Effect Of Periodic Boundary Conditions Research Articles

The noise-generating mechanism of rod-airfoil configuration and the effect of spanwise length on noise prediction

The wake-blade interaction in turbo-engines is the primary source of broadband noise. In this work, the noise mechanism of a benchmark rod-airfoil configuration is exhaustively analyzed using compressible Large Eddy Simulation (LES) and Ffowcs Williams-Hawkings (FW-H) methods. The baseline model with a spanwise length shortened by 7d (d is the rod diameter) is simulated under periodic boundary conditions, and the effect of spanwise lengths on noise prediction and correlation is investigated in combination with two other models (3d and 1d). In terms of the baseline model, the noise predicted by different FW-H surfaces is compared first, and the noise radiated from different chordwise and spanwise regions of the airfoil surface reveal the generating mechanism of the main dipolar noise and the effect of periodic boundary conditions on noise prediction, respectively. Additionally, the fluctuating surface pressure characterized at four frequency bands confirms that the wake-airfoil interaction dominates the leading-edge noise at low frequencies. The characteristic frequencies of the noise and transient velocity fields reveal that the primary broadband noise source is the interaction of large-scale vortices with the airfoil surface at the leading-edge region. Finally, the effect of spanwise length on noise prediction and correlation is studied for three models. The differences in the flow structures lead to different features of their coherence coefficients. The results presented in this work provide a comprehensive understanding of the noise-generating mechanism of wake-airfoil interaction in turbomachinery and the practical application of simplified models in the future numerical simulations.

Spectrum of light nuclei in a finite volume

Lattice quantum chromodynamics calculations of multi-baryon systems with physical quark masses would start a new age of ab initio predictions in nuclear physics. Performed on a finite grid, such calculations demand extrapolation of their finite volume numerical results to free-space physical quantities. Such extraction of the physical information can be carried out fitting effective field theories (EFTs) directly to the finite-volume results or utilizing the L\"uscher free-space formula or its generalizations for extrapolating the lattice data to infinite volume. To understand better the effect of periodic boundary conditions on the binding energy of few nucleon systems we explore here light nuclei with physical masses in a finite box and in free space. The stochastic variational method is used to solve the few-body systems. Substantial optimizations of the method are introduced to enable efficient calculations in a periodic box. With the optimized code, we perform accurate calculations of light nuclei $A \le 4$ within leading order pionless EFT. Using L\"uscher formula for the two-body system, and its generalization for 3- and 4-body systems, we examine the box effect and explore possible limitations of these formulas for the considered nuclear systems.

Lattice Boltzmann simulation of natural convection in a cavity with periodic boundary condition on both side walls

The study of the effect of periodic boundary condition is of primary interest to understand the maximum temperature of the wall and avoid overheating phenomena. Available computational fluid dynamic studies provide information on the effect of the local heat flux or local temperature. This study aims to understand the temperature and fluid flow distribution inside the cavity under such periodic thermal boundary conditions to avoid overheating phenomena. This study reports the numerical results of the natural convection inside an air-filled cavity with its horizontal walls insulated and its vertical walls at different thermal boundary conditions. The heat flux of the left wall and the temperature of the right wall are varied sinusoidally. The Lattice Boltzmann Method (LBM) is used to solve the governing equations with the related boundary conditions. The maximum temperature of the wall under the heat flux and the maximum stream function inside the cavity are investigated. Also, the effect of oscillation period on the maximum temperature of the left wall is studied. The results show that the maximum temperature of the left wall corresponds to the heat flux wavelength of 0.8. Moreover, the stream function value increases intensively by increasing the Rayleigh number.

A unified approach to two-dimensional linear advection-dispersion equation in cylindrical coordinates on a finite domain

A new analytical solution of the two-dimensional linear advection-dispersion equation (2-D LAD) in cylindrical coordinates on a finite domain is developed for solute transport in a radial porous medium and a circular source subject to steady and uniform groundwater. The analytical solutions of the 2-D LAD equation with the first- and third-type inlet boundary conditions are obtained by utilizing the finite Hankel transform and the unified transform method, also known as the Fokas method. More precisely, after employing the second kind finite Hankel transform, the 2-D LAD equation is converted to the 1-D LAD equation, which invites the Fokas method, to undertake a refined analysis. As a consequence, we derive a new exact analytical solution of the 2-D LAD equation in cylindrical coordinates with more general boundary conditions, and its physical applications are notable in several physical circumstances. In particular, it can be shown that for large Péclet number, the type of the inlet boundary conditions has no significant discrepancy between the solutions. Furthermore, it is also shown that for large Péclet number, the exit boundary of finite or infinite domain has no significant influence on the solutions. More importantly, we show that if the inlet boundary value is asymptotically t-periodic for large t, the solution of the 2-D LAD in a radial geometry is also asymptotically t-periodic for large t with the same period, which can be used to understand the effect of periodic boundary conditions for solute transport in porous media. All these analytical predictions are consistent with numerical results.

CALANIE: Anisotropic elastic correction to the total energy, to mitigate the effect of periodic boundary conditions

CALANIE (CALculation of ANIsotropic Elastic energy) computer program evaluates the elastic interaction correction to the total energy of a localized object, for example a defect in a material simulated using an ab initio or molecular statics approach, resulting from the use of periodic boundary conditions. The correction, computed using a fully elastically anisotropic Green’s function formalism, arises from the elastic interaction between a defect and its own periodically translated images. The long-range field of elastic displacements produced by the defect is described in the elastic dipole approximation. Applications of the method are illustrated by two case studies, one involving an ab initio investigation of point defects and vacancy migration in FCC gold, and another a molecular statics simulation of a dislocation loop. We explore the convergence of the method as a function of the simulation cell size, and note the significance of taking into account the elastic correction in the limit where the size of the defect is comparable with the size of the simulation cell. Program summaryProgram Title: CALANIE, version 2.0Program Files doi:http://dx.doi.org/10.17632/3h6xffk9h6.1Licensing provisions: Apache License, Version 2.0Programming language: C/C++Nature of problem: Periodic boundary conditions (PBCs) are often used in the context of ab initio and molecular statics atomic scale simulations. A localized defect in a crystalline material, simulated using PBCs, interacts elastically with its own periodically translated images, and this gives rise to a systematic error in the computed defect formation and migration energies. Evaluating the correction to the total energy resulting from effects of elastic interaction between a defect and its periodic images, to alleviate the contribution to the total energy arising from PBCs, is an essential aspect of any accurate total energy calculation performed using PBCs.Solution method: The energy of interaction between a localized defect and its periodically translated images is computed in the linear elasticity approximation. The energy of elastic interaction is expressed analytically in terms of the elastic dipole tensor of the defect and elastic Green’s function. Elements of the dipole tensor are computed as a part of the simulation evaluating the formation energy of the defect. Elastic Green’s function and its first and second derivatives are computed numerically from the elastic constants of the material. The method and the corresponding numerical procedures are implemented in the CALANIE computer program. The program evaluates matrix elements of the elastic dipole tensor of a localized defect and the elastic correction to the total energy arising from the use of periodic boundary conditions.Restrictions: The approach assumes the validity of the linear elasticity approximation. This limits the accuracy of evaluation of the elastic correction, which becomes less precise if the size of the defect is comparable with the size of the simulation cell.Unusual features: An open source code, containing full detail of the relevant theoretical concepts, algorithms and numerical implementation.

Formation of stress- and thermal-induced martensitic nanostructures in a single crystal with phase-dependent elastic properties

In the present paper, the effect of phase-dependent elastic properties on martensitic phase transformations (PTs) in a single crystal is investigated using the phase field approach. The simplest phase dependence of elastic properties is defined by different Young’s moduli for austenite (A) and martensite (M), and its effect is investigated for thermal- and stress-induced propagation of an A–M interface. The phase dependence of elastic properties is then included using the quadratic elastic energy with two constants different for A and martensitic variants. The coupled system of phase field and elasticity equations is solved using the nonlinear finite element method, and various examples of PTs are studied. A planar A–M interface propagation is studied under different thermal and mechanical loadings. It is revealed that the effect of phase-dependent elastic properties is more pronounced when thermal strain is included due to the interplay of elastic, transformational and thermal strains. The thermal-induced growth of a martensitic nucleus and the effect of periodic boundary conditions on the nucleus growth are investigated for both phase-independent (PI) and phase-dependent (PD) elastic properties with thermal strain and without it. Martensitic PTs with two variants are studied under different loadings using a one-fourth model and symmetric boundary conditions to reduce the effect of stress concentrations. Martensitic PTs with two variants are also studied in the presence of two circular holes for both the PI and PD elastic properties. This pronounces the significant effect of heterogeneous stress concentration and the size on the PTs. The effect of phase-dependent elastic properties is also studied on twinning in a martensitic grain embedded inside an austenitic matrix under overcooling. The obtained results reveal a significant effect of phase-dependent elastic properties on different types of martensitic PTs and remarkably change the interpretation of structural transformations at the nanoscale.

Lipid and Peptide Diffusion in Bilayers: The Saffman-Delbrück Model and Periodic Boundary Conditions.

The periodic Saffman-Delbrück (PSD) model, an extension of the Saffman-Delbrück model developed to describe the effects of periodic boundary conditions on the diffusion constants of lipids and proteins obtained from simulation, is tested using the coarse-grained Martini and all-atom CHARMM36 (C36) force fields. Simulations of pure Martini dipalmitoylphosphatidylcholine (DPPC) bilayers and those with one embedded gramicidin A (gA) dimer or one gA monomer with sizes ranging from 512 to 2048 lipids support the PSD model. Underestimates of D∞ (the value of the diffusion constant for an infinite system) from the 512-lipid system are 35% for DPPC, 45% for the gA monomer, and 70% for the gA dimer. Simulations of all-atom DPPC and dioleoylphosphatidylcholine (DOPC) bilayers yield diffusion constants not far from experiment. However, the PSD model predicts that diffusion constants at the sizes of the simulation should underestimate experiment by approximately a factor of 3 for DPPC and 2 for DOPC. This likely implies a deficiency in the C36 force field. A Bayesian method for extrapolating diffusion constants of lipids and proteins in membranes obtained from simulation to infinite system size is provided.

Insights on finite size effects in ab initio study of CO adsorption and dissociation on Fe 110 surface

Adsorption and dissociation of hydrocarbons on metallic surfaces represent crucial steps on the path to carburization, eventually leading to dusting corrosion. While adsorption of CO molecules on Fe surface is a barrier-less exothermic process, this is not the case for the dissociation of CO into C and O adatoms and the diffusion of C beneath the surface that are found to be associated with large energy barriers. In practice, these barriers can be affected by numerous factors that combine to favour the CO-Fe reaction such as the abundance of CO and other hydrocarbons as well as the presence of structural defects. From a numerical point of view, studying these factors is challenging and a step-by-step approach is necessary to assess, in particular, the influence of the finite box size on the reaction parameters for adsorption and dissociation of CO on metal surfaces. Here, we use density functional theory (DFT) total energy calculations with the climbing-image nudged elastic band method to estimate the adsorption energies and dissociation barriers for different CO coverages with surface supercells of different sizes. We further compute the effect of periodic boundary condition for DFT calculations and find that the contribution from van der Waals interaction in the computation of adsorption parameters is important as they contribute to correcting the finite-size error in small systems. The dissociation process involves carbon insertion into the Fe surface causing a lattice deformation that requires a larger surface system for unrestricted relaxation. We show that, in the larger surface systems associated with dilute CO-coverages, C-insertion is energetically more favourable, leading to a significant decrease in the dissociation barrier. This observation suggests that a large surface system with dilute coverage is necessary for all similar metal-hydrocarbon reactions in order to study their fundamental electronic mechanisms, as an isolated phenomenon, free from finite-size effects.

Free Surfaces Overcome Superheating in Simulated Melting of Isotactic Polypropylene

The equilibrium melting temperature Tm is a challenging experimental benchmark for molecular dynamics (MD) simulations of polymer melting and crystallization. Tm obtained from MD heating scans of α phase isotactic polypropylene (αiPP) can exhibit superheating of over 100 °C. This superheating has been attributed to the combined effects of periodic boundary conditions and ultrafast heating rates, both of which inhibit nucleation of the melt. We have developed a simple method to minimize this superheating; we replace the periodic crystal structure with a periodic array of finite thickness slabs, separated by vacuum gaps. Thermal disorder at the slab surface promotes melting by reducing the melt nucleation barrier. For experimental comparison, we synthesized and measured the melting temperatures of a series of low-molecular-weight iPP oligomers. In simulations of slab systems, melting initiates at the free surface; as the temperature rises above Tm, the interface advances into the crystal, with a velocity pr...

Boundary condition effects on the dynamic and electric properties of hydration layers.

Water solvation has a central role in several biochemical processes ranging from protein folding to biomolecular recognition and enzyme catalysis. Because of its importance, the structure and dynamics of hydration layers around biological macromolecules have been the targets of a great number of experimental and computational studies. In the present contribution, we have investigated the effects of periodic boundary conditions (PBCs), as used in conjunction with molecular dynamics (MD) simulations, on the dynamic and electric properties of water layers. In particular, we have systematically performed MD simulations of neat water and biomolecules in aqueous solutions by imposing a different external dielectric constant, a generally overlooked parameter in PBC simulations. The effect of the system size has also been addressed. Overall, our results consistently indicate that the dipole moment properties of water layers, and specifically the dipole moment fluctuations and the reorientational correlation functions, can be sensitive to the choice of the external boundary conditions, whereas other molecular properties, such as the self-diffusion coefficient and the reorientational relaxation times, are not affected. We think that our investigation may help to assess appropriate simulation conditions for modeling the aqueous environment of relevant biochemical systems and processes.

DFT simulation of vibrational properties of adenine adsorbed on gold surface: The effect of periodic boundary conditions

We performed simulations of the vibrational properties of the adenine adsorbed on Au(100) gold surface by using three different geometric models. For each geometric model we used two approximations to build partial the Hessian matrix: in the first one the gold atoms were allowed to vibrate while in the second one they are pinned to their relaxed positions. The influence of the vibrations of the first layers in the gold surface upon the vibrational bands of the adenine as well as the influence of a point-like defect in the surface is discussed in detail. The comparison with experimental results indicates that two maxima of the SERS signals are present (just above 300 and 600cm−1) while the values obtained from our calculations are 327cm−1 and 625cm−1. For the frequencies ranging between 0 and 700cm−1 we note shifts of up to 40cm−1 caused by the use of different computational models. For the frequencies larger than 700cm−1 there are practically no differences between the models used by us. The presence of a point-like defect on the surface leads to important modification of the vibrational properties of adenine in the low frequency regime. While in the case of VbDOS projected over the gold atoms the presence of the defect leads to minimal changes of the vibrational density of states, the influence of the defect of the adenine’s atoms is more important. This is clearly visible by comparing the VbDOS at low frequencies.

Effect of periodic boundary conditions on granular motion in horizontal rotating cylinders modelled using the DEM

Applying periodic boundary conditions to DEM simulations of granular motion in horizontal, rotating cylinders can result in unexpected bulk axial flow. This axial flow is shown to contribute significantly to the mean square deviation in particle position and consequently must be considered in the analysis of processes governed by slow axial motion such as axial dispersion.

Decay of magnetohydrodynamic turbulence at low magnetic Reynolds number

We report a detailed numerical investigation of homogeneous decaying turbulence in an electrically conducting fluid in the presence of a uniform constant magnetic field. The asymptotic limit of low magnetic Reynolds number is assumed. Large-eddy simulations with the dynamic Smagorinsky model are performed in a computational box sufficiently large to minimize the effect of periodic boundary conditions. The initial microscale Reynolds number is about 170 and the magnetic interaction parameter N varies between 0 and 50. We find that except for a short period of time when N = 50, the flow evolution is strongly influenced by nonlinearity and cannot be adequately described by any of the existing theoretical models. One particularly noteworthy result is the near equipartition between the rates of Joule and viscous dissipations of the kinetic energy observed at all values of N during the late stages of the decay. Further, the velocity components parallel and perpendicular to the magnetic field decay at different rates, whose value depends on the strength of the magnetic field and the stage of the decay. This leads to a complex evolution of the Reynolds stress anisotropy ellipsoid, which goes from being rod-shaped, through spherical to disc-shaped. We also discuss the possibility of the power-law decay, the comparison between computed, experimental and theoretical decay exponents, the anisotropy of small-scale fluctuations, and the evolution of the spectral energy distributions.

Phase diagrams of SU(N) gauge theories with fermions in various representations

We minimize the one-loop effective potential for SU(N) gauge theories including fermions with finite mass in the fundamental (F), adjoint (Adj), symmetric (S), and antisymmetric (AS) representations. We calculate the phase diagram on S1 ? 3 as a function of the length of the compact dimension, ?, and the fermion mass, m, for various N and Nf. We consider the effect of periodic boundary conditions [PBC(+)] on fermions as well as antiperiodic boundary conditions [ABC(-)]. With standard ABC(-) on fermions only the deconfined phase is found at one-loop for all representations considered. However, the use of PBC(+) produces a rich phase structure. These phases are distinguished by the eigenvalues of the Polyakov loop P. In the case of fundamental representation fermions [QCD(F,+)], a phase in which Re?Tr P is minimized (and negative) is favoured for all values of m?. For N odd charge conjugation () symmetry is spontaneously broken in this phase due to (1/N) effects. Minimization of the effective potential for QCD(AS/S,+) results in a phase where |Im?Tr P| is maximized, resulting in -breaking for all N and all values of m?, however, the partition function is the same up to (1/N) corrections as when ABC are applied. Therefore, regarding orientifold planar equivalence, we argue that in the one-loop approximation -breaking in QCD(AS/S,+) resulting from the application of PBC on fermions does not invalidate the large N equivalence with QCD(Adj,-). Similarly, with respect to orbifold planar equivalence, breaking of Z2 interchange symmetry resulting from application of PBC to bifundamental (BF) representation fermions does not invalidate equivalence with QCD(Adj,-) in the one-loop perturbative limit because the partition functions of QCD(BF,-) and QCD(BF,+) are the same. Of particular interest as well is the case of adjoint fermions where for 1$>Nf > 1 Majorana flavour confinement is obtained for sufficiently small m?, and deconfinement for sufficiently large m?. For N ? 3 these two phases are separated by one or more additional phases, some of which can be characterized as partially-confining phases.

Design of all-glass multilayer phase gratings for cylindrical microlenses

We introduce a design method for diffractive cylindrical microlenses fabricated with a new technology similar to the fabrication of all-solid photonic crystal fibers. Unlike conventional microlenses that are fabricated with etching methods and thus have a step-index profile, the refractive index of each layer can be individually designed. We study the transmitted field of such nonperiodic lamellar phase grating. By using the field-stitching method we can suppress the effect of periodic boundary conditions of the Fourier modal method when calculating the transmitted field of nonperiodic lamellar phase elements. We suggest an algorithm to design multilayer phase elements, which act as cylindrical lenses. We show experimental and theoretical data for a diffraction-limited lens.

CaSPA – an algorithm for calculation of the size of percolating aggregates

We present an algorithm (CaSPA) which accounts for the effects of periodic boundary conditions in the calculation of size of percolating aggregated clusters. The algorithm calculates the gyration tensor, allowing for a mixture of infinite (macroscale) and finite (microscale) principle moments. Equilibration of a triblock copolymer system from a disordered initial configuration to a hexagonal phase is examined using the algorithm.

Degradation of a nano-cutting tool: An MD simulation

We present the results of molecular-dynamics simulations of repeated interactions of a realistically-shaped cutting edge with a model, infinitely hard grain. The model edge is composed of several hundred thousand atoms of an fcc metal treated with the Sutton–Chen potential, moving with a constant speed of 20ms−1. Plastic deformations appearing upon contact with the rigid obstacle are observed and described by means of temperature fields and the atomic slip vector. Preferred slip planes are identified and the normal force experienced by the tool is investigated. The effect of periodic boundary conditions along the z (in-plane) direction is assessed by comparing the results of two simulations, differing only by the simulation box’s length along the z-axis.

Structural Properties of Carbon Nanogears

Structural stabilities of different types of carbon nanogears have been tested against temperature by means of a molecular dynamics procedure. Effects of periodic boundary conditions were also examined. It has been found that although the two types of nanogears (armchair and zigzag CNT yielding) investigated look similar in configuration, when tested against high temperatures, bond breakings and deformations occur at different regions.

Permeation of ions through a model biological channel: effect of periodic boundary conditions and cell size

The effect of the simulation cell size and periodic boundary conditions on non-equilibrium molecular dynamics simulations of the structure and dynamics with explicit water molecules and ions in and near a model channel in a biological membrane is considered. The approach seems satisfactory. In particular, the presence of image channels that often contain image ions seems to have little effect on the average structure, channel content, or current for this system.

Effect of boundary conditions on quantum-anti-dots in the presence of a magnetic field

Many people have studied the conductance properties through an array of anti-dots, especially since the observation of Weiss oscillations. In most cases, however, in which the recursive Greenźs functions are used on a spatial lattice, periodic boundary conditions are employed. In this paper, we analyse the effects of boundary conditions and magnetic field on the conductance behavior in a number of anti-dot-shaped, GaAs/AlGaAs 2DEG quantum systems. The effect of periodic boundary conditions causes a reduction in the overall conductance. The effect of changing the boundary conditions is more profound for lower numbers of anti-dots.