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.     

边界元法应用的若干近期研究

边界元法是在有限元法等其它数值方法的推动和竞争下发展的,作为一种数值方法,发展其实际应用至关重要,本文扼要介绍作者们近期近作若干工作,其中包括：二维弹性体移动接触和滚动接触,网络要群环境下边界元并行计算,以及二维三维边界元实用软件及应用等。     

BEM for simulation of a 2D elastic body with randomly distributed circular inclusions

Based on our 2D BEM software THBEM2 which can be applied to the simulation of an elastic body with randomly distributed identical circular holes, a scheme of BEM for the simulation of elastic bodies with randomly distributed circular inclusions is proposed. The numerical examples given show that the boundary element method is more accurate and more effective than the finite element method for such a problem. The scheme presented can also be successfully used to estimate the effective elastic properties of composite materials. Project supported by the National Natural Science Foundation of China (No. 19772025).     

Analysis of nonlinear large deformation problems by boundary element method

In this paper, the author deduces an approximate solution of nonlinear influence function in rate form for two-dimensional elastic problems on current configuration by the method of comoving coordinate system.Here BEM formulation of large deformation based on Chen’s theory[1] is given. The computational processes of nonlinear boundary integral equation is discussed. The author also compiles a nonlinear computing program NBEM. Numerical examples show that the results presented here is available to the solution of engineering problems.     

Study for 2D moving contact elastic body with closed crack using BEM

Using a sub-regional boundary element method, an algorithm for the two-dimensional elastic bodies with a closed crack loaded by a moving contact elastic body is proposed. Since the extent and status of the contact surface of two elastic bodies and the crack within the body are all not known in advance, a double iterative contact algorithm is used. The BEM program for solving the closed crack problems is developed, some numerical examples are calculated, and the results of the center crack cases are shown to be in good agreement with the analytical solution in the classical fracture mechanics. In the condition of friction and non-friction, some coupling computational results of the SIF for the closed crack, with different angles and loaded by a moving contact elastic body, are also obtained by a numerical computation. The project supported by the National Natural Science Foundation of China (10172053) and NJTU Foundation of China (PD-157)     

A boundary element method for calculating elastic waves scattered by axisymmetric inclusion

A Boundary Element Method (BEM) is described to compute the scattering of elastic waves by an axisymmetric inclusion in an infinite elastic medium. The boundary loads applied to the inclusion is expanded in terms of Fourier series in an infinite space. The boundary integral equation is solved in the general direction of the axisymmetric inclusion by BEM. The problem of the 3-D scattering of elastic waves is reduced to a 1-Done. According to the geometric features of the axisymmetric in clusion the ring shell elements are adopted in this method. A comparison is made with other BEM methods. The numerical results show this method can reduce the amount of calculation and enhance the speed of convergence. Supported by Foundation of Ph. D Program of State Education Commission of China     

AN EFFICIENT PREDICTION METHOD FOR THE TWO-DIMENSIONAL COUPLED STRUCTURAL-ACOUSTIC ANALYSIS

A wave number method (WNM) is proposed to deal with the two-dimensional cou-pled structural-acoustic problem. Based on an indirect Trefftz approach, the displacement andthe pressure response are approximated respectively by a set of wave functions, which exactlysatisfy the governing equations and are independent of the size of the coupled system. The wavefunctions comprise the exact solutions of the homogeneous part of the governing equations andsome particular solution functions, which arise from the external excitation. The weighting coef-ficients of the wave functions can be obtained by enforcing the pressure approximation to satisfythe boundary conditions and it is performed by applying the weighted residual formulation. Theexample is computed by the WNM and the BEM. The results show that, the WNM can attainthe same accuracy and convergence as the BEM with less degrees of freedom. structural-acoustic, Trefftz-method,BEM,weighted residual formulation     

The boundary expanding-contracting method

The boundary element method (KEM) which a used to solve the elastic problems has more advantages than other numerical methods. Especially, it can resolve rapidly varying internal stress and strain fields more accurately. However, it is of en fails in the region near the boundary because of the singularity of the solutions. Though we can increase the boundary meshes more and more, the solutions of stress on the boundary can’t be given directly; which has obstructed the applications of the HEM to some extent.In this paper we proposed the boudary expanding-contracting principle and the boundary expanding-contracting method (BECM) based on the principle. With this method, not only the solutions in the region near or on the boundary can be obtained directly, but the iterative processes can also he used conveniently to improve the accuracy of the solutions.     

Bending of Composite Plates under Static and Dynamic Loading

The stationary bending of a two-dimensional elastic system of joined rectangular plates with different mechanical properties and boundary conditions is studied. A technique for solving the corresponding problem is proposed. It is based on approximate approaches in combination with a generalization of the force method. It is established that the static strain state of the system is essentially dependent on the difference between the elastic moduli of the constituent plates. It is also shown that with certain boundary conditions, the dependence of the first resonant frequency on the relative position of plates along the contact line is nonmonotonic__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 1, pp. 77–84, January 2005.     

Boundary element analysis for elastic and elastoplastic problems of 2D orthotropic media with stress concentration

Both the orthotropy and the stress concentration are common issues in modern structural engineering. This paper introduces the boundary element method (BEM) into the elastic and elastoplastic analyses for 2D orthotropic media with stress concentration. The discretized boundary element formulations are established, and the stress formulae as well as the fundamental solutions are derived in matrix notations. The numerical procedures are proposed to analyze both elastic and elastoplastic problems of 2D orthotropic media with stress concentration. To obtain more precise stress values with fewer elements, the quadratic isoparametric element formulation is adopted in the boundary discretization and numerical procedures. Numerical examples show that there are significant stress concentrations and different elastoplastic behaviors in some orthotropic media, and some of the computational results are compared with other solutions. Good agreements are also observed, which demonstrates the efficiency and reliability of the present BEM in the stress concentration analysis for orthotropic media. The project supported by the Basic Research Foundation of Tsinghua University, the National Foundation for Excellent Doctoral Thesis (200025) and the National Natural Science Foundation of China (19902007). The English text was polished by Keren Wang.     

PROGRAM-PATTERN MULTIPOLE BOUNDARY ELEMENT METHOD FOR FRICTIONAL CONTACT

A mathematical program is proposed for the highly nonlinear problem involving frictional contact. A program-pattern using the fast multipole boundary element method (FMBEM) is given for 3-D elastic contact with friction to replace the Monte Carlo method. A newoptimized generalized minimal residual (GMRES) algorithm is presented. Numerical examples demonstrate the validity of the program-pattern optimization model for node-to-surface contact with friction. The GMRES algorithm greatly improves the computational efficiency.     

The minimal error method for the Cauchy problem in linear elasticity. Numerical implementation for two-dimensional homogeneous isotropic linear elasticity

In this paper, yet another iterative procedure, namely the minimal error method (MEM), for solving stably the Cauchy problem in linear elasticity is introduced and investigated. Furthermore, this method is compared with another two iterative algorithms, i.e. the conjugate gradient (CGM) and Landweber–Fridman methods (LFM), previously proposed by Marin et al. [Marin, L., Háo, D.N., Lesnic, D., 2002b. Conjugate gradient-boundary element method for the Cauchy problem in elasticity. Quarterly Journal of Mechanics and Applied Mathematics 55, 227–247] and Marin and Lesnic [Marin, L., Lesnic, D., 2005. Boundary element-Landweber method for the Cauchy problem in linear elasticity. IMA Journal of Applied Mathematics 18, 817–825], respectively, in the case of two-dimensional homogeneous isotropic linear elasticity. The inverse problem analysed in this paper is regularized by providing an efficient stopping criterion that ceases the iterative process in order to retrieve stable numerical solutions. The numerical implementation of the aforementioned iterative algorithms is realized by employing the boundary element method (BEM) for two-dimensional homogeneous isotropic linear elastic materials.     

Boundary element method for thermoelastic analysis of three-dimensional transversely isotropic solids

Thermal effects are well known to manifest themselves as additional volume integral terms in the direct formulation of the boundary integral equation (BIE) for linear elastic solids when using the boundary element method (BEM). This domain integral has been successfully transformed in an exact manner to surface ones only in isotropy and in 2D anisotropy, thereby restoring the BEM as a truly boundary solution technique. The difficulties with extending it to 3D general anisotropic solids lie in the mathematical complexity of the Green’s function and its derivatives for such materials. These quantities are required items in the BEM formulation. In this paper, the exact, analytical transformation of the volume integral associated with thermal effects to surface ones is achieved for a transversely isotropic material using a similar approach which the authors have previously employed for the same task in BEM for 2D general anisotropy. A numerical scheme, however, needs to be employed to evaluate some of the new terms introduced in the surface integrals that arise from this process here. The mathematical soundness of the formulation is demonstrated by a few examples; the numerical results obtained are checked by alternative means, including those obtained from the commercial FEM code, ANSYS.     

双材料界面裂纹复应力强度因子的正则化边界元法

双材料界面裂纹渐近位移和应力场表现出剧烈的振荡特性, 许多用于表征经典平方根($r^{1/2})$和负平方根($r^{-1/2})$渐近物理场的传统数值方法失效, 给界面裂纹复应力强度因子($K_{1} +{i}K_{2} )$的精确求解增加了难度. 引入一种含有复振荡因子的新型"特殊裂尖单元", 可精确表征裂纹尖端渐近位移和应力场的振荡特性, 在避免裂尖区域高密度网格剖分的情况下, 可实现双材料界面裂纹复应力强度因子的精确求解. 此外, 结合边界元法中计算近奇异积分的正则化算法, 成功求解了大尺寸比(超薄)双材料界面裂纹的断裂力学参数. 数值算例表明, 所提算法稳定, 效率高, 在不增加计算量的前提下, 显著提高了裂尖近场力学参量和断裂力学参数的求解精度和数值稳定性.      

Nonlinear dynamic analysis of Timoshenko beams by BEM. Part I: Theory and numerical implementation

In this two-part contribution, a boundary element method is developed for the nonlinear dynamic analysis of beams of arbitrary doubly symmetric simply or multiply connected constant cross section, undergoing moderate large displacements and small deformations under general boundary conditions, taking into account the effects of shear deformation and rotary inertia. Part I is devoted to the theoretical developments and their numerical implementation and Part II discusses analytical and numerical results obtained from both analytical or numerical research efforts from the literature and the proposed method. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse loading and bending moments in both directions as well as to axial loading. To account for shear deformations, the concept of shear deformation coefficients is used. Five boundary value problems are formulated with respect to the transverse displacements, to the axial displacement and to two stress functions and solved using the Analog Equation Method, a BEM based method. Application of the boundary element technique yields a nonlinear coupled system of equations of motion. The solution of this system is accomplished iteratively by employing the average acceleration method in combination with the modified Newton–Raphson method. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress functions using only boundary integration. The proposed model takes into account the coupling effects of bending and shear deformations along the member, as well as the shear forces along the span induced by the applied axial loading.

     

正交各向异性弹性问题的规则化边界元法

本文致力于平面正交各向异性弹性问题的规则化边界元法研究,提出了新的规则化边界元法的理论和方法。对问题的基本解的特性进行了研究,确立基本解的积分恒等式,提出一种基本解的分解技术,在此基础上,结合转化域积分方程为边界积分方程的极限定理,建立了新颖的规则化边界积分方程。和现有方法比,本文不必将问题变换为各向同性的去处理,从而不含反演运算,也有别于Galerkin方法,无需计算重积分,因此所提方法不仅效率高,而且程序设计简单。特别是,所建方程可计算任何边界位移梯度,进而可计算任意边界应力,而不仅限于面力。数值实施时,采用二次单元和椭圆弧精确单元来描述边界几何,使用不连续插值逼近边界函数。数值算例表明,本文算法稳定、效率高,所取得的边界量数值结果与精确解相当接近。     

On the Dynamic Interaction and Cross-Interaction of 2D Rigid Structures with Orthotropic Elastic Media Possessing General Principal Axes Orientation

A direct boundary element method (BEM) implementation for the dynamic interaction analysis in the frequency domain of 2D rigid structures with elastic orthotropic media is presented. The BEM implementation is based on non-singular full-space influence functions. The rigid structure response is obtained by applying equilibrium and kinetic compatibility conditions. The method is applied to the analysis of the dynamic response of a rigid tunnel in a half-space with various elasticity principal axes inclinations and to the analysis of two rigid rectangular galleries in a half space with various distances between them.     

Singular boundary method for solving plane strain elastostatic problems

This study documents the first attempt to apply the singular boundary method (SBM), a novel boundary only collocation method, to two-dimensional (2D) elasticity problems. Unlike the method of fundamental solutions (MFS), the source points coincide with the collocation points on the physical boundary by using an inverse interpolation technique to regularize the singularity of the fundamental solution of the equation governing the problems of interest. Three benchmark elasticity problems are tested to demonstrate the feasibility and accuracy of the proposed method through detailed comparisons with the MFS, boundary element method (BEM), and finite element method (FEM).     

Time-harmonic BEM for 2-D piezoelectricity applied to eigenvalue problems

We will derive the fundamental generalized displacement solution, using the Radon transform, and present the direct formulation of the time-harmonic boundary element method (BEM) for the two-dimensional general piezoelectric solids. The fundamental solution consists of the static singular and the dynamics regular parts; the former, evaluated analytically, is the fundamental solution for the static problem and the latter is given by a line integral along the unit circle. The static BEM is a component of the time-harmonic BEM, which is formulated following the physical interpretation of Somigliana’s identity in terms of the fundamental generalized line force and dislocation solutions obtained through the Stroh–Lekhnitskii (SL) formalism. The time-harmonic BEM is obtained by adding the boundary integrals for the dynamic regular part which, from the original double integral representation over the boundary element and the unit circle, are reduced to simple line integrals along the unit circle.The BEM will be applied to the determination of the eigen frequencies of piezoelectric resonators. The eigenvalue problem deals with full non-symmetric complex-valued matrices whose components depend non-linearly on the frequency. A comparative study will be made of non-linear eigenvalue solvers: QZ algorithm and the implicitly restarted Arnoldi method (IRAM). The FEM results whose accuracy is well established serve as the basis of the comparison. It is found that the IRAM is faster and has more control over the solution procedure than the QZ algorithm. The use of the time-harmonic fundamental solution provides a clean boundary only formulation of the BEM and, when applied to the eigenvalue problems with IRAM, provides eigen frequencies accurate enough to be used for industrial applications. It supersedes the dual reciprocity BEM and challenges to replace the FEM designed for the eigenvalue problems for piezoelectricity.