The attenuation of the higher-order cross-section modes in a duct with a thin porous layer

A numerical method for sound propagation of higher-order cross-sectional modes in a duct of arbitrary cross-section and boundary conditions with nonzero, complex acoustic admittance has been considered. This method assumes that the cross-section of the duct is uniform and that the duct is of a considerable length so that the longitudinal modes can be neglected. The problem is reduced to a two-dimensional (2D) finite element (FE) solution, from which a set of cross-sectional eigen-values and eigen-functions are determined. This result is used to obtain the modal frequencies, velocities and the attenuation coefficients. The 2D FE solution is then extended to three-dimensional via the normal mode decomposition technique. The numerical solution is validated against experimental data for sound propagation in a pipe with inner walls partially covered by coarse sand or granulated rubber. The values of the eigen-frequencies calculated from the proposed numerical model are validated against those predicted by the standard analytical solution for both a circular and rectangular pipe with rigid walls. It is shown that the considered numerical method is useful for predicting the sound pressure distribution, attenuation, and eigen-frequencies in a duct with acoustically nonrigid boundary conditions. The purpose of this work is to pave the way for the development of an efficient inverse problem solution for the remote characterization of the acoustic boundary conditions in natural and artificial waveguides.     

Quantum-hydrodynamic approach to the problem of electron exchange between atomic particles and nanosystems

The application of quantum-hydrodynamic methods for solving the problem of electron exchange between atomic particles and solid surfaces, and nanosystems has been examined. The derivation of a system of equations that is alternative to the nonstationary Schrödinger equation is given to describe the dynamics of electronic processes with variable charge and current densities. A comparison of results of solving the nonstationary Schrödinger equation and the quantum-hydrodynamic system of equations shows that both approaches give a good coincidence. The numerical solution to the system of quantum-hydrodynamic equations has a number of advantages, because it does not lead to oscillations at the boundary of the computational mesh and nor to the problem of exponential growth in numerical complexity for many-electron systems.     

运动边界产生的孤立波解

本文给出了求解劝边界Boussinesq方程的差分格式,计算了动边界以不同形式运动时所产生的波。数值结果表明了孤立波解的存在。     

Plasma-sheath transition in the magnetized plasma-wall problem for collisionless ions

A model for the collisionless plasma-wall problem under the action of an applied magnetic field is developed. The behavior of its solution is examined and found to be qualitatively consistent with experiment. The plasma and the sheath are then modeled separately to obtain the position of the quasi-neutral plasma boundary and the position of the edge of the electron-free sheath. It is shown that the plasma boundary can be specified as the point where the component of the ion velocity normal to the wall reaches the ion sound speed (Bohm criterion), and the sheath edge is specified as the point corresponding to Godyak's condition for the electric field. Studying the behavior near the plasma boundary and the sheath edge, the plasma solution and the solution of the space charge region are patched together to approximate the solution of the plasma-wall problem.     

Charge Transfer in the Lattice with an Impurity Site. Reflection and Transmission of the Wave Packet

The quantum dynamics of the propagation of the charge wave function in a uniform lattice containing a single impurity site is considered. A nonstationary problem is solved in the tight-binding approximation. The initial state is the wave function fully localized at one of the lattice sites. The coefficients of transmission of the wave packet through the impurity site and reflections from it are calculated as a function of the parameters of the problem, that is, the additional energy on the impurity and the distance between the impurity and the initial position of the charge. The problem is solved for two types of boundary conditions: an infinite and a semi-infinite lattice. Good agreement with numerical simulation is obtained.     

Fourier spectral embedded boundary solution of the Poisson’s and Laplace equations with Dirichlet boundary conditions

A Fourier spectral embedded boundary method, for solution of the Poisson’s equation with Dirichlet boundary conditions and arbitrary forcing functions (including zero forcing function), is presented in this paper. This iterative method begins by transformation of the Dirichlet boundary conditions from the physical boundaries to some corresponding regular grid points (which are called the numerical boundaries), using a second order interpolation method. Then the transformed boundary conditions and the forcing function are extended to a square, smoothly and periodically, via multiplying them by some suitable error functions. Instead of direct solution of the resulting extended Poisson’s problem, it is suggested to define and solve an equivalent transient diffusion problem on the regular domain, until achievement of the steady solution (which is considered as the solution of the original problem). Without need of any numerical time integration method, time advancement of the solution is obtained directly, from the exact solution of the transient problem in the Fourier space. Consequently, timestep sizes can be chosen without stability limitations, which it means higher rates of convergence in comparison with the classical relaxation methods. The method is presented in details for one- and two-dimensional problems, and a new emerged phenomenon (which is called the saturation state) is illustrated both in the physical and spectral spaces. The numerical experiments have been performed on the one- and two-dimensional irregular domains to show the accuracy of the method and its superiority (from the rate of convergence viewpoint) to the other classical relaxation methods. Capability of the method, in dealing with complex geometries, and in presence of discontinuity at the boundaries, has been shown via some numerical experiments on a four-leaf shape geometry.     

迭代正则化方法求解电介质中空间电荷分布

首先介绍了迭代正则化方法的理论基础,建立了含有空间电荷密度分布的Fredholm第一类积分方程的反卷积算法,利用数值实验研究了加性高斯白噪声对迭代反卷积算法的影响,以及迭代停止标准对非适定问题的数值解的影响,最后使用该方法求解电介质样品中的空间电荷分布.结果表明,在无噪或者低噪环境下,反卷积算法能够非常好地计算出非适定问题的解.当噪声影响增大,信噪比降低时,反卷积的计算结果受到明显的影响.迭代停止标准对数值解的计算精度起着明显的作用.对实际测量数据进行处理表明,迭代正则化反卷积算法能够计算出固体电介质中的空间电荷分布.     

Coaxial Waveguide Slot Bridge Cell for Liquid Substances Study

The experimental method suggested suppose the studying of energy characteristics of slot bridge (SB) cells in the multi mode operating regime based on the preliminary numerical investigation of scattering resonant characteristics of SB (solution of boundary value problem and investigation of eigen frequencies and relevant field modes). The configuration of experimental model of SB cell is computed accordingly to required frequency band and preliminary information about dielectric parameters of materials chosen for tests. Ethanol scattering characteristics of SB cell are computed by software, developed on the base of rigorous solution of the boundary value problem, providing numerical data with any required accuracy.     

Numerical difficulties with boundary element solutions of interior acoustic problems

Although boundary element methods have been applied to interior problems for many years, the numerical difficulties that can occur have not been thoroughly explored. Various authors have reported low-frequency breakdowns and artificial damping due to discretization errors. In this paper, it is shown through a simple example problem that the numerical difficulties depend on the solution formulation. When the boundary conditions are imposed directly, the solution suffers from artificial damping, which may potentially lead to erroneous predictions when boundary element methods are used to evaluate the performance of damping materials. This difficulty can be alleviated by first computing an impedance or admittance matrix, and then using its reactive component to derive the solution for the acoustic field. Numerical computations are used to demonstrate that this technique eliminates artificial damping, but does not correct errors in the reactive components of the impedance or admittance matrices, which then causes nonexistence and nonuniqueness difficulties at the interior resonance frequencies for hard-wall and pressure release boundary conditions, respectively. It is shown that the admittance formulation is better suited to boundary element computations for interior problems because the resonance frequencies for pressure release boundary conditions do not begin until the smallest dimension of the boundary surface is at least one half the acoustic wavelength. Aside from producing much more accurate predictions, the admittance matrix is also much easier to interpolate at low frequencies due to the absence of interior resonances. For the example problem considered, only the formulation using the reactive component of the admittance matrix produces accurate solutions as long as the surface element discretization satisfies the standard six-element-per-wavelength rule.     

An efficient finite volume method for electric field–space charge coupled problems

The goal of this paper is to introduce some recently developed finite volume schemes to enable numerical simulation of electric field–space charge coupled problems. The key features of this methodology are the possibility of handling problems with complex geometries and accurately capturing the charge density distribution. The total variation diminishing (TVD) scheme and the improved deferred correction (IDC) scheme are used to compute the convective and diffusive fluxes respectively. Our technique is firstly verified with the computation of hydrostatic solutions in a two coaxial cylinders configuration. The homogeneous and autonomous injection from the inner or outer electrode is considered. Comparison has been made with the analytical solution. The numerical technique is also applied to the problem of corona discharge in a blade-plane configuration. The good agreement between our numerical solution and the one obtained with a combination approach of Finite Element Method (FEM) and Method of Characteristics (MoC) is shown.     

Geometrically nonlinear flexural vibrations of plates: In-plane boundary conditions and some symmetry properties

This study is devoted to the derivation of some properties of the von Kármán equations for geometrically nonlinear models of plates, with a boundary of arbitrary shape, for applications to nonlinear vibration and buckling. An intrinsic formulation of the local partial differential equations in terms of the transverse displacement and an Airy stress function as unknowns is provided. Classical homogeneous boundary conditions—with vanishing prescribed forces and displacements—are derived in terms of the Airy stress function in the case of a boundary of arbitrary geometry. A special property of this operator, crucial for some energy-conserving numerical schemes and called “triple self-adjointness”, is derived in the case of an edge of arbitrary shape. It is shown that this property takes a simple form for some classical boundary conditions, so that the calculations in some practical cases are also simplified. The applications of this work are either semi-analytical methods of solution, using an expansion of the solution onto an eigenmode basis of the associated linear problem, or special energy-conserving numerical methods.     

Numerical Approximation of a Nonlinear 3D Heat Radiation Problem

In this paper, we are concerned with the numerical approximation of a steady-state heat radiation problem with a nonlinear Stefan-Boltzmann boundary condition in$\mathbb{R}^3$. We first derive an equivalent minimization problem and then present a finite element analysis to the solution of such a minimization problem. Moreover, we apply the Newton iterative method for solving the nonlinear equation resulting from the minimization problem. A numerical example is given to illustrate theoretical results.     

Double-deck structure of the boundary layer in the problem of flow in an axially symmetric pipe with small irregularities on the wall for large Reynolds numbers

The problem of flow of a viscous incompressible fluid in an axially symmetric pipe with small irregularities on the wall is considered. An asymptotic solution of the problem with the double-deck structure of the boundary layer and the unperturbed flow in the environment (the “core flow”) is obtained. The results of flow numerical simulation in the thin and “thick” boundary layers are given.     

On the two-component Benard problem a numerical solution

A numerical solution for the two-component Bénard problem is presented, taking into account the contribution of thermal diffusion to the total density gradient. The results are compared with the approximate solution obtained by the variational technique of the local potential introduced some years ago by Glansdorff and Prigogine. The results calculated by the two methods are in agreement when the Soret coefficient is not too large. But when the gradients become important, the exact numerical solution presented here shows a small divergence from the variational method.The critical Rayleigh number is also compared with the one extrapolated from the analytical solution obtained for free boundary conditions.     

Depolarization of the electromagnetic field scattered by a three-dimensional large-scale irregularity of the lower ionosphere

This paper develops analytical and numerical methods for the solution of three-dimensional problems of radio wave propagation. We consider a three-dimensional vector problem for the electromagnetic field of a vertical electric dipole in a planar Earth-ionosphere waveguide with a largescale local irregularity of negative characteristics at the anisotropic ionospheric boundary. The field components at the boundary surfaces obey the Leontovich boundary conditions. The problem is reduced to a system of two-dimensional integral equations taking into account the overexcitation and depolarization of the field scattered by the irregularity. Using asymptotic (with respect to the parameter kr≫1, where r is the distance from the source or receiver to the nearest point of the irregularity, k=2π/λ, and λ is the radio wavelength) integration over the direction perpendicular to the ray path, we transform this system to one-dimensional integral equations where integration contours represent the geometric contour of the irregularity. The system is numerically solved in the diagonal approximation, combining direct inversion of the Volterra integral operator and subsequent iterations. The proposed numerical algorithm reduces the computer time required for the solution of this problem and is applicable for studying both small-scale and large-scale irregularities. We obtained novel estimates for the field components that are not excited by the source but result entirely from scattering by the sample three-dimensional ionospheric irregularity.     

A short remark on the band structure of free-edge platonic crystals

A corrected version of the multipole solution for a thin plate perforated in a doubly periodic fashion is presented. It is assumed that free-edge boundary conditions are imposed at the edge of each cylindrical inclusion. The solution procedure given here exploits a well-known property of Bessel functions to obtain the solution directly, in contrast to the existing incorrect derivation. A series of band diagrams and an updated table of values are given for the resulting system (correcting known publications on the topic), which shows a spectral band at low frequency for the free-edge problem. This is in contrast to clamped-edge boundary conditions for the same biharmonic plate problem, which features a low-frequency band gap. The numerical solution procedure outlined here is also simplified relative to earlier publications, and exploits the spectral properties of complex-valued matrices to determine the band structure of the structured plate.     

Flow through a porous membrane simulated by cellular automata and by finite elements

Computational results concerning incompressible viscous flow through two channels connected by a porous membrane are presented. The example is extraordinary for its four different types of boundary conditions that are necessary to make the problem complete. The solution is accomplished by two methods: by cellular automata and by finite elements. The numerical means to satisfy the boundary conditions are given for both methods. Overall agreement is achieved, but significant differences show up in details.     

Exact boundary conditions for the initial value problem of convex conservation laws

The initial value problem of convex conservation laws, which includes the famous Burgers’ (inviscid) equation, plays an important rule not only in theoretical analysis for conservation laws, but also in numerical computations for various numerical methods. For example, the initial value problem of the Burgers’ equation is one of the most popular benchmarks in testing various numerical methods. But in all the numerical tests the initial data have to be assumed that they are either periodic or having a compact support, so that periodic boundary conditions at the periodic boundaries or two constant boundary conditions at two far apart spatial artificial boundaries can be used in practical computations. In this paper for the initial value problem with any initial data we propose exact boundary conditions at two spatial artificial boundaries, which contain a finite computational domain, by using the Lax’s exact formulas for the convex conservation laws. The well-posedness of the initial-boundary problem is discussed and the finite difference schemes applied to the artificial boundary problems are described. Numerical tests with the proposed artificial boundary conditions are carried out by using the Lax–Friedrichs monotone difference schemes.