Scalar wave propagation in 2D: a BEM formulation based on the operational quadrature method

This work presents a boundary element method formulation for the analysis of scalar wave propagation problems. The formulation presented here employs the so-called operational quadrature method, by means of which the convolution integral, presented in time-domain BEM formulations, is substituted by a quadrature formula, whose weights are computed by using the Laplace transform of the fundamental solution and a linear multistep method. Two examples are presented at the end of the article with the aim of validating the formulation.     

Computation of time and space derivatives in a CQM-based BEM formulation

This work proposes an approach for the numerical computation of time and space derivatives of the time-domain solution of scalar wave propagation problems by means of a boundary element method formulation. Here, this formulation employs the so-called convolution quadrature method. Non-homogeneous initial conditions are taken into account by means of a general procedure, known as initial condition pseudo-force procedure, which replaces the initial conditions by equivalent pseudo-forces. The boundary integral equation with initial conditions contribution is differentiated analytically and the quadrature weights of the standard formulation are transformed in order to compute time and space derivatives at interior points. Numerical examples are presented to show the efficiency of the implemented formulation.     

Scalar wave equation by the boundary element method: A D‐BEM approach with constant time‐weighting functions

A D‐BEM approach, based on time‐weighting residuals, is developed for the solution of two‐dimensional scalar wave propagation problems. The modified basic equation of the D‐BEM formulation is generated by weighting, with respect to time, the basic D‐BEM equation, under the assumption of linear and cubic time variation for the potential and for the flux. A constant time‐weighting function is adopted. The time integration reduces the order of the time‐derivative that appears in the domain integral; as a consequence, the initial conditions are directly taken into account. An assessment of the potentialities of the proposed formulation is provided by the examples included at the end of the work. Copyright © 2009 John Wiley & Sons, Ltd.     

Potential discontinuity simulations with the numerical Green’s function in steady and transient problems

This paper applies the numerical Green’s function (NGF) boundary element formulation (BEM) first in standard form to solve the Laplace equation and then, coupled to the operational quadrature method (OQM), to solve time domain problems (TD-BEM). Both involve the analysis of potential discontinuities in the respective scalar model simulation. The implementation of the associated Green’s function acting as the fundamental solution is advantageous since element discretization of actual discontinuity surfaces are no longer required. In the OQM the convolution integral is substituted by a quadrature formula, whose weights are computed using the fundamental solution in the Laplace domain, producing the direct solution to the problem in the time domain. Applications of the NGF to problems involving the Laplace equation and its transient counterpart are presented for two-dimensional potential flow examples, confirming that the formulation is stable and accurate.     

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.     

Time discontinuous linear traction approximation in time-domain BEM scalar wave propagation analysis

In the present paper the traditional BEM formulation for time-domain scalar wave propagation analysis is extended to a new class of problems. A procedure to consider linear time interpolation for boundary tractions is worked out. Time discontinuities are included by adding to the standard BEM equation the integral equation for velocities. Numerical examples are presented in order to assess the accuracy of the proposed formulation. © 1998 John Wiley & Sons, Ltd.     

A study on time schemes for DRBEM analysis of elastic impact wave

 The precise integration and differential quadrature methods are two new unconditionally stable numerical schemes to approximate time derivative with more than the second order accuracy. Recent studies showed that compared with the Houbolt and Newmark methods, they produced more accurate solutions with large time step for the problems where response is primarily dominated by low and intermediate frequency modes. This paper aims to investigate these time schemes in the context of the dual reciprocity BEM (DRBEM) formulation of various shock-excited scalar elastic wave problems, where high modes have important affect on traction response. The Houbolt method was widely recommended in many literatures for such DRBEM dynamic formulations. However, this study found that the damped Newmark algorithm was the most efficient and accurate for impact traction analysis in conjunction with the DRBEM. The precise integration and differential quadrature methods are shown inapplicable for such shock-excited problems due to the absence of numerical damping. On the other hand, we also found that to achieve the same order of accuracy, the differential quadrature method required much less computing effort than the precise integration method due to the use of the Bartels–Stewart algorithm solving the resulting Lyapunov matrix analogization equation. Received 6 November 2000     

Application of 3D time domain boundary element formulation to wave propagation in poroelastic solids

The dynamic responses of fluid-saturated semi-infinite porous continua to transient excitations such as seismic waves or ground vibrations are important in the design of soil-structure systems. Biot's theory of porous media governs the wave propagation in a porous elastic solid infiltrated with fluid. The significant difference to an elastic solid is the appearance of the so-called slow compressional wave. The most powerful methodology to tackle wave propagation in a semi-infinite homogeneous poroelastic domain is the boundary element method (BEM). To model the dynamic behavior of a poroelastic material in the time domain, the time domain fundamental solution is needed. Such solution however does not exist in closed form. The recently developed ‘convolution quadrature method’, proposed by Lubich, utilizes the existing Laplace transformed fundamental solution and makes it possible to work in the time domain. Hence, applying this quadrature formula to the time dependent boundary integral equation, a time-stepping procedure is obtained based only on the Laplace domain fundamental solution and a linear multistep method. Finally, two examples show both the accuracy of the proposed time-stepping procedure and the appearance of the slow compressional wave, additionally to the other waves known from elastodynamics.     

3D non-linear time domain FEM–BEM approach to soil–structure interaction problems

Dynamic soil–structure interaction is concerned with the study of structures supported on flexible soils and subjected to dynamic actions. Methods combining the finite element method (FEM) and the boundary element method (BEM) are well suited to address dynamic soil–structure interaction problems. Hence, FEM–BEM models have been widely used. However, non-linear contact conditions and non-linear behavior of the structures have not usually been considered in the analyses. This paper presents a 3D non-linear time domain FEM–BEM numerical model designed to address soil–structure interaction problems. The BEM formulation, based on element subdivision and the constant velocity approach, was improved by using interpolation matrices. The FEM approach was based on implicit Green's functions and non-linear contact was considered at the FEM–BEM interface. Two engineering problems were studied with the proposed methodology: the propagation of waves in an elastic foundation and the dynamic response of a structure to an incident wave field.     

A numerical study of wave structures developed on the free surface of a film flowing on inclined planes and subjected to surface shear

In this work, we determine the different patterns of possible wave structures that can be observed on a thin film flowing on an inclined plane when at the free surface a shear force (surface traction) is applied. Different wave structures are obtained dependening on the selected combination of downstream and upstream boundary conditions and initial conditions. The resulting initial boundary value problems are solved numerically using the direct BEM numerical solution of the complete two‐dimensional Stokes system of equations. In our numerical results, the initial discontinuous shock profiles joining uniform fluid depths are smoothed due to the two‐dimensional character of the Stokes formulation, including the effect of the gravitational force as well as the interfacial surface tension force. In this way, physically feasible continuous surface profiles are determined, in which the initial uniform depths are joined by smooth moving wave structures. Numerical solutions have been attained to reproduce the different patterns of possible wave structures previously reported in the literature and extended to identify some other new structures and features defining the behaviour of the surface patterns. Copyright © 2006 John Wiley & Sons, Ltd.     

Two dimensional transient BEM analysis for the scalar wave equation: Kernels

This paper carries out a discussion concerning kernels in two-dimensional BEM analysis of transient scalar wave propagation problems. Kernels obtained after performing analytical time integration are compared. An example of quadratic time variation is presented in order to illustrate some of the mathematical concepts discussed.     

Uniform bicubic B-splines applied to boundary element formulation for 3-D scalar problems

In this work, uniform bicubic B-spline functions are used to represent the surface geometry and interpolation functions in the formulation of boundary-element method (BEM) for three-dimensional problems. This is done as a natural generalization of cubic B-spline curves, introduced by Cabral et al, for two-dimensional problems. Three-dimensional scalar problems, with particular applications to Laplace and Helmholtz equations, are considered.     

Analytical time integration for BEM axisymmetric acoustic modelling

In this work, a numerical time‐domain approach to model acoustic wave propagation in axisymmetric bodies is developed. The acoustic medium is modelled by the boundary element method (BEM), whose time convolution integrals are evaluated analytically, employing the concept of finite part integrals. All singularities for space integration, present in the expressions generated by time integration, are treated adequately. Some applications are presented to demonstrate the validity of the analytical expressions generated for the BEM, and the results obtained with the present approach are compared with those generated by applying numerical time integration. Copyright © 2007 John Wiley & Sons, Ltd.     

Non‐reflecting artificial boundaries for transient scalar wave propagation in a two‐dimensional infinite homogeneous layer

This paper presents an exact non‐reflecting boundary condition for dealing with transient scalar wave propagation problems in a two‐dimensional infinite homogeneous layer. In order to model the complicated geometry and material properties in the near field, two vertical artificial boundaries are considered in the infinite layer so as to truncate the infinite domain into a finite domain. This treatment requires the appropriate boundary conditions, which are often referred to as the artificial boundary conditions, to be applied on the truncated boundaries. Since the infinite extension direction is different for these two truncated vertical boundaries, namely one extends toward x →∞ and another extends toward x→‐ ∞, the non‐reflecting boundary condition needs to be derived on these two boundaries. Applying the variable separation method to the wave equation results in a reduction in spatial variables by one. The reduced wave equation, which is a time‐dependent partial differential equation with only one spatial variable, can be further changed into a linear first‐order ordinary differential equation by using both the operator splitting method and the modal radiation function concept simultaneously. As a result, the non‐reflecting artificial boundary condition can be obtained by solving the ordinary differential equation whose stability is ensured. Some numerical examples have demonstrated that the non‐reflecting boundary condition is of high accuracy in dealing with scalar wave propagation problems in infinite and semi‐infinite media. Copyright © 2003 John Wiley & Sons, Ltd.     

A fast multi‐level convolution boundary element method for transient diffusion problems

A new algorithm is developed to evaluate the time convolution integrals that are associated with boundary element methods (BEM) for transient diffusion. This approach, which is based upon the multi‐level multi‐integration concepts of Brandt and Lubrecht, provides a fast, accurate and memory efficient time domain method for this entire class of problems. Conventional BEM approaches result in operation counts of order O(N²) for the discrete time convolution over N time steps. Here we focus on the formulation for linear problems of transient heat diffusion and demonstrate reduced computational complexity to order O(N^3/2) for three two‐dimensional model problems using the multi‐level convolution BEM. Memory requirements are also significantly reduced, while maintaining the same level of accuracy as the conventional time domain BEM approach. Copyright © 2005 John Wiley & Sons, Ltd.     

A new fast multipole boundary element method for solving large‐scale two‐dimensional elastostatic problems

A new fast multipole boundary element method (BEM) is presented in this paper for large‐scale analysis of two‐dimensional (2‐D) elastostatic problems based on the direct boundary integral equation (BIE) formulation. In this new formulation, the fundamental solution for 2‐D elasticity is written in a complex form using the two complex potential functions in 2‐D elasticity. In this way, the multipole and local expansions for 2‐D elasticity BIE are directly linked to those for 2‐D potential problems. Furthermore, their translations (moment to moment, moment to local, and local to local) turn out to be exactly the same as those in the 2‐D potential case. This formulation is thus very compact and more efficient than other fast multipole approaches for 2‐D elastostatic problems using Taylor series expansions of the fundamental solution in its original form. Several numerical examples are presented to study the accuracy and efficiency of the developed fast multipole BEM formulation and code. BEM models with more than one million equations have been solved successfully on a laptop computer. These results clearly demonstrate the potential of the developed fast multipole BEM for solving large‐scale 2‐D elastostatic problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Initial strain formulation without internal cells for elastoplastic analysis by triple‐reciprocity BEM

In general, internal cells are required to solve elastoplasticity problems using a conventional boundary element method (BEM). However, in this case, the merit of BEM, which is the easy method of preparation of data, is lost. The conventional multiple‐reciprocity boundary element method (MRBEM) cannot be used to solve the elastoplasticity problems because the distribution of initial strain or initial stress cannot be determined analytically. In this paper, we show that two‐dimensional elastoplasticity problems can be solved without the use of internal cells, by using the triple‐reciprocity boundary element method. An initial strain formulation is adopted and the initial strain distribution is interpolated using boundary integral equations. A new computer programme was developed and applied to several problems. Copyright © 2001 John Wiley & Sons, Ltd.     

A boundary spectral method for solving exterior acoustical problems with hypersingular integrals

A boundary spectral method is developed to solve acoustical problems with arbitrary boundary conditions. A formulation, originally derived by Burton and Miller, is used to overcome the non‐uniqueness problem in the high wave number range. This formulation is further modified into a globally non‐singular form to simplify the procedure of numerical quadrature when spectral methods are applied. In the present approach, generalized Fourier coefficients are determined instead of local variables at nodes as in conventional methods. The convergence of solutions is estimated through the decay of magnitude of the generalized Fourier coefficients. Several scattering and radiation problems from a sphere are demonstrated with high wave numbers in the present paper. Copyright © 1999 John Wiey & Sons, Ltd.     

Scalar wave equation by the boundary element method: a D-BEM approach with non-homogeneous initial conditions

This work is concerned with the development of a D-BEM approach to the solution of 2D scalar wave propagation problems. The time-marching process can be accomplished with the use of the Houbolt method, as usual, or with the use of the Newmark method. Special attention was devoted to the development of a procedure that allows for the computation of the initial conditions contributions. In order to verify the applicability of the Newmark method and also the correctness of the expressions concerned with the computation of the initial conditions contributions, four examples are presented and the D-BEM results are compared with the corresponding analytical solutions.     

A new visco- and elastodynamic time domain Boundary Element formulation

The usual time domain Boundary Element Method (BEM) contains fundamental solutions which are convoluted with time-dependent boundary data and integrated over the boundary surface. If the fundamental solution is known, e.g., in Elastodynamics, the temporal convolution can be performed analytically when the boundary data are approximated by polynomial shape functions in time and in the boundary elements. This formulation is well known, but the resulting time-stepping BEM procedure produces instabilities and high numerical damping, when the time step size is chosen too small and too large, respectively. Moreover, in case of viscoelastic or poroelastic domains, the fundamental solution is known only in the frequency domain such that the time history of a response can only be obtained by an inverse transformation of the frequency domain results. Here, a new approach for the evaluation of the convolution integrals, the so-called “Operational Quadrature Methods” developed by LUBICH, is presented. In this formulation, the convolution integral is numerically approximated by a quadrature formula whose weights are determined by the Laplace transform of the fundamental solution and a linear multistep method. Hence, the frequency domain fundamental solution can be used without the need of an inverse transformation. Therefore, the extension to viscoelastic problems succeeds using the elastic-viscoelastic correspondence principle.