平面问题等价边界积分方程的三次边界轮廓法

基于弹性力学平面问题等的边界积分方程,给出了三次单元的边界轮廓法。根据平面问题解的复变函数表示,构造了三次形函数。给出了对于混合边值问题求解系统方程确定的边界轮廓方程配置和三次单元界轮廓法的实施。     

弹性力学中一种新的边界轮廓法

利用基本解的特性,将面力积分方程化成仅含有Ｃａｕｃｈｙ主值积分的形式,基于这种边界积分方程,提出了一种新的边界轮廓法,对于三维问题,该方法只须计算沿边界单元界线的线积分,对二维问题,则只需计算边界单元两点的热函数之差,无须进行数值积分计算,实例计算说明该方法是有效的。     

弹性薄板弯曲问题的边界轮廓法

导出了弹性薄板弯曲问题边界积分方程的另一种形式,基于这种方程,提出了平板弯曲问题的边界轮廓法,讨论了三次边界单元边界轮廓法的计算列式,并给出了计算内力的边界轮廓法方程。该法无需进行数值积分计算,完全避免了角点问题和奇异积分计算。给出的算例,与解析解相比较,证实该方法的有效性。     

弹性力学边界积分方程的一个发展

本文讨论二维弹性力学平面问题，独立于Ｒｉｚｚｏ型边界分方程，一类新型的边界积分方程，其边界场变量包含应力分量σｉｊｔｉｔｊ（其中ｔｉ是边界切向余弦）。该应力分量可直接用数值方法解边界积分方程求出，它比常规的边界元解提高一阶精度。文末的算例表明确定论的实用性和有效性。     

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δij和εij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.自然边界积分方程的奇异性为强奇性,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.     

弹性力学平面问题的无奇异边界积分方程

将弹性力学平面问题归化成无奇异边界积分方程,避免了传统的边界元法中的柯西主值（CPV）积分和Hadamard-Finite-Parts(HFP)积分的计算,建立完整的数值求解体系。     

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。     

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。     

弹性力学平面问题可靠度分析的蒙特卡罗边界元法

以样条虚边界元法作为样本试验方法,采用蒙特卡罗法进行弹性力学平面问题可靠度分析.为了提高计算效率,引入Taylor展开和Neumann展开技术,避免在大量样本计算中直接生成影响矩阵及对其进行求逆运算,降低了单次样本计算时间;同时引入重要抽样技术,在相同精度情况下减少了蒙特卡罗法的抽取样本数.算例结果表明,该文提出的Taylor-Neumann展开重要抽样蒙特卡罗样条虚边界元法具有良好的计算精度和相当高的计算效率.     

弹性力学问题的局部边界积分方程方法

提出了弹性力学平面问题的局部边界积分方程方法。这种方法是一种无网格方法,它采用移动最小二乘近似试函数,且只包含中心在所考虑节点的局部边界上的边界积分。它易于施加本质边界条件。所得系统矩阵是一个带状稀疏矩阵。它组合了伽辽金有限元法、整体边界元法和无单元伽辽金法的优点。该方法可以容易推广到求解非线性问题以及非均匀介质的力学问题。计算了两个弹性力学平面问题的例子,给出了位移和能量的索波列夫模,所得计算结果证明：该方法是一种具有收敛快、精度高、简便有效的通用方法。     

奇异边界法模拟二维弹性力学问题

奇异边界法是一种新的边界型无网格数值离散方法.该方法使用基本解作为插值基函数,在继承传统边界型方法优点的同时,不需要费时费力的网格划分和奇异积分,数学简单,编程容易,是一个真正的无网格方法.为避免配置点与插值源点重合时带来的基本解源点奇异性,该方法提出了源点强度因子的概念,从而将边界型强格式方法的核心归结为求解源点强度因子.论文首次将该方法应用于求解平面弹性力学问题.数值算例表明,本文算法稳定,效率高,并可达到很高的计算精度.     

Boundary integral equations of unique solutions in elasticity

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Application of dual kriging to structural shape optimization based on the boundary contour method

Summary The paper presents an approach in which the coupling of dual kriging and the boundary contour method (BCM) is applied to structural shape optimization problems in mechanical engineering design. The problems consist of optimizing the shape of an elastic body, which requires minimizing an objective function subject to some given constraints, such as those of displacement, stress or manufacturing. The originality of the present work is involved with the use of two novel methods that are combined here to solve structural shape optimization problems. The first one, called dual kriging, is a general, versatile interpolation and geometric modeling tool. The second one is a new variant of the boundary element method (BEM), called the BCM, which achieves a further reduction in dimensionality of analysis problems. Based on the advantages of these two methods, the coupling approach presented here is expected to offer an effective as well as a straightforward manner for solving shape optimal design problems. Received 18 December 1997; accepted for publication 21 April 1998     

A numerical implementation using mid-node collocation for the hypersingular boundary contour method

A variant of the boundary element method, called the boundary contour method (BCM), offers a further reduction in dimensionality. Consequently, boundary contour analysis of two-dimensional problems does not require any numerical integration at all. In another development, a boundary contour implementation of a regularized hypersingular boundary integral equation (HBIE) using quadratic elements and end-node collocation was proposed and the technique is termed the hypersingular boundary contour method (HBCM). As reported in that work, the approach requires highly refined meshes in order to numerically enforce the stress continuity across boundary contour elements. This continuity requirement is very crucial since the regularized HBIE is only valid at collocation points where the stress tensor is continuous, while the computed stress at the endpoints of a boundary contour element, which is a non-conforming element, is generally not. This paper presents a new implementation of the HBCM for which the regularized HBIE is collocated at the mid-node of a boundary contour element. As the computed stress tensor is continuous at these mid-nodes, there is no need for unusually refined meshes. Some numerical tests herein show that, for the same mesh density, the HBCM using mid-node collocation has a comparable accuracy as the BCM.     

Boundary contour analysis for surface stress recovery in 2-D elasticity and Stokes flow

Summary A variant of the boundary element method, called the boundary contour method, offers a further reduction in dimensionality. Consequently, boundary contour analysis of 2-D problems does not require any numerical integration at all. In a boundary contour analysis, boundary stresses can be accurately computed using the approach proposed in Ref. [1]. However, due to singularity, this approach can be used only to calculate boundary stresses at points that do not lie at an end of a boundary element. Herein, it is shown that a technique based on the displacement/velocity shape functions can overcome this drawback. Further, the approach is much simpler to apply, requires less computational effort, and provides competitive accuracy. Numerical solutions and convergence study for some well-known problems in linear elasticity and Stokes flow are presented to show the effectiveness of the proposed approach. This research was supported in part by the 2004 Ralph E. Powe Junior Faculty Enhancement Award from Oak Ridge Associated Universities and by the University of South Alabama Research Council.     

A structural shape optimization system using the two-dimensional boundary contour method

Summary The research recently conducted has demonstrated that the Boundary Contour Method (BCM) is very competitive with the Boundary Element Method (BEM) in linear elasticity Design Sensitivity Analysis (DSA). Design Sensitivity Coefficients (DSCs), required by numerical optimization methods, can be efficiently and accurately obtained by two different approaches using the two-dimensional (2-D) BCM as presented in Refs. [1] and [2]. These approaches originate from the Boundary Integral Equation (BIE). As discussed in [2], the DSCs given by both BIE-based DSA approaches are identical, and thus the users can choose either of them in their applications. In order to show the advantages of this class of DSA in structural shape optimization, an efficient system is developed in which the BCM as well as a BIE-based DSA approach are coupled with a mathematical programming algorithm to solve optimal shape design problems. Numerical examples are presented. Received 20 July 1998; accepted for publication 7 December 1998     

Stochastic boundary element method in elasticity

The stochastic boundary element method is developed to analyze elasticity problems with random material and/or geometrical parameters and randomly perturbed boundaries. Based on the first-order Taylor series expansion, the boundary integration equations concerning the mean and deviation of the displacements are derived, respectively. It is found that the randomness of material parameters is equivalent to a random body force, so the mean and covariance matrices of unknown boundary displacements and tractions can be obtained. Furthermore, the mean and covariance of displacements and stresses at inner points can also be obtained. Numerical examples show that the proposed stochastic boundary element method gives satisfactory solutions, as compared with those obtained by theoretical analysis or other numerical methods. The project supported by the National Natural Science Foundation of China and the State Education Commission Foundation of China     

Quasi-optimal complexity of adaptive finite element method for linear elasticity problems in two dimensions

This paper introduces an adaptive finite element method (AFEM) using the newest vertex bisection and marking exclusively according to the error estimator without special treatment of oscillation. By the combination of the global lower bound and the localized upper bound of the posteriori error estimator, perturbation of oscillation, and cardinality of the marked element set, it is proved that the AFEM is quasi-optimal for linear elasticity problems in two dimensions, and this conclusion is verified by the numerical examples.     

基于p型有限元法和围线积分法计算复合型应力强度因子

传统有限元法由于采用低阶插值计算应力强度因子时,需要划分的网格数较多,收敛速度较慢,得到的应力强度因子精度不足。p型有限元法在网格确定时通过增加插值多项式的阶数来提高计算精度,具有网格划分少、收敛速度快、精度高、自适应能力强等特点。本文采用基于p型有限元法的有限元计算软件StressCheck计算得到应力场和位移场,并由围线积分法导出混合型应力强度因子(SIFs)。通过几个经典算例,分析了围线的选择对计算精度的影响,计算了不同裂纹长度、不同裂纹角度和裂纹在应力集中区域不同位置时的应力强度因子。并将数值结果、理论解与文献中其他数值计算方法所得的部分结果进行了对比分析,结果表明自由度数不大于7000时,导出的应力强度因子相对误差最大不超过1.2%,数值解表现出较高的精度及数值稳定性。