用边界元法计算声辐射时高次奇异积分的处理方法

边界元法应用于计算辐射声场时，由于奇积分的存在，会影响以计算结果的精度，本文描述了处理带有１／ｒ奇异积分和１／ｒ＾２二次奇异积分处理方法，包括数学证明和数值积分方法，计算结果表明这种方法能提高精度。     

模态边界元法

本文介绍了用有限区域弹性体代替传统的无限区域弹性体建立边界方程基本解的思想。文中以有限弹性体模态的线性组合表示稳态基本解,建立了积分与激励频率无关的稳态边界元的直接表达式和间接表达式。其次文中提出了求解弹性系统固有频率和振型的设想,导出了边界元法特征方程。最后,文中以一维固有特性问题为例对原理、方法的正确性和精确度进行了检验,得到了不同边界条件下的固有频率和振型。文中所给数据表明与理论值吻合良好。     

Helmholtz声学边界积分方程中奇异积分的计算

提出了一种非等参单元的四边形坐标变换,它将积分的曲面单元映射为另一四边形单元,通过两次坐标变换引入的雅可比行列式可以消除Helmholtz声学边界积分方程中的弱奇异型O(1／r))积分．而且利用δr／δn以及坐标变换可以同时消除坐标变换无法消除的Cauchy型(O(1／r^2))奇异积分,并给出了消除奇异性的详细证明．该方法给Helmholtz声学边界积分方程中的弱奇异积分与Cauchy奇异积分的计算以及编程提供了极大便利。     

三维粘性流体内流问题的边界元研究中求解奇异积分的一种新的数值解法

本文探讨了三维粘性流体内流问题的边界元法研究中奇异积分的一种有效的解法。对边界积分项,采用三角形极坐标来降低奇异积分的维数,从而将整体坐标系下的三角形单元转换成局部坐标系中的单位正方形单元;对域积分项,采用四面体极坐标,不但降低了奇异积分的维数,而且将整体坐标系下的四面体单元转换成局部坐标系中的单位立方体单元,从而使积分域简单化。最后利用高斯积分进行数值求解。     

三维热弹性力学问题的混合边界元法

本本文给出了三维无限大域内点热源作用下的位移、应力场基本解。采用基于虚拟热源法的间接边界元法和直接边界元法的混合边界元法求解三维有限域热弹性力学问题,有效地避免了热弹性力学问题中域内积分的处理。数值计算表明混合边界元法求热弹性力学问题具有简单方便、精度较高的优点。     

结构振动辐射声场的预估——边界积分方程中奇异积分的间接处理

本文讨论了利用边界积分方程和边界元技术计算结构的稳态外辐射声场的方法,同时,对边界元方法所固有的奇异数值积分提出了一种简单方便的间接处理方法。计算实例证明所编计算程序和奇异积分处理方法是成功的。利用该程序在已知结构表面振速分布的条件下,可以求出该结构在自由声场中的声功率、表面辐射效率以及声场中任意点的声压值和相位。对一个实际钢质空心封闭圆筒作了计算与实测的比较,结果显示了该方法可应用于实际结构或机器的前景。     

边界元法在环境声学中的应用

边界元法是边界积分方程的数值解法 ,是随着计算机技术的发展而出现的。建立声学边界积分方程分两种方法 :直接法与间接法。本文介绍了边界元法在环境声学中的应用 ,如声屏障和不同情况下道路周围的声场分布、复杂气象条件对声传播的影响的问题等。由于边界元法是半解析半数值解法。在解边界积分方程时会遇到解的存在与唯一性问题。     

一类二维奇异积分方程的解法

当|λ|≠1时,在空间L_P(GU Г),P>2,中按Hausdroff正规可解的充要条件。 本文在L_P(GUГ),P>2,中考虑奇异积分方程     

粘弹性分析的复变边界积分方程法

本文应用弹性－粘弹性对应原理提出了基于复位势基本解的二维粘弹性分析的复变边界积分方程方法，给出了所有基本关系式，编制了相应的计算程序并给出计算实例，与已有工作相比，本方法具有公式统一，程序简洁通用，边界单元数少和效率高等特点。     

AN INDIRECT BOUNDARY ELEMENT METHOD FOR PLATE BENDING ANALYSIS

An indirect Boundary Element Method is employed for the static analysis of homogeneous isotropic and linear elastic Kirchhoff plates of an arbitrary geometry. The objectives of this paper consists of a construction and a study of the resulting boundary integral equations as well as a development of stable powerful algorithms for their numerical approximation. These equations involve integrals with high-order kernel singularities. The treatment of singular and hypersingular integrals and a construction of solutions in the neighborhood of the irregular points on the boundary are discussed. Numerical examples illustrate the procedure and demonstrate its advantages. © 1997 by John Wiley & Sons, Ltd.     

THE EVALUATIONS OF CAUCHY PRINCIPAL VALUE INTEGRALS AND WEAKLY SINGULAR INTEGRALS IN BEM AND THEIR APPLICATIONS

A new technique is developed to evaluate the Cauchy principal value integrals and weakly singular integrals involved in the boundary integral equations. The boundary element method is then applied to analyse scattering of waves by cracks in a laminated composite plate. The Green's functions are obtained in discrete form through the thickness of the plate using a stiffness method. To circumvent the difficulties associated with the evaluation of hypersingular integrals due to the presence of cracks, the multidomain technique is applied. Numerical computations have shown the accuracy and reliability of the proposed technique. Scattered wave fields for a composite plate with a horizontal crack are computed. The numerical results show that the applications of the technique in non-destructive evaluation of defects is very promising.     

用基于流形元的子域奇异边界元法模拟重力坝的地震破坏

本文提出了用流形无法和子域奇异边界元法相结合模拟结构地震响应及地震破坏的方法。流形元法的独特的网格及接触的处理方式,使该方法不仅可以象有限元法那样精确地分析结构的变形和应力,还可以象DDA、DEM等不连续分析方法那样模拟不连续面的接触和块体的运动。本文将Newmark法引入流形无法,使得该方法可以直接用来分析动力学问题。与作者提出的子域奇异边界元法相结合,可以模拟裂缝沿任意方向扩展及结构的地震破坏问题。对Koyna重力坝进行了地震破坏模拟分析,很好地再现了该坝的破坏过程,模拟破坏形式及裂缝出现的位置与实际调查结果、实验结果及其他学者用模糊裂缝模型得到的破坏结果吻合良好。     

HYPERSINGULAR RESIDUALS—A NEW APPROACH FOR ERROR ESTIMATION IN THE BOUNDARY ELEMENT METHOD

This paper presents a new approach for a posteriori ‘pointwise’ error estimation in the boundary element method. The estimator relies upon evaluation of the residual of hypersingular integral equations, and is therefore intrinsic to the boundary integral equation approach. A methodology is developed for approximating the error on the boundary as well as in the interior of the domain. Extensive computational experiments have been performed for the two-dimensional Laplace equation and the numerical results indicate that the error estimates successfully track the form of the exact error curve. Moreover, a reasonable estimate of the magnitude of the actual error is also predicted.     

二维正交各向异性结构弹塑性问题的边界元分析

给出了二维正交各向异性结构弹塑性问题的边界元分析方法, 包括相应边界积分方程、内点应力公式、边界元求解格式以及弹塑性应力计算方法。在弹塑性分析中, 引入了Hill-Tsai 屈服准则, 采用初应力法和切向预测径向返回法确定实际应力状态。通过具体算例分析了二维正交各向异性结构的弹塑性应力和塑性区分布情况, 部分数值结果与已有结果进行了比较, 两者基本吻合。结果表明, 本文中给出的边界元法可以有效地用于求解二维正交各向异性结构的弹塑性问题。      

SENSITIVITY ANALYSIS FOR BOUNDARY ELEMENT ERROR ESTIMATION AND MESH REFINEMENT

The subject of this paper is the sensitivity analysis of approximate boundary element solutions with respect to the positions of the collocation points. The direct differentiation approach is considered here and the analysis is performed analytically. Since only the collocation points are perturbed, the shape of the body and the corresponding discretization remain unaltered. This aspect makes the present work quite different in spirit with respect to earlier analyses on shape sensitivities. Sensitivities of approximate BEM solutions with respect to the positions of collocation points are shown to be related to the residual of hypersingular integral equations. Numerical results confirm that the present approach can be seen as the analytical counterpart of an adaptive scheme for mesh refinement presented by the same author in some recent papers. Some other advantages of the present approach over the former one are also outlined.     

INFINITE DOMAIN CORRECTION FOR IN-PLANE BODY WAVES IN A TWO-DIMENSIONAL BOUNDARY ELEMENT ANALYSIS

A method is described in this article to correct for the error that arises with the discretization of domains that include boundaries that extend to infinity. Typically when open domains are discretized, part of the boundary is excluded from the calculation resulting in a truncated region. Of particular interest in this article are earthquake wave amplification problems through zoned media. In these type of problems, the boundary element discretization scheme typically results in truncated regions. Correction for truncation in anti-plane wave problems has already been addressed in a previous article by Heymsfield. In this article, truncation correction for in-plane body waves in a damped material will be discussed. To prove the validity of the proposed technique, the method is checked by calculating the soil amplification of a unit in-plane SV wave through a soil layer resting on a rock half-space. Since an analytic solution exists for this problem, the problem serves as a good basis to compare results with and without the corrections for truncation. Results for this particular problem compare the analytic solution with the numerical solution considering (1) no truncation correction, (2) only layer correction, and (3) both layer and half-space corrections. © 1997 by John Wiley & Sons, Ltd.     

正交各向异性平面问题边界元素法研究

边界元素法是近年来受到国内外广泛重视并得到迅速发展的一种计算方法。本文系统地研究了正交各向异性平面问题边界元素法的有关基本问题,包括基本解,C矩阵、G_ii矩阵和域内应力的表达式等,并在此基础上建立了常值边界元素和线性边界元素的计算公式。所述理论和公式适用于各类边值问题。最后,按本文所述理论和公式计算了含孔正交各向异性板的应力,数值结果与解析解相符甚好。      

真实梁板壳局部应力分析的高性能边界元法

经过数十年的发展边界元法在学术界已被看成有限元法的重要补充,但是要使这种补充成为工程界的实际需要还必须用它解决一些有限元法和其他方法难以解决的问题,这就是要充分发挥其高精度的优势,对一些复杂问题得到可靠的结果。为此作者近年通过误差分析提出了一种高精度边界元法计算方案,它在没有解析解和其他数值解做比较的情况下也能求得边界元法的收敛解。这种新方法的一个重要应用领域就是真实梁板壳结构的局部应力分析,即考虑梁板壳结构边缘实际几何、和基座或周围构件联合进行三维高精度边界元分析。该文给出了两个二维高精度边界元分析的算例,一个是真实悬臂薄板梁的横向弯曲,另一个是承受内压的无限长加肋圆柱壳,其中前一个算例揭示了真实悬臂薄板梁端部的局部应力远高于由梁弯曲理论所得到的应力。该文同时建立了悬臂薄板梁三维分析的边界元模型,其边界自由度数已经超出了在微机上用常规边界元法能够求解的规模。因此必须将高精度边界元法结合快速算法才能胜任此类分析,这就是作者提出的高性能边界元法的含义。最后作者展望了这两个相关新领域将要开展的研究内容,希望起到抛砖引玉的作用。