Energetic boundary element method analysis of wave propagation in 2D multilayered media

Abstract

The analysis of scalar wave propagation in 2D zonewise homogeneous media with vanishing initial and mixed boundary conditions is carried out. The problem is formulated in terms of time-dependent boundary integral equations, and then it is set in a weak form, based on a natural energy identity satisfied by the differential problem solution. Several numerical results have been obtained by means of the related energetic Galerkin boundary element method showing accuracy and stability of the method.A. AIMI, S. GAZZOLA AND C. GUARDASONI precise continuity and coerciveness properties [26,27]. Consequently, it can be discretized by unconditionally stable schemes with well-behaved stability constants even for large times.We remark that the idea of introducing a space-time weak formulation for the transient wave problem based on the energy identity is not new. In fact, in [16], it was exploited to get a satisfactory stability result for the acoustic wave equation with the aid of absorbing boundary condition. Unfortunately, the cases of Dirichlet and Neumann boundary conditions on a bounded time interval, as we have always considered, are much more difficult to be theoretically treated, and they have been analyzed in [26,27].The proposed procedure can be generalized to multi-domain wave propagation BIE problems as introduced in [28] within the limit of bi-domain cases. Here, the extension to n-domain case is proposed and further numerical simulations are presented and discussed: through comparisons with available (when possible) literature results [13,29], the stability and accuracy of the energetic approach are shown even in this framework. Plenty of examples, not easily findable in mathematical and even engineering papers on the subject, and numerical results included in this work illustrate the applicability and potentialities of this technique, even if theoretical analysis of stability and convergence for the multi-domain case is still to be done.H 1=2 .S/ :D fu 2 H 1=2 : i .U/ D uj @ i , i D 1, 2^ .U/ D u for some U 2 H 1 . /g, Figure 1. Example of bi-domain.

Energetic BEM–FEM coupling for wave propagation in 3D multidomains

“…All the obtained numerical results are similar to those found in , for an analogous exterior 2D bidomain, using an energetic BEM–BEM coupling.…”

“…All the obtained numerical results are similar to those found in [19], for an analogous exterior 2D bidomain, using an energetic BEM-BEM coupling. In order to test the performance of the proposed formulation in a more general setting, where, differently from previous examples, waves impinge on the interface also at oblique angles, a second set of analyses is run assuming a different loading.…”

“…This issue has somehow limited new developments. On the contrary, in this work, we start from a recently developed energetic space–time weak formulation of BIEs related to wave propagation problems defined on single and multidomains (see in particular and references therein). This technique has shown excellent stability properties, which are crucial in guaranteeing an efficient coupling with the FEM.…”

Energetic Bem for the Numerical Solution of Damped Wave Propagation Exterior Problems

“…This method has the fundamental property of not using explicitly the expression of the kernel which is instead replaced by its Laplace transform. On the contrary, the latter problem has been recently tackled in [1,5,4] where an energetic direct space-time Galerkin BEM for the discretization of retarded potential BIEs related to 1D, 2D and 3D wave propagation problems has been put forward and compared with the weak formulation due to Bamberger and Ha-Duong [9,10,17,18]. The proposed technique is based on a natural energy identity satisfied by the solution of the corresponding differential problem, which leads to a space-time weak formulation of the BIEs with precise stability properties.…”

A numerical study of energetic BEM-FEM applied to wave propagation in 2D multidomains

“…In the last decades, contributions to BEM-FEM coupling, in the context of hyperbolic problems, started to appear [1,21,23,29], especially analyzing stability issues. In this work, taking advantage of a recently developed energetic space-time weak formulation of BIEs related to wave propagation problems defined on single and multidomains (see in particular [3,4] and references therein), a coupling algorithm is presented, which allows a flexible use of FEM and BEM as local discretization techniques, in order to efficiently treat unbounded multilayered media. In principle, both the frequency-domain and time-domain BEM can be used for hyperbolic boundary value problems.…”