Scalar wave equation by the boundary element method: a D-BEM approach with non-homogeneous initial conditions (original) (raw)
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Journal of the Brazilian Society of Mechanical Sciences and Engineering
This work is concerned with the development of two boundary element method (BEM) formulations for the solution of one-dimensional scalar wave propagation problems. The first formulation is called TD-BEM, TD meaning time-domain, as it employs a time-dependent fundamental solution. The second formulation is called D-BEM, D meaning domain, and employs the fundamental solution from the static problem. The Houbolt and the Newmark methods are employed for the time-marching in the D-BEM approach. Two examples, constituted of five analyses, are included.
Analytical elements of time domain bem for two-dimensional scalar wave problems
International Journal For Numerical Methods in Engineering, 1992
An accurate and efficient time domain BEM for 2-D scalar wave problems is presented. Emphasis is on developing analytical boundary elements (explicit solutions of the element matrices). The solutions are obtained under the condition of straight line elements and by bringing the problem to a simple and genral form of double convolution equation which is then solved by the Cagniard-De Hoop method. Six kinds of elements for any combination of the spatial interpolation functions of order 0,1,2 with the temporal interpolation functions of order 0, 1 are given in a compact form. It is pointed out that if the order of temporal interpolation function is higher than 1, or if the continuity of velocity or acceleration is required, the time-stepping technique will face difficulty. A method to solve this problem is also presented. Advantages of using the analytical elements instead of a numerical integral procedure are apparent. Problems with such things as singular integrals, accuracy and stability are solved. Methodology and solutions are demonstrated by a comparative study of two example problems. Numerical solutions reveal that the computation is efficient, accurate and stable.
Transient analysis of wave propagation problems by half-plane BEM
S U M M A R Y In this paper, a half-plane time-domain boundary element method (BEM) was presented for analysing the 2-D scalar wave problems in a homogenous isotropic linear elastic medium. Using the existing transient full-plane fundamental solution and asking for the assistance of method of source image to satisfy the stress-free boundary conditions, first, a half-plane time-domain fundamental solution was obtained for displacement and traction fields. Then, the condensed closed-form of half-plane time-convoluted kernels were extracted analytically by applying the time-convolution integral on the determined half-plane fundamental solutions. After implementing the half-plane time-domain BEM in computer codes, its applicability and efficiency were verified and compared with those of the published works by analysing several practical examples. The studies showed that the proposed method had good agreement with the existing solutions. Compared to the full-plane time-domain BEM, half-plane time-domain BEM had more capability and better accuracy as well as much shorter run time. It is obvious that this method can be practically used to analyse the site response in substituting the old-style time-domain BEM formulation as well. In technical literature, different volumetric and boundary methods exist for the analysis of soil dynamics and wave propagation problems, which include finite difference method (FDM), finite element method and boundary element method (BEM). To analyse the infinite and semi-infinite continuous media using finite element or FDMs, the boundaries of the energy absorber should be defined and the considered domain needs to be discretized. In this case, the complexity of problem is doubled and the analysis time is increased. Nevertheless, to reduce 1-D in modelling and satisfy the Sommerfeld's radiation conditions of waves at infinity automatically, boundary element (BE) is a suitable method, especially to analyse various problems with infinite and semi-infinite boundaries (Beskos 1987, 1997). Although the BEM has been developed in both transformed and time domains, systematic analysis of engineering problems in the time domain has several advantages, from among which combining with other numerical methods and analysing non-linear behaviour, analysing of various problems with time-dependent geometry and obtaining the real valued can be pointed out. The first direct time-domain BE formulation was presented by Friedman & Shaw (1962) for antiplane elastodynamics and Cruse & Rizzo (1968) for in-plane elastodynamics. Cole et al. (1978) showed the first general formulation of BE in the time domain for 2-D scalar problems. Niwa et al. (1980), Manolis & Beskos (1981) and Manolis (1983) used the time-domain BE for the analysis of 2-D elastodynamics problems and their solutions were compared with responses obtained in the transformed domains. The new form of full-plane time-convoluted kernels and BE formulation was presented by Mansur (1983) for 2-D scalar and elastodynamics problems. His BEM formulation was obtained with regard to Heaviside functions and assuming a triangular shape for time interpolations so that Dominguez (1993) showed a better view for them. Antes (1985) developed the time-domain BE formulation for arbitrary initial conditions and Spyrakos & Antes (1986) was able to take them for dynamic analysis of various problems. However, Spyrakos & Beskos (1986) presented another form of time-domain BEM for plane stress or strain problems but Gallego & Dominguez (1990) showed that their obtained formulation was only a special case of the solutions of previous researchers. Regardless of the Heaviside functions in integration, Israil & Banerjee (1990a,b) were able to show the simpler and more tangible form of full-plane time-convoluted kernels for scalar and elastodynamics problems so that later Kamalian et al. (2003) modified their in-plane kernels and implemented that in time-domain BEM algorithm in order to analyse different geotechnical earthquake engineering problems as well. C
Initial conditions contribution in a BEM formulation based on the convolution quadrature method
International Journal for Numerical Methods in Engineering, 2006
This work presents a two-dimensional boundary element method (BEM) formulation for the analysis of scalar wave propagation problems. The formulation is based on the so-called convolution quadrature method (CQM) by means of which the convolution integral, presented in time-domain BEM formulations, is numerically substituted by a quadrature formula, whose weights are computed using the Laplace transform of the fundamental solution and a linear multistep method. This BEM formulation was initially developed for scalar wave propagation problems with null initial conditions. In order to overcome this limitation, this work presents a general procedure that enables one to take into account non-homogeneous initial conditions, after replacing the initial conditions by equivalent pseudo-forces. The numerical results included in this work show the accuracy of the proposed BEM formulation and its applicability to such kind of analysis.
Performance of mass matrices for the BEM dynamic analysis of wave propagation problems
Computer Methods in Applied Mechanics and Engineering, 1987
The performance of different mass matrix formulations for the dynamic analysis of wave propagation problems using the boundary element method (BEM) is investigated. It is shown that the BEM formulations have serious shortcomings if they are not Poisson-adjusted. The theoretical conclusions, which are expandable to other related problems, are sustained by 2D acoustic numerical results.
WIT Transactions on Modelling and Simulation
This work is concerned with the numerical computation of time and space derivatives of the time-domain solution of scalar wave propagation problems using the boundary element method (TD-BEM). In the present formulation, the BEM based on the so-called convolution quadrature method (CQM-BEM) is employed. The CQM-BEM takes into account non-homogeneous initial conditions by means of a general procedure, known as the initial condition pseudo-force procedure (ICPF), which replaces the initial conditions by equivalent pseudo-forces. The boundary integral equation with initial conditions contribution is derived analytically and the quadrature weights of the standard ICPF-CQM-BEM formulation are transformed in order to compute time and space derivatives. Two numerical examples are presented at the end of the work illustrating the efficacy of the implemented formulation.
A modified boundary integral evolution formulation for the wave equation
Advances in Engineering Software, 2009
We apply a modified boundary integral formulation otherwise known as the Green element method (GEM) to the solution of the two-dimensional scalar wave equation. GEM essentially combines three techniques namely: (a) finite difference approximation of the time term (b) finite element discretization of the problem domain and (c) boundary integral replication of the governing equation. These unique and advantageous characteristics of GEM facilitates a direct numerical approximation of the governing equation and obviate the need for converting the governing partial differential equation to a Helmholtz-type Laplace operator equation for an easier boundary element manipulation. C 1 continuity of the computed solutions is established by using Overhauser elements. Numerical tests show a reasonably close agreement with analytical results. Though in the case of the Overhauser GEM solutions, the level of accuracy obtained does not in all cases justify the extra numerical rigor.