W.J. Mansur

A linear θ method applied to 2D time-domain BEM

A linear θ method is used in this paper to improve the stability of the standard time-domain BEM formulation. The time-stepping procedure is similar to that of the Wilson θ method; however, unlike in the FEM, where linear time variation of acceleration (for elastodynamic problems) is assumed, here linear time variation for both potential and flux (for scalar waves) is assumed in the time interval θΔt. A comparison between numerical results obtained from the standard formulation and from the linear θ method studied here shows the latter to be more stable than the former. The effect of varying θ for different values of time steps is also studied in this paper. Copyright

Stability of Galerkin and collocation time domain boundary element methods as applied to the scalar wave equation

Abstract The Galerkin method for time domain boundary element analysis is presented in this paper and used for the first time as an alternative procedure to improve the stability of BEM formulations applied to the scalar wave equation. The paper includes a short review of the classical time-stepping BEM formulation and the modifications required to implement a Galerkin approach. No mathematical proof is given here, concerning the stability improvement of the Galerkin formulation over the point collocation method. However, strong numerical evidence of such stability improvement is provided by the second example of this paper where the Galerkin approach has shown to be stable for rather small time steps.

Iterative solution of bem equations by GMRES algorithm

Abstract This paper presents a performance study of the GMRES algorithm for the solution of non-symmetric dense systems of equations arising from the boundary element discretization of two-dimensional elasticity. Comparisons with Gauss elimination and bi-conjugate gradients show the computer effectiveness and accuracy of the preconditioned GMRES algorithm.

A more stable scheme for BEM/FEM coupling applied to two-dimensional elastodynamics

A linear θ method is used in this paper in order to improve the stability of the time domain BEM/FEM coupling formulation. To make possible the coupling of BEM approaches that employ the standard linear θ method with FEM algorithms that employ the Newmark scheme, the former has to be modified. In the standard linear θ method, the response is initially computed at the time (n+θ)Δt and subsequently the response at the time (n+1)Δt is obtained. In the modified version, integral equations are written at the time (n+θ)Δt and then modified so that unknowns at the time (n+1)Δt are introduced back in the BEM system of equations. Thus it becomes possible to write BEM/FEM coupling compatibility conditions at the time (n+1)Δt. In this way, the θ scheme, which is more stable than the standard BEM formulation, is implicitly considered. The stability improvement achieved by the formulation presented here, when applied to two-dimensional elastodynamics, is illustrated by two numerical examples. The effect of varying θ on the stability of the numerical responses is also studied in this paper.

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.

Explicit time-domain approaches based on numerical Green's functions computed by finite differences - The ExGA family

The present paper describes a new family of time stepping methods to integrate dynamic equations of motion. The scalar wave equation is considered here; however, the method can be applied to time-domain analyses of other hyperbolic (e.g., elastodynamics) or parabolic (e.g., transient diffusion) problems. The algorithms presented require the knowledge of the Greens function of mechanical systems in nodal coordinates. The finite difference method is used here to compute numerically the problem Greens function; however, any other numerical method can be employed, e.g., finite elements, finite volumes, etc. The Greens matrix and its time derivative are computed explicitly through the range [0,@Dt] with either the fourth-order Runge-Kutta algorithm or the central difference scheme. In order to improve the stability of the algorithm based on central differences, an additional matrix called step response is also calculated. The new methods become more stable and accurate when a sub-stepping procedure is adopted to obtain the Greens and step response matrices and their time derivatives at the end of the time step. Three numerical examples are presented to illustrate the high precision of the present approach.

Dynamic analysis of fluid-soil-structure interaction problems by the boundary element method

The present paper describes an iterative procedure for BEM-BEM coupling. The paper presents suitable interface conditions and algorithms for iteratively coupling sub-domains modeled by three different boundary element time-domain formulations, namely: acoustic and elastodynamic BEM formulations based on time-dependent Greens functions and non-linear time-domain approach which employs elastostatic Greens functions and therefore requires domain discretization. Two examples are analyzed and at the end of the paper conclusions of the study are presented.

Time weighting in time domain BEM

The time-weighted time domain boundary element method (TWTDBEM) is presented as an alternative approach to improve the stability of the traditional time domain BEM (TTDBEM) formulation for dynamics. A forward time prediction method is used, which does not cause significant errors as long as the time step is not too large. Numerical experiments carried out show that the time weighting method, presented for the first time here, is stable for applications in which the TTDBEM approach is not, even when only a small number of time weighting discrete points is employed. The use of only two weighting points for each time step leads to a substantial improvement in the stability of the TWTDBEM over the TTDBEM at the expense of a computational cost increase which is not significant. In applications where the time marching process is likely to present instabilities, the time weighting method should be preferred rather than the standard point collocation method.

Stress and velocity in 2D transient elastodynamic analysis by the boundary element method

This paper is mainly concerned with the development of integral equations to compute stress and velocity components in transient elastodynamic analysis by the boundary element method. All expressions required are presented explicitly. The boundary is discretized by linear isoparametric elements whereas linear and constant time interpolation are assumed, respectively, for the displacement and traction components. Time integration is carried out analytically and the resulting expressions are presented. An assessment of the accuracy of the results provided by the present formulation can be seen at the end of the article, where two examples are presented.

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

Abstract 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.