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.     

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.     

Transient dynamic elastoplastic analysis by the time-domain BEM formulation

This paper presents a time-domain BEM formulation applied to the solution of transient dynamic elastoplastic problems. The initial stress approach is adopted to solve the elastoplastic problem. Linear time variation is assumed for the displacements and initial stress components whereas traction components are assumed to have a constant time variation. Boundary discretization employs linear elements and the part of the domain where plastic deformation is expected to occur is discretized by employing linear triangular cells. Time integrals are computed analytically, boundary integrals are computed numerically and the domain integrals are computed by following a semi-analytical procedure. A numerical example is presented and the results are compared with another BEM formulation.     

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.     

Analysis of functionally graded plates by meshless method: A purely analytical formulation

Functionally graded plates under static and dynamic loads are investigated by the local integral equation method (LIEM) in this paper. Plate bending problem is described by the Reissner moderate thick plate theory. The governing equations for the functionally graded material with respect to the neutral plane are presented in the Laplace transform domain and therefore the in-plane and bending problems are uncoupled. Both isotropic and orthotropic material properties are considered. The local integral equation method is developed with the locally supported radial basis function (RBF) interpolation. As the closed forms of the local boundary integrals are obtained, there are no domain or boundary integrals to be calculated numerically in this approach. The solutions of the nodal values for the entire plate are obtained by solving a set of linear algebraic equation system with certain boundary conditions. Details of numerical procedures are presented and the accuracy and convergence characteristics of the method are examined. Several examples are presented for the functionally graded plates under static and dynamic loads and the accuracy for proposed method has been observed compared with 3D analytical solutions.     

A flexibility matrix solution of the vibration problem of plates based on the boundary element method

Summary A Boundary element method (BEM) is developed for the dynamic analysis of thin elastic plates. The method is based on the capability to establish a flexibility matrix (discrete Green's function) with respect to a set of nodal mass points using a BEM solution for the static plate problem. A lumped mass matrix is constructed from the tributary mass areas to the nodal mass points. Both free and forced vibrations are considered and numerical examples are presented to illustrate the method and its merits.     

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.     

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.     

Static and harmonic BEM solutions of gradient elasticity problems with axisymmetry

An advanced boundary element method/fast Fourier transform (BEM/FFT) methodology for treating static and time harmonic axisymmetric problems in linear elastic structures exhibiting microstructure effects, is presented. These microstructure effects are taken into account with the aid of a simple strain gradient elastic theory proposed by Aifantis and co-workers [Aifantis (1992), Altan and Aifantis (1992), Ru and Aifantis (1993)]. Boundary integral representations of both static and dynamic gradient elastic problems are employed. Boundary quantities, classical and non-classical (due to gradient terms) boundary conditions are expanded in complex Fourier series in the circumferential direction and the problem is decomposed into a series of problems, which are solved by the BEM by discretizing only the surface generator of the axisymmetric body. The BEM integrations are performed by FFT in the circumferential directions simultaneously for all Fourier coefficients and by Gauss quadrature in the generator direction. All the strongly singular integrals are computed directly by employing highly accurate three-dimensional integration techniques. The Fourier transform solution is numerically inverted by the FFT to provide the final solution. The accuracy of the proposed boundary element methodology is demonstrated by means of representative numerical examples.The authors acknowledge with thanks for the support provided by I.K.Y. through the program IKYDA 2002 (scientific cooperation between the University of Patras, Greece and the Ruhr-University Bochum, Germany).     

Inelastic transient dynamic analysis of Reissner–Mindlin plates by the D/BEM

A direct domain/boundary element method (D/BEM) for dynamic analysis of elastoplastic Reissner–Mindlin plates in bending is developed. Thus, effects of shear deformation and rotatory inertia are included in the formulation. The method employs the elastostatic fundamental solution of the problem resulting in both boundary and domain integrals due to inertia and inelasticity. Thus, a boundary as well as a domain space discretization by means of quadratic boundary and interior elements is utilized. By using an explicit time‐integration scheme employed on the incremental form of the matrix equation of motion, the history of the plate dynamic response can be obtained. Numerical results for the forced vibration of elastoplastic Reissner–Mindlin plates with smooth boundaries subjected to impulsive loading are presented for illustrating the proposed method and demonstrating its merits. Copyright © 2000 John Wiley & Sons, Ltd.     

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.     

Application of the boundary element method to three-dimensional potential problems in heterogeneous media

The present work discusses a solution procedure for heterogeneous media three-dimensional potential problems, involving nonlinear boundary conditions. The problem is represented mathematically by the Laplace equation and the adopted numerical technique is the boundary element method (BEM), here using velocity correcting fields to simulate the conductivity variation of the domain. The integral equation is discretized using surface elements for the boundary integrals and cells, for the domain integrals. The adopted strategy subdivides the discretized equations in two systems: the principal one involves the calculation of the potential in all boundary nodes and the secondary which determines the correcting field of the directional derivatives of the potential in all points. Comparisons with other numerical and analytical solutions are presented for some examples.     

Two-dimensional time-harmonic BEM for cracked anisotropic solids

A mixed time-harmonic boundary element procedure for the analysis of two-dimensional dynamic problems in cracked solids of general anisotropy is presented. To the author's knowledge, no previous BE approach for time-harmonic two-dimensional crack problems in anisotropic solids exists. In the present work, the fundamental solution is split into the static singular part plus dynamic regular terms. Hypersingular integrals associated to the singular part in the traction boundary integral equation are transformed, by means of a simple change of variable, into regular ones plus very simple singular integrals with known analytical solution. Subsequently, only regular (frequency dependent) terms have to be added to the regularized static fundamental solution in order to solve the dynamic problem. The generality of this procedure permits the use of general straight or curved quadratic boundary elements. In particular, discontinuous quarter-point elements are used to represent the crack-tip behavior. Stress intensity factors are accurately computed from the nodal crack opening displacements at discontinuous quarter-point elements. The efficiency and robustness of the present time-harmonic BEM are verified numerically by several test examples. Results are also obtained for more complex configurations, not previously studied in the literature. They include curved crack geometry.     

A recursive application of the integral equation in the boundary element method

This paper presents a recursive application of the governing integral equation aimed at improving the accuracy of numerical results of the boundary element method (BEM). Usually, only the results at internal domain points when using BEM are found using this approach, since the nodal boundary values have already been calculated. Here, it is shown that the same idea can be used to obtain better accuracy for the boundary results as well. Instead of locating the new source points inside the domain, they are positioned on the boundary, with different coordinates to the nodal points. The procedure is certainly general, but will be presented using as an example the two dimensional Laplace equation, for the sake of simplicity to point out the main concepts and numerical aspects of the method proposed, especially due to the determination of directional derivatives of the primal variable, which is part in hyper-singular BEM theory.     

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.     

Boundary-only element solutions of 2D and 3D nonlinear and nonhomogeneous elastic problems

This paper presents a robust boundary element method (BEM) that can be used to solve elastic problems with nonlinearly varying material parameters, such as the functionally graded material (FGM) and damage mechanics problems. The main feature of this method is that no internal cells are required to evaluate domain integrals appearing in the conventional integral equations derived for these problems, and very few internal points are needed to improve the computational accuracy. In addition, one of the basic field quantities used in the boundary integral equations is normalized by the material parameter. As a result, no gradients of the field quantities are involved in the integral equations. Another advantage of using the normalized quantities is that no material parameters are included in the boundary integrals, so that a unified equation form can be established for multi-region problems which have different material parameters. This is very efficient for solving composite structural problems.     

Laplacian decomposition and the boundary element method for solving Stokes problems

The solution of the equations governing the steady incompressible slow viscous fluid flow is analysed using a novel technique based on a Laplacian decomposition instead of the more traditional approaches based on the biharmonic streamfunction formulation or the velocity-pressure formulation. This results in the need to solve the Laplace equations for the pressure and other auxiliary harmonic functions which arise from the ideas of Almansi's decomposition. These equations, which become coupled through the boundary conditions, are numerically solved using the boundary element method (BEM). Results both on the boundary and inside the solution domain are presented and discussed for a simple benchmark test example and a few applications in smooth and non-smooth geometries in order to illustrate that the Laplacian decomposition in combination with BEM provides an efficient technique, in terms of accuracy and convergence, to investigate numerically a Stokes flow.     

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.     

Acoustic calculations with the hybrid boundary element method in time domain

The boundary element method (BEM) is an efficient tool for the calculation of acoustic wave propagation in fluids. Transient waves can be solved by either using a formulation in frequency domain along with an inverse Fourier transformation or a time domain formulation. To increase the efficiency for the solver and allow for an efficient coupling with finite element domains the symmetry of the system matrices is advantageous. If Hamilton's principle is used, a symmetric variational formulation can be established with the velocity potential as field variable. The single field principle is generalized as multifield principle as basis of a hybrid BEM for the calculation of acoustic fields in compressible fluids in time domain. The state variables are separated into boundary variables, which are approximated by piecewise polynomials and domain variables, which are approximated by a superposition of weighted fundamental solutions. In both approximations the time and space dependency is separated. This is why static fundamental solution can be used for the field approximation. The domain integrals are eliminated, respectively, transformed into boundary integrals and an equation of motion with symmetric mass and stiffness matrix is obtained, which can be solved by a direct time integration scheme or by mode superposition. The time derivative of the equation of motion leads to a formulation with pressure and acoustic flux on the boundary for an easier interpretation of the variables.     

Explicit boundary element method for nonlinear solid mechanics using domain integral reduction

Applied to solid mechanics problems with geometric nonlinearity, current finite element and boundary element methods face difficulties if the domain is highly distorted. Furthermore, current boundary element method (BEM) methods for geometrically nonlinear problems are implicit: the source term depends on the unknowns within the arguments of domain integrals. In the current study, a new BEM method is formulated which is explicit and whose stiffness matrices require no domain function evaluations. It exploits a rigorous incremental equilibrium equation. The method is also based on a Domain Integral Reduction Algorithm (DIRA), exploiting the Helmholtz decomposition to obviate domain function evaluations. The current version of DIRA introduces a major improvement compared to the initial version.