A singularly perturbed reaction diffusion problem forthe nonlinear boundary condition with two parameters

A class of singularly perturbed initial boundary value problems of reaction diffusion equations for the nonlinear boundary condition with two parameters is considered. Under suitable conditions, by using the theory of differential inequalities, the existence and the asymptotic behaviour of the solution for the initial boundary value problem are studied. The obtained solution indicates that there are initial and boundary layers and the thickness of the boundary layer is less than the thickness of the initial layer.     

Three-dimensional free vibration analysis of isotropic rectangular plates using the B-spline Ritz method

This paper presents three-dimensional free vibration analysis of isotropic rectangular plates with any thicknesses and arbitrary boundary conditions using the B-spline Ritz method based on the theory of elasticity. The proposed method is formulated by the Ritz procedure with a triplicate series of B-spline functions as amplitude displacement components. The geometric boundary conditions are numerically satisfied by the method of artificial spring. To demonstrate the convergence and accuracy of the present method, several examples with various boundary conditions are solved, and the results are compared with other published solutions by exact and other numerical methods based on the theory of elasticity and various plate theories. Rapid, stable convergences as well as high accuracy are obtained by the present method. The effects of geometric parameters on the vibrational behavior of cantilevered rectangular plates are also investigated. The results reported here may serve as benchmark data for finite element solutions and future developments in numerical methods.     

Free vibrations of cylindrical shells with elastic-support boundary conditions

This paper concerns the free vibrations of cylindrical shells with elastic boundary conditions. Based on the Flügge classical thin shell theory, the equations of motion for the cylindrical shells are solved by the method of wave propagations. The wave numbers are obtained by directly solving an eighth order equation. The elastic-support boundary conditions can be arbitrarily specified in terms of 8 independent sets of distributed springs. All the classical homogeneous boundary conditions can be considered as the special cases when the stiffness for each set of springs is equal to either infinity or zero. The present solutions are validated by the results previously given by other researchers and/or obtained using finite element models. The effects on the frequency parameters of elastic restraints are investigated for shells of different geometrical characteristics.     

Two-phase flow in porous media: dynamical phase transition

We study numerically the behavior of two-phase flow in porous media via the parameters capillary number and viscosity ratio, under steady-state conditions and various levels of saturation. We construct a phase diagram, where the phases are defined according to whether one or both fluids move. We establish a semi-empirical theory for the location of the phase boundaries. The steady-state conditions are obtained by implementing biperiodic boundary conditions.     

Magneto-elastic dynamics and bifurcation of rotating annular plate

In this paper, magneto-elastic dynamic behavior, bifurcation, and chaos of a rotating annular thin plate with various boundary conditions are investigated. Based on the thin plate theory and the Maxwell equations, the magneto-elastic dynamic equations of rotating annular plate are derived by means of Hamilton's principle. Bessel function as a mode shape function and the Galerkin method are used to achieve the transverse vibration differential equation of the rotating annular plate with different boundary conditions. By numerical analysis, the bifurcation diagrams with magnetic induction, amplitude and frequency of transverse excitation force as the control parameters are respectively plotted under different boundary conditions such as clamped supported sides, simply supported sides, and clamped-one-side combined with simply-anotherside. Poincare′ maps, time history charts, power spectrum charts, and phase diagrams are obtained under certain conditions,and the influence of the bifurcation parameters on the bifurcation and chaos of the system is discussed. The results show that the motion of the system is a complicated and repeated process from multi-periodic motion to quasi-period motion to chaotic motion, which is accompanied by intermittent chaos, when the bifurcation parameters change. If the amplitude of transverse excitation force is bigger or magnetic induction intensity is smaller or boundary constraints level is lower, the system can be more prone to chaos.     

Quantum brownian motion on a triangular lattice and c=2 boundary conformal field theory

We study a single particle diffusing on a triangular lattice and interacting with a heat bath, using boundary conformal field theory (CFT) and exact integrability techniques. We derive a correspondence between the phase diagram of this problem and that recently obtained for the 2-dimensional 3-state Potts model with a boundary. Exact results are obtained on phases with intermediate mobilities. These correspond to nontrivial boundary states in a conformal field theory with 2 free bosons which we explicitly construct for the first time. These conformally invariant boundary conditions are not simply products of Dirichlet and Neumann ones and unlike those trivial boundary conditions, are not invariant under a Heisenberg algebra.     

Boundary conditions to the Ginzburg-Landau equations at the twinning plane in a (<Emphasis Type="Italic">d</Emphasis>+<Emphasis Type="Italic">s</Emphasis>) superconductor

Effective boundary conditions to the Ginzburg-Landau equations at the twinning plane of an orthorhombic (d+s) superconductor are obtained on the basis of microscopic BCS theory. The range of the parameters of orthorhombic strain is found where the order parameter behaves variously on both sides of the twinning boundary.     

栅状表面液晶盒的光学属性

基于栅状表面等效锚定能公式和弹性理论,对Bryan-Brown等人提出的混和排列向列相液晶盒进行研究.通过变分理论得到指向矢倾角和扭曲角满足的方程和边界条件.根据这些方程和边界条件,利用差分迭代方法计算出指向矢倾角和扭曲角对应某一个δ/λ值在不同电压下的分布.应用琼斯矩阵方法,研究这种液晶盒的电光效应.这些结果都与栅状表面的几何参数有关,即与周期λ和振幅δ有关.     

Free vibration of rectangular nanoplates using Rayleigh–Ritz method

Vibration analysis of isotropic rectangular nanoplates based on the classical plate theory in conjunction with Eringen's nonlocal elasticity theory is considered. Nanoplates are one of the structural units that are used in nanoscale applications. In this study, Rayleigh–Ritz method with algebraic polynomial displacement function is used to solve the vibration problem of isotropic rectangular nanoplates subjected to different boundary conditions. The advantage of the method is that one can easily handle the specified boundary conditions at the edges. A comparison of the results with those available in the literature has been made. The proposed method is also validated by convergence studies. Frequency parameters are given for different nonlocality parameters, length of nanoplates and boundary conditions. The study highlights that nonlocality effects increase with the increase in mode number and the influence of nonlocal effects becomes increasingly pronounced for higher order vibration modes. Three-dimensional mode shapes for the specified nanoplates have also been presented.     

Vibration analysis of functionally graded annular plates with mixed boundary conditions in thermal environment

The free vibration analysis of functionally graded annular plates with mixed boundary conditions in thermal environment is carried out by the 3D elasticity theory and the Chebyshev–Ritz method. The material properties are assumed to be temperature dependent and graded in the thickness direction. The mixed boundary conditions which include upper and lower surfaces partially fixed, inner side partially fixed and outer side partially fixed are considered, respectively. The accuracy of the present approach for solving the free vibration of the plates with different boundary conditions is validated by comparing the present numerical results with the results available. The effects of the different mixed boundary conditions, the temperature rise, the material graded index and the geometrical parameters on the eigen-frequencies are studied.     

Axisymmetric vibrations of polar orthotropic mindlin annular plates of variable thickness

Analysis and numerical results are presented for the axisymmetric vibrations of polar orthotropic annular plates with linear variation in thickness, according to Mindlin's shear theory of plates. A chebyshev collocation technique has been employed to obtain the frequency equations for the transverse motion of such plates, for three different boundary conditions. Frequencies, mode shapes and moments for the first three modes of vibration have been computed for different plate parameters. A comparison of frequencies with the corresponding values obtained by classical plate theory leads to some interesting conclusions.     

Boundary conditions for the bending of a piezoelectric beam

For beam bending in transversely isotropic piezoelectric media, the reciprocal theorem and the general solution of piezoelasticity are applied in a novel way to obtain the appropriate stress and mixed boundary conditions accurate to all orders for the beam of general edge geometry and loadings. By generalizing the method developed by Gregory and Wan, a set of necessary conditions on the edge-data for the existence of a rapidly decaying solution is established. The prescribed edge-data of the beam must satisfy these conditions in order that they could generate a decaying state within the beam. When stress and mixed conditions are imposed on the beam edge, these decaying state conditions for the case of bending deformation of piezoelectric beam are derived explicitly. They are then used for the correct formulation of boundary conditions for the beam theory solution (or the interior solution). Besides, an analytical solution of elastic beam is formulated to verify validity of our boundary conditions. For the stress data, our boundary conditions coincide with those obtained in conventional forms of beam theories. More importantly, the appropriate boundary conditions with two sets of mixed edge-data are obtained for the first time.     

The nano scale vibration of protein microtubules based on modified strain gradient theory

This paper investigates the free vibration of protein microtubules (MTs) embedded in the cytoplasm by using linear and nonlinear Euler–Bernoulli beam model based on modified strain gradient theory. The protein microtubule is modeled as a simply support or clamped–clamped beam. Beside, the elastic medium surrounding of MTs is modeled with Pasternak foundation. Vibration equations are obtained by using Hamilton principle and these equations are solved according to boundary conditions. Finally the dependency of vibration frequencies on environmental conditions, MTs size, changes of temperature and material length scale parameters (size effects) is studied. By comparing the findings, it could be said that the MTs' frequency is greatly increased in the presence of cytoplasm and it is very dependent to material length scale parameters.     

Free vibration study of layered cylindrical shells by collocation with splines

The free vibration of circular cylindrical thin shells, made up of uniform layers of isotropic or specially orthotropic materials, is studied using point collocation method and employing spline function approximations. The equations of motion for the shell are derived by extending Love's first approximation theory. Assuming the solution in a separable form a system of coupled differential equations, in the longitudinal, circumferential and transverse displacement functions, is obtained. These functions are approximated by Bickley-type splines of suitable orders. The process of point collocation with suitable boundary conditions results in a generalized eigenvalue problem from which the values of a frequency parameter and the corresponding mode shapes of vibration, for specified values of the other parameters, are obtained. Two types of boundary conditions and four types of layers are considered. The effect of neglecting the coupling between the flexural and extensional displacements is analysed. The influences of the relative layer thickness, a length parameter and a total thickness parameter on the frequencies are studied. Both axisymmetric and asymmetric vibrations are investigated. The effect of the circumferential node number on the vibrational behaviour of the shell is also analysed.     

Analysis of Asymmetrical Coplanar Strip Lines with Finite Boundary Substrate

Analytical closed-form expressions for the quasi-TEM parameters for asymmetrical coplanar strip lines (ACPS) with finite boundary substrate are obtained using conformal mapping method. The formulations are CAD-oriented purpose. The calculated results are compared with those obtained for a corresponding ACPS assumed that the substrate is infinite. The analyses are useful for MMIC's designs and transmission line theory.     

Finite size effects in the spin-1 XXZ and supersymmetric sine-Gordon models with Dirichlet boundary conditions

Starting from the Bethe Ansatz solution of the open integrable spin-1 XXZ quantum spin chain with diagonal boundary terms, we derive a set of nonlinear integral equations (NLIEs), which we propose to describe the boundary supersymmetric sine-Gordon model BSSG⁺ with Dirichlet boundary conditions on a finite interval. We compute the corresponding boundary S matrix, and find that it coincides with the one proposed by Bajnok, Palla and Takács for the Dirichlet BSSG⁺ model. We derive a relation between the (UV) parameters in the boundary conditions and the (IR) parameters in the boundary S matrix. By computing the boundary vacuum energy, we determine a previously unknown parameter in the scattering theory. We solve the NLIEs numerically for intermediate values of the interval length, and find agreement with our analytical result for the effective central charge in the UV limit and with boundary conformal perturbation theory.     

Boundary conditions for regularized 13-moment-equations for micro-channel-flows

Boundary conditions are the major obstacle in simulations based on advanced continuum models of rarefied and micro-flows of gases. In this paper, we present a theory how to combine the regularized 13-moment-equations derived from Boltzmann’s equation with boundary conditions obtained from Maxwell’s kinetic accommodation model. While for the linear case these kinetic boundary conditions suffice, we need additional conditions in the non-linear case. These are provided by the bulk solutions obtained after properly transforming the equations while keeping their asymptotic accuracy with respect to Boltzmann’s equation.After finding a suitable set of boundary conditions and equations, a numerical method for generic shear flow problems is formulated. Several test simulations demonstrate the stable and oscillation-free performance of the new approach.     

Scattering of electromagnetic waves by a randomly rough surface as a boundary problem of laser radiation interaction with light-scattering materials and media

The light scattering by a rough surface with random Gaussian fluctuations of roughness is studied in the case of coarse roughness, whose parameters—mean deviation and correlation length—are much greater than the radiation wavelength. Closed analytical solutions of the problem are presented in terms of radiophysics for the boundary conditions of an ideal conductor and the impedance boundary conditions. These solutions are formulated in terms of a photometric scattering indicatrix. The possibility of their application to the problems of photometry and theory of radiative transfer and scattering in turbid media, in particular, in simulation of the process of boundary scattering of laser radiation by rough surfaces of biological tissues and media, is discussed.     

Nonlinear curvature-based model and resonant finite-amplitude vibrations of symmetric cross-ply laminates

Moving from a general plate theory, a modified general classical laminated plate theory (MGCLPT) exhibiting nonlinear curvatures but still allowing for some worth features of linear curvature models (von Karman) is formulated. Starting from MGCLPT partial differential equations, a minimal discretized model suitable for the analysis of resonant finite-amplitude vibrations of symmetric cross-ply laminates, with immovable or movable supports, is obtained via the Galerkin procedure. Periodic responses of a single-mode model and of a 3:1 internally resonant two-mode model excited at primary resonance are obtained via the multiple time scale method. The influence of various system parameters (thickness ratio, plate aspect, number of laminae, kind of material, mode number) is addressed, and the comparison of nonlinear vibration results as obtained with the MGCLPT and the von Karman models for different boundary conditions shows some interesting differences.