各向异性体内含任意孔洞对反平面波散射的边界元方法

本文借助于广义格林公式导出了用位移表示的各向异性介质中SH波入射时的边界积分方程.根据本文作者在文献[8]给出的基本解,求解了各向异性介质中孔洞对SH波的散射问题.边界积分方程的离散基于常数元模式.文中给出了一个圆柱、一个椭圆柱和两个椭圆柱形式的孔洞周围的位移场和应力场的数值结果.最后,对入射波频率较高时的情形作了说明.     

Application of the boundary integral equation (BIE) method to transient response analysis of inclusions in a half space

Transient scattering of elastic waves by inclusions in a half space is investigated by the boundary integral equation (BIE) method. The formulation of BIE presented here is based on the Fourier transform method, and involves the analysis of transformed problems and the reconstitution of transient solutions by Fourier inversion. After the BIE has been solved numerically in the transformed domain, the transient wave fields are obtained with the help of the fast Fourier transform (FFT) algorithm. After confirmation of the accuracy of the present method, some numerical examples are shown for various inclusions in a half space, such as a cavity, an elastic inclusion, and a fluid inclusion.     

A boundary element solution to scattering of elastic SH wave by arbitrarily shaped inclusions embedded in an infinite anisotropic medium

Scattering problems for inhomogeneous bodies are investigated by the integral equation method. The boundary integral equation (BIE) for the scattered displacement field associated with finite inhomogeneities in an anisotropic medium are derived with the help of the generalized Green's identity. The discretization of BIE is based upon the constant element, linear element and quadratic element. Several numerical examples for calculating the scattering displacement, stress and scattering cross section from a cylinder, an interface crack, and two elliptic cylinders are given. Results show that the present method can be advantageously applied to a wide range of scattering problems of elastic waves.     

Effects of material gradients on transient dynamic mode-III stress intensity factors in a FGM

This paper presents a transient dynamic crack analysis for a functionally graded material (FGM) by using a hypersingular time-domain boundary integral equation method. The spatial variations of the material parameters of the FGM are described by an exponential law. A numerical solution procedure is developed for solving the hypersingular time-domain traction BIE. To avoid the use of time-dependent Green’s functions which are not available for general FGM, a convolution quadrature formula is adopted for approximating the temporal convolution, while a Galerkin method is applied for the spatial discretization of the hypersingular time-domain traction BIE. Numerical results for the transient dynamic stress intensity factors for a finite crack in an infinite and linear elastic FGM subjected to an impact anti-plane crack-face loading are presented and discussed. The effects of the material gradients of the FGM on the transient dynamic stress intensity factors and their dynamic overshoot over the corresponding static stress intensity factors are analyzed.     

Amended influence matrix method for removal of rigid motion in the interior BVP for plane elasticity

A conventional complex variable boundary integral equation(CVBIE) in plane elasticity is provided. After using the Somigliana identity between a particular fundamental stress field and a physical stress field, an additional integral equality is obtained. By adding both sides of this integral equality to both sides of the conventional CVBIE, the amended boundary integral equation(BIE) is obtained. The method based on the discretization of the amended BIE is called the amended influence matrix method.With this method, for the Neumann boundary value problem(BVP) of an interior region,a unique solution for the displacement can be obtained. Several numerical examples are provided to prove the efficiency of the suggested method.     

Recent advances on the fast multipole accelerated boundary element method for 3D time-harmonic elastodynamics

This article is mainly devoted to a review on fast BEMs for elastodynamics, with particular attention on time-harmonic fast multipole methods (FMMs). It also includes original results that complete a very recent study on the FMM for elastodynamic problems in semi-infinite media. The main concepts underlying fast elastodynamic BEMs and the kernel-dependent elastodynamic FM-BEM based on the diagonal-form kernel decomposition are reviewed. An elastodynamic FM-BEM based on the half-space Green’s tensor suitable for semi-infinite media, and in particular on the fast evaluation of the corresponding governing double-layer integral operator involved in the BIE formulation of wave scattering by underground cavities, is then presented. Results on numerical tests for the multipole evaluation of the half-space traction Green’s tensor and the FMM treatment of a sample 3D problem involving wave scattering by an underground cavity demonstrate the accuracy of the proposed approach. The article concludes with a discussion of several topics open to further investigation, with relevant published work surveyed in the process.     

关于时域边界元法的若干研究

本文从三维弹性动力学方程的基本奇异解着手，导出了适于计算机计算的求解三弹性动力学问题的边界积分方程（ＢＩＥ），并在理论上提出了ＩＢＥ前缘系数矩阵（５）具有（１）准对角特性，（２）其各元素不随时间而变化。据此，本文给出了用时域边界元法求解弹性动力学问题的新方法。最后，数值算例验证了本文方法的正确性。     

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.     

Multiple reciprocity boundary element analysis of two-dimensional anisotropic thermoelasticity involving an internal arbitrary non-uniform volume heat source

In the direct boundary element method (BEM) formulation of anisotropic thermoelasticity, thermal loads manifest themselves as additional volume integral terms in the boundary integral equation (BIE). Conventionally, this requires internal cell discretisation throughout the whole domain. In this paper, the multiple reciprocity method in BEM analysis is employed to treat the general 2D thermoelasticity problem when the thermal loading is due to an internal non-uniform volume heat source. By successively performing the “volume-to-surface” integral transformation, the general formulation of the associated BIE for the problem is derived. The successful implementation of such a scheme is illustrated by three numerical examples.     

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。     

Two-dimensional polynomial eigenstrain formulation of boundary integral equation with numerical verification

The low-order polynomial-distributed eigenstrain formulation of the boundary integral equation (BIE) and the corresponding definition of the Eshelby tensors are proposed for the elliptical inhomogeneities in two-dimensional elastic media. Taking the results of the traditional subdomain boundary element method (BEM) as the control, the effectiveness of the present algorithm is verified for the elastic media with a single elliptical inhomogeneity. With the present computational model and algorithm, significant improvements are achieved in terms of the efficiency as compared with the traditional BEM and the accuracy as compared with the constant eigenstrain formulation of the BIE.     

The boundary element method applied to elastic contact problems with friction

In the current paper the boundary integral equations (BIE) for elastic contact problems with friction are derived from the incremental virtual work principle. After introducing contact conditions of adhesion and slip into BIE all variants on boundary are made to discretize by quadratic isoparametric boundary element. In the current paper not only an auto-increment loading law is presented but also the iterative calculation laws for open, slip and adhesion condition are given. The results of numerical examples are satisfactory.     

各向异性介质的弹性波散射问题的边界元方法

本文引用加权残数法建立了各向异性介质内含任意形式异质夹杂时的散射问题的边界积分方程式,导出了相应的辐射条件,计算了内含圆柱体,椭圆柱体、界面裂纹情形下对SH 波的散射位移场、应力场以及散射横截面.数值结果表明本方法用于解答各向异性介质的弹性波散射问题具有良好的精度和应用前景.     

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.     

弹性理论几类导数边界积分方程之间的变换关系

导数场边界积分方程通常难以应用，因为存在着超奇异主值积分的计算障碍。弹性理论中有几类不同的位移导数边界积分方程，本文采用算子δij和∈ij(排列张量)作用于这些导数边界积分方程，做一系列变换，原有的超奇异积分被正则化为强奇异积分获解。从而建立了这些位移导数边界积分方程之间的转换关系，它们均可以归结为自然边界积分方程。自然边界积分方程仅存在容易计算的Cauchy主值积分。自然边界积分方程分析可直接获得边界应力和位移导数。     

Degenerate scale for multiply connected Laplace problems

The degenerate scale in the boundary integral equation (BIE) or boundary element method (BEM) solution of multiply connected problem is studied in this paper. For the mathematical analysis, we use the null-field integral equation, degenerate kernels and Fourier series to examine the solvability of BIE for multiply connected problem in the discrete system. Two treatments, the method of adding a rigid body term and CHEEF concept (Combined Helmholtz Exterior integral Equation Formulation), are applied to remedy the non-unique solution due to the critical scale. The efficiency and accuracy of the two regularizations are also addressed. For simplicity without loss of generality, the eccentric case is considered to demonstrate the occurring mechanism of degenerate scale.     

各向异性体对瞬态SH波散射问题的边界元方法

本文利用Fourier变换和加权残数法建立了正交各向异性体反平面瞬态波散射问题的边界积分方程,构造了在边界上能满足波动方程的边界元函数,它比常用的二次元和高次元具有较好的适应性。数值结果表明利用该形函数计算高频情形下的散射问题具有很高的精度。     

A NOVEL BOUNDARY INTEGRAL EQUATION METHOD FOR LINEAR ELASTICITY--NATURAL BOUNDARY INTEGRAL EQUATION

The boundary integral equation (BIE) of displacement derivatives is put at a disadvantage for the difficulty involved in the evaluation of the hypersingular integrals. In this paper, the operators δij and εij are used to act on the derivative BIE. The boundary displacements, tractions and displacement derivatives are transformed into a set of new boundary tensors as boundary variables. A new BIE formulation termed natural boundary integral equation (NBIE) is obtained. The NBIE is applied to solving two-dimensional elasticity problems. In the NBIE only the strongly singular integrals are contained. The Cauchy principal value integrals occurring in the NBIE are evaluated. A combination of the NBIE and displacement BIE can be used to directly calculate the boundary stresses. The numerical results of several examples demonstrate the accuracy of the NBIE.