Periodic Boundary Conditions Research Articles

Edge and skin effects in rhombus reciprocal photonic crystals.

With the development of non-Hermitian physics, the non-Hermitian skin effect (NHSE) has attracted much attention. Existing research highlights the critical roles of the periodic boundary condition (PBC) spectrum, lattice symmetry, and macroscopic symmetry of the lattice in relation to the geometry-dependent skin effect (GDSE). However, the impact of macroscopic edge geometry is frequently neglected. We find that the GDSE is highly sensitive to the edge and cannot be simply determined by the symmetries. Specifically, the GDSE can emerge at trivial interfaces of rhombus photonic crystals (PhCs) with zigzag edge and bearded edge. Furthermore, we analyze the underlying mechanisms from the perspective of point-gap topology. This work underscores important, yet frequently overlooked, aspects in two-dimensional (2D) reciprocal PhC systems and can be used to enhance design flexibility, allowing the NHSE to have better applications in areas such as lasers and highly sensitive sensors.

Exploring structural variances in monatomic metallic glasses using machine learning and molecular dynamics simulation.

BCC and FCC metals have different glass-forming abilities (GFA) and exhibit different characteristics during the glass transition. However, the structural origin of their different GFAs is still not clear. Here, we explored the structures of eight monatomic metallic glasses by combining molecular dynamics (MD) simulations and machine learning (ML). Our findings reveal that, despite their common long-range disordered atomic structure, metallic glasses can be further classified into two distinct categories indicating an underlying structural order within the disorder. Using machine learning, we found that BCC liquids can sample more diverse glass states than FCC liquids. Furthermore, glasses formed from BCC metals (GFFBs) exhibit a higher degree of disorder than glasses formed from FCC metals (GFFFs). These findings highlight the inherent differences between GFFFs and GFFBs, which help explain the different glass-forming abilities of FCC and BCC metals. Additionally, our results demonstrate the promising potential of computer vision and ML methods in exploring material structures. Classical molecular dynamics simulations were employed to generate configurations of GFFBs and GFFFs, and the simulations were performed using the LAMMPS code. Inter-atomic interactions were described using a classical embedded atom model (EAM) potential. The initial configuration of the model consists of 32,000 atoms in a three-dimensional (3D) cubic box with periodic boundary conditions applied in all three directions. For machine learning, we utilized an unsupervised machine learning method along with MobileNetV2 for classifying glass structures. Image entropy and image distances were used to measure the structural differences of the metallic glasses.

Solutions of a Neutron Transport Equation with a Partly Elastic Collision Operators

In this paper, we derive sufficient conditions that guarantee an description of long-time asymptotic behavior of the solution to the Cauchy problem governed by a linear neutron transport equation with a partially elastic collision operator under periodic boundary conditions. Our strategy involves showing that the strongly continuous semigroups et(T+Ke)t≥0 and et(T+Kc+Ke)t≥0, generated by the operators T+Ke and T+Kc+Ke, respectively, have the same essential type.According to Proposition , it is sufficient to show that remainder term in the Dyson–Philips expansion is compact. Our analysis focuses on the compactness properties of the second-order remainder term in the Dyson–Phillips expansion related to the problem. We first show that R2(t) is compact on L2(Ω×V,dxdv), and, using an interpolation argument (see Proposition ), we establish the compactness of R2(t) on Lp(Ω×V,dxdv)-spaces for 1<p<+∞. To the best of our knowledge, outside the one-dimensional case, this result is known only for vaccum boundary conditions in the multidimensional setting. However, our result, Theorem , is new for periodic boundary conditions.

Quasicrystalline 30° twisted bilayer graphene: fractal patterns and electronic localization properties

The recently synthesized 30° twisted bilayer graphene (30°-TBG) systems are unique quasicrystal systems possessing dodecagonal symmetry with graphene’s relativistic properties. We employ a real-space numerical atomistic framework that respects both the dodecagonal rotational symmetry and the massless Dirac nature of the electrons to describe the local density of states of the system. The approach we employ is very efficiency for systems with very large unit cells and does not rely on periodic boundary conditions. These features allow us to address a broad class of multilayer two-dimensional crystal with incommensurate configurations, particularly TBGs. Our results reveal that the 30°-TBG electronic spectrum consist of extended states together with a set of localized wave functions. The localized states exhibit fractal patterns consistent with the quasicrystal tiling.

Global Well-Posedness for the Derivative Nonlinear Schrödinger Equation with Periodic Boundary Condition

Abstract We prove global well-posedness for the derivative nonlinear Schrödinger equation on the torus in the Sobolev space $H^{1}(\mathbb{T})$ provided that the mass of initial data is strictly less than $8\pi $.

Volumetric Drag Coefficients for Generic Urban Configurations: Insights from Canopy Flow Analysis

Alternative drag approaches for representing unresolved buildings were proposed in literature for computational fluid dynamics (CFD) simulation of macroscopic urban airflow. As a contribution, the present work derives the volumetric drag coefficient (Cd*) through canopy drag and velocity analysis and provides appropriate correlations for Cd*against urban morphological parameters. A total of 72 cases across various urban configurations are investigated, categorized by building typology, horizontal layout, height variability, and plan area density (λp, from 0.0625 to 0.57). Reynolds-Averaged Navier-Stokes (RANS) simulations with periodic boundary conditions are performed to model fully developed flows. Results for the normalized drag force and superficial velocity and their relations with λp are evaluated. Subsequent evaluation of the profiles for the sectional coefficients (Cd*(Z)) reveals four distinct types with variations in uniform-height cases and combinations in varying-height cases. A throughout correlations analysis, facilitated by data transformation, identifies the straightforward relations between Cd* and frontal area density (λf) and tortuosity (τ). The followed stepwise regression provides a recommended formula for Cd*, demonstrating a proper fit with the simulated values. These findings facilitate the understanding and appropriate estimation of Cd* and Cd*(Z), promoting the application of macroscopic turbulence models, for neighborhood-scale wind and air quality studies.

Experimental and numerical characterization of E-glass/epoxy plain woven fabric composites containing void defects

The presence of voids in woven fabric composite (WFC) significantly affects the mechanical and thermomechanical properties. Void defects are introduced during the resin infiltration process due to various controlling parameters involved in the curing process, i.e., pressure, temperature, resin flow etc. In this study, the mechanical constants of 2D woven fabric composite were experimentally determined. A finite element (FE) based representative volume element (RVE) model has been validated against these experimental results. A multiscale-based FE model has been developed, and periodic boundary conditions are applied to the RVE model to evaluate the homogenized thermomechanical properties of WFC containing void defects. Present numerical model incorporates the geometrical microstructures of the post-cured woven composite and void contents obtained from X-ray microtomography. The influence of void defect and resin infiltration have been incorporated to evaluate the thermomechanical properties of plain WFC. A parametric study has been carried out with respect to the variation of void defects on thermoelastic properties of E-glass/epoxy plain WFC. The variation of void defects has been considered in micro and mesoscale models. Monte Carlo simulations further quantified the effects of void content on the thermomechanical constants of WFC. The presence of voids has been observed to have significant influences on the thermomechanical constants of yarn and woven fabric composites.

Quantum turbulence in superfluid helium: Decay and energy spectrum

The paper presents a comprehensive numerical study of the free decay of vortex tangle in superfluid helium. The initial vortex tangle represents one of the stationary configurations of loops in a counterflow with a laminar normal fluid component. The calculations are carried out in the framework of the vortex line method using the full Biot–Savart law over a wide range of temperatures. The aim of the study is to identify the role of various factors introduced into the numerical procedures (removal of small loops and segments during reconnections, the addition of and exclusion of vortex points on loops) and to determine the evolution of energy spectrum during the decay of quantum turbulence. A statistical approach is used to calculate the kinetic energy distribution on length scales. The calculations are carried out using periodic boundary conditions in a cube. The results show that, in agreement with the Feynman–Vinen theory, initially the rates of reduction in vortex line density at different temperatures are the same. However, when the vortex structure becomes rarefied, the influence of the mutual friction force becomes apparent, in agreement with Schwarz's theory. Statistical method for determining the energy spectrum is used. The Kolmogorov spectrum is not observed during decays at any temperature.

Wind stress modification by offshore wind turbines: A numerical study of wave blocking impacts

Offshore wind energy, as a clean and renewable resource, offers numerous advantages over onshore wind energy due to higher wind speeds and greater turbine capacity. However, the inadequate representation of wave-atmosphere interaction within the marine-atmospheric boundary layer may constrain wind resource assessments and, consequently, the design and layout of offshore wind turbines. By using the third-generation spectral wave model WAVEWATCH III, high-resolution numerical experiments which are capable of resolving the wind turbine foundation have been conducted to explore the blocking impact of wind turbine foundation on downstream wind stress. The findings provided significant insights into the factors influencing this effect, including foundation diameter, sea state, water depth, and wave directional spreading. Specifically, higher model resolution enables a more detailed characterization of wind stress distribution behind wind turbines. Increasing the foundation diameter resulted in a reduction of downstream wind stress. Notably, changes in wind stress exhibit a strong dependence on the sea state. The use of double periodic boundary conditions for the simulation domain enables an approximate representation of the entire wind farm by using a single turbine. Model results indicate that after waves pass through 20 turbines, domain-averaged wind stress decreases by approximately 3%, a figure that increases to 6% after passing through 50 turbines. The findings suggest the need to explore to what extent the changes in wind stress can alter wind profiles and how to parameterize the blocking impact of turbine foundations in wave models. This could have significant implications for wind farm site selection and layout design.

One-dimension Periodic Potentials in Schrödinger Equation Solved by the Finite Difference Method

Abstract The one-dimensional Kronig-Penney potential in the Schrödinger equation, a standard periodic potential in quantum mechanics textbooks known for generating band structures, is solved by using the finite difference method with periodic boundary conditions. This method significantly improves the eigenvalue accuracy compared to existing approaches such as the filter method. The effects of the width and height of the Kronig-Penney potential on the eigenvalues and wave functions are then analyzed. As the potential height increases, the variation of eigenvalues with the wave vector slows down. Additionally, for higher-order band structures, the magnitude of the eigenvalue significantly decreases with increasing potential width. Finally, the Dirac comb potential, a periodic δ potential, is examined using the present framework. This potential corresponds to the Kronig-Penney potential’s width and height approaching zero and infinity, respectively. The numerical results obtained by the finite difference method for the Dirac comb potential are also perfectly consistent with the analytical solution.

Steady state distributions of moving particles in one dimension: with an eye towards axonal transport.

Axonal transport, propelled by motor proteins, plays a crucial role in maintaining the homeostasis of functional and structural components over time. To establish a steady-state distribution of moving particles, what conditions are necessary for axonal transport? This question is pertinent, for instance, to both neurofilaments and mitochondria, which are structural and functional cargoes of axonal transport. In this paper we prove four theorems regarding steady state distributions of moving particles in one dimension on a finite domain. Three of the theorems consider cases where particles approach a uniform distribution at large time. Two consider periodic boundary conditions and one considers reflecting boundary conditions. The other theorem considers reflecting boundary conditions where the velocity is space dependent. If the theoretical results hold in the complex setting of the cell, they would imply that the uniform distribution of neurofilaments observed under healthy conditions appears to require a continuous distribution of neurofilament velocities. Similarly, the spatial distribution of axonal mitochondria may be linked to spatially dependent transport velocities that remain invariant over time.

Two-dimensional Rayleigh–Bénard convection without boundaries

We study the effects of Prandtl number $\mathit {Pr}$ and Rayleigh number $\mathit {Ra}$ in two-dimensional Rayleigh–Bénard convection without boundaries, i.e. with periodic boundary conditions. For Prandtl numbers in the range $10^{-3} \leqslant \mathit {Pr} \leqslant 10^2$ , the viscous dissipation scales as $\epsilon _\nu \propto \mathit {Pr}^{1/2}\mathit {Ra}^{-1/4}$ , which is based on the observation that enstrophy $\langle {\omega ^2}\rangle \propto \mathit {Pr}^0 \mathit {Ra}^{1/4}$ , and the Nusselt number tends to follow the ‘ultimate’ scaling $\mathit {Nu} \propto \mathit {Pr}^{1/2}\mathit {Ra}^{1/2}$ for all values of $\mathit {Pr}$ considered. The inverse cascade of kinetic energy forms the power-law spectrum $\hat {E}_u(k) \propto k^{-2.3}$ , which is close to $k^{-11/5}$ proposed by the Bolgiano–Obukhov (BO) scaling. The potential energy flux is not constant, in contrast to one of the main assumptions underlying the BO phenomenology. So, the direct cascade of potential energy forms the power-law spectrum $\hat {E}_\theta (k) \propto k^{-1.2}$ , which deviates from the expected $k^{-7/5}$ . Finally, at $\mathit {Pr} \to 0$ and $\infty$ , we find that the dynamics is dominated by vertically oriented elevator modes that grow without bound, even at high Rayleigh numbers and with large-scale dissipation present.

A finite element based approach for nonlocal stress analysis for multi-phase materials and composites

AbstractThis study proposes a numerical method for calculating the stress fields in nano-scale multi-phase/composite materials, where the classical continuum theory is inadequate due to the small-scale effects, including intermolecular spaces. The method focuses on weakly nonlocal and inhomogeneous materials and involves post-processing the local stresses obtained using a conventional finite element approach, applying the classical continuum theory to calculate the nonlocal stresses. The capabilities of this method are demonstrated through some numerical examples, namely, a two-dimensional case with a circular inclusion and some commonly used scenarios to model nanocomposites. Representative volume elements of various nanocomposites, including epoxy-based materials reinforced with fumed silica, silica (Nanopox F700), and rubber (Albipox 1000) are subjected to uniaxial tensile deformation combined with periodic boundary conditions. The local and nonlocal stress fields are computed through numerical simulations and after post-processing are compared with each other. The results acquired through the nonlocal theory exhibit a softening effect, resulting in reduced stress concentration and less of a discontinuous behaviour. This research contributes to the literature by proposing an efficient and standardised numerical method for analysing the small-scale stress distribution in small-scale multi-phase materials, providing a method for more accurate design in the nano-scale regime. This proposed method is also easy to implement in standard finite element software that employs classical continuum theory.

A posteriori error analysis of a positivity preserving scheme for the power-law diffusion Keller–Segel model

Abstract We study a finite volume scheme approximating a parabolic-elliptic Keller–Segel system with power-law diffusion with exponent $\gamma \in [1,3]$ and periodic boundary conditions. We derive conditional a posteriori bounds for the error measured in the $L^{\infty }(0,T;H^{1}(\varOmega ))$ norm for the chemoattractant and by a quasi-norm-like quantity for the density. These results are based on stability estimates and suitable conforming reconstructions of the numerical solution. We perform numerical experiments showing that our error bounds are linear in mesh width and elucidating the behavior of the error estimator under changes of $\gamma $.

Multi-objective optimization of porous fins in closed cavity with heat source of sinusoidal periodic temperature boundary condition

Multi-objective optimization of porous fins in closed cavity with heat source of sinusoidal periodic temperature boundary condition

Dispersion and particle pressure in sedimenting suspensions with hydrodynamic interactions and particle inertia

Particle inertia is a feature of suspension dynamics that is often neglected. However, its neglect changes the order of the governing equations of particle motion. In this work, we examine the impact of the inertia of particles over their velocity fluctuations and the resulting stresses. In order to observe effects of particle inertia, we consider dusty-gas suspensions of spherical weakly Brownian microparticles sedimenting in a Newtonian fluid at low Reynolds numbers. We first investigate the motion of a single spherical particle. The simulations for this simple case allow us to define the appropriate physical parameters of the flow and to validate the numerical calculation of velocity statistics over a range of Péclet and Stokes numbers. Next, we perform Langevin dynamics simulations with periodic boundary conditions to integrate the equations governing the translational and rotational motions of N hydrodynamically interacting particles suspended in the viscous fluid. The first aim of this paper is to examine the behavior of the velocity variance of inertial particles, with small fluctuations produced by Brownian motion at the moderate Péclet numbers. The interplay between mild Brownian forces and hydrodynamic interactions decreases the anisotropy of velocity fluctuations observed in non-Brownian suspensions (Pe≫1) of sedimenting particles. The simulation results suggest that particle inertia attenuates the magnitude of velocity fluctuations produced by both Brownian motion and viscous hydrodynamic interactions. Additionally, we study the long-time behavior of the velocity fluctuations by calculating their autocorrelation functions and the particle diffusivities. The numerical simulations show clear evidence of particle pressure arising from the flow disturbance produced by hydrodynamic interactions in a suspension. From the particle velocity variance obtained in the present numerical simulations, we propose a simple model for particle-phase pressure as a linear function of the particle volume fraction ϕ⩽5%, which is usually a closure quantity required in models of more complex particulate systems such as fluidized beds.

Novel conformal structure-preserving schemes for the linearly damped nonlinear Schrödinger equation

In this work, by utilizing the Strang splitting technique, some advanced conformal structure-preserving schemes are developed for solving the damped nonlinear Schrödinger equation (DNLSE). The proposed innovative numerical approaches, namely the high-order compact conformal multi-symplectic method and the conformal momentum-preserving method, have two key computational advantages. Firstly, these two approaches excel in conserving local structures, and more importantly, they are capable of maintaining exact energy dissipation rates in any time-space region, particularly under periodic boundary conditions. To validate the theoretical properties and demonstrate the efficacy and stability of these approaches in long-time integrations, we conduct through extensive numerical simulations involving both bright and dark solitons.

Drop Tests and Numerical Analyses of Composite Beams with Corrugated Web

To evaluate the energy absorption ability of composite beams with corrugated web under impact, the drop tests were conducted to demonstrate the fracture modes and obtain the load and energy history curves. Based on the drop tests, a finite element (FE) model was proposed with a weak link designed to trigger crushing, developed using Abaqus/Explicit, which integrates cohesive elements and employs the improved Hashin criterion along with the Choi–Chang criterion to accurately predict the onset and progression of delamination. The model accounts for the anisotropy and progressive damage in composite laminates. The periodic boundary conditions (PBCs) were integrated into the single-wave corrugated web beam model, the inherent limitations of which were corrected, enabling it to simulate the crushing process with the same effectiveness as the multi-wave model. As a result, the proposed model can accurately depict the progressive crushing behavior of corrugated web beams under impact loads while maintaining high computational efficiency.

Quantum Chemical Insights into DNA Nucleobase Oxidation: Bridging Theory and Experiment.

The oxidation free energies of DNA nucleobases in aqueous solution are still matter of extensive discussion because of the contrasting results reported so far. With the aim of settling a longstanding debate about the oxidation potentials of DNA constituents, herein we report the results of state-of-the-art DFT-based molecular dynamics simulations, in which the whole solvent environment is modeled at the atomistic level, by using DFT supercell calculations, with periodic boundary conditions. Calculated vertical ionization energies are very close to those observed by photoelectron spectroscopy both in the gas phase and in solution. One-electron oxidation free energies in aqueous solution agree well with the results of differential pulse voltammetry measurements and with those inferred by photoelectron spectroscopy with the aid of theoretical computations to estimate vibrational relaxation.

On <i>q</i> Sturm Liouville Operator with Periodic Boundary Conditions

In this study, we consider q-Sturm Liouville operator with periodic boundary conditions. An asymptotic expression of the solution is obtained. With the help of this asymptotic representation, an asymptotic solution of the characteristic equation is presented. An application of the Rouche theorem, asymptotic expressions of eigenvalues are obtained.