Modal analysis of plates using the dual reciprocity boundary element method

This paper presents a new method for determining the natural frequencies and mode shapes for the free vibration of thin elastic plates using the boundary element and dual reciprocity methods. The solution to the plate's equation of motion is assumed to be of separable form. The problem is further simplified by using the fundamental solution of an infinite plate in the reciprocity theorem. Except for the inertia term, all domain integrals are transformed into boundary integrals using the reciprocity theorem. However, the inertia domain integral is evaluated in terms of the boundary nodes by using the dual reciprocity method. In this method, a set of interior points is selected and the deflection at these points is assumed to be a series of approximating functions. The reciprocity theorem is applied to reduce the domain integrals to a boundary integral. To evaluate the boundary integrals, the displacements and rotations are assumed to vary linearly along the boundary. The boundary integrals are discretized and evaluated numerically. The resulting matrix equations are significantly smaller than the finite element formulation for an equivalent problem. Mode shapes for the free vibration of circular and rectangular plates are obtained and compared with analytical and finite element results.     

A wavenumber domain boundary element method model for the simulation of vibration isolation by periodic pile rows

The wavenumber domain boundary element method (WDBEM) for the interaction between the half-space soil and periodic structures is important for the design of various periodic structures in civil engineering. In this study, a WDBEM model for the half-space soil and periodic pile rows is developed and used in the analysis of the vibration isolation via pile rows. To establish the model, the rigid-body-motion method for the estimation of the Cauchy type singular integrals involved in the WDBEM is established for the first time. In the proposed model, the half-space soil and periodic pile rows are treated as elastic media. Employing the spatial domain boundary integral equations for the half-space soil and pile rows as well as the sequence Fourier transform method, the wavenumber domain boundary integral equations for the soil and pile rows are derived. By using the obtained wavenumber domain boundary integral equations, WDBEM formulations for the half-space soil and periodic pile rows are established. Using the WDBEM formulations as well as the continuity conditions at the pile–soil interfaces, a coupled WDBEM model for the pile–soil system is derived. With the proposed WDBEM model, the influences of the pile length and the shear modulus of the half-space soil on the vibration isolation effect of pile rows are examined. Presented numerical results show that the isolation vibration effect of pile rows is enhanced considerably with increasing length of the piles. Besides, the isolation vibration effect of pile rows is weakened considerably with increasing shear modulus of the half-space soil. Moreover, as expected, multiple pile rows usually produce a better isolation vibration effect than a single pile row.     

Radial integration boundary element method for acoustic eigenvalue problems

In this paper, the radial integration boundary element method is developed to solve acoustic eigenvalue problems for the sake of eliminating the frequency dependency of the coefficient matrices in traditional boundary element method. The radial integration method is presented to transform domain integrals to boundary integrals. In this case, the unknown acoustic variable contained in domain integrals is approximated with the use of compactly supported radial basis functions and the combination of radial basis functions and global functions. As a domain integrals transformation method, the radial integration method is based on pure mathematical treatments and eliminates the dependence on particular solutions of the dual reciprocity method and the particular integral method. Eventually, the acoustic eigenvalue analysis procedure based on the radial integration method resorts to a generalized eigenvalue problem rather than an enhanced determinant search method or a standard eigenvalue analysis with matrices of large size, just like the multiple reciprocity method. Several numerical examples are presented to demonstrate the validity and accuracy of the proposed approach.     

Plane stress and plate bending coupling in BEM analysis of shallow shells

In this paper the shear deformable shallow shells are analysed by boundary element method. New boundary integral equations are derived utilizing the Betti's reciprocity principle and coupling boundary element formulation of shear deformable plate and two‐dimensional plane stress elasticity. Two techniques, direct integral method (DIM) and dual reciprocity method (DRM), are developed to transform domain integrals to boundary integrals. The force term is approximted by a set of radial basis functions. Several examples are presented to demonstrate the accuracy of the two methods. The accuracy of results obtained by using boundary element method are compared with exact solutions and the finite element method. Copyright © 2000 John Wiley & Sons, Ltd.     

Modal analysis of anisotropic plates using the boundary element method

A numerical formulation for analysis of dynamic problems of thin anisotropic plates bending is presented. The bending behavior follows Kirchhoff's hypothesis. The formulation is based on the direct boundary element method. The problem is simplified by using the elastostatic fundamental solution of an infinite plate. Domain integrals arising from inertial terms are transformed into boundary integrals using the dual reciprocity technique. Boundary integrals are discretized and evaluated numerically. Natural frequencies for free vibration are obtained and the respective mode shapes are shown. The accuracy of numerical results obtained is assured by comparison with analytical or finite element results.     

Free and forced vibrations of plates by boundary and interior elements

A direct boundary element method is developed for the dynamic analysis of thin elastic flexural plates of arbitrary planform and boundary conditions. The formulation employs the static fundamental solution of the problem and this creates not only boundary integrals but surface integrals as well owing to the presence of the inertia force. Thus the discretization consists of boundary as well as interior elements. Quadratic isoparametric elements and quadratic isoparametric or constant elements are employed for the boundary and interior discretization, respectively. Both free and forced vibrations are considered. The free vibration problem is reduced to a matrix eigenvalue problem with matrix coefficients independent of frequency. The forced vibration problem is solved with the aid of the Laplace transform with respect to time and this requires a numerical inversion of the transformed solution to obtain the plate dynamic response to arbitrary transient loading. The effect of external viscous or internal viscoelastic damping on the response is also studied. The proposed method is compared against the direct boundary element method in conjunction with the dynamic fundamental solution as well as the finite element method primarily by means of a number of numerical examples. These examples also serve to illustrate the use of the proposed method.     

A General Method for Evaluation of 2D and 3D Domain Integrals Without Domain Discretization and its Application in BEM

A robust method is presented to evaluate 2D and 3D domain integrals without domain discretization. Each domain integral is transformed into a double integral, a boundary integral and a 1D integral. Both integrals are evaluated by adaptive Simpson quadrature method. The method can be used to evaluate domain integrals over simply or multiply connected regions with any arbitrary form of integrands. As an application of the method, domain integrals produced in boundary element formulation of potential and elastostatic problems are analyzed. Several examples are provided to show the validity and accuracy of the method.     

Three-dimensional dynamic fracture analysis with the dual reciprocity method in Laplace domain

In this paper, the dual reciprocity boundary element method in the Laplace domain has been developed for the analysis of three-dimensional elastodynamic fracture mechanics mixed-mode problems. The boundary element method is used to calculate the unknowns of transformed boundary displacement and traction and the domain integrals in the elastodynamic equation are transformed into boundary integrals by the use of the dual reciprocity method. The transformed dynamic stress intensity factors are determined by the crack opening displacement (COD) directly in the Laplace domain. By using Durbin's inversion technique, the dynamic stress intensity factors in the time domain are obtained. Several numerical examples are presented to demonstrate the good agreement with existing solutions.     

无域积分的弹塑性边界元法的非线性互补方法

该文引入非线性互补方法来求解边界元法的弹塑性问题,其中方程组由内部点应力方程和反映塑性本构定律的互补函数形成。涉及的域积分采用径向积分法转化为边界积分。通过受内压的厚壁圆筒的应力、位移和荷载-位移情况表明了该算法的精度。     

二维边界元法中几乎奇异积分的解析法

边界元分析中的几乎奇异积分难题一直阻碍其在工程中应用。作者提出的半解析法有效计算了几乎奇异积分,在此基础上做进一步推演,得到线性单元和二次亚参元上几乎强奇异和超奇异积分的解析列式,摈弃了数值求积。该算式对高次单元也近似适用。这个算法使得边界元法能够分析弹性力学薄壁结构。     

Interface dynamic stiffness matrix approach for three‐dimensional transient multi‐region boundary element analysis

A novel substructuring method is developed for the coupling of boundary element and finite element subdomains in order to model three‐dimensional multi‐region elastodynamic problems in the time domain. The proposed procedure is based on the interface stiffness matrix approach for static multi‐region problems using variational principles together with the concept of Duhamel integrals. Unit impulses are applied at the boundary of each region in order to evaluate the impulse response matrices of the Duhamel (convolution) integrals. Although the method is not restricted to a special discretization technique, the regions are discretized using the boundary element method combined with the convolution quadrature method. This results in a time‐domain methodology with the advantages of performing computations in the Laplace domain, which produces very accurate and stable results as verified on test examples. In addition, the assembly of the boundary element regions and the coupling to finite elements are greatly simplified and more efficient. Finally, practical applications in the area of soil–structure interaction and tunneling problems are shown. Copyright © 2009 John Wiley & Sons, Ltd.     

Transient meshless boundary element method for prediction of chloride diffusion in concrete with time dependent nonlinear coefficients

Chloride-induced corrosion of steel reinforcements has been identified as one of the main causes of deterioration of concrete structures. A feasible numerical method is required to predict chloride penetration in concrete structures. A transient meshless boundary element method is proposed to predict chloride diffusion in concrete with time dependent nonlinear coefficient. Taking Green's function as the weighted function, the weighted residue method is adopted to transform the diffusion equation into equivalent integral equations. By the coupling of radial integral method and radial basis function approximation, the domain integrals in equivalent control equations are transformed into boundary integrals. Following the general procedure of boundary element meshing and traditional finite difference method, a set of nonlinear algebraic equations are constructed and are eventually solved with the modified Newtonian iterative method. Several numerical examples are provided to demonstrate the effectiveness and efficiency of the developed model. A comparison of the simulated chloride concentration with the corresponding reported experimental data in a real marine structure indicates the high accuracy and advantage of the time dependent coefficient and nonlinear model over the conventional constant coefficient model.     

The use of Stokes' fundamental solution for the boundary only element formulation of the three-dimensional Navier–Stokes equations for moderate Reynolds numbers

This paper presents a boundary element formulation for the permanent Navier–Stokes equations in which the well-known closed-form fundamental solution for the steady Stokes equations is employed. In this way, from the integral representation formulae for the Stokes' equations, an integral equation is found in which the original non-linear convective terms of the Navier–Stokes equations appear as a domain integral. Additionally, the method of dual reciprocity is used to transform the domain integral to boundary integrals (this method is closely related to the method of particular integrals also used in the literature to transform domain integrals to boundary integrals). Numerical results are presented for the three-dimensional internal flow in a cylindrical container with a rotating cover, in which the accuracy of the method is shown.     

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.     

Soil-structure interaction and wave propagation problems in 2D by a Duhamel integral based approach and the convolution quadrature method

In this paper, a new methodology for analyzing wave propagation problems, originally presented and checked by the authors for one-dimensional problems [18], is extended to plane strain elastodynamics. It is based on a Laplace domain boundary element formulation and Duhamel integrals in combination with the convolution quadrature method (CQM) [13], [14]. The CQM is a technique which approximates convolution integrals, in this case the Duhamel integrals, by a quadrature rule whose weights are determined by Laplace transformed fundamental solutions and a multi-step method. In order to investigate the accuracy and the stability of the proposed algorithm, some plane wave propagation and interaction problems are solved and the results are compared to analytical solutions and results from finite element calculations. Very good agreement is obtained. The results are very stable with respect to time step size. In the present work only multi-region boundary element analysis is discussed, but the presented technique can easily be extended to boundary element – finite element coupling as will be shown in subsequent publications.     

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.     

The dual reciprocity method applied to free vibrations of 2D structures using compact supported radial basis functions

This paper presents a new boundary element application for free vibration analysis of 2D elastic structures. The dual reciprocity method is applied using four compact supported radial basis functions for approximating the domain inertia terms. The eigen-problem of displacement is then solved considering the traction contribution by means of static condensation. The formulation is also extended to consider additional internal nodes to improve accuracy. Three numerical problems are studied to demonstrate the validity and accuracy of the developed formulation. The results are compared to those obtained from analytical and other numerical solutions. A parametric study is set up to demonstrate the effect of the compact support radius on the final results and on the sparsity of system matrices.     

A coupled symmetric BE–FE method for acoustic fluid–structure interaction

A coupled symmetric BE–FE method for the calculation of linear acoustic fluid–structure interaction in time and frequency domain is presented. In the coupling formulation a newly developed hybrid boundary element method (HBEM) will be used to describe the behaviour of the compressible fluid. The HBEM is based on Hamilton's principle formulated with the velocity potential. The state variables are separated into boundary variables which are approximated by piecewise polynomial functions and domain variables which are approximated by a superposition of static fundamental solutions. The domain integrals are eliminated, respectively, replaced by boundary integrals and a boundary element formulation with a symmetric mass and stiffness matrix is obtained as result. The structure is discretized by FEM. The coupling conditions fulfil C¹-continuity on the interface. The coupled formulation can also be used for eigenfrequency analyses by transforming it from time domain into frequency domain.     

Source point isolation boundary element method for solving general anisotropic potential and elastic problems with varying material properties

This paper presents a new robust boundary element method, based on a source point isolation technique, for solving general anisotropic potential and elastic problems with varying coefficients. Different types of fundamental solutions can be used to derive the basic integral equations for specific anisotropic problems, although fundamental solutions corresponding to isotropic problems are recommended and adopted in the paper. The use of isotropic fundamental solutions for anisotropic and/or varying material property problems results in domain integrals in the basic integral equations. The radial integration method is employed to transform the domain integrals into boundary integrals, resulting in a pure boundary element analysis algorithm that does not need any internal cells. Numerical examples for 2D and 3D potential and elastic problems are given to demonstrate the correctness and robustness of the proposed method.     

A meshless local boundary integral equation (LBIE) method for solving nonlinear problems

A new meshless method for solving nonlinear boundary value problems, based on the local boundary integral equation (LBIE) method and the moving least squares approximation, is proposed in the present paper. The total formulation and a rate formulation are developed for the implementation of the present method. The present method does not need domain and boundary elements to deal with the volume and boundary integrals, which will cause some difficulties for the conventional boundary element method (BEM) or the field/boundary element method (FBEM), as the volume integrals are inevitable in dealing with nonlinear boundary value problems. This is the same for the element free Galerkin (EFG) method which also needs element-like cells in the entire domain to evaluate volume integrals. The “companion fundamental solution” introduced in Zhu, Zhang and Atluri (1998) is used so that no derivatives of the shape functions are needed to construct the stiffness matrix for the interior nodes, as well as for those nodes with no parts of their local boundaries coinciding with the global boundary of the domain of the problem, where essential boundary conditions are specified. It is shown that the satisfaction of the essential as well as natural boundary conditions is quite simple, and algorithmically very efficient, in the present nonlinear LBIE approach. Numerical examples are presented for several problems, for which exact solutions are available. The present method converges fast to the final solution with reasonably accurate results for both the unknown variable and its derivatives. No post processing procedure is required to compute the derivatives of the unknown variable (as in the conventional FBEM), since the solution from the present method, using the moving least squares approximation, is already smooth enough. The numerical results in these examples show that high rates of convergence for the Sobolev norms ∥·∥₀ and ∥·∥₁ are achievable, and that the values of the unknown variable and its derivatives are quite accurate.