Alternative time-marching schemes for elastodynamic analysis with the domain boundary element method formulation

Abstract

This work presents alternative time-marching schemes for performing elastodynamic analysis by the Boundary Element Method. The use of the static fundamental solution and the maintenance of the domain integral associated to the accelerations characterize the formulation employed in this work. It is called D-BEM, D meaning domain. Time response is obtained by employing step-by-step time-marching procedures similar to those adopted in the Finite Element Method. Among all integration procedures, Houbolt scheme became the most popular used to march in time with D-BEM formulation, in spite of the presence of a high numerical damping. In order to improve the integration, this work presents alternative schemes that can be used to perform elastodynamic analysis by the BEM with a better damping control. In order to verify the accuracy of the proposed scheme, three examples are presented and discussed at the end of this work.

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.…”

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

“…As a consequence, an adequate approximation to the acceleration is of fundamental importance if reliable results are to be found; this requirement was fulfilled by the Houbolt method, Houbolt [16], successfully employed with the BEM. Although alternative timemarching schemes were recently proposed by Carrer and Mansur [10], Souza et al [17], Chien et al [18], the search for other approximations is a task that still deserves attention. Among the various schemes presented in the Finite Element Method (FEM) literature, e.g.…”

“…In the so-called D-BEM formulations (D meaning domain) this domain integral is kept in the BEM equations, e.g. Carrer and Mansur [10], Hatzigeorgiou and Beskos [11]. On the other hand, by means of suitable interpolation functions the domain integral can be transformed into boundary integrals, generating the so-called DR-BEM formulations (DR meaning double reciprocity), e.g.…”

Boundary element method formulations for the solution of the scalar wave equation in one-dimensional problems

“…For this reason, it was chosen to be implemented in the D-BEM formulation developed in this work. It is important to mention that although other time-marching schemes were recently proposed, see [5,7,23], the search for acceleration approximations is a task that still deserves attention. The presence of non-homogeneous initial conditions is taken into account by following the procedure presented by Carrer et al [6].…”

“…Note that, in the absence of initial conditions, only time approximations are required, as the boundary consists only of the two nodes at x = 0 and at x = L. The second formulation, named D-BEM, where D means domain, employs a static, or steady-state, fundamental solution instead of a time-dependent one, e.g. [5,12,21]. As a consequence, a domain integral containing the second-order time derivative of the potential, or the acceleration, appears in the BEM basic integral equation: this domain integral is responsible for the designation D-BEM, as the solution of the problem requires the discretization of the entire domain, diversely from the TD-BEM formulation, where the domain discretization is required only when non-homogeneous initial conditions appear.…”