Periodic Boundary Conditions Research Articles

A problem with periodic boundary conditions for the non-Fourier heat equation

All diffusion equations are based on the infinite velocity of potential fields, which leads to well-known paradoxes. Consequently, in non-stationary processes, the evolution of these quantities do not completely obey the above equations due to the lack of parameters in them that take into account the finite rate of potential growth.In the heat conduction theory, numerous generalizations of the Fourier law are used as a remedy for these issues. The article gives a brief overview of generalizations of the Fourier law. Some mathematical issues of well-posed boundary value problems for the Guyer-Krumhansl model are discussed. As an application, a boundary value problem for a general quasilinear equation with periodic boundary conditions is considered. Schauder-type a priori estimates are established and the uniqueness of the solution is proved.

Cross-stream oscillations in the granular flow through a vertical channel

The gravity flow of a granular material between two vertical walls separated by a width $2W$ is simulated using the discrete element method (DEM). Periodic boundary conditions are applied in the flow (vertical) and the other horizontal directions. The mass flow rate is controlled by specifying the average solids fraction $\bar {\phi }$ , the ratio of the volume of the particles to the volume of the channel. A steady fully developed state can be achieved for a narrow range of $\bar {\phi }$ , $\bar {\phi }_{max} \geq \bar {\phi } \geq \bar {\phi }_{cr}$ , and the material is in free fall for $\bar {\phi } < \bar {\phi }_{min}$ . For an intermediate range of $\bar {\phi }$ ( $\bar {\phi }_{cr} > \bar {\phi } \geq \bar {\phi }_{min}$ ), there are oscillations in the horizontal coordinate of the centre of mass, velocity components and stress. As $\bar {\phi }$ decreases in the range $\bar {\phi }_{cr} > \bar {\phi } \geq \bar {\phi }_{min}$ , the amplitude of the oscillations increases proportional to $\sqrt {\bar {\phi }_{cr} - \bar {\phi }}$ and the frequency appears to tend to a non-zero value as $\bar {\phi } \rightarrow \bar {\phi }_{cr}$ , indicating a supercritical Hopf bifurcation. The relation between the dominant frequency and the higher harmonics of the position, velocity and stress fluctuations are explained using the momentum balance. It is found that dissipation in the inter-particle and particle–wall contacts is critical for the presence of an oscillatory state.

Density Functional Theory and Density Functional Tight Binding Studies of Thiamine Hydrochloride Hydrates.

Thiamine hydrochloride (THCL), also known as vitamin B1, is an active pharmaceutical ingredient (API), present on the list of essential medicines developed by the WHO, which proves its importance for public health. THCL is highly hygroscopic and can occur in the form of hydrates with varying degrees of hydration, depending on the air humidity. Although experimental characterization of the THCL hydrates has been described in the literature, the questions raised in previously published works suggest that additional research and in-depth analysis of THCL dehydration behavior are still needed. Therefore, the main aim of this study was to characterize, by means of quantum chemical calculations, the behavior of thiamine hydrates and explain the previously obtained results, including changes in the NMR spectra, at the molecular level. To achieve this goal, a series of DFT (CASTEP) and DFTB (DFTB+) calculations under periodic boundary conditions have been performed, including molecular dynamics simulations and GIPAW NMR calculations. The obtained results explain the differences in the relative stability of the studied forms and changes in the spectra observed for the samples of various degrees of hydration. This work highlights the application of periodic DFT calculations in the analysis of various solid forms of APIs.

Role of grain boundary crystallography on void growth in FCC metals

The growth of an isolated spherical void at and near a grain boundary or a triple junction is studied by means of full-field crystal plasticity simulations with explicit representation of the void. To examine the fundamental aspects of void/grain boundary interactions in these cases, a three-dimensional large-strain elasto-viscoplastic fast-Fourier transform (LS-EVP-FFT) model with axisymmetric tensile periodic boundary conditions was developed. We employ this model to reveal the role of crystallography (grain orientations, grain boundary misorientation and inclination angle with respect to the largest component of the applied stress) and type of loading on void growth. It was found that grain orientation was the primary crystallographic feature governing void growth in strain-rate controlled simulations. For strains small enough to not significantly alter the local geometry, the overall growth of the void in bicrystal boundaries was not strongly affected by grain boundary inclination angle. The relative amount of void growth experienced within each grain in a bicrystal or tri-crystal was strongly dependent on the grain boundary inclination angle. We found a subset of orientations that exhibit faster void growth in bicrystal boundaries than inside the constituent single crystals. Last, we show that when placed primarily in one grain but near the grain boundary, void growth was similar to void growth when placed at the grain boundary. The growing void remained in the original grain and did not grow into the other grain, regardless of crystallographic character.

Stochastic cellular automaton model of culture formation.

We introduce a stochastic cellular automaton as a model for culture and border formation. The model can be conceptualized as a game where the expansion rate of cultures is quantified in terms of their area and perimeter in such a way that approximately geometrically round cultures get a competitive advantage. We first analyze the model with periodic boundary conditions, where we study how the model can end up in a fixed state, i.e., freezes. Then we implement the model on the European geography with mountains and rivers. We see how the model reproduces some qualitative features of European culture formation, namely, that rivers and mountains are more frequently borders between cultures, mountainous regions tend to have higher cultural diversity, and the central European plain has less clear cultural borders.

Entropy dissipative higher order accurate positivity preserving time-implicit discretizations for nonlinear degenerate parabolic equations

We develop entropy dissipative higher order accurate local discontinuous Galerkin (LDG) discretizations coupled with Diagonally Implicit Runge–Kutta (DIRK) methods for nonlinear degenerate parabolic equations with a gradient flow structure. Using the simple alternating numerical flux, we construct DIRK-LDG discretizations that combine the advantages of higher order accuracy, entropy dissipation and proper long-time behavior. We theoretically prove the entropy dissipation of the implicit Euler-LDG discretization without any time-step restrictions when no positivity constraint is imposed. Next, in order to ensure the positivity of the numerical solution, we use the Karush–Kuhn–Tucker (KKT) limiter, which achieves a positive solution by coupling the positivity preserving KKT conditions with higher order accurate DIRK-LDG discretizations using Lagrange multipliers. In addition, mass conservation of the positivity-limited solution is ensured by imposing a mass conservation equality constraint to the KKT equations. Under a time step restriction, the unique solvability and entropy dissipation for implicit first order accurate in time, but higher order accurate in space, positivity-preserving LDG discretizations with periodic boundary conditions are proved, which provide a first theoretical analysis of the KKT limiter. Finally, numerical results demonstrate the higher order accuracy and entropy dissipation of the positivity-preserving DIRK-LDG discretizations for problems requiring a positivity limiter. In addition, we can observe from the numerical results that the implicit time-discrete methods alleviate the time-step restrictions needed for the stability of the numerical discretizations, which improves computational efficiency.

Double reduction order method based conservative compact schemes for the Rosenau equation

In this paper, fourth-order compact difference schemes are derived, analyzed and tested at length for both one- and two-dimensional Rosenau equations under the spatial periodic boundary conditions on the basis of the double reduction order method and bilinear compact operator. We prove that these schemes satisfy mass and energy conservation law via the help of the energy method. In addition, the uniquely solvable, unconditional convergence and stability are all obtained with the convergence order four in space and order two in time under the L∞-norm. Several numerical examples are presented to support the theoretical results.

Long time dynamics of Nernst-Planck-Navier-Stokes systems

We consider the Nernst-Planck-Navier-Stokes system describing the electrodiffusion of ions in a viscous Newtonian fluid. We prove the exponential nonlinear stability of constant steady states in the case of periodic boundary conditions in any dimension of space without constraints on the number of species, valences and diffusivities. We consider also the case of two spatial dimensions, and we prove the exponential stability from arbitrary large data.

Quantum walk in stochastic environment.

We consider a quantized version of the Sinai-Derrida model for "random walk in random environment." The model is defined in terms of a Lindblad master equation. For a ring geometry (a chain with periodic boundary condition) it features a delocalization-transition as the bias in increased beyond a critical value, indicating that the relaxation becomes underdamped. Counterintuitively, the effective disorder is enhanced due to coherent hopping. We analyze in detail this enhancement and its dependence on the model parameters. The nonmonotonic dependence of the Lindbladian spectrum on the rate of the coherent transitions is highlighted.

Existence of solutions for stress-rate type strain-limiting viscoelasticity in Gevrey spaces.

In this work, we deal with a one-dimensional stress-rate type model for the response of viscoelastic materials, in relation to the strain-limiting theory. The model is based on a constitutive relation of stress-rate type. Unlike classical models in elasticity, the unknown of the model under consideration is uniquely the stress, avoiding the use of the deformation. Here, we treat the case of periodic boundary conditions for a linearized model. We determine an optimal function space that ensures the local existence of solutions to the linearized model around certain steady states. This optimal space is known as the Gevrey-class [Formula: see text], which characterizes the regularity properties of the solutions. The exponent [Formula: see text] in the Gevrey-class reflects the specific dispersion properties of the equation itself. This article is part of the theme issue 'Foundational issues, analysis and geometry in continuum mechanics'.

Advanced detached-eddy simulation of the MD 30P-30N three-element airfoil

An experimental version in the Detached-Eddy Simulation (DES) family (called Advanced DES or ADES) is introduced and tested on a geometry that is fairly complex but two-dimensional. The essential change in ADES is that the user is given control of the regions treated with full turbulence modelling (RANS) and those treated with Large-Eddy Simulation (LES). This zonal character makes the approach more powerful, but less practical, so that in its current state it is not ready for industrial CFD. The grid requirements of the two regions are very different, and multi-block grid structure is natural. Another key feature is a Volumetric Synthetic Turbulence Generator (VSTG), installed to feed the LES region with viable resolved turbulence, so that the resolved Reynolds stresses rapidly substitute for the modelled Reynolds stresses present in the RANS region. The VSTG operates in a volume, rather than on a surface and can be active in attached boundary layers, at a trailing edge, or after separation. The well-known McDonnell-Douglas 30P-30N airfoil is simulated with periodic lateral boundary conditions. The VSTG is successful, and the desired nature of simulation is obtained in each region. ADES involves zonal decisions, but appears robust. An inertial range is clearly indicated in frequency spectra. A grid-refinement study is included, as well as variations in lateral domain size and STG positions; this led to a matrix of 11 simulations. Cases are shown at four angles of attack and with three RANS models in addition to ADES. Pressure and friction distributions and velocity and shear stress profiles are compared in detail. The prospects for an evolution of ADES into a practical routine approach in the long term are discussed.

Casimir versus Helmholtz forces: Exact results

Recently, attention has turned to the issue of the ensemble dependence of fluctuation induced forces. As a noteworthy example, in O(n) systems the statistical mechanics underlying such forces can be shown to differ in the constant M→ magnetic canonical ensemble (CE) from those in the widely-studied constant h→ grand canonical ensemble (GCE). Here, the counterpart of the Casimir force in the GCE is the Helmholtz force in the CE. Given the difference between the two ensembles for finite systems, it is reasonable to anticipate that these forces will have, in general, different behavior for the same geometry and boundary conditions. Here we present some exact results for both the Casimir and the Helmholtz force in the case of the one-dimensional Ising model subject to periodic and antiperiodic boundary conditions and compare their behavior. We note that the Ising model has recently being solved in Phys.Rev. E 106 L042103 (2022), using a combinatorial approach, for the case of fixed value M of its order parameter. Here we derive exact result for the partition function of the one-dimensional Ising model of N spins and fixed value M using the transfer matrix method (TMM); earlier results obtained via the TMM were limited to M=0 and N even. As a byproduct, we derive several specific integral representations of the hypergeometric function of Gauss. Using those results, we rigorously derive that the free energies of the CE and grand GCE are related to each other via Legendre transformation in the thermodynamic limit, and establish the leading finite-size corrections for the canonical case, which turn out to be much more pronounced than the corresponding ones in the case of the GCE.

Conservative Fourier spectral method for a class of modified Zakharov system with high-order space fractional quantum correction

In this paper, we consider the Fourier spectral method and numerical investigation for a class of modified Zakharov system with high-order space fractional quantum correction. First, the numerical scheme of the system is developed with periodic boundary condition based on the Crank–Nicolson/leap-frog methods in time and the Fourier spectral method in space. Moreover, it is shown that the scheme preserves simultaneously mass and energy conservation laws. Second, we analyze stability and convergence of the numerical scheme. Last, the numerical experiments are given, and the results show the correctness of theoretical results and the efficiency of the conservative scheme.

On the realization of periodic boundary conditions for hexagonal unit cells

In the context of homogenization of micro-heterogeneous materials, the choice of the Representative Volume Element (RV E) plays a crucial role. For periodic microstructures, an RV E is an underlying unit cell with periodic boundary conditions. Nevertheless, the question of the implementation of periodic boundary conditions may arise here; for example, some of the applications of periodic boundary conditions in the literature for hexagonal cells are incorrect or they are not given in detail. In this paper, we analyze periodic boundary conditions for two-dimensional hexagonal unit cells. Periodic boundary conditions are characterized by the periodic fluctuations in the displacement fields and anti-periodic traction vectors at associated points of the boundary of the unit cell. From comparative calculations with an ensemble of unit cells, it is evident that the natural choice for vanishing fluctuations is to be set on the midpoints of the six perimeter lines of the cell. The partially applied choice of vanishing fluctuations in the six corner points of the outer edge of the unit cell leads to wrong results. The boundary conditions proposed here are analyzed on the basis of representative examples and compared to the results with the incorrect boundary conditions.

Influence of pressure disturbance wave on dynamic response characteristics of liquid film seal for multiphase pump

Slug flow or high GVF (Gas Volume Fraction) conditions can cause pressure disturbance waves and alternating loads at the boundary of mechanical seals for multiphase pumps, endangering the safety of multiphase pump units. The mechanical seal model is simplified by using periodic boundary conditions and numerical calculations are carried out based on the Zwart-Gerber-Belamri cavitation model. UDF (User Define Function) programs such as structural dynamics equations, alternating load equations, and pressure disturbance equations are embedded in numerical calculations, and the dynamic response characteristics of mechanical seal are studied using layered dynamic mesh technology. The results show that when the pressure disturbance occurs at the inlet, as the amplitude and period of the disturbance increase, the film thickness gradually decreases. And the fundamental reason for the hysteresis of the film thickness change is that the pressure in the high-pressure area cannot be restored in a timely manner. The maximum value of leakage and the minimum value of axial velocity are independent of the disturbance period and determined by the disturbance amplitude. The mutual interference between enhanced waves does not have a significant impact on the film thickness, while the front wave in the attenuated wave has a promoting effect on the subsequent film thickness changes, and the fluctuation of the liquid film cavitation rate and axial velocity under the attenuated wave condition deviates from the initial values. Compared with pressure disturbance conditions, alternating load conditions have a more significant impact on film thickness and leakage. During actual operation, it is necessary to avoid alternating load conditions in multiphase pump mechanical seals.

Preface

This Proceedings arose from 2023 International Conference on Applied Statistics, Modeling and Advanced Algorithms (ASMA 2023), which was held via hybrid form in Qingdao, China from July 28th to 30th, 2023. ASMA 2023 marks the first time that this series of conference had been successfully held.The scientific and social program for the meeting ran over three days with more than 60 delegates from over 10 countries gathering on July 28th and departing on July 30th, including plenary, keynote speeches, oral presentations, poster presentations, and academic communication.In all there were five keynote speeches during the meeting. Prof. Le Fang addressed a speech on Hexagonal Periodic Condition in the Simulation of Two-Dimensional Flows. In order to simulate numerically the isotropic turbulence, traditional methods are to use periodic boundary conditions. However, using periodic conditions will involve problems of background anisotropy at large scales, which prevents to generate the truly isotropic turbulence. According to qualitative calculations, he showed that this background anisotropy is principally related to the shape of the energy spectrum. In two-dimensional cases, he designed a fast Fourier transfer (FFT) algorithm for hexagonal periodic conditions, which can be used in pseudo-spectral methods. Using this algorithm, the hexagonal periodic conditions were involved to reduce the large-scale background anisotropy.Topics that were the focus of invited presentations and contributed papers at ASMA 2023 included: Statistical Mechanics, Data Mining, Modeling Simulation Input, Numerical Analysis, Application-driven Numerical Parallel Algorithms, etc. All submissions were gone through rigorous review and process by performing double blind review to check their quality of content, level of innovation, significance, originality and legibility.The Conference could not have occurred without the generous support of our Conference sponsors. It would also not have been possible without the hard work of the Technical Program Committee members. We are most grateful for the staff of Journal of Physics: Conference Series for their help and kindness in the preparation of this volume. Finally, we would like to thank all the attendees at ASMA 2023 for their scientific contributions to the Conference, and for their warm spirit of engagement which characterized the scientific discussions and social occasions.The Committee of ASMA 2023List of Committee Member is available in the pdf.

Investigation of the decomposition mechanism of MTNP melt-cast explosive at different temperatures and pressures through ReaxFF/lg molecular dynamics simulations.

Thermal decomposition of 1-methyl-3,4,5-trinitropyrazole (MTNP), a melt-cast explosive, was investigated at different temperatures (2500, 2750, 3000, 3250, and 3500 K) and pressures (3000 K/0.5 GPa, 3000 K/1 GPa) using the ReaxFF/lg force field. The study aimed to analyze the changes in reactant quantities, initial reaction pathways, and final product yields. The results demonstrated that complete decomposition of MTNP molecules occurred within a timeframe of 200 ps, with shorter decomposition times observed as the temperature increased. The high-temperature thermal decomposition of MTNP was found to follow two primary reaction pathways. Reaction 1 involved denitration, while reaction 2 proceeded with nitro group isomerization. DFT calculations indicated that nitro group isomerization was the most favorable reaction. During the initial stages, higher quantities of NO2, NO, and N2 were observed compared to other species. This can be attributed to the relatively higher nitrogen and oxygen content in the MTNP structure. Among the five reaction temperatures, it was observed that the quantities of small molecules followed the order of NO2 > NO > N2 > CO. Moreover, with increasing temperature, the quantities of all four small molecules increased, indicating that higher temperatures promoted the progression of the reactions. However, as the pressure increased, there was a trend of initially increasing and then decreasing to zero for the quantities of NO2 and NO. This suggests that high temperature accelerated the high-temperature thermal decomposition of NO2 and NO, leading to a significant increase in the content of N2. A 3 × 5 × 5 supercell model of MTNP was constructed in Materials Studio, consisting of 75 unit cells and 300 MTNP molecules. The model was then subjected to a 20 ps geometric optimization using the Polak-Ribiere version of the conjugate gradient (CG) algorithm in the large-scale atomic/molecular massively parallel simulator (LAMMPS) under the isothermal-isobaric (NPT) ensemble at 1 atm pressure and 300 K temperature. Following the optimization, molecular dynamics simulations were performed on the model at five temperatures (2500, 2750, 3000, 3200, and 3500 K) under 1 atm using the NPT ensemble for a total duration of 1 ns. During the simulations, atomic trajectories, as well as information on atomic and molecular species, were output every 500 steps. Subsequently, a custom script was utilized to analyze the thermal decomposition pathways and products. A time step of 0.1 fs was employed for the calculations, and periodic boundary conditions were applied to eliminate boundary effects.

The effect of segmentation depth in a helical fin tube bundle on the anisotropic turbulence state

The large eddy simulation approach was used to perform a numerical simulation of a flow around a helical segmented-fin tube bundle. This work focuses on the study of turbulence and the effect of segment depth on it. The simulation was conducted in an area away from the boundaries of the tube bundle, where the flow is fully-developed, and the use of periodic boundary conditions is possible. The computer-aided design of the helical segmented-fin tube geometry is incorporated into the computational grid from the immersed boundary technique. The Reynolds stress tensor was used to analyze the anisotropic state of turbulence. This flow is characterized by an anisotropic state generated by a flow with a preferential direction. A transfer of momentum in the inter-segment space and contiguous zones was observed. This intersegment flow generates a less anisotropic state of turbulence. This study aimed to understand a possible transformation of heat transfer with marginal geometrical changes and pressure drop increases.

Transient heat transfer in concentric cylinders using periodic boundary condition and asymmetric heat generation applied to thermal plug and abandonment of oil wells

When a well does not fulfil its objectives, it is repurposed or permanently plugged, a shift in budget from revenue to Plug and Abandonment (P&A) expenditures occurs. New technologies being developed for P&A make use of a powerful heat emitter for melting the surrounding of the well forming a solid plug after the cool down. However, there is a gap in understanding the heat transport process in the well and much need for appropriate mathematical tools and solutions for estimating the effectiveness of Thermal P&A. The thermal analysis of the process requires eigenvalues in polar coordinates, which return only real quantities due to its implicitly dependence on the angular eigenvalues. Here, a hybrid analytical/numerical method is applied to an asymmetric transient heat conduction problem. The oil well is conceived as a 2-D multi-layer disc cast in polar coordinates, where the Separation of Variables Method (SVM) was applied to achieve a closed-form solution. Asymmetric boundary conditions of first, second and third kind can be implemented utilizing the proposed framework. A Finite Volume Method (FVM) numerical solution was produced for code verification. Lastly, research results show that temperature levels arising from the thermite reaction throughout the composite cylinder domain in radial and azimuthal ranges are enough for surpassing the melting point of the production tube steel. The results evidenced the thermal efficiency of the TP&A procedure and suggested that the production column may be destroyed by fusion, thus reducing tamponing expenses significantly.

Effects of the strain gradients on the band structures of the elastic waves propagating in 1D phononic crystals: An analytical approach

In this paper, an analytical method is proposed for the analysis of the effects of the strain gradients on the frequency band structures and band-gaps of the elastic waves propagating in one-dimensional (1D) nano-sized phononic crystals (PCs). The considered 1D nano-sized PCs are made of periodic unit-cells containing a metal section and an epoxy section. The strain gradients are taken into account to obtain the governing equations by using the strain energy, the kinetic energy and the Hamilton's principle. An analytical solution based on the transfer matrix method is derived to compute the frequency dispersion curves or band structures of the elastic waves. The continuity conditions on the interface between the neighboring sections and the periodic boundary conditions for the unit-cell are satisfied using the proposed analytical method. Two band-gap characteristics, namely the starting frequency and the width of each band-gap, are defined and investigated. The effects of the small-scale parameters in the strain-gradient theory on the band structures and the band-gap characteristics are analyzed and discussed in detail. It is demonstrated that the strain gradients may significantly influence the frequency band structures and the band-gap characteristics of the elastic waves propagating in 1D nano-sized PCs.