A symmetric Galerkin formulation and dual reciprocity for 2D elastostatics

In this paper, a symmetric Galerkin boundary integral equation including body force terms is presented. The implementation of the dual reciprocity method to transfer the domain integrals to the boundary is presented in the context of the Galerkin formulation. Several numerical examples involving self-weight and centrifugal body forces are studied to demonstrate the efficiency of the method.     

Fracture mechanics analysis of cracked 2-D anisotropic media with a new formulation of the boundary element method

A new formulation of the boundary element method (BEM) is proposed in this paper to calculate stress intensity factors for cracked 2-D anisotropic materials. The most outstanding feature of this new approach is that the displacement and traction integral equations are collocated on the outside boundary of the problem (no-crack boundary) only and on one side of the crack surfaces only, respectively. Since the new BEM formulation uses displacements or tractions as unknowns on the outside boundary and displacement differences as unknowns on the crack surfaces, the formulation combines the best attributes of the traditional displacement BEM as well as the displacement discontinuity method (DDM). Compared with the recently proposed dual BEM, the present approach doesn't require dua elements and nodes on the crack surfaces, and further, it can be used for anisotropic media with cracks of any geometric shapes. Numerical examples of calculation of stress intensity factors were conducted, and excellent agreement with previously published results was obtained. The authors believe that the new BEM formulation presented in this paper will provide an alternative and yet efficient numerical technique for the study of cracked 2-D anisotropic media, and for the simulation of quasi-static crack propagation.     

A meshless hybrid boundary-node method for Helmholtz problems

By coupling the moving least squares (MLS) approximation with a modified functional, the hybrid boundary node-method (hybrid BNM) is a boundary-only, truly meshless method. Like boundary element method (BEM), an initial restriction of the present method is that non-homogeneous terms accounting for effects such as distributed loads are included in the formulation by means of domain integrals, and thus make the technique lose the attraction of its ‘boundary-only’ character.This paper presents a new boundary-type meshless method dual reciprocity-hybrid boundary node method (DR-HBNM), which is combined the hybrid BNM with the dual reciprocity method (DRM) for solving Helmholtz problems. In this method, the solution of Helmholtz problem is divided into two parts, i.e. the complementary solution and the particular solution. The complementary solution is solved by means of hybrid BNM and the particular one is obtained by DRM. The modified variational formulation is applied to form the discrete equations of hybrid BNM. The MLS is employed to approximate the boundary variables, while the domain variables are interpolated by fundamental solutions. The domain integration is interpolated by radial basis function (RBF). The proposed method in the paper retains the characteristics of the meshless method and BEM, which only requires discrete nodes constructed on the boundary of a domain, several nodes in the domain are needed just for the RBF interpolation. The parameters that influence the performance of this method are studied through numerical examples and known analytical fields. Numerical results for the solution of Helmholtz equation show that high convergence rates and high accuracy are achievable.     

Fast Fourier transform on multipoles (FFTM) algorithm for Laplace equation with direct and indirect boundary element method

In this paper, the fast Fourier transform on multipole (FFTM) algorithm is used to accelerate the matrix-vector product in the boundary element method (BEM) for solving Laplace equation. This is implemented in both the direct and indirect formulations of the BEM. A new formulation for handling the double layer kernel using the direct formulation is presented, and this is shown to be related to the method given by Yoshida (Application of fast multipole method to boundary integral equation method, Kyoto University, Japan, 2001). The FFTM algorithm shows different computational performances in direct and indirect formulations. The direct formulation tends to take more computational time due to the evaluation of an extra integral. The error of FFTM in the direct formulation is smaller than that in the indirect formulation because the direct formulation has the advantage of avoiding the calculations of the free term and the strongly singular integral explicitly. The multipole and local translations introduce approximation errors, but these are not significant compared with the discretization error in the direct or indirect BEM formulation. Several numerical examples are presented to compare the computational efficiency of the FFTM algorithm used with the direct and indirect BEM formulations.     

Adaptive fast multipole boundary element method for three-dimensional half-space acoustic wave problems

A new adaptive fast multipole boundary element method (BEM) for solving 3-D half-space acoustic wave problems is presented in this paper. The half-space Green's function is employed explicitly in the boundary integral equation (BIE) formulation so that a tree structure of the boundary elements only for the boundaries of the real domain need to be applied, instead of using a tree structure that contains both the real domain and its mirror image. This procedure simplifies the implementation of the adaptive fast multipole BEM and reduces the CPU time and memory storage by about a half for large-scale half-space problems. An improved adaptive fast multipole BEM is presented for the half-space acoustic wave problems, based on the one developed recently for the full-space problems. This new fast multipole BEM is validated using several simple half-space models first, and then applied to model 3-D sound barriers and a large-scale windmill model with five turbines. The largest BEM model with 557470 elements was solved in about an hour on a desktop PC. The accuracy and efficiency of the BEM results clearly show the potential of the adaptive fast multipole BEM for solving large-scale half-space acoustic wave problems that are of practical significance.     

Boundary element formulations for Mindlin plate on an elastic foundation with dynamic load

Boundary element method (BEM) for a shear deformable plate (Reissner/Mindline's theories) resting on an elastic foundation subjected to dynamic load is presented. Formulations for both Winkler and Pasternak foundations are presented. The boundary element formulation in Laplace domain is presented together with complete expressions for the internal point kernels (i.e. fundamental solutions). Quadratic isoparameteric boundary elements are used to discretise the boundary of plate domain. Time domain variables are obtained by the Durbin's inversion method from transform domain. Numerical examples are presented to demonstrate the accuracy of the boundary element method and the comparisons are made with other numerical technique.     

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 alternative treatment of body forces in the BEM for thick plates resting on elastic foundations

In this paper, domain integrals due to body forces or uniform loading in the BEM for thick plates resting on elastic foundations are transformed to equivalent boundary integrals. Unlike, common techniques of the transformation, which are based on the Green second identity, the present formulation employs the Green first identity. The necessary particular solutions are derived and the kernels for the computation of stress resultants at the internal points are derived and given in explicit forms. The main advantages of the present formulation are the derived kernels are not singular and simple if compared to those of the Green second identity technique. A general technique for avoiding the appearance of jump terms in the transformed boundary integrals is presented. Three numerical examples are presented to demonstrate the accuracy and the validity of the new boundary integrals.     

Fast single domain–subdomain BEM algorithm for 3D incompressible fluid flow and heat transfer

In this paper acceleration and computer memory reduction of an algorithm for the simulation of laminar viscous flows and heat transfer is presented. The algorithm solves the velocity–vorticity formulation of the incompressible Navier–Stokes equations in 3D. It is based on a combination of a subdomain boundary element method (BEM) and single domain BEM. The CPU time and storage requirements of the single domain BEM are reduced by implementing a fast multipole expansion method. The Laplace fundamental solution, which is used as a special weighting function in BEM, is expanded in terms of spherical harmonics. The computational domain and its boundary are recursively cut up forming a tree of clusters of boundary elements and domain cells. Data sparse representation is used in parts of the matrix, which correspond to boundary‐domain clusters pairs that are admissible for expansion. Significant reduction of the complexity is achieved. The paper presents results of testing of the multipole expansion algorithm by exploring its effect on the accuracy of the solution and its influence on the non‐linear convergence properties of the solver. Two 3D benchmark numerical examples are used: the lid‐driven cavity and the onset of natural convection in a differentially heated enclosure. Copyright © 2008 John Wiley & Sons, Ltd.     

A boundary element sensitivity formulation for contact problems using the implicit differentiation method

In this paper a sensitivity formulation using the boundary element method (BEM), for problems involving contact is presented. The proposed formulation is based on the implicit differentiation method (IDM), where the boundary integral equations are differentiated analytically with respect to the design variables. In the proposed formulation the design variables are defined in terms of the normal gap between the contact bodies. The analysis demonstrates that the proposed method is accurate and robust, as it does not resolve the whole system. The proposed method can be used for evaluating the sensitivities in any shape-optimisation problem involving contact.     

A BEM formulation for linear bending analysis of plates reinforced by beams considering different materials

In this work, a boundary element method (BEM) formulation to perform linear bending analysis of building floor structures where slabs and beams can be defined with different materials is presented. The proposed formulation is based on Kirchhoff's hypothesis, the building floor being modelled by a zoned plate, where the beams are treated as thin sub-regions with larger rigidities. This composed structure is treated as a single body, the equilibrium and compatibility conditions being automatically taken into account. In the final integral equation, the tractions are eliminated along the interfaces, therefore reducing the number of degrees of freedom. The displacements are approximated along the beam cross-section, leading to a model where the values remain defined on the beam skeleton line instead of their boundaries. The accuracy of the proposed model is shown by comparing the numerical results with a well-known finite element code.     

An accelerated symmetric time-domain boundary element formulation for elasticity

Wave propagation phenomena occur often in semi-infinite regions. It is well known that such problems can be handled well with the boundary element method (BEM). However, it is also known that the BEM, with its dense matrices, becomes prohibitive with respect to storage and computing time. Focusing on wave propagation problems, where a formulation in time domain is preferable, the mentioned limit of the method becomes evident. Several approaches, amongst them the adaptive cross approximation (ACA), have been developed in order to overcome these drawbacks mainly for elliptic problems.The present work focuses on time dependent elastic problems, which are indeed not elliptic. The application of the presented fast boundary element formulation on such problems is enabled by introducing the well known Convolution Quadrature Method (CQM) as time stepping scheme. Thus, the solution of the time dependent problem ends up in the solution of a system of decoupled Laplace domain problems. This detour is worth since the resulting problems are again elliptic and, therefore, the ACA can be used in its standard fashion.The main advantage of this approach of accelerating a time dependent BEM is that it can be easily applied to other fundamental solutions as, e.g., visco- or poroelasticity.     

A coupled BEM‐flexibility force method for bending analysis of internally supported plates

This paper presents a new method for the analysis of plates in bending with internal supports. The proposed method can be regarded as an extension of the well‐known force method (the flexibility matrix method) in the matrix analysis of structures. The solution is performed through two phases: the released plate phase, in which the plate is released from all internal supports and solved using the Boundary Element Method (BEM). The effect of internal supports is considered in the second phase, where a series of unit virtual loads is placed instead of the unknown redundant reactions at internal supports. The flexibility matrix is formed and compatibility of deformations at the locations of internal supports is satisfied. Hence, the corresponding system of equations is solved for the unknown redundant forces at internal supports. The final solution of the problem consists of the summation of two phases: the released plate phase and the cases of virtual unit loads phase. An efficient solution algorithm is developed to solve both phases simultaneously. The main advantages of the present formulation are: (1) the present formulation increases the versatility of the BEM as it allows the re‐usability of standard BEM codes for solution of plates in bending to be used in solving problems having internal supports, with even no modifications; and (2) the two solution phases are completely uncoupled; therefore it is easy to trace behaviour of the plate due to failure of one or more of the internal supports without re‐analysis. Several numerical examples are analysed. The results are compared to those of analytical and finite element models to demonstrate the accuracy and the validity of the present formulation. The present formulation is used also to study the differences between the finite element and boundary element modelling for building slabs. Copyright © 2002 John Wiley & Sons, Ltd.     

A fast multipole boundary element method for 2D multi-domain elastostatic problems based on a dual BIE formulation

A new fast multipole formulation for the hypersingular BIE (HBIE) for 2D elasticity is presented in this paper based on a complex-variable representation of the kernels, similar to the formulation developed earlier for the conventional BIE (CBIE). A dual BIE formulation using a linear combination of the developed CBIE and HBIE is applied to analyze multi-domain problems with thin inclusions or open cracks. Two pre-conditioners for the fast multipole boundary element method (BEM) are devised and their effectiveness and efficiencies in solving large-scale problems are discussed. Several numerical examples are presented to study the accuracy and efficiency of the developed fast multipole BEM using the dual BIE formulation. The numerical results clearly demonstrate the potentials of the fast multipole BEM for solving large-scale 2D multi-domain elasticity problems. The method can be applied to study composite materials, functionally-graded materials, and micro-electro-mechanical-systems with coupled fields, all of which often involve thin shapes or thin inclusions.     

A multipole expansion-based boundary element method for axisymmetric potential problem

The multipole expansion is an approximation technique used to evaluate the potential field due to sources located in the far field. Based on the multipole expansion, we describe a new technique to calculate the far potential field due to ring sources which are encountered in the boundary element method (BEM) formulation of axisymmetric problems. As the sources in the near field are processed by the slower conventional BEM, it is important to maximize the amount of multipole calculations taking advantage of both interior and exterior multipole expansions. Numerical results are presented for an axisymmetric potential test problem with Neumann and Dirichlet boundary conditions. The complexity of the proposed method remains O(N²), which is equal to that of the conventional BEM. However, the proposed technique coupled with an iterative solver speeds up the solution procedure. The technique is significantly advantageous when medium and large numbers of elements are present in the domain.     

Analysis of anisotropic Kirchhoff plates using a novel hypersingular BEM

In this article a hypersingular boundary element method (BEM) for bending of thin anisotropic plates is presented. A new complex variable fundamental solution is implemented in the algorithm. For spatial discretization a collocation method with discontinuous quadratic elements is adopted. The domain integrals arising from the transversely applied load are transformed analytically into boundary integrals by means of the radial integration technique. The considered numerical examples prove that the novel BEM formulation presented in this study is much more efficient than previous formulations developed for the analysis of this kind of problems.     

A hybrid boundary node method

A new variational formulation for boundary node method (BNM) using a hybrid displacement functional is presented here. The formulation is expressed in terms of domain and boundary variables, and the domain variables are interpolated by classical fundamental solution; while the boundary variables are interpolated by moving least squares (MLS). The main idea is to retain the dimensionality advantages of the BNM, and get a truly meshless method, which does not require a ‘boundary element mesh’, either for the purpose of interpolation of the solution variables, or for the integration of the ‘energy’. All integrals can be easily evaluated over regular shaped domains (in general, semi‐sphere in the 3‐D problem) and their boundaries. Numerical examples presented in this paper for the solution of Laplace's equation in 2‐D show that high rates of convergence with mesh refinement are achievable, and the computational results for unknown variables are most accurate. No further integrations are required to compute the unknown variables inside the domain as in the conventional BEM and BNM. Copyright © 2001 John Wiley & Sons, Ltd.     

A boundary/domain element method for analysis of building raft foundations

In this paper, a new boundary/domain element method is developed to analyse plates resting on elastic foundations. The developed formulation is then used in analysing building raft foundations. For more practical representation, the considered raft plate is treated as thick plate with free edge boundary conditions. The soil or the elastic foundation is represented as continuous media (follows the Winkler assumption). The boundary element method is employed to model the raft plate; whereas the soil is modelled using constant domain cells or elements. Therefore, in the present formulation both the domain and the boundary of the raft plate are discretized. The associate soil domain integral is replaced by equivalent boundary integrals along each cell contour. The necessary matrix implementation of such formulation is carried out and explained in details. The main advantage of the present formulation is the ability of analysing rafts on non-homogenous soils. Two examples are presented including raft on non-homogenous soil and raft for practical building applications. The results are compared with those obtained from other finite element and alternative boundary element methods to verify the validity and accuracy of the present formulation.     

A general exact transformation of body-force volume integral in BEM for 2D anisotropic elasticity

In a previous study (Zhang, Tan and Afagh, 1995), the present authors successfully transformed the body-force volume integrals in BEM for 2D anisotropic elasticity, to boundary ones. This restores the BEM as a truly boundary solution process for treating anisotropic bodies involving body forces. However, the formulation is valid only for problem domains which are geometrically convex and simply connected. This paper presents a general and exact transformation of the bodyforce volume integrals in BEM to line integrals for 2D anisotropic elasticity, in which the above-mentioned restriction on the geometry of the domain is eliminated. The successful implementation of the formulation is demonstrated by three practical examples.     

Generalized Kelvin solution based boundary element method for crack problems in multilayered solids

This paper presents a new boundary element method (BEM) for linear elastic fracture mechanics in three-dimensional multilayered solids. The BEM is based on a generalized Kelvin solution. The generalized Kelvin solution is the fundamental singular solution for a multilayered elastic solid subject to point concentrated body-forces. For solving three-dimensional elastic crack problems in a finite region, a multi-region method is also employed in the present BEM. For crack problems in an infinite space, a large finite body is used to approximate the infinite body. In addition, eight-node traction-singular boundary elements are used in representing the displacements and tractions in the vicinity of a crack front. The incorporation of the generalized Kelvin solution into the boundary integral formulation has the advantages in elimination of the element discretization at the interfaces of different elastic layers. Three numerical examples are presented to illustrate the proposed method for the calculation of stress intensity factors for cracks in layered solids. The results obtained using the proposed method are well compared with the existing results available in the relevant literature.