Calculation of domain integrals of two dimensional boundary element method

In this paper, a new method is applied to deal with domain integrals of boundary element method (BEM). In fact we focus to convert the domain integrals into boundary integrals for non-homogenous Laplace, Helmholtz and advection diffusion equations in two dimensional BEM. The transformation presented in this paper is based on divergence theorem. In addition, we prove the efficiency of method mathematically when the domain integrals are weakly singular. Numerical results are presented to verify the validity of this method for different geometries. Numerical implementation is done for the constant BEM, which can be implemented easily. To verify the new scheme, some test problems have been designed at end of the paper. The numerical results generally show that the new scheme has good accuracy with regards to other popular schemes.     

An advanced 3D boundary element method for characterizations of composite materials

Some recent developments in the modeling of composite materials using the boundary element method (BEM) are presented in this paper. The boundary integral equation for 3D multi-domain elasticity problems is reviewed. Difficulties in dealing with nearly-singular integrals, which arise in the BEM modeling of composite materials with closely packed fillers or of thin films, are discussed. New and improved techniques to deal with the nearly-singular integrals in the 3D elasticity BEM are presented. Numerical examples of layered thin films and composites with randomly distributed particles and fibers are studied. The advantages and limitations of the BEM approach in modeling advanced composites are also discussed. The developed BEM with multi-domain and thin-body capabilities is demonstrated to be a promising tool for simulations and characterization of various composite materials.     

正交各向异性平面问题边界元——边境元素法研究

在文献[1]中,本文作者研究了正交各向异性平面问题边界元素法的有关基本理论和计算公式,在上述工作的基础上,本文进一步研究各向异性平面问题边界邻域的应力分析。当采用边界元素法分析应力时,由于边界积分的奇异性,边界邻域应力的计算结果往往存在一定误差。为解决此问题,本文提出一个基于修正余能原理的所谓边境元素,包括四节点边境元素、八节点边境元素和三节点边境元素等。在边界元素法求解的基础上,进一步利用本文所述边境元素法,得到了非常满意的计算结果。      

Green's First Identity Method for Boundary-Only Solution of Self-Weight in BEM Formulation for Thick Slabs

The present paper develops a new technique for treatment of self-weight for building slabs in the boundary element method (BEM). Due to the use of BEM in the analysis, all defined variables are presented on the slab boundary (mesh is defined only along the slab boundary). Self-weight, however, is usually defined over slab domain, hence domain discretisation is required, which spoils the main advantage of the BEM. In this paper a new method is presented to transform self-weight domain integrals to the boundary for such slabs. The proposed method is based on using the so-called Green's first identity. All new kernels for generalized displacements, stress-resultants, and tractions are derived and listed explicitly. The present formulation is implemented into computer code and several examples are tested. Results are compared against results obtained from other numerical method to prove the accuracy and validity of the present formulation.     

Shape optimization in three-dimensional linear elasticity by the boundary contour method

A variant of the usual boundary element method (BEM), called the boundary contour method (BCM), has been presented in the literature in recent years. In the BCM in three dimensions, surface integrals on boundary elements of the usual BEM are transformed, through an application of Stokes’ theorem, into line integrals on the bounding contours of these elements. The BCM employs global shape functions with the weights, in the linear combinations of these shape functions, being defined piecewise on boundary elements. A very useful consequence of this approach is that stresses, at suitable points on the boundary of a body, can be easily obtained from a post-processing step of the standard BCM. The subject of this paper is shape optimization in three-dimensional (3D) linear elasticity by the BCM. This is achieved by coupling a 3D BCM code with a mathematical programming code based on the successive quadratic programming (SQP) algorithm. Numerical results are presented for several interesting illustrative examples.     

A fast boundary cloud method for 3D exterior electrostatic analysis

An accelerated boundary cloud method (BCM) for boundary‐only analysis of 3D electrostatic problems is presented here. BCM uses scattered points unlike the classical boundary element method (BEM) which uses boundary elements to discretize the surface of the conductors. BCM combines the weighted least‐squares approach for the construction of approximation functions with a boundary integral formulation for the governing equations. A linear base interpolating polynomial that can vary from cloud to cloud is employed. The boundary integrals are computed by using a cell structure and different schemes have been used to evaluate the weakly singular and non‐singular integrals. A singular value decomposition (SVD) based acceleration technique is employed to solve the dense linear system of equations arising in BCM. The performance of BCM is compared with BEM for several 3D examples. Copyright © 2004 John Wiley & Sons, Ltd.     

A mapping method for numerical evaluation of two-dimensional integrals with 1/r singularity

Singular integrals occur commonly in applications of the boundary element method (BEM). A simple mapping method is presented here for the numerical evaluation of two-dimensional integrals in which the integrands, at worst, are O(1/r) (r being the distance from a source to a field point). This mapping transforms such integrals over general curved triangles into regular 2-D integrals. Over flat and curved quadratic triangles, regular line integrals are obtained, and these can be easily evaluated by standard Gaussian quadrature. Numerical tests on some typical singular integrals, encountered in BEM applications, demonstrate the accuracy and efficacy of the method.     

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.     

A BEM approach to static and dynamic analysis of plates with internal supports

A boundary element approach is developed for the static and dynamic analysis of Kirchhoff's plates of arbitrary shape which, in addition to the boundary supports, are also supported inside the domain on isolated points (columns), lines (walls) or regions (patches). All kinds of boundary conditions are treated. The supports inside the domain of the plate may yield elastically. The method uses the Green's function for the static problem without the internal supports to establish an integral representation for the solution which involves the unknown internal reactions and inertia forces within the integrand of the domain integrals. The Green's function is established numerically using BEM. Subsequently, using an effective Gauss integration for the domain integrals and a BEM technique for line integrals a system of simultaneous, in general, nonlinear algebraic equations is obtained which is solved numerically. Several examples for both the static and dynamic problem are presented to illustrate the efficiency and the accuracy of the proposed method.     

A comparative study of three BEM for transient dynamic crack analysis of 2-D anisotropic solids

Three different boundary element methods (BEM) for transient dynamic crack analysis in two-dimensional (2-D), homogeneous, anisotropic and linear elastic solids are presented. Hypersingular traction boundary integral equations (BIEs) in frequency- domain, Laplace-domain and time-domain with the corresponding elastodynamic fundamental solutions are applied for this purpose. In the frequency-domain and the Laplace-domain BEM, numerical solutions are first obtained in the transformed domain for discrete frequency or Laplace-transform parameters. Time-dependent results are subsequently obtained by means of the inverse Fourier-transform and the inverse Laplace-transform algorithm of Stehfest. In the time-domain BEM, the quadrature formula of Lubich is adopted to approximate the arising convolution integrals in the time-domain BIEs. Hypersingular integrals involved in the traction BIEs are computed through a regularization process that converts the hypersingular integrals to regular integrals, which can be computed numerically, and singular integrals which can be integrated analytically. Numerical results for the dynamic stress intensity factors are presented and discussed for a finite crack in an infinite domain subjected to an impact crack-face loading.     

Analytical time integration for BEM axisymmetric acoustic modelling

In this work, a numerical time‐domain approach to model acoustic wave propagation in axisymmetric bodies is developed. The acoustic medium is modelled by the boundary element method (BEM), whose time convolution integrals are evaluated analytically, employing the concept of finite part integrals. All singularities for space integration, present in the expressions generated by time integration, are treated adequately. Some applications are presented to demonstrate the validity of the analytical expressions generated for the BEM, and the results obtained with the present approach are compared with those generated by applying numerical time integration. Copyright © 2007 John Wiley & Sons, Ltd.     

A two-dimensional time-domain boundary element method for dynamic crack problems in anisotropic solids

A time-domain boundary element method (BEM) for transient dynamic crack analysis in two-dimensional, homogeneous, anisotropic and linear elastic solids is presented in this paper. Strongly singular displacement boundary integral equations (DBIEs) are applied on the external boundary of the cracked body while hypersingular traction boundary integral equations (TBIEs) are used on the crack-faces. The present time-domain method uses the quadrature formula of Lubich for approximating the convolution integrals and a collocation method for the spatial discretization of the time-domain boundary integral equations. Strongly singular and hypersingular integrals are dealt with by a regularization technique based on a suitable variable change. Discontinuous quadratic quarter-point elements are implemented at the crack-tips to capture the local square-root-behavior of the crack-opening-displacements properly. Numerical examples for computing the dynamic stress intensity factors are presented and discussed to demonstrate the accuracy and the efficiency of the present method.     

Transient dynamic analysis of interface cracks in layered anisotropic solids under impact loading

Transient elastodynamic crack analysis in two-dimensional (2D), layered, anisotropic and linear elastic solids is presented in this paper. A time-domain boundary element method (BEM) in conjunction with a multi-domain technique is developed for this purpose. Time-domain elastodynamic fundamental solutions for homogenous, anisotropic and linear elastic solids are applied in the present time-domain BEM. The spatial discretization of the boundary integral equations is performed by a Galerkin-method, while a collocation method is adopted for the temporal discretization of the arising convolution integrals. An explicit time-stepping scheme is developed to compute the unknown boundary data and the crack-opening-displacements (CODs). To show the effects of the crack configuration, the material anisotropy, the layer combination and the dynamic loading on the dynamic stress intensity factors and the scattered elastic wave fields, several numerical examples are presented and discussed.     

A comparison of boundary element method formulations for steady state anisotropic heat conduction problems

In this paper the boundary element method (BEM) is numerically implemented in order to solve steady state anisotropic heat conduction problems. Various types of elements, namely, constant elements, continuous and discontinuous linear elements and continuous and discontinuous quadratic elements are used. The performances of these various BEM formulations are compared for both the direct well-posed Dirichlet problem and the inverse ill-posed Cauchy problem, revealing several features of the BEM. Furthermore, previously undetermined analytical solutions for the integrals associated with linear and quadratic elements are presented.     

A new method for evaluating singular integrals in stress analysis of solids by the direct boundary element method

The purpose of this paper is to report on a new and efficient method for the evaluation of singular integrals in stress analysis of elastic and elasto-plastic solids, respectively, by the direct boundary element method (BEM). Triangle polar co-ordinates are used to reduce the order of singularity of the boundary integrals by one degree and to carry out the integration over mappings of the boundary elements onto plane squares. The method was subsequently extended to the cubature of singular integrals over three-dimensional internal cells as occur in applications of the BEM to three-dimensional elasto-plasticity. For this purpose so-called tetrahedron polar co-ordinates were introduced. Singular boundary integrals stretching over either linear, triangular, or quadratic quadilateral, isoparametric boundry elements and singular volume integrals extending over either linear, tetrahedral, or quadratic, hexahedral, isoparametric internal cells are treated. In case of higher order isoparametric boundary elements and internal cells, division into a number of subelements and subcells, respectively, is necessary. The analytical investigation is followed by a numerical study restricted to the use of quadratic, quadrilateral, isoparametric boundary elements. This is justified by the fact that such elements, as opposed to linear elements, yield singular boundary integrals which cannot be integrated analytically. The results of the numerical investigation demonstrate the potential of the developed concept.     

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.     

Analytical evaluation and application of the singularities in boundary element method

In this paper, two alternative approaches to the boundary element method (BEM) are investigated; namely, the contour approach method and the direct approach method. A detailed comparison of these methods is made by evaluating the accuracy of singular boundary integrals. A complete and comprehensive exposition of the derivation leads to the correct implementation. This is in contrast to conventional numerical integration methods, which suffer from the numerical boundary layer. Singularities which are mathematical artifacts are shown to vanish when the contour approach and the direct approach methods are applied. Singularities which arise from the physics are dealt with by way of a complex mapping method in the BEM. The results of seven benchmark problems which are supportive of these conclusions are presented at the end of this article.     

Multidimensional numerical integration for meshless BEM

The conventional boundary element method (BEM) uses internal cells for the domain integrals, when nonlinear problems or problems with domain effects are solved. In the conventional BEM, however, the merit of the BEM, which is easy preparation of data, is lost. This paper presents numerical integration for a meshless BEM, which does not require internal cells. This method uses arbitrary internal points instead of internal cells. First, a multidimensional interpolation method for distribution in an arbitrary domain is shown using boundary integral equations. This method requires values on a boundary of a region and values at arbitrary internal points. In this paper, multidimensional numerical integration is proposed using the above multidimensional interpolation method. This integration is useful for inelastic problems and thermal stress analysis with arbitrary internal heat generation. This method is based on an improved multiple-reciprocity BEM (triple-reciprocity BEM) for heat conduction analysis with heat generation. In order to investigate the efficiency of this method, several numerical examples are given.     

Analytical integral algorithm in the BEM for orthotropic potential problems of thin bodies

There exist nearly singular integrals for boundary layer effect problem and thin body effect problem in the boundary element method (BEM). A new completely analytical integral algorithm is proposed and applied to evaluate the nearly singular integrals in the BEM for two-dimensional orthotropic potential problems of thin bodies. The completely analytical integral formulas are derived with integration by parts for the linear boundary interpolation. The present algorithm applies these analytical formulas to deal with the nearly singular integrals. The unknown potentials and fluxes at boundary nodes are firstly calculated accurately and then the physical quantities at the interior points are computed. Two benchmark numerical examples on heat conduction demonstrate that the present algorithm can handle thin structures with the thickness-to-length ratio down to 1.E−08. This indicates that the BEM is especially accurate and efficient for numerical analysis of thin body problems.     

An analysis of solidification problems by the boundary element method

The problem of interest in this paper is the calculation of the motion of the solid–liquid interface and the time-dependent temperature field during solidification of a pure metal. An iterative implicit algorithm has been developed for this purpose using the boundary element method (BEM) with time-dependent Green's functions and convolution integrals. The BEM approach requires discretization of only the surface of the solidifying body. Thus, the numerical method closely follows the physics of the problems and is intuitively very appealing. The formulation and the numerical scheme presented here are general and can be applied to a broad range of moving boundary problems. Emphasis is given to two-dimensional problems. Comparison with existing semi-analytical solutions and other numerical solutions from the literature reveals that the method is fast, accurate and without major time step limitations.