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

Abstract

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.

Nonlinear Modal Analysis of a One-Dimensional Bar Undergoing Unilateral Contact via the Time-Domain Boundary Element Method

“…Various types of BEM are available in literature such as Domain-BEM (D-BEM) [8], Time Domain-BEM (TD-BEM), Dual Reciprocity-BEM (DR-BEM) [2,17], Frequency Domain-BEM (FD-BEM) [10], Convolution Quadrature-BEM (CQ-BEM) [1,24]. D-BEM and DR-BEM both discretize time and space separately, hence are not of interest here.…”

“…In contrast, CQ-BEM and TD-BEM provide a formulation in space and time allowing to precisely capture traveling waves [19], at least for one-dimensional problems in space. CQ-BEM, first introduced by Lubich [18] and later used for transient analysis [1,24] differs from TD-BEM in the way the integrals are computed, i.e. using the Convolution Quadrature Method (CQM).…”

Initial conditions contribution in a BEM formulation based on the convolution quadrature method

“…The main characteristics of the CQM approach, initially introduced as 'operational quadrature method' [3][4][5], are: (i) the use of the fundamental solution of the The CQM approach, in spite of employing the fundamental solution in a transformed domain, provides the solution directly in the time domain and has already been successfully employed by Schanz and Antes [3,6] and Gaul and Schanz [7] in 3D viscoelastic and 3D elastodynamic BEM formulation, and by Schanz [8,9] in poroelastic applications. A BEM formulation for 2D scalar wave propagation problems, denoted OQM-BEM, can be found in Reference [4]; this formulation is renamed here as CQM-BEM. An application to the analysis of elastodynamic crack problems can be found in Reference [5].…”

“…In Equation (4) u * (X, t; , ) is the fundamental solution, which represents the effect at the field point X, at time t, of an impulse applied at time , at the source point ; and p * (X, t; , ) = *u * (X, t; , )/*n, see References [16,17]. The discretized version of Equation (4) for each source point i when the CQM is applied, is written as [4] …”

Time and space derivatives in a BEM formulation based on the CQM with initial conditions contribution

“…The main advantage of the CQM is that it can be applied to problems where the TD fundamental solution is not available or has a very difficult expression. The CQM-BEM was firstly applied to scalar wave propagation problems by Abreu et al [3].…”