跨声速流函数方程强隐式解及确定密度场的新方案

基于人工可压缩性对密度的修正,本文用强隐式格式快速求解了非正交曲线坐标系下跨声速流函数方程;在流函数场解出后,通过求解一个由动量方程、能量方程和连续方程组合而成的关于密度的一阶偏微分方程来获得密度场,因此流函数解法中常遇到的密度双值问题在这里已不存在;通常所讲的完成流函数场{Ψ}与密度场{ρ}间的迭代在本文便体现在流函数主方程与这个新推出的一阶微分方程间的迭代计算上;几个典型算例表明了本方法的有效性。     

Atom diffraction from magnetic gratings in the thin phase-grating approximation

The problem of atom diffraction from a reflecting magnetic diffraction grating is solved in the thin phase-grating approximation. The general problem for scalar diffraction is modelled using a semi-classical method in which the grating potential is separated into a reflecting term and a diffracting term. The trajectory of the atom in the reflecting potential is solved classically and the atom wave function in the diffracting potential found by integrating the phase change along the classical trajectory. The diffraction orders are obtained after Fourier transforming the result. This can be done independently of the grating potential resulting in a general formula for the diffraction efficiencies. The general result is applied to the problem of atom diffraction from a magnetic grating. Several approximations are required to reduce the problem to a form amenable to analytic solution. The results are compared with an accurate numerical method. Received 1st February 2001 and Received in final form 8 March 2001     

柱面光栅色散特性及其应用

 基于椭球面光栅理论,提出柱面光栅概念,并对其衍射特性和集光本领进行研究。结果表明：在掠入射条件下凹柱面光栅具有与凹球面光栅相似的衍射特性,但凹柱面光栅可以方便地沿光栅母线方向划分成若干相同部分独立工作,因此,安装水平隔板的掠入射凹柱面光栅单色仪可以用于托卡马克等离子体的真空紫外（VUV）光谱时空分辨测量。     

Modelling of water wave interaction with multiple cylinders of arbitrary shape

This paper describes the development of an efficient numerical model, namely scaled boundary finite-element method (SBFEM) for linear waves interaction with cylindrical structures of arbitrary shapes. The two-dimensional Helmholtz equation is firstly weakened in the circumferential direction, so that the governing partial differential equation is transformed to an ordinary matrix differential equation in radial direction, and is solved fully analytically. As a key element, a virtual porous circular cylinder surrounding the cylindrical structures is introduced so that the entire computational domain is partitioned along the virtual cylinder into an unbounded and several bounded sub-domains with common interfaces. The principle innovation is that, the present SBFEM model chooses Hankel function as a base solution for the unbounded sub-domain, while a power series is used for the internal bounded sub-domains. The approach discretises only the common interfaces of the sub-domains with surface finite-elements, and fewer elements are required to obtain very accurate results. Numerical simulations show that the new SBFEM model offers a considerable improvement by far in its numerical performance, as well as in the range of physical phenomena that is capable of simulating. The wave forces and run-ups are presented for a single and multiple cylindrical structures of different cross sectional shapes. Influences of the incident wave parameters and structural configurations on the hydrodynamics are examined.     

压力陡降及容积加热条件下气—液两相系统瞬态分析

用一种解析与数值相结合的方法预测饱和液体在压力陡降及具有内热源条件下的瞬态响应。     

Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials,Logan–Shepp ridge polynomials,Chebyshev–Fourier Series,cylindrical Robert functions,Bessel–Fourier expansions,square-to-disk conformal mapping and radial basis functions

We compare seven different strategies for computing spectrally-accurate approximations or differential equation solutions in a disk. Separation of variables for the Laplace operator yields an analytic solution as a Fourier–Bessel series, but this usually converges at an algebraic (sub-spectral) rate. The cylindrical Robert functions converge geometrically but are horribly ill-conditioned. The Zernike and Logan–Shepp polynomials span the same space, that of Cartesian polynomials of a given total degree, but the former allows partial factorization whereas the latter basis facilitates an efficient algorithm for solving the Poisson equation. The Zernike polynomials were independently rediscovered several times as the product of one-sided Jacobi polynomials in radius with a Fourier series in θ. Generically, the Zernike basis requires only half as many degrees of freedom to represent a complicated function on the disk as does a Chebyshev–Fourier basis, but the latter has the great advantage of being summed and interpolated entirely by the Fast Fourier Transform instead of the slower matrix multiplication transforms needed in radius by the Zernike basis. Conformally mapping a square to the disk and employing a bivariate Chebyshev expansion on the square is spectrally accurate, but clustering of grid points near the four singularities of the mapping makes this method less efficient than the rest, meritorious only as a quick-and-dirty way to adapt a solver-for-the-square to the disk. Radial basis functions can match the best other spectral methods in accuracy, but require slow non-tensor interpolation and summation methods. There is no single “best” basis for the disk, but we have laid out the merits and flaws of each spectral option.     

Study of the numerical artifacts in differential analysis of highly conducting gratings

Li's Fourier factorization rule [J. Opt. Soc. Am. A13, 1870 (1996)] was recently shown to be problematic to apply to highly conducting metallic gratings. We provide further information about the applicability of different differential methods and are concerned with the relation of observed numerical artifacts, the total number of retained space harmonics, the presence of both positive and negative permittivity inside the groove region, and the validity of Li's inverse rule. Two different cases corresponding to lossless and low-loss binary metallic gratings are considered, and it is shown that an increase in the number of retained space harmonics can relieve the presence of numerical artifacts.     

二维各向异性浮雕型光栅的矢量衍射分析

利用严格模式理论对二维电磁各向异性浮雕型介质光栅的衍射特性进行了理论分析，并对其退化情形进行讨论。文中采用反射-透射系数矩阵算法和校正傅里叶展开规则以提高数值计算的稳定性、收敛性和计算效率。通过对光栅进行数值计算表明，该方法在可靠性、稳定性和收敛性方面是令人满意的。     

A review on the solution of Grad–Shafranov equation in the cylindrical coordinates based on the Chebyshev collocation technique

Equilibrium reconstruction consists of identifying, from experimental measurements, a distribution of the plasma current density that satisfies the pressure balance constraint. Numerous methods exist to solve the Grad–Shafranov equation, describing the equilibrium of plasma confined by an axisymmetric magnetic field. In this paper, we have proposed a new numerical solution to the Grad–Shafranov equation (an axisymmetric, magnetic field transformed in cylindrical coordinates solved with the Chebyshev collocation method) when the source term (current density function) on the right-hand side is linear. The Chebyshev collocation method is a method for computing highly accurate numerical solutions of differential equations. We describe a circular cross-section of the tokamak and present numerical result of magnetic surfaces on the IR-T1 tokamak and then compare the results with an analytical solution.     

Factorization method for inverse obstacle scattering problem in three-dimensional planar acoustic waveguides

In this paper, we consider the inverse scattering problem of reconstructing a bounded obstacle in a three-dimensional planar waveguide from the scattered near-field data measured on a finite cylindrical surface containing the obstacle and corresponding to infinitely many incident point sources also placed on the measurement surface. The obstacle is allowed to be an impenetrable scatterer or a penetrable scatterer. We establish the validity of the factorization method with the nearfield data to characterize the obstacle in the planar waveguide by constructing an outgoing-to-incoming operator which is an integral operator defined on the measurement surface with the kernel given in terms of an infinite series.     

2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical shell and annular plate structures

This paper focuses on the dynamic behavior of functionally graded conical, cylindrical shells and annular plates. The last two structures are obtained as special cases of the conical shell formulation. The first-order shear deformation theory (FSDT) is used to analyze the above moderately thick structural elements. The treatment is developed within the theory of linear elasticity, when materials are assumed to be isotropic and inhomogeneous through the thickness direction. The two-constituent functionally graded shell consists of ceramic and metal that are graded through the thickness, from one surface of the shell to the other. Two different power-law distributions are considered for the ceramic volume fraction. The homogeneous isotropic material is inferred as a special case of functionally graded materials (FGM). The governing equations of motion, expressed as functions of five kinematic parameters, are discretized by means of the generalized differential quadrature (GDQ) method. The discretization of the system leads to a standard linear eigenvalue problem, where two independent variables are involved without using the Fourier modal expansion methodology. For the homogeneous isotropic special case, numerical solutions are compared with the ones obtained using commercial programs such as Abaqus, Ansys, Nastran, Straus, Pro/Mechanica. Very good agreement is observed. Furthermore, the convergence rate of natural frequencies is shown to be very fast and the stability of the numerical methodology is very good. Different typologies of non-uniform grid point distributions are considered. Finally, for the functionally graded material case numerical results illustrate the influence of the power-law exponent and of the power-law distribution choice on the mechanical behavior of shell structures.     

Performance analysis of the virtual sound barrier system with a diffracting sphere

Since a practical virtual sound barrier (VSB) system has to be implemented surrounding a human’s head, the effects of such a diffracting sphere on the performance of the system are considered in this paper. It is found that the performance of the VSB system with a diffracting sphere might be improved or decreased depending on the size of the zone surrounded by the error sensors and the noise frequency, and the noise reduction distribution around the sphere tends to be uniform due to the diffraction of the sphere. Numerical simulations show that the tendency of the control performance with respect to the configuration of the system with a diffracting sphere is similar to that without the sphere. The diffracting sphere might move in the quiet zone of the VSB system, and the performance decreases with the increase of the distance between the sphere and the center of the system. However, the noise reduction level can remain more than 10 dB even when the sphere is near the border of the quiet zone. Experiments with a 16-channel cylindrical VSB system in a normal room validate the above conclusions.     

Free vibration analysis of circular cylindrical shells

A general analytical method is presented for evaluating the free vibration characteristics of a circular cylindrical shell with classical boundary conditions of any type. The solution is obtained through a direct solution procedure in which Sanders' shell equations are used with the axial modal displacements represented as simple Fourier series expressions. Stokes' transformation is exploited to obtain correct series expressions for the derivatives of the Fourier series. An explicit expression of the exact frequency equation can be obtained for any kind of boundary conditions. The accuracy of the method is checked against available data. The method is used to find the modal characteristics of the thermal liner model of the U.S. Fast Test Reactor (FTP). The numerical results obtained are compared with finite element method solutions.     

Modelling the diffraction of laser-cooled atoms from magnetic gratings

The diffraction of laser-cooled atoms from a spatially-periodic potential is modelled using rigorous coupled-wave analysis. This numerical technique, normally applied to light-diffraction, is adapted for use with atomic de Broglie waves incident on a reflecting diffraction grating. The technique approximates the potential by a large number of constant layers and successively solves the complex eigenvalue problem in each layer, propagating the solution up to the surface of the grating. The method enables the diffraction efficiencies to be calculated for any periodic potential. The results from the numerical model are compared with the thin phase-grating approximation formulae for evanescent light-wave diffraction gratings and idealised magnetic diffraction gratings. The model is applied to the problem of diffracting Rb atoms from a grating made from an array of permanent magnets. Received 13 June 2000 and Received in final form 15 December 2000     

Numerical analysis of immersed finite cylindrical shells using a coupled BEM/FEM and spatial spectrum approach

This study numerically analyzes submerged cylindrical shells using a coupled boundary element method (BEM) with finite element method (FEM) in conjunction with the wave number theory, in which the spatial Fourier transform of surface velocity for cylinders is directly related to pressure in a far field. The acoustic loading is formulated using a symmetric complex matrix derived from a boundary integral equation where the symmetry is based on an acoustic reciprocal principle for surface acoustics. In this formulation the acoustic loading matrix is a large acoustic element whose degree of freedom is connected to the normal displacement of the vibrating structures. The coupled BEM/FEM equation is a banded, symmetric matrix, and thus its bandwidth can be minimized using a proper algorithm. This formulation significantly increases numerical efficiency. The computed normal velocity is thus transformed to wave number representation to examine acoustic radiation. A finite plane cylindrical shell, without attached stiffeners, and a shell with internal ring stiffeners are chosen to demonstrate the present analysis procedure. The far field pressure computed directly from the integral equation and predicted by wave number theory correlates closely with increasing vibrating frequency. Meanwhile, the influences of the internal ring structures on acoustic radiation are examined using the wave number theory, which helps in understanding how internal structures influence radiated noise.     

Object image correction using an X‐ray dynamical diffraction Fraunhofer hologram

Taking into account background correction and using Fourier analysis, a numerical method of an object image correction using an X‐ray dynamical diffraction Fraunhofer hologram is presented. An example of the image correction of a cylindrical beryllium wire is considered. A background correction of second‐order iteration leads to an almost precise reconstruction of the real part of the amplitude transmission coefficient and improves the imaginary part compared with that without a background correction. Using Fourier analysis of the reconstructed transmission coefficient, non‐physical oscillations can be avoided. This method can be applied for the determination of the complex amplitude transmission coefficient of amplitude as well as phase objects, and can be used in X‐ray microscopy.     

Diffraction of a two-shock configuration by a concave cylindrical surface

Diffraction of a two-shock configuration by a concave cylindrical surface with a continuously varying angle of diffraction in a perfect (inviscid and non-heat-conducting) gas is numerically investigated. Peculiarities of flow formation in the perturbed region due to a large initial angle of diffraction are revealed. The reasons for the appearance of the reverse flow over the cylindrical surface, which causes a vortex, are found. The influence of the vortex on the behavior of the loose end of the TU layer (slipstream) is studied. It is shown that the structure of the loose end of the TU layer is determined by vortex-related mixing in the gas. The nonstationary interaction of the diffracting shock wave and a sublayer shock with a concave cylindrical wall and the symmetry plane, which is accompanied by a change in the type of reflection, is examined. Published in Zhurnal Tekhnicheskoĭ Fiziki, 2007, Vol. 77, No. 10, pp. 24–33. The article was translated by the authors.     

The choosing of reproducing kernel particle shape function with mathematic proof

Many mechanical problems can be induced from differential equations with boundary conditions; there exist analytic and numerical methods for solving the differential equations. Usually it is not so easy to obtain analytic solutions. So it is necessary to give numerical solutions. The reproducing kernel particle (RKP) method is based on the Garlerkin Meshless method. According to the Sobolev space and Fourier transform, the RKP shape function is mathematically proved in this paper.     

Nonlinear traveling wave vibration of a circular cylindrical shell subjected to a moving concentrated harmonic force

This is a study of nonlinear traveling wave response of a cantilever circular cylindrical shell subjected to a concentrated harmonic force moving in a concentric circular path at a constant velocity. Donnell's shallow-shell theory is used, so that moderately large vibrations are analyzed. The problem is reduced to a system of ordinary differential equations by means of the Galerkin method. Frequency-responses for six different mode expansions are studied and compared with that for single mode to find the more contracted and accurate mode expansion investigating traveling wave vibration. The method of harmonic balance is applied to study the nonlinear dynamic response in forced oscillations of this system. Results obtained with analytical method are compared with numerical simulation, and the agreement between them bespeaks the validity of the method developed in this paper. The stability of the period solutions is also examined in detail.     

Surface fields on nonseparable geometries

The field on the surface of a structure having nonseparable geometry may be expanded in a set of eigenfunctions suitable for that particular geometry. The eigenfunctions are computed numerically using the boundary elements method to compute a matrix that relates the field on the surface of the structure to its normal derivative at vanishing frequency and diagonalizing the matrix. The field on the surface of a finite cylinder with hemisphical end-caps is expanded in such a set of eigenfunctions for both a simple cylinder and a cylinder containing frames. Results for the framed cylinder is compared to Fourier decomposition of the field on the cylindrical portion of the structure.