A Galerkin boundary contour method for two-dimensional linear elasticity

The Boundary Contour Method (BCM) is a recent variant of the Boundary Element Method (BEM) resting on the use of boundary approximations which a-priori satisfy the field equations. For two-dimensional problems, the evaluation of all the line-integrals involved in the collocation BCM reduces to function evaluations at the end-points of each element, thus completely avoiding numerical integrations. With reference to 2-D linear elasticity, this paper develops a variational version of BCM by transferring to the BCM context the ingredients which characterize the Galerkin-Symmetric BEM (GSBEM). The method proposed herein requires no numerical integrations: all the needed double line-integrals over boundary elements pairs can be evaluated by generating appropriate “potential functions” (in closed form) and computing their values at the element end-points. This holds for straight as well as curved elements; however the coefficient matrix of the equation system in the boundary unknowns turns out to be fully symmetric only when all the elements are straight. The numerical results obtained for some benchmark problems, for which analytical solutions are available, validate the proposed formulation and the corresponding solution procedure.     

The boundary contour method for two-dimensional linear elasticity with quadratic boundary elements

This paper presents a further development of the Boundary Contour Method (BCM) for two-dimensional linear elasticity. The new developments are: (a) explicit use of the rigid body motion solution to regularize the BCM and avoid computation of the corner tensor, (b) quadratic boundary elements compared to linear elements in previous work and (c) evaluation of stresses both inside and on the boundary of a body. This method allows boundary stress computations at regular points (i.e. at points where the boundary is locally smooth) inside boundary elements without the need of any special algorithms for the numerical evaluation of hypersingular integrals. Numerical solutions for illustrative examples are compared with analytical ones. The numerical results are uniformly accurate.     

The traction boundary contour method for linear elasticity

This paper presents a further development of the boundary contour method. The boundary contour method is extended to cover the traction boundary integral equation. A traction boundary contour method is proposed for linear elastostatics. The formulation of traction boundary contour method is regular for points except the ends of the boundary element and corners. The present approach only requires line integrals for three‐dimensional problems and function evaluations at the ends of boundary elements for two‐dimensional cases. The implementation of the traction boundary contour method with quadratic boundary elements is presented for two‐dimensional problems. Numerical results are given for some two‐dimensional examples, and these are compared with analytical solutions. This method is shown to give excellent results for illustrative examples. Copyright © 1999 John Wiley & Sons, Ltd.     

The meshless hypersingular boundary node method for three-dimensional potential theory and linear elasticity problems

The Boundary Node Method (BNM) represents a coupling between Boundary Integral Equations (BIEs) and Moving Least Squares (MLS) approximants. The main idea here is to retain the dimensionality advantage of the former and the meshless attribute of the latter. The result is a ‘meshfree’ method that decouples the mesh and the interpolation procedures. The BNM has been applied to solve 2-D and 3-D problems in potential theory and linear elasticity. The Hypersingular Boundary Element Method (HBEM) has diverse important applications in areas such as fracture mechanics, wave scattering, error analysis and adaptivity, and to obtain a symmetric Galerkin boundary element formulation. The present work presents a coupling of Hypersingular Boundary Integral Equations (HBIEs) with MLS approximants, to produce a new meshfree method — the Hypersingular Boundary Node Method (HBNM). Numerical results from this new method, for selected 3-D problems in potential theory and in linear elasticity, are presented and discussed in this paper.     

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.     

Error estimation using hypersingular integrals in boundary element methods for linear elasticity

A natural measure of the error in the boundary element method rests on the use of both the standard boundary integral equation (BIE) and the hypersingular BIE (HBIE). An approximate (numerical) solution can be obtained using either one of the BIEs. One expects that the residual, obtained when such an approximate solution is substituted to the other BIE is related to the error in the solution. The present work is developed for vector field problems of linear elasticity. In this context, suitable ‘hypersingular residuals’ are shown, under certain special circumstances, to be globally related to the error. Further, heuristic arguments are given for general mixed boundary value problems. The calculated residuals are used to compute element error indicators, and these error indicators are shown to compare well with actual errors in several numerical examples, for which exact errors are known. Conclusions are drawn and potential extensions of the present error estimation method are discussed.     

The dual boundary contour method for two-dimensional crack problems

This paper concerns the dual boundary contour method for solving two-dimensional crack problems. The formulation of the dual boundary contour method is presented. The crack surface is modeled by using continuous quadratic boundary elements. The traction boundary contour equation is applied for traction nodes on one of the crack surfaces and the displacement boundary contour equation is applied for displacement nodes on the opposite crack surface and noncrack boundaries. The direct calculation of the singular integrals arising in displacement BIEs is addressed. These singular integrals are accurately evaluated with potential functions. The singularity subtraction technique for determining the stress intensity factor K_I, K_II and the T-term are developed for mixed mode conditions. Some two-dimensional examples are presented and numerical results obtained by this approach are in very good agreement with the results of the previous papers. This revised version was published online in August 2006 with corrections to the Cover Date.     

On design sensitivity analysis in linear elasticity by the boundary contour method

A new derivation for design sensitivity analysis using the concept of the material derivative (or total derivative), and boundary contour conversion, is presented in this paper. This derivation is carried out by first taking the material derivative of the regularized Boundary Integral Equation (BIE) with respect to a shape design variable, and then converting the resulting equations into their boundary contour version. As expected, the final design sensitivity equations are identical to those presented in Ref. [1] in which the opposite process, namely, conversion of the BIE into a boundary contour version, followed by material differentiation, had been carried out.     

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 superelement for two-dimensional singular boundary value problems in linear elasticity

The finite element method for elliptic boundary value problems has been modified to deal with boundary singularities. We introduce a singular-super-element (SSE) which incorporates the known expansion for the singular solution explicitly over the internal region surrounding the singular point, whilst using blended trial functions over the intermediate region, which joins the internal and external regions smoothly. The SSE conforms with the mesh used in the external region, and may be easily incorporated into standard finite element programs. The calculations yield the expansion coefficients directly, as well as an accurate representation of the displacements in the vicinity of the singular point, for a crack or V-notch of any angle subject to any mode of loading. The SSE has been applied to determine stress intensity factors for two-dimensional crack and V-notch problems, including mixed mode. The computations converge rapidly, yielding results of high accuracy.     

The boundary contour method for two-dimensional Stokes flow and incompressible elastic materials

 A variant of the boundary element method, called the boundary contour method (BCM), offers a further reduction in dimensionality. Consequently, boundary contour analysis of two-dimensional (2-D) problems does not require any numerical integration at all. While the method has enjoyed many successful applications in linear elasticity, the above advantage has not been exploited for Stokes flow problems and incompressible media. In order to extend the BCM to these materials, this paper presents a development of the method based on the equations of Stokes flow and its 2-D kernel tensors. Potential functions are derived for quadratic boundary elements. Numerical solutions for some well-known examples are compared with the analytical ones to validate the development. Received 28 August 2001 / Accepted 15 January 2002     

The boundary node method for three‐dimensional linear elasticity

The Boundary Node Method (BNM) is developed in this paper for solving three‐dimensional problems in linear elasticity. The BNM represents a coupling between Boundary Integral Equations (BIE) and Moving Least‐Squares (MLS) interpolants. The main idea is to retain the dimensionality advantage of the former and the meshless attribute of the later. This results in decoupling of the ‘mesh’ and the interpolation procedure.For problems in linear elasticity, free rigid‐body modes in traction prescribed problems are typically eliminated by suitably restraining the body. However, an alternative approach developed recently for the Boundary Element Method (BEM) is extended in this work to the BNM. This approach is based on ideas from linear algebra to complete the rank of the singular stiffness matrix. Also, the BNM has been extended in the present work to solve problems with material discontinuities and a new procedure has been developed for obtaining displacements and stresses accurately at internal points close to the boundary of a body. Copyright © 1999 John Wiley & Sons, Ltd.     

The meshless regular hybrid boundary node method for 2D linear elasticity

The meshless Regular Hybrid Boundary Node Method (RHBNM) is a promising method for solving boundary value problems, and is further developed and numerically implemented for 2D linear elasticity in this paper. The present method is based on a modified functional and the Moving Least Squares (MLS) approximation, and exploits the meshless attributes of the MLS and the reduced dimensionality advantages of the BEM. As a result, the RHBNM is truly meshless, i.e. it only requires nodes constructed on the surface, and absolutely no cells are needed either for interpolation of the solution variables or for the boundary integration. All integrals can be easily evaluated over regular shaped domains and their boundaries.Numerical examples show that the high convergence rates with mesh refinement and the high accuracy with a small node number is achievable. The treatment of singularities and further integrations required for the computation of the unknown domain variables, as in the conventional BEM, can be avoided.     

B-spline approximation in boundary face method for three-dimensional linear elasticity

In this paper, basis functions generated from B-spline or Non-Uniform Rational B-spline (NURBS), are used for approximating the boundary variables to solve the 3D linear elasticity Boundary Integral Equations (BIEs). The implementation is based on the BFM framework in which both boundary integration and variable approximation are performed in the parametric spaces of the boundary surfaces to keep the exact geometric information in the BIEs. In order to reduce the influence of tensor product of B-spline and make the discretization of a body surface easier, the basis functions defined in global intervals are translated into local form. B-spline fitting function built with the local basis functions is converted into an interpolation type of function in which the nodal values of the boundary variables are used for control points. Numerical tests for 3D linear elasticity problems show that the BFM with B-spline basis functions outperforms that with the well-known Moving Least Square (MLS) approximation.     

Development of hybrid boundary node method in two-dimensional elasticity

As a truly meshless method, the Hybrid Boundary Node Method (HBNM) does not require a ‘boundary element mesh’, either for the purpose of interpolation of the solution variables or for the integration of ‘energy’. It has been applied to solve the potential problems. This paper presents a further development of the HBNM to the 2D elastic problems.In this paper, the hybrid displacement variational formulations have been coupled with the Moving Least Squares (MLS) approximation. The rigid body movement method is employed to solve the hyper-singular integrations. The ‘boundary layer effect’, which is the main drawback of the original HBNM, has been circumvented by an adaptive integration scheme.In the present method, the source points of the fundamental solution are arranged directly on the boundary. Thus, the uncertain scale factor taken in the Regular Hybrid Boundary Node Method (RHBNM) can be avoided. The parameters that influence the performance of this method are studied through several numerical examples and the known analytical solutions. The treatment of singularity and further integration has been given by a series of effective approaches. The computation results obtained by the present method are shown that good convergence and high accuracy with a small node number are achievable.     

Regularization of hypersingular boundary integral equations: a new approach for axisymmetric elasticity

A hypersingular boundary integral equation (HBIE) formulation, for axisymmetric linear elasticity, has been recently presented by de Lacerda and Wrobel [Int. J. Numer. Meth. Engng 52 (2001) 1337]. The strongly singular and hypersingular equations in this formulation are regularized by de Lacerda and Wrobel by employing the singularity subtraction technique. The present paper revisits the same problem. The axisymmetric HBIE formulation for linear elasticity is interpreted here in a ‘finite part’ sense and is then regularized by employing a ‘complete exclusion zone’. The resulting regularized equations are shown to be simpler than those by de Lacerda and Wrobel.     

A pure boundary element method approach for solving hypersingular boundary integral equations

This paper is concerned with discretization and numerical solution of a regularized version of the hypersingular boundary integral equation (HBIE) for the two-dimensional Laplace equation. This HBIE contains the primary unknown, as well as its gradient, on the boundary of a body. Traditionally, this equation has been solved by combining the boundary element method (BEM) together with tangential differentiation of the interpolated primary variable on the boundary. The present paper avoids this tangential differentiation. Instead, a “pure” BEM method is proposed for solving this class of problems. Dirichlet, Neumann and mixed problems are addressed in this paper, and some numerical examples are included in it.     

A new boundary element method formulation for three dimensional problems in linear elasticity

Summary A new boundary element method (BEM) formulation for planar problems of linear elasticity has been proposed recently [6]. This formulation uses a kernel which has a weaker singularity relative to the corresponding kernel in the standard formulation. The most important advantage of the new formulation, relative to the standard one, is that it delivers stresses accurately at internal points that are extremely close to the boundary of a body. A corresponding BEM formulation for three dimensional problems of linear elasticity is presented in this paper. This formulation is derived through the use of Stokes' theorem and has kernels which are only 1/r singular (wherer is the distance between a source and a field point) for the displacement equation. The standard BEM formulation for three-dimensional elasticity problems has a kernel which is 1/r ² singular.With 2 Figures     

T-stress solutions for two-dimensional crack problems in anisotropic elasticity using the boundary element method

The importance of a two‐parameter approach in the fracture mechanics analysis of many cracked components is increasingly being recognized in engineering industry. In addition to the stress intensity factor, the T stress is the second parameter considered in fracture assessments. In this paper, the path‐independent mutual M‐integral method to evaluate the T stress is extended to treat plane, generally anisotropic cracked bodies. It is implemented into the boundary element method for two‐dimensional elasticity. Examples are presented to demonstrate the veracity of the formulations developed and its applicability. The numerical solutions obtained show that material anisotropy can have a significant effect on the T stress for a given cracked geometry.     

The multi-domain hybrid boundary node method for 3D elasticity

In this paper, a multi-domain technique for 3D elasticity problems is derived from the hybrid boundary node method (Hybrid BNM). The Hybrid BNM is based on the modified variational principle and the Moving Least Squares (MLS) approximation. It does not require a boundary element mesh, neither for the purpose of interpolation of the solution variables nor for the integration of energy. This method can reduce the human-labor costs of meshing, especially for complex construction. This paper presents a further development of the Hybrid BNM for multi-domain analysis in 3D elasticity. Using the equilibrium and continuity conditions on the interfaces, the final algebraic equation is obtained by assembling the algebraic equation for each single sub-domain. The proposed multi-domain technique is capable to deal with interface and multi-medium problems and results in a block sparsity of the coefficient matrix. Numerical examples demonstrate the accuracy of the proposed multi-domain technique.