薄壁杆件翘曲剪应力的边界元精确积分解法

用非连续边界元对薄壁杆件的约束扭转进行了分析,推导出了求解边界点二次翘曲函数值的边界积分方程,给出了边界积分方程数值求解时积分计算的精确表达式。数值算例表明:利用边界积分方程方法分析薄壁杆件的约束扭转问题时效率和精度高,同时采用精确积分可以有效的处理"边界层效应"问题。     

Biharmonic analysis of rectilinear plates by the boundary element method

An improved boundary element formulation (BEM) for two-dimensional non-homogeneous biharmonic analysis of rectilinear plates is presented. A boundary element formulation is developed from a coupled set of Poisson-type boundary integral equations derived from the governing non-homogeneous biharmonic equation. Emphasis is given to the development of exact expressions for the piecewise rectilinear boundary integration of the fundamental solution and its derivatives over several types of isoparametric elements. Incorporation of the explicit form of the integrations into the boundary element formulation improves the computational accuracy of the solution by substantially eliminating the error introduced by numerical quadrature, particularly those errors encountered near singularities. In addition, the single iterative nature of the exact calculations reduces the time necessary to compile the boundary system matrices and also provides a more rapid evaluation of internal point values than do formulations using regular numerical quadrature techniques. The evaluation of the domain integrations associated with biharmonic forms of the non-homogeneous terms of the governing equation are transformed to an equivalent set of boundary integrals. Transformations of this type are introduced to avoid the difficulties of domain integration. The resulting set of boundary integrals describing the domain contribution is generally evaluated numerically; however, some exact expressions for several commonly encountered non-homogeneous terms are used. Several numerical solutions of the deflection of rectilinear plates using the boundary element method (BEM) are presented and compared to existing numerical or exact solutions.     

Microscale heat conduction in homogeneous anisotropic media: a dual-reciprocity boundary element method and polynomial time interpolation approach

In this paper, a dual-reciprocity boundary element method based on some polynomial interpolations to the time-dependent variables is presented for the numerical solution of a two-dimensional heat conduction problem governed by a third order partial differential equation (PDE) over a homogeneous anisotropic medium. The PDE is derived using a non-Fourier heat flux model which may account for thermal waves and/or microscopic effects. In the analysis, discontinuous linear elements are used to model the boundary and the variables along the boundary. The systems of algebraic equations are set up to solve all the unknowns. For the purpose of evaluating the proposed method, some numerical examples with known exact solutions are solved. The numerical results obtained agree well with the exact solutions.     

Accuracy and efficiency of higher-order boundary element methods for steady convective heat diffusion in three-dimensions

Higher-order boundary element methods (BEM) are presented for three-dimenisonal steady convective heat diffusion at high Peclet numbers. The boundary element formulation is facilitated by the definition of an influence domain due to convective kernels. This approach essentially localizes the surface integrations only within the domain of influence which becomes more narrowly focused as the Peclet number increases. The outcome of this phenomenon is an increased sparsity and improved conditioning of the global matrix. Therefore, iterative solvers for sparse matrices become a very efficient and robust tool for the corresponding boundary element matrices. In this paper, we consider an example problem with an exact solution and investigate the accuracy and efficiency of the higher-order BEM formulations for high Peclet numbers in the range from 1000 to 100,000. The bi-quartic boundary elements included in this study are shown to provide very efficient and extremely accurate solutions to this problem, even on a single engineering workstation.     

Direct evaluation of singular boundary integrals in two-dimensional biharmonic analysis

Development of techniques to provide rapid and accurate evaluation of the integrations required in boundary element method (BEM) formulations are receiving more attention in the literature. In this work, a series of direct expressions for surface integrals, required for a boundary element solution of the non-homogeneous biharmonic over a general two-dimensional curvilinear surface, are presented. The concept of an isoparametric representation, usually applied to the variation of the field variables and the geometry, is extended to the parametric mapping of the curvilinear geometry. The result renders the typically complicated Jacobian function into a series of polynomial expressions based on the shape function set and several discrete Jacobian values. An application of the isoparametric approximation of the Jacobian for a quadratic element representation is developed. Implementation of this approximation significantly improves the accuracy of the boundary integral solution by eliminating error associated with numerical quadrature. Overall computational efficiency is improved by reducing the time necessary to calculate individual surface integrals and evaluate field variables at internal points. A numerical solution of the boundary integral equations of phenomena governed by the biharmonic equation is presented and compared with an exact analysis.     

Design sensitivity analysis with hypersingular boundary elements

The finite difference load method for shape design sensitivity analysis requires the calculation of stress and stress gradient on the boundary. In the standard boundary element method, the basic state variables-displacement and traction are continuous, and are considered as very accurate. However, the boundary stress and stress gradient, derived from the differentiation of the state variables and Hooke's law, are discontinuous and have relatively lower accuracy than the basic state variables. The hypersingular boundary integral equation is introduced in this paper to determine the stress and stress gradient in the design sensitivity analysis. The numerical examples demonstrate the accuracy of the design sensitivity using the hypersingular boundary elements.     

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。     

The calculation of capacitance and force in electrostatic fields byusing the boundary element method

Methods of calculating the integrated parameters, capacitance and force, in electrostatic fields by using boundary element analysis are discussed. Force is calculated using Maxwell stress. It is concluded that the integrated parameters can be obtained with very high accuracy by using the boundary element method, due to its highly accurate field solutions. Both capacitance and force can be calculated from integrations performed on the boundary or in the field space, and the integration on the boundary needs much less calculation work than the integration in the field space. For capacitance, boundary integration results are quite accurate, without the need for the space integration     

一阶拟线性严格双曲型方程组弱间断解的精确能控性及能观性

在一阶拟线性双曲型方程组C1解的精确能控性及能观性的基础上,本文通过对弱间断解性质的研究,在初值和边值存在有限个弱间断点的情况下,得到一阶拟线性严格双曲型方程组混合初边值问题的半整体弱间断解的存在唯一性及相应的估计式,进而得到一阶拟线性严格双曲型方程组在弱间断解意义下相应的精确边界能控性及精确边界能观性。     

Nonlocal strain-softening model of quasi-brittle materials using boundary element method

The strain-softening localization problems have been studied intensively using the finite element methods. This paper addresses the localization using the boundary element approach. A plasticity model with yield limit degradation is implemented in a boundary element program to study the fracture behavior of quasi-brittle materials. A special integration algorithm is formulated and applied to deal with the singular integrations encountered in the volume integrals over the internal cells where strain-softening occurs. Strain-softening damage localizations are investigated. It is found that as different cell meshes are used in the analysis, the strain-softening region tends to localize into a zone of one cell width, which leads to incorrect results. A nonlocal strain-softening localization limiter is incorporated into the boundary element analysis to avoid the localization problems and attain realistic results.     

Analytic formulations for calculating nearly singular integrals in two-dimensional BEM

There exist the nearly singular integrals in the boundary integral equations when a source point is close to an integration element but not on the element, such as the field problems with thin domains. In this paper, the analytic formulations are achieved to calculate the nearly weakly singular, strongly singular and hyper-singular integrals on the straight elements for the two-dimensional (2D) boundary element methods (BEM). The algorithm is performed after the BIE are discretized by a set of boundary elements. The singular factor, which is expressed by the minimum relative distance from the source point to the closer element, is separated from the nearly singular integrands by the use of integration by parts. Thus, it results in exact integrations of the nearly singular integrals for the straight elements, instead of the numerical integration. The analytic algorithm is also used to calculate nearly singular integrals on the curved element by subdividing it into several linear or sub-parametric elements only when the nearly singular integrals need to be determined. The approach can achieve high accuracy in cases of the curved elements without increasing other computational efforts. As an application, the technique is employed to analyze the 2D elasticity problems, including the thin-walled structures. Some numerical results demonstrate the accuracy and effectiveness of the algorithm.     

The boundary element method applied to orthotropic shear deformable plates

This work presents a formulation for thick plates following Mindlin theory. The fundamental solution takes into account an assumed displacement distribution on the thickness, and was derived by means of Hormander operator and the Radon transform. To compute the inverse Radon transform of the fundamental solution, some numerical integrals need to be computed. How these integrations are carried out is a key point in the performance of the boundary element code. Two approaches to integrate fundamental solutions are discussed. Integral equations are obtained using Betti's reciprocal theorem. Domain integrals are exactly transformed into boundary integrals by the radial integration technique.     

Exact stiffness matrix of mono-symmetric composite I-beams with arbitrary lamination

The exact stiffness matrix, based on the simultaneous solution of the ordinary differential equations, for the static analysis of mono-symmetric arbitrarily laminated composite I-beams is presented herein. For this, a general thin-walled composite beam theory with arbitrary lamination including torsional warping is developed by introducing Vlasov’s assumption. The equilibrium equations and force–deformation relations are derived from energy principles. The explicit expressions for displacement parameters are then derived using the displacement state vector consisting of 14 displacement parameters, and the exact stiffness matrix is determined using the force–deformation relations. In addition, the analytical solutions for symmetrically laminated composite beams with various boundary conditions are derived as a special case. Finally, a finite element procedure based on Hermitian interpolation polynomial is developed. To demonstrate the validity and the accuracy of this study, the numerical solutions are presented and compared with the analytical solutions and the finite element results using the Hermitian beam elements and ABAQUS’s shell element.     

Taylor expansions for singular kernels in the boundary element method

The problem treated is the integration of singular functions which arise in three-dimensional isoparametric formulations of boundary integral equations. A Taylor expansion in the local parametric co-ordinates is developed for the singular integrand, so allowing singular terms to be integrated in closed form, even for curved surface elements. The remainder integral obtained by subtracting out the worst singularities is integrated by repeated Gaussian quadrature. Two groups of tests are presented. First, the accuracy of the integrations has been checked for plane parallelograms (for which exact solutions have been developed) and for curved elements on a sphere. Secondly, results from complete boundary element calculations based on point collocation have been compared with known analytical solutions to two problems; zonal surface harmonics on a sphere and the capacitance of an ellipsoid. The agreement obtained with few degrees-of-freedom suggests that errors which have previously been attributed to point collocation might have arisen in the numerical integration.     

Elastodynamic boundary element method for piecewise homogeneous media

A general higher-order formulation for the time domain elastodynamic direct boundary element method is presented for computing the transient displacements and stresses in multiply connected two-dimensional solids. The displacement and traction interpolation functions are linear in time and quadratic in space. All integrations are analytical, and are expressed in terms of twelve basic recurring integrals. Causality is ensured by integrating only over the dynamically active parts of each element, and the algorithm presented is time-marching and implicit. The use of analytical integrations allows both unbounded and bounded domain problems to be solved without having to introduce special enclosing elements. All of these improved features allow for a formulation that is very efficient and accurate. The stability and accuracy of the elastodynamic boundary element algorithm is demonstrated by solving several example problems and comparing the results with available analytical and numerical solutions. © 1998 John Wiley & Sons, Ltd.     

Numerical integration in 3D Galerkin BEM solution of HBIEs

 We consider hypersingular boundary integral equations associated with 3D problems defined on polygonal domains, whose solutions are approximated with a Galerkin boundary element method, related to a given triangulation of the boundary. At first, for linear shape functions, the most frequently used basis functions, we give explicit results of the analytical inner integrations. Then, after an analysis of the singularities arising in the whole integration process, we propose suitable quadrature schemes to evaluate integrals required to form the Galerkin matrix elements. Several numerical results are presented. Received 6 November 2000     

Efficient calculation of internal results in 2D elasticity BEM

The solution of boundary integral equations in their discretized form requires an accurate treatment of regular as well as singular integrals. The regular integrals are usually solved numerically using Gauss quadrature. Since these integrations make up the major part of the numerical work the choice of the appropriate Gauss order is essential to an accurate and efficient boundary element analysis. Thus, a considerable number of publications is dealing with the subject of choosing a Gauss order suitable to gain efficiency without loosing accuracy. The guidelines determining the choice of the appropriate Gauss order is usually called an integration criterion. This paper presents a study on this topic with emphasis on the accuracy of internal results in 2D elasticity. First the necessity for a new integration criterion is shown. Then a new criterion is derived. This new criterion and various existing criteria from the literature are applied to a standard benchmark problem. The superior performance of the novel criterion is demonstrated.     

The boundary contour method for piezoelectric media with linear boundary elements

This paper presents a development of the boundary contour method (BCM) for piezoelectric media. First, the divergence‐free property of the integrand of the piezoelectric boundary element is proved. Secondly, the boundary contour method formulation is derived and potential functions are obtained by introducing linear shape functions and Green's functions (Computer Methods in Applied Mechanics and Engineering 1998; 158 : 65) for piezoelectric media. The BCM is applied to the problem of piezoelectric media. Finally, numerical solutions for illustrative examples are compared with exact ones and those of the conventional boundary element method (BEM). The numerical results of the BCM coincide very well with the exact solution, and the feasibility and efficiency of the method are verified. Copyright © 2003 John Wiley & Sons, Ltd.     

A finite element Galerkin formulation and its numerical performance

A finite element Galerkin formulation admitting non-homogenous boundary conditions and discontinuous approximants is derived from a variational principle. The procedure results in matrix equations explicitly allowing for jump discontinuties across interelement boundaries. As an illustration, a specialization of the procedure similar to a reduced integration technique is used to solve several problems in elastostatics. The improvement in accuracy in comparison with conventional methods is remarkable.