Three-dimensional Green's functions in anisotropic magneto-electro-elastic bimaterials

In this paper, we derive three-dimensional Green's functions in anisotropic magneto-electro-elastic full space, half space, and bimaterials based on the extended Stroh formalism. While in the full space, the Green's functions are obtained in an explicit form, those in the half space and bimaterials are expressed as a sum of the full-space Green's function and a Mindlin- type complementary part, with the latter being evaluated in terms of a regular line integral over [0, p][0, \pi]. Despite the complexity involved, the current Green's function expressions are surprisingly simple. Furthermore, the piezoelectric, piezomagnetic, and purely elastic Green's functions can all be obtained from the current Green's functions by setting simply the appropriate material coefficients to zero. A special material case, to which the extended Stroh formalism cannot be applied directly, has also been identified.¶Simple numerical examples are presented for Green's functions in full space, half space, and bimaterials with fully coupled and uncoupled anisotropic magneto-electro-elastic material properties.For given material properties and fixed source and field points, the effect of magneto-electro-elastic coupling on the Green's function is discussed. In particular, we observed that magneto-electro-elastic coupling could significantly alter the magnitude of certain Green's displacement and stress components, with difference as high as 45% being noticed. This result is remarkable and should be of great interest in the material analysis and design.     

旋波媒质中谱域电并矢Green函数的分解

介绍了一种推导无耗、互易和无界旋波媒质中谱域并矢Green函数表达式的新方法·　这种方法以Hemholtz定理以及并矢Diracδ函数的无散和无旋分解为基础,首先将电矢量的并矢Green函数方程分解成无散电矢量的并矢Green函数方程和无旋电矢量的并矢Green函数方程,然后经Fourier变换导出了旋波媒质中谱域电并矢Green函数的无散分解表达式和无旋分解表达式·　用这种方法推导旋波媒质中并矢Green函数就可以避免必须用波场的分解法和并矢Green函数的本征函数展开法·     

Convective Instabilities in Anisotropic Porous Media

This paper addresses the problem of the onset of Rayleigh-Bénard convection in a porous layer using Brinkman's equation and anisotropic permeability. The critical Rayleigh number and wave number at marginal stabilities are calculated for both free and rigid boundaries. In both cases, it is noted that there exist ranges for which the stability criteria is intermediate to the low porosity Darcy approximation and to high porosity single viscous fluid. The permeability anisotropy is found to select the mode of instability.     

Ursell's approach to obtaining an a priori estimate for the solution of the Neumann problem for the Helmholtz equation

The geometric optics asymptotic expression for Green's function for the Helmholtz equation in the exterior of a bounded convex region in R³ is rigorously justified (it is assumed that the Neumann boundary condition is satisfied). An integral equation constructed by a three-dimensional analogue of Ursell's method from Green's function for a paraboloid of revolution is the basis of all considerations. Analysis of this integral equation also makes it possible to prove the exponential decay of Green's function for the Helmholtz equation in the zone of deep shadow.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 42, pp. 85–154, 1974.     

Integral Equations without a Unique Solution can be made Useful for Solving some Plane Harmonic Problems

Using Green's third identity an integral equation for a twodimensional harmonic problem is derived. For a particular exceptionalgeometry the integral equation does not have a unique solutionbut by applying Green's third identify a supplementary integralcondition is derived. When the integral equation and the integralcondition are solved simultaneously we obtain always a uniquesolution. The procedure is demonstrated by some numerical examples.     

Constructing approximate Green’s function for a vector equation for the electric field using the variational iteration method

In this work, we use He’s variational iteration method (VIM) to find approximate Green’s functions for a vector equation for the electric field with anisotropic dielectric permittivity and magnetic permeability. We present numerical examples which show that an approximate solution of an initial value problem (IVP) for a vector equation can be obtained by using these approximate Green’s functions.     

Integral equations and the interior Dirichlet potential problem

The two standard approaches for reformulating the interior Dirichlet potential problem as a boundary integral equation of the second kind are discussed. The integral equation derived from the representation of the solution as a double layer is shown to be more general than the one derived from Green's theorem. The boundary integral equation of the latter method, however, has definite analytical and numerical value. From it a new integral equation is derived whose solution can be represented as a convergent Neumann series and it is shown that the Green's function of the first kind can be obtained from it. An example is supplied to illustrate the method.     

Point force and edge dislocation in a two-phase anisotropic medium

Green's functions are derived for the in-plane deformation due to an in-plane point force and edge dislocation acting at a point in a plane of two joined semi-infinite anisotropic plates. The Lekhnitskii's complex potential approach is used, and a general expression of the solutions is obtained in closed-form. Including the case of an isotropic-anisotropic two-phase medium and the case of an isotropic-isotropic two-phase medium, the solutions are given for all possible combinations of materials with either s₁ s₂ or S₁=s₂, where si and s2 are the roots of the characteristic equation of the material.     

平面中Poisson方程的Dirichlet问题

基于向量旋转内积不变的特点,通过对Green积分公式的推广,得到与空间Green三个公式相似的平面Green公式.从而,得到平面中Poisson方程Robin问题的解和平面中Poisson方程Dirichlet问题的解.     

双空间指示函数方法在三维柱面对称波导中电磁波的反散射问题的推广

给出了双空间指示函数方法在三维柱面对称波导中电磁波的反散射问题的推广.基于这个观察:当Green函数的点源在障碍物内部时,那么远域数据的赋权积分可以很好地近似估计Green函数,但是当Green函数的点源在障碍物外部时,那么远域数据的赋权积分就不能很好地近似估计Green函数.建立一个积分方程:它的右边是点源在所重构区域的Green函数,那么我们可以知道这个积分方程的解的范数在未知障碍物的内部有极大值,而这些取得极大值的点所围成的区域恰好就是所重构的障碍物区域.方法最显著的优势在于它不依赖于未知障碍物的边界条件.     

Green's functions for elliptic and parabolic equations with random coefficients II

This paper is concerned with linear parabolic partial differential equations in divergence form and their discrete analogues. It is assumed that the coefficients of the equation are stationary random variables, random in both space and time. The Green's functions for the equations are then random variables. Regularity properties for expectation values of Green's functions are obtained. In particular, it is shown that the expectation value is a continuously differentiable function in the space variable whose derivatives are bounded by the corresponding derivatives of the Green's function for the heat equation. Similar results are obtained for the related finite difference equations. This paper generalises results of a previous paper which considered the case when the coefficients are constant in time but random in space.

     

Green's function for second-order periodic boundary value problems with piecewise constant arguments

We obtain, under suitable conditions, the Green's function to express the unique solution for a second-order functional differential equation with periodic boundary conditions and functional dependence given by a piecewise constant function. This expression is given in terms of the solutions for certain associated problems. The sign of the solution is determined taking into account the sign of that Green's function.     

Some relationships between Green's functions and invariant imbedding

The method of invariant imbedding has been used to resolve the solution of linear two-point boundary-value problems into contributions associated with the homogeneous equation with homogeneous boundary conditions, with inhomogeneous boundary conditions, and with an inhomogeneous source term in the equation. The relationship between the Green's function and the invariant imbedding equations is described, and it is shown that the Green's function can be determined from an initial-value problem. Several numerical examples are given which illustrate the efficacy of the initial-value algorithm.This work was supported by the US Atomic Energy Commission.     

重调和椭圆边值问题的正则积分方程

我们熟知,利用位势理论或由Green公式及基本解出发区域内调和及重调和边值问题可归化为边界上的积分方程。近年来冯康又提出一种更自然而直接的归化,即从Green公式及Green函数出发将微分方程边值问题化为边界上的含有广义函数意义下发散积分有限部分的奇异积分方程,这种归化在各种边界归化中占有特殊地位,被称为正则边界归化,本文将这一理论应用于重调和椭圆边值问题,研究了其正则归化的性质,并通过利用Green函数、Fourier分析及复变函数论方法等不同途径求出了在上半平面、单位圆内部、单位圆外部三种区域的Poisson积分公式及正则积分方程,其离散化可用于实际计算。 本文是在导师冯康教授指导下完成的,作者谨在此对他表示衷心的感谢。     

Linear self-adjoint multipoint boundary value problems and related approximation schemes

Summary Linear self-adjoint multipoint boundary value problems are investigated. The case of the homogeneous equation is shown to lead to spline solutions, which are then utilized to construct a Green's function for the case of homogeneous boundary conditions. An approximation scheme is described in terms of the eigen-functions of the inverse of the Green's operator and is shown to be optimal in the sense of then-widths of Kolmogorov. Convergence rates are given and generalizations to more general boundary value problems are discussed.     

An initial-value method for the determination of Green's functions

The convergence of an initial-value method for computing the Green's function of a class of second-order differential operators is established. The proof relies on an interpolation procedure which is shown to generalize the Nyström method for Fredholm integral equations. The approximate Green's function is related to the solution of a discrete summation equation. The results of Anselone and Moore on collectively compact operators are then applied.This research was partially supported by UNLV Grant No. 001-060-4573.     

Green's function and comparison principles for first order periodic differential equations with piecewise constant arguments

In this paper we obtain the expression of the Green's function related with a first order periodic differential equation with piecewise constant argument. We derive comparison results for the treated linear operator by studying the sign of the obtained Green's function.     

Effect of fiber orientation on the structure of shock waves in carbon fiber-epoxy composites

Using an analytical relation between the Hugoniot (anisotropic and isotropic) states and other thermodynamic (anisotropic and isotropic) states at high pressures, the effect of fiber orientation on the structure of shock waves in carbon fiber-epoxy composites of various symmetry is investigated. A correct nonlinear model of propagation of shock waves in anisotropic materials is proposed, which employs the conception of total generalized pressure and the pressure corresponding to the thermodynamic response, i.e., to the equation of state. The equation generalizes the nonlinear Hugoniot equation to anisotropic materials and is reduced to the classical variant in the case of isotropy. Invoking the relations of nonlinear anisotropic solids and the generalized decomposition of stress tensor, the double structure of shock waves, consisting of nonlinear anisotropic and isotropic elastic parts, is examined. The numerical calculations of Hugoniot levels of stress agree well with experimental data for a carbon fiber-epoxy composite selected.     

Ewald's Method Revisited: Rapidly Convergent Series Representations of Certain Green's Functions

We propose a justification of Ewald's method of obtaining rapidly convergent series for the Green's function of the 3-dimensional Helmholtz equation. Our point of view enables us to extend the method to Green's functions for the Helmholtz equation in certain domains of ^d with quite general boundary conditions.     

A new approach to the representation of trigonometric and hyperbolic functions by infinite products

An innovative technique is developed for obtaining infinite product representations for some elementary functions. The technique is based on the comparison of alternative expressions of Green's functions constructed by two different methods. Some standard boundary value problems are considered posed for two-dimensional Laplace equation on regions of a regular configuration. Classical closed analytic form of Green's functions for such problems are compared against those obtained by the method of images in the form of infinite products. This yields a number of new infinite product representations for trigonometric and hyperbolic functions.