Estudo de duas técnicas para a solução de problemas dinâmicos utilizando o método dos elementos de contorno: a superposição modal e a transformada de Laplace

Abstract

The search for a reliable and accurate method to convert domain integrals involving non-self-adjoint operators into boundary integrals, in accordance with the philosophy of the Boundary Element Method, remains a significant challenge. One of the most recent proposals to achieve this goal is the Direct Interpolation Technique of the Boundary Element Method (DIBEM). Already successfully employed in solving scalar problems governed by the Poisson, Helmholtz, and Advection-Diffusion equations, this work presents the results of using DIBEM in the analysis of wave propagation problems in homogeneous media. The main objective is to evaluate the integration of DIBEM with two distinct techniques for handling the time-dependent term: Modal Superposition and the Laplace Transform, two well-established strategies. In the first formulation, a modified modal superposition, which uses a correlated eigenvalue problem associated with the transpose of the dynamic matrix, is applied to decouple the dynamic equations. Time advancement is performed using the Houbolt algorithm, whose fictitious damping eliminates spurious modal contents, producing greater stability. In the second formulation, the Laplace transform is used to eliminate time dependence; DIBEM is used to solve the resulting stationary problem in terms of the transformation variable, and an inversion method is used to return to the time domain. Several typical wave propagation problems were solved using linear boundary elements.