INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2011; 86:1360–1378 Published online 24 January 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.3108 Non-reﬂecting boundary conditions for acoustic propagation in ducts with acoustic treatment and mean ﬂow E. Redon 1, 2, ∗, † , A.-S. Bonnet-Ben Dhia 3, 4 , J.-F. Mercier 3 and S. Poernomo Sari 1 1 Université de Bourgogne, 9 Avenue Alain Savary, 21000 Dijon, France 2 LVA INSA de Lyon, 25 bis Avenue Jean Capelle, 69621 Villeurbanne, France 3 POEMS (UMR 7231 CNRS-INRIA-ENSTA), 32 Boulevard Victor, 75015 Paris, France 4 CERFACS, 42 Avenue Gaspard Coriolis, 31057 Toulouse Cedex 1, France SUMMARY We consider a time-harmonic acoustic scattering problem in a 2D inﬁnite waveguide with walls covered with an absorbing material, in the presence of a mean ﬂow assumed uniform far from the source. To make this problem suitable for a ﬁnite element analysis, the inﬁnite domain is truncated. This paper concerns the derivation of a non-reﬂecting boundary condition on the artiﬁcial boundary by means of a Dirichlet-to-Neumann (DtN) map based on a modal decomposition. Compared with the hard-walled guide case, several difﬁculties are raised by the presence of both the liner and the mean ﬂow. In particular, acoustic modes are no longer orthogonal and behave asymptotically like the modes of a soft-walled guide. However, an accurate approximation of the DtN map can be derived using some bi-orthogonality relations, valid asymptotically for high-order modes. Numerical validations show the efﬁciency of the method. The inﬂuence of the liner with or without mean ﬂow is illustrated. Copyright  2011 John Wiley & Sons, Ltd. Received 22 January 2010; Revised 24 September 2010; Accepted 11 November 2010 KEY WORDS: Dirichlet-to-Neumann map; non-reﬂecting boundary condition; ﬁnite elements; acoustic waveguide; mean ﬂow; impedance boundary condition; liner 1. INTRODUCTION In this paper, we are interested in the numerical simulation of 2D acoustic scattering in locally perturbed inﬁnite lined guides with or without mean ﬂow in the frequency domain. Our aim is to model the inﬁnite domain by transparent boundary conditions suitable for use in a ﬁnite element scheme. We focus here on the method based on a Dirichlet-to-Neumann (DtN) boundary condition. This method requires the knowledge of the transverse modes of the guide: an important part of this paper is then devoted to the calculation and the description of duct modes and we expect such results to be useful for other applications or other modal methods [1]. This type of boundary condition is used to reduce as much as possible the computational domain, including cases with signiﬁcant attenuation for which one could use simpler conditions instead (rigid artiﬁcial boundaries for instance) but far enough from the source. Solving such scattering problems can be very helpful to better understand physical properties of the acoustical propagation in a focused part of a complex conﬁguration. Indeed, many industrial applications, such as turbofan aircraft engine, car mufﬂers, and ventilations shafts, contain ducts that present some localized inhomogeneity related to their shape, their wall properties, or the presence ∗ Correspondence to: E. Redon, Université de Bourgogne, 9 Avenue Alain Savary, 21000 Dijon, France. † E-mail: emredon@u-bourgogne.fr Copyright  2011 John Wiley & Sons, Ltd.