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.     

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.     

A 2D time-domain collocation-Galerkin BEM for dynamic crack analysis in piezoelectric solids

A time-domain boundary element method (TDBEM) for transient dynamic analysis of two-dimensional (2D), homogeneous, anisotropic and linear piezoelectric cracked solids is presented in this paper. The present analysis uses a combination of the strongly singular displacement boundary integral equations (BIEs) and the hypersingular traction boundary integral equations. The spatial discretization is performed by a Galerkin-method, while a collocation method is implemented for the temporal discretization. Both temporal and spatial integrations are carried out analytically. In this way, only the line integrals over a unit circle arising in the time-domain fundamental solutions are computed numerically by standard Gaussian quadrature. An explicit time-stepping scheme is developed to compute the unknown boundary data including the generalized crack-opening-displacements (CODs) numerically. Special crack-tip elements are adopted to ensure a direct and an accurate computation of the dynamic field intensity factors (IFs) from the CODs. Several numerical examples involving stationary cracks in both infinite and finite solids under impact loading are presented to show the accuracy and the efficiency of the developed hypersingular time-domain BEM.     

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.     

Application of relations of singularity intensities of tangent derivatives of boundary displacements and tractions to BEM

In order to obtain a highly accurate numerical solution for a two-dimensional elasticity problem with singular boundary conditions on the smooth surface of an elastic body by boundary element method (BEM), the continuous or the singular requirements of the boundary field variables and their derivatives at element intersections have to be satisfied. The singularity intensities of the unknown boundary field variables have been determined through the theoretical relations of singularity intensities of tangent derivatives of boundary displacements and tractions, a priori. The continuous or singular requirements of boundary field variables and their derivatives at element intersections are automatically satisfied by using single node quadratic element (SNQE) developed in this paper. An example problem with singular boundary conditions on the surface of a semi-infinite-plane is numerically studied by BEM by using traditional three-nodes isoparametric elements and SNQEs. Numerical results show that the computation precisions at singular boundary points are greatly increased by using SNQEs.     

A hypersingular time‐domain BEM for 2D dynamic crack analysis in anisotropic solids

A hypersingular time‐domain boundary element method (BEM) for transient elastodynamic crack analysis in two‐dimensional (2D), homogeneous, anisotropic, and linear elastic solids is presented in this paper. Stationary cracks in both infinite and finite anisotropic solids under impact loading are investigated. On the external boundary of the cracked solid the classical displacement boundary integral equations (BIEs) are used, while the hypersingular traction BIEs are applied to the crack‐faces. The temporal discretization is performed by a collocation method, while a Galerkin method is implemented for the spatial discretization. Both temporal and spatial integrations are carried out analytically. Special analytical techniques are developed to directly compute strongly singular and hypersingular integrals. Only the line integrals over an unit circle arising in the elastodynamic fundamental solutions need to be computed numerically by standard Gaussian quadrature. An explicit time‐stepping scheme is obtained to compute the unknown boundary data including the crack‐opening‐displacements (CODs). Special crack‐tip elements are adopted to ensure a direct and an accurate computation of the elastodynamic stress intensity factors from the CODs. Several numerical examples are given to show the accuracy and the efficiency of the present hypersingular time‐domain BEM. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

Dynamic element discretization method for solving 2D traction boundary integral equations

A sufficient condition for the existence of element singular integral of the traction boundary integral equation for elastic problems requires that the tangential derivatives of the boundary displacements are Hölder continuous at collocation points. This condition is violated if a collocation point is at the junction between two standard conforming boundary elements even if the field variables themselves are Hölder continuous there. Various methods are proposed to overcome this difficulty. Some of them are rather complicated and others are too different from the conventional boundary element method. A dynamic element discretization method to overcome this difficulty is proposed in this work. This method is novel and very simple: the form of the standard traction boundary integral equation remains the same; the standard conforming isoparametric elements are still used and all collocation points are located in the interior of elements where the continuity requirements are satisfied. For boundary elements with boundary points where the field variables themselves are singular, such as crack tips, corners and other boundary points where the stress tensors are not unique, a general procedure to construct special elements has been developed in this paper. Highly accurate numerical results for various typical examples have been obtained.     

A meshless singular boundary method for three‐dimensional elasticity problems

This study documents the first attempt to extend the singular boundary method, a novel meshless boundary collocation method, for the solution of 3D elasticity problems. The singular boundary method involves a coupling between the regularized BEM and the method of fundamental solutions. The main idea here is to fully inherit the dimensionality and stability advantages of the former and the meshless and integration‐free attributes of the later. This makes it particularly attractive for problems in complex geometries and three dimensions. Four benchmark 3D problems in linear elasticity are well studied to demonstrate the feasibility and accuracy of the proposed method. The advantages, disadvantages, and potential applications of the proposed method, as compared with the FEM, BEM, and method of fundamental solutions, are also examined and discussed. Copyright © 2015 John Wiley & Sons, Ltd.     

A 2D time-domain BEM for transient wave scattering analysis by a crack in anisotropic solids

A two-dimensional (2D) time-domain boundary element method (BEM) is presented in this paper for transient analysis of elastic wave scattering by a crack in homogeneous, anisotropic and linearly elastic solids. A traction boundary integral equation formulation is applied to solve the arising initial-boundary value problem. A numerical solution procedure is developed to solve the time-domain boundary integral equations. A collocation method is used for the temporal discretization, while a Galerkin-method is adopted for the spatial discretization of the boundary integral equations. Since the hypersingular boundary integral equations are first regularized to weakly singular ones, no special integration technique is needed in the present method. Special attention of the analysis is devoted to the computation of the scattered wave fields. Numerical examples are given to show the accuracy and the reliability of the present time-domain BEM. The effects of the material anisotropy on the transient wave scattering characteristics are investigated.     

Domain-Decomposition Singular Boundary Method for Stress Analysis in Multi-Layered Elastic Materials

This paper applies an improved singular boundary method (SBM) in conjunction with domain decomposition technique to stress analysis of layered elastic materials. For problems under consideration, the interface continuity conditions are approximated in the same manner as the boundary conditions. The multi-layered coating system is decomposed into multiple subdomains in terms of each layer, in which the solution is approximated separately by the SBM representation. The singular boundary method is a recent meshless boundary collocation method, in which the origin intensity factor plays a key role for its accuracy and efficiency. This study also introduces new strong-form regularization formulas to accurately evaluate the origin intensity factors for elasticity problem. Consequently, we dramatically improve the accuracy and convergence of SBM solution of the elastostatics problems. The proposed domain-decomposition SBM is tested on two benchmark problems. Based on numerical results, we discuss merits of the present SBM scheme over the other boundary discretization methods, such as the method of fundamental solution (MFS) and the boundary element method (BEM).     

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 time-domain collocation-Galerkin BEM for transient dynamic crack analysis in anisotropic solids

This paper presents a time-domain boundary element method (BEM) for transient elastodynamic crack analysis in homogeneous and linear elastic solids of general anisotropy. A finite crack subjected to a transient loading is investigated. Two-dimensional (2D) generalized plane-strain or plane-stress condition is considered. The initial-boundary value problem is described by a set of hypersingular time-dependent traction boundary integral equations (BIEs), in which the crack-opening displacements (CODs) are unknown quantities. The hypersingular time-domain BIEs are first regularized to weakly singular ones by using spatial Galerkin method, which transfers the derivatives of the fundamental solutions to the unknown CODs and the weight functions. To solve the time-domain BIEs numerically, a time-stepping scheme is developed. The scheme applies the collocation method for temporal discretization of the time-domain BIEs. As spatial shape-functions, two different functions are implemented. For elements away from crack-tips, linear spatial shape-function is used, while for elements near the crack-tips a special ‘crack-tip shape-function’ is applied to describe the local ‘square-root’ behavior of the CODs at the crack-tips properly. Special attention of the analysis is devoted to the numerical computation of the transient elastodynamic stress intensity factors for cracks in general anisotropic and linear elastic solids. Numerical examples are presented to verify the accuracy of the present time-domain BEM.     

Crack-tip amplification and shielding by micro-cracks in piezoelectric solids – Part I: Static case

Analysis of the crack-tip amplification and shielding by micro-cracks in an unbounded two-dimensional piezoelectric solid is presented in this paper. A boundary element method (BEM) based on the hypersingular traction boundary integral equations (BIEs) is developed for this purpose. Integrals with hypersingular kernels are analytically transformed into weakly singular and regular integrals. A collocation method is applied for the spatial discretization. Quadratic quarter-point elements are implemented at all crack-tips. The amplification ratios of the field intensity factors and mechanical strain or electrical energy release rate are defined to show the crack-tip amplification and shielding. Numerical results are compared with the analytical results for isotropic materials to verify the present BEM. The influences of various loading conditions, the location and orientation angles of micro-cracks on the amplification ratios are investigated. The contours of the amplification ratios for an arbitrarily located micro-crack are also presented to show the crack-tip amplification and shielding effects.     

Comparison of boundary collocation methods for singular and non-singular axisymmetric heat transfer problems

This paper presents a computational study of some boundary collocation solution methods for the Laplace equation in cylindrical coordinates with axisymmetry. The methods compared are (i) the direct boundary element method (BEM), (ii) the method of fundamental solutions (MFS) with fixed sources and (iii) the Trefftz method. Relative accuracy of these methods are compared for two test problems. The first problem is a simple problem of heat transfer through a cylindrical rod which is a standard benchmark problem in this field. The second problem deals with heat transfer in silicon melt for Czochralski (CZ) process which involves a singularity in the boundary conditions at the corner of the crystal–melt interface. All the three methods indicated above are highly successful for the simple (first) problem with MFS and Trefftz being simpler to implement than the BEM. However, the Trefftz method was not effective for the second problem due to the boundary singularity and the MFS showed oscillations near the singularity point. Hence the use of higher order non-conforming elements with accurate Gauss–Kronrod integration schemes in the direct BEM method was investigated. It was found that the boundary singularity does not deteriorate the accuracy of the results if this improved numerical integration procedure is used in the direct BEM. Hence higher order elements with Gauss–Kronrod integration schemes can be used for the solution of many free interface problems encountered in crystal growth.     

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.     

Meshfree boundary particle method applied to Helmholtz problems

This paper is concerned with the boundary particle method (BPM), a new boundary-only radial basis function collocation schemes. The method is developed based on the multiple reciprocity principle and applying either high-order nonsingular general solutions or singular fundamental solutions as the radial basis function. Like the multiple reciprocity BEM (MR-BEM), the BPM does not require any inner nodes for inhomogeneous problems and therefore is a truly boundary-only technique. On the other hand, unlike the MR-BEM, the BPM is meshfree, integration-free, symmetric, and mathematically simple technique. In particular, the method requires much less computational effort for the discretization than the MR-BEM. In this study, the accuracy and efficiency of the BPM are numerically demonstrated in some 2D inhomogeneous Helmholtz problems under complicated geometries.     

Performance evaluation of algorithms for mildly nonlinear elliptic problems

A number of numerical methods for mildly nonlinear elliptic boundary value problems on general domains is presented. The discretization procedures considered are: a fourth-order FFT-type method, collocation using Hermite bicubic splines and Galerkin with linear triangular as well as quadratic quadrilateral isoparametric elements. The linearized collocation and Galerkin equations are solved by various direct methods available in the ELLPACK system. A comparative study of the above equation solvers is presented for different domain geometries and compilers. The evaluation of software for the general mildly nonlinear elliptic equations is performed over 36 instances from a population of 16 parametrized problems with ‘real world’ and ‘mathematical’ behaviour. The performance data suggests that collocation is an effective method for such general problems, while Galerkin with quadratic quadrilateral isoparametric elements is uniformly superior to the one with linear elements.     

Integral and geometrical means in the analytical evaluation of the BEM integrals for a 3D Laplace equation

By using integral theorems and geometrical interpretations, the analytical formulas for the coefficients occurring in boundary element method (BEM) equations for a 3D Laplace equation were found for arbitrary planar polygonal boundary elements with constant approximation. Closed forms for the gradients of the coefficients were also obtained. In addition, an analytical formula for planar triangular boundary elements with a linear approximation of potential was given. The formulas obtained are appropriate especially in singular and nearly singular cases.     

Accurate numerical integration of singular boundary element kernels over boundaries with curvature

Use of the Green function, for the solution of boundary-value problems, frequently results in singular integral equations. Algorithms are presented for the accurate and efficient treatment of singular kernels frequently encountered in the boundary element method (BEM). They are based upon the use of appropriately weighted Gaussian quadrature formulae, together with numerical geometrical transformations of the region of integration. The use of high-order subdomain expansion functions, for interpolation over nonplanar elements, allows boundary curvature to be accommodated. In particular, the handling of Green functions with logarithmic and r^?1 behaviour are detailed. Volume integrals, with r^?2 singularity, are outlined. Operations are performed on a simplex, thus resulting in generality and ease of automation. This scheme has been incorporated into boundary element method software and successfully applied to a variety of problems.