Scattering of plane SH-waves by a cylindrical canyon of arbitrary shape in anisotropic media

The scattering and diffraction of plane SH-waves, by an arbitrary-shaped cylindrical canyon in anisotropic media is formulated here. Analytical solutions are obtained via the complex function theory, using the orthogonal property of the Hermite functions to solve the resulting set of infinite algebraic equations. Expressions for scattered displacements and scattered stresses are given. Three cross-sectional profile types are used in the numerical simulation of the two-dimensional canyon topography: (a) a semi-circular profile, (b) a semi-elliptical profile and (c) a triangular profile. The results obtained in (a) and (b) are consistent with known solutions computed by Trifunac and his co-workers [1,3] using a different method. As the exact solution for (c) is not known to exist, the result given here is believed to be new and would therefore serve as a useful check for numerical analysts working in this area.     

The scaling of impact phenomena

A basis for scaling laws for impact phenomena is the approximation of the actual problem with a point-source problem. Point-source solutions are characterized by a single scalar coupling-parameter measure aU^μδυ of the radius a, velocity U and mass density δ of the impactor. Point-source solutions that are known to exist are discussed. The implications to scaling are developed. Recent numerical code calculations that were designed to investigate the scaling and the application of the coupling parameter are presented. Finally, a summary of some recent applications of the theory is given.     

新的估算表面裂纹应力强度因子经验公式

该文给出了新的估算拉伸和纯弯曲载荷下表面裂纹应力强度因子的经验公式。根据疲劳裂纹扩展的数值模拟结果确定强度因子分布函数;利用按已知应力强度因子分布函数求裂纹形状及相应应力强度因子的方法计算给定尺寸的表面裂纹的应力强度因子;通过对数值结果的曲线回归得到估算表面裂纹应力强度因子经验公式。利用该公式对有限厚度和宽度平板内表面裂纹的应力强度因子进行了估算,并与已知的半椭圆形表面裂纹的应力强度因子解进行了比较。该文结果为估算表面裂纹应力强度因子提供了一种新的途径。     

Free convection from a vertical permeable circular cone with non-uniform surface temperature

Summary Laminar free convection from a vertical permeable circular cone maintained at nonuniform surface temperature is considered. Non-similar solutions for boundary-layer equations are found to exist when the surface temperature follows the power law variations with the distance measured from the leading edge. The numerical solutions of the transformed non-similar boundary-layer equations are obtained by using three methods, namely, (i) a finite difference method, (ii) a series solution method, and (iii) an asymptotic solution method. Solutions are obtained in terms of skin friction, heat transfer, velocity profile and temperature profile for smaller values of Prandtl number and temperature gradient are displayed in both tabular and graphical forms. Finite difference solutions are compared with the solutions obtained by perturbation and asymptotic techniques and found to be in excellent agreement.     

Scattering of a spheroidal particle illuminated by a gaussian beam

An approach to expanding a Gaussian beam in terms of the spheroidal wave functions in spheroidal coordinates is presented. The beam-shape coefficients of the Gaussian beam in spheroidal coordinates can be computed conveniently by use of the known expression for beam-shape coefficients, g(n), in spherical coordinates. The unknown expansion coefficients of scattered and internal electromagnetic fields are determined by a system of equations derived from the boundary conditions for continuity of the tangential components of the electric and magnetic vectors across the surface of the spheroid. A solution to the problem of scattering of a Gaussian beam by a homogeneous prolate (or oblate) spheroidal particle is obtained. The numerical values of the expansion coefficients and the scattered intensity distribution for incidence of an on-axis Gaussian beam are given.     

Calculation of flows using three-dimensional overlapping grids and multigrid methods

A computational methodology combining overlapping grid techniques with multigrid methods has been developed for three-dimensional flow calculations in or around complex geometries. The computational accuracy, efficiency and capability of the present approach are investigated in this paper. The incompressible Navier–Stokes equations are discretized using a finite volume method on a semi-staggered grid. The discrete problem is solved by a multigrid algorithm. Some numerical examples are chosen for evaluating numerical accuracy: (a) a straight pipe for which the exact solution is known; (b) curved pipes where previous experimental and numerical data are available; (c) an axisymmetric sudden expansion. The performance of the multigrid method on overlapping grids is assessed. Several cases of flows in stationary and time-dependent complex geometries are given to demonstrate the capability and the potential of the methods that we employ.     

The shape of a sessile drop for small and large surface tension

Asymptotic solutions for large and small surface tension are developed for the profile of a symmetric sessile drop. The problem for large surface tension (i.e., small Bond number) is a regular perturbation problem, where the solution may be written as a uniformly valid asymptotic expansion. The problem for small surface tension (i.e., large Bond number) is a singular perturbation problem with boundary-layer behaviour in the edge region. The solution is a matched asymptotic expansion, where some care is to be taken for the matching. The respective ranges of validity are established by comparing the asymptotic results with solutions obtained by numerical integration of the full equations.     

A boundary method of Trefftz type for PDEs with scattered data

This paper presents the application of a Trefftz type method for partial differential equations (PDEs) of the elliptic type with inhomogeneous term given by a set of scattered data. The method of particular solutions is used. Basis functions of a new type were introduced to approximate the scattered data. Using these basis functions, we get the approximation in the form of series over some orthogonal system of eigenfunctions. The particular case of the trigonometric eigenfunctions is considered. The corresponding approximation of the inhomogeneous term allows to get a particular solution for PDEs with constant coefficients or for the systems of such PDEs easily. We test our basis functions on recovering well-known Franke's and PEAKS functions given by scattered data. We also present results of solution Helnholtz PDE, PDE with differential operator of 4th order and system of PDEs arising in shell deflection problems. A comparison of the numerical solutions with analytic solutions is performed for all the problems.     

Three‐dimensional BEM for scattering of elastic waves in general anisotropic media

Based on the full‐space Green's functions, a three‐dimensional time‐harmonic boundary element method is presented for the scattering of elastic waves in a triclinic full space. The boundary integral equations for incident, scattered and total wave fields are given. An efficient numerical method is proposed to calculate the free terms for any geometry. The discretization of the boundary integral equation is achieved by using a linear triangular element. Applications are discussed for scattering of elastic waves by a spherical cavity in a 3D triclinic medium. The method has been tested by comparing the numerical results with the existing analytical solutions for an isotropic problem. The results show that, in addition to the frequency of the incident waves, the scattered waves strongly depend on the anisotropy of the media. Copyright © 2003 John Wiley & Sons, Ltd.     

Plane steady shear flow of a cohesionless granular material down an inclined plane. A model for flow avalanches Part II: numerical results

Summary The granular flow model proposed by Jenkins and Savage., [2], and extended by us [1] is used here to construct numerical solutions of steady chute flows thought to be typical of such flows.We briefly state the equations and boundary conditions and present numerical solutions when the following model parameters of the Senkins and Savage model are varied: (a) the coefficient of restitution of the particles under binary collision, (b) the number of particles per layer, (c) the inclination angle of the chute, and (d) the basal and free surface boundary conditions. We demonstrate that the Jenkins and Savage model may yield physically questionable results, that those of its extension differ markedly from them and are physically more reasonable in certain cases, but yield equally questionable results in others. The results are apt to redefine research directions which granular flow modellers might want to pursue in the near future.With 18 Figures     

Flexure of clamped semi-infinite orthotropic plates

Using the method of biorthogonal functions developed by Johnson and Little [8], solutions have been obtained for the flexure of semi-infinite orthotropic plates whose long edges are clamped and upon whose short edge (a) deflection and slope, or (b) deflection and moment, or (c) moment and shear force are prescribed. Some numerical results are presented when zero deflection and constant bending moment are prescribed on the short edge of the plate.     

An analysis of solidification problems by the boundary element method

The problem of interest in this paper is the calculation of the motion of the solid–liquid interface and the time-dependent temperature field during solidification of a pure metal. An iterative implicit algorithm has been developed for this purpose using the boundary element method (BEM) with time-dependent Green's functions and convolution integrals. The BEM approach requires discretization of only the surface of the solidifying body. Thus, the numerical method closely follows the physics of the problems and is intuitively very appealing. The formulation and the numerical scheme presented here are general and can be applied to a broad range of moving boundary problems. Emphasis is given to two-dimensional problems. Comparison with existing semi-analytical solutions and other numerical solutions from the literature reveals that the method is fast, accurate and without major time step limitations.     

An analytical solution for long-wave scattering by a submerged circular truncated shoal

An analytical solution of the long-wave (or shallow-water) equation in closed-form is obtained for simple harmonic waves scattered by a submerged circular truncated shoal. This analytical solution is firstly validated against Longuet-Higgins’s classical analytical solution for a submerged cylinder, and then validated against numerical solutions obtained by using the DRBEM (dual reciprocity boundary-element method) model for a submerged circular truncated cone. Finally, the analytical solution is used to investigate the changing trend of maximum wave amplification, the trace pattern of focal position of wave-energy versus the wave period and the influence of shoal submergence on wave-energy trapping.     

双模数复合材料层板分析中的一种新方法——有限层元素法

一种分析双模数复合材料层板的新方法——有限层元素法在本文中提出,与文献[1]提出的有限层法相比,精度有明显提高。与其它文献中的已有解比较,吻合较好。      

Three-dimensional beam propagation model for the optical path of light through a nematic liquid crystal

A three-dimensional beam propagation model is presented which, given a director profile, calculates the electric and magnetic fields for a light wave as it travels through a liquid crystal structure using a predictive-corrector Crank–Nicholson numerical method and adjusts for impedance effects in the direction of propagation. The results from the model are compared to both analytical solutions for simpler director profiles and other optical models for more complicated director profiles.     

Microscale heat conduction in homogeneous anisotropic media: a dual-reciprocity boundary element method and polynomial time interpolation approach

In this paper, a dual-reciprocity boundary element method based on some polynomial interpolations to the time-dependent variables is presented for the numerical solution of a two-dimensional heat conduction problem governed by a third order partial differential equation (PDE) over a homogeneous anisotropic medium. The PDE is derived using a non-Fourier heat flux model which may account for thermal waves and/or microscopic effects. In the analysis, discontinuous linear elements are used to model the boundary and the variables along the boundary. The systems of algebraic equations are set up to solve all the unknowns. For the purpose of evaluating the proposed method, some numerical examples with known exact solutions are solved. The numerical results obtained agree well with the exact solutions.     

Analytic model of ocean color

Ocean color is determined by spectral variations in reflectance at the sea surface. In the analytic model presented here, reflectance at the sea surface is estimated with the quasi-single-scattering approximation that ignores transspectral processes. The analytic solutions we obtained are valid for a vertically homogeneous water column. The solution provides a theoretical expression for the dimensionless, quasi-stable parameter (r), with a value of ~0.33, that appears in many models in which reflectance at the sea surface is expressed as a function of absorption coefficient (a) and backscattering coefficient (b(b)). In the solution this parameter is represented as a function of the mean cosines for downwelling and upwelling irradiances and as the ratio of the upward-scattering coefficient to the backscattering coefficient. Implementation of the model is discussed for two cases: (1) that in which molecular scattering is the main source of upwelling light, and (2) that in which particle scattering is responsible for all the upwelled light. Computations for the two cases are compared with Monte Carlo simulations, which accounts for processes not considered in the analytic model (multiple scattering, and consequent depth-dependent changes in apparent optical properties). The Monte Carlo models show variations in reflectance with the zenith angle of the incident light. The analytic model can be used to reproduce these variations fairly well for the case of molecular scattering. For the particle-scattering case also, the analytic and Monte Carlo models show similar variations in r with zenith angle. However, the analytic model (as implemented here) appears to underestimate r when the value of the backscattering coefficient b(b) increases relative to the absorption coefficient a. The errors also vary with the zenith angle of the incident light field, with the maximum underestimate being approximately 0.06 (equivalent to relative errors from 12 to 17%) for the range of b(b)/a studied here. One implication of this result is that the model could also be used to obtain approximate solutions for the Q factor, defined for a given look angle as the ratio of the upwelling irradiance at the surface to the upwelling radiance at the surface at that angle. This is a quantity that is important in remote-sensing applications of ocean-color models. An advantage of the model discussed here is that its implementation requires inputs that are in principle accessible only in a remote-sensing context.     

A new error indicator based on stresses for adaptive meshing with Hermite boundary elements

This work presents a new error indicator for adaptive meshing with Hermite boundary elements. The error indicator proposed here directly measures, in an approximate fashion, the error in the numerical solution for the stresses. The basic idea behind the error indicator is to compare, on an element by element basis, a solution for the stresses obtained with Hermite elements with a second, “reduced” solution. This “reduced” solution is obtained approximating the field variables inside each element using Lagrangian shape functions together with the nodal values for the displacements and tractions that were obtained from the analysis with Hermite elements. In this sense, it is assumed that the bigger the difference between the two solutions, the bigger the error in the numerical solution for the stresses corresponding to the model with Hermite elements. Since in the scheme presented here both numerical solutions are obtained from the same analysis, the cost associated with the computation of the error indicator is minimal. Due to its simplicity and reliability, this new error indicator is very convenient to lead adaptive processes. Furthermore, it can also be used in models with multiple subregions.