基于非协调边界元法的声辐射模态分析

利用非协调单元离散声学Helmholtz边界积分方程,采用极坐标变换法消除积分奇异性,通过CHIEF方法加Lagrange乘子法处理特征频率处解的不唯一性。在此基础上,应用非协调单元推导结构的声辐射功率和声辐射效率的表达式。以脉动球和辐射立方体为例,计算结构的声辐射功率、辐射效率、辐射模态、辐射模态效率等物理量,并与协调单元的计算结果做比较,取得较好的一致性。     

四边形常数单元离散下的声学非奇异BIE

奇异积分是基于Burton-Miller方程的声学边界元法实现过程的难点之一。关于三角形单元离散的积分单元的已经比较成熟,研究四边形常数单元离散下的声学边界积分方程(BIE),通过构造围绕配点的极小半球面进行积分,求得积分中的发散项,推导四边形常数单元离散下边界积分方程及其法向求导的非奇异表达式,从而得到非奇异Burton-Miller方程。运用Gauss Legendre积分公式计算BIE的S(x)的数值解,对比解析解的计算结果,得出了数值解、解析解以及二者的绝对误差、相对误差随ka的变化规律。实际应用时,当给定精度和ka的值后,可以通过改变所需要的截断项数,使得误差满足给定的精度要求。     

声学边界元拟奇异积分计算的自适应方法

提出一种自适应方法计算声学边界元中的拟奇异积分,通过单元分级细分将总积分转移到子单元上以消除拟奇异性。在此方法基础上深入研究拟奇异性,进一步提出接近度的概念,其中临界接近度可作为拟奇异积分计算的理论依据,并可用于预估拟奇异性是否存在。此方法的积分精度可调控,且不受场点位置限制,相比于已有方法更加灵活高效。数值分析表明拟奇异性强弱由场点与单元的相对位置决定,单元上远离场点的区域拟奇异性很弱,无需处理。研究结果为处理边界元法中的拟奇异性问题提供了新的选择和参考。     

具有卷积核的奇异积分方程与Riemann边值问题

本文首次提出并研究了在指数增长的函数类中,含卷积核和Cauchy核的奇异积分方程,特别就对偶型的奇异积分方程进行了讨论与求解,利用Fourier变换以及本文给出的引理,把对偶型奇异积分方程转化为直线上或平行直线上的解析函数边值问题.本文采用与经典的边值问题不同的解法,得到了方程的可解条件与一般解,因此推广了卷积型奇异积分方程理论,并为解决有关物理问题提供了理论依据.     

论Helmholtz方程的一类边界积分方程的合理性

本文导出了Helmholtz 方程超定边值问题有解的一个充要条件,和用非解析开拓法证明了文[1]中的Helmholtz 方程在外域中的解的边界积分表示式的合理性,并将此类边界积分表示式推广用于带空洞的有限域。这样就比较严密而又浅近地证明了基于该表示式建立起来的间接变量和直接变量边界积分方程的合理性。     

一种无奇异积分的边界单元法

处理基本解的奇异性是边界单元法的难题之一。本文避开奇异基本解,用非奇异基本解建立边界积分方程。非奇异基本解取自齐次微分方程的一般解和完备系,使求解边界积分方程容易。文中对边界未知量采用样条插值函数,计算精度良好。     

非协调单元在声学边界元法中的应用

利用非协调元离散Helmholtz边界积分方程,有效地解决协调元计算中的角点问题。为消除积分奇异性,提出了非协调元法中的极坐标变换方法。采用CHIEF法加Lagrange乘子法进行处理特征频率处解的不唯一性。解线性代数方程组获得结点处声压,网格点处的声压通过结点平均或单元平均的方法计算。通过计算脉动球和立方体的表面辐射声压,并将协调元和非协调元的计算结果做了比较,证明本文方法的有效性和对非光滑表面的适应性。     

运动介质中奇异边界元积分式的精确求解

采用边界元方法求解与运动介质相关声学问题时,难点之一是如何精确计算场点与源点重合所导致的奇异积分式。论文提出一种将具有奇性的单元面积分式拆分为奇性和非奇性积分部分分别进行计算的新方法。对奇性积分部分,经过严格的数学推导给出解析解;而对非奇性积分部分则通过高斯积分法处理。新方法可有效地提高边界元计算精度和效率,对运动介质中的有关声学问题的边界元数值计算具有重要意义。     

穿过界面的刚性线夹杂对SH波的散射

利用两个联结半平面中简谐集中力的格林函数,得出了穿过界面刚性线的散射场。刚性线的散射场可分解为有界部分和奇异部分。利用散射场的有界部分和奇异部分得出了刚性线的在SH波作用下的Cauchy型奇异积分方程。根据所得奇异积分方程和Cauchy型积分的端点性质,得出了确定刚性线和界面交点处奇异性阶数的特征方程。根据刚性线和界面交点处的奇性应力定义了交点处的应力奇异因子。对所得Cauchy型奇异积分方程的数值求解,可得刚性线端点和交点处的应力奇异因子。     

二维边界元法中几乎奇异积分的解析法

边界元分析中的几乎奇异积分难题一直阻碍其在工程中应用。作者提出的半解析法有效计算了几乎奇异积分,在此基础上做进一步推演,得到线性单元和二次亚参元上几乎强奇异和超奇异积分的解析列式,摈弃了数值求积。该算式对高次单元也近似适用。这个算法使得边界元法能够分析弹性力学薄壁结构。     

多角形域上第一类边界积分方程的高精度配置法

本文提出了一种多角区域上的第一类边界积分方程的高精度算法:离散之前采用特殊周期变换,消去边界积分方程未知函数在积分端点的奇异性,然后使用常元配置法求解,该方法在内点获得超收敛o(h3)。此外,通过Richardson整体外推,可进一步提高内点解的精度。实际计算结果表明,该方法优上同一问题的Galerkin方法甚至机械求积法。     

Some improvements of accuracy and efficiency in three dimensional direct boundary element method

Numerical analysis with the Boundary Element Method (BEM) has been used more and more in various engineering fields in recent years. In numerical techniques, however, there are some problems which have not been fully solved even now. The most essential one is the drop in the accuracy of results for internal points near the boundary of the structure, where the singularity of integrands in the boundary integral equation is too strong to be evaluated with the normal numerical method. For the boundary integral equation of stress, this problem became more serious, and the accuracy can be improved only partly, even though very refined boundary elements are used. In this paper, the boundary integral equation is newly formulated using a relative quantity of displacement. In this way, the singularity of boundary integrals is reduced by the order of 1/r, and the accuracy of solution is improved significantly. Furthermore, in order to integrate it more accurately, two kinds of numerical integral methods are newly developed. By using these methods, both displacement and stress can be obtained with excellent accuracy at almost any point in the structure without any numerical difficulty, although the discretization may be comparatively coarse. The generality and practicability of the present formulation and integral methods are confirmed through some examples of three dimensional elastic problems.     

Singularity extraction technique for integral equation methods with higher order basis functions on plane triangles and tetrahedra

A numerical solution of integral equations typically requires calculation of integrals with singular kernels. The integration of singular terms can be considered either by purely numerical techniques, e.g. Duffy's method, polar co‐ordinate transformation, or by singularity extraction. In the latter method the extracted singular integral is calculated in closed form and the remaining integral is calculated numerically. This method has been well established for linear and constant shape functions. In this paper we extend the method for polynomial shape functions of arbitrary order. We present recursive formulas by which we can extract any number of terms from the singular kernel defined by the fundamental solution of the Helmholtz equation, or its gradient, and integrate the extracted terms times a polynomial shape function in closed form over plane triangles or tetrahedra. The presented formulas generalize the singularity extraction technique for surface and volume integral equation methods with high‐order basis functions. Numerical experiments show that the developed method leads to a more accurate and robust integration scheme, and in many cases also a faster method than, for example, Duffy's transformation. Copyright © 2003 John Wiley & Sons, Ltd.     

Reconsiderations on singularity or crack tip elements

Various singularity finite elements proposed so far are reviewed and their correlations are discussed. The method adopted in the comparative study is based on the singularity mapping or transformation technique. It is suggested that a compact design of computational routines is materialized by using the element which embodies singularity transformation and can be compatible with the notion of using the variable-number-nodes element in the general-purpose computer programs.     

New variable transformations for evaluating nearly singular integrals in 2D boundary element method

This work presents a further development of the distance transformation technique for accurate evaluation of the nearly singular integrals arising in the 2D boundary element method (BEM). The traditional technique separates the nearly hypersingular integral into two parts: a near strong singular part and a nearly hypersingular part. The near strong singular part with the one-ordered distance transformation is evaluated by the standard Gaussian quadrature and the nearly hypersingular part still needs to be transformed into an analytical form. In this paper, the distance transformation is performed by four steps in case the source point coincides with the projection point or five steps otherwise. For each step, new transformation is proposed based on the approximate distance function, so that all steps can finally be unified into a uniform formation. With the new formulation, the nearly hypersingular integral can be dealt with directly and the near singularity separation and the cumbersome analytical deductions related to a specific fundamental solution are avoided. Numerical examples and comparisons with the existing methods on straight line elements and curved elements demonstrate that our method is accurate and effective.     

BEM analysis of thin structures for thermoelastic problems

A new boundary element method is developed for solving thin-body thermoelastic problems in this paper. Firstly, the novel regularized boundary integral equations (BIEs) containing indirect unknowns are proposed to cancel the singularity of fundamental solutions. Secondly, a general nonlinear transformation available for high-order geometry elements is introduced in order to remove or damp out the near singularity of fundamental solutions, which is crucial for accurate solutions of thin-body problems. Finally, the domain integrals arising in both displacement and its derivative integral equations, caused by the thermal loads, are regularized using a semi-analytical technique. Six benchmark examples are examined. Results indicate that the proposed method is accurate, convergent and computationally efficient. The proposed method is a competitive alternative to existing methods for solving thin-walled thermoelastic problems.     

Electromagnetic wave interactions with dielectric particles. I. Integral equation reformation

The conventional integral equation governing the electric field inside dielectric particles is reformed to bridge and to provide mathematical foundations for analytic techniques widely used to estimate such a field. The solution of the reformed equation inside a dielectric slab explained how inner-field formulations based on the Rayleigh, the Rayleigh-Gans, the quasi-static, and the Shifrin approximations can be supported by the particles. It also confirmed the approach employed to reform the integral equation. The analysis performed uncovered the differences between the depolarization tensor characterizing electrostatic fields inside the particles and the source dyadic resulting from the extraction of the singularity of the integral equation kernel.     

Generalized Duffy transformation for integrating vertex singularities

For an integrand with a 1/r vertex singularity, the Duffy transformation from a triangle (pyramid) to a square (cube) provides an accurate and efficient technique to evaluate the integral. In this paper, we generalize the Duffy transformation to power singularities of the form p(x)/r ^α , where p is a trivariate polynomial and α > 0 is the strength of the singularity. We use the map (u, v, w) → (x, y, z) : x = u ^β , y = x v, z = x w, and judiciously choose β to accurately estimate the integral. For α = 1, the Duffy transformation (β = 1) is optimal, whereas if α ≠ 1, we show that there are other values of β that prove to be substantially better. Numerical tests in two and three dimensions are presented that reveal the improved accuracy of the new transformation. Higher-order partition of unity finite element solutions for the Laplace equation with a derivative singularity at a re-entrant corner are presented to demonstrate the benefits of using the generalized Duffy transformation.     

ANALYSIS OF FREE-EDGE STRESS AND DISPLACEMENT FIELDS IN SCARF JOINTS SUBJECTED TO A UNIFORM CHANGE IN TEMPERATURE

The role of free-edge stresses in controlling the initiation of failure from the interface corner of a scarf joint subjected to a uniform change in temperature is examined. In general, the stress field can be expressed by σ _ij = Hr ^{λ −1} + σ _{ij 0} , where r is the radial distance from the interface corner, λ − 1 is the order of the stress singularity, H is the intensity of the singularity, and σ _{ij 0} is a non-singular constant stress. A combination of the finite element method and a path-independent integral is used to evaluate the magnitude of H for two joint configurations: (i) a scarf joint between two long bi-material strips; and (ii) a scarf joint consisting of a thin elastic layer sandwiched between two substrates. The magnitude of H is linearly dependent on a non-dimensional constant function a; the magnitude of a decreases with increasing level of mismatch in the elastic properties of the bonded materials. A comparison between the values of H evaluated by the path-independent integral method and the commonly used extrapolation method indicate that the extrapolation method could be in error by as much as 25%.     

Singularity Analysis and Boundary Integral Equation Method for Frictional Crack Problems in Two-Dimensional Elasticity

This study investigates the stress singularities in the neighborhood of the tip of a sliding crack with Coulomb-type frictional contact surfaces, and applies the boundary integral equation method to solve some frictional crack problems in plane elasticity. A universal approach to the determination of the complex order of stress singularity is established analytically by using the series expansion of the complex stress functions. When the cracks are open, or when no friction exists between the upper and lower crack faces, our results agree with those given by Williams. When displacement and traction are prescribed on the upper and lower crack surfaces (or vice versa), our result agrees with those by Muskhelishvili. For the case of a closed crack with frictional contact, the only nonzero stress intensity factor is that for pure shear or sliding mode. By using the boundary integral equation method, we derive analytically that the stress intensity factor due to the interaction of two colinear frictional cracks under far field biaxial compression can be expressed in terms of E(k) and K(k) (the complete elliptic integrals of the first and second kinds), where k=[1-(a/b)2]1/2 with 2a the distance between the two inner crack tips and b- a the length of the cracks. For the case of an infinite periodic colinear crack array under remote biaxial compression, the mode II stress intensity factor is found to be proportional to [2b tan(π a/2b)]1/2 where 2a and 2b are the crack length and period of the crack array. This revised version was published online in July 2006 with corrections to the Cover Date.