Periodic Boundary Conditions for Finite-Differentiation-Method Fast-Fourier-Transform Micromagnetics

We describe an accurate periodic boundary condition （PBC） called the symmetric PBC in the calculation of the magnetostatie interaction field in the finite-differentiation-method fast-Fourier-transform （FDM-FFT） micromagneties. The micromagnetic cells in the regular mesh used by the FDM-FFT method are finite-sized elements, but not geometrical points. Therefore, the key PBC operations for FDM-FFT methods are splitting and relocating the micromagnetic cell surfaces to stay symmetrically inside the box of half-total sizes with respect to the origin. The properties of the demagnetizing matrix of the split micromagnetic cells are discussed, and the sum rules of demagnetizing matrix are fulfilled by the symmetric PBC.

Integral Operator Solving Process of the Boundary Value Problem of Abstract Kinetic Equation with the First Kind of Critical Parameter and Generalized Periodic Boundary Conditions

In this paper the concepts of the boundary value problem of abstract kinetic equation with the first kind of critical parameter γ 0 and generalized periodic boundary conditions are introduced in a Lebesgue space which consists of functions with vector valued in a general Banach space, and then describe the solution of these abstract boundary value problem by the abstract linear integral operator of Volterra type. We call this process the integral operator solving process.

Applying Periodic Boundary Conditions to Predict Open Water Propeller Performance

Mathematical models of propellers were created that investigate the influence of periodic boundary conditions on predictions of a propeller's performance.Thrust and torque coefficients corresponding to different advance coefficients of DTMB 4119, 4382, and 4384 propellers were calculated.The pressure coefficient distribution of the DTMB 4119 propeller at different sections was also physically tested.Comparisons indicated good agreement between the results of experiments and the simulation.It showed that the periodic boundary condition can be used to rationally predict the open water performance of a propeller.By analyzing the three established modes for the computation, it was shown that using the spline curve method to divide the grids can meet the calculation's demands for precision better than using the rake cutting method.

SPECTRAL METHOD FOR ZAKHAROV EQUATION WITH PERIODIC BOUNDARY CONDITIONS

This paper describes the spectral method for numerically solving Zakharov equation with periodicboundary conditions. This method is spectral method for spatial variable and difference method fortime variable. We make error estimation of approximate solution and prove the convergence of spectralmethod. We had given the convergence rate. Also, we prove the stability of approximate method forinitial values.

PERIODIC SYSTEMS WITH TIME DEPENDENT MAXIMAL MONOTONE OPERATORS

We consider a first order periodic system in R^(N),involving a time dependent maximal monotone operator which need not have a full domain and a multivalued perturbation.We prove the existence theorems for both the convex and nonconvex problems.We also show the existence of extremal periodic solutions and provide a strong relaxation theorem.Finally,we provide an application to nonlinear periodic control systems.

Effects of Periodic Boundary Condition on Traffic Waves

In this paper, we use the speed-gradient model proposed by Jiang et al. [Transp. Res. B 36 （2002） 405] to study the effect of boundary condition on shock and rarefaction wave. Our numerical results show that this model can reproduce the evolution of the two traffic waves, which further proves that this model can be used to perfectly explore the consequences caused by various boundary conditions.

NUMERICAL ANALYSIS OF LONGTIME DYNAMIC BEHAVIOR IN WEAKLY DAMPED FORCED KdV EQUATION

The numerical analysis of the approximate inertial manifold in,weakly damped forced KdV equation is given. The results of numerical analysis under five models is the same as that of nonlinear spectral analysis.

A density functional theory study on parameters fitting of ultra long armchair(n,n) single walled boron nitride nanotubes

Armchair （n, n） single walled boron nitride nanotubes with n = 2-17 are studied by the density functional theory at the B3LYP/3-21G（d） level combined with the periodic boundary conditions for simulating the ultra long model. The results show that the structure parameters and the formation energies bear a strong relationship to n. The fitted analytical equations are developed with correlation coefficients larger than 0.999. The energy gaps of （2, 2） and （3, 3） tubes are indirect gaps, and the larger tubes （n = 4-17） have direct energy gaps. Results show that the armchair boron nitride nanotubes （n = 2-17） are insulators with wide energy gaps of between 5.93 eV and 6.23 eV.

Entanglement in an anisotropic spin-1 Heisenberg chain

The entanglement in an anisotropic spin-1 Heisenberg chain with a uniform magnetic field is investigated. The ground-state entanglement will undergo two different kinds of transitions when the anisotropy △ and the amplitude of the magnetic field B are varied. The thermal entanglement of the nearest neighbour always declines when B increases no matter what the value of the anisotropy is. It is very interesting to note that the entanglement of the next-nearest neighbour can increase to a maximum at a certain magnetic field. Regardless of the boundary condition, the nearestneighbour entanglement always decreases and approaches to a constant value when the size of the system is very large. The constant value of open boundary condition is much larger than that of periodic boundary condition.

THE GLOBAL ATTRACTOR FOR A VISCOUS WEAKLY DISSIPATIVE GENERALIZED TWO-COMPONENT μ-HUNTER-SAXTON SYSTEM

This article is concerned with the existence of global attractor of a weakly dissipative generalized two-component μ-Hunter-Saxton （gμHS2） system with viscous terms. Under the period boundary conditions and with the help of the Galerkin procedure and compactness method, we first investigate the existence of global solution for the viscous weakly dissipative （gμHS2） system. On the basis of some uniformly prior estimates of the solution to the viscous weakly dissipative （gμHS2） system, we show that the semi-group of the solution operator {S（t）}t≥0 has a bounded absorbing set. Moreover, we prove that the dynamical system {S（t）}t≥0 possesses a global attractor in the Sobolev space H2（S） × H2（S）.

Stress Analysis of Wire Strands by Mesoscale Mechanics

Steel wire ropes have wide application in a variety of engineering fields such as ocean engineering and civil engineering.The stress calculation for steel wire ropes is of crucial importance when conducting strength and fatigue analyses.In this study,we performed a finite element analysis of single-strand steel wire ropes.For the geometric modeling,we used an analytic geometry of space method.We established helical line equations and used the coordinates of the contact points.The finite-element model was simplified using the periodic law.Periodic boundary conditions were used to simulate a wire strand of infinite length under tensile strain,for which we calculated the cross-sectional stresses and inner forces.The results showed that bending and torsion moments emerged when the wire strand was under tensile load.In some cases,the bending stress reached 18%of the tensile stress,and the torsion stress reached 29%of the tensile stress,which means that the total stress was higher than the nominal stress.Whereas in ear-lier studies,a conservative prediction of nominal stress was not possible,the results of our strength and fatigue analyses were more conservative.

Bandgap characteristics of 2D plasma photonic crystal with oblique incidence:TM case

A novel periodic boundary condition （PBC）, that is the constant transverse wavenumber （CTW） method, is introduced to solve the time delay in the transverse plane with oblique incidence. Based on the novel PBC, the FDTD/PBC algorithm is proposed to study periodic structure consisting of plasma and vacuum. Then the reflection coefficient for the plasma slab from the FDTD/PBC algorithm is compared with the analytic results to show the validity of our technique. Finally, the reflection coefficients for the plasma photonic crystals are calculated using the FDTD/PBC algorithm to study the variation of bandgap characteristics with the incident angle and the plasma parameters. Thus it has provided the guiding sense for the actual manufacturing plasma photonic crystal.

GLOBAL WEAK SOLUTIONS TO ONE-DIMENSIONAL COMPRESSIBLE VISCOUS HYDRODYNAMIC EQUATIONS

In this article, we are concerned with the global weak solutions to the 1D com- pressible viscous hydrodynamic equations with dispersion correction δ2ρ（（φ（ρ））xxφ′（ρ））x with φ（ρ） = ρα. The model consists of viscous stabilizations because of quantum Fokker-Planck operator in the Wigner equation and is supplemented with periodic boundary and initial con- ditions. The diffusion term εuxx in the momentum equation may be interpreted as a classical conservative friction term because of particle interactions. We extend the existence result in [1] （α=1/2） to 0 〈 α ≤ 1. In addition, we perform the limit ε→0 with respect to 0 〈 α ≤1/2.

A LYAPUNOV-TYPE INEQUALITY FOR A PERIODIC BOUNDARY VALUE PROBLEM OF A FRACTIONAL DIFFERENTIAL EQUATION

In this paper, we establish a Lyapunov-type inequality for fractional differential periodic boundary-value problems. As applications, a necessary condition is obtained to ensure the existence and uniqueness of nontrivial solutions to this problem.

RRS-PBC: a molecular approach for periodic systems

Technically, when dealing with a perfect crystal, methods （PBC） in conjunction with plane-wave basis sets are widely in k-（reciprocal） space that impose periodic boundary conditions used. Chemists, however, tend to think of a solid as a giant mole- cule, which offers a molecular way to describe a solid by using a finite cluster model （FCM）. However, FCM may fail to sim- ulate a perfect crystal due to its inevitable boundary effects. We propose an RRS-PBC method that extracts the k-space infor- mation of a perfect crystalline solid out of a reduced real space （RRS） of an FCM. We show that the inevitable boundary effects in an FCM are eliminated naturally to achieve converged high-quality band structures.

EXISTENCE OF POSITIVE PERIODIC SOLUTIONS TO A SECOND-ORDER DIFFERENTIAL INCLUSION

In this paper,the existence of positive periodic solutions to a second-order differential inclusion is considered.Some existence results are established by the fixed-point theorem for multivalued operators and Sobolev constant.

A Finite Element Method Solver for Time-Dependent and Stationary Schrodinger Equations with a Generic Potential

A general finite element solution of the Schrodinger equation for a onedimensional problem is presented.The solver is applicable to both stationary and time-dependent cases with a general user-selected potential term.Furthermore,it is possible to include external magnetic or electric fields,as well as spin-orbital and spinmagnetic interactions.We use analytically soluble problems to validate the solver.The predicted numerical auto-states are compared with the analytical ones,and selected mean values are used to validate the auto-functions.In order to analyze the performance of the time-dependent Schrodinger equation,a traveling wave package benchmark was reproduced.In addition,a problem involving the scattering of a wave packet over a double potential barrier shows the performance of the solver in cases of transmission and reflection of packages.Other general problems,related to periodic potentials,are treated with the same general solver and a Lagrange multiplier method to introduce periodic boundary conditions.Some simple cases of known periodic potential solutions are reported.

Multiaxial fatigue life prediction of composite materials

In order to analyze the stress and strain fields in the fibers and the matrix in composite materials,a fiber-scale unit cell model is established and the corresponding periodical boundary conditions are introduced.Assuming matrix cracking as the failure mode of composite materials,an energy-based fatigue damage parameter and a multiaxial fatigue life prediction method are established.This method only needs the material properties of the fibers and the matrix to be known.After the relationship between the fatigue damage parameter and the fatigue life under any arbitrary test condition is established,the multiaxial fatigue life under any other load condition can be predicted.The proposed method has been verified using two different kinds of load forms.One is unidirectional laminates subjected to cyclic off-axis loading,and the other is filament wound composites subjected to cyclic tension-torsion loading.The fatigue lives predicted using the proposed model are in good agreements with the experimental results for both kinds of load forms.

Influence of aspect ratio and arrangement direction on the shear behavior of ellipsoids

This work studies the macroscopic and microscopic behaviors of ellipsoids under triaxial tests using 3D discrete element method(DEM)simulation.To avoid the boundary effect,a novel stress servo-controlled periodic boundary condition is proposed to maintain the confining pressure of samples during testing.The shape features of ellipsoids are investigated,including the aspect ratio of elongated/oblate ellipsoids and the initial arrangement directions of ellipsoids.The macroscopic properties of ellipsoidal particle samples,such as the deviatoric stress,volumetric strain,internal friction angle,as well as dilatancy angles are explored.Elongated and oblate ellipsoids with varying aspect ratios are investigated for the occurrence of stick-slips.In addition,it is demonstrated that the initial arrangement direction has a significant impact on the coordination number and contact force chains.The corresponding anisotropy coefficients of the entire contact network are analyzed to probe the microscopic roots of macroscopic behavior.

Theoretical aspects of selecting repeated unit cell model in micromechanical analysis using displacement-based finite element method

Repeated Unit Cell（RUC）is a useful tool in micromechanical analysis of composites using Displacement-based Finite Element（DFE）method,and merely applying Periodic Displacement Boundary Conditions（PDBCs）to RUC is almost a standard practice to conduct such analysis.Two basic questions arising from this practice are whether Periodic Traction Boundary Conditions（PTBCs,also known as traction continuity conditions）are guaranteed and whether the solution is independent of selection of RUCs.This paper presents the theoretical aspects to tackle these questions,which unify the strong form,weak form and DFE method of the micromechanical problem together.Specifically,the solution’s independence of selection of RUCs is dealt with on the strong form side,PTBCs are derived from the weak form as natural boundary conditions,and the validity of merely applying PDBCs in micromechanical Finite Element（FE）analysis is proved by referring to its intrinsic connection to the strong form and weak form.Key points in the theoretical aspects are demonstrated by illustrative examples,and the merits of setting micromechanical FE analysis under the background of a clear theoretical framework are highlighted in the efficient selection of RUCs for Uni Directional（UD）fiber-reinforced composites.