功能梯度材料动态断裂力学的径向积分边界元法

采用径向积分边界元法分析功能梯度材料动态断裂力学问题. 该方法使用与弹性模量无关的弹性静力学开尔文基本解作为问题的基本解,在导出的边界-域积分方程中含有由材料的非均质性和惯性项引起的域积分,通过径向积分法将域积分转化为等效的边界积分,得到只含边界积分的纯边界积分方程;从而建立只需边界离散的无内部网格边界元算法. 采用候博特方法求解关于时间二阶导数的系统离散的常微分方程组. 最后通过数值算例验证本文方法的精度和有效性.      

薄体位势问题边界元法中的解析积分算法

薄体结构的数值分析是边界元法的难点问题之一。该文导出了一种完全解析积分算法，用这种算法计算了薄体平面位势问题边界元法中出现的几乎弱奇异、强奇异和超奇异积分。当边界离散为一系列线性单元，边界积分方程离散计算的积分可归纳为三种形式。对薄体问题，源点与积分单元距离通常相距很近，这些积分产生显著几乎奇异性，直接采用常规高斯积分不能有效计算。为此该文导出了这些几乎奇异积分的全解析计算公式。按源点与单元的距离是否为零，公式分两种情况。新算法采用全解析积分公式处理几乎奇异积分，首先精确计算出薄体问题边界未知位势和法向位势梯度，然后再进一步计算了域内点的物理参量。算例表明该文算法可处理狭长比为1．E-08的薄体问题，显示了边界元法分析薄体问题具有独特的优势。     

REGULARIZATION OF NEARLY SINGULAR INTEGRALS IN THE BOUNDARY ELEMENT METHOD OF POTENTIAL PROBLEMS

IntroductionAsanimportantnumericalmethod ,BoundaryElementMethod (BEM)hasbeenappliedinmanyareas[1].However,theBEMhasthedifficultiesofcalculatingsingularintegralsatnodesonboundaryoratinteriorpointsveryclosetotheboundary .TheaccuracyoftheBEMdependsontheprecisionofthecalculatedvaluesofthesingularintegrals,toagreatdegree.Manyresearchersdevotethemselvestothetreatmentofthesingularintegrals[2～3],whicharereviewedindetailbyRef.[4] .Ageneralregularizationalgorithmofevaluatingthephysicalquantitiesa…     

Element nodal computation-based radial integration BEM for non-homogeneous problems

This paper presents a new strategy of using the radial integration boundary element method （RIBEM） to solve non-homogeneous heat conduction and thermoelasticity problems. In the method, the evaluation of the radial in-tegral which is used to transform domain integrals to equivalent boundary integrals is carried out on the basis of elemental nodes. As a result, the computational time spent in evaluating domain integrals can be saved considerably in comparison with the conventional RIBEM. Three numerical examples are given to demonstrate the correctness and computational efficiency of the proposed approach.     

导数场边界积分方程分析近边界应力分布

研究二维弹性力学问题边界积分方程,通过分部积分变换消除了常规导数边界积分方程中的超奇异积分,获得仅含强奇异积分的应力自然边界积分方程.对于近边界应力的计算,进一步运用正则化算法解析计算其中的几乎强奇异积分.较常规边界元法相比,应力自然边界积分方程可以求解离边界更加接近的内点应力值.算例证明了文中方法的可应用性和有效性.     

A new BEM formulation for transient axisymmetric poroelasticity via particular integrals

A simple particular integral formulation is presented for the first time in a purely axisymmetric poroelastic analysis. The axisymmetric elastostatic and steady-state potential flow equations are used as the complementary solution. The particular integrals for displacement, traction, pore pressure and flux are derived by integrating three-dimensional formulation along the circumferential direction leading to elliptic integrals.Numerical results for three axisymmetric problems of soil consolidation are given and compared with their analytical solutions to demonstrate the accuracy of the present formulation. Generally, agreement among all of those results is satisfactory if one uses a few interior points, in addition to the regular boundary points.     

二维热弹性力学边界元法中几乎奇异积分的正则化

针对二维热弹性力学边界元法中近边界点的几乎强奇异和超奇异积分,采用一种通用算法,将其实施正则化．该方法适用于线性单元,与近边界点邻近的单元上的积分采用正则化积分公式计算,远处单元的积分仍保持常规高斯积分．算例证明了该法的有效性和精确性．     

Variational boundary integral approach for asymmetric impinging jets of arbitrary two‐dimensional nozzle

We consider asymmetric impinging jets issuing from an arbitrary nozzle. The flow is assumed to be two‐dimensional, inviscid, incompressible, and irrotational. The impinging jet from an arbitrary nozzle has a couple of separated infinite free boundaries, which makes the problem hard to solve. We formulate this problem using the stream function represented with a specific single layer potential. This potential can be extended to the surrounding region of the jet flow, and this extension can be proved to be a bounded function. Using this fact, the formulation yields the boundary integral equations on the entire nozzle and free boundary. In addition, a boundary perturbation produces an extraordinary boundary integral equation for the boundary variation. Based on these variational boundary integral equations, we can provide an efficient algorithm that can treat with the asymmetric impinging jets having arbitrarily shaped nozzles. Particularly, the proposed algorithm uses the infinite computational domain instead of a truncated one. To show the convergence and accuracy of the numerical solution, we compare our solutions with the exact solutions of free jets. Numerical results on diverse impinging jets with nozzles of various shapes are also presented to demonstrate the applicability and reliability of the algorithm.     

非均质问题中的无网格边界单元法

介绍了一种不需要内部网格计算非均匀介质问题的边界元算法.该算法是建立在一种能将任何区域积分转换成边界积分的径向积分转换法基础上,首先用对应各向同性问题的基本解来建立以正规化位移表示的非均质问题的积分方程,然后用径向积分转换法将出现在积分方程中的区域积分转换成边界积分,从而形成不需要使用内部网格来计算区域积分的纯边界元算法.与其它无网格法相比,此方法需要很少的内部点,有些问题甚至不需要内部点都能得到满意的结果,因此,可以计算大型的三维非均匀介质工程问题.由于此方法继承了边界元和无网格算法的优点,因而具有广阔的发展前景.     

Improved non-singular local boundary integral equation method

When the source nodes are on the global boundary in the implementation of local boundary integral equation method (LBIEM),singularities in the local boundary integrals need to be treated specially. In the current paper,local integral equations are adopted for the nodes inside the domain trod moving least square approximation (MLSA) for the nodes on the global boundary,thus singularities will not occur in the new al- gorithm.At the same time,approximation errors of boundary integrals are reduced significantly.As applications and numerical tests,Laplace equation and Helmholtz equa- tion problems are considered and excellent numerical results are obtained.Furthermore, when solving the Hehnholtz problems,the modified basis functions with wave solutions are adapted to replace the usually-used monomial basis functions.Numerical results show that this treatment is simple and effective and its application is promising in solutions for the wave propagation problem with high wave number.     

Analysis of three-dimensional edge cracks under tensile loading

Three-dimensional edge cracks are analyzed using the Self-Similar Crack Expansion (SSCE) method with a boundary integral equation technique. The boundary integral equations for surface cracks in a half space are presented based on a half space Green's function (Mindlin, 1936). By using the SSCE method, the stress intensity factors are determined by the crack-opening displacement over the crack surface. In discrete boundary integral equations, the regular and singular integrals on the crack surface elements are evaluated by an analytical method, and the closed form expressions of the integrals are given for subsurface cracks and edge crakcs. This globally numerical and locally analytical method improves the solution accuracy and computational effort. Numerical results for edge cracks under tensile loading with various geometries, such as rectangular cracks, elliptical cracks, and semi-circular cracks, are presented using the SSCE method. Results for stress intensity factors of those surface breaking cracks are in good agreement with other numerical and analytical solutions.     

非连续边界元积分的精确表达式及相关问题

以二维位势问题边界元分析为例,给出了利用线性非连续边界元离散边界积分方程时系数矩阵积分计算的精确表达式,通过和利用Gauss积分方法计算系数矩阵所得数值结果的比较表明：配位点选择不同对数值计算结果精度影响的主要原因是积分计算的精度,尤其当配位因子选择较大时,存在的准奇异积分（Nearly Singular Integrals)很难利用常规Gauss积分方法准确求得。     

A simple BEM formulation for poroelasticity via particular integrals

A simple particular integral formulation is presented for poroelastic analysis. The elastostatics and steady-state potential flow equations are used as the complementary solution. A set of global shape functions is considered to approximate the pore pressure loading term in the poroelastic equation, the transient terms of pore pressure and displacements in the pore fluid flow equation to obtain the particular integrals for displacement, traction, pore pressure and flux.Numerical results for four plane problems of soil consolidation are given and compared with their analytical solutions to demonstrate the accuracy of the present formulation. Generally, agreement among all of those results is satisfactory if a few interior points are added to the usual boundary elements.     

Evaluation of strongly singular domain integrals for internal stresses in functionally graded materials analyses using RIBEM

An accurate evaluation of strongly singular domain integral appearing in the stress representation formula is a crucial problem in the stress analysis of functionally graded materials using boundary element method.To solve this problem,a singularity separation technique is presented in the paper to split the singular integral into regular and singular parts by subtracting and adding a singular term.The singular domain integral is transformed into a boundary integral using the radial integration method.Analytical expressions of the radial integrals are obtained for two commonly used shear moduli varying with spatial coordinates.The regular domain integral,after expressing the displacements in terms of the radial basis functions,is also transformed to the boundary using the radial integration method.Finally,a boundary element method without internal cells is established for computing the stresses at internal nodes of the functionally graded materials with varying shear modulus.     

三维间断位移法及强奇异和超奇异积分的处理方法

从积分方程Somigliana等式出发,导出三维状态下单位位错集度的基本解.在此基础上,建立了边界积分方程,并给出了其离散形式.对强奇异和超奇异积分,采用了Hadamard定义的有限部分积分来处理.最后,给出了计算裂纹应力强度因子的算例,并与解析解进行了比较,证实了该方法的有效性.     

各向异性位势平面问题的规则化边界元法

基于转化域方程为边界积分方程的极限定理及一个新颖的基本解分解技术, 建立间接变量规则化边界积分方程, 它有效地避免了奇异积分的直接计算. 与已有方法比,该方法不将问题变换为各向同性的问题去处理, 因而无需反演运算, 也有别于Galerkin方法, 无需计算重积分. 可计算任意边界位势梯度, 而不仅限于法向通量. 针对椭圆边界的边值问题, 提交一种精确单元来描述边界几何. 数值算例表明, 所提算法稳定且效率高, 所得数值结果与精确解吻合较好.      

Explicit formulations for evaluation of velocity gradients using boundary‐domain integral equations in 2D and 3D viscous flows

In this paper, explicit boundary‐domain integral equations for evaluating velocity gradients are derived from the basic velocity integral equations. A free term is produced in the new strongly singular integral equation, which is not included in recent formulations using the complex variable differentiation method (CVDM) to compute velocity gradients (Int. J. Numer. Meth. Fluids 2004; 45 :463–484; Int. J. Numer. Meth. Fluids 2005; 47 :19–43). The strongly singular domain integrals involved in the new integral equations are accurately evaluated using the radial integration method (RIM). Considerable computational time for evaluating integrals of velocity gradients can be saved by using present formulation than using CVDM. The formulation derived in this paper together with those presented in reference (Int. J. Numer. Meth. Fluids 2004; 45 :463–484) for 2D and in (Int. J. Numer. Meth. Fluids 2005; 47 :19–43) for 3D problems constitutes a complete boundary‐domain integral equation system for solving full Navier–Stokes equations using primitive variables. Three numerical examples for steady incompressible viscous flow are given to validate the derived formulations. Copyright © 2007 John Wiley & Sons, Ltd.     

边界元法计算近边界点参量的一个通用算法

针对边界元法存在近力界点参量计算的困难,给出了一个通用性方法,将近边界点到边界单元的距离参数通过分部积分变换到积分式之外,从而计算出二维问题近边界点参量的几乎强奇异和超奇异积分,由此,对任何近边界点参量,提出了整套计算方案,算例证明了本法的有效性。     

多洞室对矢量波散射引起半空间表面位移的边界元解

采用边界元方法研究了半空间中近表面多洞室对矢量波的散射问题,给出了以全空间格林函数为基本解且半空间表面离散的边界积分方程,在这一边界各分方程中,较好的消除了主值积分,在半空间表面进行离散时,采用无限单元与有限单元相结合的方法,大大减少了计算量,提出了精度。     

边界积分方程中近奇异积分计算的一种变量替换法

准确估计近奇异边界积分是边界元分析中一项很重要的课题,其重要性仅次于对奇异积分的处理. 近年来已发展了许多方法,都取得了一定程度的成功,但这个问题至今仍未得到彻 底的解决. 基于一种新的变量变换的思想和观点,提交了一种通用的积分变换法, 它非常有效地改善了被积函数的震荡特性,从而消除了积分的近奇异性,在不增加计算量的情况 下, 极大地改进了近奇异积分计算的精度. 数值算例表明,其算法稳定,效率高, 并可达到很高的计算精度,即使区域内点非常地靠近边界,仍可取得很理想的结果.