边界积分方程的若干基本理论问题

本文提出了边界积分方程理论体系框图,对其中研究得不多的和容易引起误解的环节,如基本解的定义、解的边界积分表示、超定问题有解的充要条件等,作了专门的讨论.     

平面Laplace外边值问题

证明平面调和函数的Dirichlet外问题解存在唯一的充要条件,在此基础上,确立外问题的等价边界积分方程,首次给出外域上的极值原理,对第一类Fredholm边界的积分方程的可解性进行了讨论。     

弹性力学问题中一个新的边界积分方程——自然边界积分方程

位移导数边界积分方程一直存在着超奇异积分计算的障碍,该文提出以符号算子δye和εye作用于位移导数边界积分方程,施用一系列变换将边界位移、面力和位移导数转成为新的边界张量,从而得到一个新的边界积分方程－－自然边界积分方程,自然边界积分方奇异性为强奇性,文中给出了相应的Cauchy主值积分算式,自然边界积分方程与位移边界积分方程联合可直接获取边界应力,几个算例表明了自然边界积分方程的正确性。     

边界积分方程—边界元法的基本理论及其在弹性力学方面的若干工程应用

本文第一部分对于直接法弹性力学边界积分方程的基本理论作了论述,全文采用内积公式以加权余量形式来建立边界积分方程.论述范围包括位势问题、弹性静力学问题和克希霍夫型平板理论的边界积分方程—边界元法.文中同时写出相应的变分格式.并讨论了非光滑边界的处理.本文第二部分简介对若干具体问题用特定的基本解进行的有关数值计算.文中介绍的研究组所获初步结果包括:迴转体的扭转、轴对称问题和弯曲问题,以及平板弯曲问题的边界积分方程—边界元法应用的具体结果.计算结果表明对于改进和扩充工程实用应力集中数据及平板计算(包括自由边界及角点问题)将是有益的.     

导数场边界积分方程分析近边界应力分布

研究二维弹性力学问题边界积分方程,通过分部积分变换消除了常规导数边界积分方程中的超奇异积分,获得仅含强奇异积分的应力自然边界积分方程.对于近边界应力的计算,进一步运用正则化算法解析计算其中的几乎强奇异积分.较常规边界元法相比,应力自然边界积分方程可以求解离边界更加接近的内点应力值.算例证明了文中方法的可应用性和有效性.     

弹性薄板弯曲问题的等价的直接变量边界积分方程

建立平面弹性薄板弯曲问题理论中具有直接变量的等价边界积分方程，传统的直接变量边界积分方程，它们都不是等价的，对此进行了深入的讨论。     

板弯曲与平面弹性问题的多类变量变分原理

进一步完善板弯曲与平面弹性问题的多类变量变分原理,给出了相关边界积分项的具体表达式．多类交量变分原理涵盖了平衡、应力函数、应力、位移一应变、协调和物性共五大类基本方程和所有边界条件,是一个具有更加广泛意义的变分原理．     

平面弹性外问题的等价的直接变量边界积分方程

用变分法证明平面弹性力学外边值问题的正确提法。在此基础之上，确立外问题的等价的直接变量边界积分方程。对传统的惯用的直接变量边界积分方程进行了深入的讨论，表明它与原边值问题不等价。     

各向异性位势平面问题的规则化边界元法

基于转化域方程为边界积分方程的极限定理及一个新颖的基本解分解技术, 建立间接变量规则化边界积分方程, 它有效地避免了奇异积分的直接计算. 与已有方法比,该方法不将问题变换为各向同性的问题去处理, 因而无需反演运算, 也有别于Galerkin方法, 无需计算重积分. 可计算任意边界位势梯度, 而不仅限于法向通量. 针对椭圆边界的边值问题, 提交一种精确单元来描述边界几何. 数值算例表明, 所提算法稳定且效率高, 所得数值结果与精确解吻合较好.      

热弹性平面问题的规则化边界积分方程

对于热弹性平面问题,过去广泛集中在直接变量边界元法研究,本文研究间接变量规则化边界元法,建立了间接变量规则化边界积分方程。和直接边界元法相比,间接法具有降低密度函数的连续性要求、位移梯度方程中的热载荷体积分具有较弱奇异性等优点。数值实施中,用精确单元描述边界几何,不连续插值函数逼近边界量。算例表明,本文方法效率高,所得数值结果与精确解相当吻合。     

Necessary and sufficient condition for the correct formulation of boundary integral equations of harmonic functions

A necessary and sufficient condition for the correct formulation of boundary integral equations of harmonic functions is established in the present paper. A super-determined problem of harmonic functions is proposed for the first time. Then a necessary and sufficient condition for the existence of solution of the super-determined problem is proved. At the same time, it is a necessary and sufficient condition for the correct formulation of boundary integral equations with direct unknown quantities. A relation between boundary integral equations and variational principles is discovered for the first time. And a one-to-one correspondence between boundary integral equations with direct and indirect unknown quantities is indicated. The concept and route of the present paper can be applied to other boundary value problems possessing variational principles.     

关于加权残值法的充要性问题及其权函数的选择

本文论证了偏微分方程初-边值混合问题的加权残值方程的充要性条件和有关定理。关于充要性条件的讨论也为正确选择解析函数作为权函数,建立充要的加权残值方程提供了理论指导。重点讨论了边界积分方程的充要性。指出了两种满足充要条件的权函数和相应的边界积分方程。     

Numerical comparison of two boundary element methods for plane harmonic functions

A direct boundary element method (BEM) has been studied in the paper based on a set of sufficient and necessary boundary integral equations (BIE) for the plane harmonic functions. The new sufficient and necessary BEM leads to accurate results while the conventional insufficient BEM will lead to inaccurate results when the conventional BIE has multiple solutions. Theoretical and numerical analyses show that it is beneficial to use the sufficient and necessary BEM, to avoid hidden dangers due to non-unique solution of the conventional BIE.     

Solution of the elastic boundary value problems for a layer with tunnel stress raisers

A novel procedure for solving three-dimensional problems for elastic layer weakened by through-thickness tunnel cracks has been developed and is presented in this paper. This procedure reduces the given boundary value problem to an infinite system of one-dimensional singular integral equations and is based on a system of homogeneous solutions for a layer. Integral representations of single- and double-layer potentials are used for metaharmonic and harmonic functions entering in the singular integral equations. These representations provide a continuous extendibility of the stress vector while allowing a jump in the displacement vector in the transition through the cut.Expanding the potential and biharmonic solutions in the Fourier series over the thickness coordinate yields the integral representations of the displacement vector and stress tensor. The problem of reducing a denumerable set of the integral equations of the given boundary value problem to one-to-one correspondence with the set of unknown densities appearing in the Fourier’s coefficient representations has been settled efficiently. Numerical investigations show a rapid convergence of the proposed reduction procedure as applied to the solution of the infinite system of one-dimensional integral equations. Numerical examples illustrate the proposed method and demonstrate its advantages.     

Low Reynolds number hydrodynamic interaction of a solid particle with a planar wall

The slow viscous flow problem of an arbitrary solid particle in motion near a planar wall is recast into a boundary integral formulation. The present formulation employs the Green function appropriate to the planar wall problem and is developed in sufficient generality to allow calculations for arbitrary particles in any base flow which satisfies Stokes equations and no-slip on the wall. The resulting integral equations are easily discretized and solved for the particle surface tractions. Calculations are performed for axisymmetric motions of a variety of ellips?ids near the planar wall. Agreement with existing theory is excellent.     

Interaction between a compliant disk-shaped inclusion and a crack upon incidence of an elastic wave

The propagation of harmonic elastic wave in an infinite three-dimensional matrix containing an interacting low-rigidity disk-shaped inclusion and a crack. The problem is reduced to a system of boundary integral equations for functions that characterize jumps of displacements on the inclusion and crack. The unknown functions are determined by numerical solution of the system of boundary integral equations. For the symmetric problem, graphs are given of the dynamic stress intensity factors in the vicinity of the circular inclusion and the crack on the wavenumber for different distances between them and different compliance parameters of the inclusion.     

THE GENERALIZED GALERKIN’S EQUATION OF THE FINITE ELEMENT,THE BOUNDARY VARIATIONAL EQUATIONS AND THE BOUNDARY INTEGRAL EQUATIONS

Based on[1 ],we have further applied the variational princi-ple of the variable boundary to investigate the discretizationanalysis of the solid system and derived the generalized Ga-lerkin’s equations of the finite element.the boundary varia-tional equations and the boundary integral equations.These e-quations indicate that the unknown functions of the solid sys-tem must satisfy the conditions in the element S_a or on the boun-dariesГ_a.These equations are applied to establishing the discreti-zation equations in order to obtain the numerical solution ofthe unknown functions.At a time these equations can be usedas the basis for the simplified calculation in various corres-ponding cases.In this paper,the results of boundary integral equationsshow that the calculation of integration is not accurate alongthe surfaceГ_a of interior element S_a by J-integral suggestedby Rice[2].     

Resolution of linear viscoelastic equations in the frequency domain using real Helmholtz boundary integral equations

Boundary integral equations are well suitable for the analysis of seismic waves propagation in unbounded domains. Formulations in elastodynamics are well developed. In contrast, for the dynamic analysis of viscoelastic media, there are very seldom formulations by boundary integral equations. In this Note, we propose a new and simple formulation of time harmonic viscoelasticity with the Zener model, which reduces to classical elastodynamics if a compatibility condition is satisfied by boundary conditions. Intermediate variables which satisfy the classical elastodynamic equations are introduced. It makes it possible to utilize existing numerical tools of time harmonic elastodynamics. To cite this article: S. Chaillat, H.D. Bui, C. R. Mecanique 335 (2007).     

用样条边界元与奇异积分方程法联合求解多裂纹柱体的扭转

通过间解的分离，本文将径向多裂纹柱体的导曲函两个调和函数表示，使问题归为解一组混混合型积分方程。针对方程的特点，本文联合使用三次样条边界法与奇异积分方程的数值方法对所得方程建立了数值法，并对裂纹相交情形作了特殊处理。最后对工程中感兴趣的一些典型的多裂纹柱体的扭转作了例题计算，结果表明，本文方法具有收敛快，精度高的特点。