Infinite Lattice Research Articles

Multiscale Modelling and Mechanical Anisotropy of Periodic Cellular Solids with Rigid-Jointed Truss-Like Microscopic Architecture

This paper investigates the macroscopic anisotropic behavior of periodic cellular solids with rigid-jointed microscopic truss-like architecture. A theoretical matrix-based procedure is presented to calculate the homogenized stiffness and strength properties of the material which is validated experimentally. The procedure consists of four main steps, namely, (i) using classical structural analysis to determine the stiffness properties of a lattice unit cell, (ii) employing the Bloch’s theorem to generate the irreducible representation of the infinite lattice, (iii) resorting to the Cauchy–Born Hypothesis to express the microscopic nodal forces and deformations in terms of a homogeneous macroscopic strain field applied to the lattice, and (iv) employing the Hill–Mandel homogenization principle to obtain the macro-stiffness properties of the lattice topologies. The presented model is used to investigate the anisotropic mechanical behavior of 13 2D periodic cellular solids. The results are documented in three set of charts that show (i) the change of the Young and Shear moduli of the material with respect to their relative density; (ii) the contribution of the bending stiffness of microscopic cell elements to the homogenized macroscopic stiffness of the material; and (iii) polar diagrams of the change of the elastic moduli of the cellular solid in response to direction of macroscopic loading. The three set of charts can be used for design purposes in assemblies involving the honeycomb structures as it may help in selecting the best lattice topology for a given functional stiffness and strength requirement. The theoretical model was experimentally validated by means of tensile tests performed in additively manufactured Lattice Material (LM) specimens, achieving good agreement between the results. It was observed that the model of rigid-joined LM (RJLM) predicts the homogenized mechanical properties of the LM with higher accuracy compared to those predicted by pin-jointed models.

A dynamic homogenization method for 3D core calculations

Three-dimensional deterministic core calculations are typically based on the classical two-step calculation scheme. The sizable number of approximations introduced in this approach can easily reduce the accuracy of the low-order operator solution, especially in high heterogeneous configurations. Several techniques have been developed in order to overcome these limitations, including direct approaches that make extensive use of computational resources, such as the 2D/1D Fusion method and direct 3D transport calculations. In this work we propose the method of Dynamic Homogenization as an alternative technique for 3D core calculations that lies between the classical and the direct approaches in terms of accuracy and performance. The methodology consists in an iterative process between core and assembly calculations that provides a set of homogenization parameters for the low-order operator, taking into account the actual environment of each assembly in the core. This is mediated defining a set of 2D reference homogenization problems that are solved with imposed core eigenvalue, fixed incoming boundary source and a simplified axial leakage expressed as a volumetric source. The methodology eliminates the infinite lattice approximation and the critical leakage model, and offers the advantage of avoiding off-line calculations, cross-section interpolations, power reconstruction techniques and expensive 3D transport calculations. The NEA “PWR MOX/UO2 Core Transient Benchmark” was chosen for our analysis where we compare the accuracy and the performance of the two-step and the Dynamic Homogenization calculations against the direct 3D transport solution. We analyzed two different core configurations: an axially uniform core and a partially rodded core.

Condensation of SIP Particles and Sticky Brownian Motion

We study the symmetric inclusion process (SIP) in the condensation regime. We obtain an explicit scaling for the variance of the density field in this regime, when initially started from a homogeneous product measure. This provides relevant new information on the coarsening dynamics of condensing interacting particle systems on the infinite lattice. We obtain our result by proving convergence to sticky Brownian motion for the difference of positions of two SIP particles in the sense of Mosco convergence of Dirichlet forms. Our approach implies the convergence of the probabilities of two SIP particles to be together at time t. This, combined with self-duality, allows us to obtain the explicit scaling for the variance of the fluctuation field.

Entropy Scaling Law and the Quantum Marginal Problem

Quantum many-body states that frequently appear in physics often obey an entropy scaling law, meaning that an entanglement entropy of a subsystem can be expressed as a sum of terms that scale linearly with its volume and area, plus a correction term that is independent of its size. We conjecture that these states have an efficient dual description in terms of a set of marginal density matrices on bounded regions, obeying the same entropy scaling law locally. We prove a restricted version of this conjecture for translationally invariant systems in two spatial dimensions. Specifically, we prove that a translationally invariant marginal obeying three non-linear constraints -- all of which follow from the entropy scaling law straightforwardly -- must be consistent with some global state on an infinite lattice. Moreover, we derive a closed-form expression for the maximum entropy density compatible with those marginals, deriving a variational upper bound on the thermodynamic free energy. Our construction's main assumptions are satisfied exactly by solvable models of topological order and approximately by finite-temperature Gibbs states of certain quantum spin Hamiltonians.

Near-Field Excitation of Bound States in the Continuum in All-Dielectric Metasurfaces through a Coupled Electric/Magnetic Dipole Model.

Resonant optical modes arising in all-dielectric metasurfaces have attracted much attention in recent years, especially when so-called bound states in the continuum (BICs) with diverging lifetimes are supported. With the aim of studying theoretically the emergence of BICs, we extend a coupled electric and magnetic dipole analytical formulation to deal with the proper metasurface Green function for the infinite lattice. Thereby, we show how to excite metasurface BICs, being able to address their near-field pattern through point-source excitation and their local density of states. We apply this formulation to fully characterize symmetry-protected BICs arising in all-dielectric metasurfaces made of Si nanospheres, revealing their near-field pattern and local density of states, and, thus, the mechanisms precluding their radiation into the continuum. This formulation provides, in turn, an insightful and fast tool to characterize BICs (and any other leaky/guided mode) near fields in all-dielectric (and also plasmonic) metasurfaces, which might be especially useful for the design of planar nanophotonic devices based on such resonant modes.

Pullback Exponential Attractors for Non-autonomous Recurrent Neural Networks with Discrete and Distributed Time-Varying Delays

First some sufficient conditions are presented for the existence and the construction of pullback exponential attractors for infinite dimensional non-autonomous dynamical systems with delays. This abstract result is then used to establish the existence of pullback exponential attractors for an infinite lattice model of non-autonomous recurrent neural networks with discrete and distributed time-varying delays.

Numerical simulation of quantum nonequilibrium phase transitions without finite-size effects

Classical $(1+1)D$ cellular automata, as for instance Domany-Kinzel cellular automata, are paradigmatic systems for the study of non-equilibrium phenomena. Such systems evolve in discrete time-steps, and are thus free of time-discretisation errors. Moreover, information about critical phenomena can be obtained by simulating the evolution of an initial seed that, at any finite time, has support only on a finite light-cone. This allows for essentially numerically exact simulations, free of finite-size errors or boundary effects. Here, we show how similar advantages can be gained in the quantum regime: The many-body critical dynamics occurring in $(1+1)D$ quantum cellular automata with an absorbing state can be studied directly on an infinite lattice when starting from seed initial conditions. This can be achieved efficiently by simulating the dynamics of an associated one-dimensional, non-unitary quantum cellular automaton using tensor networks. We apply our method to a model introduced recently and find accurate values for universal exponents, suggesting that this approach can be a powerful tool for precisely classifying non-equilibrium universal physics in quantum systems.

Statistical solutions and piecewise Liouville theorem for the impulsive reaction-diffusion equations on infinite lattices

We first verify the global well-posedness of the impulsive reaction-diffusion equations on infinite lattices. Then we establish that the generated process by the solution operators has a pullback attractor and a family of Borel invariant probability measures. Furthermore, we formulate the definition of statistical solution for the addressed impulsive system and prove the existence. Our results show that the statistical solution of the impulsive system satisfies merely the Liouville type theorem piecewise, and the Liouville type equation for impulsive system will not always hold true on the interval containing any impulsive point.

Revisiting the discrete planar Laplacian: exact results for the lattice Green function and continuum limit

The paper deals with the discrete Laplacian on a uniform infinite square lattice. The definition of its fundamental solution or lattice Green function (LGF) is clarified as the Fourier coefficients of a certain generalized periodic function g. Such a functional must be regularized and gives the LGF up to a constant equal to \({<}g{>}\), the mean value of g. For \({<}g{>}=0\), the LGF may be expressed in an exact analytic form in terms of hypergeometric and gamma functions. The continuum limit of the LGF is finally studied requiring an appropriate renormalization of \({<}g{>}\) in order to obtain the logarithmic Coulomb potential.

Bandgap characteristics and wave attenuation of metamaterials based on negative-stiffness dynamic vibration absorbers

Metamaterials with ultra-low frequency and broad bandgaps have seldom been reported. In this paper, we propose a metamaterial whose unit cell consists of a dynamic vibration absorber with negative stiffness (NSDVA). Firstly, the band structure, effective indices and stability of the metamaterial were investigated considering an infinite lattice. Unlike traditional metamaterials, the proposed metamaterial exhibits a quasi-static bandgap and negative effective stiffness due to the presence of a negative-stiffness element. The stable condition was defined by the critical value of the negative stiffness, which was derived analytically through the closed solution of the bandgap’s start frequency. Parametric studies on the quasi-static bandgap were conducted. It was found that, in the stable condition, the stopband becomes wider when the stiffness ratio increases and the mass ratio decreases. Furthermore, transmission and transient analyses were conducted considering a finite chain. The results of the parametric studies on the transmission were consistent with the relevant predictions on the bandgap. A harmonic transient analysis validated the capability of wave attenuation, and the wavenumber-frequency spectrum of the output signal excited by white noise further verified the band structure. Finally, the proposed metamaterial was used as a seismic metabarrier consisting of built-up steel barriers, resonators, RC retaining walls, and wire ropes. The investigation on the seismic performance of the metabarrier revealed its applicability in urban regional earthquake resistance.

Gaussian processes for surrogate modeling of discharged fuel nuclide compositions

Several applications such as nuclear forensics, nuclear fuel cycle simulations and sensitivity analysis require methods to quickly compute spent fuel nuclide compositions for various irradiation histories. Traditionally, this has been done by interpolating between one-group cross-sections that have been pre-computed from nuclear reactor simulations for a grid of input parameters, using fits such as Cubic Spline. We propose the use of Gaussian Processes (GP) to create surrogate models, which not only provide nuclide compositions, but also the gradient and estimates of their prediction uncertainty. The former is useful for applications such as forward and inverse optimization problems, the latter for uncertainty quantification applications. For this purpose, we compare GP-based surrogate model performance with Cubic-Spline-based interpolators based on infinite lattice simulations of a CANDU 6 nuclear reactor using the SERPENT 2 code, considering burnup and temperature as single dimensional input parameters resulting in a 2D study. Additionally, we compare the performance of uniform grid sampling to quasirandom sampling based on the Sobol sequence. We find that GP-based models perform significantly better in predicting spent fuel compositions than Cubic-Spline-based models on both sampling schemes. Uniform grid sampling performs better than quasirandom sampling in our 2D study, this cannot be generalized to higher dimensions. We have found that GP models require more time to compute a prediction. Although. A. AA a relatively small time (O ms), iterative methods used for statistical inference benefit from as-low-as-possible calculation times. Furthermore, we show that the predicted nuclide uncertainties are reasonably accurate. While in the studied two-dimensional case, grid- and quasirandom sampling provide similar results, quasirandom sampling will be a more effective strategy in higher-dimensional cases.

The lattice of algebraic closure operators on an infinite subgroup lattice

We say a lattice L is a subgroup lattice if there exists a group G such that where is the lattice of subgroups of G, ordered by inclusion. We prove that the lattice of algebraic closure operators which act on the subgroup lattice of an infinite group is itself a subgroup lattice if and only if the group is isomorphic to the Prüfer p-group.

Sensing and sensitivity: Computational chemistry of graphene‐based sensors

AbstractHighly efficient, tunable, biocompatible, and environmentally friendly electrochemical sensors featuring graphene‐based materials pose a formidable challenge for computational chemistry. In silico rationalization, optimization and, ultimately, prediction of their performance requires exploring a vast structural space of potential surface‐analyte complexes, further complicated by the presence of various defects and functionalities within the infinite graphene lattice. This immense number of systems and their periodic nature greatly limit the choice of computational tools applicable at a reasonable cost. An alternative approach using finite nanoflake models opens the doors to many more advanced and accurate electronic structure methods, while sacrificing the realism of representation. Locating the surface‐analyte complex is followed by an in‐depth in silico analysis of its energetic and electronic properties using, for example, energy decomposition schemes, as well as simulation of the signal, for example, a zero‐bias transmission spectra or a current–voltage curve, by means of the nonequilibrium Green's function method. These and other properties are examined in the context of a sensor's selectivity, sensitivity, and limit of detection with an aim to establish design principles for future devices. Herein, we analyze the advantages and limitations of diverse computational chemistry methods used at each of these steps in simulating graphene‐based electrochemical sensors. We present outstanding challenges toward predictive models and sketch possible solutions involving such contemporary techniques as multiscale simulations and high‐throughput screening.This article is categorized under: Structure and Mechanism &gt; Computational Materials Science Electronic Structure Theory &gt; Density Functional Theory Electronic Structure Theory &gt; Ab Initio Electronic Structure Methods

On a Vertex-Edge Marking Game on Graphs

The study of a variation of the marking game, in which the first player marks vertices and the second player marks edges of an undirected graph was proposed by Bartnicki et al. (Electron J Combin 15:R72, 2008). In this game, the goal of the second player is to mark as many edges around an unmarked vertex as possible, while the first player wants just the opposite. In this paper, we prove various bounds for the corresponding graph invariant, the vertex-edge coloring number \({\text {col}}_\mathrm{ve}(G)\) of a graph G. In particular, every (finite or infinite) graph G whose edges can be oriented in such a way that the maximum out-degree is bounded by an integer d has \({\text {col}}_\mathrm{ve}(G)\le d+2\). We investigate this invariant in (classes of) planar graphs, including some infinite lattices. We present a close connection between the vertex-edge coloring number of a graph G and the game coloring number of the subdivision graph S(G). In our main result, we bound the vertex-edge coloring number in complete graphs from below and from above, and while \({\text {col}}_\mathrm{ve}(K_n)\le \lceil \log _2{n}\rceil +2\), the difference between the upper and the lower bound is roughly \(\log _2(\log _2 n)\). The latter results are, in fact, true for any multigraph whose underlying graph is \(K_n\).

Random attractor for second-order stochastic delay lattice sine-Gordon equation

In this paper, we prove the existence of random mathcal{D}-attractor for the second-order stochastic delay sine-Gordon equation on infinite lattice with certain dissipative conditions, and then establish the upper bound of Kolmogorov ε-entropy for the random mathcal{D}-attractor.

Homoclinic solutions for perturbed discrete Schrödinger systems in infinite lattices: Sublinear and asymptotically linear cases

When the nonlinearities are sublinear or asymptotically linear at infinity, we obtain the existence of negative energy solutions for a class of non-periodic discrete Schrödinger systems with perturbed terms. Besides, for the asymptotically linear case, we obtain the existence of positive energy solutions. We also illustrate our results by giving several examples.

Counting walks by their last erased self-avoiding polygons using sieves

Let G be an infinite, vertex-transitive lattice with degree λ and fix a vertex on it. Consider all cycles of length exactly l from this vertex to itself on G. Erasing loops chronologically from these cycles, what is the fraction Fp∕λℓ(p) of cycles of length l whose last erased loop is some chosen self-avoiding polygon p of length ℓ(p), when l→∞ ? We use combinatorial sieves to prove an exact formula for Fp∕λℓ(p) that we evaluate explicitly. We further prove that for all self-avoiding polygons p, Fp∈Q[χ] with χ an irrational number depending on the lattice, e.g. χ=1∕π on the infinite square lattice. In stark contrast we current methods, we proceed via purely deterministic arguments relying on Viennot’s theory of heaps of pieces seen as a semi-commutative extension of number theory. Our approach also sheds light on the origin of the difference between exponents stemming from loop-erased walk and self-avoiding polygon models, and suggests a natural route to bridge the gap between both.

CORE DESIGN OF BREED &amp; BURN MOLTEN SALT FAST REACTOR

Previous designs of once-through solid-fuelled breed-and-burn (B&amp;B) reactor and the conventional molten salt reactor (MSR) concepts suffer from material limitation of neutron irradiation damage and chemical corrosion. A novel breed-and-burn molten salt reactor (BBMSR) concept uses separate molten salt fuel and coolant in a linear assembly core configuration. Similar to Moltex Energy Stable Salt Reactor (SSR) design, the configuration with fuel salt contained in fuel tubes and coolant salt in pool type reactor vessel has been previously studied. The study confirmed that breed-and-burn operation is feasible in principle, however with a low neutronic margin. The objective of this paper was to seek improvements of the neutronic margin with a metallic natural uranium blanket design. A parametric study was performed for the natural uranium blanket design. BBMSR neutronic performance simulation was modelled using Serpent, a Monte Carlo reactor physics code, with a single 3D hexagonal channel containing a single fuel tube in an infinite lattice with reflective radial and vacuum axial boundary conditions. The addition of a metallic natural uranium blanket inside the fuel tube, which increases the natural uranium metal to fuel salt ratio (ϒ) of the BBMSR, was shown to significantly increase the neutronic performance of the BBMSR.

Эволюция систем расселения в аспекте классической теории центральных мест

Considering the features of the evolutionary development of central places' systems, we establish that, contrary to the prevailing point of view, in the general case there is no pronounced relationship between the share of urban population and the number of central places of a lower hierarchy level subordinate to the central place of a given level. Moreover, the axiom of the theory, stating the heterogeneous distribution of only the urban population, should be replaced by the following: “Space is homogeneous and isotropic in all respects, with the exception of the distribution of the entire population or part of it.” Thus, the existence of central systems without rural population does not contradict the theory itself in its classic version. The evolution of central places' systems within an infinite lattice takes place by sequentially filling the hierarchy levels (increasing the value of the K -parameter for a given level), interrupted by the appearance of subsystems (increasing the number of hierarchy levels with constant K )—at least after filling the second and the fifth levels. The article substantiates the possibility of considering the isolated sections of the lattice. In this regard, the axiom of the “rational” consumer behavior can be considered excessive, and the postulate of the infinity of space is rejected. Only two initial axioms—on the polymorphism of central places' systems and on the most compact “packaging” of their parts—can be considered unshakable.

A novel chaotic system based on coupled map lattice and its application in HEVC encryption.

Video information is currently widely used in various fields. Compared with image and text data, video data has the characteristics of large data volume, strong data relevance, and large data redundancy, which makes traditional cryptographic systems no longer suitable for video encryption systems. The paper proposes a new chaotic system based on coupled map lattice (CML) and applies it to high efficiency video coding (HEVC) video encryption. The chaotic system logistic-iterative chaotic map with infinite collapses-coupled map lattice (L-ICMIC-CML), which is improved on the basis of the ICMIC system and combined with CML, generates stream ciphers and encrypts some syntax elements of HEVC. The experimental results show that the stream cipher generated by the L-ICMIC-CML system passes the SP800-22 Revla test and has strong randomness. Applying the stream cipher to the proposed HEVC encryption scheme, through the analysis of the encryption scheme's security, encryption time and encryption efficiency, it is better than other chaotic system encryption schemes. The video encryption system proposed in this paper is both safe and efficient.