Symmetric collocation BEM/FEM coupling procedure for 2-D dynamic structural–acoustic interaction problems

 This paper presents a symmetric collocation BEM (SCBEM)/FEM coupling procedure applicable to 2-D time domain structural–acoustic interaction problems. The use of symmetry for BEM not only saves memory storage but also enables the employment of efficient symmetric equation solvers, especially for BEM/FEM coupling procedure. Compared with symmetric Galerkin BEM (SGBEM) where double boundary integration should be carried out, SCBEM can reduce significantly the computing cost. Two numerical examples are included to illustrate the effectiveness and accuracy of the proposed method. Received: 2 November 2001 / Accepted: 27 May 2002     

A general 3D BEM/FEM coupling applied to elastodynamic continua/frame structures interaction analysis

This paper presents a procedure for coupling general finite element models with three‐dimensional bodies modelled by the Boundary Element Method (BEM). Shells, plates and frames are modelled by the Finite Element Method (FEM) and coupled to the BEM domain directly or by means of rigid blocks. The coupling is used for the analysis of buildings connected to half‐space by means of rigid footings, piles or plates in bending and other problems where combinations of different types of sub‐domains are required, composite domains for instance. Several numerical examples are analysed to demonstrate the robustness and accuracy of the proposed scheme. Copyright © 1999 John Wiley & Sons, Ltd.     

An advanced direct time domain BEM formulation for general 3-D elastodynamic problems

A new advanced time domain BEM formulation is presented for the study of general 3-D elastodynamic problems. The proposed method is based on the infinite space Stokes fundamental solutions, which, for the first time, are written in terms of body forces in the form of higher order B-spline time distributions. Higher order spatial and temporal discretization schemes are applied to the boundary integral equations of the elastodynamic system yielding a time marching solution for the characteristic response of the system due to an excitation with a B-spline distribution in time. This characteristic response due to a B-spline excitation forms the basis, within the framework of a general B-spline superposition scheme, for the calculation of the responses of the same elastodynamic system due to any transient forcing function.     

On the stability of the non-symmetric BEM/FEM coupling in linear elasticity

In this paper we discuss the use of single and double layer boundary integral equations for the numerical solution of linear elasticity problems with boundary conditions of mixed type, and the one-equation coupling of finite and boundary element methods to solve a free space transmission problem. In particular we present a sufficient and necessary condition which ensures stability of the coupled approach for any choice of finite and boundary elements. These results justify the coupling of collocation and Galerkin one-equation boundary element methods with finite elements as used in many engineering and industrial applications. Hence one may avoid the use of the symmetric formulation of boundary integral equations, which is, although well established from a mathematical point of view and also used in some engineering applications, not so much accepted in particular in industrial applications.     

Time-domain BEM for 3-D transient elastodynamic problems with interacting rigid movable disc-shaped inclusions

Formulation of time-domain boundary element method for elastodynamic analysis of interaction between rigid massive disc-shaped inclusions subjected to impinging elastic waves is presented. Boundary integral equations (BIEs) with time-retarded kernels are obtained by using the integral representations of displacements in a matrix in terms of interfacial stress jumps across the inhomogeneities and satisfaction of linearity conditions at the inclusion domains. The equations of motion for each inclusion complete the problem formulation. The time-stepping/collocation scheme is implemented for the discretization of the BIEs by taking into account the traveling nature of the generated wave field and local structure of the solution at the inclusion edges. Numerical results concern normal incidence of longitudinal wave onto two coplanar circular inclusions. The inertial effects are revealed by the time dependencies of inclusions’ kinematic parameters and dynamic stress intensity factors in the inclusion vicinities for different mass ratios and distances between the interacting obstacles.     

A coupled numerical scheme of dual reciprocity BEM with DQM for the transient elastodynamic problems

The two‐dimensional transient elastodynamic problems are solved numerically by using the coupling of the dual reciprocity boundary element method (DRBEM) in spatial domain with the differential quadrature method (DQM) in time domain. The DRBEM with the fundamental solution of the Laplace equation transforms the domain integrals into the boundary integrals that contain the first‐ and the second‐order time derivative terms. Thus, the application of DRBEM to elastodynamic problems results in a system of second‐order ordinary differential equations in time. This system is then discretized by the polynomial‐based DQM with respect to time, which gives a system of linear algebraic equations after the imposition of both the boundary and the initial conditions. Therefore, the solution is obtained at any required time level at one stroke without the use of an iterative scheme and without the need of very small step size in time direction. The numerical results are visualized in terms of graphics. Copyright © 2008 John Wiley & Sons, Ltd.     

Efficiency improvement of the frequency-domain BEM for rapid transient elastodynamic analysis

The frequency-domain fast boundary element method (BEM) combined with the exponential window technique leads to an efficient yet simple method for elastodynamic analysis. In this paper, the efficiency of this method is further enhanced by three strategies. Firstly, we propose to use exponential window with large damping parameter to improve the conditioning of the BEM matrices. Secondly, the frequency domain windowing technique is introduced to alleviate the severe Gibbs oscillations in time-domain responses caused by large damping parameters. Thirdly, a solution extrapolation scheme is applied to obtain better initial guesses for solving the sequential linear systems in the frequency domain. Numerical results of three typical examples with the problem size up to 0.7 million unknowns clearly show that the first and third strategies can significantly reduce the computational time. The second strategy can effectively eliminate the Gibbs oscillations and result in accurate time-domain responses.     

Time-domain transient elastodynamic analysis of 3-D solids by BEM

The numerical implementation of the Direct Boundary Element formulation for time-domain transient analysis of three-dimensional solids is presented in a most general and complete manner. The present formulation employs the space and time dependent fundamental solution (Stokes' solution) and Graffi's dynamic reciprocal theorem to derive the boundary integral equations in the time domain. A time-stepping scheme is then used to solve the boundary initial value problem by marching forward in time. Higher order shape functions are used to approximate the field quantities in space as well as in time, and a combination of analytical (time-integration) and numerical (spatial-integration) integration is carried out to form a system of linear equations. At the end of each time step, these equations are solved to obtain the unknown field quantities at that time. Finally, the accuracy and reliability of this algorithm is demonstrated by solving a number of example problems and comparing the results against the available analytical and numerical solution.     

A time domain BEM for 3-D elastodynamic analysis using the B-spline fundamental solutions

A Boundary Element for 3-D elastodynamic analysis is introduced in detail. The method uses a new generation of the Stokes fundamental solutions that utilize the B-spline family of polynomials. The integration techniques of the boundary element kernels are also discussed for both the singular and non-singular cases. A number of numerical examples are presented for the validation of the proposed methodology.     

Iterative coupling of BEM and FEM for nonlinear dynamic analyses

The present work deals with the iterative coupling of boundary element and finite element methods. First, the domain of the original problem is subdivided into two subdomains, which are separately modeled by the FEM and BEM. Thus the special features and advantages of the two methodologies can be taken into account. Then, prescribing arbitrary transient boundary conditions, a successive renewal of the variables on the interface between the two subdomains is performed through an iterative procedure until the final convergence is achieved. In the case of local nonlinearities within the finite element subdomain, it is straightforward to perform the iterative coupling together with the iterations needed to solve the nonlinear system. The procedure turns out to be very efficient. Moreover, a special formulation allows taking into account different durations of the time steps in each subdomain.The financial support by CAPES (Fundação Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) through scholarship No. BEX0788/02-3 (first author) is gratefully acknowledged.     

A semi-analytic method for solving some 2-D elastodynamic problems in semi-infinite media

 A new semi analytic method for solving two dimensional elastodynamic problems in semi-infinite medium is proposed. The elastodynamic field equations are Fourier transformed in the infinite space dimension(s), while time derivatives are approximated via finite difference, leading to a set of ordinary differential equations in the semi infinite direction(s), which are solved analytically. The method is inherently non-reflecting and no artificial boundaries are used. 1 and 2-D examples of a rigid body impact are studied and the non-oscillating characteristic of the solution, usually obtained by other methods, is examined. The accuracy is examined by comparing the results with solutions obtained by previous methods. Received 12 August 1998     

BEM–FEM coupling for 3D fracture mechanics applications

Attention is here focused on the implementation of a coupled BEM–FEM procedure employing the BE method for the modelling of the near-crack region and the FEM for the far-field, where no singularities in the stress field are expected to arise. The symmetric variational version of the BE method is utilized, allowing to obtain a final linear system endowed with a symmetric matrix. With respect to 3D linear elastic fracture mechanics, the code developed is used to evaluate stress intensity factors for some benchmarks and simulate fracture propagation. A research grant from MURST (Cofinanziamento 2000) is acknowledged.     

A numerical Green's function and dual reciprocity BEM method to solve elastodynamic crack problems

A computational model based on the numerical Green's function (NGF) and the dual reciprocity boundary element method (DR-BEM) is presented for the study of elastodynamic fracture mechanics problems. The numerical Green's function, corresponding to an embedded crack within the infinite medium, is introduced into a boundary element formulation, as the fundamental solution, to calculate the unknown external boundary displacements and tractions and in post-processing determine the crack opening displacements (COD). The domain inertial integral present in the elastodynamic equation is transformed into a boundary integral one by the use of the dual reciprocity technique. The dynamic stress intensity factors (SIF), computed through crack opening displacement values, are obtained for several numerical examples, indicating a good agreement with existing solutions.     

A two‐dimensional BEM/FEM coupling applied to viscoelastic analysis of composite domains

A coupling between the boundary‐element and finite‐element methods is studied for the viscoelastic analysis of reinforced media. The viscous behaviour of the composed body is taken into account by an alternative BEM methodology developed for the Boltzmann model. This methodology is based on differential constitutive relations for viscoelasticity. The reinforcements are modelled by finite elements and are considered elastic. The coupling is based on the sub‐region technique due to its generality and easy implementation. The resulting time‐marching process is able to represent both the instantaneous and the time‐dependent behaviour of a body subjected to general boundary conditions. The method is validated by an experimental result and its accuracy tested by comparing numerical results with analytical solutions. The generality of the method is proved by an infinite domain application. Copyright © 2003 John Wiley & Sons, Ltd.     

FEM/BEM for simulation of LSAW devices

This paper presents a modeling of the propagation of surface acoustic, leaky acoustic, and surface skimming bulk waves in piezoelectrics with a finite array of metallic electrodes over their surface. A combined method of matrix Green's function and the finite element method for computation of all acoustic wave fields is an effective tool for simulation of the propagation of acoustic waves in such structures. The proposed method is optimized in the speed of computation of all matrix Green's function components originally obtained. The Fourier transformations of Green's function from kappa-space domain to real space domain are performed by combined trapezoidal and Filon's integration methods for rapidly oscillating functions. The trapezoidal integration method is used on a distance from a point source from zero to a few wavelengths long, but the other has the advantage for a distance from some wavelength to infinity. That allows one, by selectively condensing computation grids around branch and singular points of the sharp behavior of Green's function, to maximize speed and accuracy of computation of integrals. FEM is used, modified originally to achieve acceleration without loss accuracy. Because of the simple geometry of the electrodes, unknown elastic fields are presented as a series of known eigenfunctions with unknown coefficients over the whole region of electrodes. All unknown coefficients are determined by applying the Galerkin method. There is good agreement between numerical and experimental conductances of acoustic wave transducers on materials such as lithium niobate and lithium tantalate.     

Cubic Bezier splines for BEM heat transfer analysis of the 2-D continuous casting problems

 This paper presents a two-dimensional model for identification of the phase change front in a continuous casting process. The transport phenomena encountered in the considered process are solved by Boundary Element Method (BEM). For the known set of external boundary conditions, the whole problem is solved in two subdomains separated by a phase change front whose position is updated during the iteration process. The solution scheme involves the application of a front tracking procedure based on using sensitivity coefficients to find the correct position of the phase change front modelled by Bezier splines. The main features of the developed algorithms were investigated by several numerical tests, the most important results of which are presented in this article. Received 6 November 2000     

On DR/BEM for eigenvalue analysis of 2-D acoustics

This paper discusses the efficiency of several DR/BEM formulations and other boundary techniques for the eigenvalue extraction of two-dimensional acoustic cavities. First, the paper shows that the well-known conical radial basis functions lead to extremely ill conditioning results in cases that the height of the cone is not properly chosen. Moreover, the accuracy of other known high-degree basis functions is tested. Second, the use of Pascals triangle is proposed as a better approximation of the inertial forces at least for the case of rectangular domains. Using Gordons blending-function formula, a systematic procedure is proposed for the selection of the proper monomials. Third, it is shown that the aforementioned functional set can be also used to establish an alternative boundary-type method where both inertial and static terms are treated in a consistent manner. The solution quality of these formulations is investigated by calculating the eigenvalues of a rectangular and a circular acoustic cavity where analytical solutions are known.     

BEM–FEM coupling model for the dynamic analysis of piles and pile groups

This paper shows a BEM–FEM coupling model for the time harmonic dynamic analysis of piles and pile groups embedded in an elastic half-space. Piles are modelled using finite elements (FEM) as a beam according to the Bernoulli hypothesis, while the soil is modelled using boundary elements (BEM) as a continuum, semi-infinite, isotropic, homogeneous or zoned homogeneous, linear, viscoelastic medium. It is assumed that the soil continuity is not altered by the presence of the piles, and the tractions at the pile–soil interface are considered as a load applied within the half-space. The formulation is exposed in detail. In order to validate the model, selected numerical results of time harmonic impedances of different pile groups configurations are evaluated and contrasted with other reference values taken from the literature.     

Iterative coupling of FEM and BEM in 3D transient elastodynamics

A domain decomposition approach is presented for the transient analysis of three-dimensional wave propagation problems. The subdomains are modelled using the FEM and/or the BEM, and the coupling of the subdomains is performed in an iterative manner, employing a sequential Neumann–Dirichlet interface relaxation algorithm which also allows for an independent choice of the time step length in each subdomain. The approach has been implemented for general 3D problems. In order to investigate the convergence behaviour of the proposed algorithm, using different combinations of FEM and BEM subdomains, a parametric study is performed with respect to the choice of the relaxation parameters. The validity of the proposed method is shown by means of two numerical examples, indicating the excellent accuracy and applicability of the new formulation.     

Iterative coupling of FEM and BEM in 3D transient elastodynamics

A domain decomposition approach is presented for the transient analysis of three-dimensional wave propagation problems. The subdomains are modelled using the FEM and/or the BEM, and the coupling of the subdomains is performed in an iterative manner, employing a sequential Neumann–Dirichlet interface relaxation algorithm which also allows for an independent choice of the time step length in each subdomain. The approach has been implemented for general 3D problems. In order to investigate the convergence behaviour of the proposed algorithm, using different combinations of FEM and BEM subdomains, a parametric study is performed with respect to the choice of the relaxation parameters. The validity of the proposed method is shown by means of two numerical examples, indicating the excellent accuracy and applicability of the new formulation.