Section: New Results
New absorbing boundary conditions for the numerical simulation of the scattering by elongated obstacles
Recently, a new class of absorbing boundary conditions called local approximate DtN absorbing boundary conditions (DtN) has been proposed to be applied on exterior artificial prolate spheroidal-shaped boundaries  . Unlike the standard approximate local DtN boundary conditions that are restricted to circular- or spherical-shaped boundaries (see  ,  ), the proposed conditions are applicable to exterior elliptical- or prolate spheroidal-shaped boundaries that are more suitable for surrounding elongated scatterers because they yield to smaller computational domains. These absorbing boundary conditions are designed to be exact for the first modes. They can be easily incorporated in any finite element parallel code while preserving the local structure of the algebraic system. Moreover, the analysis of the performance of these conditions in the low frequency regime, when using an On-Surface radiation condition formulation  , revealed that these conditions are very accurate regardless of the slenderness of the boundary  ,  . In addition, it has been demonstrated that the second-order local DtN condition (DtN2) outperforms the widely-used second-order absorbing boundary conditions (BGT2)  when expressed in prolate spheroidal coordinates  ,  .
We have extended the investigation of the performance of the local approximate DtN2 absorbing boundary condition to the case of the high frequency regime. More specifically, we have performed an analytical and numerical study to assess the effect of the slenderness of the exterior boundary on the accuracy and the efficiency of the proposed absorbing boundary condition. We have conducted this study using a domain-based formulation, that is the artificial boundary is located at some distance from the surface of the scatterer, since the OSRC approach is not adapted for such an analysis, as previously observed in  . We have proven that, in the high frequency regime, the reflected waves at the artificial boundary decay faster than 1/(ka)15/8 where k is the wavenumber and a is the semi-major axis of this boundary. Numerical results illustrate the accuracy and the efficiency of the proposed absorbing boundary condition when used for solving acoustic scattering problems in domain-based formulation.
The results of the year are available in a Research Report  and they have been presented in 2 peer-reviewed conferences Waves 2009 and Enumath 2009. This work has also been accepted for a regular lecture at the National Congress in Numerical Analysis (CANUM 2009).