Construction and performance analysis of local DtN absorbing boundary conditions for exterior Helmholtz problems. Part I : Elliptical shaped boundaries

We propose a new class of approximate {\it local} DtN boundary conditions to be applied on prolate spheroid-shaped exterior boundaries when solving acoustic scattering problems by elongated obstacles. These conditions are : (a) exact for the first modes, (b) easy to implement and to parallelize, (c) compatible with the local structure of the computational finite element scheme, and (d) applicable to exterior elliptical-shaped boundaries that are more suitable in terms of cost-effectiveness for surrounding elongated scatterers. Moreover, these conditions coincide with the classical local DtN condition designed for spherical-shaped boundaries. We investigate analytically and numerically the effect of the frequency regime and the slenderness of the boundary on the accuracy of these conditions when applied for solving radiators and scattering problems. We also compare their performance to the second order absorbing boundary condition (BGT2) designed by Bayliss, Gunzburger and Turkel when expressed in prolate spheroidal coordinates. The analysis reveals that, in the low frequency regime, the new second order DtN condition (DtN2) retains a good level of accuracy regardless of the slenderness of the boundary. In addition, the DtN2 boundary condition outperforms the BGT2 condition. Such superiority is clearly noticeable for large eccentricity values