). This conclusion is supported by numerical simulations and applications to real geological data, both on regular lattices and irregularly distributed points.

Grid Cells in Cognition: Mechanisms and Function.

The activity patterns of grid cells form distinctively regular triangular lattices over the explored spatial environment and are largely invariant to visual stimuli, animal movement, and environment geometry. These neurons present numerous fascinating challenges to the curious (neuro)scientist: What are the circuit mechanisms responsible for creating spatially periodic activity patterns from the monotonic input-output responses of single neurons? How and why does the brain encode a local, nonperiodic variable-the allocentric position of the animal-with a periodic, nonlocal code? And, are grid cells truly specialized for spatial computations? Otherwise, what is their role in general cognition more broadly? We review efforts in uncovering the mechanisms and functional properties of grid cells, highlighting recent progress in the experimental validation of mechanistic grid cell models, and discuss the coding properties and functional advantages of the grid code as suggested by continuous attractor network models of grid cells.

Disclosing magnetic clusters in the metallic half-Heusler ferromagnet Cr4PtGa17 with a breathing pyrochlore lattice

We report a ferromagnetic resonance (FMR) study of a single crystal of Cr4PtGa17, the first material of a half-Heusler (HH) type with a metallic ferromagnetic breathing pyrochlore structure. The spatial distribution of the magnetic free energy density derived from the FMR data appears to be of a cubic symmetry despite the lower symmetry coordination of magnetic Cr ions. This surprising finding strongly suggests that the Cr4Ga14 structural unit at the Y site of the HH general XYZ formula forms an octahedrally shaped integral magnetic cluster, likely due to strong exchange interaction between the Cr ions via covalent Cr-Ga-Cr bonds. The presence of magnetic clusters within a regular metallic lattice appears to be a new intriguing characteristics of these novel types of HH compounds whereas Cr4PtGa17, at the same time, constitutes the rare example of such metallic pyrochlore-type material.

Anomalous quantum transport in fractal lattices

Fractal lattices are self-similar structures with repeated patterns on different scales. Quantum transport through such structures is subtle due to the possible co-existence of localized and extended states. Here, we study the dynamical properties of two fractal lattices, the Sierpiński gasket and the Sierpiński carpet. While the gasket exhibits sub-diffusive behavior, sub-ballistic transport occurs in the carpet. We show that the different dynamical behavior is in line with qualitative differences of the systems’ spectral properties. Specifically, in contrast to the Sierpiński carpet, the Sierpiński gasket exhibits an inverse power-law behavior of the level spacing distribution. As a possible technological application, we discuss a memory effect in the Sierpiński gasket which allows to read off the phase information of an initial state from the spatial distribution after long evolution times. We also show that interpolating between fractal and regular lattices allows for flexible tuning between different transport regimes.

Cut‐Cell Microstructures for Two‐scale Structural Optimization

AbstractTwo‐scale topology optimization, combined with the design of microstructure families with a broad range of effective material parameters, is widely used in many fabrication applications to achieve a target deformation behavior for a variety of objects. The main idea of this approach is to optimize the distribution of material properties in the object partitioned into relatively coarse cells, and then replace each cell with microstructure geometry that mimics these material properties. In this paper, we focus on adapting this approach to complex shapes in situations when preserving the shape's surface is essential.Our approach extends any regular (i.e. defined on a regular lattice grid) microstructure family to complex shapes, by enriching it with tiles adapted to the geometry of the cut‐cell. We propose a fully automated and robust pipeline based on this approach, and we show that the performance of the regular microstructure family is only minimally affected by our extension while allowing its use on 2D and 3D shapes of high complexity.

Role of cilia activity and surrounding viscous fluid in properties of metachronal waves.

Large groups of active cilia collectively beat in a fluid medium as metachronal waves, essential for some microorganisms motility and for flow generation in mucociliary clearance. Several models can predict the emergence of metachronal waves, but what controls the properties of metachronal waves is still unclear. Here, we numerically investigate the respective impacts of active beating and viscous dissipation on the properties of metachronal waves in a collection of oscillators, using a simple model for cilia in the presence of noise on regular lattices in one and two dimensions. We characterize the wave using spatial correlation and the frequency of collective beating. Our results clearly show that the viscosity of the fluid medium does not affect the wavelength; the activity of the cilia does. These numerical results are supported by a dimensional analysis, which shows that the result of wavelength invariance is robust against the model taken for sustained beating and the structure of hydrodynamic coupling. Interestingly, the enhancement of cilia activity increases the wavelength and decreases the beating frequency, keeping the wave velocity almost unchanged. These results might have significance in understanding paramecium locomotion and mucociliary clearance diseases.

Preferential selection based on aspiration and memory in spatial evolutionary prisoner’s dilemma game

When studying the spontaneous emergence of cooperation within the framework of evolutionary game theory, the process of strategy selection plays a crucial role in evolutionary dynamics. In most previous studies, it has been assumed that individuals randomly select learning targets from their neighbors. However, by considering the heterogeneity of individuals’ influence in real-life social networks, a preference for more realistic choices should be taken into account. Here, we propose a preferential selection mechanism based on individuals’ payoff aspiration in memory-based evolutionary game on square lattice networks. For each iteration, players in the regular lattices will prioritize selecting neighbors with the highest satisfaction rate over the most recent runs. The simulation results demonstrate that both the level of aspiration and the length of memory can elevate the level of cooperation to a certain extent. For the moderate aspiration level, when the value of betrayal temptation is less than threshold, cooperation level monotonously decreases with the increment of memory length. However, when betrayal temptation is above threshold, memory mechanism can facilitates the evolution of cooperation, while the absence of such a mechanism leads collaborators toward extinction.

Deformation mechanics of generalized missing rib chiral lattice structures

The planar missing rib lattice topology is generalized to construct a family of regular and irregular tetra-chiral periodic lattices. The geometry of a lattice in the family is completely defined by a set of six parameters. Using a unified energy based approach, the constitutive model and all effective elastic properties of the resulting lattice structures are determined from the unit periodic element of the lattice. Straightforward analytical expressions of the effective properties are obtained for some special geometries, which for the general case may be lengthy. In the latter case, the compliance tensor can easily be obtained numerically. The proposed approach involves accurate determination of the strain tensor and the compliance tensor for which a minimum error norm method is used. The results from the analytical expressions are compared with those obtained from Finite Element Analysis (FEA), and a very good match is observed. To draw a parity with experimental determination of the elastic properties and address the observation of auxeticity (negative Poisson’s ratio) in such lattice structures, the concept of empirical Young’s modulus and Poisson’s ratio is introduced. Using this idea, for the conventional chiral lattice, some disagreement in the existing literature is resolved.

Combined optimization of spot positions and weights for better FLASH proton therapy

Objective. In Intensity Modulated Proton Therapy (IMPT), the weights of individual pencil-beams or spots are optimized to fulfil dosimetric constraints. Theses spots are usually located on a regular lattice and their positions are fixed during optimization. In many cases, the range of spot weights may however be limited, leading sometimes to sub-optimal plan quality. An emblematic use case is the delivery of a plan at ultra-high dose rate (FLASH-RT), for which the spot weights are typically constrained to high values. Approach. To improve further the quality of IMPT FLASH plans, we propose here a novel algorithm to optimize both the spot weights and positions directly based on the objectives defined by the treatment planner. Main results. For all cases considered, optimizing the spot positions lead to an enhanced dosimetric score, while maintaining a high dose rate. Significance. Overall, this approach resulted in a substantial plan quality improvement compared to optimizing only the spot weights, and in a similar execution time.