边界元方法的一些研究进展

本文旨在综述我们小组近二十年来在边界元方法这一领域的一些研究成果,在简要介绍边界元方法的基本思想后,主要介绍了一类非线性界面问题的有限元-边界元耦合方法、求解电磁散射问题的有限元-边界元耦合方法和超奇异积分的一类计算方法.     

Artificial Multilevel Boundary Element Preconditioners

A hierarchical multilevel preconditioner is constructed for an efficient solution of a first kind boundary integral equation with the single layer potential operator discretized by a boundary element method. This technique is based on a hierarchical clustering of all boundary elements as used in fast boundary element methods. This hierarchy is applied to define a sequence of nested boundary element spaces of piecewise constant basis functions as used in the definition of the preconditioning multilevel operator.     

一种改进边界元法在弹性扭转问题中的应用

本文用一种改进边界元法分析与计算了椭圆截面等直杆的扭转问题.并与边界元法的解进行比较,其结果极为符合.然而,改进边界元法较边界元法所需要的数据量少得多,计算时间也将大大减少了.因此,本文方法对求解Poisson方程问题是一种经济而行之有效的数值计算方法.     

一个扩散问题的自然边界元法与有限元法组合

本文讨论由Helmholtz方程描述的扩散问题的自然边界元法与有限元法的组合．取一个圆作为公共边界，用Fourier展开建立边界积分方程，将无界区域上的问题化为有界区域上的非局部边值问题．在变分方程中公共边界上的未知量只包含函数本身而不包含其法向导数，从而减少了未知数的数目，并且边界元剐度矩阵只有极少量不同的元素，有利于数值计算．这种组台方法优越于建立在直接边界元法基础上的组合方法．文中证明了变分解的唯一性，数值解的收敛性和误差估计．最后讨论了数值技术并给出一个算倒．     

Algebraic Multigrid Preconditioners for Sparse Approximations of Boundary Element Matrices

This paper presents new algebraic multigrid preconditioners for sparse representations of boundary element matrices which arise from the so‐called adaptive‐cross‐approximation to dense boundary element matrices resulting from the standard collocation, or Galerkin boundary element discretization of the single layer potential operator.     

Stabilized boundary element methods for exterior Helmholtz problems

In this paper we describe and analyze some modified boundary element methods to solve the exterior Dirichlet boundary value problem for the Helmholtz equation. As in classical combined field integral equations also the proposed approach avoids spurious modes. Moreover, the stability of related modified boundary element methods can be shown even in the case of Lipschitz boundaries. The proposed regularization is done based on boundary integral operators which are already included in standard boundary element formulations. Numerical examples are given to compare the proposed approach with other already existing regularized formulations.     

A BE based finite element for crack propagation processes

A boundary element based finite macro element for the simulation of 3D crack propagation in the framework of linear elastic fracture mechanics is presented. While the major part of the numerical model is discretized with finite elements, a small domain containing the crack is meshed with boundary elements. By means of the Symmetric Galerkin BEM a stiffness formulation for the cracked BE domain is obtained which enables a direct FEM/BEM coupling. All necessary operations for the crack propagation are carried out within this boundary element based finite macro element and exploit the potential of the boundary integral formulation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The coupling of bem and fem for the two-dimensional viscous flow problem

In this paper we propose a numerical scheme for treating the problem of sJow viscous flow past an obstacle in the plane. This scheme is a combination of boundary element and finite element methods. By introducing an auxiliary boundary curve, we divide the region under consideration into two subregions, an inner and an outer region. In the inner region, we employ a finite element method (FEM) for solving a system of simplified field equations with proper natural boundary conditions. In the outer region, the solution is expressed in the form of a simple-layer potential with density function satisfying a system of modified integral equations of the first kind. The latter are solved by a boundary element method (BEM). Both solutions are matched on the common auxiliary boundary curve. Error estimates in suitable function spaces are derived in terms of the mesh widths as well as the small parameters, the Reynolds numbers     

A NOTE ON C~1-CURVED FINITE ELEMENT

This paper shows that the C1-curved finite element developed by Bernadon in general can not satisfy the essential boundary conditions on approximate boundary. Furthermore, a modified C1-curved finite element is given, which is compatible with the element of Argyris triangle and can satisfy the homogeneous Dirichlet boundary conditions onapproximate boundary.     

Adaptive boundary element methods with convergence rates

This paper presents adaptive boundary element methods for positive, negative, as well as zero order operator equations, together with proofs that they converge at certain rates. The convergence rates are quasi-optimal in a certain sense under mild assumptions that are analogous to what is typically assumed in the theory of adaptive finite element methods. In particular, no saturation-type assumption is used. The main ingredients of the proof that constitute new findings are some results on a posteriori error estimates for boundary element methods, and an inverse-type inequality involving boundary integral operators on locally refined finite element spaces.     

Error bounds for the finite element approximation of an incompressible material in an unbounded domain

Summary. In this paper we design high-order local artificial boundary conditions and present error bounds for the finite element approximation of an incompressible elastic material in an unbounded domain. The finite element approximation is formulated in a bounded computational domain using a nonlocal approximate artificial boundary condition or a local one. In fact there are a family of nonlocal approximate artificial boundary conditions with increasing accuracy (and computational cost) and a family of local ones for a given artificial boundary. Our error bounds indicate how the errors of the finite element approximations depend on the mesh size, the terms used in the approximate artificial boundary condition and the location of the artificial boundary. Numerical examples of an incompressible elastic material outside a circle in the plane is presented. Numerical results demonstrate the performance of our error bounds. Received August 31, 1998 / Revised version received November 6, 2001 / Published online March 8, 2002     

多裂纹问题计算分析的本征COD边界积分方程方法

针对多裂纹问题，若采用常规的数值求解技术，计算效率较低.为实现多裂纹问题的大规模数值模拟，建立了本征裂纹张开位移（crack opening displacement, COD）边界积分方程及其迭代算法，并引入Eshelby矩阵的定义，将多裂纹分为近场裂纹和远场裂纹来处理裂纹间的相互影响.以采用常单元作为离散单元的快速多极边界元法为参照，对提出的计算模型和迭代算法进行了数值验证.结果表明，本征COD边界积分方程方法在处理多裂纹问题时取得较大的改进，其计算效率显著高于传统的边界元法和快速多极边界元法.     

An Iterative FEM for Solving Wave Problems of a Halfspace

An iterative finite element method for solving wave problems of a halfspace is presented in this paper. The halfspace is first truncated by introducing a fictive finite boundary on which some fictive boundary conditions must be imposed. A finite computational domain is in each iteration subjected to actual boundary conditions on real boundary and to fictive Dirichlet or Neumann boundary conditions on the fictive boundary. The radiation condition is satisfied by using DtN operator. The DtN operator is not introduce in the finite element formulation on the fictive boundary so any finite elements can be used. The method is simple and specially useful for computing higher harmonics.     

应用边界元法模拟纤维增强复合材料平面弹性问题

将含有随机分布多种夹杂相复合材料的二维弹性力学问题归结为复连通区域的边界积分方程,进而转化成矩阵方程进行求解和分析．根据同类夹杂相外在边界上的面力与位移之间关系矩阵完全相同的特点,使得最后的矩阵方程阶数得到大规模减少,这正是此处提出改进的边界元方法的主要思路．数值算例表明,对于此类问题,与常规的边界元分域解法相比更加有效．以该方法为基础,可以详细给出纤维增强复合材料二维条件下的宏观等效力学性质．     

A subregion DRBEM formulation for the dynamic analysis of two-dimensional cracks

The dual reciprocity boundary element method employing the step by step time integration technique is developed to analyse two-dimensional dynamic crack problems. In this method the equation of motion is expressed in boundary integral form using elastostatic fundamental solutions. In order to transform the domain integral into an equivalent boundary integral, a general radial basis function is used for the derivation of the particular solutions. The dual reciprocity boundary element method is combined with an efficient subregion boundary element method to overcome the difficulty of a singular system of algebraic equations in crack problems. Dynamic stress intensity factors are calculated using the discontinuous quarter-point elements. Several examples are presented to show the formulation details and to demonstrate the computational efficiency of the method.     

关于薄板的无网格局部边界积分方程方法中的友解

无网格局部边界积分方程方法是最近发展起来的一种新的数值方法,这种方法综合了伽辽金有限元、边界元和无单元伽辽金法的优点,是一种具有广阔应用前景的、真正的无网格方法．把无网格局部边界积分方程方法应用于求解薄板问题,给出了薄板无网格局部边界积分方程方法所需要的友解及其全部公式．     

Error estimates for the finite element approximation of linear elastic equations in an unbounded domain

In this paper we present error estimates for the finite element approximation of linear elastic equations in an unbounded domain. The finite element approximation is formulated on a bounded computational domain using a nonlocal approximate artificial boundary condition or a local one. In fact there are a family of nonlocal approximate boundary conditions with increasing accuracy (and computational cost) and a family of local ones for a given artificial boundary. Our error estimates show how the errors of the finite element approximations depend on the mesh size, the terms used in the approximate artificial boundary condition, and the location of the artificial boundary. A numerical example for Navier equations outside a circle in the plane is presented. Numerical results demonstrate the performance of our error estimates.

     

On the exponential convergence of the h–p version for boundary element Galerkin methods on polygons

This paper applies the technique of the h–p version to the boundary element method for boundary value problems on non-smooth, plane domains with piecewise analytic boundary and data. The exponential rate of convergence of the boundary element Galerkin solution is proved when a geometric mesh refinement towards the vertices is used.     

特征值问题协调/非协调有限元后验误差分析

本文研究对称椭圆特征值问题的有限元后验误差估计,包括协调元和非协调元,具有下列特色:(1)对协调/非协调元建立了有限元特征函数uh的误差与相应的边值问题有限元解的误差在局部能量模意义下的恒等关系式,该边值问题的右端为有限元特征值λh与uh的乘积,有限元解恰好为uh.从而边值问题有限元解在能量模意义下的局部后验误差指示子,包括残差型和重构型后验误差指示子,成为有限元特征函数在能量模意义下的局部后验误差指示子.(2)讨论了协调有限元特征函数的基于插值后处理的梯度重构型后验误差估计,对有限元特征函数的导数得到了最大模意义下的渐近准确局部后验误差指示子.     

Hermite三次样条多小波自然边界元法

针对应用自然边界元方法解上半平面的Laplace方程的Neumann边值问题时存在奇异积分的困难,本文提出了Hermite三次样条多小波自然边界元法.Hermite三次样条多小波具有较短的紧支集、很好的稳定性和显式表达式,而且它们在不同层上的导数还是相互正交的.因此,本文将它与自然边界元法相结合,利用小波伽辽金法离散自然边界积分方程,使自然边界积分方程中的强奇异积分化为弱奇异积分,从而降低了问题的复杂性.文中给出的算例表明了该方法的可行性和有效性.