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。     

Dynamic analysis of shear deformable plates using the Dual Reciprocity Method

The Dual Reciprocity Method is a popular mathematical technique to treat domain integrals in the boundary element method (BEM). This technique has been used to treat inertial integrals in the dynamic thin plate bending analysis using a direct formulation of the BEM based on the elastostatic fundamental solution of the problem. In this work, this approach was applied for the dynamic analysis of shear deformable plates based on the Reissner plate bending theory, considering the rotary inertia of the plate. Three kinds of problems: modal, harmonic and transient dynamic analysis, were analyzed. Numerical examples are presented to demonstrate the efficiency and accuracy of the proposed formulation.     

Dynamic analysis of interfacial crack problems in anisotropic bi-materials by a time-domain BEM

A time-domain boundary element method (BEM) together with the sub-domain technique is applied to study dynamic interfacial crack problems in two-dimensional (2D), piecewise homogeneous, anisotropic and linear elastic bi-materials. The bi-material system is divided into two homogeneous sub-domains along the interface and the traditional displacement boundary integral equations (BIEs) are applied on the boundary of each sub-domain. The present time-domain BEM uses a quadrature formula for the temporal discretization to approximate the convolution integrals and a collocation method for the spatial discretization. Quadratic quarter-point elements are implemented at the tips of the interface cracks. A displacement extrapolation technique is used to determine the complex dynamic stress intensity factors (SIFs). Numerical examples for computing the complex dynamic SIFs are presented and discussed to demonstrate the accuracy and the efficiency of the present time-domain BEM.     

A comparison of transformation methods for evaluating two‐dimensional weakly singular integrals

Accurate numerical evaluation of integrals arising in the boundary element method is fundamental to achieving useful results via this solution technique. In this paper, a number of techniques are considered to evaluate the weakly singular integrals which arise in the solution of Laplace's equation in three dimensions and Poisson's equation in two dimensions. Both are two‐dimensional weakly singular integrals and are evaluated using (in a product fashion) methods which have recently been used for evaluating one‐dimensional weakly singular integrals arising in the boundary element method. The methods used are based on various polynomial transformations of conventional Gaussian quadrature points where the transformation polynomial has zero Jacobian at the singular point. Methods which split the region of integration into sub‐regions are considered as well as non‐splitting methods. In particular, the newly introduced and highly accurate generalized composite subtraction of singularity and non‐linear transformation approach (GSSNT) is applied to various two‐dimensional weakly singular integrals. A study of the different methods reveals complex relationships between transformation orders, position of the singular point, integration kernel and basis function. It is concluded that the GSSNT method gives the best overall results for the two‐dimensional weakly singular integrals studied. Copyright © 2002 John Wiley & Sons, Ltd.     

Stress intensity factors for a bounded plate containing two collinear cracks

This paper presents a method of numerical analysis for the problem of two collinear cracks in a finite, linearly elastic, isotropic plate and subjected to in plane forces.

The problem is treated imagining the plate with the two cracks draws in an unbounded region. Using the analytical solution of a point force applied to an infinite plate with two collinear cracks of equal length, the boundary conditions are written by superimposing the effect of interior loading upon the effect of concentrated loads applied on a curve parallel to the outer boundary. The boundary condition are satisfied in a least square sense.

Numerical example are given for square plates with inner or edge cracks. The accuracy is discussed.     

Solution of multiple crack problem in a finite plate using an alternating method based on two kinds of integral equation

This paper investigates a solution of multiple crack problem in a finite plate using an alternating method. The finite plate with cracks is an overlapping region of two regions: namely the infinite region exterior to the cracks and the finite region interior to finite plate without cracks. It is assumed that the cracks are applied by some loading and edges of the finite plate are of traction free. Governing equations for the problem and an alternating method are suggested. In the iteration, we need to solve two boundary value problems. One is the multiple crack problem in an infinite plate, and the other is the boundary value problem for the finite plate without crack. Several numerical examples are provided to prove the effectiveness of the suggested method.     

复合材料层合板智能结构主动振动控制的边界元法

利用边界元法模拟智能结构的振动控制，推导出具有压电传感器及致动器的复合材料层合板的边界积分方程，应用负速度反馈控制律，研究了复合材料层合板智能结构主动振动控制问题，算例分析证明该方程的正确性。     

Direct evaluation of singular integrals in elastoplastic analysis by the boundary element method

This paper presents a new method for the direct and accurate evaluation of strongly singular integrals in the sense of Cauchy principal values and weakly singular integrals over quadratic boundary elements in three-dimensional stress analysis and quadratic internal cells in two-dimensional elastoplastic analysis by the boundary element method. A quadratic triangle polar co-ordinate transformation technique is applied to reduce the order of singularity of the singular integrals. Next, a form of Stokes' theorem is introduced in order to remove the singularity in the Cauchy principal value integrals; therefore, the evaluation of these integrals can be carried out by standard Gaussian quadrature. Numerical examples of 2-D elastoplastic problems and a 3-D elastic problem show the effectiveness and efficiency of the method.     

Boundary element analysis of structures with bridged interfacial cracks

The direct boundary integral equations method has been applied to analyze stresses in a fracture process zone (a crack bridged zone) and to calculate stress intensity factors module for structures with bridged interfacial cracks under mechanical loading. Bridged zones at interfacial cracks are considered as parts of these cracks with assumption that surfaces of interfacial cracks are connected by distributed spring-like bonds with given bond deformation law. For numerical analysis of piecewise structures with bridged interfacial cracks the multi-domain formulation of the boundary elements method is used. The stress intensity factors module evaluation is performed on the basis of displacements and stresses computed at nodal points of special quadratic boundary elements adjoined to a crack tip. The comparative study between the results obtained by the boundary elements method and the results obtained previously by the singular integral–differential equations method is performed and the validity of the presented numerical formulation is demonstrated. The new problem for a bridged circumferential crack between a cylindrical inclusion and a matrix in plate of finite size is also solved. Stresses distributions along the bridged zone and the stress intensity factors modulus dependencies versus the bridged zone length and bonds stiffness are presented and discussed for this problem.     

Time-domain BEM for transient interfacial crack problems in anisotropic piezoelectric bi-materials

A time-domain boundary element method (BEM) together with the sub-domain technique is applied to study transient response of interfacial cracks in piecewise homogeneous, anisotropic and linear piezoelectric bi-materials under electrical and mechanical impacts. The present time-domain BEM uses a quadrature formula for the temporal discretization to approximate the convolution integrals and a collocation method for the spatial discretization. Quadratic quarter-point elements are implemented at the tips of the interface cracks. To determine the real or complex dynamic stress intensity factors and the dynamic electrical displacement intensity factor of the interfacial cracks, an explicit extrapolating formula in a typical state of the crack plane perpendicular to the poling direction is presented in this paper. Numerical examples are presented; and the effects of the load combination and material combination on dynamic intensity factors and dynamic energy release rate are discussed.     

Modal analysis of plates using the dual reciprocity boundary element method

This paper presents a new method for determining the natural frequencies and mode shapes for the free vibration of thin elastic plates using the boundary element and dual reciprocity methods. The solution to the plate's equation of motion is assumed to be of separable form. The problem is further simplified by using the fundamental solution of an infinite plate in the reciprocity theorem. Except for the inertia term, all domain integrals are transformed into boundary integrals using the reciprocity theorem. However, the inertia domain integral is evaluated in terms of the boundary nodes by using the dual reciprocity method. In this method, a set of interior points is selected and the deflection at these points is assumed to be a series of approximating functions. The reciprocity theorem is applied to reduce the domain integrals to a boundary integral. To evaluate the boundary integrals, the displacements and rotations are assumed to vary linearly along the boundary. The boundary integrals are discretized and evaluated numerically. The resulting matrix equations are significantly smaller than the finite element formulation for an equivalent problem. Mode shapes for the free vibration of circular and rectangular plates are obtained and compared with analytical and finite element results.     

Boundary element method for shear deformable plates with combined geometric and material nonlinearities

In this paper a new boundary element formulation for shear deformable plate theory with combined geometric and material nonlinearities is presented. The material is assumed to undergo large deflection with small strains. The von Mises criteria is used to evaluate the plastic zone and an elastic perfectly plastic material behaviour is assumed. An initial stress formulation is used to formulate the boundary integral equations. The domain integrals involving geometrical and material nonlinear terms are evaluated using a cell discretization technique. A total incremental method is applied to solve the nonlinear boundary integral equations. Numerical examples are presented to demonstrate the validity and the accuracy of the proposed formulation.     

An approximate method for evaluating hypersingular integrals

This paper describes a new method for direct numerical evaluation of multidimensional hypersingular integrals assigned on smooth curves and surfaces. These integrals arise when the boundary integral equations are used to solve problems of mechanics, electrodynamics, aerodynamics, etc. The hypersingular integrals are considered, in the sense of Hadamard, as finite parts. The main advantage of the proposed method is the numerical computation of the hypersingular integrals by the direct application of the developed cubature formulas, thus requiring little analytical pre-work. The method is not restricted to the type of problem however and may be easily applied to any hypersingular integrals. The convergence of the proposed technique has been proved and error estimates are given. An illustrative example demonstrates the accuracy and efficiency of the method.     

Evaluation of T-stresses and stress intensity factors in stationary thermoelasticity by the coservation integral method

New methods for determining T-stresses and stress intensity factors in two-dimensional stationary thermoelasticity are presented. The methods utilize the path independence of J^-integral. The mutual M-integral expressed through J^-integrals provides sufficient information for determining T-stresses. The M-integral is composed of contour and domain integrals. Their integrands include the thermomechanical fields under consideration and mechanical fields corresponding to an auxiliary solution. The thermomechanical fields along the integration path and in the enclosed domain are obtained by the boundary element method. Stress intensity factors are calculated using the path independent J^-integral technique. This integral is expressed in a pure contour integral form. Numerical results for a cracked rectangular plate and thick-walled tube with internal cracks are presented. This revised version was published online in July 2006 with corrections to the Cover Date.     

Analysis of a finite composite plate with smooth rigid pin

A continuum method of analysis is presented in this paper for the problem of a smooth rigid pin in a finite composite plate subjected to uniaxial loading. The pin could be of interference, push or clearance fit. The plate is idealized to an orthotropic sheet. As the load on the plate is progressively increased, the contact along the pin-hole interface is partial above certain load levels in all three types of fit. In misfit pins (interference or clearance), such situations result in mixed boundary value problems with moving boundaries and in all of them the arc of contact and the stress and displacement fields vary nonlinearly with the applied load. In infinite domains similar problems were analysed earlier by ‘inverse formulation’ and, now, the same approach is selected for finite plates. Finite outer domains introduce analytical complexities in the satisfaction of boundary conditions. These problems are circumvented by adopting a method in which the successive integrals of boundary error functions are equated to zero. Numerical results are presented which bring out the effects of the rectangular geometry and the orthotropic property of the plate. The present solutions are the first step towards the development of special finite elements for fastener joints.     

Analyzing the interaction between collinear interfacial cracks by an efficient boundary element-free method

A new traction boundary integral equation is presented for analyzing the interaction effect of any number of collinear interface cracks in a two-dimensional bimaterial. The dislocation densities on every crack surface are expressed in the products of the characteristic terms and the weight functions, and the unknown weight functions are approximated using the moving least-squares technique based on the constructed orthogonal basis functions. An efficient numerical integral method is employed to evaluate the Cauchy principal integrals that appear in the meshless method. The boundary element-free method is established, and a series of numerical results is presented. The interaction between the collinear interfacial cracks is analyzed.     

Boundary element analysis of reinforced shear deformable shells

In this paper, stiffened shear‐deformable shells are analysed using the boundary element method. Coupled boundary integral equations are presented for describing curved shells under general loading conditions. The equations are based on boundary integral equations for plane stress and plate bending, with coupling terms arising from the curvature of the shell. Domain integrals are transformed into boundary integrals using the dual reciprocity technique. Stiffeners are modelled as curved beams, continuously attached to the shell. Numerical solutions calculated using the present method are compared with finite element results in two examples. Copyright © 2002 John Wiley & Sons, Ltd.     

一种结构静力重分析方法及其在层合板逐次失效分析中的应用

本文提出了一种结构静力重分析方法,通过引入结构刚体位移特征向量,可以消除刚度矩阵的奇异性,从而求出结构广义柔度矩阵,位移一般解可在边界条件尚未引入之前导出。对于给定载荷及边界条件的静力问题,可以通过求解一个阶数较小的线性系统来获得位移解。对于局部修改的结构,相对于新结构的广义柔度矩阵可以迅速加以修正。这种静力重分析方法可以用于载荷、边界条件及结构单元同时或分别改变时的静力重分析问题。 之后,这种重分析方法被应用于复合材料层合板逐次失效分析,用以进行受破坏结构的刚度重分析。用有限元法求得各铺层的应力之后,引入破坏准则来判断铺层失效及失效模式,在本构方程中引入三个参数来修改失效铺层的刚度。层合板的破坏扩展及极限强度通过逐渐增加载荷迭代求得。文中提供了多个算例,计算结果与实验数据及文献吻合良好。     

Free vibration and buckling analysis of laminated composite plates using the NURBS-based isogeometric finite element method

An isogeometric finite element method based on non-uniform rational B-splines (NURBS) basis functions is developed for natural frequencies and buckling analysis of thin symmetrically laminated composite plates based upon the classical plate theory (CPT). The approximation of the solution space for the deflection field of the plate and the parameterization of the geometry are performed using NURBS-based approach. The essential boundary conditions are formulated separately from the discrete system equations by the aid of Lagrange multiplier method, while an orthogonal transformation technique is also applied to impose the essential boundary conditions in the discrete eigen-value equation. The accuracy and the efficiency of the proposed method are thus demonstrated through a series of numerical experiments of laminated composite plates with different boundary conditions, fiber orientations, lay-up number, eigen-modes, etc. The obtained numerical results are then compared with either the analytical solutions or other available numerical methods, and excellent agreements are found.     

EVALUATION OF THE STRESS TENSOR IN 3-D ELASTOPLASTICITY BY DIRECT SOLVING OF HYPERSINGULAR INTEGRALS

A 3-D hypersingular Boundary Integral Equation (BIE) of elastoplasticity is derived. Using this formulation the displacement rate gradients and the complete stress tensor on the boundary can be evaluated directly as opposed to the classical approach, where the shape functions derivatives are to be calculated. The regularization of strongly singular and hypersingular boundary integrals, as well as strongly singular domain integrals for a source point positioned on the boundary is carried out in a general manner. Arbitrary types of elements and arbitrary positions of the source point with respect to continuity requirements can be used. Numerical 3-D elastoplastic examples (notch and crack problems) illustrate the advantages of the proposed method.